%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP220^1 : TPTP v9.2.1. Bugfixed v9.2.1.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n031.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Feb 28 09:05:36 AM UTC 2026
% Result : Theorem 32.64s 8.94s
% Output : Refutation 32.64s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 6
% Syntax : Number of formulae : 35 ( 11 unt; 0 typ; 2 def)
% Number of atoms : 139 ( 49 equ; 0 cnn)
% Maximal formula atoms : 3 ( 3 avg)
% Number of connectives : 425 ( 47 ~; 20 |; 0 &; 340 @)
% ( 2 <=>; 16 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Number of types : 56 ( 55 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 2340 (2337 usr; 60 con; 0-8 aty)
% Number of variables : 33 ( 0 ^; 33 !; 0 ?; 33 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
rat: $tType ).
thf(type_def_6,type,
int: $tType ).
thf(type_def_8,type,
real: $tType ).
thf(type_def_9,type,
nat: $tType ).
thf(type_def_10,type,
code_integer: $tType ).
thf(type_def_11,type,
product_prod_int_int: $tType ).
thf(type_def_12,type,
code_natural: $tType ).
thf(type_def_13,type,
product_prod_nat_nat: $tType ).
thf(type_def_14,type,
num: $tType ).
thf(type_def_15,type,
complex: $tType ).
thf(type_def_16,type,
option_num: $tType ).
thf(type_def_17,type,
product_prod_num_num: $tType ).
thf(type_def_18,type,
produc6271795597528267376eger_o: $tType ).
thf(type_def_19,type,
produc8923325533196201883nteger: $tType ).
thf(type_def_20,type,
set_real: $tType ).
thf(type_def_21,type,
set_set_nat: $tType ).
thf(type_def_22,type,
set_nat: $tType ).
thf(type_def_23,type,
filter_real: $tType ).
thf(type_def_24,type,
extended_enat: $tType ).
thf(type_def_25,type,
filter_int: $tType ).
thf(type_def_26,type,
filter_nat: $tType ).
thf(type_def_27,type,
set_o: $tType ).
thf(type_def_28,type,
set_complex: $tType ).
thf(type_def_29,type,
set_int: $tType ).
thf(type_def_30,type,
set_list_nat: $tType ).
thf(type_def_31,type,
set_Product_unit: $tType ).
thf(type_def_32,type,
set_char: $tType ).
thf(type_def_33,type,
list_nat: $tType ).
thf(type_def_34,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(type_def_35,type,
sum_sum_nat_nat: $tType ).
thf(type_def_36,type,
set_Sum_sum_nat_nat: $tType ).
thf(type_def_37,type,
char: $tType ).
thf(type_def_38,type,
literal: $tType ).
thf(type_def_39,type,
list_o: $tType ).
thf(type_def_40,type,
list_int: $tType ).
thf(type_def_41,type,
set_Pr958786334691620121nt_int: $tType ).
thf(type_def_42,type,
set_Code_integer: $tType ).
thf(type_def_43,type,
set_rat: $tType ).
thf(type_def_44,type,
list_complex: $tType ).
thf(type_def_45,type,
list_real: $tType ).
thf(type_def_46,type,
list_set_nat: $tType ).
thf(type_def_47,type,
list_VEBT_VEBT: $tType ).
thf(type_def_48,type,
set_VEBT_VEBT: $tType ).
thf(type_def_49,type,
vEBT_VEBT: $tType ).
thf(type_def_50,type,
list_list_o: $tType ).
thf(type_def_51,type,
list_list_int: $tType ).
thf(type_def_52,type,
list_list_nat: $tType ).
thf(type_def_53,type,
list_list_VEBT_VEBT: $tType ).
thf(type_def_54,type,
option4927543243414619207at_nat: $tType ).
thf(type_def_55,type,
produc7822875418678951345atural: $tType ).
thf(type_def_56,type,
produc5835291356934675326atural: $tType ).
thf(type_def_57,type,
product_prod_nat_num: $tType ).
thf(type_def_58,type,
produc6756470014200349799atural: $tType ).
thf(type_def_59,type,
produc9072475918466114483BT_nat: $tType ).
thf(type_def_60,type,
product_unit: $tType ).
thf(func_def_0,type,
produc6756470014200349799atural: $tType ).
thf(func_def_1,type,
produc5835291356934675326atural: $tType ).
thf(func_def_2,type,
produc7822875418678951345atural: $tType ).
thf(func_def_3,type,
produc8923325533196201883nteger: $tType ).
thf(func_def_4,type,
option4927543243414619207at_nat: $tType ).
thf(func_def_5,type,
produc9072475918466114483BT_nat: $tType ).
thf(func_def_6,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(func_def_7,type,
set_Pr958786334691620121nt_int: $tType ).
thf(func_def_8,type,
set_Sum_sum_nat_nat: $tType ).
thf(func_def_9,type,
list_list_VEBT_VEBT: $tType ).
thf(func_def_10,type,
produc6271795597528267376eger_o: $tType ).
thf(func_def_11,type,
product_prod_num_num: $tType ).
thf(func_def_12,type,
product_prod_nat_num: $tType ).
thf(func_def_13,type,
product_prod_nat_nat: $tType ).
thf(func_def_14,type,
product_prod_int_int: $tType ).
thf(func_def_15,type,
sum_sum_nat_nat: $tType ).
thf(func_def_16,type,
list_list_nat: $tType ).
thf(func_def_17,type,
list_list_int: $tType ).
thf(func_def_18,type,
list_VEBT_VEBT: $tType ).
thf(func_def_19,type,
set_list_nat: $tType ).
thf(func_def_20,type,
list_set_nat: $tType ).
thf(func_def_21,type,
set_VEBT_VEBT: $tType ).
thf(func_def_22,type,
set_set_nat: $tType ).
thf(func_def_23,type,
set_Code_integer: $tType ).
thf(func_def_24,type,
set_Product_unit: $tType ).
thf(func_def_25,type,
list_list_o: $tType ).
thf(func_def_26,type,
list_complex: $tType ).
thf(func_def_27,type,
set_complex: $tType ).
thf(func_def_28,type,
filter_real: $tType ).
thf(func_def_29,type,
option_num: $tType ).
thf(func_def_30,type,
filter_nat: $tType ).
thf(func_def_31,type,
filter_int: $tType ).
thf(func_def_32,type,
set_char: $tType ).
thf(func_def_33,type,
list_real: $tType ).
thf(func_def_34,type,
set_real: $tType ).
thf(func_def_35,type,
list_nat: $tType ).
thf(func_def_36,type,
list_int: $tType ).
thf(func_def_37,type,
vEBT_VEBT: $tType ).
thf(func_def_38,type,
set_rat: $tType ).
thf(func_def_39,type,
set_nat: $tType ).
thf(func_def_40,type,
set_int: $tType ).
thf(func_def_41,type,
code_natural: $tType ).
thf(func_def_42,type,
code_integer: $tType ).
thf(func_def_43,type,
product_unit: $tType ).
thf(func_def_44,type,
extended_enat: $tType ).
thf(func_def_45,type,
list_o: $tType ).
thf(func_def_46,type,
complex: $tType ).
thf(func_def_47,type,
literal: $tType ).
thf(func_def_48,type,
set_o: $tType ).
thf(func_def_49,type,
char: $tType ).
thf(func_def_50,type,
real: $tType ).
thf(func_def_51,type,
rat: $tType ).
thf(func_def_52,type,
num: $tType ).
thf(func_def_53,type,
nat: $tType ).
thf(func_def_54,type,
int: $tType ).
thf(func_def_55,type,
archim2889992004027027881ng_rat: rat > int ).
thf(func_def_56,type,
archim7802044766580827645g_real: real > int ).
thf(func_def_57,type,
archim3151403230148437115or_rat: rat > int ).
thf(func_def_58,type,
archim6058952711729229775r_real: real > int ).
thf(func_def_59,type,
archimedean_frac_rat: rat > rat ).
thf(func_def_60,type,
archim2898591450579166408c_real: real > real ).
thf(func_def_61,type,
archim7778729529865785530nd_rat: rat > int ).
thf(func_def_62,type,
archim8280529875227126926d_real: real > int ).
thf(func_def_63,type,
bNF_re1962705104956426057at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > $o ).
thf(func_def_64,type,
bNF_re895249473297799549at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ).
thf(func_def_65,type,
bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
thf(func_def_66,type,
bNF_re4695409256820837752l_real: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( real > real > real ) > $o ).
thf(func_def_67,type,
bNF_re4521903465945308077real_o: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > $o ) > ( real > $o ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( real > real > $o ) > $o ).
thf(func_def_68,type,
bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ).
thf(func_def_69,type,
bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).
thf(func_def_70,type,
bNF_re398004352372739002nteger: ( int > code_integer > $o ) > ( ( int > int ) > ( code_integer > code_integer ) > $o ) > ( int > int > int ) > ( code_integer > code_integer > code_integer ) > $o ).
thf(func_def_71,type,
bNF_re3379532845092657523nteger: ( int > code_integer > $o ) > ( int > code_integer > $o ) > ( int > int ) > ( code_integer > code_integer ) > $o ).
thf(func_def_72,type,
bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
thf(func_def_73,type,
bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).
thf(func_def_74,type,
bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).
thf(func_def_75,type,
bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
thf(func_def_76,type,
bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).
thf(func_def_77,type,
bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).
thf(func_def_78,type,
bNF_re88643428490162567atural: ( nat > code_natural > $o ) > ( ( nat > nat ) > ( code_natural > code_natural ) > $o ) > ( nat > nat > nat ) > ( code_natural > code_natural > code_natural ) > $o ).
thf(func_def_79,type,
bNF_re5252274238750452962nteger: ( nat > code_natural > $o ) > ( int > code_integer > $o ) > ( nat > int ) > ( code_natural > code_integer ) > $o ).
thf(func_def_80,type,
bNF_re3704215830270325841atural: ( nat > code_natural > $o ) > ( nat > code_natural > $o ) > ( nat > nat ) > ( code_natural > code_natural ) > $o ).
thf(func_def_81,type,
bNF_re4935368626670024657nteger: ( nat > nat > $o ) > ( ( int > int ) > ( code_integer > code_integer ) > $o ) > ( nat > int > int ) > ( nat > code_integer > code_integer ) > $o ).
thf(func_def_82,type,
bNF_re4785983289428654063nt_int: ( nat > nat > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( nat > int > int ) > ( nat > int > int ) > $o ).
thf(func_def_83,type,
bNF_re3692170009638431535atural: ( nat > nat > $o ) > ( ( nat > nat ) > ( code_natural > code_natural ) > $o ) > ( nat > nat > nat ) > ( nat > code_natural > code_natural ) > $o ).
thf(func_def_84,type,
bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
thf(func_def_85,type,
bNF_re4153400068438556298nteger: ( nat > nat > $o ) > ( int > code_integer > $o ) > ( nat > int ) > ( nat > code_integer ) > $o ).
thf(func_def_86,type,
bNF_re6650684261131312217nt_int: ( nat > nat > $o ) > ( int > int > $o ) > ( nat > int ) > ( nat > int ) > $o ).
thf(func_def_87,type,
bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
thf(func_def_88,type,
bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).
thf(func_def_89,type,
bNF_re7876454716742015248nteger: ( num > num > $o ) > ( ( num > int ) > ( num > code_integer ) > $o ) > ( num > num > int ) > ( num > num > code_integer ) > $o ).
thf(func_def_90,type,
bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
thf(func_def_91,type,
bNF_re6501075790457514782nteger: ( num > num > $o ) > ( int > code_integer > $o ) > ( num > int ) > ( num > code_integer ) > $o ).
thf(func_def_92,type,
bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
thf(func_def_93,type,
bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).
thf(func_def_94,type,
bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
thf(func_def_95,type,
bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).
thf(func_def_96,type,
bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).
thf(func_def_97,type,
bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
thf(func_def_98,type,
bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).
thf(func_def_99,type,
bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).
thf(func_def_100,type,
bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).
thf(func_def_101,type,
bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).
thf(func_def_102,type,
bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).
thf(func_def_103,type,
bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).
thf(func_def_104,type,
bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
thf(func_def_105,type,
bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).
thf(func_def_106,type,
bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
thf(func_def_107,type,
bNF_re8246922863344978751at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > $o ).
thf(func_def_108,type,
bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).
thf(func_def_109,type,
binomial: nat > nat > nat ).
thf(func_def_110,type,
gbinomial_complex: complex > nat > complex ).
thf(func_def_111,type,
gbinomial_int: int > nat > int ).
thf(func_def_112,type,
gbinomial_nat: nat > nat > nat ).
thf(func_def_113,type,
gbinomial_rat: rat > nat > rat ).
thf(func_def_114,type,
gbinomial_real: real > nat > real ).
thf(func_def_115,type,
bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_116,type,
bit_and_not_num: num > num > option_num ).
thf(func_def_117,type,
bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_118,type,
bit_concat_bit: nat > int > int > int ).
thf(func_def_119,type,
bit_or_not_num_neg: num > num > num ).
thf(func_def_120,type,
bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_121,type,
bit_ri7632146776885996613nteger: code_integer > code_integer ).
thf(func_def_122,type,
bit_ri7919022796975470100ot_int: int > int ).
thf(func_def_123,type,
bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
thf(func_def_124,type,
bit_ri631733984087533419it_int: nat > int > int ).
thf(func_def_125,type,
bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).
thf(func_def_126,type,
bit_se2773287842338716102atural: code_natural > code_natural > code_natural ).
thf(func_def_127,type,
bit_se725231765392027082nd_int: int > int > int ).
thf(func_def_128,type,
bit_se727722235901077358nd_nat: nat > nat > nat ).
thf(func_def_129,type,
bit_se3928097537394005634nteger: nat > code_integer > code_integer ).
thf(func_def_130,type,
bit_se2751692689150723149atural: nat > code_natural > code_natural ).
thf(func_def_131,type,
bit_se8568078237143864401it_int: nat > int > int ).
thf(func_def_132,type,
bit_se8570568707652914677it_nat: nat > nat > nat ).
thf(func_def_133,type,
bit_se1345352211410354436nteger: nat > code_integer > code_integer ).
thf(func_def_134,type,
bit_se168947363167071951atural: nat > code_natural > code_natural ).
thf(func_def_135,type,
bit_se2159334234014336723it_int: nat > int > int ).
thf(func_def_136,type,
bit_se2161824704523386999it_nat: nat > nat > nat ).
thf(func_def_137,type,
bit_se2119862282449309892nteger: nat > code_integer ).
thf(func_def_138,type,
bit_se2000444600071755411sk_int: nat > int ).
thf(func_def_139,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(func_def_140,type,
bit_se1080825931792720795nteger: code_integer > code_integer > code_integer ).
thf(func_def_141,type,
bit_se9127793120404214118atural: code_natural > code_natural > code_natural ).
thf(func_def_142,type,
bit_se1409905431419307370or_int: int > int > int ).
thf(func_def_143,type,
bit_se1412395901928357646or_nat: nat > nat > nat ).
thf(func_def_144,type,
bit_se7788150548672797655nteger: nat > code_integer > code_integer ).
thf(func_def_145,type,
bit_se6611745700429515170atural: nat > code_natural > code_natural ).
thf(func_def_146,type,
bit_se545348938243370406it_int: nat > int > int ).
thf(func_def_147,type,
bit_se547839408752420682it_nat: nat > nat > nat ).
thf(func_def_148,type,
bit_se2793503036327961859nteger: nat > code_integer > code_integer ).
thf(func_def_149,type,
bit_se1617098188084679374atural: nat > code_natural > code_natural ).
thf(func_def_150,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(func_def_151,type,
bit_se7882103937844011126it_nat: nat > nat > nat ).
thf(func_def_152,type,
bit_se1745604003318907178nteger: nat > code_integer > code_integer ).
thf(func_def_153,type,
bit_se569199155075624693atural: nat > code_natural > code_natural ).
thf(func_def_154,type,
bit_se2923211474154528505it_int: nat > int > int ).
thf(func_def_155,type,
bit_se2925701944663578781it_nat: nat > nat > nat ).
thf(func_def_156,type,
bit_se8260200283734997820nteger: nat > code_integer > code_integer ).
thf(func_def_157,type,
bit_se7083795435491715335atural: nat > code_natural > code_natural ).
thf(func_def_158,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(func_def_159,type,
bit_se4205575877204974255it_nat: nat > nat > nat ).
thf(func_def_160,type,
bit_se3222712562003087583nteger: code_integer > code_integer > code_integer ).
thf(func_def_161,type,
bit_se2046307713759805098atural: code_natural > code_natural > code_natural ).
thf(func_def_162,type,
bit_se6526347334894502574or_int: int > int > int ).
thf(func_def_163,type,
bit_se6528837805403552850or_nat: nat > nat > nat ).
thf(func_def_164,type,
bit_se9216721137139052372nteger: code_integer > nat > $o ).
thf(func_def_165,type,
bit_se8040316288895769887atural: code_natural > nat > $o ).
thf(func_def_166,type,
bit_se1146084159140164899it_int: int > nat > $o ).
thf(func_def_167,type,
bit_se1148574629649215175it_nat: nat > nat > $o ).
thf(func_def_168,type,
bit_take_bit_num: nat > num > option_num ).
thf(func_def_169,type,
bit_un1837492267222099188nd_num: num > num > option_num ).
thf(func_def_170,type,
bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_171,type,
bit_un2785000775030745342or_num: num > num > num ).
thf(func_def_172,type,
bit_un6909899581280750971um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_173,type,
bit_un6178654185764691216or_num: num > num > option_num ).
thf(func_def_174,type,
bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_175,type,
bit_un7362597486090784418nd_num: num > num > option_num ).
thf(func_def_176,type,
bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_177,type,
bit_un6697907153464112080or_num: num > num > num ).
thf(func_def_178,type,
bit_un4773296044027857193um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_179,type,
bit_un2480387367778600638or_num: num > num > option_num ).
thf(func_def_180,type,
bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_181,type,
boolea2445317508997433345nteger: ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer ) > code_integer > code_integer > ( code_integer > code_integer > code_integer ) > $o ).
thf(func_def_182,type,
boolea8527374999097803216ff_int: ( int > int > int ) > ( int > int > int ) > ( int > int ) > int > int > ( int > int > int ) > $o ).
thf(func_def_183,type,
code_Neg: num > code_integer ).
thf(func_def_184,type,
code_Pos: num > code_integer ).
thf(func_def_185,type,
code_Suc: code_natural > code_natural ).
thf(func_def_186,type,
code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).
thf(func_def_187,type,
code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_188,type,
code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_189,type,
code_dup: code_integer > code_integer ).
thf(func_def_190,type,
code_int_of_integer: code_integer > int ).
thf(func_def_191,type,
code_integer_of_int: int > code_integer ).
thf(func_def_192,type,
code_integer_of_nat: nat > code_integer ).
thf(func_def_193,type,
code_i5400310926305786745atural: code_natural > code_integer ).
thf(func_def_194,type,
code_integer_of_num: num > code_integer ).
thf(func_def_195,type,
code_nat_of_integer: code_integer > nat ).
thf(func_def_196,type,
code_nat_of_natural: code_natural > nat ).
thf(func_def_197,type,
code_natural_of_nat: nat > code_natural ).
thf(func_def_198,type,
code_num_of_integer: code_integer > num ).
thf(func_def_199,type,
code_pcr_integer: int > code_integer > $o ).
thf(func_def_200,type,
code_pcr_natural: nat > code_natural > $o ).
thf(func_def_201,type,
code_sub: num > num > code_integer ).
thf(func_def_202,type,
code_Target_negative: num > int ).
thf(func_def_203,type,
code_Target_positive: num > int ).
thf(func_def_204,type,
code_T6385005292777649522of_nat: nat > int ).
thf(func_def_205,type,
comple4887499456419720421f_real: set_real > real ).
thf(func_def_206,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(func_def_207,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(func_def_208,type,
comple1385675409528146559p_real: set_real > real ).
thf(func_def_209,type,
arg: complex > real ).
thf(func_def_210,type,
cis: real > complex ).
thf(func_def_211,type,
cnj: complex > complex ).
thf(func_def_212,type,
complex2: real > real > complex ).
thf(func_def_213,type,
im: complex > real ).
thf(func_def_214,type,
re: complex > real ).
thf(func_def_215,type,
csqrt: complex > complex ).
thf(func_def_216,type,
imaginary_unit: complex ).
thf(func_def_217,type,
rcis: real > real > complex ).
thf(func_def_218,type,
nth_item_rel: nat > nat > $o ).
thf(func_def_219,type,
differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).
thf(func_def_220,type,
has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).
thf(func_def_221,type,
has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
thf(func_def_222,type,
has_ve631408500373753343e_real: ( real > real ) > real > filter_real > $o ).
thf(func_def_223,type,
adjust_div: product_prod_int_int > int ).
thf(func_def_224,type,
adjust_mod: int > int > int ).
thf(func_def_225,type,
divmod_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_226,type,
eucl_rel_int: int > int > product_prod_int_int > $o ).
thf(func_def_227,type,
unique5052692396658037445od_int: num > num > product_prod_int_int ).
thf(func_def_228,type,
unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).
thf(func_def_229,type,
unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_230,type,
unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).
thf(func_def_231,type,
unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_232,type,
euclid4774559944035922753ze_int: int > nat ).
thf(func_def_233,type,
euclid4777050414544973029ze_nat: nat > nat ).
thf(func_def_234,type,
euclid3395696857347342551nt_int: int > int ).
thf(func_def_235,type,
euclid3398187327856392827nt_nat: nat > nat ).
thf(func_def_236,type,
extended_eSuc: extended_enat > extended_enat ).
thf(func_def_237,type,
extended_enat2: nat > extended_enat ).
thf(func_def_238,type,
extend3600170679010898289d_enat: ( nat > extended_enat ) > extended_enat > extended_enat > extended_enat ).
thf(func_def_239,type,
extend5688581933313929465d_enat: extended_enat ).
thf(func_def_240,type,
comm_s8582702949713902594nteger: code_integer > nat > code_integer ).
thf(func_def_241,type,
comm_s2602460028002588243omplex: complex > nat > complex ).
thf(func_def_242,type,
comm_s4660882817536571857er_int: int > nat > int ).
thf(func_def_243,type,
comm_s4663373288045622133er_nat: nat > nat > nat ).
thf(func_def_244,type,
comm_s4028243227959126397er_rat: rat > nat > rat ).
thf(func_def_245,type,
comm_s7457072308508201937r_real: real > nat > real ).
thf(func_def_246,type,
semiri3624122377584611663nteger: nat > code_integer ).
thf(func_def_247,type,
semiri2447717529341329178atural: nat > code_natural ).
thf(func_def_248,type,
semiri5044797733671781792omplex: nat > complex ).
thf(func_def_249,type,
semiri1406184849735516958ct_int: nat > int ).
thf(func_def_250,type,
semiri1408675320244567234ct_nat: nat > nat ).
thf(func_def_251,type,
semiri773545260158071498ct_rat: nat > rat ).
thf(func_def_252,type,
semiri2265585572941072030t_real: nat > real ).
thf(func_def_253,type,
invers8013647133539491842omplex: complex > complex ).
thf(func_def_254,type,
inverse_inverse_rat: rat > rat ).
thf(func_def_255,type,
inverse_inverse_real: real > real ).
thf(func_def_256,type,
at_bot_real: filter_real ).
thf(func_def_257,type,
at_top_int: filter_int ).
thf(func_def_258,type,
at_top_nat: filter_nat ).
thf(func_def_259,type,
at_top_real: filter_real ).
thf(func_def_260,type,
eventually_nat: ( nat > $o ) > filter_nat > $o ).
thf(func_def_261,type,
eventually_real: ( real > $o ) > filter_real > $o ).
thf(func_def_262,type,
filterlim_nat_int: ( nat > int ) > filter_int > filter_nat > $o ).
thf(func_def_263,type,
filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
thf(func_def_264,type,
filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
thf(func_def_265,type,
filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
thf(func_def_266,type,
filtermap_real_real: ( real > real ) > filter_real > filter_real ).
thf(func_def_267,type,
finite_card_o: set_o > nat ).
thf(func_def_268,type,
finite_card_complex: set_complex > nat ).
thf(func_def_269,type,
finite_card_int: set_int > nat ).
thf(func_def_270,type,
finite_card_list_nat: set_list_nat > nat ).
thf(func_def_271,type,
finite_card_nat: set_nat > nat ).
thf(func_def_272,type,
finite410649719033368117t_unit: set_Product_unit > nat ).
thf(func_def_273,type,
finite_card_set_nat: set_set_nat > nat ).
thf(func_def_274,type,
finite_card_char: set_char > nat ).
thf(func_def_275,type,
finite3207457112153483333omplex: set_complex > $o ).
thf(func_def_276,type,
finite_finite_int: set_int > $o ).
thf(func_def_277,type,
finite_finite_nat: set_nat > $o ).
thf(func_def_278,type,
bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
thf(func_def_279,type,
bij_betw_int_nat: ( int > nat ) > set_int > set_nat > $o ).
thf(func_def_280,type,
bij_be8532844293280997160at_nat: ( list_nat > nat ) > set_list_nat > set_nat > $o ).
thf(func_def_281,type,
bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
thf(func_def_282,type,
bij_betw_nat_int: ( nat > int ) > set_nat > set_int > $o ).
thf(func_def_283,type,
bij_be6293887246118711976st_nat: ( nat > list_nat ) > set_nat > set_list_nat > $o ).
thf(func_def_284,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(func_def_285,type,
bij_be8693218025023041337at_nat: ( nat > product_prod_nat_nat ) > set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_286,type,
bij_be4790990086886966983at_nat: ( nat > sum_sum_nat_nat ) > set_nat > set_Sum_sum_nat_nat > $o ).
thf(func_def_287,type,
bij_be5333170631980326235at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > set_nat > $o ).
thf(func_def_288,type,
bij_be5432664580149595207at_nat: ( sum_sum_nat_nat > nat ) > set_Sum_sum_nat_nat > set_nat > $o ).
thf(func_def_289,type,
comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_290,type,
comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_291,type,
comp_int_int_num: ( int > int ) > ( num > int ) > num > int ).
thf(func_def_292,type,
comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
thf(func_def_293,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(func_def_294,type,
id_o: $o > $o ).
thf(func_def_295,type,
id_nat: nat > nat ).
thf(func_def_296,type,
inj_on_int_nat: ( int > nat ) > set_int > $o ).
thf(func_def_297,type,
inj_on_list_nat_nat: ( list_nat > nat ) > set_list_nat > $o ).
thf(func_def_298,type,
inj_on_nat_int: ( nat > int ) > set_nat > $o ).
thf(func_def_299,type,
inj_on_nat_list_nat: ( nat > list_nat ) > set_nat > $o ).
thf(func_def_300,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_301,type,
inj_on5538052773655684606at_nat: ( nat > product_prod_nat_nat ) > set_nat > $o ).
thf(func_def_302,type,
inj_on_nat_char: ( nat > char ) > set_nat > $o ).
thf(func_def_303,type,
inj_on5701776251185195458at_nat: ( nat > sum_sum_nat_nat ) > set_nat > $o ).
thf(func_def_304,type,
inj_on2178005380612969504at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_305,type,
inj_on_real_real: ( real > real ) > set_real > $o ).
thf(func_def_306,type,
inj_on_set_nat_nat: ( set_nat > nat ) > set_set_nat > $o ).
thf(func_def_307,type,
inj_on6343450744447823682at_nat: ( sum_sum_nat_nat > nat ) > set_Sum_sum_nat_nat > $o ).
thf(func_def_308,type,
map_fu8272188784021352819nteger: ( code_integer > int ) > ( ( int > int ) > code_integer > code_integer ) > ( int > int > int ) > code_integer > code_integer > code_integer ).
thf(func_def_309,type,
map_fu2599414010547811884nteger: ( code_integer > int ) > ( int > code_integer ) > ( int > int ) > code_integer > code_integer ).
thf(func_def_310,type,
map_fu6549440983881763648atural: ( code_natural > nat ) > ( ( nat > nat ) > code_natural > code_natural ) > ( nat > nat > nat ) > code_natural > code_natural > code_natural ).
thf(func_def_311,type,
map_fu2787874002554666395nteger: ( code_natural > nat ) > ( int > code_integer ) > ( nat > int ) > code_natural > code_integer ).
thf(func_def_312,type,
map_fu1239815594074539274atural: ( code_natural > nat ) > ( nat > code_natural ) > ( nat > nat ) > code_natural > code_natural ).
thf(func_def_313,type,
map_fu434086159418415080_int_o: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > $o ) > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > int > int > $o ).
thf(func_def_314,type,
map_fu4960017516451851995nt_int: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > int > int > int ).
thf(func_def_315,type,
map_fu4826362097070443709at_o_o: ( int > product_prod_nat_nat ) > ( $o > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).
thf(func_def_316,type,
map_fu2345160673673942751at_nat: ( int > product_prod_nat_nat ) > ( nat > nat ) > ( product_prod_nat_nat > nat ) > int > nat ).
thf(func_def_317,type,
map_fu3667384564859982768at_int: ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int ) > ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ).
thf(func_def_318,type,
map_fu8224359078269604234nteger: ( nat > nat ) > ( ( int > int ) > code_integer > code_integer ) > ( nat > int > int ) > nat > code_integer > code_integer ).
thf(func_def_319,type,
map_fu6981160461238011112atural: ( nat > nat ) > ( ( nat > nat ) > code_natural > code_natural ) > ( nat > nat > nat ) > nat > code_natural > code_natural ).
thf(func_def_320,type,
map_fu6290471996055670595nteger: ( nat > nat ) > ( int > code_integer ) > ( nat > int ) > nat > code_integer ).
thf(func_def_321,type,
map_fu4333342158222067775at_rat: ( rat > product_prod_int_int ) > ( ( product_prod_int_int > product_prod_int_int ) > rat > rat ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > rat > rat > rat ).
thf(func_def_322,type,
map_fu898904425404107465nt_o_o: ( rat > product_prod_int_int ) > ( $o > $o ) > ( product_prod_int_int > $o ) > rat > $o ).
thf(func_def_323,type,
map_fu5673905371560938248nt_rat: ( rat > product_prod_int_int ) > ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > rat > rat ).
thf(func_def_324,type,
map_fu1532550112467129777l_real: ( real > nat > rat ) > ( ( ( nat > rat ) > nat > rat ) > real > real ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > real > real > real ).
thf(func_def_325,type,
map_fu7146612038024189824t_real: ( real > nat > rat ) > ( ( nat > rat ) > real ) > ( ( nat > rat ) > nat > rat ) > real > real ).
thf(func_def_326,type,
map_fu1856342031159181835at_o_o: ( real > nat > rat ) > ( $o > $o ) > ( ( nat > rat ) > $o ) > real > $o ).
thf(func_def_327,type,
the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
thf(func_def_328,type,
fun_is_measure_int: ( int > nat ) > $o ).
thf(func_def_329,type,
gcd_Gcd_int: set_int > int ).
thf(func_def_330,type,
gcd_Gcd_nat: set_nat > nat ).
thf(func_def_331,type,
gcd_Lcm_int: set_int > int ).
thf(func_def_332,type,
gcd_Lcm_nat: set_nat > nat ).
thf(func_def_333,type,
bezw: nat > nat > product_prod_int_int ).
thf(func_def_334,type,
bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_335,type,
gcd_gcd_Code_integer: code_integer > code_integer > code_integer ).
thf(func_def_336,type,
gcd_gcd_int: int > int > int ).
thf(func_def_337,type,
gcd_gcd_nat: nat > nat > nat ).
thf(func_def_338,type,
gcd_lcm_Code_integer: code_integer > code_integer > code_integer ).
thf(func_def_339,type,
gcd_lcm_int: int > int > int ).
thf(func_def_340,type,
gcd_lcm_nat: nat > nat > nat ).
thf(func_def_341,type,
gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_342,type,
abs_abs_Code_integer: code_integer > code_integer ).
thf(func_def_343,type,
abs_abs_complex: complex > complex ).
thf(func_def_344,type,
abs_abs_int: int > int ).
thf(func_def_345,type,
abs_abs_rat: rat > rat ).
thf(func_def_346,type,
abs_abs_real: real > real ).
thf(func_def_347,type,
comm_monoid_nat: ( nat > nat > nat ) > nat > $o ).
thf(func_def_348,type,
minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).
thf(func_def_349,type,
minus_7197305767214868737atural: code_natural > code_natural > code_natural ).
thf(func_def_350,type,
minus_minus_complex: complex > complex > complex ).
thf(func_def_351,type,
minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_352,type,
minus_minus_int: int > int > int ).
thf(func_def_353,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_354,type,
minus_minus_rat: rat > rat > rat ).
thf(func_def_355,type,
minus_minus_real: real > real > real ).
thf(func_def_356,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_357,type,
monoid_nat: ( nat > nat > nat ) > nat > $o ).
thf(func_def_358,type,
one_one_Code_integer: code_integer ).
thf(func_def_359,type,
one_one_Code_natural: code_natural ).
thf(func_def_360,type,
one_one_complex: complex ).
thf(func_def_361,type,
one_on7984719198319812577d_enat: extended_enat ).
thf(func_def_362,type,
one_one_int: int ).
thf(func_def_363,type,
one_one_nat: nat ).
thf(func_def_364,type,
one_one_rat: rat ).
thf(func_def_365,type,
one_one_real: real ).
thf(func_def_366,type,
plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).
thf(func_def_367,type,
plus_p4538020629002901425atural: code_natural > code_natural > code_natural ).
thf(func_def_368,type,
plus_plus_complex: complex > complex > complex ).
thf(func_def_369,type,
plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_370,type,
plus_plus_int: int > int > int ).
thf(func_def_371,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_372,type,
plus_plus_num: num > num > num ).
thf(func_def_373,type,
plus_plus_rat: rat > rat > rat ).
thf(func_def_374,type,
plus_plus_real: real > real > real ).
thf(func_def_375,type,
plus_plus_literal: literal > literal > literal ).
thf(func_def_376,type,
sgn_sgn_Code_integer: code_integer > code_integer ).
thf(func_def_377,type,
sgn_sgn_complex: complex > complex ).
thf(func_def_378,type,
sgn_sgn_int: int > int ).
thf(func_def_379,type,
sgn_sgn_rat: rat > rat ).
thf(func_def_380,type,
sgn_sgn_real: real > real ).
thf(func_def_381,type,
times_3573771949741848930nteger: code_integer > code_integer > code_integer ).
thf(func_def_382,type,
times_2397367101498566445atural: code_natural > code_natural > code_natural ).
thf(func_def_383,type,
times_times_complex: complex > complex > complex ).
thf(func_def_384,type,
times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_385,type,
times_times_int: int > int > int ).
thf(func_def_386,type,
times_times_nat: nat > nat > nat ).
thf(func_def_387,type,
times_times_num: num > num > num ).
thf(func_def_388,type,
times_times_rat: rat > rat > rat ).
thf(func_def_389,type,
times_times_real: real > real > real ).
thf(func_def_390,type,
uminus1351360451143612070nteger: code_integer > code_integer ).
thf(func_def_391,type,
uminus1482373934393186551omplex: complex > complex ).
thf(func_def_392,type,
uminus_uminus_int: int > int ).
thf(func_def_393,type,
uminus_uminus_rat: rat > rat ).
thf(func_def_394,type,
uminus_uminus_real: real > real ).
thf(func_def_395,type,
zero_z3403309356797280102nteger: code_integer ).
thf(func_def_396,type,
zero_z2226904508553997617atural: code_natural ).
thf(func_def_397,type,
zero_zero_complex: complex ).
thf(func_def_398,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(func_def_399,type,
zero_zero_int: int ).
thf(func_def_400,type,
zero_zero_nat: nat ).
thf(func_def_401,type,
zero_zero_rat: rat ).
thf(func_def_402,type,
zero_zero_real: real ).
thf(func_def_403,type,
zero_zero_literal: literal ).
thf(func_def_404,type,
groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
thf(func_def_405,type,
groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
thf(func_def_406,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_407,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(func_def_408,type,
groups1705073143266064639nt_int: ( int > int ) > set_int > int ).
thf(func_def_409,type,
groups705719431365010083at_int: ( nat > int ) > set_nat > int ).
thf(func_def_410,type,
groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_411,type,
groups3417619833198082522nteger: ( $o > code_integer ) > code_integer > list_o > code_integer ).
thf(func_def_412,type,
groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).
thf(func_def_413,type,
groups9119017779487936845_o_nat: ( $o > nat ) > nat > list_o > nat ).
thf(func_def_414,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(func_def_415,type,
the_int: ( int > $o ) > int ).
thf(func_def_416,type,
the_Pr4378521158711661632nt_int: ( product_prod_int_int > $o ) > product_prod_int_int ).
thf(func_def_417,type,
the_real: ( real > $o ) > real ).
thf(func_def_418,type,
if_int_int: $o > ( int > int ) > ( int > int ) > int > int ).
thf(func_def_419,type,
if_nat_int_int: $o > ( nat > int > int ) > ( nat > int > int ) > nat > int > int ).
thf(func_def_420,type,
if_nat_nat_nat: $o > ( nat > nat > nat ) > ( nat > nat > nat ) > nat > nat > nat ).
thf(func_def_421,type,
if_nat_rat: $o > ( nat > rat ) > ( nat > rat ) > nat > rat ).
thf(func_def_422,type,
if_Code_integer: $o > code_integer > code_integer > code_integer ).
thf(func_def_423,type,
if_Code_natural: $o > code_natural > code_natural > code_natural ).
thf(func_def_424,type,
if_complex: $o > complex > complex > complex ).
thf(func_def_425,type,
if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).
thf(func_def_426,type,
if_int: $o > int > int > int ).
thf(func_def_427,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(func_def_428,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(func_def_429,type,
if_nat: $o > nat > nat > nat ).
thf(func_def_430,type,
if_num: $o > num > num > num ).
thf(func_def_431,type,
if_option_num: $o > option_num > option_num > option_num ).
thf(func_def_432,type,
if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).
thf(func_def_433,type,
if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_434,type,
if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
thf(func_def_435,type,
if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_436,type,
if_rat: $o > rat > rat > rat ).
thf(func_def_437,type,
if_real: $o > real > real > real ).
thf(func_def_438,type,
if_set_int: $o > set_int > set_int > set_int ).
thf(func_def_439,type,
if_Sum_sum_nat_nat: $o > sum_sum_nat_nat > sum_sum_nat_nat > sum_sum_nat_nat ).
thf(func_def_440,type,
abs_Integ: product_prod_nat_nat > int ).
thf(func_def_441,type,
neg: num > int ).
thf(func_def_442,type,
pos: num > int ).
thf(func_def_443,type,
rep_Integ: int > product_prod_nat_nat ).
thf(func_def_444,type,
cr_int: product_prod_nat_nat > int > $o ).
thf(func_def_445,type,
dup: int > int ).
thf(func_def_446,type,
int_ge_less_than: int > set_Pr958786334691620121nt_int ).
thf(func_def_447,type,
int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
thf(func_def_448,type,
intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_449,type,
nat2: int > nat ).
thf(func_def_450,type,
pcr_int: product_prod_nat_nat > int > $o ).
thf(func_def_451,type,
power_int_real: real > int > real ).
thf(func_def_452,type,
ring_11222124179247155820nteger: set_Code_integer ).
thf(func_def_453,type,
ring_1_Ints_complex: set_complex ).
thf(func_def_454,type,
ring_1_Ints_int: set_int ).
thf(func_def_455,type,
ring_1_Ints_rat: set_rat ).
thf(func_def_456,type,
ring_1_Ints_real: set_real ).
thf(func_def_457,type,
ring_18347121197199848620nteger: int > code_integer ).
thf(func_def_458,type,
ring_17405671764205052669omplex: int > complex ).
thf(func_def_459,type,
ring_1_of_int_int: int > int ).
thf(func_def_460,type,
ring_1_of_int_rat: int > rat ).
thf(func_def_461,type,
ring_1_of_int_real: int > real ).
thf(func_def_462,type,
sub: num > num > int ).
thf(func_def_463,type,
inf_inf_int: int > int > int ).
thf(func_def_464,type,
inf_inf_nat: nat > nat > nat ).
thf(func_def_465,type,
semila9081495762789891438tr_nat: ( nat > nat > nat ) > nat > $o ).
thf(func_def_466,type,
semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(func_def_467,type,
semila1248733672344298208er_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(func_def_468,type,
sup_sup_int: int > int > int ).
thf(func_def_469,type,
sup_sup_nat: nat > nat > nat ).
thf(func_def_470,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_471,type,
sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_472,type,
lattic8263393255366662781ax_int: set_int > int ).
thf(func_def_473,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(func_def_474,type,
lattic7826324295020591184_F_nat: ( nat > nat > nat ) > nat > set_nat > nat ).
thf(func_def_475,type,
quotie3684837364556693515t_real: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > real ) > ( real > nat > rat ) > ( ( nat > rat ) > real > $o ) > $o ).
thf(func_def_476,type,
quotie1194848508323700631at_int: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > int ) > ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int > $o ) > $o ).
thf(func_def_477,type,
bfun_nat_real: ( nat > real ) > filter_nat > $o ).
thf(func_def_478,type,
at_infinity_real: filter_real ).
thf(func_def_479,type,
append_int: list_int > list_int > list_int ).
thf(func_def_480,type,
append_nat: list_nat > list_nat > list_nat ).
thf(func_def_481,type,
drop_nat: nat > list_nat > list_nat ).
thf(func_def_482,type,
fold_int_int: ( int > int > int ) > list_int > int > int ).
thf(func_def_483,type,
fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
thf(func_def_484,type,
last_nat: list_nat > nat ).
thf(func_def_485,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(func_def_486,type,
cons_int: int > list_int > list_int ).
thf(func_def_487,type,
cons_nat: nat > list_nat > list_nat ).
thf(func_def_488,type,
nil_int: list_int ).
thf(func_def_489,type,
nil_nat: list_nat ).
thf(func_def_490,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(func_def_491,type,
set_o2: list_o > set_o ).
thf(func_def_492,type,
set_complex2: list_complex > set_complex ).
thf(func_def_493,type,
set_int2: list_int > set_int ).
thf(func_def_494,type,
set_nat2: list_nat > set_nat ).
thf(func_def_495,type,
set_real2: list_real > set_real ).
thf(func_def_496,type,
set_set_nat2: list_set_nat > set_set_nat ).
thf(func_def_497,type,
set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
thf(func_def_498,type,
size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
thf(func_def_499,type,
tl_nat: list_nat > list_nat ).
thf(func_def_500,type,
nth_o: list_o > nat > $o ).
thf(func_def_501,type,
nth_complex: list_complex > nat > complex ).
thf(func_def_502,type,
nth_int: list_int > nat > int ).
thf(func_def_503,type,
nth_nat: list_nat > nat > nat ).
thf(func_def_504,type,
nth_real: list_real > nat > real ).
thf(func_def_505,type,
nth_set_nat: list_set_nat > nat > set_nat ).
thf(func_def_506,type,
nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_507,type,
replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_508,type,
rotate1_o: list_o > list_o ).
thf(func_def_509,type,
rotate1_int: list_int > list_int ).
thf(func_def_510,type,
rotate1_nat: list_nat > list_nat ).
thf(func_def_511,type,
rotate1_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_512,type,
subseqs_o: list_o > list_list_o ).
thf(func_def_513,type,
subseqs_int: list_int > list_list_int ).
thf(func_def_514,type,
subseqs_nat: list_nat > list_list_nat ).
thf(func_def_515,type,
subseqs_VEBT_VEBT: list_VEBT_VEBT > list_list_VEBT_VEBT ).
thf(func_def_516,type,
take_nat: nat > list_nat > list_nat ).
thf(func_def_517,type,
upt: nat > nat > list_nat ).
thf(func_def_518,type,
upto: int > int > list_int ).
thf(func_def_519,type,
upto_aux: int > int > list_int > list_int ).
thf(func_def_520,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_521,type,
suc: nat > nat ).
thf(func_def_522,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(func_def_523,type,
case_nat_o: $o > ( nat > $o ) > nat > $o ).
thf(func_def_524,type,
case_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(func_def_525,type,
case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
thf(func_def_526,type,
pred: nat > nat ).
thf(func_def_527,type,
semiri3842193898606819883omplex: set_complex ).
thf(func_def_528,type,
semiring_1_Nats_int: set_int ).
thf(func_def_529,type,
semiri4939895301339042750nteger: nat > code_integer ).
thf(func_def_530,type,
semiri3763490453095760265atural: nat > code_natural ).
thf(func_def_531,type,
semiri8010041392384452111omplex: nat > complex ).
thf(func_def_532,type,
semiri1314217659103216013at_int: nat > int ).
thf(func_def_533,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(func_def_534,type,
semiri681578069525770553at_rat: nat > rat ).
thf(func_def_535,type,
semiri5074537144036343181t_real: nat > real ).
thf(func_def_536,type,
size_size_list_o: list_o > nat ).
thf(func_def_537,type,
size_s3451745648224563538omplex: list_complex > nat ).
thf(func_def_538,type,
size_size_list_int: list_int > nat ).
thf(func_def_539,type,
size_s2710708370519433104list_o: list_list_o > nat ).
thf(func_def_540,type,
size_s533118279054570080st_int: list_list_int > nat ).
thf(func_def_541,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(func_def_542,type,
size_s8217280938318005548T_VEBT: list_list_VEBT_VEBT > nat ).
thf(func_def_543,type,
size_size_list_nat: list_nat > nat ).
thf(func_def_544,type,
size_size_list_real: list_real > nat ).
thf(func_def_545,type,
size_s3254054031482475050et_nat: list_set_nat > nat ).
thf(func_def_546,type,
size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
thf(func_def_547,type,
size_size_num: num > nat ).
thf(func_def_548,type,
size_size_char: char > nat ).
thf(func_def_549,type,
size_size_VEBT_VEBT: vEBT_VEBT > nat ).
thf(func_def_550,type,
nat_int_decode: nat > int ).
thf(func_def_551,type,
nat_int_encode: int > nat ).
thf(func_def_552,type,
nat_list_decode: nat > list_nat ).
thf(func_def_553,type,
nat_list_decode_rel: nat > nat > $o ).
thf(func_def_554,type,
nat_list_encode: list_nat > nat ).
thf(func_def_555,type,
nat_list_encode_rel: list_nat > list_nat > $o ).
thf(func_def_556,type,
nat_prod_decode: nat > product_prod_nat_nat ).
thf(func_def_557,type,
nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
thf(func_def_558,type,
nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_559,type,
nat_prod_encode: product_prod_nat_nat > nat ).
thf(func_def_560,type,
nat_set_decode: nat > set_nat ).
thf(func_def_561,type,
nat_set_encode: set_nat > nat ).
thf(func_def_562,type,
nat_sum_decode: nat > sum_sum_nat_nat ).
thf(func_def_563,type,
nat_sum_encode: sum_sum_nat_nat > nat ).
thf(func_def_564,type,
nat_triangle: nat > nat ).
thf(func_def_565,type,
root: nat > real > real ).
thf(func_def_566,type,
sqrt: real > real ).
thf(func_def_567,type,
bitM: num > num ).
thf(func_def_568,type,
inc: num > num ).
thf(func_def_569,type,
nat_of_num: num > nat ).
thf(func_def_570,type,
neg_nu8804712462038260780nteger: code_integer > code_integer ).
thf(func_def_571,type,
neg_nu7009210354673126013omplex: complex > complex ).
thf(func_def_572,type,
neg_numeral_dbl_int: int > int ).
thf(func_def_573,type,
neg_numeral_dbl_rat: rat > rat ).
thf(func_def_574,type,
neg_numeral_dbl_real: real > real ).
thf(func_def_575,type,
neg_nu7757733837767384882nteger: code_integer > code_integer ).
thf(func_def_576,type,
neg_nu6511756317524482435omplex: complex > complex ).
thf(func_def_577,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(func_def_578,type,
neg_nu3179335615603231917ec_rat: rat > rat ).
thf(func_def_579,type,
neg_nu6075765906172075777c_real: real > real ).
thf(func_def_580,type,
neg_nu5831290666863070958nteger: code_integer > code_integer ).
thf(func_def_581,type,
neg_nu8557863876264182079omplex: complex > complex ).
thf(func_def_582,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(func_def_583,type,
neg_nu5219082963157363817nc_rat: rat > rat ).
thf(func_def_584,type,
neg_nu8295874005876285629c_real: real > real ).
thf(func_def_585,type,
neg_nu5755505904847501662nteger: num > num > code_integer ).
thf(func_def_586,type,
neg_nu8416839295433526191omplex: num > num > complex ).
thf(func_def_587,type,
neg_numeral_sub_int: num > num > int ).
thf(func_def_588,type,
neg_numeral_sub_rat: num > num > rat ).
thf(func_def_589,type,
neg_numeral_sub_real: num > num > real ).
thf(func_def_590,type,
bit0: num > num ).
thf(func_def_591,type,
bit1: num > num ).
thf(func_def_592,type,
one: num ).
thf(func_def_593,type,
case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
thf(func_def_594,type,
size_num: num > nat ).
thf(func_def_595,type,
num_of_nat: nat > num ).
thf(func_def_596,type,
numera6620942414471956472nteger: num > code_integer ).
thf(func_def_597,type,
numera5444537566228673987atural: num > code_natural ).
thf(func_def_598,type,
numera6690914467698888265omplex: num > complex ).
thf(func_def_599,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(func_def_600,type,
numeral_numeral_int: num > int ).
thf(func_def_601,type,
numeral_numeral_nat: num > nat ).
thf(func_def_602,type,
numeral_numeral_rat: num > rat ).
thf(func_def_603,type,
numeral_numeral_real: num > real ).
thf(func_def_604,type,
pow: num > num > num ).
thf(func_def_605,type,
pred_numeral: num > nat ).
thf(func_def_606,type,
sqr: num > num ).
thf(func_def_607,type,
none_num: option_num ).
thf(func_def_608,type,
none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
thf(func_def_609,type,
some_num: num > option_num ).
thf(func_def_610,type,
some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
thf(func_def_611,type,
case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
thf(func_def_612,type,
case_option_int_num: int > ( num > int ) > option_num > int ).
thf(func_def_613,type,
case_option_num_num: num > ( num > num ) > option_num > num ).
thf(func_def_614,type,
case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
thf(func_def_615,type,
map_option_num_num: ( num > num ) > option_num > option_num ).
thf(func_def_616,type,
bot_bot_nat: nat ).
thf(func_def_617,type,
bot_bot_set_int: set_int ).
thf(func_def_618,type,
bot_bot_set_nat: set_nat ).
thf(func_def_619,type,
bot_bot_set_real: set_real ).
thf(func_def_620,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(func_def_621,type,
ord_Least_real: ( real > $o ) > real ).
thf(func_def_622,type,
ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
thf(func_def_623,type,
ord_le5570908160329646204atural: code_natural > code_natural > $o ).
thf(func_def_624,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
thf(func_def_625,type,
ord_less_int: int > int > $o ).
thf(func_def_626,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_627,type,
ord_less_num: num > num > $o ).
thf(func_def_628,type,
ord_less_rat: rat > rat > $o ).
thf(func_def_629,type,
ord_less_real: real > real > $o ).
thf(func_def_630,type,
ord_le3102999989581377725nteger: code_integer > code_integer > $o ).
thf(func_def_631,type,
ord_le1926595141338095240atural: code_natural > code_natural > $o ).
thf(func_def_632,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(func_def_633,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_634,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_635,type,
ord_less_eq_num: num > num > $o ).
thf(func_def_636,type,
ord_less_eq_rat: rat > rat > $o ).
thf(func_def_637,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_638,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_639,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(func_def_640,type,
ord_max_Code_integer: code_integer > code_integer > code_integer ).
thf(func_def_641,type,
ord_max_int: int > int > int ).
thf(func_def_642,type,
ord_max_nat: nat > nat > nat ).
thf(func_def_643,type,
ord_max_num: num > num > num ).
thf(func_def_644,type,
ord_max_rat: rat > rat > rat ).
thf(func_def_645,type,
ord_max_real: real > real > real ).
thf(func_def_646,type,
ord_min_int: int > int > int ).
thf(func_def_647,type,
ord_min_nat: nat > nat > nat ).
thf(func_def_648,type,
order_mono_nat_nat: ( nat > nat ) > $o ).
thf(func_def_649,type,
order_mono_real_real: ( real > real ) > $o ).
thf(func_def_650,type,
order_5726023648592871131at_nat: ( nat > nat ) > $o ).
thf(func_def_651,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(func_def_652,type,
top_top_set_o: set_o ).
thf(func_def_653,type,
top_top_set_int: set_int ).
thf(func_def_654,type,
top_top_set_list_nat: set_list_nat ).
thf(func_def_655,type,
top_top_set_nat: set_nat ).
thf(func_def_656,type,
top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).
thf(func_def_657,type,
top_to1996260823553986621t_unit: set_Product_unit ).
thf(func_def_658,type,
top_top_set_real: set_real ).
thf(func_def_659,type,
top_top_set_char: set_char ).
thf(func_def_660,type,
top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).
thf(func_def_661,type,
power_8256067586552552935nteger: code_integer > nat > code_integer ).
thf(func_def_662,type,
power_7079662738309270450atural: code_natural > nat > code_natural ).
thf(func_def_663,type,
power_power_complex: complex > nat > complex ).
thf(func_def_664,type,
power_power_int: int > nat > int ).
thf(func_def_665,type,
power_power_nat: nat > nat > nat ).
thf(func_def_666,type,
power_power_rat: rat > nat > rat ).
thf(func_def_667,type,
power_power_real: real > nat > real ).
thf(func_def_668,type,
produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
thf(func_def_669,type,
produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_670,type,
produc3574140220909816553atural: code_natural > code_natural > produc7822875418678951345atural ).
thf(func_def_671,type,
produc6639722614265839536atural: code_natural > produc7822875418678951345atural > produc5835291356934675326atural ).
thf(func_def_672,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(func_def_673,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_674,type,
product_Pair_nat_num: nat > num > product_prod_nat_num ).
thf(func_def_675,type,
product_Pair_num_num: num > num > product_prod_num_num ).
thf(func_def_676,type,
produc4480994950612372183atural: produc7822875418678951345atural > produc7822875418678951345atural > produc6756470014200349799atural ).
thf(func_def_677,type,
produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
thf(func_def_678,type,
produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).
thf(func_def_679,type,
produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_680,type,
produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
thf(func_def_681,type,
produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
thf(func_def_682,type,
produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
thf(func_def_683,type,
produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
thf(func_def_684,type,
produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_685,type,
produc8282080750456430313atural: ( code_natural > code_natural > produc6756470014200349799atural ) > produc7822875418678951345atural > produc6756470014200349799atural ).
thf(func_def_686,type,
produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(func_def_687,type,
produc8211389475949308722nt_int: ( int > int > int ) > product_prod_int_int > int ).
thf(func_def_688,type,
produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
thf(func_def_689,type,
produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_690,type,
produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_691,type,
produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_692,type,
produc2761476792215241774st_nat: ( nat > nat > list_nat ) > product_prod_nat_nat > list_nat ).
thf(func_def_693,type,
produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
thf(func_def_694,type,
produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_695,type,
produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
thf(func_def_696,type,
product_fst_int_int: product_prod_int_int > int ).
thf(func_def_697,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(func_def_698,type,
produc6591912806276919810atural: produc5835291356934675326atural > produc7822875418678951345atural ).
thf(func_def_699,type,
product_snd_int_int: product_prod_int_int > int ).
thf(func_def_700,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(func_def_701,type,
produc5538323210962509403atural: ( produc7822875418678951345atural > produc5835291356934675326atural ) > ( code_natural > produc7822875418678951345atural > produc5835291356934675326atural ) > produc7822875418678951345atural > produc5835291356934675326atural ).
thf(func_def_702,type,
quotie8700032322157300518t_real: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > real ) > ( real > nat > rat ) > $o ).
thf(func_def_703,type,
quotie6776551016481293500at_int: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > int ) > ( int > product_prod_nat_nat ) > $o ).
thf(func_def_704,type,
inc_shift: code_natural > code_natural > code_natural ).
thf(func_def_705,type,
iterat8892046348760725948atural: code_natural > ( code_natural > produc7822875418678951345atural > produc5835291356934675326atural ) > code_natural > produc7822875418678951345atural > produc5835291356934675326atural ).
thf(func_def_706,type,
log: code_natural > code_natural > code_natural ).
thf(func_def_707,type,
log_rel: produc7822875418678951345atural > produc7822875418678951345atural > $o ).
thf(func_def_708,type,
minus_shift: code_natural > code_natural > code_natural > code_natural ).
thf(func_def_709,type,
next: produc7822875418678951345atural > produc5835291356934675326atural ).
thf(func_def_710,type,
range: code_natural > produc7822875418678951345atural > produc5835291356934675326atural ).
thf(func_def_711,type,
split_seed: produc7822875418678951345atural > produc6756470014200349799atural ).
thf(func_def_712,type,
abs_Rat: product_prod_int_int > rat ).
thf(func_def_713,type,
fract: int > int > rat ).
thf(func_def_714,type,
frct: product_prod_int_int > rat ).
thf(func_def_715,type,
rep_Rat: rat > product_prod_int_int ).
thf(func_def_716,type,
field_5140801741446780682s_real: set_real ).
thf(func_def_717,type,
field_7254667332652039916t_real: rat > real ).
thf(func_def_718,type,
normalize: product_prod_int_int > product_prod_int_int ).
thf(func_def_719,type,
of_int: int > rat ).
thf(func_def_720,type,
pcr_rat: product_prod_int_int > rat > $o ).
thf(func_def_721,type,
positive: rat > $o ).
thf(func_def_722,type,
quotient_of: rat > product_prod_int_int ).
thf(func_def_723,type,
ratrel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_724,type,
ratreal: rat > real ).
thf(func_def_725,type,
real2: ( nat > rat ) > real ).
thf(func_def_726,type,
cauchy: ( nat > rat ) > $o ).
thf(func_def_727,type,
cr_real: ( nat > rat ) > real > $o ).
thf(func_def_728,type,
pcr_real: ( nat > rat ) > real > $o ).
thf(func_def_729,type,
positive2: real > $o ).
thf(func_def_730,type,
realrel: ( nat > rat ) > ( nat > rat ) > $o ).
thf(func_def_731,type,
rep_real: real > nat > rat ).
thf(func_def_732,type,
vanishes: ( nat > rat ) > $o ).
thf(func_def_733,type,
real_V2521375963428798218omplex: set_complex ).
thf(func_def_734,type,
real_V470468836141973256s_real: set_real ).
thf(func_def_735,type,
real_V5970128139526366754l_real: ( real > real ) > $o ).
thf(func_def_736,type,
real_V3694042436643373181omplex: complex > complex > real ).
thf(func_def_737,type,
real_V975177566351809787t_real: real > real > real ).
thf(func_def_738,type,
real_V4572627801940501177l_real: ( real > real ) > $o ).
thf(func_def_739,type,
real_V1022390504157884413omplex: complex > real ).
thf(func_def_740,type,
real_V7735802525324610683m_real: real > real ).
thf(func_def_741,type,
real_V4546457046886955230omplex: real > complex ).
thf(func_def_742,type,
real_V1803761363581548252l_real: real > real ).
thf(func_def_743,type,
real_V2046097035970521341omplex: real > complex > complex ).
thf(func_def_744,type,
real_V1485227260804924795R_real: real > real > real ).
thf(func_def_745,type,
transp_nat_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > $o ).
thf(func_def_746,type,
algebr932160517623751201me_int: int > int > $o ).
thf(func_def_747,type,
algebr934650988132801477me_nat: nat > nat > $o ).
thf(func_def_748,type,
divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
thf(func_def_749,type,
divide5121882707175180666atural: code_natural > code_natural > code_natural ).
thf(func_def_750,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(func_def_751,type,
divide_divide_int: int > int > int ).
thf(func_def_752,type,
divide_divide_nat: nat > nat > nat ).
thf(func_def_753,type,
divide_divide_rat: rat > rat > rat ).
thf(func_def_754,type,
divide_divide_real: real > real > real ).
thf(func_def_755,type,
dvd_dvd_Code_integer: code_integer > code_integer > $o ).
thf(func_def_756,type,
dvd_dvd_Code_natural: code_natural > code_natural > $o ).
thf(func_def_757,type,
dvd_dvd_complex: complex > complex > $o ).
thf(func_def_758,type,
dvd_dvd_int: int > int > $o ).
thf(func_def_759,type,
dvd_dvd_nat: nat > nat > $o ).
thf(func_def_760,type,
dvd_dvd_rat: rat > rat > $o ).
thf(func_def_761,type,
dvd_dvd_real: real > real > $o ).
thf(func_def_762,type,
modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
thf(func_def_763,type,
modulo8411746178871703098atural: code_natural > code_natural > code_natural ).
thf(func_def_764,type,
modulo_modulo_int: int > int > int ).
thf(func_def_765,type,
modulo_modulo_nat: nat > nat > nat ).
thf(func_def_766,type,
normal571682499473568032ze_int: int > int ).
thf(func_def_767,type,
normal574172969982618308ze_nat: nat > nat ).
thf(func_def_768,type,
unit_f2746056213392204926or_int: int > int ).
thf(func_def_769,type,
unit_f2748546683901255202or_nat: nat > nat ).
thf(func_def_770,type,
zero_n356916108424825756nteger: $o > code_integer ).
thf(func_def_771,type,
zero_n8403883297036319079atural: $o > code_natural ).
thf(func_def_772,type,
zero_n1201886186963655149omplex: $o > complex ).
thf(func_def_773,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(func_def_774,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(func_def_775,type,
zero_n2052037380579107095ol_rat: $o > rat ).
thf(func_def_776,type,
zero_n3304061248610475627l_real: $o > real ).
thf(func_def_777,type,
suminf_real: ( nat > real ) > real ).
thf(func_def_778,type,
summable_real: ( nat > real ) > $o ).
thf(func_def_779,type,
sums_real: ( nat > real ) > real > $o ).
thf(func_def_780,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(func_def_781,type,
collect_int: ( int > $o ) > set_int ).
thf(func_def_782,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(func_def_783,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_784,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(func_def_785,type,
collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(func_def_786,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_787,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(func_def_788,type,
pow_nat: set_nat > set_set_nat ).
thf(func_def_789,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(func_def_790,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(func_def_791,type,
image_list_nat_nat: ( list_nat > nat ) > set_list_nat > set_nat ).
thf(func_def_792,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(func_def_793,type,
image_nat_list_nat: ( nat > list_nat ) > set_nat > set_list_nat ).
thf(func_def_794,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_795,type,
image_5846123807819985514at_nat: ( nat > product_prod_nat_nat ) > set_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_796,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(func_def_797,type,
image_nat_char: ( nat > char ) > set_nat > set_char ).
thf(func_def_798,type,
image_678696785212003926at_nat: ( nat > sum_sum_nat_nat ) > set_nat > set_Sum_sum_nat_nat ).
thf(func_def_799,type,
image_2486076414777270412at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > set_nat ).
thf(func_def_800,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(func_def_801,type,
image_char_nat: ( char > nat ) > set_char > set_nat ).
thf(func_def_802,type,
image_1320371278474632150at_nat: ( sum_sum_nat_nat > nat ) > set_Sum_sum_nat_nat > set_nat ).
thf(func_def_803,type,
insert_int: int > set_int > set_int ).
thf(func_def_804,type,
insert_nat: nat > set_nat > set_nat ).
thf(func_def_805,type,
insert_real: real > set_real > set_real ).
thf(func_def_806,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_807,type,
set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
thf(func_def_808,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(func_def_809,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(func_def_810,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(func_def_811,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(func_def_812,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(func_def_813,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(func_def_814,type,
set_ord_atLeast_real: real > set_real ).
thf(func_def_815,type,
set_ord_atMost_nat: nat > set_nat ).
thf(func_def_816,type,
set_or6656581121297822940st_int: int > int > set_int ).
thf(func_def_817,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(func_def_818,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(func_def_819,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(func_def_820,type,
set_or1633881224788618240n_real: real > real > set_real ).
thf(func_def_821,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(func_def_822,type,
set_or5849166863359141190n_real: real > set_real ).
thf(func_def_823,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(func_def_824,type,
set_or5984915006950818249n_real: real > set_real ).
thf(func_def_825,type,
abort_real: literal > ( product_unit > real ) > real ).
thf(func_def_826,type,
literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
thf(func_def_827,type,
ascii_of: char > char ).
thf(func_def_828,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(func_def_829,type,
size_char: char > nat ).
thf(func_def_830,type,
comm_s629917340098488124ar_nat: char > nat ).
thf(func_def_831,type,
integer_of_char: char > code_integer ).
thf(func_def_832,type,
unique3096191561947761185of_nat: nat > char ).
thf(func_def_833,type,
sum_Inl_nat_nat: nat > sum_sum_nat_nat ).
thf(func_def_834,type,
sum_Inr_nat_nat: nat > sum_sum_nat_nat ).
thf(func_def_835,type,
sum_ca7763040182479039464nt_nat: ( nat > int ) > ( nat > int ) > sum_sum_nat_nat > int ).
thf(func_def_836,type,
sum_ca6763686470577984908at_nat: ( nat > nat ) > ( nat > nat ) > sum_sum_nat_nat > nat ).
thf(func_def_837,type,
topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
thf(func_def_838,type,
topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
thf(func_def_839,type,
topolo6980174941875973593q_real: ( nat > real ) > $o ).
thf(func_def_840,type,
topolo2177554685111907308n_real: real > set_real > filter_real ).
thf(func_def_841,type,
topolo7531315842566124627t_real: ( nat > real ) > $o ).
thf(func_def_842,type,
topolo2815343760600316023s_real: real > filter_real ).
thf(func_def_843,type,
topolo6517432010174082258omplex: ( nat > complex ) > $o ).
thf(func_def_844,type,
topolo4055970368930404560y_real: ( nat > real ) > $o ).
thf(func_def_845,type,
arccos: real > real ).
thf(func_def_846,type,
arcosh_real: real > real ).
thf(func_def_847,type,
arcsin: real > real ).
thf(func_def_848,type,
arctan: real > real ).
thf(func_def_849,type,
arsinh_real: real > real ).
thf(func_def_850,type,
artanh_real: real > real ).
thf(func_def_851,type,
cos_complex: complex > complex ).
thf(func_def_852,type,
cos_real: real > real ).
thf(func_def_853,type,
cos_coeff: nat > real ).
thf(func_def_854,type,
cosh_complex: complex > complex ).
thf(func_def_855,type,
cosh_real: real > real ).
thf(func_def_856,type,
cot_complex: complex > complex ).
thf(func_def_857,type,
cot_real: real > real ).
thf(func_def_858,type,
exp_complex: complex > complex ).
thf(func_def_859,type,
exp_real: real > real ).
thf(func_def_860,type,
ln_ln_real: real > real ).
thf(func_def_861,type,
log2: real > real > real ).
thf(func_def_862,type,
pi: real ).
thf(func_def_863,type,
powr_real: real > real > real ).
thf(func_def_864,type,
powr_real2: real > real > real ).
thf(func_def_865,type,
sin_complex: complex > complex ).
thf(func_def_866,type,
sin_real: real > real ).
thf(func_def_867,type,
sin_coeff: nat > real ).
thf(func_def_868,type,
sinh_complex: complex > complex ).
thf(func_def_869,type,
sinh_real: real > real ).
thf(func_def_870,type,
tan_complex: complex > complex ).
thf(func_def_871,type,
tan_real: real > real ).
thf(func_def_872,type,
tanh_complex: complex > complex ).
thf(func_def_873,type,
tanh_real: real > real ).
thf(func_def_874,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(func_def_875,type,
vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_876,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(func_def_877,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(func_def_878,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(func_def_879,type,
vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
thf(func_def_880,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(func_def_881,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(func_def_882,type,
vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_883,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(func_def_884,type,
vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_885,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(func_def_886,type,
vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_887,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(func_def_888,type,
vEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(func_def_889,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(func_def_890,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(func_def_891,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
thf(func_def_892,type,
accp_nat: ( nat > nat > $o ) > nat > $o ).
thf(func_def_893,type,
accp_P8126237942716283194atural: ( produc7822875418678951345atural > produc7822875418678951345atural > $o ) > produc7822875418678951345atural > $o ).
thf(func_def_894,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(func_def_895,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_896,type,
accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
thf(func_def_897,type,
accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
thf(func_def_898,type,
member_o: $o > set_o > $o ).
thf(func_def_899,type,
member_Code_integer: code_integer > set_Code_integer > $o ).
thf(func_def_900,type,
member_complex: complex > set_complex > $o ).
thf(func_def_901,type,
member_int: int > set_int > $o ).
thf(func_def_902,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(func_def_903,type,
member_nat: nat > set_nat > $o ).
thf(func_def_904,type,
member_rat: rat > set_rat > $o ).
thf(func_def_905,type,
member_real: real > set_real > $o ).
thf(func_def_906,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(func_def_907,type,
member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
thf(func_def_908,type,
va: nat ).
thf(func_def_909,type,
y: nat ).
thf(func_def_928,type,
sP0: vEBT_VEBT > nat > $o ).
thf(func_def_929,type,
sP1: nat > vEBT_VEBT > $o ).
thf(func_def_930,type,
sP2: real > real > real > $o ).
thf(func_def_931,type,
sP3: nat > ( nat > nat ) > $o ).
thf(func_def_932,type,
sP4: option_num > num > num > $o ).
thf(func_def_933,type,
sP5: num > num > option_num > $o ).
thf(func_def_934,type,
sP6: num > option_num > num > $o ).
thf(func_def_935,type,
sP7: num > option_num > num > $o ).
thf(func_def_936,type,
sP8: num > num > option_num > $o ).
thf(func_def_937,type,
sP9: option_num > num > num > $o ).
thf(func_def_938,type,
sP10: num > option_num > num > $o ).
thf(func_def_939,type,
sP11: num > option_num > num > $o ).
thf(func_def_940,type,
sP12: ( real > real ) > ( nat > real > real ) > nat > real > real > $o ).
thf(func_def_941,type,
sP13: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_942,type,
sP14: vEBT_VEBT > nat > $o ).
thf(func_def_943,type,
sP15: nat > nat > list_VEBT_VEBT > nat > nat > $o ).
thf(func_def_944,type,
sP16: list_VEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_945,type,
sP17: real > nat > ( nat > real > real ) > $o ).
thf(func_def_946,type,
sP18: $o > nat > vEBT_VEBT > option4927543243414619207at_nat > list_VEBT_VEBT > nat > $o ).
thf(func_def_947,type,
sP19: vEBT_VEBT > $o > nat > $o ).
thf(func_def_948,type,
sP20: real > real > real > $o ).
thf(func_def_949,type,
sP21: nat > nat > ( nat > int ) > int > $o ).
thf(func_def_950,type,
sP22: nat > list_nat > $o ).
thf(func_def_951,type,
sP23: vEBT_VEBT > nat > $o ).
thf(func_def_952,type,
sP24: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_953,type,
sP25: rat > rat > rat > $o ).
thf(func_def_954,type,
sP26: real > num > real > $o ).
thf(func_def_955,type,
sP27: rat > num > rat > $o ).
thf(func_def_956,type,
sP28: num > real > real > $o ).
thf(func_def_957,type,
sP29: nat > list_VEBT_VEBT > nat > nat > nat > $o ).
thf(func_def_958,type,
sP30: nat > list_VEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_959,type,
sP31: nat > nat > nat > list_VEBT_VEBT > nat > $o ).
thf(func_def_960,type,
sP32: list_VEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_961,type,
sP33: nat > vEBT_VEBT > $o ).
thf(func_def_962,type,
sP34: vEBT_VEBT > nat > $o ).
thf(func_def_963,type,
sP35: nat > vEBT_VEBT > $o ).
thf(func_def_964,type,
sP36: nat > vEBT_VEBT > $o ).
thf(func_def_965,type,
sP37: int > int > ( int > $o ) > $o ).
thf(func_def_966,type,
sP38: int > int > ( int > $o ) > $o ).
thf(func_def_967,type,
sP39: num > option_num > num > $o ).
thf(func_def_968,type,
sP40: num > num > option_num > $o ).
thf(func_def_969,type,
sP41: num > num > option_num > $o ).
thf(func_def_970,type,
sP42: num > option_num > num > $o ).
thf(func_def_971,type,
sP43: num > option_num > num > $o ).
thf(func_def_972,type,
sP44: num > num > option_num > $o ).
thf(func_def_973,type,
sP45: num > option_num > num > $o ).
thf(func_def_974,type,
sP46: num > num > option_num > $o ).
thf(func_def_975,type,
sP47: rat > rat > rat > $o ).
thf(func_def_976,type,
sP48: rat > rat > rat > $o ).
thf(func_def_977,type,
sP49: int > int > int > $o ).
thf(func_def_978,type,
sP50: nat > vEBT_VEBT > $o ).
thf(func_def_979,type,
sP51: nat > vEBT_VEBT > $o ).
thf(func_def_980,type,
sP52: ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > $o ).
thf(func_def_981,type,
sP53: rat > rat > rat > $o ).
thf(func_def_982,type,
sP54: ( int > $o ) > int > int > $o ).
thf(func_def_983,type,
sP55: ( int > $o ) > int > int > $o ).
thf(func_def_984,type,
sP56: ( int > $o ) > int > int > $o ).
thf(func_def_985,type,
sP57: rat > num > rat > $o ).
thf(func_def_986,type,
sP58: set_int > int > ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_987,type,
sP59: set_int > int > ( int > $o ) > $o ).
thf(func_def_988,type,
sP60: ( int > $o ) > set_int > int > ( int > $o ) > $o ).
thf(func_def_989,type,
sP61: set_int > int > ( int > $o ) > $o ).
thf(func_def_990,type,
sP62: real > real > real > $o ).
thf(func_def_991,type,
sP63: real > real > real > $o ).
thf(func_def_992,type,
sP64: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > $o ).
thf(func_def_993,type,
sP65: int > product_prod_int_int > int > $o ).
thf(func_def_994,type,
sP66: real > ( nat > real > real ) > real > nat > $o ).
thf(func_def_995,type,
sP67: ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > $o ).
thf(func_def_996,type,
sP68: vEBT_VEBT > nat > $o ).
thf(func_def_997,type,
sP69: real > real > real > $o ).
thf(func_def_998,type,
sP70: rat > rat > rat > $o ).
thf(func_def_999,type,
sP71: real > num > real > $o ).
thf(func_def_1000,type,
sP72: real > real > ( real > real > $o ) > $o ).
thf(func_def_1001,type,
sP73: vEBT_VEBT > nat > $o ).
thf(func_def_1002,type,
sP74: vEBT_VEBT > nat > $o ).
thf(func_def_1003,type,
sP75: list_nat > nat > $o ).
thf(func_def_1004,type,
sP76: real > num > real > $o ).
thf(func_def_1005,type,
sP77: real > real > real > $o ).
thf(func_def_1006,type,
sP78: real > real > real > $o ).
thf(func_def_1007,type,
sP79: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > $o ).
thf(func_def_1008,type,
sP80: rat > num > rat > $o ).
thf(func_def_1009,type,
sP81: num > num > num > $o ).
thf(func_def_1010,type,
sP82: num > num > num > $o ).
thf(func_def_1011,type,
sP83: num > num > num > $o ).
thf(func_def_1012,type,
sP84: num > num > num > $o ).
thf(func_def_1013,type,
sP85: num > num > num > $o ).
thf(func_def_1014,type,
sP86: num > num > num > $o ).
thf(func_def_1015,type,
sP87: num > num > num > $o ).
thf(func_def_1016,type,
sP88: num > num > num > $o ).
thf(func_def_1017,type,
sP89: num > num > num > $o ).
thf(func_def_1018,type,
sP90: num > num > num > $o ).
thf(func_def_1019,type,
sP91: num > num > num > $o ).
thf(func_def_1020,type,
sP92: num > num > num > $o ).
thf(func_def_1021,type,
sP93: num > num > num > $o ).
thf(func_def_1022,type,
sP94: num > num > num > $o ).
thf(func_def_1023,type,
sP95: num > num > num > $o ).
thf(func_def_1024,type,
sP96: num > num > num > $o ).
thf(func_def_1025,type,
sP97: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_1026,type,
sP98: nat > vEBT_VEBT > $o ).
thf(func_def_1027,type,
sP99: nat > list_VEBT_VEBT > nat > nat > nat > $o ).
thf(func_def_1028,type,
sP100: nat > nat > list_VEBT_VEBT > $o ).
thf(func_def_1029,type,
sP101: $o > $o > nat > $o ).
thf(func_def_1030,type,
sP102: $o > $o > $o > nat > $o ).
thf(func_def_1031,type,
sP103: vEBT_VEBT > $o > nat > $o ).
thf(func_def_1032,type,
sP104: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_1033,type,
sP105: real > nat > real > $o ).
thf(func_def_1034,type,
sP106: ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > $o ).
thf(func_def_1035,type,
sP107: ( num > $o ) > ( num > $o ) > ( num > $o ) > ( num > $o ) > $o ).
thf(func_def_1036,type,
sP108: real > ( real > real ) > ( real > real > real ) > real > $o ).
thf(func_def_1037,type,
sP109: real > num > real > $o ).
thf(func_def_1038,type,
sP110: $o > nat > vEBT_VEBT > $o ).
thf(func_def_1039,type,
sP111: $o > nat > vEBT_VEBT > $o ).
thf(func_def_1040,type,
sP112: $o > nat > vEBT_VEBT > $o ).
thf(func_def_1041,type,
sP113: real > real > real > $o ).
thf(func_def_1042,type,
sP114: rat > num > rat > $o ).
thf(func_def_1043,type,
sP115: vEBT_VEBT > nat > $o ).
thf(func_def_1044,type,
sP116: vEBT_VEBT > nat > $o ).
thf(func_def_1045,type,
sP117: vEBT_VEBT > nat > $o ).
thf(func_def_1046,type,
sP118: vEBT_VEBT > nat > $o ).
thf(func_def_1047,type,
sP119: nat > vEBT_VEBT > $o ).
thf(func_def_1048,type,
sP120: num > rat > rat > $o ).
thf(func_def_1049,type,
sP121: ( int > $o ) > int > int > $o ).
thf(func_def_1050,type,
sP122: int > ( int > $o ) > int > $o ).
thf(func_def_1051,type,
sP123: real > num > real > $o ).
thf(func_def_1052,type,
sP124: real > ( real > real ) > real > $o ).
thf(func_def_1053,type,
sP125: real > real > nat > ( nat > real > real ) > $o ).
thf(func_def_1054,type,
sP126: num > num > num > $o ).
thf(func_def_1055,type,
sP127: num > num > num > $o ).
thf(func_def_1056,type,
sP128: num > num > num > $o ).
thf(func_def_1057,type,
sP129: num > num > num > $o ).
thf(func_def_1058,type,
sP130: num > num > num > $o ).
thf(func_def_1059,type,
sP131: num > num > num > $o ).
thf(func_def_1060,type,
sP132: num > num > num > $o ).
thf(func_def_1061,type,
sP133: num > num > num > $o ).
thf(func_def_1062,type,
sP134: ( num > $o ) > ( num > $o ) > ( num > $o ) > ( num > $o ) > $o ).
thf(func_def_1063,type,
sP135: set_int > ( int > $o ) > ( int > $o ) > int > $o ).
thf(func_def_1064,type,
sP136: rat > rat > rat > $o ).
thf(func_def_1065,type,
sP137: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > $o ).
thf(func_def_1066,type,
sP138: rat > num > rat > $o ).
thf(func_def_1067,type,
sP139: $o > $o > nat > $o ).
thf(func_def_1068,type,
sP140: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > $o ).
thf(func_def_1069,type,
sP141: nat > nat > ( nat > $o ) > $o ).
thf(func_def_1070,type,
sP142: num > num > num > $o ).
thf(func_def_1071,type,
sP143: num > num > num > $o ).
thf(func_def_1072,type,
sP144: num > num > num > $o ).
thf(func_def_1073,type,
sP145: num > num > num > $o ).
thf(func_def_1074,type,
sP146: num > num > num > $o ).
thf(func_def_1075,type,
sP147: num > num > num > $o ).
thf(func_def_1076,type,
sP148: num > num > num > $o ).
thf(func_def_1077,type,
sP149: num > num > num > $o ).
thf(func_def_1078,type,
sP150: $o > $o > nat > $o ).
thf(func_def_1079,type,
sP151: $o > $o > nat > $o > $o ).
thf(func_def_1080,type,
sP152: $o > nat > vEBT_VEBT > $o ).
thf(func_def_1081,type,
sP153: vEBT_VEBT > nat > $o > $o ).
thf(func_def_1082,type,
sP154: vEBT_VEBT > nat > $o ).
thf(func_def_1083,type,
sP155: real > nat > ( nat > real > real ) > $o ).
thf(func_def_1084,type,
sP156: real > ( nat > real > real ) > nat > $o ).
thf(func_def_1085,type,
sP157: vEBT_VEBT > nat > $o > $o ).
thf(func_def_1086,type,
sP158: ( int > $o ) > int > int > $o ).
thf(func_def_1087,type,
sP159: ( int > $o ) > int > int > $o ).
thf(func_def_1088,type,
sP160: ( int > $o ) > int > int > $o ).
thf(func_def_1089,type,
sP161: rat > num > rat > $o ).
thf(func_def_1090,type,
sP162: num > rat > rat > $o ).
thf(func_def_1091,type,
sP163: ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > $o ).
thf(func_def_1092,type,
sP164: nat > vEBT_VEBT > $o ).
thf(func_def_1093,type,
sP165: real > num > real > $o ).
thf(func_def_1094,type,
sP166: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_1095,type,
sP167: set_int > ( int > $o ) > ( int > $o ) > int > $o ).
thf(func_def_1096,type,
sP168: rat > rat > rat > $o ).
thf(func_def_1097,type,
sP169: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_1098,type,
sP170: ( num > $o ) > ( num > $o ) > ( num > $o ) > ( num > $o ) > $o ).
thf(func_def_1099,type,
sP171: int > ( int > $o ) > $o ).
thf(func_def_1100,type,
sP172: nat > nat > ( nat > $o ) > $o ).
thf(func_def_1101,type,
sP173: real > num > real > $o ).
thf(func_def_1102,type,
sP174: ( int > $o ) > $o ).
thf(func_def_1103,type,
sP175: option_num > num > num > $o ).
thf(func_def_1104,type,
sP176: option_num > num > num > $o ).
thf(func_def_1105,type,
sP177: num > num > option_num > $o ).
thf(func_def_1106,type,
sP178: num > num > option_num > $o ).
thf(func_def_1107,type,
sP179: num > num > option_num > $o ).
thf(func_def_1108,type,
sP180: option_num > num > num > $o ).
thf(func_def_1109,type,
sP181: option_num > num > num > $o ).
thf(func_def_1110,type,
sP182: option_num > num > num > $o ).
thf(func_def_1111,type,
sP183: ( real > real ) > real > ( real > real ) > real > $o ).
thf(func_def_1112,type,
sP184: ( real > real ) > real > real > $o ).
thf(func_def_1113,type,
sP185: real > real > ( real > real ) > $o ).
thf(func_def_1114,type,
sP186: real > ( real > real ) > real > $o ).
thf(func_def_1115,type,
sP187: nat > vEBT_VEBT > $o ).
thf(func_def_1116,type,
sP188: ( num > $o ) > ( num > $o ) > ( num > $o ) > ( num > $o ) > $o ).
thf(func_def_1117,type,
sP189: real > ( real > real ) > ( real > real ) > real > $o ).
thf(func_def_1118,type,
sP190: real > real > ( real > real ) > $o ).
thf(func_def_1119,type,
sP191: ( real > real ) > ( real > real ) > real > real > $o ).
thf(func_def_1120,type,
sP192: ( real > real ) > real > real > $o ).
thf(func_def_1121,type,
sP193: ( nat > $o ) > $o ).
thf(func_def_1122,type,
sP194: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_1123,type,
sP195: option_num > num > num > $o ).
thf(func_def_1124,type,
sP196: option_num > num > num > $o ).
thf(func_def_1125,type,
sP197: option_num > num > num > $o ).
thf(func_def_1126,type,
sP198: num > option_num > num > $o ).
thf(func_def_1127,type,
sP199: option_num > num > num > $o ).
thf(func_def_1128,type,
sP200: option_num > num > num > $o ).
thf(func_def_1129,type,
sP201: option_num > num > num > $o ).
thf(func_def_1130,type,
sP202: option_num > num > num > $o ).
thf(func_def_1131,type,
sP203: num > option_num > num > $o ).
thf(func_def_1132,type,
sP204: option_num > num > num > $o ).
thf(func_def_1133,type,
sP205: num > option_num > num > $o ).
thf(func_def_1134,type,
sP206: num > num > option_num > $o ).
thf(func_def_1135,type,
sP207: num > num > option_num > $o ).
thf(func_def_1136,type,
sP208: num > option_num > num > $o ).
thf(func_def_1137,type,
sP209: num > num > option_num > $o ).
thf(func_def_1138,type,
sP210: num > option_num > num > $o ).
thf(func_def_1139,type,
sP211: ( int > $o ) > ( int > $o ) > int > set_int > $o ).
thf(func_def_1140,type,
sP212: set_int > int > ( int > $o ) > $o ).
thf(func_def_1141,type,
sP213: option_num > num > num > $o ).
thf(func_def_1142,type,
sP214: num > num > option_num > $o ).
thf(func_def_1143,type,
sP215: option_num > num > num > $o ).
thf(func_def_1144,type,
sP216: num > option_num > num > $o ).
thf(func_def_1145,type,
sP217: num > num > option_num > $o ).
thf(func_def_1146,type,
sP218: num > option_num > num > $o ).
thf(func_def_1147,type,
sP219: num > option_num > num > $o ).
thf(func_def_1148,type,
sP220: num > num > option_num > $o ).
thf(func_def_1149,type,
sP221: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_1150,type,
sP222: vEBT_VEBT > $o > nat > $o ).
thf(func_def_1151,type,
sP223: vEBT_VEBT > $o > nat > $o ).
thf(func_def_1152,type,
sP224: nat > $o > vEBT_VEBT > $o ).
thf(func_def_1153,type,
sP225: nat > $o > vEBT_VEBT > $o ).
thf(func_def_1154,type,
sP226: ( real > real ) > ( real > real ) > real > real > $o ).
thf(func_def_1155,type,
sP227: set_int > ( int > $o ) > ( int > $o ) > int > $o ).
thf(func_def_1156,type,
sP228: set_int > ( int > $o ) > int > $o ).
thf(func_def_1157,type,
sP229: rat > rat > rat > $o ).
thf(func_def_1158,type,
sK230: nat > vEBT_VEBT > nat ).
thf(func_def_1159,type,
sK231: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1160,type,
sK232: nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1161,type,
sK233: vEBT_VEBT > nat > nat ).
thf(func_def_1162,type,
sK234: vEBT_VEBT > nat > nat ).
thf(func_def_1163,type,
sK235: vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1164,type,
sK236: vEBT_VEBT > nat > nat ).
thf(func_def_1165,type,
sK237: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1166,type,
sK238: nat > vEBT_VEBT > nat ).
thf(func_def_1167,type,
sK239: nat > vEBT_VEBT > nat ).
thf(func_def_1168,type,
sK240: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1169,type,
sK241: nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1170,type,
sK242: vEBT_VEBT > $o ).
thf(func_def_1171,type,
sK243: vEBT_VEBT > $o ).
thf(func_def_1172,type,
sK244: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1173,type,
sK245: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1174,type,
sK246: ( code_integer > $o ) > code_integer > code_integer ).
thf(func_def_1175,type,
sK247: ( code_integer > $o ) > code_integer > code_integer ).
thf(func_def_1176,type,
sK248: int > ( int > $o ) > int ).
thf(func_def_1177,type,
sK249: ( nat > $o ) > nat > nat ).
thf(func_def_1178,type,
sK250: ( nat > $o ) > nat > nat ).
thf(func_def_1179,type,
sK251: rat > rat > rat ).
thf(func_def_1180,type,
sK252: real > real > real ).
thf(func_def_1181,type,
sK253: real > real > nat ).
thf(func_def_1182,type,
sK254: nat > set_nat > nat ).
thf(func_def_1183,type,
sK255: real > nat ).
thf(func_def_1184,type,
sK256: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1185,type,
sK257: ( int > $o ) > int ).
thf(func_def_1186,type,
sK258: ( int > $o ) > int > int ).
thf(func_def_1187,type,
sK259: ( int > $o ) > int > int ).
thf(func_def_1188,type,
sK260: int > int > int ).
thf(func_def_1189,type,
sK261: int > set_int > int ).
thf(func_def_1190,type,
sK262: nat > nat ).
thf(func_def_1191,type,
sK263: nat > nat ).
thf(func_def_1192,type,
sK264: real > ( real > real ) > set_real > real ).
thf(func_def_1193,type,
sK265: int > int ).
thf(func_def_1194,type,
sK266: nat > nat > nat > nat ).
thf(func_def_1195,type,
sK267: nat > ( nat > nat ) > nat ).
thf(func_def_1196,type,
sK268: nat > ( nat > nat ) > nat ).
thf(func_def_1197,type,
sK269: ( nat > nat ) > nat > nat ).
thf(func_def_1198,type,
sK270: ( nat > nat ) > nat > nat ).
thf(func_def_1199,type,
sK271: rat > int ).
thf(func_def_1200,type,
sK272: int > nat ).
thf(func_def_1201,type,
sK273: int > nat ).
thf(func_def_1202,type,
sK274: rat > nat ).
thf(func_def_1203,type,
sK275: int > int ).
thf(func_def_1204,type,
sK276: int > int ).
thf(func_def_1205,type,
sK277: int > nat ).
thf(func_def_1206,type,
sK278: int > nat ).
thf(func_def_1207,type,
sK279: rat > product_prod_int_int ).
thf(func_def_1208,type,
sK280: rat > rat ).
thf(func_def_1209,type,
sK281: num > num ).
thf(func_def_1210,type,
sK282: num > option_num > num > num ).
thf(func_def_1211,type,
sK283: num > option_num > num > num ).
thf(func_def_1212,type,
sK284: num > num ).
thf(func_def_1213,type,
sK285: num > num > option_num > num ).
thf(func_def_1214,type,
sK286: num > num > option_num > num ).
thf(func_def_1215,type,
sK287: num > option_num > num > num ).
thf(func_def_1216,type,
sK288: num > option_num > num > num ).
thf(func_def_1217,type,
sK289: num > option_num > num > num ).
thf(func_def_1218,type,
sK290: num > num > option_num > num ).
thf(func_def_1219,type,
sK291: num > num > option_num > num ).
thf(func_def_1220,type,
sK292: num > num > option_num > num ).
thf(func_def_1221,type,
sK293: int > int > int > int ).
thf(func_def_1222,type,
sK294: int > int > int > int ).
thf(func_def_1223,type,
sK295: nat > nat > nat ).
thf(func_def_1224,type,
sK296: real > real > real ).
thf(func_def_1225,type,
sK297: ( real > real ) > ( nat > real > real ) > nat > real > real > real ).
thf(func_def_1226,type,
sK298: real > ( nat > real > real ) > nat > real > real ).
thf(func_def_1227,type,
sK299: real > ( nat > real > real ) > nat > real > nat ).
thf(func_def_1228,type,
sK300: int > int > set_int > int ).
thf(func_def_1229,type,
sK301: int > int > set_int > int ).
thf(func_def_1230,type,
sK302: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1231,type,
sK303: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1232,type,
sK304: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1233,type,
sK305: code_integer > code_integer > code_integer ).
thf(func_def_1234,type,
sK306: complex > int ).
thf(func_def_1235,type,
sK307: num > num ).
thf(func_def_1236,type,
sK308: real > real > real ).
thf(func_def_1237,type,
sK309: ( nat > int ) > nat ).
thf(func_def_1238,type,
sK310: ( nat > $o ) > nat > nat ).
thf(func_def_1239,type,
sK311: nat > nat > nat ).
thf(func_def_1240,type,
sK312: nat > nat > nat ).
thf(func_def_1241,type,
sK313: nat > nat > nat ).
thf(func_def_1242,type,
sK314: nat > ( nat > $o ) > nat ).
thf(func_def_1243,type,
sK315: nat > ( nat > $o ) > nat ).
thf(func_def_1244,type,
sK316: nat > nat > nat ).
thf(func_def_1245,type,
sK317: vEBT_VEBT > nat > nat ).
thf(func_def_1246,type,
sK318: ( int > int > $o ) > int ).
thf(func_def_1247,type,
sK319: ( int > int > $o ) > int ).
thf(func_def_1248,type,
sK320: ( real > real > $o ) > real ).
thf(func_def_1249,type,
sK321: ( nat > rat ) > rat ).
thf(func_def_1250,type,
sK322: nat > ( nat > rat ) > nat ).
thf(func_def_1251,type,
sK323: real > real > real ).
thf(func_def_1252,type,
sK324: rat > rat ).
thf(func_def_1253,type,
sK325: real > int ).
thf(func_def_1254,type,
sK326: int > int > set_int > int ).
thf(func_def_1255,type,
sK327: int > int > set_int > int ).
thf(func_def_1256,type,
sK328: ( nat > rat ) > nat ).
thf(func_def_1257,type,
sK329: real > real ).
thf(func_def_1258,type,
sK330: ( nat > $o ) > nat ).
thf(func_def_1259,type,
sK331: real > real ).
thf(func_def_1260,type,
sK332: nat > complex > nat ).
thf(func_def_1261,type,
sK333: int > int > int ).
thf(func_def_1262,type,
sK334: nat > rat > nat ).
thf(func_def_1263,type,
sK335: ( nat > real ) > nat ).
thf(func_def_1264,type,
sK336: nat > nat > nat > nat ).
thf(func_def_1265,type,
sK337: nat > nat > nat > nat ).
thf(func_def_1266,type,
sK338: nat > nat > nat > nat ).
thf(func_def_1267,type,
sK339: rat > ( nat > rat ) > ( nat > rat ) > nat ).
thf(func_def_1268,type,
sK340: ( nat > rat ) > ( nat > rat ) > rat ).
thf(func_def_1269,type,
sK341: nat > ( nat > rat ) > ( nat > rat ) > nat ).
thf(func_def_1270,type,
sK342: rat > rat ).
thf(func_def_1271,type,
sK343: nat > nat > nat ).
thf(func_def_1272,type,
sK344: rat > int ).
thf(func_def_1273,type,
sK345: rat > int ).
thf(func_def_1274,type,
sK346: set_real > real > ( real > real ) > real ).
thf(func_def_1275,type,
sK347: num > num ).
thf(func_def_1276,type,
sK348: rat > ( nat > rat ) > nat ).
thf(func_def_1277,type,
sK349: list_o > ( $o > $o ) > $o ).
thf(func_def_1278,type,
sK350: int > ( nat > $o ) > nat ).
thf(func_def_1279,type,
sK351: ( nat > real ) > nat ).
thf(func_def_1280,type,
sK352: ( nat > real ) > nat ).
thf(func_def_1281,type,
sK353: code_integer > code_integer > code_integer ).
thf(func_def_1282,type,
sK354: nat > nat > nat ).
thf(func_def_1283,type,
sK355: list_VEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1284,type,
sK356: list_VEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1285,type,
sK357: nat > nat > list_VEBT_VEBT > nat > nat > nat ).
thf(func_def_1286,type,
sK358: nat > nat > list_VEBT_VEBT > nat > nat > nat ).
thf(func_def_1287,type,
sK359: list_VEBT_VEBT > vEBT_VEBT ).
thf(func_def_1288,type,
sK360: list_VEBT_VEBT > nat ).
thf(func_def_1289,type,
sK361: list_VEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1290,type,
sK362: rat > rat ).
thf(func_def_1291,type,
sK363: ( nat > nat > $o ) > nat ).
thf(func_def_1292,type,
sK364: ( nat > nat > $o ) > nat ).
thf(func_def_1293,type,
sK365: ( nat > nat > $o ) > nat ).
thf(func_def_1294,type,
sK366: ( nat > nat > $o ) > nat ).
thf(func_def_1295,type,
sK367: ( nat > nat > $o ) > nat ).
thf(func_def_1296,type,
sK368: real > nat > ( nat > real > real ) > real ).
thf(func_def_1297,type,
sK369: real > nat > ( nat > real > real ) > nat ).
thf(func_def_1298,type,
sK370: nat > ( real > real ) > ( nat > real > real ) > real > real ).
thf(func_def_1299,type,
sK371: real > real ).
thf(func_def_1300,type,
sK372: nat > nat ).
thf(func_def_1301,type,
sK373: real > real ).
thf(func_def_1302,type,
sK374: real > real > real ).
thf(func_def_1303,type,
sK375: real > nat ).
thf(func_def_1304,type,
sK376: rat > ( nat > rat ) > nat ).
thf(func_def_1305,type,
sK377: rat > int ).
thf(func_def_1306,type,
sK378: nat > nat > ( nat > $o ) > nat ).
thf(func_def_1307,type,
sK379: vEBT_VEBT > $o > nat > $o ).
thf(func_def_1308,type,
sK380: vEBT_VEBT > $o > nat > $o ).
thf(func_def_1309,type,
sK381: list_VEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1310,type,
sK382: nat > vEBT_VEBT > $o > option4927543243414619207at_nat ).
thf(func_def_1311,type,
sK383: nat > vEBT_VEBT > $o > nat ).
thf(func_def_1312,type,
sK384: nat > vEBT_VEBT > $o > vEBT_VEBT ).
thf(func_def_1313,type,
sK385: nat > vEBT_VEBT > $o > list_VEBT_VEBT ).
thf(func_def_1314,type,
sK386: real > real ).
thf(func_def_1315,type,
sK387: ( int > int > $o ) > int > int > int ).
thf(func_def_1316,type,
sK388: ( int > int > $o ) > int > int > int ).
thf(func_def_1317,type,
sK389: ( int > int > $o ) > int ).
thf(func_def_1318,type,
sK390: rat > rat ).
thf(func_def_1319,type,
sK391: real > nat ).
thf(func_def_1320,type,
sK392: code_integer > code_integer > code_integer ).
thf(func_def_1321,type,
sK393: set_real > real ).
thf(func_def_1322,type,
sK394: real > set_real > real ).
thf(func_def_1323,type,
sK395: real > set_real > real ).
thf(func_def_1324,type,
sK396: rat > rat ).
thf(func_def_1325,type,
sK397: nat > nat ).
thf(func_def_1326,type,
sK398: nat > nat > ( nat > int ) > int > nat ).
thf(func_def_1327,type,
sK399: nat > ( nat > int ) > nat > nat ).
thf(func_def_1328,type,
sK400: ( nat > nat ) > nat ).
thf(func_def_1329,type,
sK401: ( nat > nat ) > nat ).
thf(func_def_1330,type,
sK402: nat > ( nat > int ) > nat ).
thf(func_def_1331,type,
sK403: nat > int > ( nat > int ) > nat ).
thf(func_def_1332,type,
sK404: int > nat ).
thf(func_def_1333,type,
sK405: int > nat ).
thf(func_def_1334,type,
sK406: ( nat > real ) > nat > int ).
thf(func_def_1335,type,
sK407: nat > list_nat > nat ).
thf(func_def_1336,type,
sK408: vEBT_VEBT > nat > $o ).
thf(func_def_1337,type,
sK409: vEBT_VEBT > nat > $o ).
thf(func_def_1338,type,
sK410: vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1339,type,
sK411: vEBT_VEBT > nat > option4927543243414619207at_nat ).
thf(func_def_1340,type,
sK412: vEBT_VEBT > nat > nat ).
thf(func_def_1341,type,
sK413: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1342,type,
sK414: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1343,type,
sK415: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1344,type,
sK416: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1345,type,
sK417: real > real > real ).
thf(func_def_1346,type,
sK418: nat > nat ).
thf(func_def_1347,type,
sK419: ( num > $o ) > num ).
thf(func_def_1348,type,
sK420: ( nat > $o ) > nat ).
thf(func_def_1349,type,
sK421: ( nat > $o ) > $o ).
thf(func_def_1350,type,
sK422: ( nat > $o ) > nat ).
thf(func_def_1351,type,
sK423: ( nat > real > real ) > real > nat > nat ).
thf(func_def_1352,type,
sK424: ( nat > real > real ) > real > nat > real ).
thf(func_def_1353,type,
sK425: int > int > set_int > int ).
thf(func_def_1354,type,
sK426: int > int > set_int > int ).
thf(func_def_1355,type,
sK427: nat > nat > nat ).
thf(func_def_1356,type,
sK428: nat > nat > nat ).
thf(func_def_1357,type,
sK429: rat > nat ).
thf(func_def_1358,type,
sK430: nat > nat > nat > nat ).
thf(func_def_1359,type,
sK431: nat > nat > nat > nat ).
thf(func_def_1360,type,
sK432: ( rat > $o ) > rat > rat ).
thf(func_def_1361,type,
sK433: ( rat > $o ) > rat > rat ).
thf(func_def_1362,type,
sK434: ( rat > $o ) > rat > rat ).
thf(func_def_1363,type,
sK435: ( rat > $o ) > rat > rat ).
thf(func_def_1364,type,
sK436: real > real > real > real ).
thf(func_def_1365,type,
sK437: set_nat > nat > nat ).
thf(func_def_1366,type,
sK438: int > int ).
thf(func_def_1367,type,
sK439: rat > rat ).
thf(func_def_1368,type,
sK440: code_integer > code_integer > code_integer ).
thf(func_def_1369,type,
sK441: nat > vEBT_VEBT > nat ).
thf(func_def_1370,type,
sK442: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1371,type,
sK443: nat > vEBT_VEBT > nat ).
thf(func_def_1372,type,
sK444: nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1373,type,
sK445: nat > vEBT_VEBT > nat ).
thf(func_def_1374,type,
sK446: nat > vEBT_VEBT > nat ).
thf(func_def_1375,type,
sK447: nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1376,type,
sK448: nat > vEBT_VEBT > nat ).
thf(func_def_1377,type,
sK449: nat > vEBT_VEBT > nat ).
thf(func_def_1378,type,
sK450: nat > vEBT_VEBT > nat ).
thf(func_def_1379,type,
sK451: nat > vEBT_VEBT > nat ).
thf(func_def_1380,type,
sK452: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1381,type,
sK453: vEBT_VEBT > nat > nat ).
thf(func_def_1382,type,
sK454: vEBT_VEBT > nat > nat ).
thf(func_def_1383,type,
sK455: vEBT_VEBT > nat > nat ).
thf(func_def_1384,type,
sK456: vEBT_VEBT > nat > nat ).
thf(func_def_1385,type,
sK457: vEBT_VEBT > nat > nat ).
thf(func_def_1386,type,
sK458: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1387,type,
sK459: vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1388,type,
sK460: nat > vEBT_VEBT > nat ).
thf(func_def_1389,type,
sK461: nat > vEBT_VEBT > nat ).
thf(func_def_1390,type,
sK462: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1391,type,
sK463: nat > vEBT_VEBT > nat ).
thf(func_def_1392,type,
sK464: nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1393,type,
sK465: nat > list_VEBT_VEBT > nat ).
thf(func_def_1394,type,
sK466: nat > list_VEBT_VEBT > nat ).
thf(func_def_1395,type,
sK467: vEBT_VEBT > $o ).
thf(func_def_1396,type,
sK468: vEBT_VEBT > $o ).
thf(func_def_1397,type,
sK469: produc9072475918466114483BT_nat > nat ).
thf(func_def_1398,type,
sK470: produc9072475918466114483BT_nat > list_VEBT_VEBT ).
thf(func_def_1399,type,
sK471: produc9072475918466114483BT_nat > nat ).
thf(func_def_1400,type,
sK472: produc9072475918466114483BT_nat > vEBT_VEBT ).
thf(func_def_1401,type,
sK473: produc9072475918466114483BT_nat > nat ).
thf(func_def_1402,type,
sK474: produc9072475918466114483BT_nat > vEBT_VEBT ).
thf(func_def_1403,type,
sK475: produc9072475918466114483BT_nat > list_VEBT_VEBT ).
thf(func_def_1404,type,
sK476: produc9072475918466114483BT_nat > nat ).
thf(func_def_1405,type,
sK477: produc9072475918466114483BT_nat > nat ).
thf(func_def_1406,type,
sK478: produc9072475918466114483BT_nat > vEBT_VEBT ).
thf(func_def_1407,type,
sK479: produc9072475918466114483BT_nat > nat ).
thf(func_def_1408,type,
sK480: produc9072475918466114483BT_nat > nat ).
thf(func_def_1409,type,
sK481: produc9072475918466114483BT_nat > nat ).
thf(func_def_1410,type,
sK482: produc9072475918466114483BT_nat > nat ).
thf(func_def_1411,type,
sK483: produc9072475918466114483BT_nat > list_VEBT_VEBT ).
thf(func_def_1412,type,
sK484: produc9072475918466114483BT_nat > $o ).
thf(func_def_1413,type,
sK485: produc9072475918466114483BT_nat > nat ).
thf(func_def_1414,type,
sK486: produc9072475918466114483BT_nat > $o ).
thf(func_def_1415,type,
sK487: produc9072475918466114483BT_nat > nat ).
thf(func_def_1416,type,
sK488: produc9072475918466114483BT_nat > list_VEBT_VEBT ).
thf(func_def_1417,type,
sK489: produc9072475918466114483BT_nat > vEBT_VEBT ).
thf(func_def_1418,type,
sK490: int > int > ( int > $o ) > int ).
thf(func_def_1419,type,
sK491: int > int > ( int > $o ) > int ).
thf(func_def_1420,type,
sK492: int > int > ( int > $o ) > int ).
thf(func_def_1421,type,
sK493: int > int > ( int > $o ) > int ).
thf(func_def_1422,type,
sK494: num > option_num > num ).
thf(func_def_1423,type,
sK495: num > option_num > num > num ).
thf(func_def_1424,type,
sK496: num > option_num > num > num ).
thf(func_def_1425,type,
sK497: num > num > option_num > num ).
thf(func_def_1426,type,
sK498: num > option_num > num ).
thf(func_def_1427,type,
sK499: num > option_num > num > num ).
thf(func_def_1428,type,
sK500: num > option_num > num > num ).
thf(func_def_1429,type,
sK501: num > num > option_num > num ).
thf(func_def_1430,type,
sK502: num > num > option_num > num ).
thf(func_def_1431,type,
sK503: num > num > option_num > num ).
thf(func_def_1432,type,
sK504: num > num > option_num > num ).
thf(func_def_1433,type,
sK505: num > option_num > num > num ).
thf(func_def_1434,type,
sK506: ( nat > rat ) > rat ).
thf(func_def_1435,type,
sK507: nat > ( nat > rat ) > nat ).
thf(func_def_1436,type,
sK508: nat > ( nat > rat ) > nat ).
thf(func_def_1437,type,
sK509: ( nat > real ) > nat ).
thf(func_def_1438,type,
sK510: ( nat > $o ) > nat > nat ).
thf(func_def_1439,type,
sK511: ( nat > $o ) > nat > nat ).
thf(func_def_1440,type,
sK512: int > nat ).
thf(func_def_1441,type,
sK513: list_int > ( int > $o ) > int ).
thf(func_def_1442,type,
sK514: ( nat > nat > $o ) > nat ).
thf(func_def_1443,type,
sK515: ( nat > nat > $o ) > nat ).
thf(func_def_1444,type,
sK516: ( nat > nat > $o ) > nat ).
thf(func_def_1445,type,
sK517: ( real > $o ) > int ).
thf(func_def_1446,type,
sK518: rat > nat ).
thf(func_def_1447,type,
sK519: rat > rat ).
thf(func_def_1448,type,
sK520: nat > real > real ).
thf(func_def_1449,type,
sK521: real > nat ).
thf(func_def_1450,type,
sK522: real > nat ).
thf(func_def_1451,type,
sK523: int > nat ).
thf(func_def_1452,type,
sK524: int > nat ).
thf(func_def_1453,type,
sK525: rat > ( nat > rat ) > nat ).
thf(func_def_1454,type,
sK526: ( nat > rat ) > rat ).
thf(func_def_1455,type,
sK527: nat > ( nat > rat ) > nat ).
thf(func_def_1456,type,
sK528: num > num ).
thf(func_def_1457,type,
sK529: real > real > real ).
thf(func_def_1458,type,
sK530: nat > nat > nat > nat ).
thf(func_def_1459,type,
sK531: ( rat > $o ) > int ).
thf(func_def_1460,type,
sK532: rat > rat > rat ).
thf(func_def_1461,type,
sK533: ( int > $o ) > real > int ).
thf(func_def_1462,type,
sK534: real > ( nat > real ) > nat ).
thf(func_def_1463,type,
sK535: real > ( nat > real ) > real ).
thf(func_def_1464,type,
sK536: ( nat > real ) > nat ).
thf(func_def_1465,type,
sK537: ( real > real ) > real > real ).
thf(func_def_1466,type,
sK538: nat > code_natural > code_natural ).
thf(func_def_1467,type,
sK539: int > int > int > int ).
thf(func_def_1468,type,
sK540: real > real ).
thf(func_def_1469,type,
sK541: nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1470,type,
sK542: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1471,type,
sK543: nat > vEBT_VEBT > nat ).
thf(func_def_1472,type,
sK544: nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1473,type,
sK545: nat > vEBT_VEBT > nat ).
thf(func_def_1474,type,
sK546: nat > vEBT_VEBT > nat ).
thf(func_def_1475,type,
sK547: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1476,type,
sK548: nat > vEBT_VEBT > nat ).
thf(func_def_1477,type,
sK549: nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1478,type,
sK550: nat > vEBT_VEBT > nat ).
thf(func_def_1479,type,
sK551: nat > vEBT_VEBT > nat ).
thf(func_def_1480,type,
sK552: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1481,type,
sK553: real > real ).
thf(func_def_1482,type,
sK554: ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1483,type,
sK555: rat > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1484,type,
sK556: rat > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1485,type,
sK557: int > nat ).
thf(func_def_1486,type,
sK558: real > int ).
thf(func_def_1487,type,
sK559: list_VEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1488,type,
sK560: int > nat ).
thf(func_def_1489,type,
sK561: int > nat ).
thf(func_def_1490,type,
sK562: nat > ( nat > $o ) > nat ).
thf(func_def_1491,type,
sK563: nat > ( nat > $o ) > nat ).
thf(func_def_1492,type,
sK564: complex > real ).
thf(func_def_1493,type,
sK565: num > num ).
thf(func_def_1494,type,
sK566: real > real > nat ).
thf(func_def_1495,type,
sK567: rat > int ).
thf(func_def_1496,type,
sK568: rat > int ).
thf(func_def_1497,type,
sK569: ( nat > rat ) > real > nat ).
thf(func_def_1498,type,
sK570: ( nat > set_nat ) > nat ).
thf(func_def_1499,type,
sK571: set_nat > nat ).
thf(func_def_1500,type,
sK572: set_nat > nat ).
thf(func_def_1501,type,
sK573: nat > nat > nat > nat ).
thf(func_def_1502,type,
sK574: set_int > int > ( int > $o ) > int ).
thf(func_def_1503,type,
sK575: int > set_int > int > int ).
thf(func_def_1504,type,
sK576: int > set_int > int > int ).
thf(func_def_1505,type,
sK577: ( int > $o ) > set_int > int > int ).
thf(func_def_1506,type,
sK578: set_int > int > ( int > $o ) > int ).
thf(func_def_1507,type,
sK579: int > set_int > int > int ).
thf(func_def_1508,type,
sK580: int > set_int > int > int ).
thf(func_def_1509,type,
sK581: ( int > $o ) > set_int > int > int ).
thf(func_def_1510,type,
sK582: nat > nat ).
thf(func_def_1511,type,
sK583: real > real ).
thf(func_def_1512,type,
sK584: list_nat > nat ).
thf(func_def_1513,type,
sK585: list_nat > list_nat ).
thf(func_def_1514,type,
sK586: real > ( real > $o ) > real ).
thf(func_def_1515,type,
sK587: real > ( real > $o ) > real ).
thf(func_def_1516,type,
sK588: code_integer > nat ).
thf(func_def_1517,type,
sK589: real > nat ).
thf(func_def_1518,type,
sK590: real > int ).
thf(func_def_1519,type,
sK591: rat > int ).
thf(func_def_1520,type,
sK592: num > num ).
thf(func_def_1521,type,
sK593: ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1522,type,
sK594: rat > int ).
thf(func_def_1523,type,
sK595: int > ( int > $o ) > int ).
thf(func_def_1524,type,
sK596: ( nat > nat > $o ) > nat ).
thf(func_def_1525,type,
sK597: ( nat > nat > $o ) > nat ).
thf(func_def_1526,type,
sK598: ( nat > nat > $o ) > nat ).
thf(func_def_1527,type,
sK599: ( nat > nat > $o ) > nat ).
thf(func_def_1528,type,
sK600: vEBT_VEBT > nat ).
thf(func_def_1529,type,
sK601: list_VEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1530,type,
sK602: list_VEBT_VEBT > vEBT_VEBT ).
thf(func_def_1531,type,
sK603: list_VEBT_VEBT > nat ).
thf(func_def_1532,type,
sK604: real > ( real > $o ) > real ).
thf(func_def_1533,type,
sK605: real > ( real > $o ) > real ).
thf(func_def_1534,type,
sK606: ( real > $o ) > real > real ).
thf(func_def_1535,type,
sK607: ( real > $o ) > real > real ).
thf(func_def_1536,type,
sK608: nat > ( nat > $o ) > nat ).
thf(func_def_1537,type,
sK609: real > int ).
thf(func_def_1538,type,
sK610: code_integer > code_integer > code_integer ).
thf(func_def_1539,type,
sK611: real > nat > real ).
thf(func_def_1540,type,
sK612: int > int > set_int > int ).
thf(func_def_1541,type,
sK613: int > int > set_int > int ).
thf(func_def_1542,type,
sK614: nat > nat > nat ).
thf(func_def_1543,type,
sK615: real > real > real ).
thf(func_def_1544,type,
sK616: real > real > real ).
thf(func_def_1545,type,
sK617: extended_enat > extended_enat > nat > nat ).
thf(func_def_1546,type,
sK618: extended_enat > extended_enat > nat > nat ).
thf(func_def_1547,type,
sK619: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > vEBT_VEBT ).
thf(func_def_1548,type,
sK620: ( nat > nat ) > nat ).
thf(func_def_1549,type,
sK621: rat > nat ).
thf(func_def_1550,type,
sK622: ( nat > $o ) > nat ).
thf(func_def_1551,type,
sK623: num > num ).
thf(func_def_1552,type,
sK624: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1553,type,
sK625: real > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1554,type,
sK626: real > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1555,type,
sK627: ( nat > $o ) > nat > nat ).
thf(func_def_1556,type,
sK628: nat > nat > nat ).
thf(func_def_1557,type,
sK629: int > product_prod_int_int > int > int ).
thf(func_def_1558,type,
sK630: int > product_prod_int_int > int > int ).
thf(func_def_1559,type,
sK631: product_prod_int_int > int > int > int ).
thf(func_def_1560,type,
sK632: real > ( nat > real > real ) > real > nat > nat ).
thf(func_def_1561,type,
sK633: real > ( nat > real > real ) > real > nat > real ).
thf(func_def_1562,type,
sK634: nat > real > real > ( real > real ) > ( nat > real > real ) > real ).
thf(func_def_1563,type,
sK635: list_nat > nat > nat ).
thf(func_def_1564,type,
sK636: list_nat > nat > list_nat ).
thf(func_def_1565,type,
sK637: ( nat > rat ) > rat ).
thf(func_def_1566,type,
sK638: ( nat > rat ) > nat ).
thf(func_def_1567,type,
sK639: int > ( int > $o ) > int ).
thf(func_def_1568,type,
sK640: ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1569,type,
sK641: rat > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1570,type,
sK642: rat > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1571,type,
sK643: rat > rat > rat ).
thf(func_def_1572,type,
sK644: int > nat ).
thf(func_def_1573,type,
sK645: code_integer > code_integer > code_integer ).
thf(func_def_1574,type,
sK646: ( nat > $o ) > nat ).
thf(func_def_1575,type,
sK647: ( nat > $o ) > nat ).
thf(func_def_1576,type,
sK648: ( nat > $o ) > nat > nat > nat ).
thf(func_def_1577,type,
sK649: ( nat > nat > $o ) > nat ).
thf(func_def_1578,type,
sK650: ( nat > nat > $o ) > nat ).
thf(func_def_1579,type,
sK651: ( nat > nat > $o ) > nat ).
thf(func_def_1580,type,
sK652: ( nat > nat > $o ) > nat ).
thf(func_def_1581,type,
sK653: ( nat > nat > $o ) > nat ).
thf(func_def_1582,type,
sK654: ( complex > $o ) > list_complex > complex ).
thf(func_def_1583,type,
sK655: int > int ).
thf(func_def_1584,type,
sK656: ( real > real ) > real > set_real > real ).
thf(func_def_1585,type,
sK657: ( nat > $o ) > nat ).
thf(func_def_1586,type,
sK658: ( nat > real ) > real > nat ).
thf(func_def_1587,type,
sK659: rat > rat > rat ).
thf(func_def_1588,type,
sK660: nat > nat > ( nat > $o ) > nat ).
thf(func_def_1589,type,
sK661: real > real ).
thf(func_def_1590,type,
sK662: int > ( int > $o ) > int ).
thf(func_def_1591,type,
sK663: int > ( int > $o ) > int ).
thf(func_def_1592,type,
sK664: int > ( int > $o ) > int ).
thf(func_def_1593,type,
sK665: int > ( int > $o ) > int ).
thf(func_def_1594,type,
sK666: ( nat > real ) > real ).
thf(func_def_1595,type,
sK667: nat > ( nat > real ) > nat ).
thf(func_def_1596,type,
sK668: nat > ( nat > real ) > nat ).
thf(func_def_1597,type,
sK669: ( set_nat > $o ) > list_set_nat > set_nat ).
thf(func_def_1598,type,
sK670: rat > int ).
thf(func_def_1599,type,
sK671: code_integer > code_integer ).
thf(func_def_1600,type,
sK672: nat > nat > nat > nat ).
thf(func_def_1601,type,
sK673: nat > nat > nat > nat ).
thf(func_def_1602,type,
sK674: nat > nat > nat ).
thf(func_def_1603,type,
sK675: vEBT_VEBT > nat > option4927543243414619207at_nat ).
thf(func_def_1604,type,
sK676: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1605,type,
sK677: vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1606,type,
sK678: vEBT_VEBT > nat > nat ).
thf(func_def_1607,type,
sK679: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1608,type,
sK680: nat > vEBT_VEBT > $o ).
thf(func_def_1609,type,
sK681: nat > vEBT_VEBT > $o ).
thf(func_def_1610,type,
sK682: ( nat > rat ) > nat ).
thf(func_def_1611,type,
sK683: ( nat > $o ) > nat > nat ).
thf(func_def_1612,type,
sK684: ( nat > $o ) > nat > nat ).
thf(func_def_1613,type,
sK685: nat > nat ).
thf(func_def_1614,type,
sK686: int > int ).
thf(func_def_1615,type,
sK687: real > real > ( real > real > $o ) > real ).
thf(func_def_1616,type,
sK688: real > real > real > ( real > real > $o ) > real ).
thf(func_def_1617,type,
sK689: real > real > real > ( real > real > $o ) > real ).
thf(func_def_1618,type,
sK690: ( real > real > $o ) > real ).
thf(func_def_1619,type,
sK691: ( real > real > $o ) > real ).
thf(func_def_1620,type,
sK692: ( real > real > $o ) > real ).
thf(func_def_1621,type,
sK693: vEBT_VEBT > nat > nat ).
thf(func_def_1622,type,
sK694: ( nat > $o ) > nat > nat ).
thf(func_def_1623,type,
sK695: ( nat > $o ) > nat > nat ).
thf(func_def_1624,type,
sK696: ( nat > real ) > nat ).
thf(func_def_1625,type,
sK697: ( nat > real ) > nat ).
thf(func_def_1626,type,
sK698: ( real > real ) > set_real > real ).
thf(func_def_1627,type,
sK699: list_nat > nat > nat ).
thf(func_def_1628,type,
sK700: list_nat > nat > list_nat ).
thf(func_def_1629,type,
sK701: nat > nat ).
thf(func_def_1630,type,
sK702: nat > ( nat > $o ) > nat > nat ).
thf(func_def_1631,type,
sK703: ( nat > $o ) > nat ).
thf(func_def_1632,type,
sK704: nat > nat ).
thf(func_def_1633,type,
sK705: real > real ).
thf(func_def_1634,type,
sK706: code_integer > code_integer > code_integer > code_integer ).
thf(func_def_1635,type,
sK707: nat > nat ).
thf(func_def_1636,type,
sK708: nat > real > real ).
thf(func_def_1637,type,
sK709: real > real > real ).
thf(func_def_1638,type,
sK710: ( nat > $o ) > nat > nat ).
thf(func_def_1639,type,
sK711: ( nat > $o ) > nat > nat ).
thf(func_def_1640,type,
sK712: int > int > nat ).
thf(func_def_1641,type,
sK713: nat > nat > nat ).
thf(func_def_1642,type,
sK714: real > nat ).
thf(func_def_1643,type,
sK715: real > nat ).
thf(func_def_1644,type,
sK716: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1645,type,
sK717: real > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1646,type,
sK718: real > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1647,type,
sK719: int > nat ).
thf(func_def_1648,type,
sK720: nat > nat ).
thf(func_def_1649,type,
sK721: ( real > $o ) > real > nat > real ).
thf(func_def_1650,type,
sK722: int > int > int ).
thf(func_def_1651,type,
sK723: num > num > num ).
thf(func_def_1652,type,
sK724: num > num > num > num ).
thf(func_def_1653,type,
sK725: num > num > num > num ).
thf(func_def_1654,type,
sK726: num > num > num > num ).
thf(func_def_1655,type,
sK727: num > num > num > num ).
thf(func_def_1656,type,
sK728: num > num > num > num ).
thf(func_def_1657,type,
sK729: num > num > num > num ).
thf(func_def_1658,type,
sK730: num > num > num > num ).
thf(func_def_1659,type,
sK731: num > num > num ).
thf(func_def_1660,type,
sK732: num > num > num > num ).
thf(func_def_1661,type,
sK733: num > num > num > num ).
thf(func_def_1662,type,
sK734: num > num > num > num ).
thf(func_def_1663,type,
sK735: real > int ).
thf(func_def_1664,type,
sK736: nat > real > real ).
thf(func_def_1665,type,
sK737: list_real > ( real > $o ) > real ).
thf(func_def_1666,type,
sK738: num > num > num > num ).
thf(func_def_1667,type,
sK739: num > num > num > num ).
thf(func_def_1668,type,
sK740: num > num > num ).
thf(func_def_1669,type,
sK741: num > num ).
thf(func_def_1670,type,
sK742: num > num > num > num ).
thf(func_def_1671,type,
sK743: num > num > num > num ).
thf(func_def_1672,type,
sK744: num > num > num ).
thf(func_def_1673,type,
sK745: num > num ).
thf(func_def_1674,type,
sK746: num > num > num > num ).
thf(func_def_1675,type,
sK747: num > num > num > num ).
thf(func_def_1676,type,
sK748: num > num > num > num ).
thf(func_def_1677,type,
sK749: num > num > num > num ).
thf(func_def_1678,type,
sK750: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1679,type,
sK751: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1680,type,
sK752: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1681,type,
sK753: rat > rat ).
thf(func_def_1682,type,
sK754: nat > real > ( nat > real ) > ( real > real ) > real ).
thf(func_def_1683,type,
sK755: product_prod_nat_nat > nat ).
thf(func_def_1684,type,
sK756: product_prod_nat_nat > nat ).
thf(func_def_1685,type,
sK757: nat > real > real ).
thf(func_def_1686,type,
sK758: ( rat > $o ) > int ).
thf(func_def_1687,type,
sK759: ( rat > $o ) > int ).
thf(func_def_1688,type,
sK760: nat > vEBT_VEBT > $o ).
thf(func_def_1689,type,
sK761: nat > vEBT_VEBT > $o ).
thf(func_def_1690,type,
sK762: vEBT_VEBT > nat > option4927543243414619207at_nat ).
thf(func_def_1691,type,
sK763: vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1692,type,
sK764: vEBT_VEBT > nat > nat ).
thf(func_def_1693,type,
sK765: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1694,type,
sK766: num > num ).
thf(func_def_1695,type,
sK767: nat > nat ).
thf(func_def_1696,type,
sK768: ( int > int > $o ) > int ).
thf(func_def_1697,type,
sK769: ( int > int > $o ) > int ).
thf(func_def_1698,type,
sK770: set_nat > nat ).
thf(func_def_1699,type,
sK771: set_nat > nat ).
thf(func_def_1700,type,
sK772: list_VEBT_VEBT > vEBT_VEBT ).
thf(func_def_1701,type,
sK773: list_VEBT_VEBT > nat ).
thf(func_def_1702,type,
sK774: nat > list_VEBT_VEBT > nat > nat > nat > nat ).
thf(func_def_1703,type,
sK775: nat > list_VEBT_VEBT > nat > nat > nat > nat ).
thf(func_def_1704,type,
sK776: vEBT_VEBT > nat > list_VEBT_VEBT > nat ).
thf(func_def_1705,type,
sK777: vEBT_VEBT > nat > list_VEBT_VEBT > nat ).
thf(func_def_1706,type,
sK778: nat > list_VEBT_VEBT > vEBT_VEBT ).
thf(func_def_1707,type,
sK779: real > nat > real ).
thf(func_def_1708,type,
sK780: num > num ).
thf(func_def_1709,type,
sK781: vEBT_VEBT > $o > nat > list_VEBT_VEBT ).
thf(func_def_1710,type,
sK782: vEBT_VEBT > $o > nat > option4927543243414619207at_nat ).
thf(func_def_1711,type,
sK783: vEBT_VEBT > $o > nat > nat ).
thf(func_def_1712,type,
sK784: vEBT_VEBT > $o > nat > vEBT_VEBT ).
thf(func_def_1713,type,
sK785: vEBT_VEBT > $o > nat > $o ).
thf(func_def_1714,type,
sK786: vEBT_VEBT > $o > nat > $o ).
thf(func_def_1715,type,
sK787: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1716,type,
sK788: vEBT_VEBT > option4927543243414619207at_nat ).
thf(func_def_1717,type,
sK789: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1718,type,
sK790: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1719,type,
sK791: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1720,type,
sK792: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1721,type,
sK793: int > nat ).
thf(func_def_1722,type,
sK794: nat > nat ).
thf(func_def_1723,type,
sK795: real > ( real > real ) > real ).
thf(func_def_1724,type,
sK796: ( real > real ) > real > real > real ).
thf(func_def_1725,type,
sK797: ( nat > rat ) > rat ).
thf(func_def_1726,type,
sK798: real > ( int > $o ) > int ).
thf(func_def_1727,type,
sK799: rat > ( rat > $o ) > rat ).
thf(func_def_1728,type,
sK800: rat > ( rat > $o ) > rat ).
thf(func_def_1729,type,
sK801: int > nat ).
thf(func_def_1730,type,
sK802: int > int ).
thf(func_def_1731,type,
sK803: ( nat > num ) > nat ).
thf(func_def_1732,type,
sK804: ( real > real ) > real > real ).
thf(func_def_1733,type,
sK805: ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1734,type,
sK806: rat > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1735,type,
sK807: rat > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_1736,type,
sK808: ( num > $o ) > ( num > $o ) > ( num > $o ) > ( num > $o ) > num ).
thf(func_def_1737,type,
sK809: num > ( num > $o ) > ( num > $o ) > num ).
thf(func_def_1738,type,
sK810: num > ( num > $o ) > ( num > $o ) > num ).
thf(func_def_1739,type,
sK811: int > nat ).
thf(func_def_1740,type,
sK812: int > int > int ).
thf(func_def_1741,type,
sK813: int > int > nat ).
thf(func_def_1742,type,
sK814: real > ( real > real ) > ( real > real > real ) > real > real ).
thf(func_def_1743,type,
sK815: real > real > ( real > real > real ) > ( real > real ) > real ).
thf(func_def_1744,type,
sK816: ( nat > rat ) > nat ).
thf(func_def_1745,type,
sK817: rat > rat ).
thf(func_def_1746,type,
sK818: real > nat > real ).
thf(func_def_1747,type,
sK819: nat > nat ).
thf(func_def_1748,type,
sK820: ( nat > real ) > nat ).
thf(func_def_1749,type,
sK821: ( nat > real ) > nat ).
thf(func_def_1750,type,
sK822: ( nat > real ) > real ).
thf(func_def_1751,type,
sK823: num > num ).
thf(func_def_1752,type,
sK824: ( nat > complex ) > real ).
thf(func_def_1753,type,
sK825: nat > ( nat > complex ) > nat ).
thf(func_def_1754,type,
sK826: nat > ( nat > complex ) > nat ).
thf(func_def_1755,type,
sK827: int > int ).
thf(func_def_1756,type,
sK828: real > ( nat > real ) > real ).
thf(func_def_1757,type,
sK829: int > int > int ).
thf(func_def_1758,type,
sK830: int > int > int ).
thf(func_def_1759,type,
sK831: real > real ).
thf(func_def_1760,type,
sK832: ( code_integer > code_integer > $o ) > code_integer ).
thf(func_def_1761,type,
sK833: $o > nat > vEBT_VEBT > nat ).
thf(func_def_1762,type,
sK834: $o > nat > vEBT_VEBT > nat ).
thf(func_def_1763,type,
sK835: $o > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1764,type,
sK836: $o > nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1765,type,
sK837: $o > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1766,type,
sK838: $o > nat > vEBT_VEBT > nat ).
thf(func_def_1767,type,
sK839: $o > nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1768,type,
sK840: $o > nat > vEBT_VEBT > nat ).
thf(func_def_1769,type,
sK841: $o > nat > vEBT_VEBT > nat ).
thf(func_def_1770,type,
sK842: $o > nat > vEBT_VEBT > nat ).
thf(func_def_1771,type,
sK843: $o > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1772,type,
sK844: $o > nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1773,type,
sK845: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1774,type,
sK846: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1775,type,
sK847: vEBT_VEBT > $o ).
thf(func_def_1776,type,
sK848: vEBT_VEBT > $o ).
thf(func_def_1777,type,
sK849: num > num ).
thf(func_def_1778,type,
sK850: ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_1779,type,
sK851: ( real > real ) > real ).
thf(func_def_1780,type,
sK852: nat > ( nat > $o ) > nat ).
thf(func_def_1781,type,
sK853: real > int ).
thf(func_def_1782,type,
sK854: rat > rat > rat ).
thf(func_def_1783,type,
sK855: int > set_int > int > int ).
thf(func_def_1784,type,
sK856: int > set_int > int > int ).
thf(func_def_1785,type,
sK857: nat > nat > nat ).
thf(func_def_1786,type,
sK858: int > set_int > int > int ).
thf(func_def_1787,type,
sK859: int > set_int > int > int ).
thf(func_def_1788,type,
sK860: real > real ).
thf(func_def_1789,type,
sK861: real > nat > real ).
thf(func_def_1790,type,
sK862: vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1791,type,
sK863: vEBT_VEBT > nat > nat ).
thf(func_def_1792,type,
sK864: vEBT_VEBT > nat > nat ).
thf(func_def_1793,type,
sK865: vEBT_VEBT > nat > nat ).
thf(func_def_1794,type,
sK866: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1795,type,
sK867: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1796,type,
sK868: nat > vEBT_VEBT > nat ).
thf(func_def_1797,type,
sK869: nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1798,type,
sK870: nat > vEBT_VEBT > nat ).
thf(func_def_1799,type,
sK871: nat > vEBT_VEBT > nat ).
thf(func_def_1800,type,
sK872: nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1801,type,
sK873: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1802,type,
sK874: ( nat > real > real ) > nat > ( real > real ) > real > real ).
thf(func_def_1803,type,
sK875: ( nat > real > real ) > nat ).
thf(func_def_1804,type,
sK876: ( nat > real > real ) > real ).
thf(func_def_1805,type,
sK877: ( int > $o ) > rat > int ).
thf(func_def_1806,type,
sK878: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1807,type,
sK879: vEBT_VEBT > nat > nat ).
thf(func_def_1808,type,
sK880: vEBT_VEBT > nat > nat ).
thf(func_def_1809,type,
sK881: vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1810,type,
sK882: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1811,type,
sK883: vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1812,type,
sK884: vEBT_VEBT > nat > nat ).
thf(func_def_1813,type,
sK885: vEBT_VEBT > nat > nat ).
thf(func_def_1814,type,
sK886: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1815,type,
sK887: vEBT_VEBT > nat > nat ).
thf(func_def_1816,type,
sK888: vEBT_VEBT > nat > nat ).
thf(func_def_1817,type,
sK889: vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1818,type,
sK890: nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1819,type,
sK891: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1820,type,
sK892: nat > vEBT_VEBT > $o ).
thf(func_def_1821,type,
sK893: nat > vEBT_VEBT > $o ).
thf(func_def_1822,type,
sK894: real > real > nat ).
thf(func_def_1823,type,
sK895: vEBT_VEBT > $o ).
thf(func_def_1824,type,
sK896: vEBT_VEBT > $o ).
thf(func_def_1825,type,
sK897: nat > vEBT_VEBT > option4927543243414619207at_nat ).
thf(func_def_1826,type,
sK898: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1827,type,
sK899: nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1828,type,
sK900: nat > vEBT_VEBT > nat ).
thf(func_def_1829,type,
sK901: nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1830,type,
sK902: num > num ).
thf(func_def_1831,type,
sK903: real > nat ).
thf(func_def_1832,type,
sK904: int > ( int > $o ) > int ).
thf(func_def_1833,type,
sK905: nat > vEBT_VEBT > $o ).
thf(func_def_1834,type,
sK906: nat > vEBT_VEBT > $o ).
thf(func_def_1835,type,
sK907: vEBT_VEBT > nat > option4927543243414619207at_nat ).
thf(func_def_1836,type,
sK908: vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1837,type,
sK909: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1838,type,
sK910: vEBT_VEBT > nat > nat ).
thf(func_def_1839,type,
sK911: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1840,type,
sK912: vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1841,type,
sK913: vEBT_VEBT > nat > option4927543243414619207at_nat ).
thf(func_def_1842,type,
sK914: real > ( real > real ) > real ).
thf(func_def_1843,type,
sK915: rat > rat > rat ).
thf(func_def_1844,type,
sK916: real > ( real > $o ) > real ).
thf(func_def_1845,type,
sK917: real > ( real > $o ) > real ).
thf(func_def_1846,type,
sK918: real > ( real > $o ) > real ).
thf(func_def_1847,type,
sK919: real > ( real > $o ) > real ).
thf(func_def_1848,type,
sK920: int > set_int > int ).
thf(func_def_1849,type,
sK921: rat > int ).
thf(func_def_1850,type,
sK922: int > ( int > $o ) > int > int ).
thf(func_def_1851,type,
sK923: int > ( int > $o ) > int > int ).
thf(func_def_1852,type,
sK924: ( int > $o ) > int > int > int ).
thf(func_def_1853,type,
sK925: ( int > $o ) > int > int > int ).
thf(func_def_1854,type,
sK926: ( nat > $o ) > nat ).
thf(func_def_1855,type,
sK927: nat > rat > ( nat > rat ) > nat ).
thf(func_def_1856,type,
sK928: ( nat > rat ) > rat ).
thf(func_def_1857,type,
sK929: ( nat > rat ) > nat ).
thf(func_def_1858,type,
sK930: nat > nat > nat ).
thf(func_def_1859,type,
sK931: int > int > int > int ).
thf(func_def_1860,type,
sK932: rat > rat > nat ).
thf(func_def_1861,type,
sK933: num > num ).
thf(func_def_1862,type,
sK934: num > num ).
thf(func_def_1863,type,
sK935: real > int ).
thf(func_def_1864,type,
sK936: ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1865,type,
sK937: real > real > nat ).
thf(func_def_1866,type,
sK938: set_nat > nat ).
thf(func_def_1867,type,
sK939: set_nat > nat ).
thf(func_def_1868,type,
sK940: real > real ).
thf(func_def_1869,type,
sK941: real > ( real > real ) > real > real ).
thf(func_def_1870,type,
sK942: real > ( real > real ) > real > real ).
thf(func_def_1871,type,
sK943: ( real > real ) > real > real > real ).
thf(func_def_1872,type,
sK944: real > real > nat > ( nat > real > real ) > real ).
thf(func_def_1873,type,
sK945: real > real > nat > ( nat > real > real ) > nat ).
thf(func_def_1874,type,
sK946: real > nat > real > ( real > real ) > ( nat > real > real ) > real ).
thf(func_def_1875,type,
sK947: ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1876,type,
sK948: ( int > $o ) > product_prod_nat_nat ).
thf(func_def_1877,type,
sK949: int > int > int > int ).
thf(func_def_1878,type,
sK950: int > int > int > int ).
thf(func_def_1879,type,
sK951: num > num > num ).
thf(func_def_1880,type,
sK952: num > num > num > num ).
thf(func_def_1881,type,
sK953: num > num > num > num ).
thf(func_def_1882,type,
sK954: num > num > num > num ).
thf(func_def_1883,type,
sK955: num > num > num > num ).
thf(func_def_1884,type,
sK956: num > num > num > num ).
thf(func_def_1885,type,
sK957: num > num > num > num ).
thf(func_def_1886,type,
sK958: num > num > num ).
thf(func_def_1887,type,
sK959: num > num > num > num ).
thf(func_def_1888,type,
sK960: num > num > num > num ).
thf(func_def_1889,type,
sK961: num > num > num > num ).
thf(func_def_1890,type,
sK962: num > num > num > num ).
thf(func_def_1891,type,
sK963: complex > nat ).
thf(func_def_1892,type,
sK964: int > set_int > int > int ).
thf(func_def_1893,type,
sK965: int > set_int > int > int ).
thf(func_def_1894,type,
sK966: ( nat > num ) > nat ).
thf(func_def_1895,type,
sK967: ( num > $o ) > ( num > $o ) > ( num > $o ) > ( num > $o ) > num ).
thf(func_def_1896,type,
sK968: num > ( num > $o ) > ( num > $o ) > num ).
thf(func_def_1897,type,
sK969: num > ( num > $o ) > ( num > $o ) > num ).
thf(func_def_1898,type,
sK970: nat > nat ).
thf(func_def_1899,type,
sK971: real > set_real > ( real > real ) > real ).
thf(func_def_1900,type,
sK972: nat > extended_enat > nat ).
thf(func_def_1901,type,
sK973: int > int > set_int > int ).
thf(func_def_1902,type,
sK974: int > int > set_int > int ).
thf(func_def_1903,type,
sK975: ( complex > $o ) > int ).
thf(func_def_1904,type,
sK976: rat > rat ).
thf(func_def_1905,type,
sK977: int > nat ).
thf(func_def_1906,type,
sK978: ( int > $o ) > int ).
thf(func_def_1907,type,
sK979: ( int > $o ) > int > int ).
thf(func_def_1908,type,
sK980: set_int > ( int > $o ) > int > int ).
thf(func_def_1909,type,
sK981: set_int > ( int > $o ) > int > int ).
thf(func_def_1910,type,
sK982: int > set_int > ( int > $o ) > int ).
thf(func_def_1911,type,
sK983: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_1912,type,
sK984: int > ( int > $o ) > int ).
thf(func_def_1913,type,
sK985: int > ( int > $o ) > int ).
thf(func_def_1914,type,
sK986: real > nat ).
thf(func_def_1915,type,
sK987: ( int > $o ) > int > int ).
thf(func_def_1916,type,
sK988: ( nat > $o ) > nat ).
thf(func_def_1917,type,
sK989: ( nat > $o ) > nat ).
thf(func_def_1918,type,
sK990: ( nat > nat > $o ) > nat ).
thf(func_def_1919,type,
sK991: ( nat > nat > $o ) > nat ).
thf(func_def_1920,type,
sK992: ( nat > nat > $o ) > nat ).
thf(func_def_1921,type,
sK993: ( nat > nat > $o ) > nat ).
thf(func_def_1922,type,
sK994: ( real > real ) > set_real > real ).
thf(func_def_1923,type,
sK995: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1924,type,
sK996: real > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1925,type,
sK997: real > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1926,type,
sK998: int > ( int > $o ) > int ).
thf(func_def_1927,type,
sK999: int > ( int > $o ) > int ).
thf(func_def_1928,type,
sK1000: int > int ).
thf(func_def_1929,type,
sK1001: ( real > $o ) > nat ).
thf(func_def_1930,type,
sK1002: ( real > $o ) > real ).
thf(func_def_1931,type,
sK1003: ( real > $o ) > real ).
thf(func_def_1932,type,
sK1004: real > int ).
thf(func_def_1933,type,
sK1005: real > int ).
thf(func_def_1934,type,
sK1006: ( nat > $o ) > nat ).
thf(func_def_1935,type,
sK1007: real > nat > real ).
thf(func_def_1936,type,
sK1008: real > ( nat > rat ) > nat ).
thf(func_def_1937,type,
sK1009: rat > rat ).
thf(func_def_1938,type,
sK1010: nat > ( nat > nat ) > nat ).
thf(func_def_1939,type,
sK1011: ( nat > nat ) > nat > nat ).
thf(func_def_1940,type,
sK1012: int > int > nat ).
thf(func_def_1941,type,
sK1013: nat > real > nat ).
thf(func_def_1942,type,
sK1014: real > real ).
thf(func_def_1943,type,
sK1015: ( list_nat > $o ) > ( list_nat > $o ) > list_nat ).
thf(func_def_1944,type,
sK1016: ( rat > rat > $o ) > rat ).
thf(func_def_1945,type,
sK1017: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1946,type,
sK1018: real > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1947,type,
sK1019: real > ( real > $o ) > ( real > $o ) > real ).
thf(func_def_1948,type,
sK1020: ( real > real ) > real > real ).
thf(func_def_1949,type,
sK1021: nat > nat ).
thf(func_def_1950,type,
sK1022: nat > nat > ( nat > $o ) > nat ).
thf(func_def_1951,type,
sK1023: nat > nat > ( nat > $o ) > nat ).
thf(func_def_1952,type,
sK1024: real > real > ( real > real ) > real ).
thf(func_def_1953,type,
sK1025: ( real > real ) > real > real > ( real > real ) > real ).
thf(func_def_1954,type,
sK1026: real > real ).
thf(func_def_1955,type,
sK1027: num > num > num > num ).
thf(func_def_1956,type,
sK1028: num > num > num > num ).
thf(func_def_1957,type,
sK1029: num > num > num > num ).
thf(func_def_1958,type,
sK1030: num > num > num > num ).
thf(func_def_1959,type,
sK1031: num > num > num ).
thf(func_def_1960,type,
sK1032: num > num > num > num ).
thf(func_def_1961,type,
sK1033: num > num > num > num ).
thf(func_def_1962,type,
sK1034: num > num > num > num ).
thf(func_def_1963,type,
sK1035: num > num > num > num ).
thf(func_def_1964,type,
sK1036: num > num > num ).
thf(func_def_1965,type,
sK1037: num > num > num > num ).
thf(func_def_1966,type,
sK1038: num > num > num > num ).
thf(func_def_1967,type,
sK1039: nat > nat > nat ).
thf(func_def_1968,type,
sK1040: nat > nat > nat ).
thf(func_def_1969,type,
sK1041: real > nat > real ).
thf(func_def_1970,type,
sK1042: vEBT_VEBT > nat > $o > $o ).
thf(func_def_1971,type,
sK1043: vEBT_VEBT > nat > $o > $o ).
thf(func_def_1972,type,
sK1044: $o > nat > vEBT_VEBT > nat ).
thf(func_def_1973,type,
sK1045: $o > nat > vEBT_VEBT > option4927543243414619207at_nat ).
thf(func_def_1974,type,
sK1046: $o > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1975,type,
sK1047: $o > nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1976,type,
sK1048: nat > $o > vEBT_VEBT > option4927543243414619207at_nat ).
thf(func_def_1977,type,
sK1049: nat > $o > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1978,type,
sK1050: nat > $o > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1979,type,
sK1051: ( int > $o ) > nat ).
thf(func_def_1980,type,
sK1052: ( int > $o ) > nat ).
thf(func_def_1981,type,
sK1053: real > real ).
thf(func_def_1982,type,
sK1054: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1983,type,
sK1055: vEBT_VEBT > nat > option4927543243414619207at_nat ).
thf(func_def_1984,type,
sK1056: vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1985,type,
sK1057: vEBT_VEBT > nat > nat ).
thf(func_def_1986,type,
sK1058: vEBT_VEBT > nat > $o ).
thf(func_def_1987,type,
sK1059: vEBT_VEBT > nat > $o ).
thf(func_def_1988,type,
sK1060: real > real ).
thf(func_def_1989,type,
sK1061: nat > nat > nat ).
thf(func_def_1990,type,
sK1062: nat > nat > nat ).
thf(func_def_1991,type,
sK1063: int > int > set_int > int ).
thf(func_def_1992,type,
sK1064: int > int > set_int > int ).
thf(func_def_1993,type,
sK1065: real > real > rat ).
thf(func_def_1994,type,
sK1066: ( nat > rat ) > nat ).
thf(func_def_1995,type,
sK1067: int > int > set_int > int ).
thf(func_def_1996,type,
sK1068: int > int > set_int > int ).
thf(func_def_1997,type,
sK1069: ( code_integer > $o ) > code_integer ).
thf(func_def_1998,type,
sK1070: ( code_integer > $o ) > code_integer ).
thf(func_def_1999,type,
sK1071: ( code_integer > $o ) > $o ).
thf(func_def_2000,type,
sK1072: nat > ( nat > $o ) > nat ).
thf(func_def_2001,type,
sK1073: nat > ( nat > $o ) > nat ).
thf(func_def_2002,type,
sK1074: ( nat > nat ) > nat ).
thf(func_def_2003,type,
sK1075: ( nat > nat ) > nat ).
thf(func_def_2004,type,
sK1076: ( nat > num ) > nat ).
thf(func_def_2005,type,
sK1077: real > real > real ).
thf(func_def_2006,type,
sK1078: ( nat > int ) > nat ).
thf(func_def_2007,type,
sK1079: int > int > int ).
thf(func_def_2008,type,
sK1080: real > nat > ( nat > real > real ) > real ).
thf(func_def_2009,type,
sK1081: real > nat > ( nat > real > real ) > nat ).
thf(func_def_2010,type,
sK1082: ( nat > real > real ) > nat > real > ( real > real ) > real ).
thf(func_def_2011,type,
sK1083: nat > nat ).
thf(func_def_2012,type,
sK1084: real > ( nat > real > real ) > nat > nat ).
thf(func_def_2013,type,
sK1085: real > ( nat > real > real ) > nat > real ).
thf(func_def_2014,type,
sK1086: ( real > real ) > ( nat > real > real ) > real > nat > real ).
thf(func_def_2015,type,
sK1087: real > nat ).
thf(func_def_2016,type,
sK1088: real > nat ).
thf(func_def_2017,type,
sK1089: nat > nat ).
thf(func_def_2018,type,
sK1090: vEBT_VEBT > nat > $o > option4927543243414619207at_nat ).
thf(func_def_2019,type,
sK1091: vEBT_VEBT > nat > $o > nat ).
thf(func_def_2020,type,
sK1092: vEBT_VEBT > nat > $o > list_VEBT_VEBT ).
thf(func_def_2021,type,
sK1093: vEBT_VEBT > nat > $o > vEBT_VEBT ).
thf(func_def_2022,type,
sK1094: vEBT_VEBT > nat > $o > vEBT_VEBT ).
thf(func_def_2023,type,
sK1095: vEBT_VEBT > $o ).
thf(func_def_2024,type,
sK1096: vEBT_VEBT > $o ).
thf(func_def_2025,type,
sK1097: int > int ).
thf(func_def_2026,type,
sK1098: nat > nat > nat ).
thf(func_def_2027,type,
sK1099: real > real ).
thf(func_def_2028,type,
sK1100: int > int ).
thf(func_def_2029,type,
sK1101: ( real > $o ) > nat ).
thf(func_def_2030,type,
sK1102: ( real > $o ) > real ).
thf(func_def_2031,type,
sK1103: ( real > $o ) > real ).
thf(func_def_2032,type,
sK1104: extended_enat > nat > nat ).
thf(func_def_2033,type,
sK1105: int > nat ).
thf(func_def_2034,type,
sK1106: int > nat ).
thf(func_def_2035,type,
sK1107: ( nat > int ) > nat ).
thf(func_def_2036,type,
sK1108: nat > nat > nat ).
thf(func_def_2037,type,
sK1109: nat > nat > nat ).
thf(func_def_2038,type,
sK1110: nat > nat > nat ).
thf(func_def_2039,type,
sK1111: real > real > real ).
thf(func_def_2040,type,
sK1112: rat > int ).
thf(func_def_2041,type,
sK1113: real > int ).
thf(func_def_2042,type,
sK1114: ( nat > real ) > real > nat > int ).
thf(func_def_2043,type,
sK1115: real > real > nat ).
thf(func_def_2044,type,
sK1116: ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_2045,type,
sK1117: rat > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_2046,type,
sK1118: rat > ( rat > $o ) > ( rat > $o ) > rat ).
thf(func_def_2047,type,
sK1119: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_2048,type,
sK1120: ( int > $o ) > int ).
thf(func_def_2049,type,
sK1121: int > ( int > $o ) > int ).
thf(func_def_2050,type,
sK1122: int > ( int > $o ) > int ).
thf(func_def_2051,type,
sK1123: rat > rat > rat ).
thf(func_def_2052,type,
sK1124: int > int > int ).
thf(func_def_2053,type,
sK1125: real > int ).
thf(func_def_2054,type,
sK1126: real > real ).
thf(func_def_2055,type,
sK1127: ( real > nat > real ) > real > ( real > nat > real ) > nat ).
thf(func_def_2056,type,
sK1128: ( real > nat > real ) > real > real > real ).
thf(func_def_2057,type,
sK1129: ( nat > real ) > ( real > nat > real ) > real > real > real ).
thf(func_def_2058,type,
sK1130: ( nat > real ) > ( real > nat > real ) > real > real > real ).
thf(func_def_2059,type,
sK1131: ( nat > real ) > ( real > nat > real ) > real > real > nat ).
thf(func_def_2060,type,
sK1132: nat > vEBT_VEBT > $o ).
thf(func_def_2061,type,
sK1133: nat > vEBT_VEBT > $o ).
thf(func_def_2062,type,
sK1134: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_2063,type,
sK1135: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_2064,type,
sK1136: vEBT_VEBT > option4927543243414619207at_nat ).
thf(func_def_2065,type,
sK1137: vEBT_VEBT > nat > option4927543243414619207at_nat ).
thf(func_def_2066,type,
sK1138: vEBT_VEBT > nat > nat ).
thf(func_def_2067,type,
sK1139: vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_2068,type,
sK1140: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_2069,type,
sK1141: rat > rat ).
thf(func_def_2070,type,
sK1142: rat > rat > rat ).
thf(func_def_2071,type,
sK1143: int > int ).
thf(func_def_2072,type,
sK1144: code_integer > code_integer ).
thf(func_def_2073,type,
sK1145: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_2074,type,
sK1146: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_2075,type,
sK1147: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_2076,type,
sK1148: int > int > set_int > int ).
thf(func_def_2077,type,
sK1149: int > int > set_int > int ).
thf(func_def_2078,type,
sK1150: nat > nat > nat ).
thf(func_def_2079,type,
sK1151: int > int > int ).
thf(func_def_2080,type,
sK1152: real > int ).
thf(func_def_2081,type,
sK1153: nat > nat > nat ).
thf(func_def_2082,type,
sK1154: list_nat > ( nat > $o ) > nat ).
thf(func_def_2083,type,
sK1155: int > int > int > int ).
thf(func_def_2084,type,
sK1156: ( int > $o ) > int ).
thf(func_def_2085,type,
sK1157: ( int > $o ) > int > int ).
thf(func_def_2086,type,
sK1158: set_int > ( int > $o ) > int > int ).
thf(func_def_2087,type,
sK1159: set_int > ( int > $o ) > int > int ).
thf(func_def_2088,type,
sK1160: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_2089,type,
sK1161: int > ( int > $o ) > set_int > int ).
thf(func_def_2090,type,
sK1162: ( int > $o ) > int > int ).
thf(func_def_2091,type,
sK1163: ( int > $o ) > int > int ).
thf(func_def_2092,type,
sK1164: ( rat > $o ) > int ).
thf(func_def_2093,type,
sK1165: ( rat > $o ) > int ).
thf(func_def_2094,type,
sK1166: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_2095,type,
sK1167: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_2096,type,
sK1168: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_2097,type,
sK1169: int > nat ).
thf(func_def_2098,type,
sK1170: nat > nat ).
thf(func_def_2099,type,
sK1171: nat > nat > nat ).
thf(func_def_2100,type,
sK1172: nat > nat > nat ).
thf(func_def_2101,type,
sK1173: nat > nat > nat ).
thf(func_def_2102,type,
sK1174: real > nat > real ).
thf(func_def_2103,type,
sK1175: int > ( int > $o ) > int ).
thf(func_def_2104,type,
sK1176: int > ( int > $o ) > int ).
thf(func_def_2105,type,
sK1177: int > ( int > $o ) > int ).
thf(func_def_2106,type,
sK1178: int > ( int > $o ) > int ).
thf(func_def_2107,type,
sK1179: real > real ).
thf(func_def_2108,type,
sK1180: real > int ).
thf(func_def_2109,type,
sK1181: nat > rat > ( nat > rat ) > nat ).
thf(func_def_2110,type,
sK1182: ( nat > rat ) > rat ).
thf(func_def_2111,type,
sK1183: ( nat > rat ) > nat ).
thf(func_def_2112,type,
sK1184: rat > nat ).
thf(func_def_2113,type,
sK1185: int > nat ).
thf(func_def_2114,type,
sK1186: nat > nat > nat ).
thf(func_def_2115,type,
sK1187: ( num > $o ) > ( num > $o ) > ( num > $o ) > ( num > $o ) > num ).
thf(func_def_2116,type,
sK1188: num > ( num > $o ) > ( num > $o ) > num ).
thf(func_def_2117,type,
sK1189: num > ( num > $o ) > ( num > $o ) > num ).
thf(func_def_2118,type,
sK1190: complex > complex ).
thf(func_def_2119,type,
sK1191: complex > real ).
thf(func_def_2120,type,
sK1192: product_prod_num_num > num ).
thf(func_def_2121,type,
sK1193: product_prod_num_num > num ).
thf(func_def_2122,type,
sK1194: product_prod_num_num > num ).
thf(func_def_2123,type,
sK1195: product_prod_num_num > num ).
thf(func_def_2124,type,
sK1196: product_prod_num_num > num ).
thf(func_def_2125,type,
sK1197: product_prod_num_num > num ).
thf(func_def_2126,type,
sK1198: product_prod_num_num > num ).
thf(func_def_2127,type,
sK1199: product_prod_num_num > num ).
thf(func_def_2128,type,
sK1200: product_prod_num_num > num ).
thf(func_def_2129,type,
sK1201: product_prod_num_num > num ).
thf(func_def_2130,type,
sK1202: product_prod_num_num > num ).
thf(func_def_2131,type,
sK1203: product_prod_num_num > num ).
thf(func_def_2132,type,
sK1204: int > int > int ).
thf(func_def_2133,type,
sK1205: vEBT_VEBT > $o ).
thf(func_def_2134,type,
sK1206: vEBT_VEBT > $o ).
thf(func_def_2135,type,
sK1207: vEBT_VEBT > option4927543243414619207at_nat ).
thf(func_def_2136,type,
sK1208: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_2137,type,
sK1209: vEBT_VEBT > nat ).
thf(func_def_2138,type,
sK1210: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_2139,type,
sK1211: nat > code_integer > code_integer ).
thf(func_def_2140,type,
sK1212: int > ( int > $o ) > int ).
thf(func_def_2141,type,
sK1213: ( int > $o ) > int > int ).
thf(func_def_2142,type,
sK1214: ( nat > $o ) > nat ).
thf(func_def_2143,type,
sK1215: nat > nat > ( nat > $o ) > nat ).
thf(func_def_2144,type,
sK1216: nat > nat > ( nat > $o ) > nat ).
thf(func_def_2145,type,
sK1217: ( nat > int ) > nat ).
thf(func_def_2146,type,
sK1218: code_integer > code_integer > code_integer ).
thf(func_def_2147,type,
sK1219: ( int > $o ) > int ).
thf(func_def_2148,type,
sK1220: ( int > $o ) > int ).
thf(func_def_2149,type,
sK1221: nat > ( nat > $o ) > nat ).
thf(func_def_2150,type,
sK1222: nat > ( nat > $o ) > nat ).
thf(func_def_2151,type,
sK1223: nat > nat > nat ).
thf(func_def_2152,type,
sK1224: ( nat > set_nat ) > nat ).
thf(func_def_2153,type,
sK1225: ( nat > real ) > nat ).
thf(func_def_2154,type,
sK1226: ( nat > real ) > nat ).
thf(func_def_2155,type,
sK1227: ( int > $o ) > int ).
thf(func_def_2156,type,
sK1228: ( int > $o ) > int > int ).
thf(func_def_2157,type,
sK1229: ( int > $o ) > int > int ).
thf(func_def_2158,type,
sK1230: ( int > $o ) > int > int ).
thf(func_def_2159,type,
sK1231: int > nat > ( nat > int ) > nat ).
thf(func_def_2160,type,
sK1232: nat > ( nat > int ) > nat ).
thf(func_def_2161,type,
sK1233: option_num > num > num > num ).
thf(func_def_2162,type,
sK1234: option_num > num > num > num ).
thf(func_def_2163,type,
sK1235: option_num > num > num > num ).
thf(func_def_2164,type,
sK1236: option_num > num > num > num ).
thf(func_def_2165,type,
sK1237: num > num > option_num > num ).
thf(func_def_2166,type,
sK1238: num > num > option_num > num ).
thf(func_def_2167,type,
sK1239: num > num > option_num > num ).
thf(func_def_2168,type,
sK1240: num > option_num > num ).
thf(func_def_2169,type,
sK1241: option_num > num > num ).
thf(func_def_2170,type,
sK1242: option_num > num > num > num ).
thf(func_def_2171,type,
sK1243: num > num > option_num > num ).
thf(func_def_2172,type,
sK1244: num > num > option_num > num ).
thf(func_def_2173,type,
sK1245: int > int > set_int > int ).
thf(func_def_2174,type,
sK1246: int > int > set_int > int ).
thf(func_def_2175,type,
sK1247: ( nat > $o ) > nat ).
thf(func_def_2176,type,
sK1248: ( nat > $o ) > nat > nat ).
thf(func_def_2177,type,
sK1249: real > real ).
thf(func_def_2178,type,
sK1250: int > int ).
thf(func_def_2179,type,
sK1251: ( nat > complex ) > real > nat ).
thf(func_def_2180,type,
sK1252: real > ( real > real ) > real > real ).
thf(func_def_2181,type,
sK1253: real > real > ( real > real ) > real ).
thf(func_def_2182,type,
sK1254: ( real > real ) > real > real > real ).
thf(func_def_2183,type,
sK1255: ( real > real ) > real > ( real > real ) > real > real ).
thf(func_def_2184,type,
sK1256: ( real > real ) > real > ( real > real ) > real > real ).
thf(func_def_2185,type,
sK1257: ( real > real ) > real > ( real > real ) > real > real ).
thf(func_def_2186,type,
sK1258: ( real > real ) > real > real > real ).
thf(func_def_2187,type,
sK1259: int > int > set_int > int ).
thf(func_def_2188,type,
sK1260: int > int > set_int > int ).
thf(func_def_2189,type,
sK1261: nat > vEBT_VEBT > nat ).
thf(func_def_2190,type,
sK1262: nat > vEBT_VEBT > option4927543243414619207at_nat ).
thf(func_def_2191,type,
sK1263: nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_2192,type,
sK1264: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_2193,type,
sK1265: vEBT_VEBT > $o ).
thf(func_def_2194,type,
sK1266: vEBT_VEBT > $o ).
thf(func_def_2195,type,
sK1267: ( num > $o ) > ( num > $o ) > ( num > $o ) > ( num > $o ) > num ).
thf(func_def_2196,type,
sK1268: num > ( num > $o ) > ( num > $o ) > num ).
thf(func_def_2197,type,
sK1269: num > ( num > $o ) > ( num > $o ) > num ).
thf(func_def_2198,type,
sK1270: ( int > $o ) > int ).
thf(func_def_2199,type,
sK1271: ( int > $o ) > $o ).
thf(func_def_2200,type,
sK1272: ( int > $o ) > int ).
thf(func_def_2201,type,
sK1273: nat > nat ).
thf(func_def_2202,type,
sK1274: code_integer > nat ).
thf(func_def_2203,type,
sK1275: real > ( real > real ) > ( nat > real > real ) > nat > real ).
thf(func_def_2204,type,
sK1276: real > ( nat > real > real ) > nat > real ).
thf(func_def_2205,type,
sK1277: real > ( nat > real > real ) > nat > nat ).
thf(func_def_2206,type,
sK1278: ( real > real ) > real > real > real ).
thf(func_def_2207,type,
sK1279: ( real > real ) > ( real > real ) > real > real > real ).
thf(func_def_2208,type,
sK1280: real > real > ( real > real ) > real ).
thf(func_def_2209,type,
sK1281: real > ( real > real ) > ( real > real ) > real > real ).
thf(func_def_2210,type,
sK1282: ( real > real ) > ( real > real ) > ( real > real ) > ( real > real ) > real > real > real ).
thf(func_def_2211,type,
sK1283: real > nat ).
thf(func_def_2212,type,
sK1284: nat > nat > nat ).
thf(func_def_2213,type,
sK1285: complex > real ).
thf(func_def_2214,type,
sK1286: complex > real ).
thf(func_def_2215,type,
sK1287: int > nat ).
thf(func_def_2216,type,
sK1288: real ).
thf(func_def_2217,type,
sK1289: real > ( real > real ) > real ).
thf(func_def_2218,type,
sK1290: produc9072475918466114483BT_nat > nat ).
thf(func_def_2219,type,
sK1291: produc9072475918466114483BT_nat > option4927543243414619207at_nat ).
thf(func_def_2220,type,
sK1292: produc9072475918466114483BT_nat > vEBT_VEBT ).
thf(func_def_2221,type,
sK1293: produc9072475918466114483BT_nat > nat ).
thf(func_def_2222,type,
sK1294: produc9072475918466114483BT_nat > list_VEBT_VEBT ).
thf(func_def_2223,type,
sK1295: produc9072475918466114483BT_nat > nat ).
thf(func_def_2224,type,
sK1296: produc9072475918466114483BT_nat > $o ).
thf(func_def_2225,type,
sK1297: produc9072475918466114483BT_nat > $o ).
thf(func_def_2226,type,
sK1298: produc9072475918466114483BT_nat > vEBT_VEBT ).
thf(func_def_2227,type,
sK1299: produc9072475918466114483BT_nat > nat ).
thf(func_def_2228,type,
sK1300: produc9072475918466114483BT_nat > option4927543243414619207at_nat ).
thf(func_def_2229,type,
sK1301: produc9072475918466114483BT_nat > list_VEBT_VEBT ).
thf(func_def_2230,type,
sK1302: set_nat > nat > nat ).
thf(func_def_2231,type,
sK1303: ( nat > $o ) > nat ).
thf(func_def_2232,type,
sK1304: rat > ( nat > rat ) > nat ).
thf(func_def_2233,type,
sK1305: nat > nat > nat > nat ).
thf(func_def_2234,type,
sK1306: real > real > real ).
thf(func_def_2235,type,
sK1307: ( real > real ) > real > real ).
thf(func_def_2236,type,
sK1308: ( int > $o ) > int > int ).
thf(func_def_2237,type,
sK1309: rat > rat ).
thf(func_def_2238,type,
sK1310: rat > rat ).
thf(func_def_2239,type,
sK1311: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_2240,type,
sK1312: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_2241,type,
sK1313: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_2242,type,
sK1314: rat > ( int > $o ) > int ).
thf(func_def_2243,type,
sK1315: vEBT_VEBT > nat ).
thf(func_def_2244,type,
sK1316: list_VEBT_VEBT > vEBT_VEBT ).
thf(func_def_2245,type,
sK1317: list_VEBT_VEBT > nat ).
thf(func_def_2246,type,
sK1318: list_VEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_2247,type,
sK1319: nat > nat > nat ).
thf(func_def_2248,type,
sK1320: ( nat > real > real ) > real ).
thf(func_def_2249,type,
sK1321: ( nat > real > real ) > nat ).
thf(func_def_2250,type,
sK1322: ( nat > real > real ) > nat > ( real > real ) > real > real ).
thf(func_def_2251,type,
sK1323: ( nat > real ) > nat ).
thf(func_def_2252,type,
sK1324: ( nat > real ) > nat ).
thf(func_def_2253,type,
sK1325: nat > ( nat > $o ) > nat ).
thf(func_def_2254,type,
sK1326: nat > ( nat > $o ) > nat ).
thf(func_def_2255,type,
sK1327: real > ( nat > rat ) > nat ).
thf(func_def_2256,type,
sK1328: real > real ).
thf(func_def_2257,type,
sK1329: ( nat > real > real ) > real ).
thf(func_def_2258,type,
sK1330: ( nat > real > real ) > nat ).
thf(func_def_2259,type,
sK1331: ( nat > real > real ) > ( real > real ) > real > nat > real ).
thf(func_def_2260,type,
sK1332: ( nat > real ) > nat > nat ).
thf(func_def_2261,type,
sK1333: ( nat > real ) > nat > nat ).
thf(func_def_2262,type,
sK1334: ( real > real ) > real > real ).
thf(func_def_2263,type,
sK1335: nat > nat ).
thf(func_def_2264,type,
sK1336: nat > nat ).
thf(func_def_2265,type,
sK1337: ( nat > real ) > nat > nat ).
thf(func_def_2266,type,
sK1338: ( nat > real ) > nat > nat ).
thf(func_def_2267,type,
sK1339: int > set_int > int > int ).
thf(func_def_2268,type,
sK1340: int > set_int > int > int ).
thf(func_def_2269,type,
sK1341: nat > int > int ).
thf(func_def_2270,type,
sK1342: num > num ).
thf(func_def_2271,type,
sK1343: option_num > num > num > num ).
thf(func_def_2272,type,
sK1344: option_num > num > num > num ).
thf(func_def_2273,type,
sK1345: option_num > num > num > num ).
thf(func_def_2274,type,
sK1346: option_num > num > num > num ).
thf(func_def_2275,type,
sK1347: num > num ).
thf(func_def_2276,type,
sK1348: num > option_num > num > num ).
thf(func_def_2277,type,
sK1349: num > option_num > num > num ).
thf(func_def_2278,type,
sK1350: num > num ).
thf(func_def_2279,type,
sK1351: num > num ).
thf(func_def_2280,type,
sK1352: option_num > num > num > num ).
thf(func_def_2281,type,
sK1353: option_num > num > num > num ).
thf(func_def_2282,type,
sK1354: ( real > $o ) > nat > rat ).
thf(func_def_2283,type,
sK1355: ( nat > $o ) > nat ).
thf(func_def_2284,type,
sK1356: nat > nat > nat > nat ).
thf(func_def_2285,type,
sK1357: nat > nat > nat > nat ).
thf(func_def_2286,type,
sK1358: int > int ).
thf(func_def_2287,type,
sK1359: num > option_num > num > num ).
thf(func_def_2288,type,
sK1360: num > option_num > num > num ).
thf(func_def_2289,type,
sK1361: num > num > option_num > num ).
thf(func_def_2290,type,
sK1362: option_num > num > num ).
thf(func_def_2291,type,
sK1363: num > option_num > num ).
thf(func_def_2292,type,
sK1364: num > num > option_num > num ).
thf(func_def_2293,type,
sK1365: num > option_num > num > num ).
thf(func_def_2294,type,
sK1366: num > option_num > num > num ).
thf(func_def_2295,type,
sK1367: option_num > num > num > num ).
thf(func_def_2296,type,
sK1368: option_num > num > num > num ).
thf(func_def_2297,type,
sK1369: option_num > num > num > num ).
thf(func_def_2298,type,
sK1370: option_num > num > num > num ).
thf(func_def_2299,type,
sK1371: ( int > $o ) > int ).
thf(func_def_2300,type,
sK1372: ( nat > rat ) > rat ).
thf(func_def_2301,type,
sK1373: ( nat > rat ) > nat ).
thf(func_def_2302,type,
sK1374: int > int > int ).
thf(func_def_2303,type,
sK1375: set_int > int > ( int > $o ) > int ).
thf(func_def_2304,type,
sK1376: int > int > set_int > int ).
thf(func_def_2305,type,
sK1377: int > int > set_int > int ).
thf(func_def_2306,type,
sK1378: int > set_int > ( int > $o ) > int ).
thf(func_def_2307,type,
sK1379: int > ( int > int > $o ) > int > int ).
thf(func_def_2308,type,
sK1380: int > ( int > int > $o ) > int > int ).
thf(func_def_2309,type,
sK1381: int > int ).
thf(func_def_2310,type,
sK1382: ( nat > real ) > nat > nat ).
thf(func_def_2311,type,
sK1383: ( nat > nat ) > nat ).
thf(func_def_2312,type,
sK1384: nat > nat ).
thf(func_def_2313,type,
sK1385: nat > nat ).
thf(func_def_2314,type,
sK1386: real > real ).
thf(func_def_2315,type,
sK1387: ( nat > real ) > ( nat > real ) > real ).
thf(func_def_2316,type,
sK1388: ( nat > real ) > ( nat > real ) > nat ).
thf(func_def_2317,type,
sK1389: ( nat > real ) > nat ).
thf(func_def_2318,type,
sK1390: ( nat > real ) > nat ).
thf(func_def_2319,type,
sK1391: int > set_int > int > int ).
thf(func_def_2320,type,
sK1392: int > set_int > int > int ).
thf(func_def_2321,type,
sK1393: ( set_nat > $o ) > ( set_nat > $o ) > set_nat ).
thf(func_def_2322,type,
sK1394: rat > ( rat > $o ) > rat ).
thf(func_def_2323,type,
sK1395: rat > ( rat > $o ) > rat ).
thf(func_def_2324,type,
sK1396: rat > ( rat > $o ) > rat ).
thf(func_def_2325,type,
sK1397: rat > ( rat > $o ) > rat ).
thf(func_def_2326,type,
sK1398: ( nat > real ) > nat ).
thf(func_def_2327,type,
sK1399: ( nat > real ) > nat ).
thf(func_def_2328,type,
sK1400: real > real ).
thf(func_def_2329,type,
sK1401: real > ( real > real ) > real ).
thf(func_def_2330,type,
sK1402: num > option_num > num > num ).
thf(func_def_2331,type,
sK1403: num > option_num > num > num ).
thf(func_def_2332,type,
sK1404: num > option_num > num > num ).
thf(func_def_2333,type,
sK1405: num > option_num > num > num ).
thf(func_def_2334,type,
sK1406: num > option_num > num ).
thf(func_def_2335,type,
sK1407: num > option_num > num > num ).
thf(func_def_2336,type,
sK1408: option_num > num > num ).
thf(func_def_2337,type,
sK1409: num > num > option_num > num ).
thf(func_def_2338,type,
sK1410: num > num > option_num > num ).
thf(func_def_2339,type,
sK1411: option_num > num > num > num ).
thf(func_def_2340,type,
sK1412: option_num > num > num > num ).
thf(func_def_2341,type,
sK1413: num > num > option_num > num ).
thf(func_def_2342,type,
sK1414: real > real ).
thf(func_def_2343,type,
sK1415: nat > ( nat > $o ) > nat > nat ).
thf(func_def_2344,type,
sK1416: rat > int ).
thf(func_def_2345,type,
sK1417: rat > rat ).
thf(func_def_2346,type,
sK1418: rat > rat ).
thf(func_def_2347,type,
sK1419: num > num ).
thf(func_def_2348,type,
sK1420: ( real > real ) > real > real ).
thf(func_def_2349,type,
sK1421: real > real > real > real ).
thf(func_def_2350,type,
sK1422: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_2351,type,
sK1423: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_2352,type,
sK1424: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_2353,type,
sK1425: produc9072475918466114483BT_nat > $o ).
thf(func_def_2354,type,
sK1426: produc9072475918466114483BT_nat > nat ).
thf(func_def_2355,type,
sK1427: produc9072475918466114483BT_nat > $o ).
thf(func_def_2356,type,
sK1428: produc9072475918466114483BT_nat > nat ).
thf(func_def_2357,type,
sK1429: produc9072475918466114483BT_nat > option4927543243414619207at_nat ).
thf(func_def_2358,type,
sK1430: produc9072475918466114483BT_nat > nat ).
thf(func_def_2359,type,
sK1431: produc9072475918466114483BT_nat > vEBT_VEBT ).
thf(func_def_2360,type,
sK1432: produc9072475918466114483BT_nat > list_VEBT_VEBT ).
thf(func_def_2361,type,
sK1433: nat > $o > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_2362,type,
sK1434: nat > $o > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_2363,type,
sK1435: nat > $o > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_2364,type,
sK1436: nat > $o > vEBT_VEBT > nat ).
thf(func_def_2365,type,
sK1437: nat > $o > vEBT_VEBT > nat ).
thf(func_def_2366,type,
sK1438: nat > $o > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_2367,type,
sK1439: vEBT_VEBT > $o > nat > vEBT_VEBT ).
thf(func_def_2368,type,
sK1440: vEBT_VEBT > $o > nat > list_VEBT_VEBT ).
thf(func_def_2369,type,
sK1441: vEBT_VEBT > $o > nat > nat ).
thf(func_def_2370,type,
sK1442: vEBT_VEBT > $o > nat > nat ).
thf(func_def_2371,type,
sK1443: vEBT_VEBT > $o > nat > nat ).
thf(func_def_2372,type,
sK1444: vEBT_VEBT > $o > nat > list_VEBT_VEBT ).
thf(func_def_2373,type,
sK1445: vEBT_VEBT > $o > nat > vEBT_VEBT ).
thf(func_def_2374,type,
sK1446: vEBT_VEBT > $o > nat > nat ).
thf(func_def_2375,type,
sK1447: $o > nat > vEBT_VEBT > $o ).
thf(func_def_2376,type,
sK1448: $o > nat > vEBT_VEBT > $o ).
thf(func_def_2377,type,
sK1449: int > int ).
thf(func_def_2378,type,
sK1450: ( real > real ) > real > real > real ).
thf(func_def_2379,type,
sK1451: nat > nat ).
thf(func_def_2380,type,
sK1452: nat > nat > nat ).
thf(func_def_2381,type,
sK1453: ( nat > real ) > nat ).
thf(func_def_2382,type,
sK1454: ( nat > real ) > nat ).
thf(func_def_2383,type,
sK1455: real > nat ).
thf(func_def_2384,type,
sK1456: real > nat ).
thf(func_def_2385,type,
sK1457: real > real > real ).
thf(func_def_2386,type,
sK1458: ( nat > num ) > nat ).
thf(func_def_2387,type,
sK1459: ( real > real ) > ( real > real ) > real > real > real ).
thf(func_def_2388,type,
sK1460: ( real > real ) > real > real > ( real > real ) > real ).
thf(func_def_2389,type,
sK1461: set_int > ( int > $o ) > int > int ).
thf(func_def_2390,type,
sK1462: int > set_int > int > int ).
thf(func_def_2391,type,
sK1463: int > set_int > int > int ).
thf(func_def_2392,type,
sK1464: ( int > $o ) > int > set_int > int ).
thf(func_def_2393,type,
sK1465: ( nat > $o ) > nat ).
thf(func_def_2394,type,
sK1466: real > real > real ).
thf(func_def_2395,type,
sK1467: int > set_int > int > int ).
thf(func_def_2396,type,
sK1468: int > set_int > int > int ).
thf(func_def_2397,type,
sK1469: ( real > $o ) > nat > rat ).
thf(func_def_2398,type,
sK1470: ( nat > nat ) > nat ).
thf(func_def_2399,type,
sK1471: nat > nat > nat ).
thf(func_def_2400,type,
sK1472: real > nat ).
thf(func_def_2401,type,
sK1473: nat > nat > nat ).
thf(func_def_2402,type,
sK1474: nat > nat > nat ).
thf(func_def_2403,type,
sK1475: ( nat > nat ) > nat ).
thf(func_def_2404,type,
sK1476: real > nat > real ).
thf(func_def_2405,type,
sK1477: nat > list_nat > nat ).
thf(func_def_2406,type,
sK1478: real > real > ( real > real ) > real ).
thf(func_def_2408,type,
ph1480:
!>[X0: $tType] : X0 ).
thf(f58174,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f55010,f55104,f58173]) ).
thf(f58173,plain,
~ spl1479_44,
inference(avatar_contradiction_clause,[status(thm)],[f58172]) ).
thf(f58172,plain,
( $false
| ~ spl1479_44 ),
inference(trivial_inequality_removal,[status(thm)],[f58171]) ).
thf(f58171,plain,
( ( $true != $true )
| ~ spl1479_44 ),
inference(superposition,[status(thm)],[f58101,f55163]) ).
thf(f55163,plain,
( ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( suc @ one_one_nat ) ) ) @ y )
= $true ),
inference(backward_demodulation,[status(thm)],[f41975,f49513]) ).
thf(f49513,plain,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ one_one_nat ) ),
inference(cnf_transformation,[status(thm)],[f526]) ).
thf(f526,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ one_one_nat ) ),
file('/export/starexec/sandbox/tmp/tmp.iMRIUser6P/DTF2THF_27261.p',fact_525_Suc__1) ).
thf(f41975,plain,
( ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ y )
= $true ),
inference(cnf_transformation,[status(thm)],[f23686]) ).
thf(f23686,plain,
( ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ y )
= $true ),
inference(flattening,[status(thm)],[f16165]) ).
thf(f16165,plain,
~ ( ( ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ y )
!= $true ) ),
inference(fool_elimination,[status(thm)],[f16164]) ).
thf(f16164,plain,
~ ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ y ),
inference(rectify,[status(thm)],[f10194]) ).
thf(f10194,negated_conjecture,
~ ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ y ),
inference(negated_conjecture,[status(cth)],[f10193]) ).
thf(f10193,conjecture,
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ y ),
file('/export/starexec/sandbox/tmp/tmp.iMRIUser6P/DTF2THF_27261.p',conj_0) ).
thf(f58101,plain,
( ! [X1: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( suc @ one_one_nat ) ) ) @ X1 )
!= $true )
| ~ spl1479_44 ),
inference(forward_demodulation,[status(thm)],[f55009,f49513]) ).
thf(f55009,plain,
( ! [X1: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ X1 )
!= $true )
| ~ spl1479_44 ),
inference(avatar_component_clause,[status(thm)],[f55008]) ).
thf(f55008,definition,
( spl1479_44
<=> ! [X1: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ X1 )
!= $true ) ),
introduced(definition,[new_symbols(naming,[spl1479_44])],[avatar_definition]) ).
thf(f55104,plain,
( spl1479_44
| ~ spl1479_43 ),
inference(avatar_split_clause,[status(thm)],[f51020,f55004,f55008]) ).
thf(f55004,definition,
( spl1479_43
<=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl1479_43])],[avatar_definition]) ).
thf(f51020,plain,
! [X0: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
!= $true )
| ( ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ X0 )
!= $true ) ),
inference(equality_resolution,[status(thm)],[f40894]) ).
thf(f40894,plain,
! [X0: nat,X1: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
!= $true )
| ( $true
!= ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X1 ) @ X0 ) )
| ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= X1 ) ),
inference(cnf_transformation,[status(thm)],[f26632]) ).
thf(f26632,plain,
! [X0: nat,X1: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
!= $true )
| ( $true
!= ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X1 ) @ X0 ) )
| ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= X1 ) ),
inference(flattening,[status(thm)],[f26631]) ).
thf(f26631,plain,
! [X0: nat,X1: nat] :
( ( $true
!= ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X1 ) @ X0 ) )
| ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= X1 )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f24942]) ).
thf(f24942,plain,
! [X0: nat,X1: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
= $true )
=> ( ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X1 )
=> ( $true
!= ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X1 ) @ X0 ) ) ) ),
inference(flattening,[status(thm)],[f19337]) ).
thf(f19337,plain,
! [X0: nat,X1: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
= $true )
=> ( ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X1 )
=> ( $true
!= ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X1 ) @ X0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f19336]) ).
thf(f19336,plain,
! [X0: nat,X1: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
=> ( ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X1 )
=> ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X1 ) @ X0 ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
! [X1: nat,X0: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
=> ( ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X0 )
=> ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X0 ) @ X1 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.iMRIUser6P/DTF2THF_27261.p',fact_1__C3_OIH_C_I1_J) ).
thf(f55010,plain,
( spl1479_43
| spl1479_44 ),
inference(avatar_split_clause,[status(thm)],[f51563,f55008,f55004]) ).
thf(f51563,plain,
! [X1: nat] :
( ( ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ X1 )
!= $true )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
= $true ) ),
inference(equality_resolution,[status(thm)],[f44576]) ).
thf(f44576,plain,
! [X0: nat,X1: nat] :
( ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= X0 )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
= $true )
| ( ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X0 ) @ X1 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f26380]) ).
thf(f26380,plain,
! [X0: nat,X1: nat] :
( ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= X0 )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
= $true )
| ( ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X0 ) @ X1 )
!= $true ) ),
inference(flattening,[status(thm)],[f26379]) ).
thf(f26379,plain,
! [X0: nat,X1: nat] :
( ( ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X0 ) @ X1 )
!= $true )
| ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= X0 )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f25716]) ).
thf(f25716,plain,
! [X0: nat,X1: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
!= $true )
=> ( ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X0 )
=> ( ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X0 ) @ X1 )
!= $true ) ) ),
inference(flattening,[status(thm)],[f21360]) ).
thf(f21360,plain,
! [X0: nat,X1: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
!= $true )
=> ( ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X0 )
=> ( ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X0 ) @ X1 )
!= $true ) ) ),
inference(fool_elimination,[status(thm)],[f21359]) ).
thf(f21359,plain,
! [X0: nat,X1: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
=> ( ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X0 )
=> ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X0 ) @ X1 ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
! [X0: nat,X1: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
=> ( ( ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X0 )
=> ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X0 ) @ X1 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.iMRIUser6P/DTF2THF_27261.p',fact_3__C3_OIH_C_I3_J) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP220^1 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.12/0.14 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.35 % Computer : n031.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Thu Feb 26 06:50:35 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.32/0.47 ---- Original DTF file ---
% 0.32/0.47 thf(spec,logic,$$dhol).
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 % File : ITP220^1 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.32/0.47 % Domain : Interactive Theorem Proving
% 0.32/0.47 % Problem : Sledgehammer problem VEBT_Definitions 00318_012244
% 0.32/0.47 % Version : [Des22] axioms.
% 0.32/0.47 % English :
% 0.32/0.47
% 0.32/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.32/0.47 % : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% 0.32/0.47 % Source : [Des22]
% 0.32/0.47 % Names : 0063_VEBT_Definitions_00318_012244 [Des22]
% 0.32/0.47
% 0.32/0.47 % Status : Theorem
% 0.32/0.47 % Rating : ? v9.2.1
% 0.32/0.47 % Syntax : Number of formulae : 11103 (6718 unt; 910 typ; 0 def)
% 0.32/0.47 % Number of atoms : 24749 (12111 equ; 0 cnn)
% 0.32/0.47 % Maximal formula atoms : 71 ( 2 avg)
% 0.32/0.47 % Number of connectives : 97635 (2153 ~; 451 |;1340 &;86640 @)
% 0.32/0.47 % ( 0 <=>;7051 =>; 0 <=; 0 <~>)
% 0.32/0.47 % Maximal formula depth : 39 ( 5 avg)
% 0.32/0.47 % Number of types : 56 ( 55 usr)
% 0.32/0.47 % Number of type conns : 2901 (2901 >; 0 *; 0 +; 0 <<)
% 0.32/0.47 % Number of symbols : 858 ( 855 usr; 57 con; 0-8 aty)
% 0.32/0.47 % Number of variables : 21152 (1643 ^;18983 !; 526 ?;21152 :)
% 0.32/0.47 % SPC : TH0_THM_EQU_NAR
% 0.32/0.47
% 0.32/0.47 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.32/0.47 % from the van Emde Boas Trees session in the Archive of Formal
% 0.32/0.47 % proofs -
% 0.32/0.47 % www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 0.32/0.47 % 2022-02-17 17:22:39.020
% 0.32/0.47 % Bugfixes : Added ()s around negated sides of equations.
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 % Could-be-implicit typings (55)
% 0.32/0.47 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
% 0.32/0.47 produc6756470014200349799atural: $tType ).
% 0.32/0.47
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% 0.32/0.47 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% 0.32/0.47 thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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% 0.32/0.47 thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
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% 0.32/0.47 thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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% 0.32/0.47
% 0.32/0.47 thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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% 0.32/0.47 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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% 0.32/0.47 thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
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% 0.32/0.47 thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
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% 0.32/0.47 thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
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% 0.32/0.47
% 0.32/0.47 thf(ty_n_t__Filter__Ofilter_It__Int__Oint_J,type,
% 0.32/0.47 filter_int: $tType ).
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% 0.32/0.47 set_char: $tType ).
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% 0.32/0.47 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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% 0.32/0.47 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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% 0.32/0.47
% 0.32/0.47 thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
% 0.32/0.47 list_int: $tType ).
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% 0.32/0.47
% 0.32/0.47 thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
% 0.32/0.47 set_rat: $tType ).
% 0.32/0.47
% 0.32/0.47 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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% 0.32/0.47 thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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% 0.32/0.47 thf(ty_n_t__Complex__Ocomplex,type,
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% 0.32/0.47 thf(ty_n_t__String__Oliteral,type,
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% 0.32/0.47 thf(ty_n_t__String__Ochar,type,
% 0.32/0.47 char: $tType ).
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% 0.32/0.47 thf(ty_n_t__Real__Oreal,type,
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% 0.32/0.47 thf(ty_n_t__Rat__Orat,type,
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% 0.32/0.47 thf(ty_n_t__Num__Onum,type,
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% 0.32/0.47 thf(ty_n_t__Nat__Onat,type,
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% 0.32/0.47 thf(ty_n_t__Int__Oint,type,
% 0.32/0.47 int: $tType ).
% 0.32/0.47
% 0.32/0.47 % Explicit typings (855)
% 0.32/0.47 thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
% 0.32/0.47 archim2889992004027027881ng_rat: rat > int ).
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% 0.32/0.47 thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
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% 0.32/0.47 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
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% 0.32/0.47 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
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% 0.32/0.47 thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
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% 0.32/0.47 thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
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% 0.32/0.47 thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
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% 0.32/0.47 thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
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% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
% 0.32/0.47 bNF_re1962705104956426057at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
% 0.32/0.47 bNF_re895249473297799549at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
% 0.32/0.47 bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.32/0.47 bNF_re4695409256820837752l_real: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( real > real > real ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_Eo_J_001_062_It__Real__Oreal_M_Eo_J,type,
% 0.32/0.47 bNF_re4521903465945308077real_o: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > $o ) > ( real > $o ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( real > real > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
% 0.32/0.47 bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_Eo_001_Eo,type,
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% 0.32/0.47 bNF_re398004352372739002nteger: ( int > code_integer > $o ) > ( ( int > int ) > ( code_integer > code_integer ) > $o ) > ( int > int > int ) > ( code_integer > code_integer > code_integer ) > $o ).
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% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.32/0.47 bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.32/0.47 bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Rat__Orat_J,type,
% 0.32/0.47 bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 0.32/0.47 bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.32/0.47 bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 0.32/0.47 bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
% 0.32/0.47 bNF_re88643428490162567atural: ( nat > code_natural > $o ) > ( ( nat > nat ) > ( code_natural > code_natural ) > $o ) > ( nat > nat > nat ) > ( code_natural > code_natural > code_natural ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 bNF_re5252274238750452962nteger: ( nat > code_natural > $o ) > ( int > code_integer > $o ) > ( nat > int ) > ( code_natural > code_integer ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
% 0.32/0.47 bNF_re3704215830270325841atural: ( nat > code_natural > $o ) > ( nat > code_natural > $o ) > ( nat > nat ) > ( code_natural > code_natural ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.32/0.47 bNF_re4935368626670024657nteger: ( nat > nat > $o ) > ( ( int > int ) > ( code_integer > code_integer ) > $o ) > ( nat > int > int ) > ( nat > code_integer > code_integer ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.32/0.47 bNF_re4785983289428654063nt_int: ( nat > nat > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( nat > int > int ) > ( nat > int > int ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
% 0.32/0.47 bNF_re3692170009638431535atural: ( nat > nat > $o ) > ( ( nat > nat ) > ( code_natural > code_natural ) > $o ) > ( nat > nat > nat ) > ( nat > code_natural > code_natural ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.47 bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 bNF_re4153400068438556298nteger: ( nat > nat > $o ) > ( int > code_integer > $o ) > ( nat > int ) > ( nat > code_integer ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Int__Oint,type,
% 0.32/0.47 bNF_re6650684261131312217nt_int: ( nat > nat > $o ) > ( int > int > $o ) > ( nat > int ) > ( nat > int ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.47 bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 0.32/0.47 bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J,type,
% 0.32/0.47 bNF_re7876454716742015248nteger: ( num > num > $o ) > ( ( num > int ) > ( num > code_integer ) > $o ) > ( num > num > int ) > ( num > num > code_integer ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Int__Oint_J,type,
% 0.32/0.47 bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 bNF_re6501075790457514782nteger: ( num > num > $o ) > ( int > code_integer > $o ) > ( num > int ) > ( num > code_integer ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
% 0.32/0.47 bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.32/0.47 bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
% 0.32/0.47 bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.32/0.47 bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
% 0.32/0.47 bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
% 0.32/0.47 bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 0.32/0.47 bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
% 0.32/0.47 bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.32/0.47 bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
% 0.32/0.47 bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.47 bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 0.32/0.47 bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 0.32/0.47 bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.32/0.47 bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
% 0.32/0.47 bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.47 bNF_re8246922863344978751at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.47 bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Binomial_Obinomial,type,
% 0.32/0.47 binomial: nat > nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
% 0.32/0.47 gbinomial_complex: complex > nat > complex ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
% 0.32/0.47 gbinomial_int: int > nat > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
% 0.32/0.47 gbinomial_nat: nat > nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
% 0.32/0.47 gbinomial_rat: rat > nat > rat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
% 0.32/0.47 gbinomial_real: real > nat > real ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Oand__int__rel,type,
% 0.32/0.47 bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Oand__not__num,type,
% 0.32/0.47 bit_and_not_num: num > num > option_num ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
% 0.32/0.47 bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Oconcat__bit,type,
% 0.32/0.47 bit_concat_bit: nat > int > int > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
% 0.32/0.47 bit_or_not_num_neg: num > num > num ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
% 0.32/0.47 bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 bit_ri7632146776885996613nteger: code_integer > code_integer ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
% 0.32/0.47 bit_ri7919022796975470100ot_int: int > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
% 0.32/0.47 bit_ri631733984087533419it_int: nat > int > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Onatural,type,
% 0.32/0.47 bit_se2773287842338716102atural: code_natural > code_natural > code_natural ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
% 0.32/0.47 bit_se725231765392027082nd_int: int > int > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
% 0.32/0.47 bit_se727722235901077358nd_nat: nat > nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 bit_se3928097537394005634nteger: nat > code_integer > code_integer ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Onatural,type,
% 0.32/0.47 bit_se2751692689150723149atural: nat > code_natural > code_natural ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
% 0.32/0.47 bit_se8568078237143864401it_int: nat > int > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
% 0.32/0.47 bit_se8570568707652914677it_nat: nat > nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 bit_se1345352211410354436nteger: nat > code_integer > code_integer ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Onatural,type,
% 0.32/0.47 bit_se168947363167071951atural: nat > code_natural > code_natural ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
% 0.32/0.47 bit_se2159334234014336723it_int: nat > int > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
% 0.32/0.47 bit_se2161824704523386999it_nat: nat > nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 bit_se2119862282449309892nteger: nat > code_integer ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
% 0.32/0.47 bit_se2000444600071755411sk_int: nat > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
% 0.32/0.47 bit_se2002935070580805687sk_nat: nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 bit_se1080825931792720795nteger: code_integer > code_integer > code_integer ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Code____Numeral__Onatural,type,
% 0.32/0.47 bit_se9127793120404214118atural: code_natural > code_natural > code_natural ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
% 0.32/0.47 bit_se1409905431419307370or_int: int > int > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
% 0.32/0.47 bit_se1412395901928357646or_nat: nat > nat > nat ).
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% 0.32/0.47 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
% 0.32/0.47 comm_s7457072308508201937r_real: real > nat > real ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 semiri3624122377584611663nteger: nat > code_integer ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Onatural,type,
% 0.32/0.47 semiri2447717529341329178atural: nat > code_natural ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
% 0.32/0.47 semiri5044797733671781792omplex: nat > complex ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
% 0.32/0.47 semiri1406184849735516958ct_int: nat > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
% 0.32/0.47 semiri1408675320244567234ct_nat: nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
% 0.32/0.47 semiri773545260158071498ct_rat: nat > rat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
% 0.32/0.47 semiri2265585572941072030t_real: nat > real ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
% 0.32/0.47 invers8013647133539491842omplex: complex > complex ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
% 0.32/0.47 inverse_inverse_rat: rat > rat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
% 0.32/0.47 inverse_inverse_real: real > real ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
% 0.32/0.47 at_bot_real: filter_real ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Filter_Oat__top_001t__Int__Oint,type,
% 0.32/0.47 at_top_int: filter_int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
% 0.32/0.47 at_top_nat: filter_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
% 0.32/0.47 at_top_real: filter_real ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
% 0.32/0.47 eventually_nat: ( nat > $o ) > filter_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
% 0.32/0.47 eventually_real: ( real > $o ) > filter_real > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Int__Oint,type,
% 0.32/0.47 filterlim_nat_int: ( nat > int ) > filter_int > filter_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.47 filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.32/0.47 filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.32/0.47 filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Filter_Ofiltermap_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.32/0.47 filtermap_real_real: ( real > real ) > filter_real > filter_real ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Finite__Set_Ocard_001_Eo,type,
% 0.32/0.47 finite_card_o: set_o > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
% 0.32/0.47 finite_card_complex: set_complex > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
% 0.32/0.47 finite_card_int: set_int > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
% 0.32/0.47 finite_card_list_nat: set_list_nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 0.32/0.47 finite_card_nat: set_nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
% 0.32/0.47 finite410649719033368117t_unit: set_Product_unit > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.32/0.47 finite_card_set_nat: set_set_nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
% 0.32/0.47 finite_card_char: set_char > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
% 0.32/0.47 finite3207457112153483333omplex: set_complex > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
% 0.32/0.47 finite_finite_int: set_int > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 0.32/0.47 finite_finite_nat: set_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 0.32/0.47 bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Obij__betw_001t__Int__Oint_001t__Nat__Onat,type,
% 0.32/0.47 bij_betw_int_nat: ( int > nat ) > set_int > set_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Obij__betw_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
% 0.32/0.47 bij_be8532844293280997160at_nat: ( list_nat > nat ) > set_list_nat > set_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 0.32/0.47 bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Int__Oint,type,
% 0.32/0.47 bij_betw_nat_int: ( nat > int ) > set_nat > set_int > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
% 0.32/0.47 bij_be6293887246118711976st_nat: ( nat > list_nat ) > set_nat > set_list_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.47 bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.47 bij_be8693218025023041337at_nat: ( nat > product_prod_nat_nat ) > set_nat > set_Pr1261947904930325089at_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.47 bij_be4790990086886966983at_nat: ( nat > sum_sum_nat_nat ) > set_nat > set_Sum_sum_nat_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Obij__betw_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 0.32/0.47 bij_be5333170631980326235at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > set_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Obij__betw_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 0.32/0.47 bij_be5432664580149595207at_nat: ( sum_sum_nat_nat > nat ) > set_Sum_sum_nat_nat > set_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Num__Onum,type,
% 0.32/0.47 comp_int_int_num: ( int > int ) > ( num > int ) > num > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
% 0.32/0.47 comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.47 comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oid_001_Eo,type,
% 0.32/0.47 id_o: $o > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
% 0.32/0.47 id_nat: nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Nat__Onat,type,
% 0.32/0.47 inj_on_int_nat: ( int > nat ) > set_int > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
% 0.32/0.47 inj_on_list_nat_nat: ( list_nat > nat ) > set_list_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Int__Oint,type,
% 0.32/0.47 inj_on_nat_int: ( nat > int ) > set_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
% 0.32/0.47 inj_on_nat_list_nat: ( nat > list_nat ) > set_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.47 inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.47 inj_on5538052773655684606at_nat: ( nat > product_prod_nat_nat ) > set_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
% 0.32/0.47 inj_on_nat_char: ( nat > char ) > set_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.47 inj_on5701776251185195458at_nat: ( nat > sum_sum_nat_nat ) > set_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 0.32/0.47 inj_on2178005380612969504at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.32/0.47 inj_on_real_real: ( real > real ) > set_real > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
% 0.32/0.47 inj_on_set_nat_nat: ( set_nat > nat ) > set_set_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oinj__on_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 0.32/0.47 inj_on6343450744447823682at_nat: ( sum_sum_nat_nat > nat ) > set_Sum_sum_nat_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Omap__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.32/0.47 map_fu8272188784021352819nteger: ( code_integer > int ) > ( ( int > int ) > code_integer > code_integer ) > ( int > int > int ) > code_integer > code_integer > code_integer ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Omap__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 map_fu2599414010547811884nteger: ( code_integer > int ) > ( int > code_integer ) > ( int > int ) > code_integer > code_integer ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Omap__fun_001t__Code____Numeral__Onatural_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
% 0.32/0.47 map_fu6549440983881763648atural: ( code_natural > nat ) > ( ( nat > nat ) > code_natural > code_natural ) > ( nat > nat > nat ) > code_natural > code_natural > code_natural ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Omap__fun_001t__Code____Numeral__Onatural_001t__Nat__Onat_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 map_fu2787874002554666395nteger: ( code_natural > nat ) > ( int > code_integer ) > ( nat > int ) > code_natural > code_integer ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Omap__fun_001t__Code____Numeral__Onatural_001t__Nat__Onat_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
% 0.32/0.47 map_fu1239815594074539274atural: ( code_natural > nat ) > ( nat > code_natural ) > ( nat > nat ) > code_natural > code_natural ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
% 0.32/0.47 map_fu434086159418415080_int_o: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > $o ) > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > int > int > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.32/0.47 map_fu4960017516451851995nt_int: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > int > int > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
% 0.32/0.47 map_fu4826362097070443709at_o_o: ( int > product_prod_nat_nat ) > ( $o > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.47 map_fu2345160673673942751at_nat: ( int > product_prod_nat_nat ) > ( nat > nat ) > ( product_prod_nat_nat > nat ) > int > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 0.32/0.47 map_fu3667384564859982768at_int: ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int ) > ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Omap__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.32/0.47 map_fu8224359078269604234nteger: ( nat > nat ) > ( ( int > int ) > code_integer > code_integer ) > ( nat > int > int ) > nat > code_integer > code_integer ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Omap__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
% 0.32/0.47 map_fu6981160461238011112atural: ( nat > nat ) > ( ( nat > nat ) > code_natural > code_natural ) > ( nat > nat > nat ) > nat > code_natural > code_natural ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Omap__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 map_fu6290471996055670595nteger: ( nat > nat ) > ( int > code_integer ) > ( nat > int ) > nat > code_integer ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
% 0.32/0.47 map_fu4333342158222067775at_rat: ( rat > product_prod_int_int ) > ( ( product_prod_int_int > product_prod_int_int ) > rat > rat ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > rat > rat > rat ).
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% 0.32/0.47 thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 if_Code_integer: $o > code_integer > code_integer > code_integer ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__Code____Numeral__Onatural,type,
% 0.32/0.47 if_Code_natural: $o > code_natural > code_natural > code_natural ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__Complex__Ocomplex,type,
% 0.32/0.47 if_complex: $o > complex > complex > complex ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__Extended____Nat__Oenat,type,
% 0.32/0.47 if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__Int__Oint,type,
% 0.32/0.47 if_int: $o > int > int > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
% 0.32/0.47 if_list_int: $o > list_int > list_int > list_int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
% 0.32/0.47 if_list_nat: $o > list_nat > list_nat > list_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__Nat__Onat,type,
% 0.32/0.47 if_nat: $o > nat > nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__Num__Onum,type,
% 0.32/0.47 if_num: $o > num > num > num ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.32/0.47 if_option_num: $o > option_num > option_num > option_num ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 0.32/0.47 if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.32/0.47 if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.32/0.47 if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.47 if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__Rat__Orat,type,
% 0.32/0.47 if_rat: $o > rat > rat > rat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__Real__Oreal,type,
% 0.32/0.47 if_real: $o > real > real > real ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
% 0.32/0.47 if_set_int: $o > set_int > set_int > set_int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.47 if_Sum_sum_nat_nat: $o > sum_sum_nat_nat > sum_sum_nat_nat > sum_sum_nat_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_OAbs__Integ,type,
% 0.32/0.47 abs_Integ: product_prod_nat_nat > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_ONeg,type,
% 0.32/0.47 neg: num > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_OPos,type,
% 0.32/0.47 pos: num > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_ORep__Integ,type,
% 0.32/0.47 rep_Integ: int > product_prod_nat_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Ocr__int,type,
% 0.32/0.47 cr_int: product_prod_nat_nat > int > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Odup,type,
% 0.32/0.47 dup: int > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Oint__ge__less__than,type,
% 0.32/0.47 int_ge_less_than: int > set_Pr958786334691620121nt_int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Oint__ge__less__than2,type,
% 0.32/0.47 int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Ointrel,type,
% 0.32/0.47 intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Onat,type,
% 0.32/0.47 nat2: int > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Opcr__int,type,
% 0.32/0.47 pcr_int: product_prod_nat_nat > int > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Opower__int_001t__Real__Oreal,type,
% 0.32/0.47 power_int_real: real > int > real ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Oring__1__class_OInts_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 ring_11222124179247155820nteger: set_Code_integer ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Oring__1__class_OInts_001t__Complex__Ocomplex,type,
% 0.32/0.47 ring_1_Ints_complex: set_complex ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
% 0.32/0.47 ring_1_Ints_int: set_int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Oring__1__class_OInts_001t__Rat__Orat,type,
% 0.32/0.47 ring_1_Ints_rat: set_rat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
% 0.32/0.47 ring_1_Ints_real: set_real ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
% 0.32/0.47 ring_18347121197199848620nteger: int > code_integer ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
% 0.32/0.47 ring_17405671764205052669omplex: int > complex ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 0.32/0.47 ring_1_of_int_int: int > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
% 0.32/0.47 ring_1_of_int_rat: int > rat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 0.32/0.47 ring_1_of_int_real: int > real ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Int_Osub,type,
% 0.32/0.47 sub: num > num > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
% 0.32/0.47 inf_inf_int: int > int > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
% 0.32/0.47 inf_inf_nat: nat > nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Lattices_Osemilattice__neutr_001t__Nat__Onat,type,
% 0.32/0.47 semila9081495762789891438tr_nat: ( nat > nat > nat ) > nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
% 0.32/0.47 semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Lattices_Osemilattice__order_001t__Nat__Onat,type,
% 0.32/0.47 semila1248733672344298208er_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
% 0.32/0.47 sup_sup_int: int > int > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
% 0.32/0.47 sup_sup_nat: nat > nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.32/0.47 sup_sup_set_nat: set_nat > set_nat > set_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.32/0.47 sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
% 0.32/0.47 lattic8263393255366662781ax_int: set_int > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
% 0.32/0.47 lattic8265883725875713057ax_nat: set_nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF_001t__Nat__Onat,type,
% 0.32/0.47 lattic7826324295020591184_F_nat: ( nat > nat > nat ) > nat > set_nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Lifting_OQuotient_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
% 0.32/0.47 quotie3684837364556693515t_real: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > real ) > ( real > nat > rat ) > ( ( nat > rat ) > real > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Lifting_OQuotient_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 0.32/0.47 quotie1194848508323700631at_int: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > int ) > ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.32/0.47 bfun_nat_real: ( nat > real ) > filter_nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
% 0.32/0.47 at_infinity_real: filter_real ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Oappend_001t__Int__Oint,type,
% 0.32/0.47 append_int: list_int > list_int > list_int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 0.32/0.47 append_nat: list_nat > list_nat > list_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Odrop_001t__Nat__Onat,type,
% 0.32/0.47 drop_nat: nat > list_nat > list_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Ofold_001t__Int__Oint_001t__Int__Oint,type,
% 0.32/0.47 fold_int_int: ( int > int > int ) > list_int > int > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.47 fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Olast_001t__Nat__Onat,type,
% 0.32/0.47 last_nat: list_nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
% 0.32/0.47 linord2614967742042102400et_nat: set_nat > list_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
% 0.32/0.47 cons_int: int > list_int > list_int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
% 0.32/0.47 cons_nat: nat > list_nat > list_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
% 0.32/0.47 nil_int: list_int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
% 0.32/0.47 nil_nat: list_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.47 map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Olist_Oset_001_Eo,type,
% 0.32/0.47 set_o2: list_o > set_o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
% 0.32/0.47 set_complex2: list_complex > set_complex ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
% 0.32/0.47 set_int2: list_int > set_int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 0.32/0.47 set_nat2: list_nat > set_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
% 0.32/0.47 set_real2: list_real > set_real ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.32/0.47 set_set_nat2: list_set_nat > set_set_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
% 0.32/0.47 set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 0.32/0.47 size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
% 0.32/0.47 tl_nat: list_nat > list_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Onth_001_Eo,type,
% 0.32/0.47 nth_o: list_o > nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
% 0.32/0.47 nth_complex: list_complex > nat > complex ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Onth_001t__Int__Oint,type,
% 0.32/0.47 nth_int: list_int > nat > int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 0.32/0.47 nth_nat: list_nat > nat > nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 0.32/0.47 nth_real: list_real > nat > real ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.32/0.47 nth_set_nat: list_set_nat > nat > set_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 0.32/0.47 nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 0.32/0.47 replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Orotate1_001_Eo,type,
% 0.32/0.47 rotate1_o: list_o > list_o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Orotate1_001t__Int__Oint,type,
% 0.32/0.47 rotate1_int: list_int > list_int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
% 0.32/0.47 rotate1_nat: list_nat > list_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Orotate1_001t__VEBT____Definitions__OVEBT,type,
% 0.32/0.47 rotate1_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Osubseqs_001_Eo,type,
% 0.32/0.47 subseqs_o: list_o > list_list_o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Osubseqs_001t__Int__Oint,type,
% 0.32/0.47 subseqs_int: list_int > list_list_int ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
% 0.32/0.47 subseqs_nat: list_nat > list_list_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Osubseqs_001t__VEBT____Definitions__OVEBT,type,
% 0.32/0.47 subseqs_VEBT_VEBT: list_VEBT_VEBT > list_list_VEBT_VEBT ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 0.32/0.47 take_nat: nat > list_nat > list_nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_List_Oupt,type,
% 0.32/0.47 upt: nat > nat > list_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Oupto,type,
% 0.32/0.48 upto: int > int > list_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Oupto__aux,type,
% 0.32/0.48 upto_aux: int > int > list_int > list_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Oupto__rel,type,
% 0.32/0.48 upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_OSuc,type,
% 0.32/0.48 suc: nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.48 compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 0.32/0.48 case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 0.32/0.48 case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.32/0.48 case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Onat_Opred,type,
% 0.32/0.48 pred: nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Complex__Ocomplex,type,
% 0.32/0.48 semiri3842193898606819883omplex: set_complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
% 0.32/0.48 semiring_1_Nats_int: set_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 semiri4939895301339042750nteger: nat > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Onatural,type,
% 0.32/0.48 semiri3763490453095760265atural: nat > code_natural ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 0.32/0.48 semiri8010041392384452111omplex: nat > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.32/0.48 semiri1314217659103216013at_int: nat > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.32/0.48 semiri1316708129612266289at_nat: nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
% 0.32/0.48 semiri681578069525770553at_rat: nat > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 0.32/0.48 semiri5074537144036343181t_real: nat > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 0.32/0.48 size_size_list_o: list_o > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 0.32/0.48 size_s3451745648224563538omplex: list_complex > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 0.32/0.48 size_size_list_int: list_int > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
% 0.32/0.48 size_s2710708370519433104list_o: list_list_o > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
% 0.32/0.48 size_s533118279054570080st_int: list_list_int > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 0.32/0.48 size_s3023201423986296836st_nat: list_list_nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
% 0.32/0.48 size_s8217280938318005548T_VEBT: list_list_VEBT_VEBT > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 0.32/0.48 size_size_list_nat: list_nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 0.32/0.48 size_size_list_real: list_real > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.32/0.48 size_s3254054031482475050et_nat: list_set_nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.32/0.48 size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 0.32/0.48 size_size_num: num > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
% 0.32/0.48 size_size_char: char > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
% 0.32/0.48 size_size_VEBT_VEBT: vEBT_VEBT > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Oint__decode,type,
% 0.32/0.48 nat_int_decode: nat > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Oint__encode,type,
% 0.32/0.48 nat_int_encode: int > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Olist__decode,type,
% 0.32/0.48 nat_list_decode: nat > list_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
% 0.32/0.48 nat_list_decode_rel: nat > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Olist__encode,type,
% 0.32/0.48 nat_list_encode: list_nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 0.32/0.48 nat_list_encode_rel: list_nat > list_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Oprod__decode,type,
% 0.32/0.48 nat_prod_decode: nat > product_prod_nat_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 0.32/0.48 nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 0.32/0.48 nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 0.32/0.48 nat_prod_encode: product_prod_nat_nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Oset__decode,type,
% 0.32/0.48 nat_set_decode: nat > set_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Oset__encode,type,
% 0.32/0.48 nat_set_encode: set_nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Osum__decode,type,
% 0.32/0.48 nat_sum_decode: nat > sum_sum_nat_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Osum__encode,type,
% 0.32/0.48 nat_sum_encode: sum_sum_nat_nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Otriangle,type,
% 0.32/0.48 nat_triangle: nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_NthRoot_Oroot,type,
% 0.32/0.48 root: nat > real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_NthRoot_Osqrt,type,
% 0.32/0.48 sqrt: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_OBitM,type,
% 0.32/0.48 bitM: num > num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oinc,type,
% 0.32/0.48 inc: num > num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Onat__of__num,type,
% 0.32/0.48 nat_of_num: num > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 neg_nu8804712462038260780nteger: code_integer > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
% 0.32/0.48 neg_nu7009210354673126013omplex: complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
% 0.32/0.48 neg_numeral_dbl_int: int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
% 0.32/0.48 neg_numeral_dbl_rat: rat > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
% 0.32/0.48 neg_numeral_dbl_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 neg_nu7757733837767384882nteger: code_integer > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
% 0.32/0.48 neg_nu6511756317524482435omplex: complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 0.32/0.48 neg_nu3811975205180677377ec_int: int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 0.32/0.48 neg_nu3179335615603231917ec_rat: rat > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
% 0.32/0.48 neg_nu6075765906172075777c_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 neg_nu5831290666863070958nteger: code_integer > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
% 0.32/0.48 neg_nu8557863876264182079omplex: complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
% 0.32/0.48 neg_nu5851722552734809277nc_int: int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
% 0.32/0.48 neg_nu5219082963157363817nc_rat: rat > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
% 0.32/0.48 neg_nu8295874005876285629c_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 neg_nu5755505904847501662nteger: num > num > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Complex__Ocomplex,type,
% 0.32/0.48 neg_nu8416839295433526191omplex: num > num > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
% 0.32/0.48 neg_numeral_sub_int: num > num > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Rat__Orat,type,
% 0.32/0.48 neg_numeral_sub_rat: num > num > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Real__Oreal,type,
% 0.32/0.48 neg_numeral_sub_real: num > num > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Onum_OBit0,type,
% 0.32/0.48 bit0: num > num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Onum_OBit1,type,
% 0.32/0.48 bit1: num > num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Onum_OOne,type,
% 0.32/0.48 one: num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.32/0.48 case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Onum_Osize__num,type,
% 0.32/0.48 size_num: num > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Onum__of__nat,type,
% 0.32/0.48 num_of_nat: nat > num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 numera6620942414471956472nteger: num > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Onatural,type,
% 0.32/0.48 numera5444537566228673987atural: num > code_natural ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
% 0.32/0.48 numera6690914467698888265omplex: num > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
% 0.32/0.48 numera1916890842035813515d_enat: num > extended_enat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
% 0.32/0.48 numeral_numeral_int: num > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 0.32/0.48 numeral_numeral_nat: num > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
% 0.32/0.48 numeral_numeral_rat: num > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 0.32/0.48 numeral_numeral_real: num > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Opow,type,
% 0.32/0.48 pow: num > num > num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Opred__numeral,type,
% 0.32/0.48 pred_numeral: num > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Osqr,type,
% 0.32/0.48 sqr: num > num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
% 0.32/0.48 none_num: option_num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.48 none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
% 0.32/0.48 some_num: num > option_num ).
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% 0.32/0.48 thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.48 some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.48 case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
% 0.32/0.48 case_option_int_num: int > ( num > int ) > option_num > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
% 0.32/0.48 case_option_num_num: num > ( num > num ) > option_num > num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
% 0.32/0.48 case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
% 0.32/0.48 map_option_num_num: ( num > num ) > option_num > option_num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
% 0.32/0.48 bot_bot_nat: nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
% 0.32/0.48 bot_bot_set_int: set_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.32/0.48 bot_bot_set_nat: set_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.32/0.48 bot_bot_set_real: set_real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
% 0.32/0.48 ord_Least_nat: ( nat > $o ) > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Oord__class_OLeast_001t__Real__Oreal,type,
% 0.32/0.48 ord_Least_real: ( real > $o ) > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Onatural,type,
% 0.32/0.48 ord_le5570908160329646204atural: code_natural > code_natural > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
% 0.32/0.48 ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 0.32/0.48 ord_less_int: int > int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.32/0.48 ord_less_nat: nat > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
% 0.32/0.48 ord_less_num: num > num > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
% 0.32/0.48 ord_less_rat: rat > rat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 0.32/0.48 ord_less_real: real > real > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 ord_le3102999989581377725nteger: code_integer > code_integer > $o ).
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% 0.32/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Onatural,type,
% 0.32/0.48 ord_le1926595141338095240atural: code_natural > code_natural > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
% 0.32/0.48 ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 0.32/0.48 ord_less_eq_int: int > int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.32/0.48 ord_less_eq_nat: nat > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
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% 0.32/0.48 thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
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% 0.32/0.48 thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
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% 0.32/0.48 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Onatural_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
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% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Quotient_OQuotient3_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Random_Olog,type,
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% 0.32/0.48 thf(sy_c_Rat_OFrct,type,
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% 0.32/0.48 thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
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% 0.32/0.48 thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
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% 0.32/0.48 thf(sy_c_Rat_Onormalize,type,
% 0.32/0.48 normalize: product_prod_int_int > product_prod_int_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rat_Oof__int,type,
% 0.32/0.48 of_int: int > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rat_Opcr__rat,type,
% 0.32/0.48 pcr_rat: product_prod_int_int > rat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rat_Opositive,type,
% 0.32/0.48 positive: rat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rat_Oquotient__of,type,
% 0.32/0.48 quotient_of: rat > product_prod_int_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rat_Oratrel,type,
% 0.32/0.48 ratrel: product_prod_int_int > product_prod_int_int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_ORatreal,type,
% 0.32/0.48 ratreal: rat > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_OReal,type,
% 0.32/0.48 real2: ( nat > rat ) > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_Ocauchy,type,
% 0.32/0.48 cauchy: ( nat > rat ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_Ocr__real,type,
% 0.32/0.48 cr_real: ( nat > rat ) > real > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_Opcr__real,type,
% 0.32/0.48 pcr_real: ( nat > rat ) > real > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_Opositive,type,
% 0.32/0.48 positive2: real > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_Orealrel,type,
% 0.32/0.48 realrel: ( nat > rat ) > ( nat > rat ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_Orep__real,type,
% 0.32/0.48 rep_real: real > nat > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_Ovanishes,type,
% 0.32/0.48 vanishes: ( nat > rat ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
% 0.32/0.48 real_V2521375963428798218omplex: set_complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_OReals_001t__Real__Oreal,type,
% 0.32/0.48 real_V470468836141973256s_real: set_real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.32/0.48 real_V5970128139526366754l_real: ( real > real ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
% 0.32/0.48 real_V3694042436643373181omplex: complex > complex > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
% 0.32/0.48 real_V975177566351809787t_real: real > real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_Olinear_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.32/0.48 real_V4572627801940501177l_real: ( real > real ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
% 0.32/0.48 real_V1022390504157884413omplex: complex > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
% 0.32/0.48 real_V7735802525324610683m_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
% 0.32/0.48 real_V4546457046886955230omplex: real > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
% 0.32/0.48 real_V1803761363581548252l_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
% 0.32/0.48 real_V2046097035970521341omplex: real > complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
% 0.32/0.48 real_V1485227260804924795R_real: real > real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Relation_Otransp_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
% 0.32/0.48 transp_nat_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
% 0.32/0.48 algebr932160517623751201me_int: int > int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
% 0.32/0.48 algebr934650988132801477me_nat: nat > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Onatural,type,
% 0.32/0.48 divide5121882707175180666atural: code_natural > code_natural > code_natural ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
% 0.32/0.48 divide1717551699836669952omplex: complex > complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
% 0.32/0.48 divide_divide_int: int > int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 0.32/0.48 divide_divide_nat: nat > nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
% 0.32/0.48 divide_divide_rat: rat > rat > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
% 0.32/0.48 divide_divide_real: real > real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 dvd_dvd_Code_integer: code_integer > code_integer > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Onatural,type,
% 0.32/0.48 dvd_dvd_Code_natural: code_natural > code_natural > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
% 0.32/0.48 dvd_dvd_complex: complex > complex > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
% 0.32/0.48 dvd_dvd_int: int > int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
% 0.32/0.48 dvd_dvd_nat: nat > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
% 0.32/0.48 dvd_dvd_rat: rat > rat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
% 0.32/0.48 dvd_dvd_real: real > real > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Onatural,type,
% 0.32/0.48 modulo8411746178871703098atural: code_natural > code_natural > code_natural ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
% 0.32/0.48 modulo_modulo_int: int > int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
% 0.32/0.48 modulo_modulo_nat: nat > nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Onormalization__semidom__class_Onormalize_001t__Int__Oint,type,
% 0.32/0.48 normal571682499473568032ze_int: int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Onormalization__semidom__class_Onormalize_001t__Nat__Onat,type,
% 0.32/0.48 normal574172969982618308ze_nat: nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Ounit__factor__class_Ounit__factor_001t__Int__Oint,type,
% 0.32/0.48 unit_f2746056213392204926or_int: int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Ounit__factor__class_Ounit__factor_001t__Nat__Onat,type,
% 0.32/0.48 unit_f2748546683901255202or_nat: nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 zero_n356916108424825756nteger: $o > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Onatural,type,
% 0.32/0.48 zero_n8403883297036319079atural: $o > code_natural ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
% 0.32/0.48 zero_n1201886186963655149omplex: $o > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
% 0.32/0.48 zero_n2684676970156552555ol_int: $o > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
% 0.32/0.48 zero_n2687167440665602831ol_nat: $o > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
% 0.32/0.48 zero_n2052037380579107095ol_rat: $o > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
% 0.32/0.48 zero_n3304061248610475627l_real: $o > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
% 0.32/0.48 suminf_real: ( nat > real ) > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
% 0.32/0.48 summable_real: ( nat > real ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Series_Osums_001t__Real__Oreal,type,
% 0.32/0.48 sums_real: ( nat > real ) > real > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_OCollect_001t__Int__Oint,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
% 0.32/0.48 collect_list_nat: ( list_nat > $o ) > set_list_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 0.32/0.48 collect_nat: ( nat > $o ) > set_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.32/0.48 collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.32/0.48 collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_OPow_001t__Nat__Onat,type,
% 0.32/0.48 pow_nat: set_nat > set_set_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
% 0.32/0.48 image_int_int: ( int > int ) > set_int > set_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
% 0.32/0.48 image_int_nat: ( int > nat ) > set_int > set_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
% 0.32/0.48 image_list_nat_nat: ( list_nat > nat ) > set_list_nat > set_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
% 0.32/0.48 image_nat_list_nat: ( nat > list_nat ) > set_nat > set_list_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.32/0.48
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% 0.32/0.48
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
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% 0.32/0.48
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
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% 0.32/0.48
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% 0.32/0.48
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% 0.32/0.48
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% 0.32/0.48
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% 0.32/0.48
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% 0.32/0.48
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% 0.32/0.48
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% 0.32/0.48
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
% 0.32/0.48 set_or1210151606488870762an_nat: nat > set_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
% 0.32/0.48 set_or5849166863359141190n_real: real > set_real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 0.32/0.48 set_ord_lessThan_nat: nat > set_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
% 0.32/0.48 set_or5984915006950818249n_real: real > set_real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_String_OCode_Oabort_001t__Real__Oreal,type,
% 0.32/0.48 abort_real: literal > ( product_unit > real ) > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_String_OLiteral,type,
% 0.32/0.48 literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_String_Oascii__of,type,
% 0.32/0.48 ascii_of: char > char ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_String_Ochar_OChar,type,
% 0.32/0.48 char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_String_Ochar_Osize__char,type,
% 0.32/0.48 size_char: char > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
% 0.32/0.48 comm_s629917340098488124ar_nat: char > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_String_Ointeger__of__char,type,
% 0.32/0.48 integer_of_char: char > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
% 0.32/0.48 unique3096191561947761185of_nat: nat > char ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Sum__Type_OInl_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.48 sum_Inl_nat_nat: nat > sum_sum_nat_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.48 sum_Inr_nat_nat: nat > sum_sum_nat_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
% 0.32/0.48 sum_ca7763040182479039464nt_nat: ( nat > int ) > ( nat > int ) > sum_sum_nat_nat > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.48 sum_ca6763686470577984908at_nat: ( nat > nat ) > ( nat > nat ) > sum_sum_nat_nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.32/0.48 topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.32/0.48 topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 0.32/0.48 topolo6980174941875973593q_real: ( nat > real ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 0.32/0.48 topolo2177554685111907308n_real: real > set_real > filter_real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Real__Oreal,type,
% 0.32/0.48 topolo7531315842566124627t_real: ( nat > real ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 0.32/0.48 topolo2815343760600316023s_real: real > filter_real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
% 0.32/0.48 topolo6517432010174082258omplex: ( nat > complex ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
% 0.32/0.48 topolo4055970368930404560y_real: ( nat > real ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oarccos,type,
% 0.32/0.48 arccos: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 0.32/0.48 arcosh_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oarcsin,type,
% 0.32/0.48 arcsin: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oarctan,type,
% 0.32/0.48 arctan: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 0.32/0.48 arsinh_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 0.32/0.48 artanh_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
% 0.32/0.48 cos_complex: complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
% 0.32/0.48 cos_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Ocos__coeff,type,
% 0.32/0.48 cos_coeff: nat > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
% 0.32/0.48 cosh_complex: complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 0.32/0.48 cosh_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
% 0.32/0.48 cot_complex: complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
% 0.32/0.48 cot_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
% 0.32/0.48 exp_complex: complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
% 0.32/0.48 exp_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 0.32/0.48 ln_ln_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Olog,type,
% 0.32/0.48 log2: real > real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Opi,type,
% 0.32/0.48 pi: real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 0.32/0.48 powr_real: real > real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Opowr__real,type,
% 0.32/0.48 powr_real2: real > real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
% 0.32/0.48 sin_complex: complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 0.32/0.48 sin_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Osin__coeff,type,
% 0.32/0.48 sin_coeff: nat > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
% 0.32/0.48 sinh_complex: complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 0.32/0.48 sinh_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
% 0.32/0.48 tan_complex: complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 0.32/0.48 tan_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
% 0.32/0.48 tanh_complex: complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 0.32/0.48 tanh_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 0.32/0.48 vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 0.32/0.48 vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 0.32/0.48 vEBT_size_VEBT: vEBT_VEBT > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 0.32/0.48 vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 0.32/0.48 vEBT_VEBT_high: nat > nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 0.32/0.48 vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 0.32/0.48 vEBT_VEBT_low: nat > nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 0.32/0.48 vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 0.32/0.48 vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 0.32/0.48 vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 0.32/0.48 vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 0.32/0.48 vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 0.32/0.48 vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 0.32/0.48 vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 0.32/0.48 vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 0.32/0.48 vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 0.32/0.48 vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
% 0.32/0.48 accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
% 0.32/0.48 accp_nat: ( nat > nat > $o ) > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
% 0.32/0.48 accp_P8126237942716283194atural: ( produc7822875418678951345atural > produc7822875418678951345atural > $o ) > produc7822875418678951345atural > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.32/0.48 accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.48 accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 0.32/0.48 accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.32/0.48 accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001_Eo,type,
% 0.32/0.48 member_o: $o > set_o > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 member_Code_integer: code_integer > set_Code_integer > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__Complex__Ocomplex,type,
% 0.32/0.48 member_complex: complex > set_complex > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__Int__Oint,type,
% 0.32/0.48 member_int: int > set_int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
% 0.32/0.48 member_list_nat: list_nat > set_list_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__Nat__Onat,type,
% 0.32/0.48 member_nat: nat > set_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__Rat__Orat,type,
% 0.32/0.48 member_rat: rat > set_rat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__Real__Oreal,type,
% 0.32/0.48 member_real: real > set_real > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.32/0.48 member_set_nat: set_nat > set_set_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 0.32/0.48 member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_v_va____,type,
% 0.32/0.48 va: nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_v_y____,type,
% 0.32/0.48 y: nat ).
% 0.32/0.48
% 0.32/0.48 % Relevant facts (10147)
% 0.32/0.48 thf(fact_0_True,axiom,
% 0.32/0.48 dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) ).
% 0.32/0.48
% 0.32/0.48 % True
% 0.32/0.48 thf(fact_1__C3_OIH_C_I1_J,axiom,
% 0.32/0.48 ! [X: nat,Xa: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
% 0.32/0.48 => ( ( X
% 0.32/0.48 = ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.48 => ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X ) @ Xa ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % "3.IH"(1)
% 0.32/0.48 thf(fact_2__C3_OIH_C_I4_J,axiom,
% 0.32/0.48 ! [X: nat,Xa: nat] :
% 0.32/0.48 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
% 0.32/0.48 => ( ( X
% 0.32/0.48 = ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.48 => ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ ( suc @ X ) ) @ Xa ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % "3.IH"(4)
% 0.32/0.48 thf(fact_3__C3_OIH_C_I3_J,axiom,
% 0.32/0.48 ! [X: nat,Xa: nat] :
% 0.32/0.48 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
% 0.32/0.48 => ( ( X
% 0.32/0.48 = ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.48 => ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X ) @ Xa ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % "3.IH"(3)
% 0.32/0.48 thf(fact_4_div2__Suc__Suc,axiom,
% 0.32/0.48 ! [M: nat] :
% 0.32/0.48 ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.48 = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % div2_Suc_Suc
% 0.32/0.48 thf(fact_5_semiring__norm_I85_J,axiom,
% 0.32/0.48 ! [M: num] :
% 0.32/0.48 ( ( bit0 @ M )
% 0.32/0.48 != one ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_norm(85)
% 0.32/0.48 thf(fact_6_semiring__norm_I83_J,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( one
% 0.32/0.48 != ( bit0 @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_norm(83)
% 0.32/0.48 thf(fact_7_buildup__nothing__in__leaf,axiom,
% 0.32/0.48 ! [N: nat,X: nat] :
% 0.32/0.48 ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 0.32/0.48
% 0.32/0.48 % buildup_nothing_in_leaf
% 0.32/0.48 thf(fact_8_numeral__Bit0__div__2,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.48 = ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % numeral_Bit0_div_2
% 0.32/0.48 thf(fact_9_numeral__Bit0__div__2,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.48 = ( numeral_numeral_int @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % numeral_Bit0_div_2
% 0.32/0.48 thf(fact_10_odd__Suc__div__two,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.48 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.48 = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % odd_Suc_div_two
% 0.32/0.48 thf(fact_11_even__Suc__div__two,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.48 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.48 = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_Suc_div_two
% 0.32/0.48 thf(fact_12_even__Suc,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
% 0.32/0.48 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_Suc
% 0.32/0.48 thf(fact_13_even__Suc__Suc__iff,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
% 0.32/0.48 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_Suc_Suc_iff
% 0.32/0.48 thf(fact_14_divide__numeral__1,axiom,
% 0.32/0.48 ! [A: complex] :
% 0.32/0.48 ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % divide_numeral_1
% 0.32/0.48 thf(fact_15_divide__numeral__1,axiom,
% 0.32/0.48 ! [A: real] :
% 0.32/0.48 ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % divide_numeral_1
% 0.32/0.48 thf(fact_16_divide__numeral__1,axiom,
% 0.32/0.48 ! [A: rat] :
% 0.32/0.48 ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % divide_numeral_1
% 0.32/0.48 thf(fact_17_nat_Oinject,axiom,
% 0.32/0.48 ! [X2: nat,Y2: nat] :
% 0.32/0.48 ( ( ( suc @ X2 )
% 0.32/0.48 = ( suc @ Y2 ) )
% 0.32/0.48 = ( X2 = Y2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % nat.inject
% 0.32/0.48 thf(fact_18_old_Onat_Oinject,axiom,
% 0.32/0.48 ! [Nat: nat,Nat2: nat] :
% 0.32/0.48 ( ( ( suc @ Nat )
% 0.32/0.48 = ( suc @ Nat2 ) )
% 0.32/0.48 = ( Nat = Nat2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % old.nat.inject
% 0.32/0.48 thf(fact_19_numeral__eq__iff,axiom,
% 0.32/0.48 ! [M: num,N: num] :
% 0.32/0.48 ( ( ( numera6690914467698888265omplex @ M )
% 0.32/0.48 = ( numera6690914467698888265omplex @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % numeral_eq_iff
% 0.32/0.48 thf(fact_20_numeral__eq__iff,axiom,
% 0.32/0.48 ! [M: num,N: num] :
% 0.32/0.48 ( ( ( numeral_numeral_real @ M )
% 0.32/0.48 = ( numeral_numeral_real @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % numeral_eq_iff
% 0.32/0.48 thf(fact_21_numeral__eq__iff,axiom,
% 0.32/0.48 ! [M: num,N: num] :
% 0.32/0.48 ( ( ( numeral_numeral_rat @ M )
% 0.32/0.48 = ( numeral_numeral_rat @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % numeral_eq_iff
% 0.32/0.48 thf(fact_22_numeral__eq__iff,axiom,
% 0.32/0.48 ! [M: num,N: num] :
% 0.32/0.48 ( ( ( numeral_numeral_nat @ M )
% 0.32/0.48 = ( numeral_numeral_nat @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % numeral_eq_iff
% 0.32/0.48 thf(fact_23_numeral__eq__iff,axiom,
% 0.32/0.48 ! [M: num,N: num] :
% 0.32/0.48 ( ( ( numeral_numeral_int @ M )
% 0.32/0.48 = ( numeral_numeral_int @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % numeral_eq_iff
% 0.32/0.48 thf(fact_24_semiring__norm_I87_J,axiom,
% 0.32/0.48 ! [M: num,N: num] :
% 0.32/0.48 ( ( ( bit0 @ M )
% 0.32/0.48 = ( bit0 @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_norm(87)
% 0.32/0.48 thf(fact_25_dvd__antisym,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ M @ N )
% 0.32/0.48 => ( ( dvd_dvd_nat @ N @ M )
% 0.32/0.48 => ( M = N ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_antisym
% 0.32/0.48 thf(fact_26_n__not__Suc__n,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( N
% 0.32/0.48 != ( suc @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % n_not_Suc_n
% 0.32/0.48 thf(fact_27_Suc__inject,axiom,
% 0.32/0.48 ! [X: nat,Y: nat] :
% 0.32/0.48 ( ( ( suc @ X )
% 0.32/0.48 = ( suc @ Y ) )
% 0.32/0.48 => ( X = Y ) ) ).
% 0.32/0.48
% 0.32/0.48 % Suc_inject
% 0.32/0.48 thf(fact_28_even__numeral,axiom,
% 0.32/0.48 ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_numeral
% 0.32/0.48 thf(fact_29_even__numeral,axiom,
% 0.32/0.48 ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_numeral
% 0.32/0.48 thf(fact_30_even__numeral,axiom,
% 0.32/0.48 ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_numeral
% 0.32/0.48 thf(fact_31_div2__even__ext__nat,axiom,
% 0.32/0.48 ! [X: nat,Y: nat] :
% 0.32/0.48 ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.48 = ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.48 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
% 0.32/0.48 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
% 0.32/0.48 => ( X = Y ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % div2_even_ext_nat
% 0.32/0.48 thf(fact_32_div__dvd__div,axiom,
% 0.32/0.48 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.48 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.48 => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.32/0.48 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 0.32/0.48 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % div_dvd_div
% 0.32/0.48 thf(fact_33_div__dvd__div,axiom,
% 0.32/0.48 ! [A: nat,B: nat,C: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.48 => ( ( dvd_dvd_nat @ A @ C )
% 0.32/0.48 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
% 0.32/0.48 = ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % div_dvd_div
% 0.32/0.48 thf(fact_34_div__dvd__div,axiom,
% 0.32/0.48 ! [A: int,B: int,C: int] :
% 0.32/0.48 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.48 => ( ( dvd_dvd_int @ A @ C )
% 0.32/0.48 => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
% 0.32/0.48 = ( dvd_dvd_int @ B @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % div_dvd_div
% 0.32/0.48 thf(fact_35_bit__eq__rec,axiom,
% 0.32/0.48 ( ( ^ [Y3: code_integer,Z: code_integer] : ( Y3 = Z ) )
% 0.32/0.48 = ( ^ [A2: code_integer,B2: code_integer] :
% 0.32/0.48 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
% 0.32/0.48 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
% 0.32/0.48 & ( ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.48 = ( divide6298287555418463151nteger @ B2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % bit_eq_rec
% 0.32/0.48 thf(fact_36_bit__eq__rec,axiom,
% 0.32/0.48 ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 0.32/0.48 = ( ^ [A2: nat,B2: nat] :
% 0.32/0.48 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
% 0.32/0.48 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
% 0.32/0.48 & ( ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.48 = ( divide_divide_nat @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % bit_eq_rec
% 0.32/0.48 thf(fact_37_bit__eq__rec,axiom,
% 0.32/0.48 ( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 0.32/0.48 = ( ^ [A2: int,B2: int] :
% 0.32/0.48 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
% 0.32/0.48 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
% 0.32/0.48 & ( ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.48 = ( divide_divide_int @ B2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % bit_eq_rec
% 0.32/0.48 thf(fact_38_zdiv__numeral__Bit0,axiom,
% 0.32/0.48 ! [V: num,W: num] :
% 0.32/0.48 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.32/0.48 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % zdiv_numeral_Bit0
% 0.32/0.48 thf(fact_39_verit__eq__simplify_I8_J,axiom,
% 0.32/0.48 ! [X2: num,Y2: num] :
% 0.32/0.48 ( ( ( bit0 @ X2 )
% 0.32/0.48 = ( bit0 @ Y2 ) )
% 0.32/0.48 = ( X2 = Y2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % verit_eq_simplify(8)
% 0.32/0.48 thf(fact_40_even__of__nat,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
% 0.32/0.48 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_of_nat
% 0.32/0.48 thf(fact_41_even__of__nat,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.48 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_of_nat
% 0.32/0.48 thf(fact_42_even__of__nat,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.48 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_of_nat
% 0.32/0.48 thf(fact_43_div__div__div__same,axiom,
% 0.32/0.48 ! [D: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.48 ( ( dvd_dvd_Code_integer @ D @ B )
% 0.32/0.48 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.48 => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
% 0.32/0.48 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % div_div_div_same
% 0.32/0.48 thf(fact_44_div__div__div__same,axiom,
% 0.32/0.48 ! [D: nat,B: nat,A: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ D @ B )
% 0.32/0.48 => ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.48 => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
% 0.32/0.48 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % div_div_div_same
% 0.32/0.48 thf(fact_45_div__div__div__same,axiom,
% 0.32/0.48 ! [D: int,B: int,A: int] :
% 0.32/0.48 ( ( dvd_dvd_int @ D @ B )
% 0.32/0.48 => ( ( dvd_dvd_int @ B @ A )
% 0.32/0.48 => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
% 0.32/0.48 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % div_div_div_same
% 0.32/0.48 thf(fact_46_dvd__div__eq__cancel,axiom,
% 0.32/0.48 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.48 ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.32/0.48 = ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.48 => ( ( dvd_dvd_Code_integer @ C @ A )
% 0.32/0.48 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.48 => ( A = B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_cancel
% 0.32/0.48 thf(fact_47_dvd__div__eq__cancel,axiom,
% 0.32/0.48 ! [A: complex,C: complex,B: complex] :
% 0.32/0.48 ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.32/0.48 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.48 => ( ( dvd_dvd_complex @ C @ A )
% 0.32/0.48 => ( ( dvd_dvd_complex @ C @ B )
% 0.32/0.48 => ( A = B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_cancel
% 0.32/0.48 thf(fact_48_dvd__div__eq__cancel,axiom,
% 0.32/0.48 ! [A: real,C: real,B: real] :
% 0.32/0.48 ( ( ( divide_divide_real @ A @ C )
% 0.32/0.48 = ( divide_divide_real @ B @ C ) )
% 0.32/0.48 => ( ( dvd_dvd_real @ C @ A )
% 0.32/0.48 => ( ( dvd_dvd_real @ C @ B )
% 0.32/0.48 => ( A = B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_cancel
% 0.32/0.48 thf(fact_49_dvd__div__eq__cancel,axiom,
% 0.32/0.48 ! [A: rat,C: rat,B: rat] :
% 0.32/0.48 ( ( ( divide_divide_rat @ A @ C )
% 0.32/0.48 = ( divide_divide_rat @ B @ C ) )
% 0.32/0.48 => ( ( dvd_dvd_rat @ C @ A )
% 0.32/0.48 => ( ( dvd_dvd_rat @ C @ B )
% 0.32/0.48 => ( A = B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_cancel
% 0.32/0.48 thf(fact_50_dvd__div__eq__cancel,axiom,
% 0.32/0.48 ! [A: nat,C: nat,B: nat] :
% 0.32/0.48 ( ( ( divide_divide_nat @ A @ C )
% 0.32/0.48 = ( divide_divide_nat @ B @ C ) )
% 0.32/0.48 => ( ( dvd_dvd_nat @ C @ A )
% 0.32/0.48 => ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.48 => ( A = B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_cancel
% 0.32/0.48 thf(fact_51_dvd__div__eq__cancel,axiom,
% 0.32/0.48 ! [A: int,C: int,B: int] :
% 0.32/0.48 ( ( ( divide_divide_int @ A @ C )
% 0.32/0.48 = ( divide_divide_int @ B @ C ) )
% 0.32/0.48 => ( ( dvd_dvd_int @ C @ A )
% 0.32/0.48 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.48 => ( A = B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_cancel
% 0.32/0.48 thf(fact_52_dvd__div__eq__iff,axiom,
% 0.32/0.48 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.48 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.32/0.48 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.48 => ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.32/0.48 = ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.48 = ( A = B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_iff
% 0.32/0.48 thf(fact_53_dvd__div__eq__iff,axiom,
% 0.32/0.48 ! [C: complex,A: complex,B: complex] :
% 0.32/0.48 ( ( dvd_dvd_complex @ C @ A )
% 0.32/0.48 => ( ( dvd_dvd_complex @ C @ B )
% 0.32/0.48 => ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.32/0.48 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.48 = ( A = B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_iff
% 0.32/0.48 thf(fact_54_dvd__div__eq__iff,axiom,
% 0.32/0.48 ! [C: real,A: real,B: real] :
% 0.32/0.48 ( ( dvd_dvd_real @ C @ A )
% 0.32/0.48 => ( ( dvd_dvd_real @ C @ B )
% 0.32/0.48 => ( ( ( divide_divide_real @ A @ C )
% 0.32/0.48 = ( divide_divide_real @ B @ C ) )
% 0.32/0.48 = ( A = B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_iff
% 0.32/0.48 thf(fact_55_dvd__div__eq__iff,axiom,
% 0.32/0.48 ! [C: rat,A: rat,B: rat] :
% 0.32/0.48 ( ( dvd_dvd_rat @ C @ A )
% 0.32/0.48 => ( ( dvd_dvd_rat @ C @ B )
% 0.32/0.48 => ( ( ( divide_divide_rat @ A @ C )
% 0.32/0.48 = ( divide_divide_rat @ B @ C ) )
% 0.32/0.48 = ( A = B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_iff
% 0.32/0.48 thf(fact_56_dvd__div__eq__iff,axiom,
% 0.32/0.48 ! [C: nat,A: nat,B: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ C @ A )
% 0.32/0.48 => ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.48 => ( ( ( divide_divide_nat @ A @ C )
% 0.32/0.48 = ( divide_divide_nat @ B @ C ) )
% 0.32/0.48 = ( A = B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_iff
% 0.32/0.48 thf(fact_57_dvd__div__eq__iff,axiom,
% 0.32/0.48 ! [C: int,A: int,B: int] :
% 0.32/0.48 ( ( dvd_dvd_int @ C @ A )
% 0.32/0.48 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.48 => ( ( ( divide_divide_int @ A @ C )
% 0.32/0.48 = ( divide_divide_int @ B @ C ) )
% 0.32/0.48 = ( A = B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_iff
% 0.32/0.48 thf(fact_58_verit__eq__simplify_I10_J,axiom,
% 0.32/0.48 ! [X2: num] :
% 0.32/0.48 ( one
% 0.32/0.48 != ( bit0 @ X2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % verit_eq_simplify(10)
% 0.32/0.48 thf(fact_59_set__decode__Suc,axiom,
% 0.32/0.48 ! [N: nat,X: nat] :
% 0.32/0.48 ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
% 0.32/0.48 = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % set_decode_Suc
% 0.32/0.48 thf(fact_60_of__nat__eq__iff,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( ( semiri8010041392384452111omplex @ M )
% 0.32/0.48 = ( semiri8010041392384452111omplex @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_eq_iff
% 0.32/0.48 thf(fact_61_of__nat__eq__iff,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( ( semiri5074537144036343181t_real @ M )
% 0.32/0.48 = ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_eq_iff
% 0.32/0.48 thf(fact_62_of__nat__eq__iff,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( ( semiri681578069525770553at_rat @ M )
% 0.32/0.48 = ( semiri681578069525770553at_rat @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_eq_iff
% 0.32/0.48 thf(fact_63_of__nat__eq__iff,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( ( semiri1316708129612266289at_nat @ M )
% 0.32/0.48 = ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_eq_iff
% 0.32/0.48 thf(fact_64_of__nat__eq__iff,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( ( semiri1314217659103216013at_int @ M )
% 0.32/0.48 = ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_eq_iff
% 0.32/0.48 thf(fact_65_div__0,axiom,
% 0.32/0.48 ! [A: complex] :
% 0.32/0.48 ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 0.32/0.48 = zero_zero_complex ) ).
% 0.32/0.48
% 0.32/0.48 % div_0
% 0.32/0.48 thf(fact_66_div__0,axiom,
% 0.32/0.48 ! [A: real] :
% 0.32/0.48 ( ( divide_divide_real @ zero_zero_real @ A )
% 0.32/0.48 = zero_zero_real ) ).
% 0.32/0.48
% 0.32/0.48 % div_0
% 0.32/0.48 thf(fact_67_div__0,axiom,
% 0.32/0.48 ! [A: rat] :
% 0.32/0.48 ( ( divide_divide_rat @ zero_zero_rat @ A )
% 0.32/0.48 = zero_zero_rat ) ).
% 0.32/0.48
% 0.32/0.48 % div_0
% 0.32/0.48 thf(fact_68_div__0,axiom,
% 0.32/0.48 ! [A: nat] :
% 0.32/0.48 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.32/0.48 = zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % div_0
% 0.32/0.48 thf(fact_69_div__0,axiom,
% 0.32/0.48 ! [A: int] :
% 0.32/0.48 ( ( divide_divide_int @ zero_zero_int @ A )
% 0.32/0.48 = zero_zero_int ) ).
% 0.32/0.48
% 0.32/0.48 % div_0
% 0.32/0.48 thf(fact_70_div__by__0,axiom,
% 0.32/0.48 ! [A: complex] :
% 0.32/0.48 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.32/0.48 = zero_zero_complex ) ).
% 0.32/0.48
% 0.32/0.48 % div_by_0
% 0.32/0.48 thf(fact_71_div__by__0,axiom,
% 0.32/0.48 ! [A: real] :
% 0.32/0.48 ( ( divide_divide_real @ A @ zero_zero_real )
% 0.32/0.48 = zero_zero_real ) ).
% 0.32/0.48
% 0.32/0.48 % div_by_0
% 0.32/0.48 thf(fact_72_div__by__0,axiom,
% 0.32/0.48 ! [A: rat] :
% 0.32/0.48 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.32/0.48 = zero_zero_rat ) ).
% 0.32/0.48
% 0.32/0.48 % div_by_0
% 0.32/0.48 thf(fact_73_div__by__0,axiom,
% 0.32/0.48 ! [A: nat] :
% 0.32/0.48 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.32/0.48 = zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % div_by_0
% 0.32/0.48 thf(fact_74_div__by__0,axiom,
% 0.32/0.48 ! [A: int] :
% 0.32/0.48 ( ( divide_divide_int @ A @ zero_zero_int )
% 0.32/0.48 = zero_zero_int ) ).
% 0.32/0.48
% 0.32/0.48 % div_by_0
% 0.32/0.48 thf(fact_75_bits__div__0,axiom,
% 0.32/0.48 ! [A: nat] :
% 0.32/0.48 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.32/0.48 = zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % bits_div_0
% 0.32/0.48 thf(fact_76_bits__div__0,axiom,
% 0.32/0.48 ! [A: int] :
% 0.32/0.48 ( ( divide_divide_int @ zero_zero_int @ A )
% 0.32/0.48 = zero_zero_int ) ).
% 0.32/0.48
% 0.32/0.48 % bits_div_0
% 0.32/0.48 thf(fact_77_bits__div__by__0,axiom,
% 0.32/0.48 ! [A: nat] :
% 0.32/0.48 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.32/0.48 = zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % bits_div_by_0
% 0.32/0.48 thf(fact_78_bits__div__by__0,axiom,
% 0.32/0.48 ! [A: int] :
% 0.32/0.48 ( ( divide_divide_int @ A @ zero_zero_int )
% 0.32/0.48 = zero_zero_int ) ).
% 0.32/0.48
% 0.32/0.48 % bits_div_by_0
% 0.32/0.48 thf(fact_79_dvd__0__left__iff,axiom,
% 0.32/0.48 ! [A: code_integer] :
% 0.32/0.48 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.32/0.48 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_0_left_iff
% 0.32/0.48 thf(fact_80_dvd__0__left__iff,axiom,
% 0.32/0.48 ! [A: real] :
% 0.32/0.48 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.32/0.48 = ( A = zero_zero_real ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_0_left_iff
% 0.32/0.48 thf(fact_81_dvd__0__left__iff,axiom,
% 0.32/0.48 ! [A: rat] :
% 0.32/0.48 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.32/0.48 = ( A = zero_zero_rat ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_0_left_iff
% 0.32/0.48 thf(fact_82_dvd__0__left__iff,axiom,
% 0.32/0.48 ! [A: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.32/0.48 = ( A = zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_0_left_iff
% 0.32/0.48 thf(fact_83_dvd__0__left__iff,axiom,
% 0.32/0.48 ! [A: int] :
% 0.32/0.48 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.32/0.48 = ( A = zero_zero_int ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_0_left_iff
% 0.32/0.48 thf(fact_84_dvd__0__right,axiom,
% 0.32/0.48 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_0_right
% 0.32/0.48 thf(fact_85_dvd__0__right,axiom,
% 0.32/0.48 ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_0_right
% 0.32/0.48 thf(fact_86_dvd__0__right,axiom,
% 0.32/0.48 ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_0_right
% 0.32/0.48 thf(fact_87_dvd__0__right,axiom,
% 0.32/0.48 ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_0_right
% 0.32/0.48 thf(fact_88_dvd__0__right,axiom,
% 0.32/0.48 ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_0_right
% 0.32/0.48 thf(fact_89_of__nat__0,axiom,
% 0.32/0.48 ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 0.32/0.48 = zero_zero_complex ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_0
% 0.32/0.48 thf(fact_90_of__nat__0,axiom,
% 0.32/0.48 ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 0.32/0.48 = zero_zero_real ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_0
% 0.32/0.48 thf(fact_91_of__nat__0,axiom,
% 0.32/0.48 ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 0.32/0.48 = zero_zero_rat ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_0
% 0.32/0.48 thf(fact_92_of__nat__0,axiom,
% 0.32/0.48 ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 0.32/0.48 = zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_0
% 0.32/0.48 thf(fact_93_of__nat__0,axiom,
% 0.32/0.48 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.32/0.48 = zero_zero_int ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_0
% 0.32/0.48 thf(fact_94_mem__Collect__eq,axiom,
% 0.32/0.48 ! [A: complex,P: complex > $o] :
% 0.32/0.48 ( ( member_complex @ A @ ( collect_complex @ P ) )
% 0.32/0.48 = ( P @ A ) ) ).
% 0.32/0.48
% 0.32/0.48 % mem_Collect_eq
% 0.32/0.48 thf(fact_95_mem__Collect__eq,axiom,
% 0.32/0.48 ! [A: real,P: real > $o] :
% 0.32/0.48 ( ( member_real @ A @ ( collect_real @ P ) )
% 0.32/0.48 = ( P @ A ) ) ).
% 0.32/0.48
% 0.32/0.48 % mem_Collect_eq
% 0.32/0.48 thf(fact_96_mem__Collect__eq,axiom,
% 0.32/0.48 ! [A: list_nat,P: list_nat > $o] :
% 0.32/0.48 ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
% 0.32/0.48 = ( P @ A ) ) ).
% 0.32/0.48
% 0.32/0.48 % mem_Collect_eq
% 0.32/0.48 thf(fact_97_mem__Collect__eq,axiom,
% 0.32/0.48 ! [A: set_nat,P: set_nat > $o] :
% 0.32/0.48 ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
% 0.32/0.48 = ( P @ A ) ) ).
% 0.32/0.48
% 0.32/0.48 % mem_Collect_eq
% 0.32/0.48 thf(fact_98_mem__Collect__eq,axiom,
% 0.32/0.48 ! [A: nat,P: nat > $o] :
% 0.32/0.48 ( ( member_nat @ A @ ( collect_nat @ P ) )
% 0.32/0.48 = ( P @ A ) ) ).
% 0.32/0.48
% 0.32/0.48 % mem_Collect_eq
% 0.32/0.48 thf(fact_99_mem__Collect__eq,axiom,
% 0.32/0.48 ! [A: int,P: int > $o] :
% 0.32/0.48 ( ( member_int @ A @ ( collect_int @ P ) )
% 0.32/0.48 = ( P @ A ) ) ).
% 0.32/0.48
% 0.32/0.48 % mem_Collect_eq
% 0.32/0.48 thf(fact_100_Collect__mem__eq,axiom,
% 0.32/0.48 ! [A3: set_complex] :
% 0.32/0.48 ( ( collect_complex
% 0.32/0.48 @ ^ [X3: complex] : ( member_complex @ X3 @ A3 ) )
% 0.32/0.48 = A3 ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_mem_eq
% 0.32/0.48 thf(fact_101_Collect__mem__eq,axiom,
% 0.32/0.48 ! [A3: set_real] :
% 0.32/0.48 ( ( collect_real
% 0.32/0.48 @ ^ [X3: real] : ( member_real @ X3 @ A3 ) )
% 0.32/0.48 = A3 ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_mem_eq
% 0.32/0.48 thf(fact_102_Collect__mem__eq,axiom,
% 0.32/0.48 ! [A3: set_list_nat] :
% 0.32/0.48 ( ( collect_list_nat
% 0.32/0.48 @ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A3 ) )
% 0.32/0.48 = A3 ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_mem_eq
% 0.32/0.48 thf(fact_103_Collect__mem__eq,axiom,
% 0.32/0.48 ! [A3: set_set_nat] :
% 0.32/0.48 ( ( collect_set_nat
% 0.32/0.48 @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A3 ) )
% 0.32/0.48 = A3 ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_mem_eq
% 0.32/0.48 thf(fact_104_Collect__mem__eq,axiom,
% 0.32/0.48 ! [A3: set_nat] :
% 0.32/0.48 ( ( collect_nat
% 0.32/0.48 @ ^ [X3: nat] : ( member_nat @ X3 @ A3 ) )
% 0.32/0.48 = A3 ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_mem_eq
% 0.32/0.48 thf(fact_105_Collect__mem__eq,axiom,
% 0.32/0.48 ! [A3: set_int] :
% 0.32/0.48 ( ( collect_int
% 0.32/0.48 @ ^ [X3: int] : ( member_int @ X3 @ A3 ) )
% 0.32/0.48 = A3 ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_mem_eq
% 0.32/0.48 thf(fact_106_Collect__cong,axiom,
% 0.32/0.48 ! [P: real > $o,Q: real > $o] :
% 0.32/0.48 ( ! [X4: real] :
% 0.32/0.48 ( ( P @ X4 )
% 0.32/0.48 = ( Q @ X4 ) )
% 0.32/0.48 => ( ( collect_real @ P )
% 0.32/0.48 = ( collect_real @ Q ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_cong
% 0.32/0.48 thf(fact_107_Collect__cong,axiom,
% 0.32/0.48 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.32/0.48 ( ! [X4: list_nat] :
% 0.32/0.48 ( ( P @ X4 )
% 0.32/0.48 = ( Q @ X4 ) )
% 0.32/0.48 => ( ( collect_list_nat @ P )
% 0.32/0.48 = ( collect_list_nat @ Q ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_cong
% 0.32/0.48 thf(fact_108_Collect__cong,axiom,
% 0.32/0.48 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.32/0.48 ( ! [X4: set_nat] :
% 0.32/0.48 ( ( P @ X4 )
% 0.32/0.48 = ( Q @ X4 ) )
% 0.32/0.48 => ( ( collect_set_nat @ P )
% 0.32/0.48 = ( collect_set_nat @ Q ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_cong
% 0.32/0.48 thf(fact_109_Collect__cong,axiom,
% 0.32/0.48 ! [P: nat > $o,Q: nat > $o] :
% 0.32/0.48 ( ! [X4: nat] :
% 0.32/0.48 ( ( P @ X4 )
% 0.32/0.48 = ( Q @ X4 ) )
% 0.32/0.48 => ( ( collect_nat @ P )
% 0.32/0.48 = ( collect_nat @ Q ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_cong
% 0.32/0.48 thf(fact_110_Collect__cong,axiom,
% 0.32/0.48 ! [P: int > $o,Q: int > $o] :
% 0.32/0.48 ( ! [X4: int] :
% 0.32/0.48 ( ( P @ X4 )
% 0.32/0.48 = ( Q @ X4 ) )
% 0.32/0.48 => ( ( collect_int @ P )
% 0.32/0.48 = ( collect_int @ Q ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Collect_cong
% 0.32/0.48 thf(fact_111_of__nat__0__eq__iff,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( zero_zero_complex
% 0.32/0.48 = ( semiri8010041392384452111omplex @ N ) )
% 0.32/0.48 = ( zero_zero_nat = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_0_eq_iff
% 0.32/0.48 thf(fact_112_of__nat__0__eq__iff,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( zero_zero_real
% 0.32/0.48 = ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.48 = ( zero_zero_nat = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_0_eq_iff
% 0.32/0.48 thf(fact_113_of__nat__0__eq__iff,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( zero_zero_rat
% 0.32/0.48 = ( semiri681578069525770553at_rat @ N ) )
% 0.32/0.48 = ( zero_zero_nat = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_0_eq_iff
% 0.32/0.48 thf(fact_114_of__nat__0__eq__iff,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( zero_zero_nat
% 0.32/0.48 = ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.48 = ( zero_zero_nat = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_0_eq_iff
% 0.32/0.48 thf(fact_115_of__nat__0__eq__iff,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( zero_zero_int
% 0.32/0.48 = ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.48 = ( zero_zero_nat = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_0_eq_iff
% 0.32/0.48 thf(fact_116_of__nat__eq__0__iff,axiom,
% 0.32/0.48 ! [M: nat] :
% 0.32/0.48 ( ( ( semiri8010041392384452111omplex @ M )
% 0.32/0.48 = zero_zero_complex )
% 0.32/0.48 = ( M = zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_eq_0_iff
% 0.32/0.48 thf(fact_117_of__nat__eq__0__iff,axiom,
% 0.32/0.48 ! [M: nat] :
% 0.32/0.48 ( ( ( semiri5074537144036343181t_real @ M )
% 0.32/0.48 = zero_zero_real )
% 0.32/0.48 = ( M = zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_eq_0_iff
% 0.32/0.48 thf(fact_118_of__nat__eq__0__iff,axiom,
% 0.32/0.48 ! [M: nat] :
% 0.32/0.48 ( ( ( semiri681578069525770553at_rat @ M )
% 0.32/0.48 = zero_zero_rat )
% 0.32/0.48 = ( M = zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_eq_0_iff
% 0.32/0.48 thf(fact_119_of__nat__eq__0__iff,axiom,
% 0.32/0.48 ! [M: nat] :
% 0.32/0.48 ( ( ( semiri1316708129612266289at_nat @ M )
% 0.32/0.48 = zero_zero_nat )
% 0.32/0.48 = ( M = zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_eq_0_iff
% 0.32/0.48 thf(fact_120_of__nat__eq__0__iff,axiom,
% 0.32/0.48 ! [M: nat] :
% 0.32/0.48 ( ( ( semiri1314217659103216013at_int @ M )
% 0.32/0.48 = zero_zero_int )
% 0.32/0.48 = ( M = zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_eq_0_iff
% 0.32/0.48 thf(fact_121_of__nat__numeral,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
% 0.32/0.48 = ( numera6690914467698888265omplex @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_numeral
% 0.32/0.48 thf(fact_122_of__nat__numeral,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
% 0.32/0.48 = ( numeral_numeral_real @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_numeral
% 0.32/0.48 thf(fact_123_of__nat__numeral,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
% 0.32/0.48 = ( numeral_numeral_rat @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_numeral
% 0.32/0.48 thf(fact_124_of__nat__numeral,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
% 0.32/0.48 = ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_numeral
% 0.32/0.48 thf(fact_125_of__nat__numeral,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 0.32/0.48 = ( numeral_numeral_int @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_numeral
% 0.32/0.48 thf(fact_126_dvd__1__left,axiom,
% 0.32/0.48 ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_1_left
% 0.32/0.48 thf(fact_127_dvd__1__iff__1,axiom,
% 0.32/0.48 ! [M: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.32/0.48 = ( M
% 0.32/0.48 = ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_1_iff_1
% 0.32/0.48 thf(fact_128_div__by__Suc__0,axiom,
% 0.32/0.48 ! [M: nat] :
% 0.32/0.48 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.32/0.48 = M ) ).
% 0.32/0.48
% 0.32/0.48 % div_by_Suc_0
% 0.32/0.48 thf(fact_129_set__decode__0,axiom,
% 0.32/0.48 ! [X: nat] :
% 0.32/0.48 ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
% 0.32/0.48 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % set_decode_0
% 0.32/0.48 thf(fact_130_of__nat__neq__0,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
% 0.32/0.48 != zero_zero_complex ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_neq_0
% 0.32/0.48 thf(fact_131_of__nat__neq__0,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 0.32/0.48 != zero_zero_real ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_neq_0
% 0.32/0.48 thf(fact_132_of__nat__neq__0,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 0.32/0.48 != zero_zero_rat ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_neq_0
% 0.32/0.48 thf(fact_133_of__nat__neq__0,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 0.32/0.48 != zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_neq_0
% 0.32/0.48 thf(fact_134_of__nat__neq__0,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 0.32/0.48 != zero_zero_int ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_neq_0
% 0.32/0.48 thf(fact_135_int__ops_I3_J,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 0.32/0.48 = ( numeral_numeral_int @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % int_ops(3)
% 0.32/0.48 thf(fact_136_zdiv__int,axiom,
% 0.32/0.48 ! [A: nat,B: nat] :
% 0.32/0.48 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
% 0.32/0.48 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % zdiv_int
% 0.32/0.48 thf(fact_137_list__decode_Ocases,axiom,
% 0.32/0.48 ! [X: nat] :
% 0.32/0.48 ( ( X != zero_zero_nat )
% 0.32/0.48 => ~ ! [N2: nat] :
% 0.32/0.48 ( X
% 0.32/0.48 != ( suc @ N2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % list_decode.cases
% 0.32/0.48 thf(fact_138_dvd__0__left,axiom,
% 0.32/0.48 ! [A: code_integer] :
% 0.32/0.48 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.32/0.48 => ( A = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_0_left
% 0.32/0.48 thf(fact_139_dvd__0__left,axiom,
% 0.32/0.48 ! [A: real] :
% 0.32/0.48 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.32/0.48 => ( A = zero_zero_real ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_0_left
% 0.32/0.48 thf(fact_140_dvd__0__left,axiom,
% 0.32/0.48 ! [A: rat] :
% 0.32/0.48 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.32/0.48 => ( A = zero_zero_rat ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_0_left
% 0.32/0.48 thf(fact_141_dvd__0__left,axiom,
% 0.32/0.48 ! [A: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.32/0.48 => ( A = zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_0_left
% 0.32/0.48 thf(fact_142_dvd__0__left,axiom,
% 0.32/0.48 ! [A: int] :
% 0.32/0.48 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.32/0.48 => ( A = zero_zero_int ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_0_left
% 0.32/0.48 thf(fact_143_zero__neq__numeral,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( zero_zero_complex
% 0.32/0.48 != ( numera6690914467698888265omplex @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % zero_neq_numeral
% 0.32/0.48 thf(fact_144_zero__neq__numeral,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( zero_zero_real
% 0.32/0.48 != ( numeral_numeral_real @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % zero_neq_numeral
% 0.32/0.48 thf(fact_145_zero__neq__numeral,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( zero_zero_rat
% 0.32/0.48 != ( numeral_numeral_rat @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % zero_neq_numeral
% 0.32/0.48 thf(fact_146_zero__neq__numeral,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( zero_zero_nat
% 0.32/0.48 != ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % zero_neq_numeral
% 0.32/0.48 thf(fact_147_zero__neq__numeral,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( zero_zero_int
% 0.32/0.48 != ( numeral_numeral_int @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % zero_neq_numeral
% 0.32/0.48 thf(fact_148_vebt__buildup_Ocases,axiom,
% 0.32/0.48 ! [X: nat] :
% 0.32/0.48 ( ( X != zero_zero_nat )
% 0.32/0.48 => ( ( X
% 0.32/0.48 != ( suc @ zero_zero_nat ) )
% 0.32/0.48 => ~ ! [Va: nat] :
% 0.32/0.48 ( X
% 0.32/0.48 != ( suc @ ( suc @ Va ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % vebt_buildup.cases
% 0.32/0.48 thf(fact_149_nat_Odistinct_I1_J,axiom,
% 0.32/0.48 ! [X2: nat] :
% 0.32/0.48 ( zero_zero_nat
% 0.32/0.48 != ( suc @ X2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % nat.distinct(1)
% 0.32/0.48 thf(fact_150_old_Onat_Odistinct_I2_J,axiom,
% 0.32/0.48 ! [Nat2: nat] :
% 0.32/0.48 ( ( suc @ Nat2 )
% 0.32/0.48 != zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % old.nat.distinct(2)
% 0.32/0.48 thf(fact_151_old_Onat_Odistinct_I1_J,axiom,
% 0.32/0.48 ! [Nat2: nat] :
% 0.32/0.48 ( zero_zero_nat
% 0.32/0.48 != ( suc @ Nat2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % old.nat.distinct(1)
% 0.32/0.48 thf(fact_152_nat_OdiscI,axiom,
% 0.32/0.48 ! [Nat: nat,X2: nat] :
% 0.32/0.48 ( ( Nat
% 0.32/0.48 = ( suc @ X2 ) )
% 0.32/0.48 => ( Nat != zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % nat.discI
% 0.32/0.48 thf(fact_153_old_Onat_Oexhaust,axiom,
% 0.32/0.48 ! [Y: nat] :
% 0.32/0.48 ( ( Y != zero_zero_nat )
% 0.32/0.48 => ~ ! [Nat3: nat] :
% 0.32/0.48 ( Y
% 0.32/0.48 != ( suc @ Nat3 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % old.nat.exhaust
% 0.32/0.48 thf(fact_154_nat__induct,axiom,
% 0.32/0.48 ! [P: nat > $o,N: nat] :
% 0.32/0.48 ( ( P @ zero_zero_nat )
% 0.32/0.48 => ( ! [N2: nat] :
% 0.32/0.48 ( ( P @ N2 )
% 0.32/0.48 => ( P @ ( suc @ N2 ) ) )
% 0.32/0.48 => ( P @ N ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % nat_induct
% 0.32/0.48 thf(fact_155_diff__induct,axiom,
% 0.32/0.48 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.32/0.48 ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
% 0.32/0.48 => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
% 0.32/0.48 => ( ! [X4: nat,Y4: nat] :
% 0.32/0.48 ( ( P @ X4 @ Y4 )
% 0.32/0.48 => ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
% 0.32/0.48 => ( P @ M @ N ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_induct
% 0.32/0.48 thf(fact_156_zero__induct,axiom,
% 0.32/0.48 ! [P: nat > $o,K: nat] :
% 0.32/0.48 ( ( P @ K )
% 0.32/0.48 => ( ! [N2: nat] :
% 0.32/0.48 ( ( P @ ( suc @ N2 ) )
% 0.32/0.48 => ( P @ N2 ) )
% 0.32/0.48 => ( P @ zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % zero_induct
% 0.32/0.48 thf(fact_157_Suc__neq__Zero,axiom,
% 0.32/0.48 ! [M: nat] :
% 0.32/0.48 ( ( suc @ M )
% 0.32/0.48 != zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % Suc_neq_Zero
% 0.32/0.48 thf(fact_158_Zero__neq__Suc,axiom,
% 0.32/0.48 ! [M: nat] :
% 0.32/0.48 ( zero_zero_nat
% 0.32/0.48 != ( suc @ M ) ) ).
% 0.32/0.48
% 0.32/0.48 % Zero_neq_Suc
% 0.32/0.48 thf(fact_159_Zero__not__Suc,axiom,
% 0.32/0.48 ! [M: nat] :
% 0.32/0.48 ( zero_zero_nat
% 0.32/0.48 != ( suc @ M ) ) ).
% 0.32/0.48
% 0.32/0.48 % Zero_not_Suc
% 0.32/0.48 thf(fact_160_not0__implies__Suc,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( N != zero_zero_nat )
% 0.32/0.48 => ? [M2: nat] :
% 0.32/0.48 ( N
% 0.32/0.48 = ( suc @ M2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % not0_implies_Suc
% 0.32/0.48 thf(fact_161_dvd__div__eq__0__iff,axiom,
% 0.32/0.48 ! [B: code_integer,A: code_integer] :
% 0.32/0.48 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.48 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.32/0.48 = zero_z3403309356797280102nteger )
% 0.32/0.48 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_0_iff
% 0.32/0.48 thf(fact_162_dvd__div__eq__0__iff,axiom,
% 0.32/0.48 ! [B: complex,A: complex] :
% 0.32/0.48 ( ( dvd_dvd_complex @ B @ A )
% 0.32/0.48 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.32/0.48 = zero_zero_complex )
% 0.32/0.48 = ( A = zero_zero_complex ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_0_iff
% 0.32/0.48 thf(fact_163_dvd__div__eq__0__iff,axiom,
% 0.32/0.48 ! [B: real,A: real] :
% 0.32/0.48 ( ( dvd_dvd_real @ B @ A )
% 0.32/0.48 => ( ( ( divide_divide_real @ A @ B )
% 0.32/0.48 = zero_zero_real )
% 0.32/0.48 = ( A = zero_zero_real ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_0_iff
% 0.32/0.48 thf(fact_164_dvd__div__eq__0__iff,axiom,
% 0.32/0.48 ! [B: rat,A: rat] :
% 0.32/0.48 ( ( dvd_dvd_rat @ B @ A )
% 0.32/0.48 => ( ( ( divide_divide_rat @ A @ B )
% 0.32/0.48 = zero_zero_rat )
% 0.32/0.48 = ( A = zero_zero_rat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_0_iff
% 0.32/0.48 thf(fact_165_dvd__div__eq__0__iff,axiom,
% 0.32/0.48 ! [B: nat,A: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.48 => ( ( ( divide_divide_nat @ A @ B )
% 0.32/0.48 = zero_zero_nat )
% 0.32/0.48 = ( A = zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_0_iff
% 0.32/0.48 thf(fact_166_dvd__div__eq__0__iff,axiom,
% 0.32/0.48 ! [B: int,A: int] :
% 0.32/0.48 ( ( dvd_dvd_int @ B @ A )
% 0.32/0.48 => ( ( ( divide_divide_int @ A @ B )
% 0.32/0.48 = zero_zero_int )
% 0.32/0.48 = ( A = zero_zero_int ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_div_eq_0_iff
% 0.32/0.48 thf(fact_167_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
% 0.32/0.48 = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.32/0.48 thf(fact_168_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
% 0.32/0.48 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.32/0.48 thf(fact_169_of__nat__dvd__iff,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
% 0.32/0.48 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_dvd_iff
% 0.32/0.48 thf(fact_170_of__nat__dvd__iff,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.48 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_dvd_iff
% 0.32/0.48 thf(fact_171_of__nat__dvd__iff,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.48 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % of_nat_dvd_iff
% 0.32/0.48 thf(fact_172_dvd__trans,axiom,
% 0.32/0.48 ! [A: nat,B: nat,C: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.48 => ( ( dvd_dvd_nat @ B @ C )
% 0.32/0.48 => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_trans
% 0.32/0.48 thf(fact_173_dvd__trans,axiom,
% 0.32/0.48 ! [A: int,B: int,C: int] :
% 0.32/0.48 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.48 => ( ( dvd_dvd_int @ B @ C )
% 0.32/0.48 => ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_trans
% 0.32/0.48 thf(fact_174_dvd__trans,axiom,
% 0.32/0.48 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.48 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.48 => ( ( dvd_dvd_Code_integer @ B @ C )
% 0.32/0.48 => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_trans
% 0.32/0.48 thf(fact_175_dvd__refl,axiom,
% 0.32/0.48 ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_refl
% 0.32/0.48 thf(fact_176_dvd__refl,axiom,
% 0.32/0.48 ! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_refl
% 0.32/0.48 thf(fact_177_dvd__refl,axiom,
% 0.32/0.48 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_refl
% 0.32/0.48 thf(fact_178_numeral__1__eq__Suc__0,axiom,
% 0.32/0.48 ( ( numeral_numeral_nat @ one )
% 0.32/0.48 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % numeral_1_eq_Suc_0
% 0.32/0.48 thf(fact_179_even__zero,axiom,
% 0.32/0.48 dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 0.32/0.48
% 0.32/0.48 % even_zero
% 0.32/0.48 thf(fact_180_even__zero,axiom,
% 0.32/0.48 dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 0.32/0.48
% 0.32/0.48 % even_zero
% 0.32/0.48 thf(fact_181_even__zero,axiom,
% 0.32/0.48 dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 0.32/0.48
% 0.32/0.48 % even_zero
% 0.32/0.48 thf(fact_182_numeral__2__eq__2,axiom,
% 0.32/0.48 ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.32/0.48 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % numeral_2_eq_2
% 0.32/0.48 thf(fact_183_int__eq__iff__numeral,axiom,
% 0.32/0.48 ! [M: nat,V: num] :
% 0.32/0.48 ( ( ( semiri1314217659103216013at_int @ M )
% 0.32/0.48 = ( numeral_numeral_int @ V ) )
% 0.32/0.48 = ( M
% 0.32/0.48 = ( numeral_numeral_nat @ V ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % int_eq_iff_numeral
% 0.32/0.48 thf(fact_184_divide__eq__0__iff,axiom,
% 0.32/0.48 ! [A: complex,B: complex] :
% 0.32/0.48 ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.32/0.48 = zero_zero_complex )
% 0.32/0.48 = ( ( A = zero_zero_complex )
% 0.32/0.48 | ( B = zero_zero_complex ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % divide_eq_0_iff
% 0.32/0.48 thf(fact_185_divide__eq__0__iff,axiom,
% 0.32/0.48 ! [A: real,B: real] :
% 0.32/0.48 ( ( ( divide_divide_real @ A @ B )
% 0.32/0.48 = zero_zero_real )
% 0.32/0.48 = ( ( A = zero_zero_real )
% 0.32/0.48 | ( B = zero_zero_real ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % divide_eq_0_iff
% 0.32/0.48 thf(fact_186_divide__eq__0__iff,axiom,
% 0.32/0.48 ! [A: rat,B: rat] :
% 0.32/0.48 ( ( ( divide_divide_rat @ A @ B )
% 0.32/0.48 = zero_zero_rat )
% 0.32/0.48 = ( ( A = zero_zero_rat )
% 0.32/0.48 | ( B = zero_zero_rat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % divide_eq_0_iff
% 0.32/0.48 thf(fact_187_divide__cancel__left,axiom,
% 0.32/0.48 ! [C: complex,A: complex,B: complex] :
% 0.32/0.48 ( ( ( divide1717551699836669952omplex @ C @ A )
% 0.32/0.48 = ( divide1717551699836669952omplex @ C @ B ) )
% 0.32/0.48 = ( ( C = zero_zero_complex )
% 0.32/0.48 | ( A = B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % divide_cancel_left
% 0.32/0.48 thf(fact_188_divide__cancel__left,axiom,
% 0.32/0.48 ! [C: real,A: real,B: real] :
% 0.32/0.48 ( ( ( divide_divide_real @ C @ A )
% 0.32/0.48 = ( divide_divide_real @ C @ B ) )
% 0.32/0.48 = ( ( C = zero_zero_real )
% 0.32/0.48 | ( A = B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % divide_cancel_left
% 0.32/0.48 thf(fact_189_divide__cancel__left,axiom,
% 0.32/0.48 ! [C: rat,A: rat,B: rat] :
% 0.32/0.48 ( ( ( divide_divide_rat @ C @ A )
% 0.32/0.48 = ( divide_divide_rat @ C @ B ) )
% 0.32/0.48 = ( ( C = zero_zero_rat )
% 0.32/0.48 | ( A = B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % divide_cancel_left
% 0.32/0.48 thf(fact_190_divide__cancel__right,axiom,
% 0.32/0.48 ! [A: complex,C: complex,B: complex] :
% 0.32/0.48 ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.32/0.48 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.48 = ( ( C = zero_zero_complex )
% 0.32/0.48 | ( A = B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % divide_cancel_right
% 0.32/0.48 thf(fact_191_divide__cancel__right,axiom,
% 0.32/0.48 ! [A: real,C: real,B: real] :
% 0.32/0.48 ( ( ( divide_divide_real @ A @ C )
% 0.32/0.48 = ( divide_divide_real @ B @ C ) )
% 0.32/0.48 = ( ( C = zero_zero_real )
% 0.32/0.48 | ( A = B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % divide_cancel_right
% 0.32/0.48 thf(fact_192_divide__cancel__right,axiom,
% 0.32/0.48 ! [A: rat,C: rat,B: rat] :
% 0.32/0.48 ( ( ( divide_divide_rat @ A @ C )
% 0.32/0.48 = ( divide_divide_rat @ B @ C ) )
% 0.32/0.48 = ( ( C = zero_zero_rat )
% 0.32/0.48 | ( A = B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % divide_cancel_right
% 0.32/0.48 thf(fact_193_division__ring__divide__zero,axiom,
% 0.32/0.48 ! [A: complex] :
% 0.32/0.48 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.32/0.48 = zero_zero_complex ) ).
% 0.32/0.48
% 0.32/0.48 % division_ring_divide_zero
% 0.32/0.48 thf(fact_194_division__ring__divide__zero,axiom,
% 0.32/0.48 ! [A: real] :
% 0.32/0.48 ( ( divide_divide_real @ A @ zero_zero_real )
% 0.32/0.48 = zero_zero_real ) ).
% 0.32/0.48
% 0.32/0.48 % division_ring_divide_zero
% 0.32/0.48 thf(fact_195_division__ring__divide__zero,axiom,
% 0.32/0.48 ! [A: rat] :
% 0.32/0.48 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.32/0.48 = zero_zero_rat ) ).
% 0.32/0.48
% 0.32/0.48 % division_ring_divide_zero
% 0.32/0.48 thf(fact_196_even__set__bit__iff,axiom,
% 0.32/0.48 ! [M: nat,A: code_integer] :
% 0.32/0.48 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 0.32/0.48 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.48 & ( M != zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_set_bit_iff
% 0.32/0.48 thf(fact_197_even__set__bit__iff,axiom,
% 0.32/0.48 ! [M: nat,A: int] :
% 0.32/0.48 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.32/0.48 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.48 & ( M != zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_set_bit_iff
% 0.32/0.48 thf(fact_198_even__set__bit__iff,axiom,
% 0.32/0.48 ! [M: nat,A: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.32/0.48 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.48 & ( M != zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_set_bit_iff
% 0.32/0.48 thf(fact_199_even__flip__bit__iff,axiom,
% 0.32/0.48 ! [M: nat,A: code_integer] :
% 0.32/0.48 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 0.32/0.48 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.48 != ( M = zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_flip_bit_iff
% 0.32/0.48 thf(fact_200_even__flip__bit__iff,axiom,
% 0.32/0.48 ! [M: nat,A: int] :
% 0.32/0.48 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.32/0.48 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.48 != ( M = zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_flip_bit_iff
% 0.32/0.48 thf(fact_201_even__flip__bit__iff,axiom,
% 0.32/0.48 ! [M: nat,A: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.32/0.48 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.48 != ( M = zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_flip_bit_iff
% 0.32/0.48 thf(fact_202_even__unset__bit__iff,axiom,
% 0.32/0.48 ! [M: nat,A: code_integer] :
% 0.32/0.48 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 0.32/0.48 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.48 | ( M = zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_unset_bit_iff
% 0.32/0.48 thf(fact_203_even__unset__bit__iff,axiom,
% 0.32/0.48 ! [M: nat,A: int] :
% 0.32/0.48 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.32/0.48 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.48 | ( M = zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_unset_bit_iff
% 0.32/0.48 thf(fact_204_even__unset__bit__iff,axiom,
% 0.32/0.48 ! [M: nat,A: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.32/0.48 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.48 | ( M = zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_unset_bit_iff
% 0.32/0.48 thf(fact_205_int__dvd__int__iff,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.48 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % int_dvd_int_iff
% 0.32/0.48 thf(fact_206_odd__Suc__minus__one,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.48 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.32/0.48 = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % odd_Suc_minus_one
% 0.32/0.48 thf(fact_207_real__of__nat__div,axiom,
% 0.32/0.48 ! [D: nat,N: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ D @ N )
% 0.32/0.48 => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
% 0.32/0.48 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % real_of_nat_div
% 0.32/0.48 thf(fact_208_diff__self,axiom,
% 0.32/0.48 ! [A: real] :
% 0.32/0.48 ( ( minus_minus_real @ A @ A )
% 0.32/0.48 = zero_zero_real ) ).
% 0.32/0.48
% 0.32/0.48 % diff_self
% 0.32/0.48 thf(fact_209_diff__self,axiom,
% 0.32/0.48 ! [A: rat] :
% 0.32/0.48 ( ( minus_minus_rat @ A @ A )
% 0.32/0.48 = zero_zero_rat ) ).
% 0.32/0.48
% 0.32/0.48 % diff_self
% 0.32/0.48 thf(fact_210_diff__self,axiom,
% 0.32/0.48 ! [A: int] :
% 0.32/0.48 ( ( minus_minus_int @ A @ A )
% 0.32/0.48 = zero_zero_int ) ).
% 0.32/0.48
% 0.32/0.48 % diff_self
% 0.32/0.48 thf(fact_211_diff__0__right,axiom,
% 0.32/0.48 ! [A: real] :
% 0.32/0.48 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % diff_0_right
% 0.32/0.48 thf(fact_212_diff__0__right,axiom,
% 0.32/0.48 ! [A: rat] :
% 0.32/0.48 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % diff_0_right
% 0.32/0.48 thf(fact_213_diff__0__right,axiom,
% 0.32/0.48 ! [A: int] :
% 0.32/0.48 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % diff_0_right
% 0.32/0.48 thf(fact_214_zero__diff,axiom,
% 0.32/0.48 ! [A: nat] :
% 0.32/0.48 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.32/0.48 = zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % zero_diff
% 0.32/0.48 thf(fact_215_diff__zero,axiom,
% 0.32/0.48 ! [A: real] :
% 0.32/0.48 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % diff_zero
% 0.32/0.48 thf(fact_216_diff__zero,axiom,
% 0.32/0.48 ! [A: rat] :
% 0.32/0.48 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % diff_zero
% 0.32/0.48 thf(fact_217_diff__zero,axiom,
% 0.32/0.48 ! [A: nat] :
% 0.32/0.48 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % diff_zero
% 0.32/0.48 thf(fact_218_diff__zero,axiom,
% 0.32/0.48 ! [A: int] :
% 0.32/0.48 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % diff_zero
% 0.32/0.48 thf(fact_219_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.32/0.48 ! [A: real] :
% 0.32/0.48 ( ( minus_minus_real @ A @ A )
% 0.32/0.48 = zero_zero_real ) ).
% 0.32/0.48
% 0.32/0.48 % cancel_comm_monoid_add_class.diff_cancel
% 0.32/0.48 thf(fact_220_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.32/0.48 ! [A: rat] :
% 0.32/0.48 ( ( minus_minus_rat @ A @ A )
% 0.32/0.48 = zero_zero_rat ) ).
% 0.32/0.48
% 0.32/0.48 % cancel_comm_monoid_add_class.diff_cancel
% 0.32/0.48 thf(fact_221_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.32/0.48 ! [A: nat] :
% 0.32/0.48 ( ( minus_minus_nat @ A @ A )
% 0.32/0.48 = zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % cancel_comm_monoid_add_class.diff_cancel
% 0.32/0.48 thf(fact_222_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.32/0.48 ! [A: int] :
% 0.32/0.48 ( ( minus_minus_int @ A @ A )
% 0.32/0.48 = zero_zero_int ) ).
% 0.32/0.48
% 0.32/0.48 % cancel_comm_monoid_add_class.diff_cancel
% 0.32/0.48 thf(fact_223_Suc__diff__diff,axiom,
% 0.32/0.48 ! [M: nat,N: nat,K: nat] :
% 0.32/0.48 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.32/0.48 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 0.32/0.48
% 0.32/0.48 % Suc_diff_diff
% 0.32/0.48 thf(fact_224_diff__Suc__Suc,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.32/0.48 = ( minus_minus_nat @ M @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_Suc_Suc
% 0.32/0.48 thf(fact_225_diff__self__eq__0,axiom,
% 0.32/0.48 ! [M: nat] :
% 0.32/0.48 ( ( minus_minus_nat @ M @ M )
% 0.32/0.48 = zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % diff_self_eq_0
% 0.32/0.48 thf(fact_226_diff__0__eq__0,axiom,
% 0.32/0.48 ! [N: nat] :
% 0.32/0.48 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.32/0.48 = zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % diff_0_eq_0
% 0.32/0.48 thf(fact_227_div__diff,axiom,
% 0.32/0.48 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.48 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.32/0.48 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.48 => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 0.32/0.48 = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % div_diff
% 0.32/0.48 thf(fact_228_div__diff,axiom,
% 0.32/0.48 ! [C: int,A: int,B: int] :
% 0.32/0.48 ( ( dvd_dvd_int @ C @ A )
% 0.32/0.48 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.48 => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.32/0.48 = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % div_diff
% 0.32/0.48 thf(fact_229_int__int__eq,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( ( semiri1314217659103216013at_int @ M )
% 0.32/0.48 = ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % int_int_eq
% 0.32/0.48 thf(fact_230_int__if,axiom,
% 0.32/0.48 ! [P: $o,A: nat,B: nat] :
% 0.32/0.48 ( ( P
% 0.32/0.48 => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
% 0.32/0.48 = ( semiri1314217659103216013at_int @ A ) ) )
% 0.32/0.48 & ( ~ P
% 0.32/0.48 => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
% 0.32/0.48 = ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % int_if
% 0.32/0.48 thf(fact_231_nat__int__comparison_I1_J,axiom,
% 0.32/0.48 ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 0.32/0.48 = ( ^ [A2: nat,B2: nat] :
% 0.32/0.48 ( ( semiri1314217659103216013at_int @ A2 )
% 0.32/0.48 = ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % nat_int_comparison(1)
% 0.32/0.48 thf(fact_232_diff__eq__diff__eq,axiom,
% 0.32/0.48 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.48 ( ( ( minus_minus_real @ A @ B )
% 0.32/0.48 = ( minus_minus_real @ C @ D ) )
% 0.32/0.48 => ( ( A = B )
% 0.32/0.48 = ( C = D ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_eq_diff_eq
% 0.32/0.48 thf(fact_233_diff__eq__diff__eq,axiom,
% 0.32/0.48 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.48 ( ( ( minus_minus_rat @ A @ B )
% 0.32/0.48 = ( minus_minus_rat @ C @ D ) )
% 0.32/0.48 => ( ( A = B )
% 0.32/0.48 = ( C = D ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_eq_diff_eq
% 0.32/0.48 thf(fact_234_diff__eq__diff__eq,axiom,
% 0.32/0.48 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.48 ( ( ( minus_minus_int @ A @ B )
% 0.32/0.48 = ( minus_minus_int @ C @ D ) )
% 0.32/0.48 => ( ( A = B )
% 0.32/0.48 = ( C = D ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_eq_diff_eq
% 0.32/0.48 thf(fact_235_diff__right__commute,axiom,
% 0.32/0.48 ! [A: real,C: real,B: real] :
% 0.32/0.48 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 0.32/0.48 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_right_commute
% 0.32/0.48 thf(fact_236_diff__right__commute,axiom,
% 0.32/0.48 ! [A: rat,C: rat,B: rat] :
% 0.32/0.48 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
% 0.32/0.48 = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_right_commute
% 0.32/0.48 thf(fact_237_diff__right__commute,axiom,
% 0.32/0.48 ! [A: nat,C: nat,B: nat] :
% 0.32/0.48 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.32/0.48 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_right_commute
% 0.32/0.48 thf(fact_238_diff__right__commute,axiom,
% 0.32/0.48 ! [A: int,C: int,B: int] :
% 0.32/0.48 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 0.32/0.48 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_right_commute
% 0.32/0.48 thf(fact_239_diff__commute,axiom,
% 0.32/0.48 ! [I: nat,J: nat,K: nat] :
% 0.32/0.48 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.32/0.48 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_commute
% 0.32/0.48 thf(fact_240_eq__iff__diff__eq__0,axiom,
% 0.32/0.48 ( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
% 0.32/0.48 = ( ^ [A2: real,B2: real] :
% 0.32/0.48 ( ( minus_minus_real @ A2 @ B2 )
% 0.32/0.48 = zero_zero_real ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % eq_iff_diff_eq_0
% 0.32/0.48 thf(fact_241_eq__iff__diff__eq__0,axiom,
% 0.32/0.48 ( ( ^ [Y3: rat,Z: rat] : ( Y3 = Z ) )
% 0.32/0.48 = ( ^ [A2: rat,B2: rat] :
% 0.32/0.48 ( ( minus_minus_rat @ A2 @ B2 )
% 0.32/0.48 = zero_zero_rat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % eq_iff_diff_eq_0
% 0.32/0.48 thf(fact_242_eq__iff__diff__eq__0,axiom,
% 0.32/0.48 ( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 0.32/0.48 = ( ^ [A2: int,B2: int] :
% 0.32/0.48 ( ( minus_minus_int @ A2 @ B2 )
% 0.32/0.48 = zero_zero_int ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % eq_iff_diff_eq_0
% 0.32/0.48 thf(fact_243_diff__divide__distrib,axiom,
% 0.32/0.48 ! [A: complex,B: complex,C: complex] :
% 0.32/0.48 ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
% 0.32/0.48 = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_divide_distrib
% 0.32/0.48 thf(fact_244_diff__divide__distrib,axiom,
% 0.32/0.48 ! [A: real,B: real,C: real] :
% 0.32/0.48 ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.32/0.48 = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_divide_distrib
% 0.32/0.48 thf(fact_245_diff__divide__distrib,axiom,
% 0.32/0.48 ! [A: rat,B: rat,C: rat] :
% 0.32/0.48 ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.32/0.48 = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_divide_distrib
% 0.32/0.48 thf(fact_246_dvd__diff__commute,axiom,
% 0.32/0.48 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.48 ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
% 0.32/0.48 = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_diff_commute
% 0.32/0.48 thf(fact_247_dvd__diff__commute,axiom,
% 0.32/0.48 ! [A: int,C: int,B: int] :
% 0.32/0.48 ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.32/0.48 = ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_diff_commute
% 0.32/0.48 thf(fact_248_dvd__diff,axiom,
% 0.32/0.48 ! [X: code_integer,Y: code_integer,Z2: code_integer] :
% 0.32/0.48 ( ( dvd_dvd_Code_integer @ X @ Y )
% 0.32/0.48 => ( ( dvd_dvd_Code_integer @ X @ Z2 )
% 0.32/0.48 => ( dvd_dvd_Code_integer @ X @ ( minus_8373710615458151222nteger @ Y @ Z2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_diff
% 0.32/0.48 thf(fact_249_dvd__diff,axiom,
% 0.32/0.48 ! [X: real,Y: real,Z2: real] :
% 0.32/0.48 ( ( dvd_dvd_real @ X @ Y )
% 0.32/0.48 => ( ( dvd_dvd_real @ X @ Z2 )
% 0.32/0.48 => ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_diff
% 0.32/0.48 thf(fact_250_dvd__diff,axiom,
% 0.32/0.48 ! [X: rat,Y: rat,Z2: rat] :
% 0.32/0.48 ( ( dvd_dvd_rat @ X @ Y )
% 0.32/0.48 => ( ( dvd_dvd_rat @ X @ Z2 )
% 0.32/0.48 => ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y @ Z2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_diff
% 0.32/0.48 thf(fact_251_dvd__diff,axiom,
% 0.32/0.48 ! [X: int,Y: int,Z2: int] :
% 0.32/0.48 ( ( dvd_dvd_int @ X @ Y )
% 0.32/0.48 => ( ( dvd_dvd_int @ X @ Z2 )
% 0.32/0.48 => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_diff
% 0.32/0.48 thf(fact_252_zero__induct__lemma,axiom,
% 0.32/0.48 ! [P: nat > $o,K: nat,I: nat] :
% 0.32/0.48 ( ( P @ K )
% 0.32/0.48 => ( ! [N2: nat] :
% 0.32/0.48 ( ( P @ ( suc @ N2 ) )
% 0.32/0.48 => ( P @ N2 ) )
% 0.32/0.48 => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % zero_induct_lemma
% 0.32/0.48 thf(fact_253_diffs0__imp__equal,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( ( minus_minus_nat @ M @ N )
% 0.32/0.48 = zero_zero_nat )
% 0.32/0.48 => ( ( ( minus_minus_nat @ N @ M )
% 0.32/0.48 = zero_zero_nat )
% 0.32/0.48 => ( M = N ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % diffs0_imp_equal
% 0.32/0.48 thf(fact_254_minus__nat_Odiff__0,axiom,
% 0.32/0.48 ! [M: nat] :
% 0.32/0.48 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.32/0.48 = M ) ).
% 0.32/0.48
% 0.32/0.48 % minus_nat.diff_0
% 0.32/0.48 thf(fact_255_dvd__diff__nat,axiom,
% 0.32/0.48 ! [K: nat,M: nat,N: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ K @ M )
% 0.32/0.48 => ( ( dvd_dvd_nat @ K @ N )
% 0.32/0.48 => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_diff_nat
% 0.32/0.48 thf(fact_256_int__ops_I1_J,axiom,
% 0.32/0.48 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.32/0.48 = zero_zero_int ) ).
% 0.32/0.48
% 0.32/0.48 % int_ops(1)
% 0.32/0.48 thf(fact_257_zero__reorient,axiom,
% 0.32/0.48 ! [X: literal] :
% 0.32/0.48 ( ( zero_zero_literal = X )
% 0.32/0.48 = ( X = zero_zero_literal ) ) ).
% 0.32/0.48
% 0.32/0.48 % zero_reorient
% 0.32/0.48 thf(fact_258_zero__reorient,axiom,
% 0.32/0.48 ! [X: real] :
% 0.32/0.48 ( ( zero_zero_real = X )
% 0.32/0.48 = ( X = zero_zero_real ) ) ).
% 0.32/0.48
% 0.32/0.48 % zero_reorient
% 0.32/0.48 thf(fact_259_zero__reorient,axiom,
% 0.32/0.48 ! [X: rat] :
% 0.32/0.48 ( ( zero_zero_rat = X )
% 0.32/0.48 = ( X = zero_zero_rat ) ) ).
% 0.32/0.48
% 0.32/0.48 % zero_reorient
% 0.32/0.48 thf(fact_260_zero__reorient,axiom,
% 0.32/0.48 ! [X: nat] :
% 0.32/0.48 ( ( zero_zero_nat = X )
% 0.32/0.48 = ( X = zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % zero_reorient
% 0.32/0.48 thf(fact_261_zero__reorient,axiom,
% 0.32/0.48 ! [X: int] :
% 0.32/0.48 ( ( zero_zero_int = X )
% 0.32/0.48 = ( X = zero_zero_int ) ) ).
% 0.32/0.48
% 0.32/0.48 % zero_reorient
% 0.32/0.48 thf(fact_262_dvd__field__iff,axiom,
% 0.32/0.48 ( dvd_dvd_real
% 0.32/0.48 = ( ^ [A2: real,B2: real] :
% 0.32/0.48 ( ( A2 = zero_zero_real )
% 0.32/0.48 => ( B2 = zero_zero_real ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_field_iff
% 0.32/0.48 thf(fact_263_dvd__field__iff,axiom,
% 0.32/0.48 ( dvd_dvd_rat
% 0.32/0.48 = ( ^ [A2: rat,B2: rat] :
% 0.32/0.48 ( ( A2 = zero_zero_rat )
% 0.32/0.48 => ( B2 = zero_zero_rat ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_field_iff
% 0.32/0.48 thf(fact_264_unset__bit__0,axiom,
% 0.32/0.48 ! [A: int] :
% 0.32/0.48 ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 0.32/0.48 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % unset_bit_0
% 0.32/0.48 thf(fact_265_unset__bit__0,axiom,
% 0.32/0.48 ! [A: nat] :
% 0.32/0.48 ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 0.32/0.48 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % unset_bit_0
% 0.32/0.48 thf(fact_266_gcd__nat_Oextremum,axiom,
% 0.32/0.48 ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.extremum
% 0.32/0.48 thf(fact_267_gcd__nat_Oextremum__strict,axiom,
% 0.32/0.48 ! [A: nat] :
% 0.32/0.48 ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.32/0.48 & ( zero_zero_nat != A ) ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.extremum_strict
% 0.32/0.48 thf(fact_268_gcd__nat_Oextremum__unique,axiom,
% 0.32/0.48 ! [A: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.32/0.48 = ( A = zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.extremum_unique
% 0.32/0.48 thf(fact_269_gcd__nat_Onot__eq__extremum,axiom,
% 0.32/0.48 ! [A: nat] :
% 0.32/0.48 ( ( A != zero_zero_nat )
% 0.32/0.48 = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
% 0.32/0.48 & ( A != zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.not_eq_extremum
% 0.32/0.48 thf(fact_270_gcd__nat_Oextremum__uniqueI,axiom,
% 0.32/0.48 ! [A: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.32/0.48 => ( A = zero_zero_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.extremum_uniqueI
% 0.32/0.48 thf(fact_271_exists__least__lemma,axiom,
% 0.32/0.48 ! [P: nat > $o] :
% 0.32/0.48 ( ~ ( P @ zero_zero_nat )
% 0.32/0.48 => ( ? [X_1: nat] : ( P @ X_1 )
% 0.32/0.48 => ? [N2: nat] :
% 0.32/0.48 ( ~ ( P @ N2 )
% 0.32/0.48 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % exists_least_lemma
% 0.32/0.48 thf(fact_272_one__div__two__eq__zero,axiom,
% 0.32/0.48 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.48 = zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % one_div_two_eq_zero
% 0.32/0.48 thf(fact_273_one__div__two__eq__zero,axiom,
% 0.32/0.48 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.48 = zero_zero_int ) ).
% 0.32/0.48
% 0.32/0.48 % one_div_two_eq_zero
% 0.32/0.48 thf(fact_274_bits__1__div__2,axiom,
% 0.32/0.48 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.48 = zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % bits_1_div_2
% 0.32/0.48 thf(fact_275_bits__1__div__2,axiom,
% 0.32/0.48 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.48 = zero_zero_int ) ).
% 0.32/0.48
% 0.32/0.48 % bits_1_div_2
% 0.32/0.48 thf(fact_276_even__push__bit__iff,axiom,
% 0.32/0.48 ! [N: nat,A: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 0.32/0.48 = ( ( N != zero_zero_nat )
% 0.32/0.48 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_push_bit_iff
% 0.32/0.48 thf(fact_277_even__push__bit__iff,axiom,
% 0.32/0.48 ! [N: nat,A: int] :
% 0.32/0.48 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se545348938243370406it_int @ N @ A ) )
% 0.32/0.48 = ( ( N != zero_zero_nat )
% 0.32/0.48 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_push_bit_iff
% 0.32/0.48 thf(fact_278_even__push__bit__iff,axiom,
% 0.32/0.48 ! [N: nat,A: code_integer] :
% 0.32/0.48 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se7788150548672797655nteger @ N @ A ) )
% 0.32/0.48 = ( ( N != zero_zero_nat )
% 0.32/0.48 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_push_bit_iff
% 0.32/0.48 thf(fact_279_even__push__bit__iff,axiom,
% 0.32/0.48 ! [N: nat,A: code_natural] :
% 0.32/0.48 ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se6611745700429515170atural @ N @ A ) )
% 0.32/0.48 = ( ( N != zero_zero_nat )
% 0.32/0.48 | ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % even_push_bit_iff
% 0.32/0.48 thf(fact_280_semiring__norm_I13_J,axiom,
% 0.32/0.48 ! [M: num,N: num] :
% 0.32/0.48 ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.32/0.48 = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_norm(13)
% 0.32/0.48 thf(fact_281_semiring__norm_I12_J,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( ( times_times_num @ one @ N )
% 0.32/0.48 = N ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_norm(12)
% 0.32/0.48 thf(fact_282_semiring__norm_I11_J,axiom,
% 0.32/0.48 ! [M: num] :
% 0.32/0.48 ( ( times_times_num @ M @ one )
% 0.32/0.48 = M ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_norm(11)
% 0.32/0.48 thf(fact_283_mult__is__0,axiom,
% 0.32/0.48 ! [M: nat,N: nat] :
% 0.32/0.48 ( ( ( times_times_nat @ M @ N )
% 0.32/0.48 = zero_zero_nat )
% 0.32/0.48 = ( ( M = zero_zero_nat )
% 0.32/0.48 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult_is_0
% 0.32/0.48 thf(fact_284_mult__0__right,axiom,
% 0.32/0.48 ! [M: nat] :
% 0.32/0.48 ( ( times_times_nat @ M @ zero_zero_nat )
% 0.32/0.48 = zero_zero_nat ) ).
% 0.32/0.48
% 0.32/0.48 % mult_0_right
% 0.32/0.48 thf(fact_285_mult__cancel1,axiom,
% 0.32/0.48 ! [K: nat,M: nat,N: nat] :
% 0.32/0.48 ( ( ( times_times_nat @ K @ M )
% 0.32/0.48 = ( times_times_nat @ K @ N ) )
% 0.32/0.48 = ( ( M = N )
% 0.32/0.48 | ( K = zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult_cancel1
% 0.32/0.48 thf(fact_286_mult__cancel2,axiom,
% 0.32/0.48 ! [M: nat,K: nat,N: nat] :
% 0.32/0.48 ( ( ( times_times_nat @ M @ K )
% 0.32/0.48 = ( times_times_nat @ N @ K ) )
% 0.32/0.48 = ( ( M = N )
% 0.32/0.48 | ( K = zero_zero_nat ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult_cancel2
% 0.32/0.48 thf(fact_287_nat__dvd__1__iff__1,axiom,
% 0.32/0.48 ! [M: nat] :
% 0.32/0.48 ( ( dvd_dvd_nat @ M @ one_one_nat )
% 0.32/0.48 = ( M = one_one_nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % nat_dvd_1_iff_1
% 0.32/0.48 thf(fact_288_mult__cancel__right,axiom,
% 0.32/0.48 ! [A: real,C: real,B: real] :
% 0.32/0.48 ( ( ( times_times_real @ A @ C )
% 0.32/0.48 = ( times_times_real @ B @ C ) )
% 0.32/0.48 = ( ( C = zero_zero_real )
% 0.32/0.48 | ( A = B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult_cancel_right
% 0.32/0.48 thf(fact_289_mult__cancel__right,axiom,
% 0.32/0.48 ! [A: rat,C: rat,B: rat] :
% 0.32/0.48 ( ( ( times_times_rat @ A @ C )
% 0.32/0.48 = ( times_times_rat @ B @ C ) )
% 0.32/0.48 = ( ( C = zero_zero_rat )
% 0.32/0.48 | ( A = B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult_cancel_right
% 0.32/0.48 thf(fact_290_mult__cancel__right,axiom,
% 0.32/0.48 ! [A: nat,C: nat,B: nat] :
% 0.32/0.48 ( ( ( times_times_nat @ A @ C )
% 0.32/0.48 = ( times_times_nat @ B @ C ) )
% 0.32/0.48 = ( ( C = zero_zero_nat )
% 0.32/0.48 | ( A = B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult_cancel_right
% 0.32/0.48 thf(fact_291_mult__cancel__right,axiom,
% 0.32/0.48 ! [A: int,C: int,B: int] :
% 0.32/0.48 ( ( ( times_times_int @ A @ C )
% 0.32/0.48 = ( times_times_int @ B @ C ) )
% 0.32/0.48 = ( ( C = zero_zero_int )
% 0.32/0.48 | ( A = B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult_cancel_right
% 0.32/0.48 thf(fact_292_mult__cancel__left,axiom,
% 0.32/0.48 ! [C: real,A: real,B: real] :
% 0.32/0.48 ( ( ( times_times_real @ C @ A )
% 0.32/0.48 = ( times_times_real @ C @ B ) )
% 0.32/0.48 = ( ( C = zero_zero_real )
% 0.32/0.48 | ( A = B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult_cancel_left
% 0.32/0.48 thf(fact_293_mult__cancel__left,axiom,
% 0.32/0.48 ! [C: rat,A: rat,B: rat] :
% 0.32/0.48 ( ( ( times_times_rat @ C @ A )
% 0.32/0.48 = ( times_times_rat @ C @ B ) )
% 0.32/0.48 = ( ( C = zero_zero_rat )
% 0.32/0.48 | ( A = B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult_cancel_left
% 0.32/0.48 thf(fact_294_mult__cancel__left,axiom,
% 0.32/0.48 ! [C: nat,A: nat,B: nat] :
% 0.32/0.48 ( ( ( times_times_nat @ C @ A )
% 0.32/0.48 = ( times_times_nat @ C @ B ) )
% 0.32/0.48 = ( ( C = zero_zero_nat )
% 0.32/0.48 | ( A = B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult_cancel_left
% 0.32/0.48 thf(fact_295_mult__cancel__left,axiom,
% 0.32/0.48 ! [C: int,A: int,B: int] :
% 0.32/0.48 ( ( ( times_times_int @ C @ A )
% 0.32/0.49 = ( times_times_int @ C @ B ) )
% 0.32/0.49 = ( ( C = zero_zero_int )
% 0.32/0.49 | ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_left
% 0.32/0.49 thf(fact_296_mult__eq__0__iff,axiom,
% 0.32/0.49 ! [A: real,B: real] :
% 0.32/0.49 ( ( ( times_times_real @ A @ B )
% 0.32/0.49 = zero_zero_real )
% 0.32/0.49 = ( ( A = zero_zero_real )
% 0.32/0.49 | ( B = zero_zero_real ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_eq_0_iff
% 0.32/0.49 thf(fact_297_mult__eq__0__iff,axiom,
% 0.32/0.49 ! [A: rat,B: rat] :
% 0.32/0.49 ( ( ( times_times_rat @ A @ B )
% 0.32/0.49 = zero_zero_rat )
% 0.32/0.49 = ( ( A = zero_zero_rat )
% 0.32/0.49 | ( B = zero_zero_rat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_eq_0_iff
% 0.32/0.49 thf(fact_298_mult__eq__0__iff,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( ( times_times_nat @ A @ B )
% 0.32/0.49 = zero_zero_nat )
% 0.32/0.49 = ( ( A = zero_zero_nat )
% 0.32/0.49 | ( B = zero_zero_nat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_eq_0_iff
% 0.32/0.49 thf(fact_299_mult__eq__0__iff,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( ( times_times_int @ A @ B )
% 0.32/0.49 = zero_zero_int )
% 0.32/0.49 = ( ( A = zero_zero_int )
% 0.32/0.49 | ( B = zero_zero_int ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_eq_0_iff
% 0.32/0.49 thf(fact_300_mult__zero__right,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( times_times_real @ A @ zero_zero_real )
% 0.32/0.49 = zero_zero_real ) ).
% 0.32/0.49
% 0.32/0.49 % mult_zero_right
% 0.32/0.49 thf(fact_301_mult__zero__right,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ( ( times_times_rat @ A @ zero_zero_rat )
% 0.32/0.49 = zero_zero_rat ) ).
% 0.32/0.49
% 0.32/0.49 % mult_zero_right
% 0.32/0.49 thf(fact_302_mult__zero__right,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( times_times_nat @ A @ zero_zero_nat )
% 0.32/0.49 = zero_zero_nat ) ).
% 0.32/0.49
% 0.32/0.49 % mult_zero_right
% 0.32/0.49 thf(fact_303_mult__zero__right,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( times_times_int @ A @ zero_zero_int )
% 0.32/0.49 = zero_zero_int ) ).
% 0.32/0.49
% 0.32/0.49 % mult_zero_right
% 0.32/0.49 thf(fact_304_mult__zero__left,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( times_times_real @ zero_zero_real @ A )
% 0.32/0.49 = zero_zero_real ) ).
% 0.32/0.49
% 0.32/0.49 % mult_zero_left
% 0.32/0.49 thf(fact_305_mult__zero__left,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ( ( times_times_rat @ zero_zero_rat @ A )
% 0.32/0.49 = zero_zero_rat ) ).
% 0.32/0.49
% 0.32/0.49 % mult_zero_left
% 0.32/0.49 thf(fact_306_mult__zero__left,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( times_times_nat @ zero_zero_nat @ A )
% 0.32/0.49 = zero_zero_nat ) ).
% 0.32/0.49
% 0.32/0.49 % mult_zero_left
% 0.32/0.49 thf(fact_307_mult__zero__left,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( times_times_int @ zero_zero_int @ A )
% 0.32/0.49 = zero_zero_int ) ).
% 0.32/0.49
% 0.32/0.49 % mult_zero_left
% 0.32/0.49 thf(fact_308_mult__numeral__left__semiring__numeral,axiom,
% 0.32/0.49 ! [V: num,W: num,Z2: complex] :
% 0.32/0.49 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z2 ) )
% 0.32/0.49 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_left_semiring_numeral
% 0.32/0.49 thf(fact_309_mult__numeral__left__semiring__numeral,axiom,
% 0.32/0.49 ! [V: num,W: num,Z2: real] :
% 0.32/0.49 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
% 0.32/0.49 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_left_semiring_numeral
% 0.32/0.49 thf(fact_310_mult__numeral__left__semiring__numeral,axiom,
% 0.32/0.49 ! [V: num,W: num,Z2: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z2 ) )
% 0.32/0.49 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_left_semiring_numeral
% 0.32/0.49 thf(fact_311_mult__numeral__left__semiring__numeral,axiom,
% 0.32/0.49 ! [V: num,W: num,Z2: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
% 0.32/0.49 = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_left_semiring_numeral
% 0.32/0.49 thf(fact_312_mult__numeral__left__semiring__numeral,axiom,
% 0.32/0.49 ! [V: num,W: num,Z2: int] :
% 0.32/0.49 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
% 0.32/0.49 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_left_semiring_numeral
% 0.32/0.49 thf(fact_313_numeral__times__numeral,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 0.32/0.49 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_times_numeral
% 0.32/0.49 thf(fact_314_numeral__times__numeral,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.32/0.49 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_times_numeral
% 0.32/0.49 thf(fact_315_numeral__times__numeral,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.32/0.49 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_times_numeral
% 0.32/0.49 thf(fact_316_numeral__times__numeral,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.49 = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_times_numeral
% 0.32/0.49 thf(fact_317_numeral__times__numeral,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.49 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_times_numeral
% 0.32/0.49 thf(fact_318_mult_Oright__neutral,axiom,
% 0.32/0.49 ! [A: complex] :
% 0.32/0.49 ( ( times_times_complex @ A @ one_one_complex )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult.right_neutral
% 0.32/0.49 thf(fact_319_mult_Oright__neutral,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( times_times_real @ A @ one_one_real )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult.right_neutral
% 0.32/0.49 thf(fact_320_mult_Oright__neutral,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ( ( times_times_rat @ A @ one_one_rat )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult.right_neutral
% 0.32/0.49 thf(fact_321_mult_Oright__neutral,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( times_times_nat @ A @ one_one_nat )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult.right_neutral
% 0.32/0.49 thf(fact_322_mult_Oright__neutral,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( times_times_int @ A @ one_one_int )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult.right_neutral
% 0.32/0.49 thf(fact_323_mult__1,axiom,
% 0.32/0.49 ! [A: complex] :
% 0.32/0.49 ( ( times_times_complex @ one_one_complex @ A )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult_1
% 0.32/0.49 thf(fact_324_mult__1,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( times_times_real @ one_one_real @ A )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult_1
% 0.32/0.49 thf(fact_325_mult__1,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ( ( times_times_rat @ one_one_rat @ A )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult_1
% 0.32/0.49 thf(fact_326_mult__1,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( times_times_nat @ one_one_nat @ A )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult_1
% 0.32/0.49 thf(fact_327_mult__1,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( times_times_int @ one_one_int @ A )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult_1
% 0.32/0.49 thf(fact_328_num__double,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( times_times_num @ ( bit0 @ one ) @ N )
% 0.32/0.49 = ( bit0 @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % num_double
% 0.32/0.49 thf(fact_329_times__divide__eq__left,axiom,
% 0.32/0.49 ! [B: complex,C: complex,A: complex] :
% 0.32/0.49 ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
% 0.32/0.49 = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % times_divide_eq_left
% 0.32/0.49 thf(fact_330_times__divide__eq__left,axiom,
% 0.32/0.49 ! [B: real,C: real,A: real] :
% 0.32/0.49 ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.49 = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % times_divide_eq_left
% 0.32/0.49 thf(fact_331_times__divide__eq__left,axiom,
% 0.32/0.49 ! [B: rat,C: rat,A: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.49 = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % times_divide_eq_left
% 0.32/0.49 thf(fact_332_divide__divide__eq__left,axiom,
% 0.32/0.49 ! [A: complex,B: complex,C: complex] :
% 0.32/0.49 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 0.32/0.49 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_divide_eq_left
% 0.32/0.49 thf(fact_333_divide__divide__eq__left,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 0.32/0.49 = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_divide_eq_left
% 0.32/0.49 thf(fact_334_divide__divide__eq__left,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 0.32/0.49 = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_divide_eq_left
% 0.32/0.49 thf(fact_335_divide__divide__eq__right,axiom,
% 0.32/0.49 ! [A: complex,B: complex,C: complex] :
% 0.32/0.49 ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.49 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_divide_eq_right
% 0.32/0.49 thf(fact_336_divide__divide__eq__right,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.49 = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_divide_eq_right
% 0.32/0.49 thf(fact_337_divide__divide__eq__right,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.49 = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_divide_eq_right
% 0.32/0.49 thf(fact_338_times__divide__eq__right,axiom,
% 0.32/0.49 ! [A: complex,B: complex,C: complex] :
% 0.32/0.49 ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.49 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % times_divide_eq_right
% 0.32/0.49 thf(fact_339_times__divide__eq__right,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.49 = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % times_divide_eq_right
% 0.32/0.49 thf(fact_340_times__divide__eq__right,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.49 = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % times_divide_eq_right
% 0.32/0.49 thf(fact_341_div__by__1,axiom,
% 0.32/0.49 ! [A: complex] :
% 0.32/0.49 ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % div_by_1
% 0.32/0.49 thf(fact_342_div__by__1,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( divide_divide_real @ A @ one_one_real )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % div_by_1
% 0.32/0.49 thf(fact_343_div__by__1,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ( ( divide_divide_rat @ A @ one_one_rat )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % div_by_1
% 0.32/0.49 thf(fact_344_div__by__1,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( divide_divide_nat @ A @ one_one_nat )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % div_by_1
% 0.32/0.49 thf(fact_345_div__by__1,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( divide_divide_int @ A @ one_one_int )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % div_by_1
% 0.32/0.49 thf(fact_346_bits__div__by__1,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( divide_divide_nat @ A @ one_one_nat )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % bits_div_by_1
% 0.32/0.49 thf(fact_347_bits__div__by__1,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( divide_divide_int @ A @ one_one_int )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % bits_div_by_1
% 0.32/0.49 thf(fact_348_of__nat__mult,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
% 0.32/0.49 = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_mult
% 0.32/0.49 thf(fact_349_of__nat__mult,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 0.32/0.49 = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_mult
% 0.32/0.49 thf(fact_350_of__nat__mult,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 0.32/0.49 = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_mult
% 0.32/0.49 thf(fact_351_of__nat__mult,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 0.32/0.49 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_mult
% 0.32/0.49 thf(fact_352_of__nat__mult,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 0.32/0.49 = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_mult
% 0.32/0.49 thf(fact_353_of__nat__1,axiom,
% 0.32/0.49 ( ( semiri8010041392384452111omplex @ one_one_nat )
% 0.32/0.49 = one_one_complex ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_1
% 0.32/0.49 thf(fact_354_of__nat__1,axiom,
% 0.32/0.49 ( ( semiri5074537144036343181t_real @ one_one_nat )
% 0.32/0.49 = one_one_real ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_1
% 0.32/0.49 thf(fact_355_of__nat__1,axiom,
% 0.32/0.49 ( ( semiri681578069525770553at_rat @ one_one_nat )
% 0.32/0.49 = one_one_rat ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_1
% 0.32/0.49 thf(fact_356_of__nat__1,axiom,
% 0.32/0.49 ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 0.32/0.49 = one_one_nat ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_1
% 0.32/0.49 thf(fact_357_of__nat__1,axiom,
% 0.32/0.49 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.32/0.49 = one_one_int ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_1
% 0.32/0.49 thf(fact_358_of__nat__1__eq__iff,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( one_one_complex
% 0.32/0.49 = ( semiri8010041392384452111omplex @ N ) )
% 0.32/0.49 = ( N = one_one_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_1_eq_iff
% 0.32/0.49 thf(fact_359_of__nat__1__eq__iff,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( one_one_real
% 0.32/0.49 = ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.49 = ( N = one_one_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_1_eq_iff
% 0.32/0.49 thf(fact_360_of__nat__1__eq__iff,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( one_one_rat
% 0.32/0.49 = ( semiri681578069525770553at_rat @ N ) )
% 0.32/0.49 = ( N = one_one_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_1_eq_iff
% 0.32/0.49 thf(fact_361_of__nat__1__eq__iff,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( one_one_nat
% 0.32/0.49 = ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.49 = ( N = one_one_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_1_eq_iff
% 0.32/0.49 thf(fact_362_of__nat__1__eq__iff,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( one_one_int
% 0.32/0.49 = ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.49 = ( N = one_one_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_1_eq_iff
% 0.32/0.49 thf(fact_363_of__nat__eq__1__iff,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( ( semiri8010041392384452111omplex @ N )
% 0.32/0.49 = one_one_complex )
% 0.32/0.49 = ( N = one_one_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_eq_1_iff
% 0.32/0.49 thf(fact_364_of__nat__eq__1__iff,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( ( semiri5074537144036343181t_real @ N )
% 0.32/0.49 = one_one_real )
% 0.32/0.49 = ( N = one_one_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_eq_1_iff
% 0.32/0.49 thf(fact_365_of__nat__eq__1__iff,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( ( semiri681578069525770553at_rat @ N )
% 0.32/0.49 = one_one_rat )
% 0.32/0.49 = ( N = one_one_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_eq_1_iff
% 0.32/0.49 thf(fact_366_of__nat__eq__1__iff,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( ( semiri1316708129612266289at_nat @ N )
% 0.32/0.49 = one_one_nat )
% 0.32/0.49 = ( N = one_one_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_eq_1_iff
% 0.32/0.49 thf(fact_367_of__nat__eq__1__iff,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( ( semiri1314217659103216013at_int @ N )
% 0.32/0.49 = one_one_int )
% 0.32/0.49 = ( N = one_one_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_eq_1_iff
% 0.32/0.49 thf(fact_368_one__eq__mult__iff,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( ( suc @ zero_zero_nat )
% 0.32/0.49 = ( times_times_nat @ M @ N ) )
% 0.32/0.49 = ( ( M
% 0.32/0.49 = ( suc @ zero_zero_nat ) )
% 0.32/0.49 & ( N
% 0.32/0.49 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_eq_mult_iff
% 0.32/0.49 thf(fact_369_mult__eq__1__iff,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( ( times_times_nat @ M @ N )
% 0.32/0.49 = ( suc @ zero_zero_nat ) )
% 0.32/0.49 = ( ( M
% 0.32/0.49 = ( suc @ zero_zero_nat ) )
% 0.32/0.49 & ( N
% 0.32/0.49 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_eq_1_iff
% 0.32/0.49 thf(fact_370_diff__Suc__1,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 0.32/0.49 = N ) ).
% 0.32/0.49
% 0.32/0.49 % diff_Suc_1
% 0.32/0.49 thf(fact_371_nat__mult__dvd__cancel__disj,axiom,
% 0.32/0.49 ! [K: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.49 = ( ( K = zero_zero_nat )
% 0.32/0.49 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_mult_dvd_cancel_disj
% 0.32/0.49 thf(fact_372_nat__mult__div__cancel__disj,axiom,
% 0.32/0.49 ! [K: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ( K = zero_zero_nat )
% 0.32/0.49 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.49 = zero_zero_nat ) )
% 0.32/0.49 & ( ( K != zero_zero_nat )
% 0.32/0.49 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.49 = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_mult_div_cancel_disj
% 0.32/0.49 thf(fact_373_push__bit__eq__0__iff,axiom,
% 0.32/0.49 ! [N: nat,A: nat] :
% 0.32/0.49 ( ( ( bit_se547839408752420682it_nat @ N @ A )
% 0.32/0.49 = zero_zero_nat )
% 0.32/0.49 = ( A = zero_zero_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_eq_0_iff
% 0.32/0.49 thf(fact_374_push__bit__eq__0__iff,axiom,
% 0.32/0.49 ! [N: nat,A: int] :
% 0.32/0.49 ( ( ( bit_se545348938243370406it_int @ N @ A )
% 0.32/0.49 = zero_zero_int )
% 0.32/0.49 = ( A = zero_zero_int ) ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_eq_0_iff
% 0.32/0.49 thf(fact_375_push__bit__eq__0__iff,axiom,
% 0.32/0.49 ! [N: nat,A: code_integer] :
% 0.32/0.49 ( ( ( bit_se7788150548672797655nteger @ N @ A )
% 0.32/0.49 = zero_z3403309356797280102nteger )
% 0.32/0.49 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_eq_0_iff
% 0.32/0.49 thf(fact_376_push__bit__eq__0__iff,axiom,
% 0.32/0.49 ! [N: nat,A: code_natural] :
% 0.32/0.49 ( ( ( bit_se6611745700429515170atural @ N @ A )
% 0.32/0.49 = zero_z2226904508553997617atural )
% 0.32/0.49 = ( A = zero_z2226904508553997617atural ) ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_eq_0_iff
% 0.32/0.49 thf(fact_377_push__bit__of__0,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( bit_se547839408752420682it_nat @ N @ zero_zero_nat )
% 0.32/0.49 = zero_zero_nat ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_of_0
% 0.32/0.49 thf(fact_378_push__bit__of__0,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( bit_se545348938243370406it_int @ N @ zero_zero_int )
% 0.32/0.49 = zero_zero_int ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_of_0
% 0.32/0.49 thf(fact_379_push__bit__of__0,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( bit_se7788150548672797655nteger @ N @ zero_z3403309356797280102nteger )
% 0.32/0.49 = zero_z3403309356797280102nteger ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_of_0
% 0.32/0.49 thf(fact_380_push__bit__of__0,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( bit_se6611745700429515170atural @ N @ zero_z2226904508553997617atural )
% 0.32/0.49 = zero_z2226904508553997617atural ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_of_0
% 0.32/0.49 thf(fact_381_mult__cancel__right2,axiom,
% 0.32/0.49 ! [A: complex,C: complex] :
% 0.32/0.49 ( ( ( times_times_complex @ A @ C )
% 0.32/0.49 = C )
% 0.32/0.49 = ( ( C = zero_zero_complex )
% 0.32/0.49 | ( A = one_one_complex ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_right2
% 0.32/0.49 thf(fact_382_mult__cancel__right2,axiom,
% 0.32/0.49 ! [A: real,C: real] :
% 0.32/0.49 ( ( ( times_times_real @ A @ C )
% 0.32/0.49 = C )
% 0.32/0.49 = ( ( C = zero_zero_real )
% 0.32/0.49 | ( A = one_one_real ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_right2
% 0.32/0.49 thf(fact_383_mult__cancel__right2,axiom,
% 0.32/0.49 ! [A: rat,C: rat] :
% 0.32/0.49 ( ( ( times_times_rat @ A @ C )
% 0.32/0.49 = C )
% 0.32/0.49 = ( ( C = zero_zero_rat )
% 0.32/0.49 | ( A = one_one_rat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_right2
% 0.32/0.49 thf(fact_384_mult__cancel__right2,axiom,
% 0.32/0.49 ! [A: int,C: int] :
% 0.32/0.49 ( ( ( times_times_int @ A @ C )
% 0.32/0.49 = C )
% 0.32/0.49 = ( ( C = zero_zero_int )
% 0.32/0.49 | ( A = one_one_int ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_right2
% 0.32/0.49 thf(fact_385_mult__cancel__right1,axiom,
% 0.32/0.49 ! [C: complex,B: complex] :
% 0.32/0.49 ( ( C
% 0.32/0.49 = ( times_times_complex @ B @ C ) )
% 0.32/0.49 = ( ( C = zero_zero_complex )
% 0.32/0.49 | ( B = one_one_complex ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_right1
% 0.32/0.49 thf(fact_386_mult__cancel__right1,axiom,
% 0.32/0.49 ! [C: real,B: real] :
% 0.32/0.49 ( ( C
% 0.32/0.49 = ( times_times_real @ B @ C ) )
% 0.32/0.49 = ( ( C = zero_zero_real )
% 0.32/0.49 | ( B = one_one_real ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_right1
% 0.32/0.49 thf(fact_387_mult__cancel__right1,axiom,
% 0.32/0.49 ! [C: rat,B: rat] :
% 0.32/0.49 ( ( C
% 0.32/0.49 = ( times_times_rat @ B @ C ) )
% 0.32/0.49 = ( ( C = zero_zero_rat )
% 0.32/0.49 | ( B = one_one_rat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_right1
% 0.32/0.49 thf(fact_388_mult__cancel__right1,axiom,
% 0.32/0.49 ! [C: int,B: int] :
% 0.32/0.49 ( ( C
% 0.32/0.49 = ( times_times_int @ B @ C ) )
% 0.32/0.49 = ( ( C = zero_zero_int )
% 0.32/0.49 | ( B = one_one_int ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_right1
% 0.32/0.49 thf(fact_389_mult__cancel__left2,axiom,
% 0.32/0.49 ! [C: complex,A: complex] :
% 0.32/0.49 ( ( ( times_times_complex @ C @ A )
% 0.32/0.49 = C )
% 0.32/0.49 = ( ( C = zero_zero_complex )
% 0.32/0.49 | ( A = one_one_complex ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_left2
% 0.32/0.49 thf(fact_390_mult__cancel__left2,axiom,
% 0.32/0.49 ! [C: real,A: real] :
% 0.32/0.49 ( ( ( times_times_real @ C @ A )
% 0.32/0.49 = C )
% 0.32/0.49 = ( ( C = zero_zero_real )
% 0.32/0.49 | ( A = one_one_real ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_left2
% 0.32/0.49 thf(fact_391_mult__cancel__left2,axiom,
% 0.32/0.49 ! [C: rat,A: rat] :
% 0.32/0.49 ( ( ( times_times_rat @ C @ A )
% 0.32/0.49 = C )
% 0.32/0.49 = ( ( C = zero_zero_rat )
% 0.32/0.49 | ( A = one_one_rat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_left2
% 0.32/0.49 thf(fact_392_mult__cancel__left2,axiom,
% 0.32/0.49 ! [C: int,A: int] :
% 0.32/0.49 ( ( ( times_times_int @ C @ A )
% 0.32/0.49 = C )
% 0.32/0.49 = ( ( C = zero_zero_int )
% 0.32/0.49 | ( A = one_one_int ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_left2
% 0.32/0.49 thf(fact_393_mult__cancel__left1,axiom,
% 0.32/0.49 ! [C: complex,B: complex] :
% 0.32/0.49 ( ( C
% 0.32/0.49 = ( times_times_complex @ C @ B ) )
% 0.32/0.49 = ( ( C = zero_zero_complex )
% 0.32/0.49 | ( B = one_one_complex ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_left1
% 0.32/0.49 thf(fact_394_mult__cancel__left1,axiom,
% 0.32/0.49 ! [C: real,B: real] :
% 0.32/0.49 ( ( C
% 0.32/0.49 = ( times_times_real @ C @ B ) )
% 0.32/0.49 = ( ( C = zero_zero_real )
% 0.32/0.49 | ( B = one_one_real ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_left1
% 0.32/0.49 thf(fact_395_mult__cancel__left1,axiom,
% 0.32/0.49 ! [C: rat,B: rat] :
% 0.32/0.49 ( ( C
% 0.32/0.49 = ( times_times_rat @ C @ B ) )
% 0.32/0.49 = ( ( C = zero_zero_rat )
% 0.32/0.49 | ( B = one_one_rat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_left1
% 0.32/0.49 thf(fact_396_mult__cancel__left1,axiom,
% 0.32/0.49 ! [C: int,B: int] :
% 0.32/0.49 ( ( C
% 0.32/0.49 = ( times_times_int @ C @ B ) )
% 0.32/0.49 = ( ( C = zero_zero_int )
% 0.32/0.49 | ( B = one_one_int ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_cancel_left1
% 0.32/0.49 thf(fact_397_diff__numeral__special_I9_J,axiom,
% 0.32/0.49 ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 0.32/0.49 = zero_zero_complex ) ).
% 0.32/0.49
% 0.32/0.49 % diff_numeral_special(9)
% 0.32/0.49 thf(fact_398_diff__numeral__special_I9_J,axiom,
% 0.32/0.49 ( ( minus_minus_real @ one_one_real @ one_one_real )
% 0.32/0.49 = zero_zero_real ) ).
% 0.32/0.49
% 0.32/0.49 % diff_numeral_special(9)
% 0.32/0.49 thf(fact_399_diff__numeral__special_I9_J,axiom,
% 0.32/0.49 ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 0.32/0.49 = zero_zero_rat ) ).
% 0.32/0.49
% 0.32/0.49 % diff_numeral_special(9)
% 0.32/0.49 thf(fact_400_diff__numeral__special_I9_J,axiom,
% 0.32/0.49 ( ( minus_minus_int @ one_one_int @ one_one_int )
% 0.32/0.49 = zero_zero_int ) ).
% 0.32/0.49
% 0.32/0.49 % diff_numeral_special(9)
% 0.32/0.49 thf(fact_401_nonzero__mult__div__cancel__left,axiom,
% 0.32/0.49 ! [A: complex,B: complex] :
% 0.32/0.49 ( ( A != zero_zero_complex )
% 0.32/0.49 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
% 0.32/0.49 = B ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_div_cancel_left
% 0.32/0.49 thf(fact_402_nonzero__mult__div__cancel__left,axiom,
% 0.32/0.49 ! [A: real,B: real] :
% 0.32/0.49 ( ( A != zero_zero_real )
% 0.32/0.49 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
% 0.32/0.49 = B ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_div_cancel_left
% 0.32/0.49 thf(fact_403_nonzero__mult__div__cancel__left,axiom,
% 0.32/0.49 ! [A: rat,B: rat] :
% 0.32/0.49 ( ( A != zero_zero_rat )
% 0.32/0.49 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
% 0.32/0.49 = B ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_div_cancel_left
% 0.32/0.49 thf(fact_404_nonzero__mult__div__cancel__left,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( A != zero_zero_nat )
% 0.32/0.49 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 0.32/0.49 = B ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_div_cancel_left
% 0.32/0.49 thf(fact_405_nonzero__mult__div__cancel__left,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( A != zero_zero_int )
% 0.32/0.49 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
% 0.32/0.49 = B ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_div_cancel_left
% 0.32/0.49 thf(fact_406_nonzero__mult__div__cancel__right,axiom,
% 0.32/0.49 ! [B: complex,A: complex] :
% 0.32/0.49 ( ( B != zero_zero_complex )
% 0.32/0.49 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
% 0.32/0.49 = A ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_div_cancel_right
% 0.32/0.49 thf(fact_407_nonzero__mult__div__cancel__right,axiom,
% 0.32/0.49 ! [B: real,A: real] :
% 0.32/0.49 ( ( B != zero_zero_real )
% 0.32/0.49 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
% 0.32/0.49 = A ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_div_cancel_right
% 0.32/0.49 thf(fact_408_nonzero__mult__div__cancel__right,axiom,
% 0.32/0.49 ! [B: rat,A: rat] :
% 0.32/0.49 ( ( B != zero_zero_rat )
% 0.32/0.49 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
% 0.32/0.49 = A ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_div_cancel_right
% 0.32/0.49 thf(fact_409_nonzero__mult__div__cancel__right,axiom,
% 0.32/0.49 ! [B: nat,A: nat] :
% 0.32/0.49 ( ( B != zero_zero_nat )
% 0.32/0.49 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.32/0.49 = A ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_div_cancel_right
% 0.32/0.49 thf(fact_410_nonzero__mult__div__cancel__right,axiom,
% 0.32/0.49 ! [B: int,A: int] :
% 0.32/0.49 ( ( B != zero_zero_int )
% 0.32/0.49 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
% 0.32/0.49 = A ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_div_cancel_right
% 0.32/0.49 thf(fact_411_div__mult__mult1__if,axiom,
% 0.32/0.49 ! [C: nat,A: nat,B: nat] :
% 0.32/0.49 ( ( ( C = zero_zero_nat )
% 0.32/0.49 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.32/0.49 = zero_zero_nat ) )
% 0.32/0.49 & ( ( C != zero_zero_nat )
% 0.32/0.49 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.32/0.49 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_mult_mult1_if
% 0.32/0.49 thf(fact_412_div__mult__mult1__if,axiom,
% 0.32/0.49 ! [C: int,A: int,B: int] :
% 0.32/0.49 ( ( ( C = zero_zero_int )
% 0.32/0.49 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.49 = zero_zero_int ) )
% 0.32/0.49 & ( ( C != zero_zero_int )
% 0.32/0.49 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.49 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_mult_mult1_if
% 0.32/0.49 thf(fact_413_div__mult__mult2,axiom,
% 0.32/0.49 ! [C: nat,A: nat,B: nat] :
% 0.32/0.49 ( ( C != zero_zero_nat )
% 0.32/0.49 => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.32/0.49 = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_mult_mult2
% 0.32/0.49 thf(fact_414_div__mult__mult2,axiom,
% 0.32/0.49 ! [C: int,A: int,B: int] :
% 0.32/0.49 ( ( C != zero_zero_int )
% 0.32/0.49 => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.49 = ( divide_divide_int @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_mult_mult2
% 0.32/0.49 thf(fact_415_div__mult__mult1,axiom,
% 0.32/0.49 ! [C: nat,A: nat,B: nat] :
% 0.32/0.49 ( ( C != zero_zero_nat )
% 0.32/0.49 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.32/0.49 = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_mult_mult1
% 0.32/0.49 thf(fact_416_div__mult__mult1,axiom,
% 0.32/0.49 ! [C: int,A: int,B: int] :
% 0.32/0.49 ( ( C != zero_zero_int )
% 0.32/0.49 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.49 = ( divide_divide_int @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_mult_mult1
% 0.32/0.49 thf(fact_417_mult__divide__mult__cancel__left__if,axiom,
% 0.32/0.49 ! [C: complex,A: complex,B: complex] :
% 0.32/0.49 ( ( ( C = zero_zero_complex )
% 0.32/0.49 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.32/0.49 = zero_zero_complex ) )
% 0.32/0.49 & ( ( C != zero_zero_complex )
% 0.32/0.49 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.32/0.49 = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_divide_mult_cancel_left_if
% 0.32/0.49 thf(fact_418_mult__divide__mult__cancel__left__if,axiom,
% 0.32/0.49 ! [C: real,A: real,B: real] :
% 0.32/0.49 ( ( ( C = zero_zero_real )
% 0.32/0.49 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.49 = zero_zero_real ) )
% 0.32/0.49 & ( ( C != zero_zero_real )
% 0.32/0.49 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.49 = ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_divide_mult_cancel_left_if
% 0.32/0.49 thf(fact_419_mult__divide__mult__cancel__left__if,axiom,
% 0.32/0.49 ! [C: rat,A: rat,B: rat] :
% 0.32/0.49 ( ( ( C = zero_zero_rat )
% 0.32/0.49 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.49 = zero_zero_rat ) )
% 0.32/0.49 & ( ( C != zero_zero_rat )
% 0.32/0.49 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.49 = ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_divide_mult_cancel_left_if
% 0.32/0.49 thf(fact_420_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.32/0.49 ! [C: complex,A: complex,B: complex] :
% 0.32/0.49 ( ( C != zero_zero_complex )
% 0.32/0.49 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.32/0.49 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_divide_mult_cancel_left
% 0.32/0.49 thf(fact_421_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.32/0.49 ! [C: real,A: real,B: real] :
% 0.32/0.49 ( ( C != zero_zero_real )
% 0.32/0.49 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.49 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_divide_mult_cancel_left
% 0.32/0.49 thf(fact_422_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.32/0.49 ! [C: rat,A: rat,B: rat] :
% 0.32/0.49 ( ( C != zero_zero_rat )
% 0.32/0.49 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.49 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_divide_mult_cancel_left
% 0.32/0.49 thf(fact_423_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.32/0.49 ! [C: complex,A: complex,B: complex] :
% 0.32/0.49 ( ( C != zero_zero_complex )
% 0.32/0.49 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
% 0.32/0.49 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_divide_mult_cancel_left2
% 0.32/0.49 thf(fact_424_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.32/0.49 ! [C: real,A: real,B: real] :
% 0.32/0.49 ( ( C != zero_zero_real )
% 0.32/0.49 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
% 0.32/0.49 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_divide_mult_cancel_left2
% 0.32/0.49 thf(fact_425_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.32/0.49 ! [C: rat,A: rat,B: rat] :
% 0.32/0.49 ( ( C != zero_zero_rat )
% 0.32/0.49 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.49 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_divide_mult_cancel_left2
% 0.32/0.49 thf(fact_426_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.32/0.49 ! [C: complex,A: complex,B: complex] :
% 0.32/0.49 ( ( C != zero_zero_complex )
% 0.32/0.49 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 0.32/0.49 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_divide_mult_cancel_right
% 0.32/0.49 thf(fact_427_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.32/0.49 ! [C: real,A: real,B: real] :
% 0.32/0.49 ( ( C != zero_zero_real )
% 0.32/0.49 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.32/0.49 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_divide_mult_cancel_right
% 0.32/0.49 thf(fact_428_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.32/0.49 ! [C: rat,A: rat,B: rat] :
% 0.32/0.49 ( ( C != zero_zero_rat )
% 0.32/0.49 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.49 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_divide_mult_cancel_right
% 0.32/0.49 thf(fact_429_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.32/0.49 ! [C: complex,A: complex,B: complex] :
% 0.32/0.49 ( ( C != zero_zero_complex )
% 0.32/0.49 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
% 0.32/0.49 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_divide_mult_cancel_right2
% 0.32/0.49 thf(fact_430_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.32/0.49 ! [C: real,A: real,B: real] :
% 0.32/0.49 ( ( C != zero_zero_real )
% 0.32/0.49 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
% 0.32/0.49 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_divide_mult_cancel_right2
% 0.32/0.49 thf(fact_431_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.32/0.49 ! [C: rat,A: rat,B: rat] :
% 0.32/0.49 ( ( C != zero_zero_rat )
% 0.32/0.49 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.49 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_mult_divide_mult_cancel_right2
% 0.32/0.49 thf(fact_432_div__self,axiom,
% 0.32/0.49 ! [A: complex] :
% 0.32/0.49 ( ( A != zero_zero_complex )
% 0.32/0.49 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.32/0.49 = one_one_complex ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_self
% 0.32/0.49 thf(fact_433_div__self,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( A != zero_zero_real )
% 0.32/0.49 => ( ( divide_divide_real @ A @ A )
% 0.32/0.49 = one_one_real ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_self
% 0.32/0.49 thf(fact_434_div__self,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ( ( A != zero_zero_rat )
% 0.32/0.49 => ( ( divide_divide_rat @ A @ A )
% 0.32/0.49 = one_one_rat ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_self
% 0.32/0.49 thf(fact_435_div__self,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( A != zero_zero_nat )
% 0.32/0.49 => ( ( divide_divide_nat @ A @ A )
% 0.32/0.49 = one_one_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_self
% 0.32/0.49 thf(fact_436_div__self,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( A != zero_zero_int )
% 0.32/0.49 => ( ( divide_divide_int @ A @ A )
% 0.32/0.49 = one_one_int ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_self
% 0.32/0.49 thf(fact_437_divide__eq__1__iff,axiom,
% 0.32/0.49 ! [A: complex,B: complex] :
% 0.32/0.49 ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.32/0.49 = one_one_complex )
% 0.32/0.49 = ( ( B != zero_zero_complex )
% 0.32/0.49 & ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_eq_1_iff
% 0.32/0.49 thf(fact_438_divide__eq__1__iff,axiom,
% 0.32/0.49 ! [A: real,B: real] :
% 0.32/0.49 ( ( ( divide_divide_real @ A @ B )
% 0.32/0.49 = one_one_real )
% 0.32/0.49 = ( ( B != zero_zero_real )
% 0.32/0.49 & ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_eq_1_iff
% 0.32/0.49 thf(fact_439_divide__eq__1__iff,axiom,
% 0.32/0.49 ! [A: rat,B: rat] :
% 0.32/0.49 ( ( ( divide_divide_rat @ A @ B )
% 0.32/0.49 = one_one_rat )
% 0.32/0.49 = ( ( B != zero_zero_rat )
% 0.32/0.49 & ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_eq_1_iff
% 0.32/0.49 thf(fact_440_one__eq__divide__iff,axiom,
% 0.32/0.49 ! [A: complex,B: complex] :
% 0.32/0.49 ( ( one_one_complex
% 0.32/0.49 = ( divide1717551699836669952omplex @ A @ B ) )
% 0.32/0.49 = ( ( B != zero_zero_complex )
% 0.32/0.49 & ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_eq_divide_iff
% 0.32/0.49 thf(fact_441_one__eq__divide__iff,axiom,
% 0.32/0.49 ! [A: real,B: real] :
% 0.32/0.49 ( ( one_one_real
% 0.32/0.49 = ( divide_divide_real @ A @ B ) )
% 0.32/0.49 = ( ( B != zero_zero_real )
% 0.32/0.49 & ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_eq_divide_iff
% 0.32/0.49 thf(fact_442_one__eq__divide__iff,axiom,
% 0.32/0.49 ! [A: rat,B: rat] :
% 0.32/0.49 ( ( one_one_rat
% 0.32/0.49 = ( divide_divide_rat @ A @ B ) )
% 0.32/0.49 = ( ( B != zero_zero_rat )
% 0.32/0.49 & ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_eq_divide_iff
% 0.32/0.49 thf(fact_443_divide__self,axiom,
% 0.32/0.49 ! [A: complex] :
% 0.32/0.49 ( ( A != zero_zero_complex )
% 0.32/0.49 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.32/0.49 = one_one_complex ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_self
% 0.32/0.49 thf(fact_444_divide__self,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( A != zero_zero_real )
% 0.32/0.49 => ( ( divide_divide_real @ A @ A )
% 0.32/0.49 = one_one_real ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_self
% 0.32/0.49 thf(fact_445_divide__self,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ( ( A != zero_zero_rat )
% 0.32/0.49 => ( ( divide_divide_rat @ A @ A )
% 0.32/0.49 = one_one_rat ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_self
% 0.32/0.49 thf(fact_446_divide__self__if,axiom,
% 0.32/0.49 ! [A: complex] :
% 0.32/0.49 ( ( ( A = zero_zero_complex )
% 0.32/0.49 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.32/0.49 = zero_zero_complex ) )
% 0.32/0.49 & ( ( A != zero_zero_complex )
% 0.32/0.49 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.32/0.49 = one_one_complex ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_self_if
% 0.32/0.49 thf(fact_447_divide__self__if,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( ( A = zero_zero_real )
% 0.32/0.49 => ( ( divide_divide_real @ A @ A )
% 0.32/0.49 = zero_zero_real ) )
% 0.32/0.49 & ( ( A != zero_zero_real )
% 0.32/0.49 => ( ( divide_divide_real @ A @ A )
% 0.32/0.49 = one_one_real ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_self_if
% 0.32/0.49 thf(fact_448_divide__self__if,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ( ( ( A = zero_zero_rat )
% 0.32/0.49 => ( ( divide_divide_rat @ A @ A )
% 0.32/0.49 = zero_zero_rat ) )
% 0.32/0.49 & ( ( A != zero_zero_rat )
% 0.32/0.49 => ( ( divide_divide_rat @ A @ A )
% 0.32/0.49 = one_one_rat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_self_if
% 0.32/0.49 thf(fact_449_divide__eq__eq__1,axiom,
% 0.32/0.49 ! [B: real,A: real] :
% 0.32/0.49 ( ( ( divide_divide_real @ B @ A )
% 0.32/0.49 = one_one_real )
% 0.32/0.49 = ( ( A != zero_zero_real )
% 0.32/0.49 & ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_eq_eq_1
% 0.32/0.49 thf(fact_450_divide__eq__eq__1,axiom,
% 0.32/0.49 ! [B: rat,A: rat] :
% 0.32/0.49 ( ( ( divide_divide_rat @ B @ A )
% 0.32/0.49 = one_one_rat )
% 0.32/0.49 = ( ( A != zero_zero_rat )
% 0.32/0.49 & ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_eq_eq_1
% 0.32/0.49 thf(fact_451_eq__divide__eq__1,axiom,
% 0.32/0.49 ! [B: real,A: real] :
% 0.32/0.49 ( ( one_one_real
% 0.32/0.49 = ( divide_divide_real @ B @ A ) )
% 0.32/0.49 = ( ( A != zero_zero_real )
% 0.32/0.49 & ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % eq_divide_eq_1
% 0.32/0.49 thf(fact_452_eq__divide__eq__1,axiom,
% 0.32/0.49 ! [B: rat,A: rat] :
% 0.32/0.49 ( ( one_one_rat
% 0.32/0.49 = ( divide_divide_rat @ B @ A ) )
% 0.32/0.49 = ( ( A != zero_zero_rat )
% 0.32/0.49 & ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % eq_divide_eq_1
% 0.32/0.49 thf(fact_453_one__divide__eq__0__iff,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( ( divide_divide_real @ one_one_real @ A )
% 0.32/0.49 = zero_zero_real )
% 0.32/0.49 = ( A = zero_zero_real ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_divide_eq_0_iff
% 0.32/0.49 thf(fact_454_one__divide__eq__0__iff,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ( ( ( divide_divide_rat @ one_one_rat @ A )
% 0.32/0.49 = zero_zero_rat )
% 0.32/0.49 = ( A = zero_zero_rat ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_divide_eq_0_iff
% 0.32/0.49 thf(fact_455_zero__eq__1__divide__iff,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( zero_zero_real
% 0.32/0.49 = ( divide_divide_real @ one_one_real @ A ) )
% 0.32/0.49 = ( A = zero_zero_real ) ) ).
% 0.32/0.49
% 0.32/0.49 % zero_eq_1_divide_iff
% 0.32/0.49 thf(fact_456_zero__eq__1__divide__iff,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ( ( zero_zero_rat
% 0.32/0.49 = ( divide_divide_rat @ one_one_rat @ A ) )
% 0.32/0.49 = ( A = zero_zero_rat ) ) ).
% 0.32/0.49
% 0.32/0.49 % zero_eq_1_divide_iff
% 0.32/0.49 thf(fact_457_right__diff__distrib__numeral,axiom,
% 0.32/0.49 ! [V: num,B: complex,C: complex] :
% 0.32/0.49 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
% 0.32/0.49 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib_numeral
% 0.32/0.49 thf(fact_458_right__diff__distrib__numeral,axiom,
% 0.32/0.49 ! [V: num,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
% 0.32/0.49 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib_numeral
% 0.32/0.49 thf(fact_459_right__diff__distrib__numeral,axiom,
% 0.32/0.49 ! [V: num,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
% 0.32/0.49 = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib_numeral
% 0.32/0.49 thf(fact_460_right__diff__distrib__numeral,axiom,
% 0.32/0.49 ! [V: num,B: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
% 0.32/0.49 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib_numeral
% 0.32/0.49 thf(fact_461_left__diff__distrib__numeral,axiom,
% 0.32/0.49 ! [A: complex,B: complex,V: num] :
% 0.32/0.49 ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 0.32/0.49 = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib_numeral
% 0.32/0.49 thf(fact_462_left__diff__distrib__numeral,axiom,
% 0.32/0.49 ! [A: real,B: real,V: num] :
% 0.32/0.49 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 0.32/0.49 = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib_numeral
% 0.32/0.49 thf(fact_463_left__diff__distrib__numeral,axiom,
% 0.32/0.49 ! [A: rat,B: rat,V: num] :
% 0.32/0.49 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 0.32/0.49 = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib_numeral
% 0.32/0.49 thf(fact_464_left__diff__distrib__numeral,axiom,
% 0.32/0.49 ! [A: int,B: int,V: num] :
% 0.32/0.49 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 0.32/0.49 = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib_numeral
% 0.32/0.49 thf(fact_465_numeral__eq__one__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ( numera6690914467698888265omplex @ N )
% 0.32/0.49 = one_one_complex )
% 0.32/0.49 = ( N = one ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_eq_one_iff
% 0.32/0.49 thf(fact_466_numeral__eq__one__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ( numeral_numeral_real @ N )
% 0.32/0.49 = one_one_real )
% 0.32/0.49 = ( N = one ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_eq_one_iff
% 0.32/0.49 thf(fact_467_numeral__eq__one__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ( numeral_numeral_rat @ N )
% 0.32/0.49 = one_one_rat )
% 0.32/0.49 = ( N = one ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_eq_one_iff
% 0.32/0.49 thf(fact_468_numeral__eq__one__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ( numeral_numeral_nat @ N )
% 0.32/0.49 = one_one_nat )
% 0.32/0.49 = ( N = one ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_eq_one_iff
% 0.32/0.49 thf(fact_469_numeral__eq__one__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ( numeral_numeral_int @ N )
% 0.32/0.49 = one_one_int )
% 0.32/0.49 = ( N = one ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_eq_one_iff
% 0.32/0.49 thf(fact_470_one__eq__numeral__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( one_one_complex
% 0.32/0.49 = ( numera6690914467698888265omplex @ N ) )
% 0.32/0.49 = ( one = N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_eq_numeral_iff
% 0.32/0.49 thf(fact_471_one__eq__numeral__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( one_one_real
% 0.32/0.49 = ( numeral_numeral_real @ N ) )
% 0.32/0.49 = ( one = N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_eq_numeral_iff
% 0.32/0.49 thf(fact_472_one__eq__numeral__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( one_one_rat
% 0.32/0.49 = ( numeral_numeral_rat @ N ) )
% 0.32/0.49 = ( one = N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_eq_numeral_iff
% 0.32/0.49 thf(fact_473_one__eq__numeral__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( one_one_nat
% 0.32/0.49 = ( numeral_numeral_nat @ N ) )
% 0.32/0.49 = ( one = N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_eq_numeral_iff
% 0.32/0.49 thf(fact_474_one__eq__numeral__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( one_one_int
% 0.32/0.49 = ( numeral_numeral_int @ N ) )
% 0.32/0.49 = ( one = N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_eq_numeral_iff
% 0.32/0.49 thf(fact_475_dvd__mult__cancel__left,axiom,
% 0.32/0.49 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.32/0.49 = ( ( C = zero_z3403309356797280102nteger )
% 0.32/0.49 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_cancel_left
% 0.32/0.49 thf(fact_476_dvd__mult__cancel__left,axiom,
% 0.32/0.49 ! [C: real,A: real,B: real] :
% 0.32/0.49 ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.49 = ( ( C = zero_zero_real )
% 0.32/0.49 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_cancel_left
% 0.32/0.49 thf(fact_477_dvd__mult__cancel__left,axiom,
% 0.32/0.49 ! [C: rat,A: rat,B: rat] :
% 0.32/0.49 ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.49 = ( ( C = zero_zero_rat )
% 0.32/0.49 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_cancel_left
% 0.32/0.49 thf(fact_478_dvd__mult__cancel__left,axiom,
% 0.32/0.49 ! [C: int,A: int,B: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.49 = ( ( C = zero_zero_int )
% 0.32/0.49 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_cancel_left
% 0.32/0.49 thf(fact_479_dvd__mult__cancel__right,axiom,
% 0.32/0.49 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.49 = ( ( C = zero_z3403309356797280102nteger )
% 0.32/0.49 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_cancel_right
% 0.32/0.49 thf(fact_480_dvd__mult__cancel__right,axiom,
% 0.32/0.49 ! [A: real,C: real,B: real] :
% 0.32/0.49 ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.32/0.49 = ( ( C = zero_zero_real )
% 0.32/0.49 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_cancel_right
% 0.32/0.49 thf(fact_481_dvd__mult__cancel__right,axiom,
% 0.32/0.49 ! [A: rat,C: rat,B: rat] :
% 0.32/0.49 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.49 = ( ( C = zero_zero_rat )
% 0.32/0.49 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_cancel_right
% 0.32/0.49 thf(fact_482_dvd__mult__cancel__right,axiom,
% 0.32/0.49 ! [A: int,C: int,B: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.49 = ( ( C = zero_zero_int )
% 0.32/0.49 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_cancel_right
% 0.32/0.49 thf(fact_483_dvd__times__left__cancel__iff,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.49 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.49 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 0.32/0.49 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_times_left_cancel_iff
% 0.32/0.49 thf(fact_484_dvd__times__left__cancel__iff,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( A != zero_zero_nat )
% 0.32/0.49 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
% 0.32/0.49 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_times_left_cancel_iff
% 0.32/0.49 thf(fact_485_dvd__times__left__cancel__iff,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( A != zero_zero_int )
% 0.32/0.49 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
% 0.32/0.49 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_times_left_cancel_iff
% 0.32/0.49 thf(fact_486_dvd__times__right__cancel__iff,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.49 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.49 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 0.32/0.49 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_times_right_cancel_iff
% 0.32/0.49 thf(fact_487_dvd__times__right__cancel__iff,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( A != zero_zero_nat )
% 0.32/0.49 => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
% 0.32/0.49 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_times_right_cancel_iff
% 0.32/0.49 thf(fact_488_dvd__times__right__cancel__iff,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( A != zero_zero_int )
% 0.32/0.49 => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
% 0.32/0.49 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_times_right_cancel_iff
% 0.32/0.49 thf(fact_489_unit__prod,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.49 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.49 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_prod
% 0.32/0.49 thf(fact_490_unit__prod,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.49 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.49 => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_prod
% 0.32/0.49 thf(fact_491_unit__prod,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.49 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.49 => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_prod
% 0.32/0.49 thf(fact_492_dvd__div__mult__self,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.49 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 0.32/0.49 = B ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_div_mult_self
% 0.32/0.49 thf(fact_493_dvd__div__mult__self,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.49 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 0.32/0.49 = B ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_div_mult_self
% 0.32/0.49 thf(fact_494_dvd__div__mult__self,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.49 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 0.32/0.49 = B ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_div_mult_self
% 0.32/0.49 thf(fact_495_dvd__mult__div__cancel,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.49 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
% 0.32/0.49 = B ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_div_cancel
% 0.32/0.49 thf(fact_496_dvd__mult__div__cancel,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.49 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
% 0.32/0.49 = B ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_div_cancel
% 0.32/0.49 thf(fact_497_dvd__mult__div__cancel,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.49 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
% 0.32/0.49 = B ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_div_cancel
% 0.32/0.49 thf(fact_498_unit__div,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.49 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.49 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div
% 0.32/0.49 thf(fact_499_unit__div,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.49 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.49 => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div
% 0.32/0.49 thf(fact_500_unit__div,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.49 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.49 => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div
% 0.32/0.49 thf(fact_501_unit__div__1__unit,axiom,
% 0.32/0.49 ! [A: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.49 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div_1_unit
% 0.32/0.49 thf(fact_502_unit__div__1__unit,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.49 => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div_1_unit
% 0.32/0.49 thf(fact_503_unit__div__1__unit,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.49 => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div_1_unit
% 0.32/0.49 thf(fact_504_unit__div__1__div__1,axiom,
% 0.32/0.49 ! [A: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.49 => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.32/0.49 = A ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div_1_div_1
% 0.32/0.49 thf(fact_505_unit__div__1__div__1,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.49 => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.32/0.49 = A ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div_1_div_1
% 0.32/0.49 thf(fact_506_unit__div__1__div__1,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.49 => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 0.32/0.49 = A ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div_1_div_1
% 0.32/0.49 thf(fact_507_divide__eq__eq__numeral1_I1_J,axiom,
% 0.32/0.49 ! [B: complex,W: num,A: complex] :
% 0.32/0.49 ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.49 = A )
% 0.32/0.49 = ( ( ( ( numera6690914467698888265omplex @ W )
% 0.32/0.49 != zero_zero_complex )
% 0.32/0.49 => ( B
% 0.32/0.49 = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
% 0.32/0.49 & ( ( ( numera6690914467698888265omplex @ W )
% 0.32/0.49 = zero_zero_complex )
% 0.32/0.49 => ( A = zero_zero_complex ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_eq_eq_numeral1(1)
% 0.32/0.49 thf(fact_508_divide__eq__eq__numeral1_I1_J,axiom,
% 0.32/0.49 ! [B: real,W: num,A: real] :
% 0.32/0.49 ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
% 0.32/0.49 = A )
% 0.32/0.49 = ( ( ( ( numeral_numeral_real @ W )
% 0.32/0.49 != zero_zero_real )
% 0.32/0.49 => ( B
% 0.32/0.49 = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
% 0.32/0.49 & ( ( ( numeral_numeral_real @ W )
% 0.32/0.49 = zero_zero_real )
% 0.32/0.49 => ( A = zero_zero_real ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_eq_eq_numeral1(1)
% 0.32/0.49 thf(fact_509_divide__eq__eq__numeral1_I1_J,axiom,
% 0.32/0.49 ! [B: rat,W: num,A: rat] :
% 0.32/0.49 ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
% 0.32/0.49 = A )
% 0.32/0.49 = ( ( ( ( numeral_numeral_rat @ W )
% 0.32/0.49 != zero_zero_rat )
% 0.32/0.49 => ( B
% 0.32/0.49 = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
% 0.32/0.49 & ( ( ( numeral_numeral_rat @ W )
% 0.32/0.49 = zero_zero_rat )
% 0.32/0.49 => ( A = zero_zero_rat ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_eq_eq_numeral1(1)
% 0.32/0.49 thf(fact_510_eq__divide__eq__numeral1_I1_J,axiom,
% 0.32/0.49 ! [A: complex,B: complex,W: num] :
% 0.32/0.49 ( ( A
% 0.32/0.49 = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
% 0.32/0.49 = ( ( ( ( numera6690914467698888265omplex @ W )
% 0.32/0.49 != zero_zero_complex )
% 0.32/0.49 => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.49 = B ) )
% 0.32/0.49 & ( ( ( numera6690914467698888265omplex @ W )
% 0.32/0.49 = zero_zero_complex )
% 0.32/0.49 => ( A = zero_zero_complex ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % eq_divide_eq_numeral1(1)
% 0.32/0.49 thf(fact_511_eq__divide__eq__numeral1_I1_J,axiom,
% 0.32/0.49 ! [A: real,B: real,W: num] :
% 0.32/0.49 ( ( A
% 0.32/0.49 = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.49 = ( ( ( ( numeral_numeral_real @ W )
% 0.32/0.49 != zero_zero_real )
% 0.32/0.49 => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
% 0.32/0.49 = B ) )
% 0.32/0.49 & ( ( ( numeral_numeral_real @ W )
% 0.32/0.49 = zero_zero_real )
% 0.32/0.49 => ( A = zero_zero_real ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % eq_divide_eq_numeral1(1)
% 0.32/0.49 thf(fact_512_eq__divide__eq__numeral1_I1_J,axiom,
% 0.32/0.49 ! [A: rat,B: rat,W: num] :
% 0.32/0.49 ( ( A
% 0.32/0.49 = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.32/0.49 = ( ( ( ( numeral_numeral_rat @ W )
% 0.32/0.49 != zero_zero_rat )
% 0.32/0.49 => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
% 0.32/0.49 = B ) )
% 0.32/0.49 & ( ( ( numeral_numeral_rat @ W )
% 0.32/0.49 = zero_zero_rat )
% 0.32/0.49 => ( A = zero_zero_rat ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % eq_divide_eq_numeral1(1)
% 0.32/0.49 thf(fact_513_nonzero__divide__mult__cancel__left,axiom,
% 0.32/0.49 ! [A: complex,B: complex] :
% 0.32/0.49 ( ( A != zero_zero_complex )
% 0.32/0.49 => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
% 0.32/0.49 = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_divide_mult_cancel_left
% 0.32/0.49 thf(fact_514_nonzero__divide__mult__cancel__left,axiom,
% 0.32/0.49 ! [A: real,B: real] :
% 0.32/0.49 ( ( A != zero_zero_real )
% 0.32/0.49 => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
% 0.32/0.49 = ( divide_divide_real @ one_one_real @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_divide_mult_cancel_left
% 0.32/0.49 thf(fact_515_nonzero__divide__mult__cancel__left,axiom,
% 0.32/0.49 ! [A: rat,B: rat] :
% 0.32/0.49 ( ( A != zero_zero_rat )
% 0.32/0.49 => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
% 0.32/0.49 = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_divide_mult_cancel_left
% 0.32/0.49 thf(fact_516_nonzero__divide__mult__cancel__right,axiom,
% 0.32/0.49 ! [B: complex,A: complex] :
% 0.32/0.49 ( ( B != zero_zero_complex )
% 0.32/0.49 => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
% 0.32/0.49 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_divide_mult_cancel_right
% 0.32/0.49 thf(fact_517_nonzero__divide__mult__cancel__right,axiom,
% 0.32/0.49 ! [B: real,A: real] :
% 0.32/0.49 ( ( B != zero_zero_real )
% 0.32/0.49 => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
% 0.32/0.49 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_divide_mult_cancel_right
% 0.32/0.49 thf(fact_518_nonzero__divide__mult__cancel__right,axiom,
% 0.32/0.49 ! [B: rat,A: rat] :
% 0.32/0.49 ( ( B != zero_zero_rat )
% 0.32/0.49 => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
% 0.32/0.49 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nonzero_divide_mult_cancel_right
% 0.32/0.49 thf(fact_519_unit__div__mult__self,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.49 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 0.32/0.49 = B ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div_mult_self
% 0.32/0.49 thf(fact_520_unit__div__mult__self,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.49 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 0.32/0.49 = B ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div_mult_self
% 0.32/0.49 thf(fact_521_unit__div__mult__self,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.49 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 0.32/0.49 = B ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div_mult_self
% 0.32/0.49 thf(fact_522_unit__mult__div__div,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.49 => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.32/0.49 = ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_mult_div_div
% 0.32/0.49 thf(fact_523_unit__mult__div__div,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.49 => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.32/0.49 = ( divide_divide_nat @ B @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_mult_div_div
% 0.32/0.49 thf(fact_524_unit__mult__div__div,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.49 => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
% 0.32/0.49 = ( divide_divide_int @ B @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_mult_div_div
% 0.32/0.49 thf(fact_525_Suc__1,axiom,
% 0.32/0.49 ( ( suc @ one_one_nat )
% 0.32/0.49 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % Suc_1
% 0.32/0.49 thf(fact_526_push__bit__Suc__numeral,axiom,
% 0.32/0.49 ! [N: nat,K: num] :
% 0.32/0.49 ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.32/0.49 = ( bit_se547839408752420682it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_Suc_numeral
% 0.32/0.49 thf(fact_527_push__bit__Suc__numeral,axiom,
% 0.32/0.49 ! [N: nat,K: num] :
% 0.32/0.49 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( numeral_numeral_int @ K ) )
% 0.32/0.49 = ( bit_se545348938243370406it_int @ N @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_Suc_numeral
% 0.32/0.49 thf(fact_528_push__bit__Suc__numeral,axiom,
% 0.32/0.49 ! [N: nat,K: num] :
% 0.32/0.49 ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ ( numera6620942414471956472nteger @ K ) )
% 0.32/0.49 = ( bit_se7788150548672797655nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_Suc_numeral
% 0.32/0.49 thf(fact_529_push__bit__Suc__numeral,axiom,
% 0.32/0.49 ! [N: nat,K: num] :
% 0.32/0.49 ( ( bit_se6611745700429515170atural @ ( suc @ N ) @ ( numera5444537566228673987atural @ K ) )
% 0.32/0.49 = ( bit_se6611745700429515170atural @ N @ ( numera5444537566228673987atural @ ( bit0 @ K ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_Suc_numeral
% 0.32/0.49 thf(fact_530_even__mult__iff,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.32/0.49 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.49 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % even_mult_iff
% 0.32/0.49 thf(fact_531_even__mult__iff,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
% 0.32/0.49 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.49 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % even_mult_iff
% 0.32/0.49 thf(fact_532_even__mult__iff,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
% 0.32/0.49 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.49 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % even_mult_iff
% 0.32/0.49 thf(fact_533_push__bit__Suc,axiom,
% 0.32/0.49 ! [N: nat,A: nat] :
% 0.32/0.49 ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ A )
% 0.32/0.49 = ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_Suc
% 0.32/0.49 thf(fact_534_push__bit__Suc,axiom,
% 0.32/0.49 ! [N: nat,A: int] :
% 0.32/0.49 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ A )
% 0.32/0.49 = ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_Suc
% 0.32/0.49 thf(fact_535_push__bit__Suc,axiom,
% 0.32/0.49 ! [N: nat,A: code_integer] :
% 0.32/0.49 ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ A )
% 0.32/0.49 = ( bit_se7788150548672797655nteger @ N @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_Suc
% 0.32/0.49 thf(fact_536_push__bit__Suc,axiom,
% 0.32/0.49 ! [N: nat,A: code_natural] :
% 0.32/0.49 ( ( bit_se6611745700429515170atural @ ( suc @ N ) @ A )
% 0.32/0.49 = ( bit_se6611745700429515170atural @ N @ ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_Suc
% 0.32/0.49 thf(fact_537_odd__two__times__div__two__nat,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.49 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.49 = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % odd_two_times_div_two_nat
% 0.32/0.49 thf(fact_538_times__int__code_I1_J,axiom,
% 0.32/0.49 ! [K: int] :
% 0.32/0.49 ( ( times_times_int @ K @ zero_zero_int )
% 0.32/0.49 = zero_zero_int ) ).
% 0.32/0.49
% 0.32/0.49 % times_int_code(1)
% 0.32/0.49 thf(fact_539_times__int__code_I2_J,axiom,
% 0.32/0.49 ! [L: int] :
% 0.32/0.49 ( ( times_times_int @ zero_zero_int @ L )
% 0.32/0.49 = zero_zero_int ) ).
% 0.32/0.49
% 0.32/0.49 % times_int_code(2)
% 0.32/0.49 thf(fact_540_minus__int__code_I1_J,axiom,
% 0.32/0.49 ! [K: int] :
% 0.32/0.49 ( ( minus_minus_int @ K @ zero_zero_int )
% 0.32/0.49 = K ) ).
% 0.32/0.49
% 0.32/0.49 % minus_int_code(1)
% 0.32/0.49 thf(fact_541_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.32/0.49 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % ab_semigroup_mult_class.mult_ac(1)
% 0.32/0.49 thf(fact_542_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.32/0.49 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % ab_semigroup_mult_class.mult_ac(1)
% 0.32/0.49 thf(fact_543_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.49 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % ab_semigroup_mult_class.mult_ac(1)
% 0.32/0.49 thf(fact_544_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.49 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % ab_semigroup_mult_class.mult_ac(1)
% 0.32/0.49 thf(fact_545_comm__monoid__mult__class_Omult__1,axiom,
% 0.32/0.49 ! [A: complex] :
% 0.32/0.49 ( ( times_times_complex @ one_one_complex @ A )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % comm_monoid_mult_class.mult_1
% 0.32/0.49 thf(fact_546_comm__monoid__mult__class_Omult__1,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( times_times_real @ one_one_real @ A )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % comm_monoid_mult_class.mult_1
% 0.32/0.49 thf(fact_547_comm__monoid__mult__class_Omult__1,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ( ( times_times_rat @ one_one_rat @ A )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % comm_monoid_mult_class.mult_1
% 0.32/0.49 thf(fact_548_comm__monoid__mult__class_Omult__1,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( times_times_nat @ one_one_nat @ A )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % comm_monoid_mult_class.mult_1
% 0.32/0.49 thf(fact_549_comm__monoid__mult__class_Omult__1,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( times_times_int @ one_one_int @ A )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % comm_monoid_mult_class.mult_1
% 0.32/0.49 thf(fact_550_mult_Oassoc,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.32/0.49 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult.assoc
% 0.32/0.49 thf(fact_551_mult_Oassoc,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.32/0.49 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult.assoc
% 0.32/0.49 thf(fact_552_mult_Oassoc,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.49 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult.assoc
% 0.32/0.49 thf(fact_553_mult_Oassoc,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.49 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult.assoc
% 0.32/0.49 thf(fact_554_mult_Ocommute,axiom,
% 0.32/0.49 ( times_times_real
% 0.32/0.49 = ( ^ [A2: real,B2: real] : ( times_times_real @ B2 @ A2 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult.commute
% 0.32/0.49 thf(fact_555_mult_Ocommute,axiom,
% 0.32/0.49 ( times_times_rat
% 0.32/0.49 = ( ^ [A2: rat,B2: rat] : ( times_times_rat @ B2 @ A2 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult.commute
% 0.32/0.49 thf(fact_556_mult_Ocommute,axiom,
% 0.32/0.49 ( times_times_nat
% 0.32/0.49 = ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult.commute
% 0.32/0.49 thf(fact_557_mult_Ocommute,axiom,
% 0.32/0.49 ( times_times_int
% 0.32/0.49 = ( ^ [A2: int,B2: int] : ( times_times_int @ B2 @ A2 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult.commute
% 0.32/0.49 thf(fact_558_mult_Ocomm__neutral,axiom,
% 0.32/0.49 ! [A: complex] :
% 0.32/0.49 ( ( times_times_complex @ A @ one_one_complex )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult.comm_neutral
% 0.32/0.49 thf(fact_559_mult_Ocomm__neutral,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( times_times_real @ A @ one_one_real )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult.comm_neutral
% 0.32/0.49 thf(fact_560_mult_Ocomm__neutral,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ( ( times_times_rat @ A @ one_one_rat )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult.comm_neutral
% 0.32/0.49 thf(fact_561_mult_Ocomm__neutral,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( times_times_nat @ A @ one_one_nat )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult.comm_neutral
% 0.32/0.49 thf(fact_562_mult_Ocomm__neutral,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( times_times_int @ A @ one_one_int )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult.comm_neutral
% 0.32/0.49 thf(fact_563_mult_Oleft__commute,axiom,
% 0.32/0.49 ! [B: real,A: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 0.32/0.49 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult.left_commute
% 0.32/0.49 thf(fact_564_mult_Oleft__commute,axiom,
% 0.32/0.49 ! [B: rat,A: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
% 0.32/0.49 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult.left_commute
% 0.32/0.49 thf(fact_565_mult_Oleft__commute,axiom,
% 0.32/0.49 ! [B: nat,A: nat,C: nat] :
% 0.32/0.49 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 0.32/0.49 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult.left_commute
% 0.32/0.49 thf(fact_566_mult_Oleft__commute,axiom,
% 0.32/0.49 ! [B: int,A: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
% 0.32/0.49 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult.left_commute
% 0.32/0.49 thf(fact_567_one__reorient,axiom,
% 0.32/0.49 ! [X: complex] :
% 0.32/0.49 ( ( one_one_complex = X )
% 0.32/0.49 = ( X = one_one_complex ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_reorient
% 0.32/0.49 thf(fact_568_one__reorient,axiom,
% 0.32/0.49 ! [X: real] :
% 0.32/0.49 ( ( one_one_real = X )
% 0.32/0.49 = ( X = one_one_real ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_reorient
% 0.32/0.49 thf(fact_569_one__reorient,axiom,
% 0.32/0.49 ! [X: rat] :
% 0.32/0.49 ( ( one_one_rat = X )
% 0.32/0.49 = ( X = one_one_rat ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_reorient
% 0.32/0.49 thf(fact_570_one__reorient,axiom,
% 0.32/0.49 ! [X: nat] :
% 0.32/0.49 ( ( one_one_nat = X )
% 0.32/0.49 = ( X = one_one_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_reorient
% 0.32/0.49 thf(fact_571_one__reorient,axiom,
% 0.32/0.49 ! [X: int] :
% 0.32/0.49 ( ( one_one_int = X )
% 0.32/0.49 = ( X = one_one_int ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_reorient
% 0.32/0.49 thf(fact_572_push__bit__of__nat,axiom,
% 0.32/0.49 ! [N: nat,M: nat] :
% 0.32/0.49 ( ( bit_se547839408752420682it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 0.32/0.49 = ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_of_nat
% 0.32/0.49 thf(fact_573_push__bit__of__nat,axiom,
% 0.32/0.49 ! [N: nat,M: nat] :
% 0.32/0.49 ( ( bit_se545348938243370406it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 0.32/0.49 = ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_of_nat
% 0.32/0.49 thf(fact_574_push__bit__of__nat,axiom,
% 0.32/0.49 ! [N: nat,M: nat] :
% 0.32/0.49 ( ( bit_se7788150548672797655nteger @ N @ ( semiri4939895301339042750nteger @ M ) )
% 0.32/0.49 = ( semiri4939895301339042750nteger @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_of_nat
% 0.32/0.49 thf(fact_575_push__bit__of__nat,axiom,
% 0.32/0.49 ! [N: nat,M: nat] :
% 0.32/0.49 ( ( bit_se6611745700429515170atural @ N @ ( semiri3763490453095760265atural @ M ) )
% 0.32/0.49 = ( semiri3763490453095760265atural @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % push_bit_of_nat
% 0.32/0.49 thf(fact_576_of__nat__push__bit,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 0.32/0.49 = ( bit_se547839408752420682it_nat @ M @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_push_bit
% 0.32/0.49 thf(fact_577_of__nat__push__bit,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 0.32/0.49 = ( bit_se545348938243370406it_int @ M @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_push_bit
% 0.32/0.49 thf(fact_578_of__nat__push__bit,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( semiri4939895301339042750nteger @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 0.32/0.49 = ( bit_se7788150548672797655nteger @ M @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_push_bit
% 0.32/0.49 thf(fact_579_of__nat__push__bit,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( semiri3763490453095760265atural @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 0.32/0.49 = ( bit_se6611745700429515170atural @ M @ ( semiri3763490453095760265atural @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % of_nat_push_bit
% 0.32/0.49 thf(fact_580_is__unit__mult__iff,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
% 0.32/0.49 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.49 & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % is_unit_mult_iff
% 0.32/0.49 thf(fact_581_is__unit__mult__iff,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
% 0.32/0.49 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.49 & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % is_unit_mult_iff
% 0.32/0.49 thf(fact_582_is__unit__mult__iff,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
% 0.32/0.49 = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.49 & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % is_unit_mult_iff
% 0.32/0.49 thf(fact_583_dvd__mult__unit__iff,axiom,
% 0.32/0.49 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.49 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.32/0.49 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_unit_iff
% 0.32/0.49 thf(fact_584_dvd__mult__unit__iff,axiom,
% 0.32/0.49 ! [B: nat,A: nat,C: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.49 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
% 0.32/0.49 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_unit_iff
% 0.32/0.49 thf(fact_585_dvd__mult__unit__iff,axiom,
% 0.32/0.49 ! [B: int,A: int,C: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.49 => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
% 0.32/0.49 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_unit_iff
% 0.32/0.49 thf(fact_586_mult__unit__dvd__iff,axiom,
% 0.32/0.49 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.49 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.32/0.49 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_unit_dvd_iff
% 0.32/0.49 thf(fact_587_mult__unit__dvd__iff,axiom,
% 0.32/0.49 ! [B: nat,A: nat,C: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.49 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.49 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_unit_dvd_iff
% 0.32/0.49 thf(fact_588_mult__unit__dvd__iff,axiom,
% 0.32/0.49 ! [B: int,A: int,C: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.49 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.49 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_unit_dvd_iff
% 0.32/0.49 thf(fact_589_dvd__mult__unit__iff_H,axiom,
% 0.32/0.49 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.49 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.49 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_unit_iff'
% 0.32/0.49 thf(fact_590_dvd__mult__unit__iff_H,axiom,
% 0.32/0.49 ! [B: nat,A: nat,C: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.49 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.49 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_unit_iff'
% 0.32/0.49 thf(fact_591_dvd__mult__unit__iff_H,axiom,
% 0.32/0.49 ! [B: int,A: int,C: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.49 => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.49 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_unit_iff'
% 0.32/0.49 thf(fact_592_mult__unit__dvd__iff_H,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.49 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.32/0.49 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_unit_dvd_iff'
% 0.32/0.49 thf(fact_593_mult__unit__dvd__iff_H,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.49 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.49 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_unit_dvd_iff'
% 0.32/0.49 thf(fact_594_mult__unit__dvd__iff_H,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.49 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.49 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_unit_dvd_iff'
% 0.32/0.49 thf(fact_595_unit__mult__left__cancel,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.49 => ( ( ( times_3573771949741848930nteger @ A @ B )
% 0.32/0.49 = ( times_3573771949741848930nteger @ A @ C ) )
% 0.32/0.49 = ( B = C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_mult_left_cancel
% 0.32/0.49 thf(fact_596_unit__mult__left__cancel,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.49 => ( ( ( times_times_nat @ A @ B )
% 0.32/0.49 = ( times_times_nat @ A @ C ) )
% 0.32/0.49 = ( B = C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_mult_left_cancel
% 0.32/0.49 thf(fact_597_unit__mult__left__cancel,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.49 => ( ( ( times_times_int @ A @ B )
% 0.32/0.49 = ( times_times_int @ A @ C ) )
% 0.32/0.49 = ( B = C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_mult_left_cancel
% 0.32/0.49 thf(fact_598_unit__mult__right__cancel,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.49 => ( ( ( times_3573771949741848930nteger @ B @ A )
% 0.32/0.49 = ( times_3573771949741848930nteger @ C @ A ) )
% 0.32/0.49 = ( B = C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_mult_right_cancel
% 0.32/0.49 thf(fact_599_unit__mult__right__cancel,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.49 => ( ( ( times_times_nat @ B @ A )
% 0.32/0.49 = ( times_times_nat @ C @ A ) )
% 0.32/0.49 = ( B = C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_mult_right_cancel
% 0.32/0.49 thf(fact_600_unit__mult__right__cancel,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.49 => ( ( ( times_times_int @ B @ A )
% 0.32/0.49 = ( times_times_int @ C @ A ) )
% 0.32/0.49 = ( B = C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_mult_right_cancel
% 0.32/0.49 thf(fact_601_mult__eq__self__implies__10,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( M
% 0.32/0.49 = ( times_times_nat @ M @ N ) )
% 0.32/0.49 => ( ( N = one_one_nat )
% 0.32/0.49 | ( M = zero_zero_nat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_eq_self_implies_10
% 0.32/0.49 thf(fact_602_int__ops_I7_J,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
% 0.32/0.49 = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % int_ops(7)
% 0.32/0.49 thf(fact_603_int__ops_I2_J,axiom,
% 0.32/0.49 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.32/0.49 = one_one_int ) ).
% 0.32/0.49
% 0.32/0.49 % int_ops(2)
% 0.32/0.49 thf(fact_604_division__decomp,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.49 => ? [B3: nat,C2: nat] :
% 0.32/0.49 ( ( A
% 0.32/0.49 = ( times_times_nat @ B3 @ C2 ) )
% 0.32/0.49 & ( dvd_dvd_nat @ B3 @ B )
% 0.32/0.49 & ( dvd_dvd_nat @ C2 @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % division_decomp
% 0.32/0.49 thf(fact_605_division__decomp,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.49 => ? [B3: int,C2: int] :
% 0.32/0.49 ( ( A
% 0.32/0.49 = ( times_times_int @ B3 @ C2 ) )
% 0.32/0.49 & ( dvd_dvd_int @ B3 @ B )
% 0.32/0.49 & ( dvd_dvd_int @ C2 @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % division_decomp
% 0.32/0.49 thf(fact_606_dvd__productE,axiom,
% 0.32/0.49 ! [P2: nat,A: nat,B: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ P2 @ ( times_times_nat @ A @ B ) )
% 0.32/0.49 => ~ ! [X4: nat,Y4: nat] :
% 0.32/0.49 ( ( P2
% 0.32/0.49 = ( times_times_nat @ X4 @ Y4 ) )
% 0.32/0.49 => ( ( dvd_dvd_nat @ X4 @ A )
% 0.32/0.49 => ~ ( dvd_dvd_nat @ Y4 @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_productE
% 0.32/0.49 thf(fact_607_dvd__productE,axiom,
% 0.32/0.49 ! [P2: int,A: int,B: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ P2 @ ( times_times_int @ A @ B ) )
% 0.32/0.49 => ~ ! [X4: int,Y4: int] :
% 0.32/0.49 ( ( P2
% 0.32/0.49 = ( times_times_int @ X4 @ Y4 ) )
% 0.32/0.49 => ( ( dvd_dvd_int @ X4 @ A )
% 0.32/0.49 => ~ ( dvd_dvd_int @ Y4 @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_productE
% 0.32/0.49 thf(fact_608_unit__dvdE,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.49 => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.49 => ! [C3: code_integer] :
% 0.32/0.49 ( B
% 0.32/0.49 != ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_dvdE
% 0.32/0.49 thf(fact_609_unit__dvdE,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.49 => ~ ( ( A != zero_zero_nat )
% 0.32/0.49 => ! [C3: nat] :
% 0.32/0.49 ( B
% 0.32/0.49 != ( times_times_nat @ A @ C3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_dvdE
% 0.32/0.49 thf(fact_610_unit__dvdE,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.49 => ~ ( ( A != zero_zero_int )
% 0.32/0.49 => ! [C3: int] :
% 0.32/0.49 ( B
% 0.32/0.49 != ( times_times_int @ A @ C3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_dvdE
% 0.32/0.49 thf(fact_611_is__unit__div__mult2__eq,axiom,
% 0.32/0.49 ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.49 => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.32/0.49 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.49 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % is_unit_div_mult2_eq
% 0.32/0.49 thf(fact_612_is__unit__div__mult2__eq,axiom,
% 0.32/0.49 ! [B: nat,C: nat,A: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.49 => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.32/0.49 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.49 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % is_unit_div_mult2_eq
% 0.32/0.49 thf(fact_613_is__unit__div__mult2__eq,axiom,
% 0.32/0.49 ! [B: int,C: int,A: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.49 => ( ( dvd_dvd_int @ C @ one_one_int )
% 0.32/0.49 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.49 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % is_unit_div_mult2_eq
% 0.32/0.49 thf(fact_614_unit__div__mult__swap,axiom,
% 0.32/0.49 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.32/0.49 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.49 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div_mult_swap
% 0.32/0.49 thf(fact_615_unit__div__mult__swap,axiom,
% 0.32/0.49 ! [C: nat,A: nat,B: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.32/0.49 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.32/0.49 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div_mult_swap
% 0.32/0.49 thf(fact_616_unit__div__mult__swap,axiom,
% 0.32/0.49 ! [C: int,A: int,B: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ C @ one_one_int )
% 0.32/0.49 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.32/0.49 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div_mult_swap
% 0.32/0.49 thf(fact_617_unit__div__commute,axiom,
% 0.32/0.49 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.49 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.32/0.49 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div_commute
% 0.32/0.49 thf(fact_618_unit__div__commute,axiom,
% 0.32/0.49 ! [B: nat,A: nat,C: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.49 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.32/0.49 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div_commute
% 0.32/0.49 thf(fact_619_unit__div__commute,axiom,
% 0.32/0.49 ! [B: int,A: int,C: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.49 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.32/0.49 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_div_commute
% 0.32/0.49 thf(fact_620_div__mult__unit2,axiom,
% 0.32/0.49 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.32/0.49 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.49 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.49 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_mult_unit2
% 0.32/0.49 thf(fact_621_div__mult__unit2,axiom,
% 0.32/0.49 ! [C: nat,B: nat,A: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.32/0.49 => ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.49 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.49 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_mult_unit2
% 0.32/0.49 thf(fact_622_div__mult__unit2,axiom,
% 0.32/0.49 ! [C: int,B: int,A: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ C @ one_one_int )
% 0.32/0.49 => ( ( dvd_dvd_int @ B @ A )
% 0.32/0.49 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.49 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_mult_unit2
% 0.32/0.49 thf(fact_623_unit__eq__div2,axiom,
% 0.32/0.49 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.49 => ( ( A
% 0.32/0.49 = ( divide6298287555418463151nteger @ C @ B ) )
% 0.32/0.49 = ( ( times_3573771949741848930nteger @ A @ B )
% 0.32/0.49 = C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_eq_div2
% 0.32/0.49 thf(fact_624_unit__eq__div2,axiom,
% 0.32/0.49 ! [B: nat,A: nat,C: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.49 => ( ( A
% 0.32/0.49 = ( divide_divide_nat @ C @ B ) )
% 0.32/0.49 = ( ( times_times_nat @ A @ B )
% 0.32/0.49 = C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_eq_div2
% 0.32/0.49 thf(fact_625_unit__eq__div2,axiom,
% 0.32/0.49 ! [B: int,A: int,C: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.49 => ( ( A
% 0.32/0.49 = ( divide_divide_int @ C @ B ) )
% 0.32/0.49 = ( ( times_times_int @ A @ B )
% 0.32/0.49 = C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_eq_div2
% 0.32/0.49 thf(fact_626_unit__eq__div1,axiom,
% 0.32/0.49 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.49 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.32/0.49 = C )
% 0.32/0.49 = ( A
% 0.32/0.49 = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_eq_div1
% 0.32/0.49 thf(fact_627_unit__eq__div1,axiom,
% 0.32/0.49 ! [B: nat,A: nat,C: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.49 => ( ( ( divide_divide_nat @ A @ B )
% 0.32/0.49 = C )
% 0.32/0.49 = ( A
% 0.32/0.49 = ( times_times_nat @ C @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_eq_div1
% 0.32/0.49 thf(fact_628_unit__eq__div1,axiom,
% 0.32/0.49 ! [B: int,A: int,C: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.49 => ( ( ( divide_divide_int @ A @ B )
% 0.32/0.49 = C )
% 0.32/0.49 = ( A
% 0.32/0.49 = ( times_times_int @ C @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_eq_div1
% 0.32/0.49 thf(fact_629_mult__right__cancel,axiom,
% 0.32/0.49 ! [C: real,A: real,B: real] :
% 0.32/0.49 ( ( C != zero_zero_real )
% 0.32/0.49 => ( ( ( times_times_real @ A @ C )
% 0.32/0.49 = ( times_times_real @ B @ C ) )
% 0.32/0.49 = ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_right_cancel
% 0.32/0.49 thf(fact_630_mult__right__cancel,axiom,
% 0.32/0.49 ! [C: rat,A: rat,B: rat] :
% 0.32/0.49 ( ( C != zero_zero_rat )
% 0.32/0.49 => ( ( ( times_times_rat @ A @ C )
% 0.32/0.49 = ( times_times_rat @ B @ C ) )
% 0.32/0.49 = ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_right_cancel
% 0.32/0.49 thf(fact_631_mult__right__cancel,axiom,
% 0.32/0.49 ! [C: nat,A: nat,B: nat] :
% 0.32/0.49 ( ( C != zero_zero_nat )
% 0.32/0.49 => ( ( ( times_times_nat @ A @ C )
% 0.32/0.49 = ( times_times_nat @ B @ C ) )
% 0.32/0.49 = ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_right_cancel
% 0.32/0.49 thf(fact_632_mult__right__cancel,axiom,
% 0.32/0.49 ! [C: int,A: int,B: int] :
% 0.32/0.49 ( ( C != zero_zero_int )
% 0.32/0.49 => ( ( ( times_times_int @ A @ C )
% 0.32/0.49 = ( times_times_int @ B @ C ) )
% 0.32/0.49 = ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_right_cancel
% 0.32/0.49 thf(fact_633_mult__left__cancel,axiom,
% 0.32/0.49 ! [C: real,A: real,B: real] :
% 0.32/0.49 ( ( C != zero_zero_real )
% 0.32/0.49 => ( ( ( times_times_real @ C @ A )
% 0.32/0.49 = ( times_times_real @ C @ B ) )
% 0.32/0.49 = ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_left_cancel
% 0.32/0.49 thf(fact_634_mult__left__cancel,axiom,
% 0.32/0.49 ! [C: rat,A: rat,B: rat] :
% 0.32/0.49 ( ( C != zero_zero_rat )
% 0.32/0.49 => ( ( ( times_times_rat @ C @ A )
% 0.32/0.49 = ( times_times_rat @ C @ B ) )
% 0.32/0.49 = ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_left_cancel
% 0.32/0.49 thf(fact_635_mult__left__cancel,axiom,
% 0.32/0.49 ! [C: nat,A: nat,B: nat] :
% 0.32/0.49 ( ( C != zero_zero_nat )
% 0.32/0.49 => ( ( ( times_times_nat @ C @ A )
% 0.32/0.49 = ( times_times_nat @ C @ B ) )
% 0.32/0.49 = ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_left_cancel
% 0.32/0.49 thf(fact_636_mult__left__cancel,axiom,
% 0.32/0.49 ! [C: int,A: int,B: int] :
% 0.32/0.49 ( ( C != zero_zero_int )
% 0.32/0.49 => ( ( ( times_times_int @ C @ A )
% 0.32/0.49 = ( times_times_int @ C @ B ) )
% 0.32/0.49 = ( A = B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_left_cancel
% 0.32/0.49 thf(fact_637_no__zero__divisors,axiom,
% 0.32/0.49 ! [A: real,B: real] :
% 0.32/0.49 ( ( A != zero_zero_real )
% 0.32/0.49 => ( ( B != zero_zero_real )
% 0.32/0.49 => ( ( times_times_real @ A @ B )
% 0.32/0.49 != zero_zero_real ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % no_zero_divisors
% 0.32/0.49 thf(fact_638_no__zero__divisors,axiom,
% 0.32/0.49 ! [A: rat,B: rat] :
% 0.32/0.49 ( ( A != zero_zero_rat )
% 0.32/0.49 => ( ( B != zero_zero_rat )
% 0.32/0.49 => ( ( times_times_rat @ A @ B )
% 0.32/0.49 != zero_zero_rat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % no_zero_divisors
% 0.32/0.49 thf(fact_639_no__zero__divisors,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( A != zero_zero_nat )
% 0.32/0.49 => ( ( B != zero_zero_nat )
% 0.32/0.49 => ( ( times_times_nat @ A @ B )
% 0.32/0.49 != zero_zero_nat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % no_zero_divisors
% 0.32/0.49 thf(fact_640_no__zero__divisors,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( A != zero_zero_int )
% 0.32/0.49 => ( ( B != zero_zero_int )
% 0.32/0.49 => ( ( times_times_int @ A @ B )
% 0.32/0.49 != zero_zero_int ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % no_zero_divisors
% 0.32/0.49 thf(fact_641_divisors__zero,axiom,
% 0.32/0.49 ! [A: real,B: real] :
% 0.32/0.49 ( ( ( times_times_real @ A @ B )
% 0.32/0.49 = zero_zero_real )
% 0.32/0.49 => ( ( A = zero_zero_real )
% 0.32/0.49 | ( B = zero_zero_real ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divisors_zero
% 0.32/0.49 thf(fact_642_divisors__zero,axiom,
% 0.32/0.49 ! [A: rat,B: rat] :
% 0.32/0.49 ( ( ( times_times_rat @ A @ B )
% 0.32/0.49 = zero_zero_rat )
% 0.32/0.49 => ( ( A = zero_zero_rat )
% 0.32/0.49 | ( B = zero_zero_rat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divisors_zero
% 0.32/0.49 thf(fact_643_divisors__zero,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( ( times_times_nat @ A @ B )
% 0.32/0.49 = zero_zero_nat )
% 0.32/0.49 => ( ( A = zero_zero_nat )
% 0.32/0.49 | ( B = zero_zero_nat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divisors_zero
% 0.32/0.49 thf(fact_644_divisors__zero,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( ( times_times_int @ A @ B )
% 0.32/0.49 = zero_zero_int )
% 0.32/0.49 => ( ( A = zero_zero_int )
% 0.32/0.49 | ( B = zero_zero_int ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divisors_zero
% 0.32/0.49 thf(fact_645_mult__not__zero,axiom,
% 0.32/0.49 ! [A: real,B: real] :
% 0.32/0.49 ( ( ( times_times_real @ A @ B )
% 0.32/0.49 != zero_zero_real )
% 0.32/0.49 => ( ( A != zero_zero_real )
% 0.32/0.49 & ( B != zero_zero_real ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_not_zero
% 0.32/0.49 thf(fact_646_mult__not__zero,axiom,
% 0.32/0.49 ! [A: rat,B: rat] :
% 0.32/0.49 ( ( ( times_times_rat @ A @ B )
% 0.32/0.49 != zero_zero_rat )
% 0.32/0.49 => ( ( A != zero_zero_rat )
% 0.32/0.49 & ( B != zero_zero_rat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_not_zero
% 0.32/0.49 thf(fact_647_mult__not__zero,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( ( times_times_nat @ A @ B )
% 0.32/0.49 != zero_zero_nat )
% 0.32/0.49 => ( ( A != zero_zero_nat )
% 0.32/0.49 & ( B != zero_zero_nat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_not_zero
% 0.32/0.49 thf(fact_648_mult__not__zero,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( ( times_times_int @ A @ B )
% 0.32/0.49 != zero_zero_int )
% 0.32/0.49 => ( ( A != zero_zero_int )
% 0.32/0.49 & ( B != zero_zero_int ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_not_zero
% 0.32/0.49 thf(fact_649_left__diff__distrib,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.32/0.49 = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib
% 0.32/0.49 thf(fact_650_left__diff__distrib,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.32/0.49 = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib
% 0.32/0.49 thf(fact_651_left__diff__distrib,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.32/0.49 = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib
% 0.32/0.49 thf(fact_652_right__diff__distrib,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.32/0.49 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib
% 0.32/0.49 thf(fact_653_right__diff__distrib,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.32/0.49 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib
% 0.32/0.49 thf(fact_654_right__diff__distrib,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.32/0.49 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib
% 0.32/0.49 thf(fact_655_left__diff__distrib_H,axiom,
% 0.32/0.49 ! [B: real,C: real,A: real] :
% 0.32/0.49 ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
% 0.32/0.49 = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib'
% 0.32/0.49 thf(fact_656_left__diff__distrib_H,axiom,
% 0.32/0.49 ! [B: rat,C: rat,A: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
% 0.32/0.49 = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib'
% 0.32/0.49 thf(fact_657_left__diff__distrib_H,axiom,
% 0.32/0.49 ! [B: nat,C: nat,A: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
% 0.32/0.49 = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib'
% 0.32/0.49 thf(fact_658_left__diff__distrib_H,axiom,
% 0.32/0.49 ! [B: int,C: int,A: int] :
% 0.32/0.49 ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
% 0.32/0.49 = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib'
% 0.32/0.49 thf(fact_659_right__diff__distrib_H,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.32/0.49 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib'
% 0.32/0.49 thf(fact_660_right__diff__distrib_H,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.32/0.49 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib'
% 0.32/0.49 thf(fact_661_right__diff__distrib_H,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
% 0.32/0.49 = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib'
% 0.32/0.49 thf(fact_662_right__diff__distrib_H,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.32/0.49 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib'
% 0.32/0.49 thf(fact_663_times__divide__times__eq,axiom,
% 0.32/0.49 ! [X: complex,Y: complex,Z2: complex,W: complex] :
% 0.32/0.49 ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z2 @ W ) )
% 0.32/0.49 = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ Y @ W ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % times_divide_times_eq
% 0.32/0.49 thf(fact_664_times__divide__times__eq,axiom,
% 0.32/0.49 ! [X: real,Y: real,Z2: real,W: real] :
% 0.32/0.49 ( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z2 @ W ) )
% 0.32/0.49 = ( divide_divide_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ W ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % times_divide_times_eq
% 0.32/0.49 thf(fact_665_times__divide__times__eq,axiom,
% 0.32/0.49 ! [X: rat,Y: rat,Z2: rat,W: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z2 @ W ) )
% 0.32/0.49 = ( divide_divide_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ W ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % times_divide_times_eq
% 0.32/0.49 thf(fact_666_divide__divide__times__eq,axiom,
% 0.32/0.49 ! [X: complex,Y: complex,Z2: complex,W: complex] :
% 0.32/0.49 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z2 @ W ) )
% 0.32/0.49 = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y @ Z2 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_divide_times_eq
% 0.32/0.49 thf(fact_667_divide__divide__times__eq,axiom,
% 0.32/0.49 ! [X: real,Y: real,Z2: real,W: real] :
% 0.32/0.49 ( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z2 @ W ) )
% 0.32/0.49 = ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z2 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_divide_times_eq
% 0.32/0.49 thf(fact_668_divide__divide__times__eq,axiom,
% 0.32/0.49 ! [X: rat,Y: rat,Z2: rat,W: rat] :
% 0.32/0.49 ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z2 @ W ) )
% 0.32/0.49 = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y @ Z2 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_divide_times_eq
% 0.32/0.49 thf(fact_669_divide__divide__eq__left_H,axiom,
% 0.32/0.49 ! [A: complex,B: complex,C: complex] :
% 0.32/0.49 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 0.32/0.49 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_divide_eq_left'
% 0.32/0.49 thf(fact_670_divide__divide__eq__left_H,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 0.32/0.49 = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_divide_eq_left'
% 0.32/0.49 thf(fact_671_divide__divide__eq__left_H,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 0.32/0.49 = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_divide_eq_left'
% 0.32/0.49 thf(fact_672_Suc__mult__cancel1,axiom,
% 0.32/0.49 ! [K: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 0.32/0.49 = ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.32/0.49 = ( M = N ) ) ).
% 0.32/0.49
% 0.32/0.49 % Suc_mult_cancel1
% 0.32/0.49 thf(fact_673_dvd__triv__right,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_triv_right
% 0.32/0.49 thf(fact_674_dvd__triv__right,axiom,
% 0.32/0.49 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_triv_right
% 0.32/0.49 thf(fact_675_dvd__triv__right,axiom,
% 0.32/0.49 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_triv_right
% 0.32/0.49 thf(fact_676_dvd__triv__right,axiom,
% 0.32/0.49 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_triv_right
% 0.32/0.49 thf(fact_677_dvd__triv__right,axiom,
% 0.32/0.49 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_triv_right
% 0.32/0.49 thf(fact_678_dvd__mult__right,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.32/0.49 => ( dvd_dvd_Code_integer @ B @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_right
% 0.32/0.49 thf(fact_679_dvd__mult__right,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 0.32/0.49 => ( dvd_dvd_real @ B @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_right
% 0.32/0.49 thf(fact_680_dvd__mult__right,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.32/0.49 => ( dvd_dvd_rat @ B @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_right
% 0.32/0.49 thf(fact_681_dvd__mult__right,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.49 => ( dvd_dvd_nat @ B @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_right
% 0.32/0.49 thf(fact_682_dvd__mult__right,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.49 => ( dvd_dvd_int @ B @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_right
% 0.32/0.49 thf(fact_683_mult__dvd__mono,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.49 => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.32/0.49 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_dvd_mono
% 0.32/0.49 thf(fact_684_mult__dvd__mono,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.49 ( ( dvd_dvd_real @ A @ B )
% 0.32/0.49 => ( ( dvd_dvd_real @ C @ D )
% 0.32/0.49 => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_dvd_mono
% 0.32/0.49 thf(fact_685_mult__dvd__mono,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.49 ( ( dvd_dvd_rat @ A @ B )
% 0.32/0.49 => ( ( dvd_dvd_rat @ C @ D )
% 0.32/0.49 => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_dvd_mono
% 0.32/0.49 thf(fact_686_mult__dvd__mono,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.49 => ( ( dvd_dvd_nat @ C @ D )
% 0.32/0.49 => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_dvd_mono
% 0.32/0.49 thf(fact_687_mult__dvd__mono,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.49 => ( ( dvd_dvd_int @ C @ D )
% 0.32/0.49 => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_dvd_mono
% 0.32/0.49 thf(fact_688_dvd__triv__left,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_triv_left
% 0.32/0.49 thf(fact_689_dvd__triv__left,axiom,
% 0.32/0.49 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_triv_left
% 0.32/0.49 thf(fact_690_dvd__triv__left,axiom,
% 0.32/0.49 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_triv_left
% 0.32/0.49 thf(fact_691_dvd__triv__left,axiom,
% 0.32/0.49 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_triv_left
% 0.32/0.49 thf(fact_692_dvd__triv__left,axiom,
% 0.32/0.49 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_triv_left
% 0.32/0.49 thf(fact_693_dvd__mult__left,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.32/0.49 => ( dvd_dvd_Code_integer @ A @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_left
% 0.32/0.49 thf(fact_694_dvd__mult__left,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 0.32/0.49 => ( dvd_dvd_real @ A @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_left
% 0.32/0.49 thf(fact_695_dvd__mult__left,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.32/0.49 => ( dvd_dvd_rat @ A @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_left
% 0.32/0.49 thf(fact_696_dvd__mult__left,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.49 => ( dvd_dvd_nat @ A @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_left
% 0.32/0.49 thf(fact_697_dvd__mult__left,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.49 => ( dvd_dvd_int @ A @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult_left
% 0.32/0.49 thf(fact_698_dvd__mult2,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.49 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult2
% 0.32/0.49 thf(fact_699_dvd__mult2,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( dvd_dvd_real @ A @ B )
% 0.32/0.49 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult2
% 0.32/0.49 thf(fact_700_dvd__mult2,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( dvd_dvd_rat @ A @ B )
% 0.32/0.49 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult2
% 0.32/0.49 thf(fact_701_dvd__mult2,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.49 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult2
% 0.32/0.49 thf(fact_702_dvd__mult2,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.49 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult2
% 0.32/0.49 thf(fact_703_dvd__mult,axiom,
% 0.32/0.49 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ A @ C )
% 0.32/0.49 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult
% 0.32/0.49 thf(fact_704_dvd__mult,axiom,
% 0.32/0.49 ! [A: real,C: real,B: real] :
% 0.32/0.49 ( ( dvd_dvd_real @ A @ C )
% 0.32/0.49 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult
% 0.32/0.49 thf(fact_705_dvd__mult,axiom,
% 0.32/0.49 ! [A: rat,C: rat,B: rat] :
% 0.32/0.49 ( ( dvd_dvd_rat @ A @ C )
% 0.32/0.49 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult
% 0.32/0.49 thf(fact_706_dvd__mult,axiom,
% 0.32/0.49 ! [A: nat,C: nat,B: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ C )
% 0.32/0.49 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult
% 0.32/0.49 thf(fact_707_dvd__mult,axiom,
% 0.32/0.49 ! [A: int,C: int,B: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ C )
% 0.32/0.49 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_mult
% 0.32/0.49 thf(fact_708_dvd__def,axiom,
% 0.32/0.49 ( dvd_dvd_Code_integer
% 0.32/0.49 = ( ^ [B2: code_integer,A2: code_integer] :
% 0.32/0.49 ? [K2: code_integer] :
% 0.32/0.49 ( A2
% 0.32/0.49 = ( times_3573771949741848930nteger @ B2 @ K2 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_def
% 0.32/0.49 thf(fact_709_dvd__def,axiom,
% 0.32/0.49 ( dvd_dvd_real
% 0.32/0.49 = ( ^ [B2: real,A2: real] :
% 0.32/0.49 ? [K2: real] :
% 0.32/0.49 ( A2
% 0.32/0.49 = ( times_times_real @ B2 @ K2 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_def
% 0.32/0.49 thf(fact_710_dvd__def,axiom,
% 0.32/0.49 ( dvd_dvd_rat
% 0.32/0.49 = ( ^ [B2: rat,A2: rat] :
% 0.32/0.49 ? [K2: rat] :
% 0.32/0.49 ( A2
% 0.32/0.49 = ( times_times_rat @ B2 @ K2 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_def
% 0.32/0.49 thf(fact_711_dvd__def,axiom,
% 0.32/0.49 ( dvd_dvd_nat
% 0.32/0.49 = ( ^ [B2: nat,A2: nat] :
% 0.32/0.49 ? [K2: nat] :
% 0.32/0.49 ( A2
% 0.32/0.49 = ( times_times_nat @ B2 @ K2 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_def
% 0.32/0.49 thf(fact_712_dvd__def,axiom,
% 0.32/0.49 ( dvd_dvd_int
% 0.32/0.49 = ( ^ [B2: int,A2: int] :
% 0.32/0.49 ? [K2: int] :
% 0.32/0.49 ( A2
% 0.32/0.49 = ( times_times_int @ B2 @ K2 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_def
% 0.32/0.49 thf(fact_713_dvdI,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer,K: code_integer] :
% 0.32/0.49 ( ( A
% 0.32/0.49 = ( times_3573771949741848930nteger @ B @ K ) )
% 0.32/0.49 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvdI
% 0.32/0.49 thf(fact_714_dvdI,axiom,
% 0.32/0.49 ! [A: real,B: real,K: real] :
% 0.32/0.49 ( ( A
% 0.32/0.49 = ( times_times_real @ B @ K ) )
% 0.32/0.49 => ( dvd_dvd_real @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvdI
% 0.32/0.49 thf(fact_715_dvdI,axiom,
% 0.32/0.49 ! [A: rat,B: rat,K: rat] :
% 0.32/0.49 ( ( A
% 0.32/0.49 = ( times_times_rat @ B @ K ) )
% 0.32/0.49 => ( dvd_dvd_rat @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvdI
% 0.32/0.49 thf(fact_716_dvdI,axiom,
% 0.32/0.49 ! [A: nat,B: nat,K: nat] :
% 0.32/0.49 ( ( A
% 0.32/0.49 = ( times_times_nat @ B @ K ) )
% 0.32/0.49 => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvdI
% 0.32/0.49 thf(fact_717_dvdI,axiom,
% 0.32/0.49 ! [A: int,B: int,K: int] :
% 0.32/0.49 ( ( A
% 0.32/0.49 = ( times_times_int @ B @ K ) )
% 0.32/0.49 => ( dvd_dvd_int @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvdI
% 0.32/0.49 thf(fact_718_dvdE,axiom,
% 0.32/0.49 ! [B: code_integer,A: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.49 => ~ ! [K3: code_integer] :
% 0.32/0.49 ( A
% 0.32/0.49 != ( times_3573771949741848930nteger @ B @ K3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvdE
% 0.32/0.49 thf(fact_719_dvdE,axiom,
% 0.32/0.49 ! [B: real,A: real] :
% 0.32/0.49 ( ( dvd_dvd_real @ B @ A )
% 0.32/0.49 => ~ ! [K3: real] :
% 0.32/0.49 ( A
% 0.32/0.49 != ( times_times_real @ B @ K3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvdE
% 0.32/0.49 thf(fact_720_dvdE,axiom,
% 0.32/0.49 ! [B: rat,A: rat] :
% 0.32/0.49 ( ( dvd_dvd_rat @ B @ A )
% 0.32/0.49 => ~ ! [K3: rat] :
% 0.32/0.49 ( A
% 0.32/0.49 != ( times_times_rat @ B @ K3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvdE
% 0.32/0.49 thf(fact_721_dvdE,axiom,
% 0.32/0.49 ! [B: nat,A: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.49 => ~ ! [K3: nat] :
% 0.32/0.49 ( A
% 0.32/0.49 != ( times_times_nat @ B @ K3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvdE
% 0.32/0.49 thf(fact_722_dvdE,axiom,
% 0.32/0.49 ! [B: int,A: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ B @ A )
% 0.32/0.49 => ~ ! [K3: int] :
% 0.32/0.49 ( A
% 0.32/0.49 != ( times_times_int @ B @ K3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvdE
% 0.32/0.49 thf(fact_723_mult__0,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( times_times_nat @ zero_zero_nat @ N )
% 0.32/0.49 = zero_zero_nat ) ).
% 0.32/0.49
% 0.32/0.49 % mult_0
% 0.32/0.49 thf(fact_724_nat__mult__eq__cancel__disj,axiom,
% 0.32/0.49 ! [K: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ( times_times_nat @ K @ M )
% 0.32/0.49 = ( times_times_nat @ K @ N ) )
% 0.32/0.49 = ( ( K = zero_zero_nat )
% 0.32/0.49 | ( M = N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_mult_eq_cancel_disj
% 0.32/0.49 thf(fact_725_mult__of__nat__commute,axiom,
% 0.32/0.49 ! [X: nat,Y: complex] :
% 0.32/0.49 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
% 0.32/0.49 = ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_of_nat_commute
% 0.32/0.49 thf(fact_726_mult__of__nat__commute,axiom,
% 0.32/0.49 ! [X: nat,Y: real] :
% 0.32/0.49 ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
% 0.32/0.49 = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_of_nat_commute
% 0.32/0.49 thf(fact_727_mult__of__nat__commute,axiom,
% 0.32/0.49 ! [X: nat,Y: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
% 0.32/0.49 = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_of_nat_commute
% 0.32/0.49 thf(fact_728_mult__of__nat__commute,axiom,
% 0.32/0.49 ! [X: nat,Y: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
% 0.32/0.49 = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_of_nat_commute
% 0.32/0.49 thf(fact_729_mult__of__nat__commute,axiom,
% 0.32/0.49 ! [X: nat,Y: int] :
% 0.32/0.49 ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
% 0.32/0.49 = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_of_nat_commute
% 0.32/0.49 thf(fact_730_zero__neq__one,axiom,
% 0.32/0.49 zero_zero_complex != one_one_complex ).
% 0.32/0.49
% 0.32/0.49 % zero_neq_one
% 0.32/0.49 thf(fact_731_zero__neq__one,axiom,
% 0.32/0.49 zero_zero_real != one_one_real ).
% 0.32/0.49
% 0.32/0.49 % zero_neq_one
% 0.32/0.49 thf(fact_732_zero__neq__one,axiom,
% 0.32/0.49 zero_zero_rat != one_one_rat ).
% 0.32/0.49
% 0.32/0.49 % zero_neq_one
% 0.32/0.49 thf(fact_733_zero__neq__one,axiom,
% 0.32/0.49 zero_zero_nat != one_one_nat ).
% 0.32/0.49
% 0.32/0.49 % zero_neq_one
% 0.32/0.49 thf(fact_734_zero__neq__one,axiom,
% 0.32/0.49 zero_zero_int != one_one_int ).
% 0.32/0.49
% 0.32/0.49 % zero_neq_one
% 0.32/0.49 thf(fact_735_int__diff__cases,axiom,
% 0.32/0.49 ! [Z2: int] :
% 0.32/0.49 ~ ! [M2: nat,N2: nat] :
% 0.32/0.49 ( Z2
% 0.32/0.49 != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % int_diff_cases
% 0.32/0.49 thf(fact_736_diff__mult__distrib,axiom,
% 0.32/0.49 ! [M: nat,N: nat,K: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 0.32/0.49 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % diff_mult_distrib
% 0.32/0.49 thf(fact_737_diff__mult__distrib2,axiom,
% 0.32/0.49 ! [K: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.49 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % diff_mult_distrib2
% 0.32/0.49 thf(fact_738_dvd__unit__imp__unit,axiom,
% 0.32/0.49 ! [A: code_integer,B: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.49 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.49 => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_unit_imp_unit
% 0.32/0.49 thf(fact_739_dvd__unit__imp__unit,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.49 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.49 => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_unit_imp_unit
% 0.32/0.49 thf(fact_740_dvd__unit__imp__unit,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.49 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.49 => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % dvd_unit_imp_unit
% 0.32/0.49 thf(fact_741_unit__imp__dvd,axiom,
% 0.32/0.49 ! [B: code_integer,A: code_integer] :
% 0.32/0.49 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.49 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_imp_dvd
% 0.32/0.49 thf(fact_742_unit__imp__dvd,axiom,
% 0.32/0.49 ! [B: nat,A: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.49 => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_imp_dvd
% 0.32/0.49 thf(fact_743_unit__imp__dvd,axiom,
% 0.32/0.49 ! [B: int,A: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.49 => ( dvd_dvd_int @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % unit_imp_dvd
% 0.32/0.49 thf(fact_744_one__dvd,axiom,
% 0.32/0.49 ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 0.32/0.49
% 0.32/0.49 % one_dvd
% 0.32/0.49 thf(fact_745_one__dvd,axiom,
% 0.32/0.49 ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 0.32/0.49
% 0.32/0.49 % one_dvd
% 0.32/0.49 thf(fact_746_one__dvd,axiom,
% 0.32/0.49 ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 0.32/0.49
% 0.32/0.49 % one_dvd
% 0.32/0.49 thf(fact_747_one__dvd,axiom,
% 0.32/0.49 ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 0.32/0.49
% 0.32/0.49 % one_dvd
% 0.32/0.49 thf(fact_748_one__dvd,axiom,
% 0.32/0.49 ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 0.32/0.49
% 0.32/0.49 % one_dvd
% 0.32/0.49 thf(fact_749_one__dvd,axiom,
% 0.32/0.49 ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 0.32/0.49
% 0.32/0.49 % one_dvd
% 0.32/0.49 thf(fact_750_zdvd__zdiffD,axiom,
% 0.32/0.49 ! [K: int,M: int,N: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
% 0.32/0.49 => ( ( dvd_dvd_int @ K @ N )
% 0.32/0.49 => ( dvd_dvd_int @ K @ M ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % zdvd_zdiffD
% 0.32/0.49 thf(fact_751_div__mult2__eq,axiom,
% 0.32/0.49 ! [M: nat,N: nat,Q2: nat] :
% 0.32/0.49 ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 0.32/0.49 = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_mult2_eq
% 0.32/0.49 thf(fact_752_zdvd__mult__cancel,axiom,
% 0.32/0.49 ! [K: int,M: int,N: int] :
% 0.32/0.49 ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
% 0.32/0.49 => ( ( K != zero_zero_int )
% 0.32/0.49 => ( dvd_dvd_int @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % zdvd_mult_cancel
% 0.32/0.49 thf(fact_753_bezout1__nat,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ? [D2: nat,X4: nat,Y4: nat] :
% 0.32/0.49 ( ( dvd_dvd_nat @ D2 @ A )
% 0.32/0.49 & ( dvd_dvd_nat @ D2 @ B )
% 0.32/0.49 & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y4 ) )
% 0.32/0.49 = D2 )
% 0.32/0.49 | ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y4 ) )
% 0.32/0.49 = D2 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % bezout1_nat
% 0.32/0.49 thf(fact_754_push__bit__double,axiom,
% 0.32/0.49 ! [N: nat,A: nat] :
% 0.32/0.49 ( ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.49 = ( times_times_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % push_bit_double
% 0.32/0.50 thf(fact_755_push__bit__double,axiom,
% 0.32/0.50 ! [N: nat,A: int] :
% 0.32/0.50 ( ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.32/0.50 = ( times_times_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % push_bit_double
% 0.32/0.50 thf(fact_756_push__bit__double,axiom,
% 0.32/0.50 ! [N: nat,A: code_integer] :
% 0.32/0.50 ( ( bit_se7788150548672797655nteger @ N @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.32/0.50 = ( times_3573771949741848930nteger @ ( bit_se7788150548672797655nteger @ N @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % push_bit_double
% 0.32/0.50 thf(fact_757_push__bit__double,axiom,
% 0.32/0.50 ! [N: nat,A: code_natural] :
% 0.32/0.50 ( ( bit_se6611745700429515170atural @ N @ ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
% 0.32/0.50 = ( times_2397367101498566445atural @ ( bit_se6611745700429515170atural @ N @ A ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % push_bit_double
% 0.32/0.50 thf(fact_758_is__unit__div__mult__cancel__right,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.50 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.50 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
% 0.32/0.50 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % is_unit_div_mult_cancel_right
% 0.32/0.50 thf(fact_759_is__unit__div__mult__cancel__right,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( A != zero_zero_nat )
% 0.32/0.50 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.50 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
% 0.32/0.50 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % is_unit_div_mult_cancel_right
% 0.32/0.50 thf(fact_760_is__unit__div__mult__cancel__right,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( A != zero_zero_int )
% 0.32/0.50 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.50 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
% 0.32/0.50 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % is_unit_div_mult_cancel_right
% 0.32/0.50 thf(fact_761_is__unit__div__mult__cancel__left,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.50 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.50 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.32/0.50 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % is_unit_div_mult_cancel_left
% 0.32/0.50 thf(fact_762_is__unit__div__mult__cancel__left,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( A != zero_zero_nat )
% 0.32/0.50 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.50 => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
% 0.32/0.50 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % is_unit_div_mult_cancel_left
% 0.32/0.50 thf(fact_763_is__unit__div__mult__cancel__left,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( A != zero_zero_int )
% 0.32/0.50 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.50 => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
% 0.32/0.50 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % is_unit_div_mult_cancel_left
% 0.32/0.50 thf(fact_764_is__unitE,axiom,
% 0.32/0.50 ! [A: code_integer,C: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.50 => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.50 => ! [B4: code_integer] :
% 0.32/0.50 ( ( B4 != zero_z3403309356797280102nteger )
% 0.32/0.50 => ( ( dvd_dvd_Code_integer @ B4 @ one_one_Code_integer )
% 0.32/0.50 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 0.32/0.50 = B4 )
% 0.32/0.50 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B4 )
% 0.32/0.50 = A )
% 0.32/0.50 => ( ( ( times_3573771949741848930nteger @ A @ B4 )
% 0.32/0.50 = one_one_Code_integer )
% 0.32/0.50 => ( ( divide6298287555418463151nteger @ C @ A )
% 0.32/0.50 != ( times_3573771949741848930nteger @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % is_unitE
% 0.32/0.50 thf(fact_765_is__unitE,axiom,
% 0.32/0.50 ! [A: nat,C: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.50 => ~ ( ( A != zero_zero_nat )
% 0.32/0.50 => ! [B4: nat] :
% 0.32/0.50 ( ( B4 != zero_zero_nat )
% 0.32/0.50 => ( ( dvd_dvd_nat @ B4 @ one_one_nat )
% 0.32/0.50 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 0.32/0.50 = B4 )
% 0.32/0.50 => ( ( ( divide_divide_nat @ one_one_nat @ B4 )
% 0.32/0.50 = A )
% 0.32/0.50 => ( ( ( times_times_nat @ A @ B4 )
% 0.32/0.50 = one_one_nat )
% 0.32/0.50 => ( ( divide_divide_nat @ C @ A )
% 0.32/0.50 != ( times_times_nat @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % is_unitE
% 0.32/0.50 thf(fact_766_is__unitE,axiom,
% 0.32/0.50 ! [A: int,C: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.50 => ~ ( ( A != zero_zero_int )
% 0.32/0.50 => ! [B4: int] :
% 0.32/0.50 ( ( B4 != zero_zero_int )
% 0.32/0.50 => ( ( dvd_dvd_int @ B4 @ one_one_int )
% 0.32/0.50 => ( ( ( divide_divide_int @ one_one_int @ A )
% 0.32/0.50 = B4 )
% 0.32/0.50 => ( ( ( divide_divide_int @ one_one_int @ B4 )
% 0.32/0.50 = A )
% 0.32/0.50 => ( ( ( times_times_int @ A @ B4 )
% 0.32/0.50 = one_one_int )
% 0.32/0.50 => ( ( divide_divide_int @ C @ A )
% 0.32/0.50 != ( times_times_int @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % is_unitE
% 0.32/0.50 thf(fact_767_frac__eq__eq,axiom,
% 0.32/0.50 ! [Y: complex,Z2: complex,X: complex,W: complex] :
% 0.32/0.50 ( ( Y != zero_zero_complex )
% 0.32/0.50 => ( ( Z2 != zero_zero_complex )
% 0.32/0.50 => ( ( ( divide1717551699836669952omplex @ X @ Y )
% 0.32/0.50 = ( divide1717551699836669952omplex @ W @ Z2 ) )
% 0.32/0.50 = ( ( times_times_complex @ X @ Z2 )
% 0.32/0.50 = ( times_times_complex @ W @ Y ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % frac_eq_eq
% 0.32/0.50 thf(fact_768_frac__eq__eq,axiom,
% 0.32/0.50 ! [Y: real,Z2: real,X: real,W: real] :
% 0.32/0.50 ( ( Y != zero_zero_real )
% 0.32/0.50 => ( ( Z2 != zero_zero_real )
% 0.32/0.50 => ( ( ( divide_divide_real @ X @ Y )
% 0.32/0.50 = ( divide_divide_real @ W @ Z2 ) )
% 0.32/0.50 = ( ( times_times_real @ X @ Z2 )
% 0.32/0.50 = ( times_times_real @ W @ Y ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % frac_eq_eq
% 0.32/0.50 thf(fact_769_frac__eq__eq,axiom,
% 0.32/0.50 ! [Y: rat,Z2: rat,X: rat,W: rat] :
% 0.32/0.50 ( ( Y != zero_zero_rat )
% 0.32/0.50 => ( ( Z2 != zero_zero_rat )
% 0.32/0.50 => ( ( ( divide_divide_rat @ X @ Y )
% 0.32/0.50 = ( divide_divide_rat @ W @ Z2 ) )
% 0.32/0.50 = ( ( times_times_rat @ X @ Z2 )
% 0.32/0.50 = ( times_times_rat @ W @ Y ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % frac_eq_eq
% 0.32/0.50 thf(fact_770_divide__eq__eq,axiom,
% 0.32/0.50 ! [B: complex,C: complex,A: complex] :
% 0.32/0.50 ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.32/0.50 = A )
% 0.32/0.50 = ( ( ( C != zero_zero_complex )
% 0.32/0.50 => ( B
% 0.32/0.50 = ( times_times_complex @ A @ C ) ) )
% 0.32/0.50 & ( ( C = zero_zero_complex )
% 0.32/0.50 => ( A = zero_zero_complex ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_eq_eq
% 0.32/0.50 thf(fact_771_divide__eq__eq,axiom,
% 0.32/0.50 ! [B: real,C: real,A: real] :
% 0.32/0.50 ( ( ( divide_divide_real @ B @ C )
% 0.32/0.50 = A )
% 0.32/0.50 = ( ( ( C != zero_zero_real )
% 0.32/0.50 => ( B
% 0.32/0.50 = ( times_times_real @ A @ C ) ) )
% 0.32/0.50 & ( ( C = zero_zero_real )
% 0.32/0.50 => ( A = zero_zero_real ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_eq_eq
% 0.32/0.50 thf(fact_772_divide__eq__eq,axiom,
% 0.32/0.50 ! [B: rat,C: rat,A: rat] :
% 0.32/0.50 ( ( ( divide_divide_rat @ B @ C )
% 0.32/0.50 = A )
% 0.32/0.50 = ( ( ( C != zero_zero_rat )
% 0.32/0.50 => ( B
% 0.32/0.50 = ( times_times_rat @ A @ C ) ) )
% 0.32/0.50 & ( ( C = zero_zero_rat )
% 0.32/0.50 => ( A = zero_zero_rat ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_eq_eq
% 0.32/0.50 thf(fact_773_eq__divide__eq,axiom,
% 0.32/0.50 ! [A: complex,B: complex,C: complex] :
% 0.32/0.50 ( ( A
% 0.32/0.50 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.50 = ( ( ( C != zero_zero_complex )
% 0.32/0.50 => ( ( times_times_complex @ A @ C )
% 0.32/0.50 = B ) )
% 0.32/0.50 & ( ( C = zero_zero_complex )
% 0.32/0.50 => ( A = zero_zero_complex ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % eq_divide_eq
% 0.32/0.50 thf(fact_774_eq__divide__eq,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( A
% 0.32/0.50 = ( divide_divide_real @ B @ C ) )
% 0.32/0.50 = ( ( ( C != zero_zero_real )
% 0.32/0.50 => ( ( times_times_real @ A @ C )
% 0.32/0.50 = B ) )
% 0.32/0.50 & ( ( C = zero_zero_real )
% 0.32/0.50 => ( A = zero_zero_real ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % eq_divide_eq
% 0.32/0.50 thf(fact_775_eq__divide__eq,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( A
% 0.32/0.50 = ( divide_divide_rat @ B @ C ) )
% 0.32/0.50 = ( ( ( C != zero_zero_rat )
% 0.32/0.50 => ( ( times_times_rat @ A @ C )
% 0.32/0.50 = B ) )
% 0.32/0.50 & ( ( C = zero_zero_rat )
% 0.32/0.50 => ( A = zero_zero_rat ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % eq_divide_eq
% 0.32/0.50 thf(fact_776_divide__eq__imp,axiom,
% 0.32/0.50 ! [C: complex,B: complex,A: complex] :
% 0.32/0.50 ( ( C != zero_zero_complex )
% 0.32/0.50 => ( ( B
% 0.32/0.50 = ( times_times_complex @ A @ C ) )
% 0.32/0.50 => ( ( divide1717551699836669952omplex @ B @ C )
% 0.32/0.50 = A ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_eq_imp
% 0.32/0.50 thf(fact_777_divide__eq__imp,axiom,
% 0.32/0.50 ! [C: real,B: real,A: real] :
% 0.32/0.50 ( ( C != zero_zero_real )
% 0.32/0.50 => ( ( B
% 0.32/0.50 = ( times_times_real @ A @ C ) )
% 0.32/0.50 => ( ( divide_divide_real @ B @ C )
% 0.32/0.50 = A ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_eq_imp
% 0.32/0.50 thf(fact_778_divide__eq__imp,axiom,
% 0.32/0.50 ! [C: rat,B: rat,A: rat] :
% 0.32/0.50 ( ( C != zero_zero_rat )
% 0.32/0.50 => ( ( B
% 0.32/0.50 = ( times_times_rat @ A @ C ) )
% 0.32/0.50 => ( ( divide_divide_rat @ B @ C )
% 0.32/0.50 = A ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_eq_imp
% 0.32/0.50 thf(fact_779_eq__divide__imp,axiom,
% 0.32/0.50 ! [C: complex,A: complex,B: complex] :
% 0.32/0.50 ( ( C != zero_zero_complex )
% 0.32/0.50 => ( ( ( times_times_complex @ A @ C )
% 0.32/0.50 = B )
% 0.32/0.50 => ( A
% 0.32/0.50 = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % eq_divide_imp
% 0.32/0.50 thf(fact_780_eq__divide__imp,axiom,
% 0.32/0.50 ! [C: real,A: real,B: real] :
% 0.32/0.50 ( ( C != zero_zero_real )
% 0.32/0.50 => ( ( ( times_times_real @ A @ C )
% 0.32/0.50 = B )
% 0.32/0.50 => ( A
% 0.32/0.50 = ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % eq_divide_imp
% 0.32/0.50 thf(fact_781_eq__divide__imp,axiom,
% 0.32/0.50 ! [C: rat,A: rat,B: rat] :
% 0.32/0.50 ( ( C != zero_zero_rat )
% 0.32/0.50 => ( ( ( times_times_rat @ A @ C )
% 0.32/0.50 = B )
% 0.32/0.50 => ( A
% 0.32/0.50 = ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % eq_divide_imp
% 0.32/0.50 thf(fact_782_nonzero__divide__eq__eq,axiom,
% 0.32/0.50 ! [C: complex,B: complex,A: complex] :
% 0.32/0.50 ( ( C != zero_zero_complex )
% 0.32/0.50 => ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.32/0.50 = A )
% 0.32/0.50 = ( B
% 0.32/0.50 = ( times_times_complex @ A @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % nonzero_divide_eq_eq
% 0.32/0.50 thf(fact_783_nonzero__divide__eq__eq,axiom,
% 0.32/0.50 ! [C: real,B: real,A: real] :
% 0.32/0.50 ( ( C != zero_zero_real )
% 0.32/0.50 => ( ( ( divide_divide_real @ B @ C )
% 0.32/0.50 = A )
% 0.32/0.50 = ( B
% 0.32/0.50 = ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % nonzero_divide_eq_eq
% 0.32/0.50 thf(fact_784_nonzero__divide__eq__eq,axiom,
% 0.32/0.50 ! [C: rat,B: rat,A: rat] :
% 0.32/0.50 ( ( C != zero_zero_rat )
% 0.32/0.50 => ( ( ( divide_divide_rat @ B @ C )
% 0.32/0.50 = A )
% 0.32/0.50 = ( B
% 0.32/0.50 = ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % nonzero_divide_eq_eq
% 0.32/0.50 thf(fact_785_nonzero__eq__divide__eq,axiom,
% 0.32/0.50 ! [C: complex,A: complex,B: complex] :
% 0.32/0.50 ( ( C != zero_zero_complex )
% 0.32/0.50 => ( ( A
% 0.32/0.50 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.50 = ( ( times_times_complex @ A @ C )
% 0.32/0.50 = B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % nonzero_eq_divide_eq
% 0.32/0.50 thf(fact_786_nonzero__eq__divide__eq,axiom,
% 0.32/0.50 ! [C: real,A: real,B: real] :
% 0.32/0.50 ( ( C != zero_zero_real )
% 0.32/0.50 => ( ( A
% 0.32/0.50 = ( divide_divide_real @ B @ C ) )
% 0.32/0.50 = ( ( times_times_real @ A @ C )
% 0.32/0.50 = B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % nonzero_eq_divide_eq
% 0.32/0.50 thf(fact_787_nonzero__eq__divide__eq,axiom,
% 0.32/0.50 ! [C: rat,A: rat,B: rat] :
% 0.32/0.50 ( ( C != zero_zero_rat )
% 0.32/0.50 => ( ( A
% 0.32/0.50 = ( divide_divide_rat @ B @ C ) )
% 0.32/0.50 = ( ( times_times_rat @ A @ C )
% 0.32/0.50 = B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % nonzero_eq_divide_eq
% 0.32/0.50 thf(fact_788_mult__numeral__1,axiom,
% 0.32/0.50 ! [A: complex] :
% 0.32/0.50 ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_numeral_1
% 0.32/0.50 thf(fact_789_mult__numeral__1,axiom,
% 0.32/0.50 ! [A: real] :
% 0.32/0.50 ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_numeral_1
% 0.32/0.50 thf(fact_790_mult__numeral__1,axiom,
% 0.32/0.50 ! [A: rat] :
% 0.32/0.50 ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_numeral_1
% 0.32/0.50 thf(fact_791_mult__numeral__1,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_numeral_1
% 0.32/0.50 thf(fact_792_mult__numeral__1,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_numeral_1
% 0.32/0.50 thf(fact_793_mult__numeral__1__right,axiom,
% 0.32/0.50 ! [A: complex] :
% 0.32/0.50 ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_numeral_1_right
% 0.32/0.50 thf(fact_794_mult__numeral__1__right,axiom,
% 0.32/0.50 ! [A: real] :
% 0.32/0.50 ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_numeral_1_right
% 0.32/0.50 thf(fact_795_mult__numeral__1__right,axiom,
% 0.32/0.50 ! [A: rat] :
% 0.32/0.50 ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_numeral_1_right
% 0.32/0.50 thf(fact_796_mult__numeral__1__right,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_numeral_1_right
% 0.32/0.50 thf(fact_797_mult__numeral__1__right,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_numeral_1_right
% 0.32/0.50 thf(fact_798_div__mult2__numeral__eq,axiom,
% 0.32/0.50 ! [A: nat,K: num,L: num] :
% 0.32/0.50 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
% 0.32/0.50 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult2_numeral_eq
% 0.32/0.50 thf(fact_799_div__mult2__numeral__eq,axiom,
% 0.32/0.50 ! [A: int,K: num,L: num] :
% 0.32/0.50 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
% 0.32/0.50 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult2_numeral_eq
% 0.32/0.50 thf(fact_800_dvd__div__mult,axiom,
% 0.32/0.50 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.50 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
% 0.32/0.50 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_mult
% 0.32/0.50 thf(fact_801_dvd__div__mult,axiom,
% 0.32/0.50 ! [C: nat,B: nat,A: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.50 => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
% 0.32/0.50 = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_mult
% 0.32/0.50 thf(fact_802_dvd__div__mult,axiom,
% 0.32/0.50 ! [C: int,B: int,A: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.50 => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
% 0.32/0.50 = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_mult
% 0.32/0.50 thf(fact_803_div__mult__swap,axiom,
% 0.32/0.50 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.50 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.50 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_swap
% 0.32/0.50 thf(fact_804_div__mult__swap,axiom,
% 0.32/0.50 ! [C: nat,B: nat,A: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.50 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.32/0.50 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_swap
% 0.32/0.50 thf(fact_805_div__mult__swap,axiom,
% 0.32/0.50 ! [C: int,B: int,A: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.50 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.32/0.50 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_swap
% 0.32/0.50 thf(fact_806_div__div__eq__right,axiom,
% 0.32/0.50 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.50 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.50 => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.50 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_div_eq_right
% 0.32/0.50 thf(fact_807_div__div__eq__right,axiom,
% 0.32/0.50 ! [C: nat,B: nat,A: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.50 => ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.50 => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.32/0.50 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_div_eq_right
% 0.32/0.50 thf(fact_808_div__div__eq__right,axiom,
% 0.32/0.50 ! [C: int,B: int,A: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.50 => ( ( dvd_dvd_int @ B @ A )
% 0.32/0.50 => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.32/0.50 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_div_eq_right
% 0.32/0.50 thf(fact_809_dvd__div__mult2__eq,axiom,
% 0.32/0.50 ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
% 0.32/0.50 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.50 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_mult2_eq
% 0.32/0.50 thf(fact_810_dvd__div__mult2__eq,axiom,
% 0.32/0.50 ! [B: nat,C: nat,A: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
% 0.32/0.50 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.50 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_mult2_eq
% 0.32/0.50 thf(fact_811_dvd__div__mult2__eq,axiom,
% 0.32/0.50 ! [B: int,C: int,A: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
% 0.32/0.50 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.50 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_mult2_eq
% 0.32/0.50 thf(fact_812_dvd__mult__imp__div,axiom,
% 0.32/0.50 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
% 0.32/0.50 => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_mult_imp_div
% 0.32/0.50 thf(fact_813_dvd__mult__imp__div,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
% 0.32/0.50 => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_mult_imp_div
% 0.32/0.50 thf(fact_814_dvd__mult__imp__div,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
% 0.32/0.50 => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_mult_imp_div
% 0.32/0.50 thf(fact_815_div__mult__div__if__dvd,axiom,
% 0.32/0.50 ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.50 => ( ( dvd_dvd_Code_integer @ D @ C )
% 0.32/0.50 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 0.32/0.50 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_div_if_dvd
% 0.32/0.50 thf(fact_816_div__mult__div__if__dvd,axiom,
% 0.32/0.50 ! [B: nat,A: nat,D: nat,C: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.50 => ( ( dvd_dvd_nat @ D @ C )
% 0.32/0.50 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
% 0.32/0.50 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_div_if_dvd
% 0.32/0.50 thf(fact_817_div__mult__div__if__dvd,axiom,
% 0.32/0.50 ! [B: int,A: int,D: int,C: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ B @ A )
% 0.32/0.50 => ( ( dvd_dvd_int @ D @ C )
% 0.32/0.50 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
% 0.32/0.50 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_div_if_dvd
% 0.32/0.50 thf(fact_818_div__mult2__eq_H,axiom,
% 0.32/0.50 ! [A: nat,M: nat,N: nat] :
% 0.32/0.50 ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 0.32/0.50 = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult2_eq'
% 0.32/0.50 thf(fact_819_div__mult2__eq_H,axiom,
% 0.32/0.50 ! [A: int,M: nat,N: nat] :
% 0.32/0.50 ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.32/0.50 = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult2_eq'
% 0.32/0.50 thf(fact_820_right__inverse__eq,axiom,
% 0.32/0.50 ! [B: complex,A: complex] :
% 0.32/0.50 ( ( B != zero_zero_complex )
% 0.32/0.50 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.32/0.50 = one_one_complex )
% 0.32/0.50 = ( A = B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % right_inverse_eq
% 0.32/0.50 thf(fact_821_right__inverse__eq,axiom,
% 0.32/0.50 ! [B: real,A: real] :
% 0.32/0.50 ( ( B != zero_zero_real )
% 0.32/0.50 => ( ( ( divide_divide_real @ A @ B )
% 0.32/0.50 = one_one_real )
% 0.32/0.50 = ( A = B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % right_inverse_eq
% 0.32/0.50 thf(fact_822_right__inverse__eq,axiom,
% 0.32/0.50 ! [B: rat,A: rat] :
% 0.32/0.50 ( ( B != zero_zero_rat )
% 0.32/0.50 => ( ( ( divide_divide_rat @ A @ B )
% 0.32/0.50 = one_one_rat )
% 0.32/0.50 = ( A = B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % right_inverse_eq
% 0.32/0.50 thf(fact_823_numeral__One,axiom,
% 0.32/0.50 ( ( numera6690914467698888265omplex @ one )
% 0.32/0.50 = one_one_complex ) ).
% 0.32/0.50
% 0.32/0.50 % numeral_One
% 0.32/0.50 thf(fact_824_numeral__One,axiom,
% 0.32/0.50 ( ( numeral_numeral_real @ one )
% 0.32/0.50 = one_one_real ) ).
% 0.32/0.50
% 0.32/0.50 % numeral_One
% 0.32/0.50 thf(fact_825_numeral__One,axiom,
% 0.32/0.50 ( ( numeral_numeral_rat @ one )
% 0.32/0.50 = one_one_rat ) ).
% 0.32/0.50
% 0.32/0.50 % numeral_One
% 0.32/0.50 thf(fact_826_numeral__One,axiom,
% 0.32/0.50 ( ( numeral_numeral_nat @ one )
% 0.32/0.50 = one_one_nat ) ).
% 0.32/0.50
% 0.32/0.50 % numeral_One
% 0.32/0.50 thf(fact_827_numeral__One,axiom,
% 0.32/0.50 ( ( numeral_numeral_int @ one )
% 0.32/0.50 = one_one_int ) ).
% 0.32/0.50
% 0.32/0.50 % numeral_One
% 0.32/0.50 thf(fact_828_not__is__unit__0,axiom,
% 0.32/0.50 ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 0.32/0.50
% 0.32/0.50 % not_is_unit_0
% 0.32/0.50 thf(fact_829_not__is__unit__0,axiom,
% 0.32/0.50 ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 0.32/0.50
% 0.32/0.50 % not_is_unit_0
% 0.32/0.50 thf(fact_830_not__is__unit__0,axiom,
% 0.32/0.50 ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 0.32/0.50
% 0.32/0.50 % not_is_unit_0
% 0.32/0.50 thf(fact_831_unit__div__cancel,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.50 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.32/0.50 = ( divide6298287555418463151nteger @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % unit_div_cancel
% 0.32/0.50 thf(fact_832_unit__div__cancel,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.50 => ( ( ( divide_divide_nat @ B @ A )
% 0.32/0.50 = ( divide_divide_nat @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % unit_div_cancel
% 0.32/0.50 thf(fact_833_unit__div__cancel,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.50 => ( ( ( divide_divide_int @ B @ A )
% 0.32/0.50 = ( divide_divide_int @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % unit_div_cancel
% 0.32/0.50 thf(fact_834_div__unit__dvd__iff,axiom,
% 0.32/0.50 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.50 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.32/0.50 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_unit_dvd_iff
% 0.32/0.50 thf(fact_835_div__unit__dvd__iff,axiom,
% 0.32/0.50 ! [B: nat,A: nat,C: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.50 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.32/0.50 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_unit_dvd_iff
% 0.32/0.50 thf(fact_836_div__unit__dvd__iff,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.50 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.32/0.50 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_unit_dvd_iff
% 0.32/0.50 thf(fact_837_dvd__div__unit__iff,axiom,
% 0.32/0.50 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.50 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
% 0.32/0.50 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_unit_iff
% 0.32/0.50 thf(fact_838_dvd__div__unit__iff,axiom,
% 0.32/0.50 ! [B: nat,A: nat,C: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.50 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
% 0.32/0.50 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_unit_iff
% 0.32/0.50 thf(fact_839_dvd__div__unit__iff,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.50 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
% 0.32/0.50 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_unit_iff
% 0.32/0.50 thf(fact_840_numerals_I1_J,axiom,
% 0.32/0.50 ( ( numeral_numeral_nat @ one )
% 0.32/0.50 = one_one_nat ) ).
% 0.32/0.50
% 0.32/0.50 % numerals(1)
% 0.32/0.50 thf(fact_841_One__nat__def,axiom,
% 0.32/0.50 ( one_one_nat
% 0.32/0.50 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.50
% 0.32/0.50 % One_nat_def
% 0.32/0.50 thf(fact_842_diff__Suc__eq__diff__pred,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.32/0.50 = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_Suc_eq_diff_pred
% 0.32/0.50 thf(fact_843_divide__eq__eq__numeral_I1_J,axiom,
% 0.32/0.50 ! [B: complex,C: complex,W: num] :
% 0.32/0.50 ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.32/0.50 = ( numera6690914467698888265omplex @ W ) )
% 0.32/0.50 = ( ( ( C != zero_zero_complex )
% 0.32/0.50 => ( B
% 0.32/0.50 = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
% 0.32/0.50 & ( ( C = zero_zero_complex )
% 0.32/0.50 => ( ( numera6690914467698888265omplex @ W )
% 0.32/0.50 = zero_zero_complex ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_eq_eq_numeral(1)
% 0.32/0.50 thf(fact_844_divide__eq__eq__numeral_I1_J,axiom,
% 0.32/0.50 ! [B: real,C: real,W: num] :
% 0.32/0.50 ( ( ( divide_divide_real @ B @ C )
% 0.32/0.50 = ( numeral_numeral_real @ W ) )
% 0.32/0.50 = ( ( ( C != zero_zero_real )
% 0.32/0.50 => ( B
% 0.32/0.50 = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.32/0.50 & ( ( C = zero_zero_real )
% 0.32/0.50 => ( ( numeral_numeral_real @ W )
% 0.32/0.50 = zero_zero_real ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_eq_eq_numeral(1)
% 0.32/0.50 thf(fact_845_divide__eq__eq__numeral_I1_J,axiom,
% 0.32/0.50 ! [B: rat,C: rat,W: num] :
% 0.32/0.50 ( ( ( divide_divide_rat @ B @ C )
% 0.32/0.50 = ( numeral_numeral_rat @ W ) )
% 0.32/0.50 = ( ( ( C != zero_zero_rat )
% 0.32/0.50 => ( B
% 0.32/0.50 = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.32/0.50 & ( ( C = zero_zero_rat )
% 0.32/0.50 => ( ( numeral_numeral_rat @ W )
% 0.32/0.50 = zero_zero_rat ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_eq_eq_numeral(1)
% 0.32/0.50 thf(fact_846_eq__divide__eq__numeral_I1_J,axiom,
% 0.32/0.50 ! [W: num,B: complex,C: complex] :
% 0.32/0.50 ( ( ( numera6690914467698888265omplex @ W )
% 0.32/0.50 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.50 = ( ( ( C != zero_zero_complex )
% 0.32/0.50 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
% 0.32/0.50 = B ) )
% 0.32/0.50 & ( ( C = zero_zero_complex )
% 0.32/0.50 => ( ( numera6690914467698888265omplex @ W )
% 0.32/0.50 = zero_zero_complex ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % eq_divide_eq_numeral(1)
% 0.32/0.50 thf(fact_847_eq__divide__eq__numeral_I1_J,axiom,
% 0.32/0.50 ! [W: num,B: real,C: real] :
% 0.32/0.50 ( ( ( numeral_numeral_real @ W )
% 0.32/0.50 = ( divide_divide_real @ B @ C ) )
% 0.32/0.50 = ( ( ( C != zero_zero_real )
% 0.32/0.50 => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
% 0.32/0.50 = B ) )
% 0.32/0.50 & ( ( C = zero_zero_real )
% 0.32/0.50 => ( ( numeral_numeral_real @ W )
% 0.32/0.50 = zero_zero_real ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % eq_divide_eq_numeral(1)
% 0.32/0.50 thf(fact_848_eq__divide__eq__numeral_I1_J,axiom,
% 0.32/0.50 ! [W: num,B: rat,C: rat] :
% 0.32/0.50 ( ( ( numeral_numeral_rat @ W )
% 0.32/0.50 = ( divide_divide_rat @ B @ C ) )
% 0.32/0.50 = ( ( ( C != zero_zero_rat )
% 0.32/0.50 => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
% 0.32/0.50 = B ) )
% 0.32/0.50 & ( ( C = zero_zero_rat )
% 0.32/0.50 => ( ( numeral_numeral_rat @ W )
% 0.32/0.50 = zero_zero_rat ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % eq_divide_eq_numeral(1)
% 0.32/0.50 thf(fact_849_add__divide__eq__if__simps_I4_J,axiom,
% 0.32/0.50 ! [Z2: complex,A: complex,B: complex] :
% 0.32/0.50 ( ( ( Z2 = zero_zero_complex )
% 0.32/0.50 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 0.32/0.50 = A ) )
% 0.32/0.50 & ( ( Z2 != zero_zero_complex )
% 0.32/0.50 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 0.32/0.50 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_divide_eq_if_simps(4)
% 0.32/0.50 thf(fact_850_add__divide__eq__if__simps_I4_J,axiom,
% 0.32/0.50 ! [Z2: real,A: real,B: real] :
% 0.32/0.50 ( ( ( Z2 = zero_zero_real )
% 0.32/0.50 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 0.32/0.50 = A ) )
% 0.32/0.50 & ( ( Z2 != zero_zero_real )
% 0.32/0.50 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 0.32/0.50 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_divide_eq_if_simps(4)
% 0.32/0.50 thf(fact_851_add__divide__eq__if__simps_I4_J,axiom,
% 0.32/0.50 ! [Z2: rat,A: rat,B: rat] :
% 0.32/0.50 ( ( ( Z2 = zero_zero_rat )
% 0.32/0.50 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 0.32/0.50 = A ) )
% 0.32/0.50 & ( ( Z2 != zero_zero_rat )
% 0.32/0.50 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 0.32/0.50 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_divide_eq_if_simps(4)
% 0.32/0.50 thf(fact_852_diff__frac__eq,axiom,
% 0.32/0.50 ! [Y: complex,Z2: complex,X: complex,W: complex] :
% 0.32/0.50 ( ( Y != zero_zero_complex )
% 0.32/0.50 => ( ( Z2 != zero_zero_complex )
% 0.32/0.50 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z2 ) )
% 0.32/0.50 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_frac_eq
% 0.32/0.50 thf(fact_853_diff__frac__eq,axiom,
% 0.32/0.50 ! [Y: real,Z2: real,X: real,W: real] :
% 0.32/0.50 ( ( Y != zero_zero_real )
% 0.32/0.50 => ( ( Z2 != zero_zero_real )
% 0.32/0.50 => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
% 0.32/0.50 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_frac_eq
% 0.32/0.50 thf(fact_854_diff__frac__eq,axiom,
% 0.32/0.50 ! [Y: rat,Z2: rat,X: rat,W: rat] :
% 0.32/0.50 ( ( Y != zero_zero_rat )
% 0.32/0.50 => ( ( Z2 != zero_zero_rat )
% 0.32/0.50 => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z2 ) )
% 0.32/0.50 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_frac_eq
% 0.32/0.50 thf(fact_855_diff__divide__eq__iff,axiom,
% 0.32/0.50 ! [Z2: complex,X: complex,Y: complex] :
% 0.32/0.50 ( ( Z2 != zero_zero_complex )
% 0.32/0.50 => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
% 0.32/0.50 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_divide_eq_iff
% 0.32/0.50 thf(fact_856_diff__divide__eq__iff,axiom,
% 0.32/0.50 ! [Z2: real,X: real,Y: real] :
% 0.32/0.50 ( ( Z2 != zero_zero_real )
% 0.32/0.50 => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z2 ) )
% 0.32/0.50 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_divide_eq_iff
% 0.32/0.50 thf(fact_857_diff__divide__eq__iff,axiom,
% 0.32/0.50 ! [Z2: rat,X: rat,Y: rat] :
% 0.32/0.50 ( ( Z2 != zero_zero_rat )
% 0.32/0.50 => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z2 ) )
% 0.32/0.50 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_divide_eq_iff
% 0.32/0.50 thf(fact_858_divide__diff__eq__iff,axiom,
% 0.32/0.50 ! [Z2: complex,X: complex,Y: complex] :
% 0.32/0.50 ( ( Z2 != zero_zero_complex )
% 0.32/0.50 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z2 ) @ Y )
% 0.32/0.50 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_diff_eq_iff
% 0.32/0.50 thf(fact_859_divide__diff__eq__iff,axiom,
% 0.32/0.50 ! [Z2: real,X: real,Y: real] :
% 0.32/0.50 ( ( Z2 != zero_zero_real )
% 0.32/0.50 => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z2 ) @ Y )
% 0.32/0.50 = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_diff_eq_iff
% 0.32/0.50 thf(fact_860_divide__diff__eq__iff,axiom,
% 0.32/0.50 ! [Z2: rat,X: rat,Y: rat] :
% 0.32/0.50 ( ( Z2 != zero_zero_rat )
% 0.32/0.50 => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y )
% 0.32/0.50 = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_diff_eq_iff
% 0.32/0.50 thf(fact_861_dvd__div__div__eq__mult,axiom,
% 0.32/0.50 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 0.32/0.50 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.50 => ( ( C != zero_z3403309356797280102nteger )
% 0.32/0.50 => ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.50 => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.32/0.50 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.32/0.50 = ( divide6298287555418463151nteger @ D @ C ) )
% 0.32/0.50 = ( ( times_3573771949741848930nteger @ B @ C )
% 0.32/0.50 = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_div_eq_mult
% 0.32/0.50 thf(fact_862_dvd__div__div__eq__mult,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat,D: nat] :
% 0.32/0.50 ( ( A != zero_zero_nat )
% 0.32/0.50 => ( ( C != zero_zero_nat )
% 0.32/0.50 => ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.50 => ( ( dvd_dvd_nat @ C @ D )
% 0.32/0.50 => ( ( ( divide_divide_nat @ B @ A )
% 0.32/0.50 = ( divide_divide_nat @ D @ C ) )
% 0.32/0.50 = ( ( times_times_nat @ B @ C )
% 0.32/0.50 = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_div_eq_mult
% 0.32/0.50 thf(fact_863_dvd__div__div__eq__mult,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int,D: int] :
% 0.32/0.50 ( ( A != zero_zero_int )
% 0.32/0.50 => ( ( C != zero_zero_int )
% 0.32/0.50 => ( ( dvd_dvd_int @ A @ B )
% 0.32/0.50 => ( ( dvd_dvd_int @ C @ D )
% 0.32/0.50 => ( ( ( divide_divide_int @ B @ A )
% 0.32/0.50 = ( divide_divide_int @ D @ C ) )
% 0.32/0.50 = ( ( times_times_int @ B @ C )
% 0.32/0.50 = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_div_eq_mult
% 0.32/0.50 thf(fact_864_dvd__div__iff__mult,axiom,
% 0.32/0.50 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.50 ( ( C != zero_z3403309356797280102nteger )
% 0.32/0.50 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.50 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.50 = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_iff_mult
% 0.32/0.50 thf(fact_865_dvd__div__iff__mult,axiom,
% 0.32/0.50 ! [C: nat,B: nat,A: nat] :
% 0.32/0.50 ( ( C != zero_zero_nat )
% 0.32/0.50 => ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.50 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.32/0.50 = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_iff_mult
% 0.32/0.50 thf(fact_866_dvd__div__iff__mult,axiom,
% 0.32/0.50 ! [C: int,B: int,A: int] :
% 0.32/0.50 ( ( C != zero_zero_int )
% 0.32/0.50 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.50 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.32/0.50 = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_iff_mult
% 0.32/0.50 thf(fact_867_div__dvd__iff__mult,axiom,
% 0.32/0.50 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.50 ( ( B != zero_z3403309356797280102nteger )
% 0.32/0.50 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.50 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.32/0.50 = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_dvd_iff_mult
% 0.32/0.50 thf(fact_868_div__dvd__iff__mult,axiom,
% 0.32/0.50 ! [B: nat,A: nat,C: nat] :
% 0.32/0.50 ( ( B != zero_zero_nat )
% 0.32/0.50 => ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.50 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.32/0.50 = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_dvd_iff_mult
% 0.32/0.50 thf(fact_869_div__dvd__iff__mult,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( B != zero_zero_int )
% 0.32/0.50 => ( ( dvd_dvd_int @ B @ A )
% 0.32/0.50 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.32/0.50 = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_dvd_iff_mult
% 0.32/0.50 thf(fact_870_dvd__div__eq__mult,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.50 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.50 => ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.50 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.32/0.50 = C )
% 0.32/0.50 = ( B
% 0.32/0.50 = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_eq_mult
% 0.32/0.50 thf(fact_871_dvd__div__eq__mult,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( A != zero_zero_nat )
% 0.32/0.50 => ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.50 => ( ( ( divide_divide_nat @ B @ A )
% 0.32/0.50 = C )
% 0.32/0.50 = ( B
% 0.32/0.50 = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_eq_mult
% 0.32/0.50 thf(fact_872_dvd__div__eq__mult,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( A != zero_zero_int )
% 0.32/0.50 => ( ( dvd_dvd_int @ A @ B )
% 0.32/0.50 => ( ( ( divide_divide_int @ B @ A )
% 0.32/0.50 = C )
% 0.32/0.50 = ( B
% 0.32/0.50 = ( times_times_int @ C @ A ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_div_eq_mult
% 0.32/0.50 thf(fact_873_unit__div__eq__0__iff,axiom,
% 0.32/0.50 ! [B: code_integer,A: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.50 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.32/0.50 = zero_z3403309356797280102nteger )
% 0.32/0.50 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % unit_div_eq_0_iff
% 0.32/0.50 thf(fact_874_unit__div__eq__0__iff,axiom,
% 0.32/0.50 ! [B: nat,A: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.50 => ( ( ( divide_divide_nat @ A @ B )
% 0.32/0.50 = zero_zero_nat )
% 0.32/0.50 = ( A = zero_zero_nat ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % unit_div_eq_0_iff
% 0.32/0.50 thf(fact_875_unit__div__eq__0__iff,axiom,
% 0.32/0.50 ! [B: int,A: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.50 => ( ( ( divide_divide_int @ A @ B )
% 0.32/0.50 = zero_zero_int )
% 0.32/0.50 = ( A = zero_zero_int ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % unit_div_eq_0_iff
% 0.32/0.50 thf(fact_876_gcd__nat_Onot__eq__order__implies__strict,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( A != B )
% 0.32/0.50 => ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.50 => ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.50 & ( A != B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % gcd_nat.not_eq_order_implies_strict
% 0.32/0.50 thf(fact_877_gcd__nat_Ostrict__implies__not__eq,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.50 & ( A != B ) )
% 0.32/0.50 => ( A != B ) ) ).
% 0.32/0.50
% 0.32/0.50 % gcd_nat.strict_implies_not_eq
% 0.32/0.50 thf(fact_878_gcd__nat_Ostrict__implies__order,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.50 & ( A != B ) )
% 0.32/0.50 => ( dvd_dvd_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % gcd_nat.strict_implies_order
% 0.32/0.50 thf(fact_879_gcd__nat_Ostrict__iff__order,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.50 & ( A != B ) )
% 0.32/0.50 = ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.50 & ( A != B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % gcd_nat.strict_iff_order
% 0.32/0.50 thf(fact_880_gcd__nat_Oorder__iff__strict,axiom,
% 0.32/0.50 ( dvd_dvd_nat
% 0.32/0.50 = ( ^ [A2: nat,B2: nat] :
% 0.32/0.50 ( ( ( dvd_dvd_nat @ A2 @ B2 )
% 0.32/0.50 & ( A2 != B2 ) )
% 0.32/0.50 | ( A2 = B2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % gcd_nat.order_iff_strict
% 0.32/0.50 thf(fact_881_gcd__nat_Ostrict__iff__not,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.50 & ( A != B ) )
% 0.32/0.50 = ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.50 & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % gcd_nat.strict_iff_not
% 0.32/0.50 thf(fact_882_gcd__nat_Ostrict__trans2,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.50 & ( A != B ) )
% 0.32/0.50 => ( ( dvd_dvd_nat @ B @ C )
% 0.32/0.50 => ( ( dvd_dvd_nat @ A @ C )
% 0.32/0.50 & ( A != C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % gcd_nat.strict_trans2
% 0.32/0.50 thf(fact_883_gcd__nat_Ostrict__trans1,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.50 => ( ( ( dvd_dvd_nat @ B @ C )
% 0.32/0.50 & ( B != C ) )
% 0.32/0.50 => ( ( dvd_dvd_nat @ A @ C )
% 0.32/0.50 & ( A != C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % gcd_nat.strict_trans1
% 0.32/0.50 thf(fact_884_gcd__nat_Ostrict__trans,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.50 & ( A != B ) )
% 0.32/0.50 => ( ( ( dvd_dvd_nat @ B @ C )
% 0.32/0.50 & ( B != C ) )
% 0.32/0.50 => ( ( dvd_dvd_nat @ A @ C )
% 0.32/0.50 & ( A != C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % gcd_nat.strict_trans
% 0.32/0.50 thf(fact_885_gcd__nat_Oantisym,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.50 => ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.50 => ( A = B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % gcd_nat.antisym
% 0.32/0.50 thf(fact_886_gcd__nat_Oirrefl,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ~ ( ( dvd_dvd_nat @ A @ A )
% 0.32/0.50 & ( A != A ) ) ).
% 0.32/0.50
% 0.32/0.50 % gcd_nat.irrefl
% 0.32/0.50 thf(fact_887_gcd__nat_Oeq__iff,axiom,
% 0.32/0.50 ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 0.32/0.50 = ( ^ [A2: nat,B2: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ A2 @ B2 )
% 0.32/0.50 & ( dvd_dvd_nat @ B2 @ A2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % gcd_nat.eq_iff
% 0.32/0.50 thf(fact_888_gcd__nat_Otrans,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.50 => ( ( dvd_dvd_nat @ B @ C )
% 0.32/0.50 => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % gcd_nat.trans
% 0.32/0.50 thf(fact_889_gcd__nat_Orefl,axiom,
% 0.32/0.50 ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 0.32/0.50
% 0.32/0.50 % gcd_nat.refl
% 0.32/0.50 thf(fact_890_gcd__nat_Oasym,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.50 & ( A != B ) )
% 0.32/0.50 => ~ ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.50 & ( B != A ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % gcd_nat.asym
% 0.32/0.50 thf(fact_891_evenE,axiom,
% 0.32/0.50 ! [A: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ~ ! [B4: code_integer] :
% 0.32/0.50 ( A
% 0.32/0.50 != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % evenE
% 0.32/0.50 thf(fact_892_evenE,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ~ ! [B4: nat] :
% 0.32/0.50 ( A
% 0.32/0.50 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % evenE
% 0.32/0.50 thf(fact_893_evenE,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ~ ! [B4: int] :
% 0.32/0.50 ( A
% 0.32/0.50 != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % evenE
% 0.32/0.50 thf(fact_894_Suc__double__not__eq__double,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.50 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.50
% 0.32/0.50 % Suc_double_not_eq_double
% 0.32/0.50 thf(fact_895_double__not__eq__Suc__double,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.32/0.50 != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % double_not_eq_Suc_double
% 0.32/0.50 thf(fact_896_odd__one,axiom,
% 0.32/0.50 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% 0.32/0.50
% 0.32/0.50 % odd_one
% 0.32/0.50 thf(fact_897_odd__one,axiom,
% 0.32/0.50 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% 0.32/0.50
% 0.32/0.50 % odd_one
% 0.32/0.50 thf(fact_898_odd__one,axiom,
% 0.32/0.50 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% 0.32/0.50
% 0.32/0.50 % odd_one
% 0.32/0.50 thf(fact_899_even__two__times__div__two,axiom,
% 0.32/0.50 ! [A: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.32/0.50 = A ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_two_times_div_two
% 0.32/0.50 thf(fact_900_even__two__times__div__two,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.50 = A ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_two_times_div_two
% 0.32/0.50 thf(fact_901_even__two__times__div__two,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.32/0.50 = A ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_two_times_div_two
% 0.32/0.50 thf(fact_902_dbl__simps_I3_J,axiom,
% 0.32/0.50 ( ( neg_nu7009210354673126013omplex @ one_one_complex )
% 0.32/0.50 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dbl_simps(3)
% 0.32/0.50 thf(fact_903_dbl__simps_I3_J,axiom,
% 0.32/0.50 ( ( neg_numeral_dbl_real @ one_one_real )
% 0.32/0.50 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dbl_simps(3)
% 0.32/0.50 thf(fact_904_dbl__simps_I3_J,axiom,
% 0.32/0.50 ( ( neg_numeral_dbl_rat @ one_one_rat )
% 0.32/0.50 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dbl_simps(3)
% 0.32/0.50 thf(fact_905_dbl__simps_I3_J,axiom,
% 0.32/0.50 ( ( neg_numeral_dbl_int @ one_one_int )
% 0.32/0.50 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dbl_simps(3)
% 0.32/0.50 thf(fact_906_triangle__def,axiom,
% 0.32/0.50 ( nat_triangle
% 0.32/0.50 = ( ^ [N3: nat] : ( divide_divide_nat @ ( times_times_nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % triangle_def
% 0.32/0.50 thf(fact_907_set__bit__0,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 0.32/0.50 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % set_bit_0
% 0.32/0.50 thf(fact_908_set__bit__0,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 0.32/0.50 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % set_bit_0
% 0.32/0.50 thf(fact_909_real__divide__square__eq,axiom,
% 0.32/0.50 ! [R: real,A: real] :
% 0.32/0.50 ( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
% 0.32/0.50 = ( divide_divide_real @ A @ R ) ) ).
% 0.32/0.50
% 0.32/0.50 % real_divide_square_eq
% 0.32/0.50 thf(fact_910_odd__two__times__div__two__succ,axiom,
% 0.32/0.50 ! [A: code_integer] :
% 0.32/0.50 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 0.32/0.50 = A ) ) ).
% 0.32/0.50
% 0.32/0.50 % odd_two_times_div_two_succ
% 0.32/0.50 thf(fact_911_odd__two__times__div__two__succ,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 0.32/0.50 = A ) ) ).
% 0.32/0.50
% 0.32/0.50 % odd_two_times_div_two_succ
% 0.32/0.50 thf(fact_912_odd__two__times__div__two__succ,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 0.32/0.50 = A ) ) ).
% 0.32/0.50
% 0.32/0.50 % odd_two_times_div_two_succ
% 0.32/0.50 thf(fact_913_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.32/0.50 ! [N: nat] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
% 0.32/0.50 = ( N = zero_zero_nat ) ) ).
% 0.32/0.50
% 0.32/0.50 % semiring_parity_class.even_mask_iff
% 0.32/0.50 thf(fact_914_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.32/0.50 ! [N: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
% 0.32/0.50 = ( N = zero_zero_nat ) ) ).
% 0.32/0.50
% 0.32/0.50 % semiring_parity_class.even_mask_iff
% 0.32/0.50 thf(fact_915_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.32/0.50 ! [N: nat] :
% 0.32/0.50 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 0.32/0.50 = ( N = zero_zero_nat ) ) ).
% 0.32/0.50
% 0.32/0.50 % semiring_parity_class.even_mask_iff
% 0.32/0.50 thf(fact_916_zdvd__mono,axiom,
% 0.32/0.50 ! [K: int,M: int,T: int] :
% 0.32/0.50 ( ( K != zero_zero_int )
% 0.32/0.50 => ( ( dvd_dvd_int @ M @ T )
% 0.32/0.50 = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % zdvd_mono
% 0.32/0.50 thf(fact_917_even__succ__div__2,axiom,
% 0.32/0.50 ! [A: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.50 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_succ_div_2
% 0.32/0.50 thf(fact_918_even__succ__div__2,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.50 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_succ_div_2
% 0.32/0.50 thf(fact_919_even__succ__div__2,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.50 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_succ_div_2
% 0.32/0.50 thf(fact_920_odd__succ__div__two,axiom,
% 0.32/0.50 ! [A: code_integer] :
% 0.32/0.50 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.50 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % odd_succ_div_two
% 0.32/0.50 thf(fact_921_odd__succ__div__two,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.50 = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % odd_succ_div_two
% 0.32/0.50 thf(fact_922_odd__succ__div__two,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.50 = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % odd_succ_div_two
% 0.32/0.50 thf(fact_923_even__succ__div__two,axiom,
% 0.32/0.50 ! [A: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.50 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_succ_div_two
% 0.32/0.50 thf(fact_924_even__succ__div__two,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.50 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_succ_div_two
% 0.32/0.50 thf(fact_925_even__succ__div__two,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.50 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_succ_div_two
% 0.32/0.50 thf(fact_926_inf__period_I1_J,axiom,
% 0.32/0.50 ! [P: real > $o,D3: real,Q: real > $o] :
% 0.32/0.50 ( ! [X4: real,K3: real] :
% 0.32/0.50 ( ( P @ X4 )
% 0.32/0.50 = ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D3 ) ) ) )
% 0.32/0.50 => ( ! [X4: real,K3: real] :
% 0.32/0.50 ( ( Q @ X4 )
% 0.32/0.50 = ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D3 ) ) ) )
% 0.32/0.50 => ! [X5: real,K4: real] :
% 0.32/0.50 ( ( ( P @ X5 )
% 0.32/0.50 & ( Q @ X5 ) )
% 0.32/0.50 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) )
% 0.32/0.50 & ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % inf_period(1)
% 0.32/0.50 thf(fact_927_inf__period_I1_J,axiom,
% 0.32/0.50 ! [P: rat > $o,D3: rat,Q: rat > $o] :
% 0.32/0.50 ( ! [X4: rat,K3: rat] :
% 0.32/0.50 ( ( P @ X4 )
% 0.32/0.50 = ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) ) )
% 0.32/0.50 => ( ! [X4: rat,K3: rat] :
% 0.32/0.50 ( ( Q @ X4 )
% 0.32/0.50 = ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) ) )
% 0.32/0.50 => ! [X5: rat,K4: rat] :
% 0.32/0.50 ( ( ( P @ X5 )
% 0.32/0.50 & ( Q @ X5 ) )
% 0.32/0.50 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) )
% 0.32/0.50 & ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % inf_period(1)
% 0.32/0.50 thf(fact_928_inf__period_I1_J,axiom,
% 0.32/0.50 ! [P: int > $o,D3: int,Q: int > $o] :
% 0.32/0.50 ( ! [X4: int,K3: int] :
% 0.32/0.50 ( ( P @ X4 )
% 0.32/0.50 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
% 0.32/0.50 => ( ! [X4: int,K3: int] :
% 0.32/0.50 ( ( Q @ X4 )
% 0.32/0.50 = ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
% 0.32/0.50 => ! [X5: int,K4: int] :
% 0.32/0.50 ( ( ( P @ X5 )
% 0.32/0.50 & ( Q @ X5 ) )
% 0.32/0.50 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
% 0.32/0.50 & ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % inf_period(1)
% 0.32/0.50 thf(fact_929_even__odd__cases,axiom,
% 0.32/0.50 ! [X: nat] :
% 0.32/0.50 ( ! [N2: nat] :
% 0.32/0.50 ( X
% 0.32/0.50 != ( plus_plus_nat @ N2 @ N2 ) )
% 0.32/0.50 => ~ ! [N2: nat] :
% 0.32/0.50 ( X
% 0.32/0.50 != ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_odd_cases
% 0.32/0.50 thf(fact_930_add__right__cancel,axiom,
% 0.32/0.50 ! [B: real,A: real,C: real] :
% 0.32/0.50 ( ( ( plus_plus_real @ B @ A )
% 0.32/0.50 = ( plus_plus_real @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_cancel
% 0.32/0.50 thf(fact_931_add__right__cancel,axiom,
% 0.32/0.50 ! [B: rat,A: rat,C: rat] :
% 0.32/0.50 ( ( ( plus_plus_rat @ B @ A )
% 0.32/0.50 = ( plus_plus_rat @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_cancel
% 0.32/0.50 thf(fact_932_add__right__cancel,axiom,
% 0.32/0.50 ! [B: nat,A: nat,C: nat] :
% 0.32/0.50 ( ( ( plus_plus_nat @ B @ A )
% 0.32/0.50 = ( plus_plus_nat @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_cancel
% 0.32/0.50 thf(fact_933_add__right__cancel,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( ( plus_plus_int @ B @ A )
% 0.32/0.50 = ( plus_plus_int @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_cancel
% 0.32/0.50 thf(fact_934_add__left__cancel,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( ( plus_plus_real @ A @ B )
% 0.32/0.50 = ( plus_plus_real @ A @ C ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_cancel
% 0.32/0.50 thf(fact_935_add__left__cancel,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( ( plus_plus_rat @ A @ B )
% 0.32/0.50 = ( plus_plus_rat @ A @ C ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_cancel
% 0.32/0.50 thf(fact_936_add__left__cancel,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ( plus_plus_nat @ A @ B )
% 0.32/0.50 = ( plus_plus_nat @ A @ C ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_cancel
% 0.32/0.50 thf(fact_937_add__left__cancel,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( ( plus_plus_int @ A @ B )
% 0.32/0.50 = ( plus_plus_int @ A @ C ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_cancel
% 0.32/0.50 thf(fact_938_semiring__norm_I6_J,axiom,
% 0.32/0.50 ! [M: num,N: num] :
% 0.32/0.50 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.32/0.50 = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % semiring_norm(6)
% 0.32/0.50 thf(fact_939_add__Suc__right,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 0.32/0.50 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_Suc_right
% 0.32/0.50 thf(fact_940_Nat_Oadd__0__right,axiom,
% 0.32/0.50 ! [M: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 0.32/0.50 = M ) ).
% 0.32/0.50
% 0.32/0.50 % Nat.add_0_right
% 0.32/0.50 thf(fact_941_add__is__0,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ( ( plus_plus_nat @ M @ N )
% 0.32/0.50 = zero_zero_nat )
% 0.32/0.50 = ( ( M = zero_zero_nat )
% 0.32/0.50 & ( N = zero_zero_nat ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_is_0
% 0.32/0.50 thf(fact_942_diff__diff__left,axiom,
% 0.32/0.50 ! [I: nat,J: nat,K: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.32/0.50 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_diff_left
% 0.32/0.50 thf(fact_943_push__bit__push__bit,axiom,
% 0.32/0.50 ! [M: nat,N: nat,A: nat] :
% 0.32/0.50 ( ( bit_se547839408752420682it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 0.32/0.50 = ( bit_se547839408752420682it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 0.32/0.50
% 0.32/0.50 % push_bit_push_bit
% 0.32/0.50 thf(fact_944_push__bit__push__bit,axiom,
% 0.32/0.50 ! [M: nat,N: nat,A: int] :
% 0.32/0.50 ( ( bit_se545348938243370406it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
% 0.32/0.50 = ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 0.32/0.50
% 0.32/0.50 % push_bit_push_bit
% 0.32/0.50 thf(fact_945_push__bit__push__bit,axiom,
% 0.32/0.50 ! [M: nat,N: nat,A: code_integer] :
% 0.32/0.50 ( ( bit_se7788150548672797655nteger @ M @ ( bit_se7788150548672797655nteger @ N @ A ) )
% 0.32/0.50 = ( bit_se7788150548672797655nteger @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 0.32/0.50
% 0.32/0.50 % push_bit_push_bit
% 0.32/0.50 thf(fact_946_push__bit__push__bit,axiom,
% 0.32/0.50 ! [M: nat,N: nat,A: code_natural] :
% 0.32/0.50 ( ( bit_se6611745700429515170atural @ M @ ( bit_se6611745700429515170atural @ N @ A ) )
% 0.32/0.50 = ( bit_se6611745700429515170atural @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 0.32/0.50
% 0.32/0.50 % push_bit_push_bit
% 0.32/0.50 thf(fact_947_add_Oright__neutral,axiom,
% 0.32/0.50 ! [A: literal] :
% 0.32/0.50 ( ( plus_plus_literal @ A @ zero_zero_literal )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add.right_neutral
% 0.32/0.50 thf(fact_948_add_Oright__neutral,axiom,
% 0.32/0.50 ! [A: real] :
% 0.32/0.50 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add.right_neutral
% 0.32/0.50 thf(fact_949_add_Oright__neutral,axiom,
% 0.32/0.50 ! [A: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add.right_neutral
% 0.32/0.50 thf(fact_950_add_Oright__neutral,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add.right_neutral
% 0.32/0.50 thf(fact_951_add_Oright__neutral,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add.right_neutral
% 0.32/0.50 thf(fact_952_double__zero__sym,axiom,
% 0.32/0.50 ! [A: real] :
% 0.32/0.50 ( ( zero_zero_real
% 0.32/0.50 = ( plus_plus_real @ A @ A ) )
% 0.32/0.50 = ( A = zero_zero_real ) ) ).
% 0.32/0.50
% 0.32/0.50 % double_zero_sym
% 0.32/0.50 thf(fact_953_double__zero__sym,axiom,
% 0.32/0.50 ! [A: rat] :
% 0.32/0.50 ( ( zero_zero_rat
% 0.32/0.50 = ( plus_plus_rat @ A @ A ) )
% 0.32/0.50 = ( A = zero_zero_rat ) ) ).
% 0.32/0.50
% 0.32/0.50 % double_zero_sym
% 0.32/0.50 thf(fact_954_double__zero__sym,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( zero_zero_int
% 0.32/0.50 = ( plus_plus_int @ A @ A ) )
% 0.32/0.50 = ( A = zero_zero_int ) ) ).
% 0.32/0.50
% 0.32/0.50 % double_zero_sym
% 0.32/0.50 thf(fact_955_add__cancel__left__left,axiom,
% 0.32/0.50 ! [B: real,A: real] :
% 0.32/0.50 ( ( ( plus_plus_real @ B @ A )
% 0.32/0.50 = A )
% 0.32/0.50 = ( B = zero_zero_real ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_cancel_left_left
% 0.32/0.50 thf(fact_956_add__cancel__left__left,axiom,
% 0.32/0.50 ! [B: rat,A: rat] :
% 0.32/0.50 ( ( ( plus_plus_rat @ B @ A )
% 0.32/0.50 = A )
% 0.32/0.50 = ( B = zero_zero_rat ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_cancel_left_left
% 0.32/0.50 thf(fact_957_add__cancel__left__left,axiom,
% 0.32/0.50 ! [B: nat,A: nat] :
% 0.32/0.50 ( ( ( plus_plus_nat @ B @ A )
% 0.32/0.50 = A )
% 0.32/0.50 = ( B = zero_zero_nat ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_cancel_left_left
% 0.32/0.50 thf(fact_958_add__cancel__left__left,axiom,
% 0.32/0.50 ! [B: int,A: int] :
% 0.32/0.50 ( ( ( plus_plus_int @ B @ A )
% 0.32/0.50 = A )
% 0.32/0.50 = ( B = zero_zero_int ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_cancel_left_left
% 0.32/0.50 thf(fact_959_add__cancel__left__right,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( ( plus_plus_real @ A @ B )
% 0.32/0.50 = A )
% 0.32/0.50 = ( B = zero_zero_real ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_cancel_left_right
% 0.32/0.50 thf(fact_960_add__cancel__left__right,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ( ( plus_plus_rat @ A @ B )
% 0.32/0.50 = A )
% 0.32/0.50 = ( B = zero_zero_rat ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_cancel_left_right
% 0.32/0.50 thf(fact_961_add__cancel__left__right,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( ( plus_plus_nat @ A @ B )
% 0.32/0.50 = A )
% 0.32/0.50 = ( B = zero_zero_nat ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_cancel_left_right
% 0.32/0.50 thf(fact_962_add__cancel__left__right,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( ( plus_plus_int @ A @ B )
% 0.32/0.50 = A )
% 0.32/0.50 = ( B = zero_zero_int ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_cancel_left_right
% 0.32/0.50 thf(fact_963_add__cancel__right__left,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( A
% 0.32/0.50 = ( plus_plus_real @ B @ A ) )
% 0.32/0.50 = ( B = zero_zero_real ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_cancel_right_left
% 0.32/0.50 thf(fact_964_add__cancel__right__left,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ( A
% 0.32/0.50 = ( plus_plus_rat @ B @ A ) )
% 0.32/0.50 = ( B = zero_zero_rat ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_cancel_right_left
% 0.32/0.50 thf(fact_965_add__cancel__right__left,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( A
% 0.32/0.50 = ( plus_plus_nat @ B @ A ) )
% 0.32/0.50 = ( B = zero_zero_nat ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_cancel_right_left
% 0.32/0.50 thf(fact_966_add__cancel__right__left,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( A
% 0.32/0.50 = ( plus_plus_int @ B @ A ) )
% 0.32/0.50 = ( B = zero_zero_int ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_cancel_right_left
% 0.32/0.50 thf(fact_967_add__cancel__right__right,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( A
% 0.32/0.50 = ( plus_plus_real @ A @ B ) )
% 0.32/0.50 = ( B = zero_zero_real ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_cancel_right_right
% 0.32/0.50 thf(fact_968_add__cancel__right__right,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ( A
% 0.32/0.50 = ( plus_plus_rat @ A @ B ) )
% 0.32/0.50 = ( B = zero_zero_rat ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_cancel_right_right
% 0.32/0.50 thf(fact_969_add__cancel__right__right,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( A
% 0.32/0.50 = ( plus_plus_nat @ A @ B ) )
% 0.32/0.50 = ( B = zero_zero_nat ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_cancel_right_right
% 0.32/0.50 thf(fact_970_add__cancel__right__right,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( A
% 0.32/0.50 = ( plus_plus_int @ A @ B ) )
% 0.32/0.50 = ( B = zero_zero_int ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_cancel_right_right
% 0.32/0.50 thf(fact_971_add__eq__0__iff__both__eq__0,axiom,
% 0.32/0.50 ! [X: nat,Y: nat] :
% 0.32/0.50 ( ( ( plus_plus_nat @ X @ Y )
% 0.32/0.50 = zero_zero_nat )
% 0.32/0.50 = ( ( X = zero_zero_nat )
% 0.32/0.50 & ( Y = zero_zero_nat ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_eq_0_iff_both_eq_0
% 0.32/0.50 thf(fact_972_zero__eq__add__iff__both__eq__0,axiom,
% 0.32/0.50 ! [X: nat,Y: nat] :
% 0.32/0.50 ( ( zero_zero_nat
% 0.32/0.50 = ( plus_plus_nat @ X @ Y ) )
% 0.32/0.50 = ( ( X = zero_zero_nat )
% 0.32/0.50 & ( Y = zero_zero_nat ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % zero_eq_add_iff_both_eq_0
% 0.32/0.50 thf(fact_973_add__0,axiom,
% 0.32/0.50 ! [A: literal] :
% 0.32/0.50 ( ( plus_plus_literal @ zero_zero_literal @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_0
% 0.32/0.50 thf(fact_974_add__0,axiom,
% 0.32/0.50 ! [A: real] :
% 0.32/0.50 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_0
% 0.32/0.50 thf(fact_975_add__0,axiom,
% 0.32/0.50 ! [A: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_0
% 0.32/0.50 thf(fact_976_add__0,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_0
% 0.32/0.50 thf(fact_977_add__0,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_0
% 0.32/0.50 thf(fact_978_double__eq__0__iff,axiom,
% 0.32/0.50 ! [A: real] :
% 0.32/0.50 ( ( ( plus_plus_real @ A @ A )
% 0.32/0.50 = zero_zero_real )
% 0.32/0.50 = ( A = zero_zero_real ) ) ).
% 0.32/0.50
% 0.32/0.50 % double_eq_0_iff
% 0.32/0.50 thf(fact_979_double__eq__0__iff,axiom,
% 0.32/0.50 ! [A: rat] :
% 0.32/0.50 ( ( ( plus_plus_rat @ A @ A )
% 0.32/0.50 = zero_zero_rat )
% 0.32/0.50 = ( A = zero_zero_rat ) ) ).
% 0.32/0.50
% 0.32/0.50 % double_eq_0_iff
% 0.32/0.50 thf(fact_980_double__eq__0__iff,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( ( plus_plus_int @ A @ A )
% 0.32/0.50 = zero_zero_int )
% 0.32/0.50 = ( A = zero_zero_int ) ) ).
% 0.32/0.50
% 0.32/0.50 % double_eq_0_iff
% 0.32/0.50 thf(fact_981_numeral__plus__numeral,axiom,
% 0.32/0.50 ! [M: num,N: num] :
% 0.32/0.50 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 0.32/0.50 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % numeral_plus_numeral
% 0.32/0.50 thf(fact_982_numeral__plus__numeral,axiom,
% 0.32/0.50 ! [M: num,N: num] :
% 0.32/0.50 ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.32/0.50 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % numeral_plus_numeral
% 0.32/0.50 thf(fact_983_numeral__plus__numeral,axiom,
% 0.32/0.50 ! [M: num,N: num] :
% 0.32/0.50 ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.32/0.50 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % numeral_plus_numeral
% 0.32/0.50 thf(fact_984_numeral__plus__numeral,axiom,
% 0.32/0.50 ! [M: num,N: num] :
% 0.32/0.50 ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.50 = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % numeral_plus_numeral
% 0.32/0.50 thf(fact_985_numeral__plus__numeral,axiom,
% 0.32/0.50 ! [M: num,N: num] :
% 0.32/0.50 ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.50 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % numeral_plus_numeral
% 0.32/0.50 thf(fact_986_add__numeral__left,axiom,
% 0.32/0.50 ! [V: num,W: num,Z2: complex] :
% 0.32/0.50 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z2 ) )
% 0.32/0.50 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_numeral_left
% 0.32/0.50 thf(fact_987_add__numeral__left,axiom,
% 0.32/0.50 ! [V: num,W: num,Z2: real] :
% 0.32/0.50 ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
% 0.32/0.50 = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_numeral_left
% 0.32/0.50 thf(fact_988_add__numeral__left,axiom,
% 0.32/0.50 ! [V: num,W: num,Z2: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z2 ) )
% 0.32/0.50 = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_numeral_left
% 0.32/0.50 thf(fact_989_add__numeral__left,axiom,
% 0.32/0.50 ! [V: num,W: num,Z2: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
% 0.32/0.50 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_numeral_left
% 0.32/0.50 thf(fact_990_add__numeral__left,axiom,
% 0.32/0.50 ! [V: num,W: num,Z2: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
% 0.32/0.50 = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_numeral_left
% 0.32/0.50 thf(fact_991_semiring__norm_I2_J,axiom,
% 0.32/0.50 ( ( plus_plus_num @ one @ one )
% 0.32/0.50 = ( bit0 @ one ) ) ).
% 0.32/0.50
% 0.32/0.50 % semiring_norm(2)
% 0.32/0.50 thf(fact_992_add__diff__cancel,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel
% 0.32/0.50 thf(fact_993_add__diff__cancel,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel
% 0.32/0.50 thf(fact_994_add__diff__cancel,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel
% 0.32/0.50 thf(fact_995_diff__add__cancel,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % diff_add_cancel
% 0.32/0.50 thf(fact_996_diff__add__cancel,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % diff_add_cancel
% 0.32/0.50 thf(fact_997_diff__add__cancel,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % diff_add_cancel
% 0.32/0.50 thf(fact_998_add__diff__cancel__left,axiom,
% 0.32/0.50 ! [C: real,A: real,B: real] :
% 0.32/0.50 ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.32/0.50 = ( minus_minus_real @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_left
% 0.32/0.50 thf(fact_999_add__diff__cancel__left,axiom,
% 0.32/0.50 ! [C: rat,A: rat,B: rat] :
% 0.32/0.50 ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.32/0.50 = ( minus_minus_rat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_left
% 0.32/0.50 thf(fact_1000_add__diff__cancel__left,axiom,
% 0.32/0.50 ! [C: nat,A: nat,B: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.32/0.50 = ( minus_minus_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_left
% 0.32/0.50 thf(fact_1001_add__diff__cancel__left,axiom,
% 0.32/0.50 ! [C: int,A: int,B: int] :
% 0.32/0.50 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.32/0.50 = ( minus_minus_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_left
% 0.32/0.50 thf(fact_1002_add__diff__cancel__left_H,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 0.32/0.50 = B ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_left'
% 0.32/0.50 thf(fact_1003_add__diff__cancel__left_H,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
% 0.32/0.50 = B ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_left'
% 0.32/0.50 thf(fact_1004_add__diff__cancel__left_H,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 0.32/0.50 = B ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_left'
% 0.32/0.50 thf(fact_1005_add__diff__cancel__left_H,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 0.32/0.50 = B ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_left'
% 0.32/0.50 thf(fact_1006_add__diff__cancel__right,axiom,
% 0.32/0.50 ! [A: real,C: real,B: real] :
% 0.32/0.50 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.32/0.50 = ( minus_minus_real @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_right
% 0.32/0.50 thf(fact_1007_add__diff__cancel__right,axiom,
% 0.32/0.50 ! [A: rat,C: rat,B: rat] :
% 0.32/0.50 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.50 = ( minus_minus_rat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_right
% 0.32/0.50 thf(fact_1008_add__diff__cancel__right,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.50 = ( minus_minus_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_right
% 0.32/0.50 thf(fact_1009_add__diff__cancel__right,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.32/0.50 = ( minus_minus_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_right
% 0.32/0.50 thf(fact_1010_add__diff__cancel__right_H,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_right'
% 0.32/0.50 thf(fact_1011_add__diff__cancel__right_H,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_right'
% 0.32/0.50 thf(fact_1012_add__diff__cancel__right_H,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_right'
% 0.32/0.50 thf(fact_1013_add__diff__cancel__right_H,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_right'
% 0.32/0.50 thf(fact_1014_dvd__add__triv__left__iff,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.32/0.50 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_triv_left_iff
% 0.32/0.50 thf(fact_1015_dvd__add__triv__left__iff,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.32/0.50 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_triv_left_iff
% 0.32/0.50 thf(fact_1016_dvd__add__triv__left__iff,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.50 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_triv_left_iff
% 0.32/0.50 thf(fact_1017_dvd__add__triv__left__iff,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.50 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_triv_left_iff
% 0.32/0.50 thf(fact_1018_dvd__add__triv__left__iff,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.32/0.50 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_triv_left_iff
% 0.32/0.50 thf(fact_1019_dvd__add__triv__right__iff,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
% 0.32/0.50 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_triv_right_iff
% 0.32/0.50 thf(fact_1020_dvd__add__triv__right__iff,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.32/0.50 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_triv_right_iff
% 0.32/0.50 thf(fact_1021_dvd__add__triv__right__iff,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.32/0.50 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_triv_right_iff
% 0.32/0.50 thf(fact_1022_dvd__add__triv__right__iff,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.32/0.50 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_triv_right_iff
% 0.32/0.50 thf(fact_1023_dvd__add__triv__right__iff,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.32/0.50 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_triv_right_iff
% 0.32/0.50 thf(fact_1024_of__nat__add,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.50 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % of_nat_add
% 0.32/0.50 thf(fact_1025_of__nat__add,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.50 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % of_nat_add
% 0.32/0.50 thf(fact_1026_of__nat__add,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.50 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % of_nat_add
% 0.32/0.50 thf(fact_1027_of__nat__add,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.50 = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % of_nat_add
% 0.32/0.50 thf(fact_1028_of__nat__add,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.50 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % of_nat_add
% 0.32/0.50 thf(fact_1029_mult__Suc__right,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ( times_times_nat @ M @ ( suc @ N ) )
% 0.32/0.50 = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mult_Suc_right
% 0.32/0.50 thf(fact_1030_nat__1__eq__mult__iff,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ( one_one_nat
% 0.32/0.50 = ( times_times_nat @ M @ N ) )
% 0.32/0.50 = ( ( M = one_one_nat )
% 0.32/0.50 & ( N = one_one_nat ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % nat_1_eq_mult_iff
% 0.32/0.50 thf(fact_1031_nat__mult__eq__1__iff,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ( ( times_times_nat @ M @ N )
% 0.32/0.50 = one_one_nat )
% 0.32/0.50 = ( ( M = one_one_nat )
% 0.32/0.50 & ( N = one_one_nat ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % nat_mult_eq_1_iff
% 0.32/0.50 thf(fact_1032_dbl__simps_I2_J,axiom,
% 0.32/0.50 ( ( neg_numeral_dbl_real @ zero_zero_real )
% 0.32/0.50 = zero_zero_real ) ).
% 0.32/0.50
% 0.32/0.50 % dbl_simps(2)
% 0.32/0.50 thf(fact_1033_dbl__simps_I2_J,axiom,
% 0.32/0.50 ( ( neg_numeral_dbl_rat @ zero_zero_rat )
% 0.32/0.50 = zero_zero_rat ) ).
% 0.32/0.50
% 0.32/0.50 % dbl_simps(2)
% 0.32/0.50 thf(fact_1034_dbl__simps_I2_J,axiom,
% 0.32/0.50 ( ( neg_numeral_dbl_int @ zero_zero_int )
% 0.32/0.50 = zero_zero_int ) ).
% 0.32/0.50
% 0.32/0.50 % dbl_simps(2)
% 0.32/0.50 thf(fact_1035_triangle__Suc,axiom,
% 0.32/0.50 ! [N: nat] :
% 0.32/0.50 ( ( nat_triangle @ ( suc @ N ) )
% 0.32/0.50 = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % triangle_Suc
% 0.32/0.50 thf(fact_1036_triangle__0,axiom,
% 0.32/0.50 ( ( nat_triangle @ zero_zero_nat )
% 0.32/0.50 = zero_zero_nat ) ).
% 0.32/0.50
% 0.32/0.50 % triangle_0
% 0.32/0.50 thf(fact_1037_diff__add__zero,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.50 = zero_zero_nat ) ).
% 0.32/0.50
% 0.32/0.50 % diff_add_zero
% 0.32/0.50 thf(fact_1038_distrib__left__numeral,axiom,
% 0.32/0.50 ! [V: num,B: complex,C: complex] :
% 0.32/0.50 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
% 0.32/0.50 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_left_numeral
% 0.32/0.50 thf(fact_1039_distrib__left__numeral,axiom,
% 0.32/0.50 ! [V: num,B: real,C: real] :
% 0.32/0.50 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
% 0.32/0.50 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_left_numeral
% 0.32/0.50 thf(fact_1040_distrib__left__numeral,axiom,
% 0.32/0.50 ! [V: num,B: rat,C: rat] :
% 0.32/0.50 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.50 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_left_numeral
% 0.32/0.50 thf(fact_1041_distrib__left__numeral,axiom,
% 0.32/0.50 ! [V: num,B: nat,C: nat] :
% 0.32/0.50 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.50 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_left_numeral
% 0.32/0.50 thf(fact_1042_distrib__left__numeral,axiom,
% 0.32/0.50 ! [V: num,B: int,C: int] :
% 0.32/0.50 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
% 0.32/0.50 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_left_numeral
% 0.32/0.50 thf(fact_1043_distrib__right__numeral,axiom,
% 0.32/0.50 ! [A: complex,B: complex,V: num] :
% 0.32/0.50 ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 0.32/0.50 = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_right_numeral
% 0.32/0.50 thf(fact_1044_distrib__right__numeral,axiom,
% 0.32/0.50 ! [A: real,B: real,V: num] :
% 0.32/0.50 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 0.32/0.50 = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_right_numeral
% 0.32/0.50 thf(fact_1045_distrib__right__numeral,axiom,
% 0.32/0.50 ! [A: rat,B: rat,V: num] :
% 0.32/0.50 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 0.32/0.50 = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_right_numeral
% 0.32/0.50 thf(fact_1046_distrib__right__numeral,axiom,
% 0.32/0.50 ! [A: nat,B: nat,V: num] :
% 0.32/0.50 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
% 0.32/0.50 = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_right_numeral
% 0.32/0.50 thf(fact_1047_distrib__right__numeral,axiom,
% 0.32/0.50 ! [A: int,B: int,V: num] :
% 0.32/0.50 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 0.32/0.50 = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_right_numeral
% 0.32/0.50 thf(fact_1048_dvd__add__times__triv__right__iff,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 0.32/0.50 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_times_triv_right_iff
% 0.32/0.50 thf(fact_1049_dvd__add__times__triv__right__iff,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
% 0.32/0.50 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_times_triv_right_iff
% 0.32/0.50 thf(fact_1050_dvd__add__times__triv__right__iff,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
% 0.32/0.50 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_times_triv_right_iff
% 0.32/0.50 thf(fact_1051_dvd__add__times__triv__right__iff,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
% 0.32/0.50 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_times_triv_right_iff
% 0.32/0.50 thf(fact_1052_dvd__add__times__triv__right__iff,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
% 0.32/0.50 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_times_triv_right_iff
% 0.32/0.50 thf(fact_1053_dvd__add__times__triv__left__iff,axiom,
% 0.32/0.50 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
% 0.32/0.50 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_times_triv_left_iff
% 0.32/0.50 thf(fact_1054_dvd__add__times__triv__left__iff,axiom,
% 0.32/0.50 ! [A: real,C: real,B: real] :
% 0.32/0.50 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
% 0.32/0.50 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_times_triv_left_iff
% 0.32/0.50 thf(fact_1055_dvd__add__times__triv__left__iff,axiom,
% 0.32/0.50 ! [A: rat,C: rat,B: rat] :
% 0.32/0.50 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
% 0.32/0.50 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_times_triv_left_iff
% 0.32/0.50 thf(fact_1056_dvd__add__times__triv__left__iff,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
% 0.32/0.50 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_times_triv_left_iff
% 0.32/0.50 thf(fact_1057_dvd__add__times__triv__left__iff,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
% 0.32/0.50 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % dvd_add_times_triv_left_iff
% 0.32/0.50 thf(fact_1058_div__add,axiom,
% 0.32/0.50 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.32/0.50 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.50 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.32/0.50 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_add
% 0.32/0.50 thf(fact_1059_div__add,axiom,
% 0.32/0.50 ! [C: nat,A: nat,B: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ C @ A )
% 0.32/0.50 => ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.50 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_add
% 0.32/0.50 thf(fact_1060_div__add,axiom,
% 0.32/0.50 ! [C: int,A: int,B: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ C @ A )
% 0.32/0.50 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.50 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_add
% 0.32/0.50 thf(fact_1061_Suc__numeral,axiom,
% 0.32/0.50 ! [N: num] :
% 0.32/0.50 ( ( suc @ ( numeral_numeral_nat @ N ) )
% 0.32/0.50 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % Suc_numeral
% 0.32/0.50 thf(fact_1062_dbl__simps_I5_J,axiom,
% 0.32/0.50 ! [K: num] :
% 0.32/0.50 ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.32/0.50 = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dbl_simps(5)
% 0.32/0.50 thf(fact_1063_dbl__simps_I5_J,axiom,
% 0.32/0.50 ! [K: num] :
% 0.32/0.50 ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 0.32/0.50 = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dbl_simps(5)
% 0.32/0.50 thf(fact_1064_dbl__simps_I5_J,axiom,
% 0.32/0.50 ! [K: num] :
% 0.32/0.50 ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
% 0.32/0.50 = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dbl_simps(5)
% 0.32/0.50 thf(fact_1065_dbl__simps_I5_J,axiom,
% 0.32/0.50 ! [K: num] :
% 0.32/0.50 ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 0.32/0.50 = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dbl_simps(5)
% 0.32/0.50 thf(fact_1066_div__mult__self4,axiom,
% 0.32/0.50 ! [B: nat,C: nat,A: nat] :
% 0.32/0.50 ( ( B != zero_zero_nat )
% 0.32/0.50 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 0.32/0.50 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_self4
% 0.32/0.50 thf(fact_1067_div__mult__self4,axiom,
% 0.32/0.50 ! [B: int,C: int,A: int] :
% 0.32/0.50 ( ( B != zero_zero_int )
% 0.32/0.50 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 0.32/0.50 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_self4
% 0.32/0.50 thf(fact_1068_div__mult__self3,axiom,
% 0.32/0.50 ! [B: nat,C: nat,A: nat] :
% 0.32/0.50 ( ( B != zero_zero_nat )
% 0.32/0.50 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 0.32/0.50 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_self3
% 0.32/0.50 thf(fact_1069_div__mult__self3,axiom,
% 0.32/0.50 ! [B: int,C: int,A: int] :
% 0.32/0.50 ( ( B != zero_zero_int )
% 0.32/0.50 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 0.32/0.50 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_self3
% 0.32/0.50 thf(fact_1070_div__mult__self2,axiom,
% 0.32/0.50 ! [B: nat,A: nat,C: nat] :
% 0.32/0.50 ( ( B != zero_zero_nat )
% 0.32/0.50 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 0.32/0.50 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_self2
% 0.32/0.50 thf(fact_1071_div__mult__self2,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( B != zero_zero_int )
% 0.32/0.50 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 0.32/0.50 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_self2
% 0.32/0.50 thf(fact_1072_div__mult__self1,axiom,
% 0.32/0.50 ! [B: nat,A: nat,C: nat] :
% 0.32/0.50 ( ( B != zero_zero_nat )
% 0.32/0.50 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 0.32/0.50 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_self1
% 0.32/0.50 thf(fact_1073_div__mult__self1,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( B != zero_zero_int )
% 0.32/0.50 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 0.32/0.50 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_self1
% 0.32/0.50 thf(fact_1074_one__plus__numeral,axiom,
% 0.32/0.50 ! [N: num] :
% 0.32/0.50 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
% 0.32/0.50 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % one_plus_numeral
% 0.32/0.50 thf(fact_1075_one__plus__numeral,axiom,
% 0.32/0.50 ! [N: num] :
% 0.32/0.50 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.32/0.50 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % one_plus_numeral
% 0.32/0.50 thf(fact_1076_one__plus__numeral,axiom,
% 0.32/0.50 ! [N: num] :
% 0.32/0.50 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.32/0.50 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % one_plus_numeral
% 0.32/0.50 thf(fact_1077_one__plus__numeral,axiom,
% 0.32/0.50 ! [N: num] :
% 0.32/0.50 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 0.32/0.50 = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % one_plus_numeral
% 0.32/0.50 thf(fact_1078_one__plus__numeral,axiom,
% 0.32/0.50 ! [N: num] :
% 0.32/0.50 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.32/0.50 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % one_plus_numeral
% 0.32/0.50 thf(fact_1079_numeral__plus__one,axiom,
% 0.32/0.50 ! [N: num] :
% 0.32/0.50 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
% 0.32/0.50 = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % numeral_plus_one
% 0.32/0.50 thf(fact_1080_numeral__plus__one,axiom,
% 0.32/0.50 ! [N: num] :
% 0.32/0.50 ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.32/0.50 = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % numeral_plus_one
% 0.32/0.50 thf(fact_1081_numeral__plus__one,axiom,
% 0.32/0.50 ! [N: num] :
% 0.32/0.50 ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 0.32/0.50 = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % numeral_plus_one
% 0.32/0.50 thf(fact_1082_numeral__plus__one,axiom,
% 0.32/0.50 ! [N: num] :
% 0.32/0.50 ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.32/0.50 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % numeral_plus_one
% 0.32/0.50 thf(fact_1083_numeral__plus__one,axiom,
% 0.32/0.50 ! [N: num] :
% 0.32/0.50 ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.32/0.50 = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % numeral_plus_one
% 0.32/0.50 thf(fact_1084_add__2__eq__Suc_H,axiom,
% 0.32/0.50 ! [N: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.50 = ( suc @ ( suc @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_2_eq_Suc'
% 0.32/0.50 thf(fact_1085_add__2__eq__Suc,axiom,
% 0.32/0.50 ! [N: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.50 = ( suc @ ( suc @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_2_eq_Suc
% 0.32/0.50 thf(fact_1086_of__nat__Suc,axiom,
% 0.32/0.50 ! [M: nat] :
% 0.32/0.50 ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 0.32/0.50 = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % of_nat_Suc
% 0.32/0.50 thf(fact_1087_of__nat__Suc,axiom,
% 0.32/0.50 ! [M: nat] :
% 0.32/0.50 ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 0.32/0.50 = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % of_nat_Suc
% 0.32/0.50 thf(fact_1088_of__nat__Suc,axiom,
% 0.32/0.50 ! [M: nat] :
% 0.32/0.50 ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 0.32/0.50 = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % of_nat_Suc
% 0.32/0.50 thf(fact_1089_of__nat__Suc,axiom,
% 0.32/0.50 ! [M: nat] :
% 0.32/0.50 ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 0.32/0.50 = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % of_nat_Suc
% 0.32/0.50 thf(fact_1090_of__nat__Suc,axiom,
% 0.32/0.50 ! [M: nat] :
% 0.32/0.50 ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 0.32/0.50 = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % of_nat_Suc
% 0.32/0.50 thf(fact_1091_add__self__div__2,axiom,
% 0.32/0.50 ! [M: nat] :
% 0.32/0.50 ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.50 = M ) ).
% 0.32/0.50
% 0.32/0.50 % add_self_div_2
% 0.32/0.50 thf(fact_1092_one__add__one,axiom,
% 0.32/0.50 ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 0.32/0.50 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % one_add_one
% 0.32/0.50 thf(fact_1093_one__add__one,axiom,
% 0.32/0.50 ( ( plus_plus_real @ one_one_real @ one_one_real )
% 0.32/0.50 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % one_add_one
% 0.32/0.50 thf(fact_1094_one__add__one,axiom,
% 0.32/0.50 ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 0.32/0.50 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % one_add_one
% 0.32/0.50 thf(fact_1095_one__add__one,axiom,
% 0.32/0.50 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 0.32/0.50 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % one_add_one
% 0.32/0.50 thf(fact_1096_one__add__one,axiom,
% 0.32/0.50 ( ( plus_plus_int @ one_one_int @ one_one_int )
% 0.32/0.50 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % one_add_one
% 0.32/0.50 thf(fact_1097_even__add,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.32/0.50 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_add
% 0.32/0.50 thf(fact_1098_even__add,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.50 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_add
% 0.32/0.50 thf(fact_1099_even__add,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
% 0.32/0.50 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.50 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_add
% 0.32/0.50 thf(fact_1100_odd__add,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
% 0.32/0.50 = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.50 != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % odd_add
% 0.32/0.50 thf(fact_1101_odd__add,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
% 0.32/0.50 = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.50 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % odd_add
% 0.32/0.50 thf(fact_1102_odd__add,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
% 0.32/0.50 = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.50 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % odd_add
% 0.32/0.50 thf(fact_1103_even__plus__one__iff,axiom,
% 0.32/0.50 ! [A: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 0.32/0.50 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_plus_one_iff
% 0.32/0.50 thf(fact_1104_even__plus__one__iff,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 0.32/0.50 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_plus_one_iff
% 0.32/0.50 thf(fact_1105_even__plus__one__iff,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 0.32/0.50 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_plus_one_iff
% 0.32/0.50 thf(fact_1106_even__diff,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.32/0.50 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_diff
% 0.32/0.50 thf(fact_1107_even__diff,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
% 0.32/0.50 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % even_diff
% 0.32/0.50 thf(fact_1108_push__bit__of__1,axiom,
% 0.32/0.50 ! [N: nat] :
% 0.32/0.50 ( ( bit_se547839408752420682it_nat @ N @ one_one_nat )
% 0.32/0.50 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.50
% 0.32/0.50 % push_bit_of_1
% 0.32/0.50 thf(fact_1109_push__bit__of__1,axiom,
% 0.32/0.50 ! [N: nat] :
% 0.32/0.50 ( ( bit_se545348938243370406it_int @ N @ one_one_int )
% 0.32/0.50 = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.50
% 0.32/0.50 % push_bit_of_1
% 0.32/0.50 thf(fact_1110_push__bit__of__1,axiom,
% 0.32/0.50 ! [N: nat] :
% 0.32/0.50 ( ( bit_se7788150548672797655nteger @ N @ one_one_Code_integer )
% 0.32/0.50 = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.50
% 0.32/0.50 % push_bit_of_1
% 0.32/0.50 thf(fact_1111_push__bit__of__1,axiom,
% 0.32/0.50 ! [N: nat] :
% 0.32/0.50 ( ( bit_se6611745700429515170atural @ N @ one_one_Code_natural )
% 0.32/0.50 = ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.50
% 0.32/0.50 % push_bit_of_1
% 0.32/0.50 thf(fact_1112_push__bit__of__Suc__0,axiom,
% 0.32/0.50 ! [N: nat] :
% 0.32/0.50 ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.32/0.50 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.50
% 0.32/0.50 % push_bit_of_Suc_0
% 0.32/0.50 thf(fact_1113_int__distrib_I4_J,axiom,
% 0.32/0.50 ! [W: int,Z1: int,Z22: int] :
% 0.32/0.50 ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 0.32/0.50 = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % int_distrib(4)
% 0.32/0.50 thf(fact_1114_int__distrib_I3_J,axiom,
% 0.32/0.50 ! [Z1: int,Z22: int,W: int] :
% 0.32/0.50 ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 0.32/0.50 = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % int_distrib(3)
% 0.32/0.50 thf(fact_1115_int__distrib_I2_J,axiom,
% 0.32/0.50 ! [W: int,Z1: int,Z22: int] :
% 0.32/0.50 ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
% 0.32/0.50 = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % int_distrib(2)
% 0.32/0.50 thf(fact_1116_int__distrib_I1_J,axiom,
% 0.32/0.50 ! [Z1: int,Z22: int,W: int] :
% 0.32/0.50 ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
% 0.32/0.50 = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % int_distrib(1)
% 0.32/0.50 thf(fact_1117_nat__mult__1,axiom,
% 0.32/0.50 ! [N: nat] :
% 0.32/0.50 ( ( times_times_nat @ one_one_nat @ N )
% 0.32/0.50 = N ) ).
% 0.32/0.50
% 0.32/0.50 % nat_mult_1
% 0.32/0.50 thf(fact_1118_add__mult__distrib,axiom,
% 0.32/0.50 ! [M: nat,N: nat,K: nat] :
% 0.32/0.50 ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 0.32/0.50 = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mult_distrib
% 0.32/0.50 thf(fact_1119_nat__mult__1__right,axiom,
% 0.32/0.50 ! [N: nat] :
% 0.32/0.50 ( ( times_times_nat @ N @ one_one_nat )
% 0.32/0.50 = N ) ).
% 0.32/0.50
% 0.32/0.50 % nat_mult_1_right
% 0.32/0.50 thf(fact_1120_add__mult__distrib2,axiom,
% 0.32/0.50 ! [K: nat,M: nat,N: nat] :
% 0.32/0.50 ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.50 = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mult_distrib2
% 0.32/0.50 thf(fact_1121_left__add__mult__distrib,axiom,
% 0.32/0.50 ! [I: nat,U: nat,J: nat,K: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 0.32/0.50 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% 0.32/0.50
% 0.32/0.50 % left_add_mult_distrib
% 0.32/0.50 thf(fact_1122_dbl__def,axiom,
% 0.32/0.50 ( neg_numeral_dbl_real
% 0.32/0.50 = ( ^ [X3: real] : ( plus_plus_real @ X3 @ X3 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dbl_def
% 0.32/0.50 thf(fact_1123_dbl__def,axiom,
% 0.32/0.50 ( neg_numeral_dbl_rat
% 0.32/0.50 = ( ^ [X3: rat] : ( plus_plus_rat @ X3 @ X3 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dbl_def
% 0.32/0.50 thf(fact_1124_dbl__def,axiom,
% 0.32/0.50 ( neg_numeral_dbl_int
% 0.32/0.50 = ( ^ [X3: int] : ( plus_plus_int @ X3 @ X3 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % dbl_def
% 0.32/0.50 thf(fact_1125_is__num__normalize_I1_J,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % is_num_normalize(1)
% 0.32/0.50 thf(fact_1126_is__num__normalize_I1_J,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % is_num_normalize(1)
% 0.32/0.50 thf(fact_1127_is__num__normalize_I1_J,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % is_num_normalize(1)
% 0.32/0.50 thf(fact_1128_add__right__imp__eq,axiom,
% 0.32/0.50 ! [B: real,A: real,C: real] :
% 0.32/0.50 ( ( ( plus_plus_real @ B @ A )
% 0.32/0.50 = ( plus_plus_real @ C @ A ) )
% 0.32/0.50 => ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_imp_eq
% 0.32/0.50 thf(fact_1129_add__right__imp__eq,axiom,
% 0.32/0.50 ! [B: rat,A: rat,C: rat] :
% 0.32/0.50 ( ( ( plus_plus_rat @ B @ A )
% 0.32/0.50 = ( plus_plus_rat @ C @ A ) )
% 0.32/0.50 => ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_imp_eq
% 0.32/0.50 thf(fact_1130_add__right__imp__eq,axiom,
% 0.32/0.50 ! [B: nat,A: nat,C: nat] :
% 0.32/0.50 ( ( ( plus_plus_nat @ B @ A )
% 0.32/0.50 = ( plus_plus_nat @ C @ A ) )
% 0.32/0.50 => ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_imp_eq
% 0.32/0.50 thf(fact_1131_add__right__imp__eq,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( ( plus_plus_int @ B @ A )
% 0.32/0.50 = ( plus_plus_int @ C @ A ) )
% 0.32/0.50 => ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_imp_eq
% 0.32/0.50 thf(fact_1132_add__left__imp__eq,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( ( plus_plus_real @ A @ B )
% 0.32/0.50 = ( plus_plus_real @ A @ C ) )
% 0.32/0.50 => ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_imp_eq
% 0.32/0.50 thf(fact_1133_add__left__imp__eq,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( ( plus_plus_rat @ A @ B )
% 0.32/0.50 = ( plus_plus_rat @ A @ C ) )
% 0.32/0.50 => ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_imp_eq
% 0.32/0.50 thf(fact_1134_add__left__imp__eq,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ( plus_plus_nat @ A @ B )
% 0.32/0.50 = ( plus_plus_nat @ A @ C ) )
% 0.32/0.50 => ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_imp_eq
% 0.32/0.50 thf(fact_1135_add__left__imp__eq,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( ( plus_plus_int @ A @ B )
% 0.32/0.50 = ( plus_plus_int @ A @ C ) )
% 0.32/0.50 => ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_imp_eq
% 0.32/0.50 thf(fact_1136_add_Oleft__commute,axiom,
% 0.32/0.50 ! [B: real,A: real,C: real] :
% 0.32/0.50 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 0.32/0.50 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.left_commute
% 0.32/0.50 thf(fact_1137_add_Oleft__commute,axiom,
% 0.32/0.50 ! [B: rat,A: rat,C: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
% 0.32/0.50 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.left_commute
% 0.32/0.50 thf(fact_1138_add_Oleft__commute,axiom,
% 0.32/0.50 ! [B: nat,A: nat,C: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 0.32/0.50 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.left_commute
% 0.32/0.50 thf(fact_1139_add_Oleft__commute,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 0.32/0.50 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.left_commute
% 0.32/0.50 thf(fact_1140_add_Ocommute,axiom,
% 0.32/0.50 ( plus_plus_real
% 0.32/0.50 = ( ^ [A2: real,B2: real] : ( plus_plus_real @ B2 @ A2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.commute
% 0.32/0.50 thf(fact_1141_add_Ocommute,axiom,
% 0.32/0.50 ( plus_plus_rat
% 0.32/0.50 = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ B2 @ A2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.commute
% 0.32/0.50 thf(fact_1142_add_Ocommute,axiom,
% 0.32/0.50 ( plus_plus_nat
% 0.32/0.50 = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.commute
% 0.32/0.50 thf(fact_1143_add_Ocommute,axiom,
% 0.32/0.50 ( plus_plus_int
% 0.32/0.50 = ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.commute
% 0.32/0.50 thf(fact_1144_add_Oright__cancel,axiom,
% 0.32/0.50 ! [B: real,A: real,C: real] :
% 0.32/0.50 ( ( ( plus_plus_real @ B @ A )
% 0.32/0.50 = ( plus_plus_real @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.right_cancel
% 0.32/0.50 thf(fact_1145_add_Oright__cancel,axiom,
% 0.32/0.50 ! [B: rat,A: rat,C: rat] :
% 0.32/0.50 ( ( ( plus_plus_rat @ B @ A )
% 0.32/0.50 = ( plus_plus_rat @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.right_cancel
% 0.32/0.50 thf(fact_1146_add_Oright__cancel,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( ( plus_plus_int @ B @ A )
% 0.32/0.50 = ( plus_plus_int @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.right_cancel
% 0.32/0.50 thf(fact_1147_add_Oleft__cancel,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( ( plus_plus_real @ A @ B )
% 0.32/0.50 = ( plus_plus_real @ A @ C ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.left_cancel
% 0.32/0.50 thf(fact_1148_add_Oleft__cancel,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( ( plus_plus_rat @ A @ B )
% 0.32/0.50 = ( plus_plus_rat @ A @ C ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.left_cancel
% 0.32/0.50 thf(fact_1149_add_Oleft__cancel,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( ( plus_plus_int @ A @ B )
% 0.32/0.50 = ( plus_plus_int @ A @ C ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.left_cancel
% 0.32/0.50 thf(fact_1150_add_Oassoc,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.assoc
% 0.32/0.50 thf(fact_1151_add_Oassoc,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.assoc
% 0.32/0.50 thf(fact_1152_add_Oassoc,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.assoc
% 0.32/0.50 thf(fact_1153_add_Oassoc,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.assoc
% 0.32/0.50 thf(fact_1154_group__cancel_Oadd2,axiom,
% 0.32/0.50 ! [B5: real,K: real,B: real,A: real] :
% 0.32/0.50 ( ( B5
% 0.32/0.50 = ( plus_plus_real @ K @ B ) )
% 0.32/0.50 => ( ( plus_plus_real @ A @ B5 )
% 0.32/0.50 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.add2
% 0.32/0.50 thf(fact_1155_group__cancel_Oadd2,axiom,
% 0.32/0.50 ! [B5: rat,K: rat,B: rat,A: rat] :
% 0.32/0.50 ( ( B5
% 0.32/0.50 = ( plus_plus_rat @ K @ B ) )
% 0.32/0.50 => ( ( plus_plus_rat @ A @ B5 )
% 0.32/0.50 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.add2
% 0.32/0.50 thf(fact_1156_group__cancel_Oadd2,axiom,
% 0.32/0.50 ! [B5: nat,K: nat,B: nat,A: nat] :
% 0.32/0.50 ( ( B5
% 0.32/0.50 = ( plus_plus_nat @ K @ B ) )
% 0.32/0.50 => ( ( plus_plus_nat @ A @ B5 )
% 0.32/0.50 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.add2
% 0.32/0.50 thf(fact_1157_group__cancel_Oadd2,axiom,
% 0.32/0.50 ! [B5: int,K: int,B: int,A: int] :
% 0.32/0.50 ( ( B5
% 0.32/0.50 = ( plus_plus_int @ K @ B ) )
% 0.32/0.50 => ( ( plus_plus_int @ A @ B5 )
% 0.32/0.50 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.add2
% 0.32/0.50 thf(fact_1158_group__cancel_Oadd1,axiom,
% 0.32/0.50 ! [A3: real,K: real,A: real,B: real] :
% 0.32/0.50 ( ( A3
% 0.32/0.50 = ( plus_plus_real @ K @ A ) )
% 0.32/0.50 => ( ( plus_plus_real @ A3 @ B )
% 0.32/0.50 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.add1
% 0.32/0.50 thf(fact_1159_group__cancel_Oadd1,axiom,
% 0.32/0.50 ! [A3: rat,K: rat,A: rat,B: rat] :
% 0.32/0.50 ( ( A3
% 0.32/0.50 = ( plus_plus_rat @ K @ A ) )
% 0.32/0.50 => ( ( plus_plus_rat @ A3 @ B )
% 0.32/0.50 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.add1
% 0.32/0.50 thf(fact_1160_group__cancel_Oadd1,axiom,
% 0.32/0.50 ! [A3: nat,K: nat,A: nat,B: nat] :
% 0.32/0.50 ( ( A3
% 0.32/0.50 = ( plus_plus_nat @ K @ A ) )
% 0.32/0.50 => ( ( plus_plus_nat @ A3 @ B )
% 0.32/0.50 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.add1
% 0.32/0.50 thf(fact_1161_group__cancel_Oadd1,axiom,
% 0.32/0.50 ! [A3: int,K: int,A: int,B: int] :
% 0.32/0.50 ( ( A3
% 0.32/0.50 = ( plus_plus_int @ K @ A ) )
% 0.32/0.50 => ( ( plus_plus_int @ A3 @ B )
% 0.32/0.50 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.add1
% 0.32/0.50 thf(fact_1162_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.32/0.50 ! [I: real,J: real,K: real,L: real] :
% 0.32/0.50 ( ( ( I = J )
% 0.32/0.50 & ( K = L ) )
% 0.32/0.50 => ( ( plus_plus_real @ I @ K )
% 0.32/0.50 = ( plus_plus_real @ J @ L ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(4)
% 0.32/0.50 thf(fact_1163_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.32/0.50 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.32/0.50 ( ( ( I = J )
% 0.32/0.50 & ( K = L ) )
% 0.32/0.50 => ( ( plus_plus_rat @ I @ K )
% 0.32/0.50 = ( plus_plus_rat @ J @ L ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(4)
% 0.32/0.50 thf(fact_1164_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.32/0.50 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.50 ( ( ( I = J )
% 0.32/0.50 & ( K = L ) )
% 0.32/0.50 => ( ( plus_plus_nat @ I @ K )
% 0.32/0.50 = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(4)
% 0.32/0.50 thf(fact_1165_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.32/0.50 ! [I: int,J: int,K: int,L: int] :
% 0.32/0.50 ( ( ( I = J )
% 0.32/0.50 & ( K = L ) )
% 0.32/0.50 => ( ( plus_plus_int @ I @ K )
% 0.32/0.50 = ( plus_plus_int @ J @ L ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(4)
% 0.32/0.50 thf(fact_1166_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ab_semigroup_add_class.add_ac(1)
% 0.32/0.50 thf(fact_1167_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ab_semigroup_add_class.add_ac(1)
% 0.32/0.50 thf(fact_1168_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ab_semigroup_add_class.add_ac(1)
% 0.32/0.50 thf(fact_1169_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ab_semigroup_add_class.add_ac(1)
% 0.32/0.50 thf(fact_1170_zadd__int__left,axiom,
% 0.32/0.50 ! [M: nat,N: nat,Z2: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
% 0.32/0.50 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% 0.32/0.50
% 0.32/0.50 % zadd_int_left
% 0.32/0.50 thf(fact_1171_int__plus,axiom,
% 0.32/0.50 ! [N: nat,M: nat] :
% 0.32/0.50 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
% 0.32/0.50 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % int_plus
% 0.32/0.50 thf(fact_1172_int__ops_I5_J,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.50 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % int_ops(5)
% 0.32/0.50 thf(fact_1173_comm__monoid__add__class_Oadd__0,axiom,
% 0.32/0.50 ! [A: real] :
% 0.32/0.50 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % comm_monoid_add_class.add_0
% 0.32/0.50 thf(fact_1174_comm__monoid__add__class_Oadd__0,axiom,
% 0.32/0.50 ! [A: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % comm_monoid_add_class.add_0
% 0.32/0.50 thf(fact_1175_comm__monoid__add__class_Oadd__0,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % comm_monoid_add_class.add_0
% 0.32/0.50 thf(fact_1176_comm__monoid__add__class_Oadd__0,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % comm_monoid_add_class.add_0
% 0.32/0.50 thf(fact_1177_add_Ocomm__neutral,axiom,
% 0.32/0.50 ! [A: real] :
% 0.32/0.50 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add.comm_neutral
% 0.32/0.50 thf(fact_1178_add_Ocomm__neutral,axiom,
% 0.32/0.50 ! [A: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add.comm_neutral
% 0.32/0.50 thf(fact_1179_add_Ocomm__neutral,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add.comm_neutral
% 0.32/0.50 thf(fact_1180_add_Ocomm__neutral,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add.comm_neutral
% 0.32/0.50 thf(fact_1181_add_Ogroup__left__neutral,axiom,
% 0.32/0.50 ! [A: real] :
% 0.32/0.50 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add.group_left_neutral
% 0.32/0.50 thf(fact_1182_add_Ogroup__left__neutral,axiom,
% 0.32/0.50 ! [A: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add.group_left_neutral
% 0.32/0.50 thf(fact_1183_add_Ogroup__left__neutral,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add.group_left_neutral
% 0.32/0.50 thf(fact_1184_verit__sum__simplify,axiom,
% 0.32/0.50 ! [A: real] :
% 0.32/0.50 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % verit_sum_simplify
% 0.32/0.50 thf(fact_1185_verit__sum__simplify,axiom,
% 0.32/0.50 ! [A: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % verit_sum_simplify
% 0.32/0.50 thf(fact_1186_verit__sum__simplify,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % verit_sum_simplify
% 0.32/0.50 thf(fact_1187_verit__sum__simplify,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % verit_sum_simplify
% 0.32/0.50 thf(fact_1188_combine__common__factor,axiom,
% 0.32/0.50 ! [A: real,E: real,B: real,C: real] :
% 0.32/0.50 ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
% 0.32/0.50 = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % combine_common_factor
% 0.32/0.50 thf(fact_1189_combine__common__factor,axiom,
% 0.32/0.50 ! [A: rat,E: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
% 0.32/0.50 = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % combine_common_factor
% 0.32/0.50 thf(fact_1190_combine__common__factor,axiom,
% 0.32/0.50 ! [A: nat,E: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
% 0.32/0.50 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % combine_common_factor
% 0.32/0.50 thf(fact_1191_combine__common__factor,axiom,
% 0.32/0.50 ! [A: int,E: int,B: int,C: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
% 0.32/0.50 = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % combine_common_factor
% 0.32/0.50 thf(fact_1192_distrib__right,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_right
% 0.32/0.50 thf(fact_1193_distrib__right,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_right
% 0.32/0.50 thf(fact_1194_distrib__right,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_right
% 0.32/0.50 thf(fact_1195_distrib__right,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_right
% 0.32/0.50 thf(fact_1196_distrib__left,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.32/0.50 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_left
% 0.32/0.50 thf(fact_1197_distrib__left,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.50 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_left
% 0.32/0.50 thf(fact_1198_distrib__left,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.50 = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_left
% 0.32/0.50 thf(fact_1199_distrib__left,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.32/0.50 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % distrib_left
% 0.32/0.50 thf(fact_1200_comm__semiring__class_Odistrib,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % comm_semiring_class.distrib
% 0.32/0.50 thf(fact_1201_comm__semiring__class_Odistrib,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % comm_semiring_class.distrib
% 0.32/0.50 thf(fact_1202_comm__semiring__class_Odistrib,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % comm_semiring_class.distrib
% 0.32/0.50 thf(fact_1203_comm__semiring__class_Odistrib,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % comm_semiring_class.distrib
% 0.32/0.50 thf(fact_1204_ring__class_Oring__distribs_I1_J,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.32/0.50 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ring_class.ring_distribs(1)
% 0.32/0.50 thf(fact_1205_ring__class_Oring__distribs_I1_J,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.50 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ring_class.ring_distribs(1)
% 0.32/0.50 thf(fact_1206_ring__class_Oring__distribs_I1_J,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.32/0.51 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % ring_class.ring_distribs(1)
% 0.32/0.51 thf(fact_1207_ring__class_Oring__distribs_I2_J,axiom,
% 0.32/0.51 ! [A: real,B: real,C: real] :
% 0.32/0.51 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.32/0.51 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % ring_class.ring_distribs(2)
% 0.32/0.51 thf(fact_1208_ring__class_Oring__distribs_I2_J,axiom,
% 0.32/0.51 ! [A: rat,B: rat,C: rat] :
% 0.32/0.51 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.32/0.51 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % ring_class.ring_distribs(2)
% 0.32/0.51 thf(fact_1209_ring__class_Oring__distribs_I2_J,axiom,
% 0.32/0.51 ! [A: int,B: int,C: int] :
% 0.32/0.51 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.51 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % ring_class.ring_distribs(2)
% 0.32/0.51 thf(fact_1210_group__cancel_Osub1,axiom,
% 0.32/0.51 ! [A3: real,K: real,A: real,B: real] :
% 0.32/0.51 ( ( A3
% 0.32/0.51 = ( plus_plus_real @ K @ A ) )
% 0.32/0.51 => ( ( minus_minus_real @ A3 @ B )
% 0.32/0.51 = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % group_cancel.sub1
% 0.32/0.51 thf(fact_1211_group__cancel_Osub1,axiom,
% 0.32/0.51 ! [A3: rat,K: rat,A: rat,B: rat] :
% 0.32/0.51 ( ( A3
% 0.32/0.51 = ( plus_plus_rat @ K @ A ) )
% 0.32/0.51 => ( ( minus_minus_rat @ A3 @ B )
% 0.32/0.51 = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % group_cancel.sub1
% 0.32/0.51 thf(fact_1212_group__cancel_Osub1,axiom,
% 0.32/0.51 ! [A3: int,K: int,A: int,B: int] :
% 0.32/0.51 ( ( A3
% 0.32/0.51 = ( plus_plus_int @ K @ A ) )
% 0.32/0.51 => ( ( minus_minus_int @ A3 @ B )
% 0.32/0.51 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % group_cancel.sub1
% 0.32/0.51 thf(fact_1213_diff__eq__eq,axiom,
% 0.32/0.51 ! [A: real,B: real,C: real] :
% 0.32/0.51 ( ( ( minus_minus_real @ A @ B )
% 0.32/0.51 = C )
% 0.32/0.51 = ( A
% 0.32/0.51 = ( plus_plus_real @ C @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_eq_eq
% 0.32/0.51 thf(fact_1214_diff__eq__eq,axiom,
% 0.32/0.51 ! [A: rat,B: rat,C: rat] :
% 0.32/0.51 ( ( ( minus_minus_rat @ A @ B )
% 0.32/0.51 = C )
% 0.32/0.51 = ( A
% 0.32/0.51 = ( plus_plus_rat @ C @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_eq_eq
% 0.32/0.51 thf(fact_1215_diff__eq__eq,axiom,
% 0.32/0.51 ! [A: int,B: int,C: int] :
% 0.32/0.51 ( ( ( minus_minus_int @ A @ B )
% 0.32/0.51 = C )
% 0.32/0.51 = ( A
% 0.32/0.51 = ( plus_plus_int @ C @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_eq_eq
% 0.32/0.51 thf(fact_1216_eq__diff__eq,axiom,
% 0.32/0.51 ! [A: real,C: real,B: real] :
% 0.32/0.51 ( ( A
% 0.32/0.51 = ( minus_minus_real @ C @ B ) )
% 0.32/0.51 = ( ( plus_plus_real @ A @ B )
% 0.32/0.51 = C ) ) ).
% 0.32/0.51
% 0.32/0.51 % eq_diff_eq
% 0.32/0.51 thf(fact_1217_eq__diff__eq,axiom,
% 0.32/0.51 ! [A: rat,C: rat,B: rat] :
% 0.32/0.51 ( ( A
% 0.32/0.51 = ( minus_minus_rat @ C @ B ) )
% 0.32/0.51 = ( ( plus_plus_rat @ A @ B )
% 0.32/0.51 = C ) ) ).
% 0.32/0.51
% 0.32/0.51 % eq_diff_eq
% 0.32/0.51 thf(fact_1218_eq__diff__eq,axiom,
% 0.32/0.51 ! [A: int,C: int,B: int] :
% 0.32/0.51 ( ( A
% 0.32/0.51 = ( minus_minus_int @ C @ B ) )
% 0.32/0.51 = ( ( plus_plus_int @ A @ B )
% 0.32/0.51 = C ) ) ).
% 0.32/0.51
% 0.32/0.51 % eq_diff_eq
% 0.32/0.51 thf(fact_1219_add__diff__eq,axiom,
% 0.32/0.51 ! [A: real,B: real,C: real] :
% 0.32/0.51 ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.32/0.51 = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_diff_eq
% 0.32/0.51 thf(fact_1220_add__diff__eq,axiom,
% 0.32/0.51 ! [A: rat,B: rat,C: rat] :
% 0.32/0.51 ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.32/0.51 = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_diff_eq
% 0.32/0.51 thf(fact_1221_add__diff__eq,axiom,
% 0.32/0.51 ! [A: int,B: int,C: int] :
% 0.32/0.51 ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.32/0.51 = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_diff_eq
% 0.32/0.51 thf(fact_1222_diff__diff__eq2,axiom,
% 0.32/0.51 ! [A: real,B: real,C: real] :
% 0.32/0.51 ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.32/0.51 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_diff_eq2
% 0.32/0.51 thf(fact_1223_diff__diff__eq2,axiom,
% 0.32/0.51 ! [A: rat,B: rat,C: rat] :
% 0.32/0.51 ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.32/0.51 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_diff_eq2
% 0.32/0.51 thf(fact_1224_diff__diff__eq2,axiom,
% 0.32/0.51 ! [A: int,B: int,C: int] :
% 0.32/0.51 ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.32/0.51 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_diff_eq2
% 0.32/0.51 thf(fact_1225_diff__add__eq,axiom,
% 0.32/0.51 ! [A: real,B: real,C: real] :
% 0.32/0.51 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.32/0.51 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_add_eq
% 0.32/0.51 thf(fact_1226_diff__add__eq,axiom,
% 0.32/0.51 ! [A: rat,B: rat,C: rat] :
% 0.32/0.51 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.32/0.51 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_add_eq
% 0.32/0.51 thf(fact_1227_diff__add__eq,axiom,
% 0.32/0.51 ! [A: int,B: int,C: int] :
% 0.32/0.51 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.32/0.51 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_add_eq
% 0.32/0.51 thf(fact_1228_diff__add__eq__diff__diff__swap,axiom,
% 0.32/0.51 ! [A: real,B: real,C: real] :
% 0.32/0.51 ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.32/0.51 = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_add_eq_diff_diff_swap
% 0.32/0.51 thf(fact_1229_diff__add__eq__diff__diff__swap,axiom,
% 0.32/0.51 ! [A: rat,B: rat,C: rat] :
% 0.32/0.51 ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.51 = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_add_eq_diff_diff_swap
% 0.32/0.51 thf(fact_1230_diff__add__eq__diff__diff__swap,axiom,
% 0.32/0.51 ! [A: int,B: int,C: int] :
% 0.32/0.51 ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.32/0.51 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_add_eq_diff_diff_swap
% 0.32/0.51 thf(fact_1231_add__implies__diff,axiom,
% 0.32/0.51 ! [C: real,B: real,A: real] :
% 0.32/0.51 ( ( ( plus_plus_real @ C @ B )
% 0.32/0.51 = A )
% 0.32/0.51 => ( C
% 0.32/0.51 = ( minus_minus_real @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_implies_diff
% 0.32/0.51 thf(fact_1232_add__implies__diff,axiom,
% 0.32/0.51 ! [C: rat,B: rat,A: rat] :
% 0.32/0.51 ( ( ( plus_plus_rat @ C @ B )
% 0.32/0.51 = A )
% 0.32/0.51 => ( C
% 0.32/0.51 = ( minus_minus_rat @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_implies_diff
% 0.32/0.51 thf(fact_1233_add__implies__diff,axiom,
% 0.32/0.51 ! [C: nat,B: nat,A: nat] :
% 0.32/0.51 ( ( ( plus_plus_nat @ C @ B )
% 0.32/0.51 = A )
% 0.32/0.51 => ( C
% 0.32/0.51 = ( minus_minus_nat @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_implies_diff
% 0.32/0.51 thf(fact_1234_add__implies__diff,axiom,
% 0.32/0.51 ! [C: int,B: int,A: int] :
% 0.32/0.51 ( ( ( plus_plus_int @ C @ B )
% 0.32/0.51 = A )
% 0.32/0.51 => ( C
% 0.32/0.51 = ( minus_minus_int @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_implies_diff
% 0.32/0.51 thf(fact_1235_add__diff__add,axiom,
% 0.32/0.51 ! [A: real,C: real,B: real,D: real] :
% 0.32/0.51 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
% 0.32/0.51 = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_diff_add
% 0.32/0.51 thf(fact_1236_add__diff__add,axiom,
% 0.32/0.51 ! [A: rat,C: rat,B: rat,D: rat] :
% 0.32/0.51 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
% 0.32/0.51 = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_diff_add
% 0.32/0.51 thf(fact_1237_add__diff__add,axiom,
% 0.32/0.51 ! [A: int,C: int,B: int,D: int] :
% 0.32/0.51 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
% 0.32/0.51 = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_diff_add
% 0.32/0.51 thf(fact_1238_diff__diff__eq,axiom,
% 0.32/0.51 ! [A: real,B: real,C: real] :
% 0.32/0.51 ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.32/0.51 = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_diff_eq
% 0.32/0.51 thf(fact_1239_diff__diff__eq,axiom,
% 0.32/0.51 ! [A: rat,B: rat,C: rat] :
% 0.32/0.51 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.32/0.51 = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_diff_eq
% 0.32/0.51 thf(fact_1240_diff__diff__eq,axiom,
% 0.32/0.51 ! [A: nat,B: nat,C: nat] :
% 0.32/0.51 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 0.32/0.51 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_diff_eq
% 0.32/0.51 thf(fact_1241_diff__diff__eq,axiom,
% 0.32/0.51 ! [A: int,B: int,C: int] :
% 0.32/0.51 ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.32/0.51 = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_diff_eq
% 0.32/0.51 thf(fact_1242_add__One__commute,axiom,
% 0.32/0.51 ! [N: num] :
% 0.32/0.51 ( ( plus_plus_num @ one @ N )
% 0.32/0.51 = ( plus_plus_num @ N @ one ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_One_commute
% 0.32/0.51 thf(fact_1243_add__divide__distrib,axiom,
% 0.32/0.51 ! [A: complex,B: complex,C: complex] :
% 0.32/0.51 ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
% 0.32/0.51 = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_divide_distrib
% 0.32/0.51 thf(fact_1244_add__divide__distrib,axiom,
% 0.32/0.51 ! [A: real,B: real,C: real] :
% 0.32/0.51 ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.32/0.51 = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_divide_distrib
% 0.32/0.51 thf(fact_1245_add__divide__distrib,axiom,
% 0.32/0.51 ! [A: rat,B: rat,C: rat] :
% 0.32/0.51 ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.32/0.51 = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_divide_distrib
% 0.32/0.51 thf(fact_1246_dvd__add,axiom,
% 0.32/0.51 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.51 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.51 => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.32/0.51 => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_add
% 0.32/0.51 thf(fact_1247_dvd__add,axiom,
% 0.32/0.51 ! [A: real,B: real,C: real] :
% 0.32/0.51 ( ( dvd_dvd_real @ A @ B )
% 0.32/0.51 => ( ( dvd_dvd_real @ A @ C )
% 0.32/0.51 => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_add
% 0.32/0.51 thf(fact_1248_dvd__add,axiom,
% 0.32/0.51 ! [A: rat,B: rat,C: rat] :
% 0.32/0.51 ( ( dvd_dvd_rat @ A @ B )
% 0.32/0.51 => ( ( dvd_dvd_rat @ A @ C )
% 0.32/0.51 => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_add
% 0.32/0.51 thf(fact_1249_dvd__add,axiom,
% 0.32/0.51 ! [A: nat,B: nat,C: nat] :
% 0.32/0.51 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.51 => ( ( dvd_dvd_nat @ A @ C )
% 0.32/0.51 => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_add
% 0.32/0.51 thf(fact_1250_dvd__add,axiom,
% 0.32/0.51 ! [A: int,B: int,C: int] :
% 0.32/0.51 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.51 => ( ( dvd_dvd_int @ A @ C )
% 0.32/0.51 => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_add
% 0.32/0.51 thf(fact_1251_dvd__add__left__iff,axiom,
% 0.32/0.51 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.51 ( ( dvd_dvd_Code_integer @ A @ C )
% 0.32/0.51 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.32/0.51 = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_add_left_iff
% 0.32/0.51 thf(fact_1252_dvd__add__left__iff,axiom,
% 0.32/0.51 ! [A: real,C: real,B: real] :
% 0.32/0.51 ( ( dvd_dvd_real @ A @ C )
% 0.32/0.51 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.32/0.51 = ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_add_left_iff
% 0.32/0.51 thf(fact_1253_dvd__add__left__iff,axiom,
% 0.32/0.51 ! [A: rat,C: rat,B: rat] :
% 0.32/0.51 ( ( dvd_dvd_rat @ A @ C )
% 0.32/0.51 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.51 = ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_add_left_iff
% 0.32/0.51 thf(fact_1254_dvd__add__left__iff,axiom,
% 0.32/0.51 ! [A: nat,C: nat,B: nat] :
% 0.32/0.51 ( ( dvd_dvd_nat @ A @ C )
% 0.32/0.51 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.51 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_add_left_iff
% 0.32/0.51 thf(fact_1255_dvd__add__left__iff,axiom,
% 0.32/0.51 ! [A: int,C: int,B: int] :
% 0.32/0.51 ( ( dvd_dvd_int @ A @ C )
% 0.32/0.51 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.32/0.51 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_add_left_iff
% 0.32/0.51 thf(fact_1256_dvd__add__right__iff,axiom,
% 0.32/0.51 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.51 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.51 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.32/0.51 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_add_right_iff
% 0.32/0.51 thf(fact_1257_dvd__add__right__iff,axiom,
% 0.32/0.51 ! [A: real,B: real,C: real] :
% 0.32/0.51 ( ( dvd_dvd_real @ A @ B )
% 0.32/0.51 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.32/0.51 = ( dvd_dvd_real @ A @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_add_right_iff
% 0.32/0.51 thf(fact_1258_dvd__add__right__iff,axiom,
% 0.32/0.51 ! [A: rat,B: rat,C: rat] :
% 0.32/0.51 ( ( dvd_dvd_rat @ A @ B )
% 0.32/0.51 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.51 = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_add_right_iff
% 0.32/0.51 thf(fact_1259_dvd__add__right__iff,axiom,
% 0.32/0.51 ! [A: nat,B: nat,C: nat] :
% 0.32/0.51 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.51 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.51 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_add_right_iff
% 0.32/0.51 thf(fact_1260_dvd__add__right__iff,axiom,
% 0.32/0.51 ! [A: int,B: int,C: int] :
% 0.32/0.51 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.51 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.32/0.51 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_add_right_iff
% 0.32/0.51 thf(fact_1261_add__Suc__shift,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.32/0.51 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_Suc_shift
% 0.32/0.51 thf(fact_1262_add__Suc,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.32/0.51 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_Suc
% 0.32/0.51 thf(fact_1263_nat__arith_Osuc1,axiom,
% 0.32/0.51 ! [A3: nat,K: nat,A: nat] :
% 0.32/0.51 ( ( A3
% 0.32/0.51 = ( plus_plus_nat @ K @ A ) )
% 0.32/0.51 => ( ( suc @ A3 )
% 0.32/0.51 = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nat_arith.suc1
% 0.32/0.51 thf(fact_1264_Suc__eq__plus1,axiom,
% 0.32/0.51 ( suc
% 0.32/0.51 = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % Suc_eq_plus1
% 0.32/0.51 thf(fact_1265_plus__1__eq__Suc,axiom,
% 0.32/0.51 ( ( plus_plus_nat @ one_one_nat )
% 0.32/0.51 = suc ) ).
% 0.32/0.51
% 0.32/0.51 % plus_1_eq_Suc
% 0.32/0.51 thf(fact_1266_Suc__eq__plus1__left,axiom,
% 0.32/0.51 ( suc
% 0.32/0.51 = ( plus_plus_nat @ one_one_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % Suc_eq_plus1_left
% 0.32/0.51 thf(fact_1267_mult__Suc,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( times_times_nat @ ( suc @ M ) @ N )
% 0.32/0.51 = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_Suc
% 0.32/0.51 thf(fact_1268_add__eq__self__zero,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( ( plus_plus_nat @ M @ N )
% 0.32/0.51 = M )
% 0.32/0.51 => ( N = zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_eq_self_zero
% 0.32/0.51 thf(fact_1269_plus__nat_Oadd__0,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 0.32/0.51 = N ) ).
% 0.32/0.51
% 0.32/0.51 % plus_nat.add_0
% 0.32/0.51 thf(fact_1270_Euclid__induct,axiom,
% 0.32/0.51 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.32/0.51 ( ! [A4: nat,B4: nat] :
% 0.32/0.51 ( ( P @ A4 @ B4 )
% 0.32/0.51 = ( P @ B4 @ A4 ) )
% 0.32/0.51 => ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
% 0.32/0.51 => ( ! [A4: nat,B4: nat] :
% 0.32/0.51 ( ( P @ A4 @ B4 )
% 0.32/0.51 => ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
% 0.32/0.51 => ( P @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % Euclid_induct
% 0.32/0.51 thf(fact_1271_plus__int__code_I2_J,axiom,
% 0.32/0.51 ! [L: int] :
% 0.32/0.51 ( ( plus_plus_int @ zero_zero_int @ L )
% 0.32/0.51 = L ) ).
% 0.32/0.51
% 0.32/0.51 % plus_int_code(2)
% 0.32/0.51 thf(fact_1272_plus__int__code_I1_J,axiom,
% 0.32/0.51 ! [K: int] :
% 0.32/0.51 ( ( plus_plus_int @ K @ zero_zero_int )
% 0.32/0.51 = K ) ).
% 0.32/0.51
% 0.32/0.51 % plus_int_code(1)
% 0.32/0.51 thf(fact_1273_odd__nonzero,axiom,
% 0.32/0.51 ! [Z2: int] :
% 0.32/0.51 ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
% 0.32/0.51 != zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % odd_nonzero
% 0.32/0.51 thf(fact_1274_diff__add__inverse2,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.32/0.51 = M ) ).
% 0.32/0.51
% 0.32/0.51 % diff_add_inverse2
% 0.32/0.51 thf(fact_1275_diff__add__inverse,axiom,
% 0.32/0.51 ! [N: nat,M: nat] :
% 0.32/0.51 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.32/0.51 = M ) ).
% 0.32/0.51
% 0.32/0.51 % diff_add_inverse
% 0.32/0.51 thf(fact_1276_diff__cancel2,axiom,
% 0.32/0.51 ! [M: nat,K: nat,N: nat] :
% 0.32/0.51 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.32/0.51 = ( minus_minus_nat @ M @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_cancel2
% 0.32/0.51 thf(fact_1277_Nat_Odiff__cancel,axiom,
% 0.32/0.51 ! [K: nat,M: nat,N: nat] :
% 0.32/0.51 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.32/0.51 = ( minus_minus_nat @ M @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % Nat.diff_cancel
% 0.32/0.51 thf(fact_1278_bezout__lemma__nat,axiom,
% 0.32/0.51 ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
% 0.32/0.51 ( ( dvd_dvd_nat @ D @ A )
% 0.32/0.51 => ( ( dvd_dvd_nat @ D @ B )
% 0.32/0.51 => ( ( ( ( times_times_nat @ A @ X )
% 0.32/0.51 = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
% 0.32/0.51 | ( ( times_times_nat @ B @ X )
% 0.32/0.51 = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
% 0.32/0.51 => ? [X4: nat,Y4: nat] :
% 0.32/0.51 ( ( dvd_dvd_nat @ D @ A )
% 0.32/0.51 & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.51 & ( ( ( times_times_nat @ A @ X4 )
% 0.32/0.51 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y4 ) @ D ) )
% 0.32/0.51 | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X4 )
% 0.32/0.51 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % bezout_lemma_nat
% 0.32/0.51 thf(fact_1279_bezout__add__nat,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ? [D2: nat,X4: nat,Y4: nat] :
% 0.32/0.51 ( ( dvd_dvd_nat @ D2 @ A )
% 0.32/0.51 & ( dvd_dvd_nat @ D2 @ B )
% 0.32/0.51 & ( ( ( times_times_nat @ A @ X4 )
% 0.32/0.51 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D2 ) )
% 0.32/0.51 | ( ( times_times_nat @ B @ X4 )
% 0.32/0.51 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D2 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % bezout_add_nat
% 0.32/0.51 thf(fact_1280_zdvd__period,axiom,
% 0.32/0.51 ! [A: int,D: int,X: int,T: int,C: int] :
% 0.32/0.51 ( ( dvd_dvd_int @ A @ D )
% 0.32/0.51 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
% 0.32/0.51 = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % zdvd_period
% 0.32/0.51 thf(fact_1281_zdvd__reduce,axiom,
% 0.32/0.51 ! [K: int,N: int,M: int] :
% 0.32/0.51 ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
% 0.32/0.51 = ( dvd_dvd_int @ K @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % zdvd_reduce
% 0.32/0.51 thf(fact_1282_push__bit__add,axiom,
% 0.32/0.51 ! [N: nat,A: nat,B: nat] :
% 0.32/0.51 ( ( bit_se547839408752420682it_nat @ N @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.51 = ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % push_bit_add
% 0.32/0.51 thf(fact_1283_push__bit__add,axiom,
% 0.32/0.51 ! [N: nat,A: int,B: int] :
% 0.32/0.51 ( ( bit_se545348938243370406it_int @ N @ ( plus_plus_int @ A @ B ) )
% 0.32/0.51 = ( plus_plus_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % push_bit_add
% 0.32/0.51 thf(fact_1284_push__bit__add,axiom,
% 0.32/0.51 ! [N: nat,A: code_integer,B: code_integer] :
% 0.32/0.51 ( ( bit_se7788150548672797655nteger @ N @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.32/0.51 = ( plus_p5714425477246183910nteger @ ( bit_se7788150548672797655nteger @ N @ A ) @ ( bit_se7788150548672797655nteger @ N @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % push_bit_add
% 0.32/0.51 thf(fact_1285_push__bit__add,axiom,
% 0.32/0.51 ! [N: nat,A: code_natural,B: code_natural] :
% 0.32/0.51 ( ( bit_se6611745700429515170atural @ N @ ( plus_p4538020629002901425atural @ A @ B ) )
% 0.32/0.51 = ( plus_p4538020629002901425atural @ ( bit_se6611745700429515170atural @ N @ A ) @ ( bit_se6611745700429515170atural @ N @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % push_bit_add
% 0.32/0.51 thf(fact_1286_Suc__nat__number__of__add,axiom,
% 0.32/0.51 ! [V: num,N: nat] :
% 0.32/0.51 ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
% 0.32/0.51 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % Suc_nat_number_of_add
% 0.32/0.51 thf(fact_1287_exp__add__not__zero__imp__right,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.51 != zero_zero_nat )
% 0.32/0.51 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.51 != zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % exp_add_not_zero_imp_right
% 0.32/0.51 thf(fact_1288_exp__add__not__zero__imp__right,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.51 != zero_zero_int )
% 0.32/0.51 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.32/0.51 != zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % exp_add_not_zero_imp_right
% 0.32/0.51 thf(fact_1289_exp__add__not__zero__imp__left,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.51 != zero_zero_nat )
% 0.32/0.51 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.32/0.51 != zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % exp_add_not_zero_imp_left
% 0.32/0.51 thf(fact_1290_exp__add__not__zero__imp__left,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.51 != zero_zero_int )
% 0.32/0.51 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.32/0.51 != zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % exp_add_not_zero_imp_left
% 0.32/0.51 thf(fact_1291_div__exp__eq,axiom,
% 0.32/0.51 ! [A: nat,M: nat,N: nat] :
% 0.32/0.51 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.51 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_exp_eq
% 0.32/0.51 thf(fact_1292_div__exp__eq,axiom,
% 0.32/0.51 ! [A: int,M: nat,N: nat] :
% 0.32/0.51 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.51 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_exp_eq
% 0.32/0.51 thf(fact_1293_unity__coeff__ex,axiom,
% 0.32/0.51 ! [P: code_integer > $o,L: code_integer] :
% 0.32/0.51 ( ( ? [X3: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X3 ) ) )
% 0.32/0.51 = ( ? [X3: code_integer] :
% 0.32/0.51 ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X3 @ zero_z3403309356797280102nteger ) )
% 0.32/0.51 & ( P @ X3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % unity_coeff_ex
% 0.32/0.51 thf(fact_1294_unity__coeff__ex,axiom,
% 0.32/0.51 ! [P: real > $o,L: real] :
% 0.32/0.51 ( ( ? [X3: real] : ( P @ ( times_times_real @ L @ X3 ) ) )
% 0.32/0.51 = ( ? [X3: real] :
% 0.32/0.51 ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X3 @ zero_zero_real ) )
% 0.32/0.51 & ( P @ X3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % unity_coeff_ex
% 0.32/0.51 thf(fact_1295_unity__coeff__ex,axiom,
% 0.32/0.51 ! [P: rat > $o,L: rat] :
% 0.32/0.51 ( ( ? [X3: rat] : ( P @ ( times_times_rat @ L @ X3 ) ) )
% 0.32/0.51 = ( ? [X3: rat] :
% 0.32/0.51 ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X3 @ zero_zero_rat ) )
% 0.32/0.51 & ( P @ X3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % unity_coeff_ex
% 0.32/0.51 thf(fact_1296_unity__coeff__ex,axiom,
% 0.32/0.51 ! [P: nat > $o,L: nat] :
% 0.32/0.51 ( ( ? [X3: nat] : ( P @ ( times_times_nat @ L @ X3 ) ) )
% 0.32/0.51 = ( ? [X3: nat] :
% 0.32/0.51 ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X3 @ zero_zero_nat ) )
% 0.32/0.51 & ( P @ X3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % unity_coeff_ex
% 0.32/0.51 thf(fact_1297_unity__coeff__ex,axiom,
% 0.32/0.51 ! [P: int > $o,L: int] :
% 0.32/0.51 ( ( ? [X3: int] : ( P @ ( times_times_int @ L @ X3 ) ) )
% 0.32/0.51 = ( ? [X3: int] :
% 0.32/0.51 ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X3 @ zero_zero_int ) )
% 0.32/0.51 & ( P @ X3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % unity_coeff_ex
% 0.32/0.51 thf(fact_1298_four__x__squared,axiom,
% 0.32/0.51 ! [X: real] :
% 0.32/0.51 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % four_x_squared
% 0.32/0.51 thf(fact_1299_inf__period_I4_J,axiom,
% 0.32/0.51 ! [D: code_integer,D3: code_integer,T: code_integer] :
% 0.32/0.51 ( ( dvd_dvd_Code_integer @ D @ D3 )
% 0.32/0.51 => ! [X5: code_integer,K4: code_integer] :
% 0.32/0.51 ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) ) )
% 0.32/0.51 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % inf_period(4)
% 0.32/0.51 thf(fact_1300_inf__period_I4_J,axiom,
% 0.32/0.51 ! [D: real,D3: real,T: real] :
% 0.32/0.51 ( ( dvd_dvd_real @ D @ D3 )
% 0.32/0.51 => ! [X5: real,K4: real] :
% 0.32/0.51 ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
% 0.32/0.51 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % inf_period(4)
% 0.32/0.51 thf(fact_1301_inf__period_I4_J,axiom,
% 0.32/0.51 ! [D: rat,D3: rat,T: rat] :
% 0.32/0.51 ( ( dvd_dvd_rat @ D @ D3 )
% 0.32/0.51 => ! [X5: rat,K4: rat] :
% 0.32/0.51 ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
% 0.32/0.51 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % inf_period(4)
% 0.32/0.51 thf(fact_1302_inf__period_I4_J,axiom,
% 0.32/0.51 ! [D: int,D3: int,T: int] :
% 0.32/0.51 ( ( dvd_dvd_int @ D @ D3 )
% 0.32/0.51 => ! [X5: int,K4: int] :
% 0.32/0.51 ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
% 0.32/0.51 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % inf_period(4)
% 0.32/0.51 thf(fact_1303_inf__period_I3_J,axiom,
% 0.32/0.51 ! [D: code_integer,D3: code_integer,T: code_integer] :
% 0.32/0.51 ( ( dvd_dvd_Code_integer @ D @ D3 )
% 0.32/0.51 => ! [X5: code_integer,K4: code_integer] :
% 0.32/0.51 ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) )
% 0.32/0.51 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % inf_period(3)
% 0.32/0.51 thf(fact_1304_inf__period_I3_J,axiom,
% 0.32/0.51 ! [D: real,D3: real,T: real] :
% 0.32/0.51 ( ( dvd_dvd_real @ D @ D3 )
% 0.32/0.51 => ! [X5: real,K4: real] :
% 0.32/0.51 ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
% 0.32/0.51 = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % inf_period(3)
% 0.32/0.51 thf(fact_1305_inf__period_I3_J,axiom,
% 0.32/0.51 ! [D: rat,D3: rat,T: rat] :
% 0.32/0.51 ( ( dvd_dvd_rat @ D @ D3 )
% 0.32/0.51 => ! [X5: rat,K4: rat] :
% 0.32/0.51 ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
% 0.32/0.51 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % inf_period(3)
% 0.32/0.51 thf(fact_1306_inf__period_I3_J,axiom,
% 0.32/0.51 ! [D: int,D3: int,T: int] :
% 0.32/0.51 ( ( dvd_dvd_int @ D @ D3 )
% 0.32/0.51 => ! [X5: int,K4: int] :
% 0.32/0.51 ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.32/0.51 = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % inf_period(3)
% 0.32/0.51 thf(fact_1307_one__plus__numeral__commute,axiom,
% 0.32/0.51 ! [X: num] :
% 0.32/0.51 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
% 0.32/0.51 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 0.32/0.51
% 0.32/0.51 % one_plus_numeral_commute
% 0.32/0.51 thf(fact_1308_one__plus__numeral__commute,axiom,
% 0.32/0.51 ! [X: num] :
% 0.32/0.51 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 0.32/0.51 = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % one_plus_numeral_commute
% 0.32/0.51 thf(fact_1309_one__plus__numeral__commute,axiom,
% 0.32/0.51 ! [X: num] :
% 0.32/0.51 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 0.32/0.51 = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % one_plus_numeral_commute
% 0.32/0.51 thf(fact_1310_one__plus__numeral__commute,axiom,
% 0.32/0.51 ! [X: num] :
% 0.32/0.51 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 0.32/0.51 = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % one_plus_numeral_commute
% 0.32/0.51 thf(fact_1311_one__plus__numeral__commute,axiom,
% 0.32/0.51 ! [X: num] :
% 0.32/0.51 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 0.32/0.51 = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % one_plus_numeral_commute
% 0.32/0.51 thf(fact_1312_numeral__Bit0,axiom,
% 0.32/0.51 ! [N: num] :
% 0.32/0.51 ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 0.32/0.51 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_Bit0
% 0.32/0.51 thf(fact_1313_numeral__Bit0,axiom,
% 0.32/0.51 ! [N: num] :
% 0.32/0.51 ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 0.32/0.51 = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_Bit0
% 0.32/0.51 thf(fact_1314_numeral__Bit0,axiom,
% 0.32/0.51 ! [N: num] :
% 0.32/0.51 ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 0.32/0.51 = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_Bit0
% 0.32/0.51 thf(fact_1315_numeral__Bit0,axiom,
% 0.32/0.51 ! [N: num] :
% 0.32/0.51 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.32/0.51 = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_Bit0
% 0.32/0.51 thf(fact_1316_numeral__Bit0,axiom,
% 0.32/0.51 ! [N: num] :
% 0.32/0.51 ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 0.32/0.51 = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_Bit0
% 0.32/0.51 thf(fact_1317_square__diff__square__factored,axiom,
% 0.32/0.51 ! [X: real,Y: real] :
% 0.32/0.51 ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
% 0.32/0.51 = ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % square_diff_square_factored
% 0.32/0.51 thf(fact_1318_square__diff__square__factored,axiom,
% 0.32/0.51 ! [X: rat,Y: rat] :
% 0.32/0.51 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 0.32/0.51 = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % square_diff_square_factored
% 0.32/0.51 thf(fact_1319_square__diff__square__factored,axiom,
% 0.32/0.51 ! [X: int,Y: int] :
% 0.32/0.51 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 0.32/0.51 = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % square_diff_square_factored
% 0.32/0.51 thf(fact_1320_eq__add__iff2,axiom,
% 0.32/0.51 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.32/0.51 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 0.32/0.51 = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.32/0.51 = ( C
% 0.32/0.51 = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % eq_add_iff2
% 0.32/0.51 thf(fact_1321_eq__add__iff2,axiom,
% 0.32/0.51 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.32/0.51 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 0.32/0.51 = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.32/0.51 = ( C
% 0.32/0.51 = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % eq_add_iff2
% 0.32/0.51 thf(fact_1322_eq__add__iff2,axiom,
% 0.32/0.51 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.32/0.51 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 0.32/0.51 = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.32/0.51 = ( C
% 0.32/0.51 = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % eq_add_iff2
% 0.32/0.51 thf(fact_1323_eq__add__iff1,axiom,
% 0.32/0.51 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.32/0.51 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 0.32/0.51 = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.32/0.51 = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
% 0.32/0.51 = D ) ) ).
% 0.32/0.51
% 0.32/0.51 % eq_add_iff1
% 0.32/0.51 thf(fact_1324_eq__add__iff1,axiom,
% 0.32/0.51 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.32/0.51 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 0.32/0.51 = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.32/0.51 = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
% 0.32/0.51 = D ) ) ).
% 0.32/0.51
% 0.32/0.51 % eq_add_iff1
% 0.32/0.51 thf(fact_1325_eq__add__iff1,axiom,
% 0.32/0.51 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.32/0.51 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 0.32/0.51 = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.32/0.51 = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
% 0.32/0.51 = D ) ) ).
% 0.32/0.51
% 0.32/0.51 % eq_add_iff1
% 0.32/0.51 thf(fact_1326_mult__diff__mult,axiom,
% 0.32/0.51 ! [X: real,Y: real,A: real,B: real] :
% 0.32/0.51 ( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
% 0.32/0.51 = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_diff_mult
% 0.32/0.51 thf(fact_1327_mult__diff__mult,axiom,
% 0.32/0.51 ! [X: rat,Y: rat,A: rat,B: rat] :
% 0.32/0.51 ( ( minus_minus_rat @ ( times_times_rat @ X @ Y ) @ ( times_times_rat @ A @ B ) )
% 0.32/0.51 = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_diff_mult
% 0.32/0.51 thf(fact_1328_mult__diff__mult,axiom,
% 0.32/0.51 ! [X: int,Y: int,A: int,B: int] :
% 0.32/0.51 ( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
% 0.32/0.51 = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_diff_mult
% 0.32/0.51 thf(fact_1329_div__plus__div__distrib__dvd__right,axiom,
% 0.32/0.51 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.51 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.51 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.32/0.51 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_plus_div_distrib_dvd_right
% 0.32/0.51 thf(fact_1330_div__plus__div__distrib__dvd__right,axiom,
% 0.32/0.51 ! [C: nat,B: nat,A: nat] :
% 0.32/0.51 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.51 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.51 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_plus_div_distrib_dvd_right
% 0.32/0.51 thf(fact_1331_div__plus__div__distrib__dvd__right,axiom,
% 0.32/0.51 ! [C: int,B: int,A: int] :
% 0.32/0.51 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.51 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.51 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_plus_div_distrib_dvd_right
% 0.32/0.51 thf(fact_1332_div__plus__div__distrib__dvd__left,axiom,
% 0.32/0.51 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.51 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.32/0.51 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.32/0.51 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_plus_div_distrib_dvd_left
% 0.32/0.51 thf(fact_1333_div__plus__div__distrib__dvd__left,axiom,
% 0.32/0.51 ! [C: nat,A: nat,B: nat] :
% 0.32/0.51 ( ( dvd_dvd_nat @ C @ A )
% 0.32/0.51 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.51 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_plus_div_distrib_dvd_left
% 0.32/0.51 thf(fact_1334_div__plus__div__distrib__dvd__left,axiom,
% 0.32/0.51 ! [C: int,A: int,B: int] :
% 0.32/0.51 ( ( dvd_dvd_int @ C @ A )
% 0.32/0.51 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.51 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_plus_div_distrib_dvd_left
% 0.32/0.51 thf(fact_1335_add__is__1,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( ( plus_plus_nat @ M @ N )
% 0.32/0.51 = ( suc @ zero_zero_nat ) )
% 0.32/0.51 = ( ( ( M
% 0.32/0.51 = ( suc @ zero_zero_nat ) )
% 0.32/0.51 & ( N = zero_zero_nat ) )
% 0.32/0.51 | ( ( M = zero_zero_nat )
% 0.32/0.51 & ( N
% 0.32/0.51 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_is_1
% 0.32/0.51 thf(fact_1336_one__is__add,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( ( suc @ zero_zero_nat )
% 0.32/0.51 = ( plus_plus_nat @ M @ N ) )
% 0.32/0.51 = ( ( ( M
% 0.32/0.51 = ( suc @ zero_zero_nat ) )
% 0.32/0.51 & ( N = zero_zero_nat ) )
% 0.32/0.51 | ( ( M = zero_zero_nat )
% 0.32/0.51 & ( N
% 0.32/0.51 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % one_is_add
% 0.32/0.51 thf(fact_1337_diff__add__0,axiom,
% 0.32/0.51 ! [N: nat,M: nat] :
% 0.32/0.51 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % diff_add_0
% 0.32/0.51 thf(fact_1338_mult__eq__if,axiom,
% 0.32/0.51 ( times_times_nat
% 0.32/0.51 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_eq_if
% 0.32/0.51 thf(fact_1339_int__ops_I4_J,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 0.32/0.51 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % int_ops(4)
% 0.32/0.51 thf(fact_1340_int__Suc,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 0.32/0.51 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % int_Suc
% 0.32/0.51 thf(fact_1341_bezout__add__strong__nat,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( A != zero_zero_nat )
% 0.32/0.51 => ? [D2: nat,X4: nat,Y4: nat] :
% 0.32/0.51 ( ( dvd_dvd_nat @ D2 @ A )
% 0.32/0.51 & ( dvd_dvd_nat @ D2 @ B )
% 0.32/0.51 & ( ( times_times_nat @ A @ X4 )
% 0.32/0.51 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D2 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % bezout_add_strong_nat
% 0.32/0.51 thf(fact_1342_divide__add__eq__iff,axiom,
% 0.32/0.51 ! [Z2: complex,X: complex,Y: complex] :
% 0.32/0.51 ( ( Z2 != zero_zero_complex )
% 0.32/0.51 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z2 ) @ Y )
% 0.32/0.51 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_add_eq_iff
% 0.32/0.51 thf(fact_1343_divide__add__eq__iff,axiom,
% 0.32/0.51 ! [Z2: real,X: real,Y: real] :
% 0.32/0.51 ( ( Z2 != zero_zero_real )
% 0.32/0.51 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z2 ) @ Y )
% 0.32/0.51 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_add_eq_iff
% 0.32/0.51 thf(fact_1344_divide__add__eq__iff,axiom,
% 0.32/0.51 ! [Z2: rat,X: rat,Y: rat] :
% 0.32/0.51 ( ( Z2 != zero_zero_rat )
% 0.32/0.51 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y )
% 0.32/0.51 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_add_eq_iff
% 0.32/0.51 thf(fact_1345_add__divide__eq__iff,axiom,
% 0.32/0.51 ! [Z2: complex,X: complex,Y: complex] :
% 0.32/0.51 ( ( Z2 != zero_zero_complex )
% 0.32/0.51 => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
% 0.32/0.51 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_divide_eq_iff
% 0.32/0.51 thf(fact_1346_add__divide__eq__iff,axiom,
% 0.32/0.51 ! [Z2: real,X: real,Y: real] :
% 0.32/0.51 ( ( Z2 != zero_zero_real )
% 0.32/0.51 => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z2 ) )
% 0.32/0.51 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_divide_eq_iff
% 0.32/0.51 thf(fact_1347_add__divide__eq__iff,axiom,
% 0.32/0.51 ! [Z2: rat,X: rat,Y: rat] :
% 0.32/0.51 ( ( Z2 != zero_zero_rat )
% 0.32/0.51 => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z2 ) )
% 0.32/0.51 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_divide_eq_iff
% 0.32/0.51 thf(fact_1348_add__num__frac,axiom,
% 0.32/0.51 ! [Y: complex,Z2: complex,X: complex] :
% 0.32/0.51 ( ( Y != zero_zero_complex )
% 0.32/0.51 => ( ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ X @ Y ) )
% 0.32/0.51 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_num_frac
% 0.32/0.51 thf(fact_1349_add__num__frac,axiom,
% 0.32/0.51 ! [Y: real,Z2: real,X: real] :
% 0.32/0.51 ( ( Y != zero_zero_real )
% 0.32/0.51 => ( ( plus_plus_real @ Z2 @ ( divide_divide_real @ X @ Y ) )
% 0.32/0.51 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_num_frac
% 0.32/0.51 thf(fact_1350_add__num__frac,axiom,
% 0.32/0.51 ! [Y: rat,Z2: rat,X: rat] :
% 0.32/0.51 ( ( Y != zero_zero_rat )
% 0.32/0.51 => ( ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) )
% 0.32/0.51 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_num_frac
% 0.32/0.51 thf(fact_1351_add__frac__num,axiom,
% 0.32/0.51 ! [Y: complex,X: complex,Z2: complex] :
% 0.32/0.51 ( ( Y != zero_zero_complex )
% 0.32/0.51 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z2 )
% 0.32/0.51 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_frac_num
% 0.32/0.51 thf(fact_1352_add__frac__num,axiom,
% 0.32/0.51 ! [Y: real,X: real,Z2: real] :
% 0.32/0.51 ( ( Y != zero_zero_real )
% 0.32/0.51 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z2 )
% 0.32/0.51 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_frac_num
% 0.32/0.51 thf(fact_1353_add__frac__num,axiom,
% 0.32/0.51 ! [Y: rat,X: rat,Z2: rat] :
% 0.32/0.51 ( ( Y != zero_zero_rat )
% 0.32/0.51 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 )
% 0.32/0.51 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_frac_num
% 0.32/0.51 thf(fact_1354_add__frac__eq,axiom,
% 0.32/0.51 ! [Y: complex,Z2: complex,X: complex,W: complex] :
% 0.32/0.51 ( ( Y != zero_zero_complex )
% 0.32/0.51 => ( ( Z2 != zero_zero_complex )
% 0.32/0.51 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z2 ) )
% 0.32/0.51 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_frac_eq
% 0.32/0.51 thf(fact_1355_add__frac__eq,axiom,
% 0.32/0.51 ! [Y: real,Z2: real,X: real,W: real] :
% 0.32/0.51 ( ( Y != zero_zero_real )
% 0.32/0.51 => ( ( Z2 != zero_zero_real )
% 0.32/0.51 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
% 0.32/0.51 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_frac_eq
% 0.32/0.51 thf(fact_1356_add__frac__eq,axiom,
% 0.32/0.51 ! [Y: rat,Z2: rat,X: rat,W: rat] :
% 0.32/0.51 ( ( Y != zero_zero_rat )
% 0.32/0.51 => ( ( Z2 != zero_zero_rat )
% 0.32/0.51 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z2 ) )
% 0.32/0.51 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_frac_eq
% 0.32/0.51 thf(fact_1357_add__divide__eq__if__simps_I1_J,axiom,
% 0.32/0.51 ! [Z2: complex,A: complex,B: complex] :
% 0.32/0.51 ( ( ( Z2 = zero_zero_complex )
% 0.32/0.51 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 0.32/0.51 = A ) )
% 0.32/0.51 & ( ( Z2 != zero_zero_complex )
% 0.32/0.51 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 0.32/0.51 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_divide_eq_if_simps(1)
% 0.32/0.51 thf(fact_1358_add__divide__eq__if__simps_I1_J,axiom,
% 0.32/0.51 ! [Z2: real,A: real,B: real] :
% 0.32/0.51 ( ( ( Z2 = zero_zero_real )
% 0.32/0.51 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 0.32/0.51 = A ) )
% 0.32/0.51 & ( ( Z2 != zero_zero_real )
% 0.32/0.51 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 0.32/0.51 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_divide_eq_if_simps(1)
% 0.32/0.51 thf(fact_1359_add__divide__eq__if__simps_I1_J,axiom,
% 0.32/0.51 ! [Z2: rat,A: rat,B: rat] :
% 0.32/0.51 ( ( ( Z2 = zero_zero_rat )
% 0.32/0.51 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 0.32/0.51 = A ) )
% 0.32/0.51 & ( ( Z2 != zero_zero_rat )
% 0.32/0.51 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 0.32/0.51 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_divide_eq_if_simps(1)
% 0.32/0.51 thf(fact_1360_add__divide__eq__if__simps_I2_J,axiom,
% 0.32/0.51 ! [Z2: complex,A: complex,B: complex] :
% 0.32/0.51 ( ( ( Z2 = zero_zero_complex )
% 0.32/0.51 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 0.32/0.51 = B ) )
% 0.32/0.51 & ( ( Z2 != zero_zero_complex )
% 0.32/0.51 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 0.32/0.51 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_divide_eq_if_simps(2)
% 0.32/0.51 thf(fact_1361_add__divide__eq__if__simps_I2_J,axiom,
% 0.32/0.51 ! [Z2: real,A: real,B: real] :
% 0.32/0.51 ( ( ( Z2 = zero_zero_real )
% 0.32/0.51 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 0.32/0.51 = B ) )
% 0.32/0.51 & ( ( Z2 != zero_zero_real )
% 0.32/0.51 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 0.32/0.51 = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_divide_eq_if_simps(2)
% 0.32/0.51 thf(fact_1362_add__divide__eq__if__simps_I2_J,axiom,
% 0.32/0.51 ! [Z2: rat,A: rat,B: rat] :
% 0.32/0.51 ( ( ( Z2 = zero_zero_rat )
% 0.32/0.51 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 0.32/0.51 = B ) )
% 0.32/0.51 & ( ( Z2 != zero_zero_rat )
% 0.32/0.51 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 0.32/0.51 = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_divide_eq_if_simps(2)
% 0.32/0.51 thf(fact_1363_div__add__self2,axiom,
% 0.32/0.51 ! [B: nat,A: nat] :
% 0.32/0.51 ( ( B != zero_zero_nat )
% 0.32/0.51 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.32/0.51 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_add_self2
% 0.32/0.51 thf(fact_1364_div__add__self2,axiom,
% 0.32/0.51 ! [B: int,A: int] :
% 0.32/0.51 ( ( B != zero_zero_int )
% 0.32/0.51 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.32/0.51 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_add_self2
% 0.32/0.51 thf(fact_1365_div__add__self1,axiom,
% 0.32/0.51 ! [B: nat,A: nat] :
% 0.32/0.51 ( ( B != zero_zero_nat )
% 0.32/0.51 => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.32/0.51 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_add_self1
% 0.32/0.51 thf(fact_1366_div__add__self1,axiom,
% 0.32/0.51 ! [B: int,A: int] :
% 0.32/0.51 ( ( B != zero_zero_int )
% 0.32/0.51 => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.32/0.51 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_add_self1
% 0.32/0.51 thf(fact_1367_square__diff__one__factored,axiom,
% 0.32/0.51 ! [X: complex] :
% 0.32/0.51 ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
% 0.32/0.51 = ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % square_diff_one_factored
% 0.32/0.51 thf(fact_1368_square__diff__one__factored,axiom,
% 0.32/0.51 ! [X: real] :
% 0.32/0.51 ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
% 0.32/0.51 = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % square_diff_one_factored
% 0.32/0.51 thf(fact_1369_square__diff__one__factored,axiom,
% 0.32/0.51 ! [X: rat] :
% 0.32/0.51 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
% 0.32/0.51 = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % square_diff_one_factored
% 0.32/0.51 thf(fact_1370_square__diff__one__factored,axiom,
% 0.32/0.51 ! [X: int] :
% 0.32/0.51 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
% 0.32/0.51 = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % square_diff_one_factored
% 0.32/0.51 thf(fact_1371_push__bit__nat__def,axiom,
% 0.32/0.51 ( bit_se547839408752420682it_nat
% 0.32/0.51 = ( ^ [N3: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % push_bit_nat_def
% 0.32/0.51 thf(fact_1372_push__bit__int__def,axiom,
% 0.32/0.51 ( bit_se545348938243370406it_int
% 0.32/0.51 = ( ^ [N3: nat,K2: int] : ( times_times_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % push_bit_int_def
% 0.32/0.51 thf(fact_1373_nat__1__add__1,axiom,
% 0.32/0.51 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 0.32/0.51 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nat_1_add_1
% 0.32/0.51 thf(fact_1374_add__eq__if,axiom,
% 0.32/0.51 ( plus_plus_nat
% 0.32/0.51 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_eq_if
% 0.32/0.51 thf(fact_1375_left__add__twice,axiom,
% 0.32/0.51 ! [A: complex,B: complex] :
% 0.32/0.51 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
% 0.32/0.51 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % left_add_twice
% 0.32/0.51 thf(fact_1376_left__add__twice,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.32/0.51 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % left_add_twice
% 0.32/0.51 thf(fact_1377_left__add__twice,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.51 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % left_add_twice
% 0.32/0.51 thf(fact_1378_left__add__twice,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.51 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % left_add_twice
% 0.32/0.51 thf(fact_1379_left__add__twice,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.32/0.51 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % left_add_twice
% 0.32/0.51 thf(fact_1380_mult__2__right,axiom,
% 0.32/0.51 ! [Z2: complex] :
% 0.32/0.51 ( ( times_times_complex @ Z2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( plus_plus_complex @ Z2 @ Z2 ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_2_right
% 0.32/0.51 thf(fact_1381_mult__2__right,axiom,
% 0.32/0.51 ! [Z2: real] :
% 0.32/0.51 ( ( times_times_real @ Z2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( plus_plus_real @ Z2 @ Z2 ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_2_right
% 0.32/0.51 thf(fact_1382_mult__2__right,axiom,
% 0.32/0.51 ! [Z2: rat] :
% 0.32/0.51 ( ( times_times_rat @ Z2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( plus_plus_rat @ Z2 @ Z2 ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_2_right
% 0.32/0.51 thf(fact_1383_mult__2__right,axiom,
% 0.32/0.51 ! [Z2: nat] :
% 0.32/0.51 ( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_2_right
% 0.32/0.51 thf(fact_1384_mult__2__right,axiom,
% 0.32/0.51 ! [Z2: int] :
% 0.32/0.51 ( ( times_times_int @ Z2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( plus_plus_int @ Z2 @ Z2 ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_2_right
% 0.32/0.51 thf(fact_1385_mult__2,axiom,
% 0.32/0.51 ! [Z2: complex] :
% 0.32/0.51 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 )
% 0.32/0.51 = ( plus_plus_complex @ Z2 @ Z2 ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_2
% 0.32/0.51 thf(fact_1386_mult__2,axiom,
% 0.32/0.51 ! [Z2: real] :
% 0.32/0.51 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 )
% 0.32/0.51 = ( plus_plus_real @ Z2 @ Z2 ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_2
% 0.32/0.51 thf(fact_1387_mult__2,axiom,
% 0.32/0.51 ! [Z2: rat] :
% 0.32/0.51 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 )
% 0.32/0.51 = ( plus_plus_rat @ Z2 @ Z2 ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_2
% 0.32/0.51 thf(fact_1388_mult__2,axiom,
% 0.32/0.51 ! [Z2: nat] :
% 0.32/0.51 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
% 0.32/0.51 = ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_2
% 0.32/0.51 thf(fact_1389_mult__2,axiom,
% 0.32/0.51 ! [Z2: int] :
% 0.32/0.51 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z2 )
% 0.32/0.51 = ( plus_plus_int @ Z2 @ Z2 ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_2
% 0.32/0.51 thf(fact_1390_field__sum__of__halves,axiom,
% 0.32/0.51 ! [X: real] :
% 0.32/0.51 ( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = X ) ).
% 0.32/0.51
% 0.32/0.51 % field_sum_of_halves
% 0.32/0.51 thf(fact_1391_field__sum__of__halves,axiom,
% 0.32/0.51 ! [X: rat] :
% 0.32/0.51 ( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = X ) ).
% 0.32/0.51
% 0.32/0.51 % field_sum_of_halves
% 0.32/0.51 thf(fact_1392_odd__even__add,axiom,
% 0.32/0.51 ! [A: code_integer,B: code_integer] :
% 0.32/0.51 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.51 => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
% 0.32/0.51 => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % odd_even_add
% 0.32/0.51 thf(fact_1393_odd__even__add,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.51 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.32/0.51 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % odd_even_add
% 0.32/0.51 thf(fact_1394_odd__even__add,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.51 => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
% 0.32/0.51 => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % odd_even_add
% 0.32/0.51 thf(fact_1395_nat__induct2,axiom,
% 0.32/0.51 ! [P: nat > $o,N: nat] :
% 0.32/0.51 ( ( P @ zero_zero_nat )
% 0.32/0.51 => ( ( P @ one_one_nat )
% 0.32/0.51 => ( ! [N2: nat] :
% 0.32/0.51 ( ( P @ N2 )
% 0.32/0.51 => ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.51 => ( P @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nat_induct2
% 0.32/0.51 thf(fact_1396_even__diff__iff,axiom,
% 0.32/0.51 ! [K: int,L: int] :
% 0.32/0.51 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
% 0.32/0.51 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % even_diff_iff
% 0.32/0.51 thf(fact_1397_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.32/0.51 ! [N: nat,M: nat] :
% 0.32/0.51 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.51 != zero_zero_nat )
% 0.32/0.51 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.32/0.51 != zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % exp_not_zero_imp_exp_diff_not_zero
% 0.32/0.51 thf(fact_1398_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.32/0.51 ! [N: nat,M: nat] :
% 0.32/0.51 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.32/0.51 != zero_zero_int )
% 0.32/0.51 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.32/0.51 != zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % exp_not_zero_imp_exp_diff_not_zero
% 0.32/0.51 thf(fact_1399_push__bit__eq__mult,axiom,
% 0.32/0.51 ( bit_se547839408752420682it_nat
% 0.32/0.51 = ( ^ [N3: nat,A2: nat] : ( times_times_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % push_bit_eq_mult
% 0.32/0.51 thf(fact_1400_push__bit__eq__mult,axiom,
% 0.32/0.51 ( bit_se545348938243370406it_int
% 0.32/0.51 = ( ^ [N3: nat,A2: int] : ( times_times_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % push_bit_eq_mult
% 0.32/0.51 thf(fact_1401_push__bit__eq__mult,axiom,
% 0.32/0.51 ( bit_se7788150548672797655nteger
% 0.32/0.51 = ( ^ [N3: nat,A2: code_integer] : ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % push_bit_eq_mult
% 0.32/0.51 thf(fact_1402_push__bit__eq__mult,axiom,
% 0.32/0.51 ( bit_se6611745700429515170atural
% 0.32/0.51 = ( ^ [N3: nat,A2: code_natural] : ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % push_bit_eq_mult
% 0.32/0.51 thf(fact_1403_exp__dvdE,axiom,
% 0.32/0.51 ! [N: nat,A: nat] :
% 0.32/0.51 ( ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A )
% 0.32/0.51 => ~ ! [B4: nat] :
% 0.32/0.51 ( A
% 0.32/0.51 != ( bit_se547839408752420682it_nat @ N @ B4 ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % exp_dvdE
% 0.32/0.51 thf(fact_1404_exp__dvdE,axiom,
% 0.32/0.51 ! [N: nat,A: int] :
% 0.32/0.51 ( ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A )
% 0.32/0.51 => ~ ! [B4: int] :
% 0.32/0.51 ( A
% 0.32/0.51 != ( bit_se545348938243370406it_int @ N @ B4 ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % exp_dvdE
% 0.32/0.51 thf(fact_1405_exp__dvdE,axiom,
% 0.32/0.51 ! [N: nat,A: code_integer] :
% 0.32/0.51 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A )
% 0.32/0.51 => ~ ! [B4: code_integer] :
% 0.32/0.51 ( A
% 0.32/0.51 != ( bit_se7788150548672797655nteger @ N @ B4 ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % exp_dvdE
% 0.32/0.51 thf(fact_1406_exp__dvdE,axiom,
% 0.32/0.51 ! [N: nat,A: code_natural] :
% 0.32/0.51 ( ( dvd_dvd_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) @ A )
% 0.32/0.51 => ~ ! [B4: code_natural] :
% 0.32/0.51 ( A
% 0.32/0.51 != ( bit_se6611745700429515170atural @ N @ B4 ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % exp_dvdE
% 0.32/0.51 thf(fact_1407_oddE,axiom,
% 0.32/0.51 ! [A: code_integer] :
% 0.32/0.51 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.51 => ~ ! [B4: code_integer] :
% 0.32/0.51 ( A
% 0.32/0.51 != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) @ one_one_Code_integer ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % oddE
% 0.32/0.51 thf(fact_1408_oddE,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.51 => ~ ! [B4: nat] :
% 0.32/0.51 ( A
% 0.32/0.51 != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) @ one_one_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % oddE
% 0.32/0.51 thf(fact_1409_oddE,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.51 => ~ ! [B4: int] :
% 0.32/0.51 ( A
% 0.32/0.51 != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) @ one_one_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % oddE
% 0.32/0.51 thf(fact_1410_inf__period_I2_J,axiom,
% 0.32/0.51 ! [P: real > $o,D3: real,Q: real > $o] :
% 0.32/0.51 ( ! [X4: real,K3: real] :
% 0.32/0.51 ( ( P @ X4 )
% 0.32/0.51 = ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D3 ) ) ) )
% 0.32/0.51 => ( ! [X4: real,K3: real] :
% 0.32/0.51 ( ( Q @ X4 )
% 0.32/0.51 = ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D3 ) ) ) )
% 0.32/0.51 => ! [X5: real,K4: real] :
% 0.32/0.51 ( ( ( P @ X5 )
% 0.32/0.51 | ( Q @ X5 ) )
% 0.32/0.51 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) )
% 0.32/0.51 | ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % inf_period(2)
% 0.32/0.51 thf(fact_1411_inf__period_I2_J,axiom,
% 0.32/0.51 ! [P: rat > $o,D3: rat,Q: rat > $o] :
% 0.32/0.51 ( ! [X4: rat,K3: rat] :
% 0.32/0.51 ( ( P @ X4 )
% 0.32/0.51 = ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) ) )
% 0.32/0.51 => ( ! [X4: rat,K3: rat] :
% 0.32/0.51 ( ( Q @ X4 )
% 0.32/0.51 = ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) ) )
% 0.32/0.51 => ! [X5: rat,K4: rat] :
% 0.32/0.51 ( ( ( P @ X5 )
% 0.32/0.51 | ( Q @ X5 ) )
% 0.32/0.51 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) )
% 0.32/0.51 | ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % inf_period(2)
% 0.32/0.51 thf(fact_1412_inf__period_I2_J,axiom,
% 0.32/0.51 ! [P: int > $o,D3: int,Q: int > $o] :
% 0.32/0.51 ( ! [X4: int,K3: int] :
% 0.32/0.51 ( ( P @ X4 )
% 0.32/0.51 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
% 0.32/0.51 => ( ! [X4: int,K3: int] :
% 0.32/0.51 ( ( Q @ X4 )
% 0.32/0.51 = ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
% 0.32/0.51 => ! [X5: int,K4: int] :
% 0.32/0.51 ( ( ( P @ X5 )
% 0.32/0.51 | ( Q @ X5 ) )
% 0.32/0.51 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
% 0.32/0.51 | ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % inf_period(2)
% 0.32/0.51 thf(fact_1413_sum__power2__eq__zero__iff,axiom,
% 0.32/0.51 ! [X: rat,Y: rat] :
% 0.32/0.51 ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = zero_zero_rat )
% 0.32/0.51 = ( ( X = zero_zero_rat )
% 0.32/0.51 & ( Y = zero_zero_rat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % sum_power2_eq_zero_iff
% 0.32/0.51 thf(fact_1414_sum__power2__eq__zero__iff,axiom,
% 0.32/0.51 ! [X: real,Y: real] :
% 0.32/0.51 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = zero_zero_real )
% 0.32/0.51 = ( ( X = zero_zero_real )
% 0.32/0.51 & ( Y = zero_zero_real ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % sum_power2_eq_zero_iff
% 0.32/0.51 thf(fact_1415_sum__power2__eq__zero__iff,axiom,
% 0.32/0.51 ! [X: int,Y: int] :
% 0.32/0.51 ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = zero_zero_int )
% 0.32/0.51 = ( ( X = zero_zero_int )
% 0.32/0.51 & ( Y = zero_zero_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % sum_power2_eq_zero_iff
% 0.32/0.51 thf(fact_1416_zero__eq__power2,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_rat )
% 0.32/0.51 = ( A = zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_eq_power2
% 0.32/0.51 thf(fact_1417_zero__eq__power2,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_nat )
% 0.32/0.51 = ( A = zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_eq_power2
% 0.32/0.51 thf(fact_1418_zero__eq__power2,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_real )
% 0.32/0.51 = ( A = zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_eq_power2
% 0.32/0.51 thf(fact_1419_zero__eq__power2,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_int )
% 0.32/0.51 = ( A = zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_eq_power2
% 0.32/0.51 thf(fact_1420_zero__eq__power2,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_complex )
% 0.32/0.51 = ( A = zero_zero_complex ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_eq_power2
% 0.32/0.51 thf(fact_1421_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.32/0.51 ! [Y: nat,X: num,N: nat] :
% 0.32/0.51 ( ( ( semiri8010041392384452111omplex @ Y )
% 0.32/0.51 = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
% 0.32/0.51 = ( Y
% 0.32/0.51 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % real_of_nat_eq_numeral_power_cancel_iff
% 0.32/0.51 thf(fact_1422_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.32/0.51 ! [Y: nat,X: num,N: nat] :
% 0.32/0.51 ( ( ( semiri5074537144036343181t_real @ Y )
% 0.32/0.51 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.32/0.51 = ( Y
% 0.32/0.51 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % real_of_nat_eq_numeral_power_cancel_iff
% 0.32/0.51 thf(fact_1423_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.32/0.51 ! [Y: nat,X: num,N: nat] :
% 0.32/0.51 ( ( ( semiri681578069525770553at_rat @ Y )
% 0.32/0.51 = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.32/0.51 = ( Y
% 0.32/0.51 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % real_of_nat_eq_numeral_power_cancel_iff
% 0.32/0.51 thf(fact_1424_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.32/0.51 ! [Y: nat,X: num,N: nat] :
% 0.32/0.51 ( ( ( semiri1316708129612266289at_nat @ Y )
% 0.32/0.51 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.32/0.51 = ( Y
% 0.32/0.51 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % real_of_nat_eq_numeral_power_cancel_iff
% 0.32/0.51 thf(fact_1425_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.32/0.51 ! [Y: nat,X: num,N: nat] :
% 0.32/0.51 ( ( ( semiri1314217659103216013at_int @ Y )
% 0.32/0.51 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.32/0.51 = ( Y
% 0.32/0.51 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % real_of_nat_eq_numeral_power_cancel_iff
% 0.32/0.51 thf(fact_1426_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.51 ! [X: num,N: nat,Y: nat] :
% 0.32/0.51 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
% 0.32/0.51 = ( semiri8010041392384452111omplex @ Y ) )
% 0.32/0.51 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/0.51 = Y ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_power_eq_of_nat_cancel_iff
% 0.32/0.51 thf(fact_1427_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.51 ! [X: num,N: nat,Y: nat] :
% 0.32/0.51 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 0.32/0.51 = ( semiri5074537144036343181t_real @ Y ) )
% 0.32/0.51 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/0.51 = Y ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_power_eq_of_nat_cancel_iff
% 0.32/0.51 thf(fact_1428_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.51 ! [X: num,N: nat,Y: nat] :
% 0.32/0.51 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
% 0.32/0.51 = ( semiri681578069525770553at_rat @ Y ) )
% 0.32/0.51 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/0.51 = Y ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_power_eq_of_nat_cancel_iff
% 0.32/0.51 thf(fact_1429_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.51 ! [X: num,N: nat,Y: nat] :
% 0.32/0.51 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/0.51 = ( semiri1316708129612266289at_nat @ Y ) )
% 0.32/0.51 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/0.51 = Y ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_power_eq_of_nat_cancel_iff
% 0.32/0.51 thf(fact_1430_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.51 ! [X: num,N: nat,Y: nat] :
% 0.32/0.51 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.32/0.51 = ( semiri1314217659103216013at_int @ Y ) )
% 0.32/0.51 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/0.51 = Y ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_power_eq_of_nat_cancel_iff
% 0.32/0.51 thf(fact_1431_power__mult__numeral,axiom,
% 0.32/0.51 ! [A: nat,M: num,N: num] :
% 0.32/0.51 ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.51 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mult_numeral
% 0.32/0.51 thf(fact_1432_power__mult__numeral,axiom,
% 0.32/0.51 ! [A: real,M: num,N: num] :
% 0.32/0.51 ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.51 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mult_numeral
% 0.32/0.51 thf(fact_1433_power__mult__numeral,axiom,
% 0.32/0.51 ! [A: int,M: num,N: num] :
% 0.32/0.51 ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.51 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mult_numeral
% 0.32/0.51 thf(fact_1434_power__mult__numeral,axiom,
% 0.32/0.51 ! [A: complex,M: num,N: num] :
% 0.32/0.51 ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.51 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mult_numeral
% 0.32/0.51 thf(fact_1435_power2__diff,axiom,
% 0.32/0.51 ! [X: complex,Y: complex] :
% 0.32/0.51 ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_diff
% 0.32/0.51 thf(fact_1436_power2__diff,axiom,
% 0.32/0.51 ! [X: real,Y: real] :
% 0.32/0.51 ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_diff
% 0.32/0.51 thf(fact_1437_power2__diff,axiom,
% 0.32/0.51 ! [X: rat,Y: rat] :
% 0.32/0.51 ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_diff
% 0.32/0.51 thf(fact_1438_power2__diff,axiom,
% 0.32/0.51 ! [X: int,Y: int] :
% 0.32/0.51 ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_diff
% 0.32/0.51 thf(fact_1439_real__average__minus__first,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 0.32/0.51 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % real_average_minus_first
% 0.32/0.51 thf(fact_1440_real__average__minus__second,axiom,
% 0.32/0.51 ! [B: real,A: real] :
% 0.32/0.51 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 0.32/0.51 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % real_average_minus_second
% 0.32/0.51 thf(fact_1441_power__Suc0__right,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % power_Suc0_right
% 0.32/0.51 thf(fact_1442_power__Suc0__right,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % power_Suc0_right
% 0.32/0.51 thf(fact_1443_power__Suc0__right,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % power_Suc0_right
% 0.32/0.51 thf(fact_1444_power__Suc0__right,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % power_Suc0_right
% 0.32/0.51 thf(fact_1445_power__add__numeral,axiom,
% 0.32/0.51 ! [A: complex,M: num,N: num] :
% 0.32/0.51 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.32/0.51 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_add_numeral
% 0.32/0.51 thf(fact_1446_power__add__numeral,axiom,
% 0.32/0.51 ! [A: real,M: num,N: num] :
% 0.32/0.51 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.32/0.51 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_add_numeral
% 0.32/0.51 thf(fact_1447_power__add__numeral,axiom,
% 0.32/0.51 ! [A: rat,M: num,N: num] :
% 0.32/0.51 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.32/0.51 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_add_numeral
% 0.32/0.51 thf(fact_1448_power__add__numeral,axiom,
% 0.32/0.51 ! [A: nat,M: num,N: num] :
% 0.32/0.51 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.32/0.51 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_add_numeral
% 0.32/0.51 thf(fact_1449_power__add__numeral,axiom,
% 0.32/0.51 ! [A: int,M: num,N: num] :
% 0.32/0.51 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.32/0.51 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_add_numeral
% 0.32/0.51 thf(fact_1450_power__add__numeral2,axiom,
% 0.32/0.51 ! [A: complex,M: num,N: num,B: complex] :
% 0.32/0.51 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.32/0.51 = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_add_numeral2
% 0.32/0.51 thf(fact_1451_power__add__numeral2,axiom,
% 0.32/0.51 ! [A: real,M: num,N: num,B: real] :
% 0.32/0.51 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.32/0.51 = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_add_numeral2
% 0.32/0.51 thf(fact_1452_power__add__numeral2,axiom,
% 0.32/0.51 ! [A: rat,M: num,N: num,B: rat] :
% 0.32/0.51 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.32/0.51 = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_add_numeral2
% 0.32/0.51 thf(fact_1453_power__add__numeral2,axiom,
% 0.32/0.51 ! [A: nat,M: num,N: num,B: nat] :
% 0.32/0.51 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.32/0.51 = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_add_numeral2
% 0.32/0.51 thf(fact_1454_power__add__numeral2,axiom,
% 0.32/0.51 ! [A: int,M: num,N: num,B: int] :
% 0.32/0.51 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.32/0.51 = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_add_numeral2
% 0.32/0.51 thf(fact_1455_power__one,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( power_power_rat @ one_one_rat @ N )
% 0.32/0.51 = one_one_rat ) ).
% 0.32/0.51
% 0.32/0.51 % power_one
% 0.32/0.51 thf(fact_1456_power__one,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( power_power_nat @ one_one_nat @ N )
% 0.32/0.51 = one_one_nat ) ).
% 0.32/0.51
% 0.32/0.51 % power_one
% 0.32/0.51 thf(fact_1457_power__one,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( power_power_real @ one_one_real @ N )
% 0.32/0.51 = one_one_real ) ).
% 0.32/0.51
% 0.32/0.51 % power_one
% 0.32/0.51 thf(fact_1458_power__one,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( power_power_int @ one_one_int @ N )
% 0.32/0.51 = one_one_int ) ).
% 0.32/0.51
% 0.32/0.51 % power_one
% 0.32/0.51 thf(fact_1459_power__one,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( power_power_complex @ one_one_complex @ N )
% 0.32/0.51 = one_one_complex ) ).
% 0.32/0.51
% 0.32/0.51 % power_one
% 0.32/0.51 thf(fact_1460_power__one__right,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( power_power_nat @ A @ one_one_nat )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % power_one_right
% 0.32/0.51 thf(fact_1461_power__one__right,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( power_power_real @ A @ one_one_nat )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % power_one_right
% 0.32/0.51 thf(fact_1462_power__one__right,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( power_power_int @ A @ one_one_nat )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % power_one_right
% 0.32/0.51 thf(fact_1463_power__one__right,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( power_power_complex @ A @ one_one_nat )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % power_one_right
% 0.32/0.51 thf(fact_1464_sum__squares__eq__zero__iff,axiom,
% 0.32/0.51 ! [X: real,Y: real] :
% 0.32/0.51 ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
% 0.32/0.51 = zero_zero_real )
% 0.32/0.51 = ( ( X = zero_zero_real )
% 0.32/0.51 & ( Y = zero_zero_real ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % sum_squares_eq_zero_iff
% 0.32/0.51 thf(fact_1465_sum__squares__eq__zero__iff,axiom,
% 0.32/0.51 ! [X: rat,Y: rat] :
% 0.32/0.51 ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 0.32/0.51 = zero_zero_rat )
% 0.32/0.51 = ( ( X = zero_zero_rat )
% 0.32/0.51 & ( Y = zero_zero_rat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % sum_squares_eq_zero_iff
% 0.32/0.51 thf(fact_1466_sum__squares__eq__zero__iff,axiom,
% 0.32/0.51 ! [X: int,Y: int] :
% 0.32/0.51 ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 0.32/0.51 = zero_zero_int )
% 0.32/0.51 = ( ( X = zero_zero_int )
% 0.32/0.51 & ( Y = zero_zero_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % sum_squares_eq_zero_iff
% 0.32/0.51 thf(fact_1467_power__0__Suc,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 0.32/0.51 = zero_zero_rat ) ).
% 0.32/0.51
% 0.32/0.51 % power_0_Suc
% 0.32/0.51 thf(fact_1468_power__0__Suc,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % power_0_Suc
% 0.32/0.51 thf(fact_1469_power__0__Suc,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 0.32/0.51 = zero_zero_real ) ).
% 0.32/0.51
% 0.32/0.51 % power_0_Suc
% 0.32/0.51 thf(fact_1470_power__0__Suc,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % power_0_Suc
% 0.32/0.51 thf(fact_1471_power__0__Suc,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 0.32/0.51 = zero_zero_complex ) ).
% 0.32/0.51
% 0.32/0.51 % power_0_Suc
% 0.32/0.51 thf(fact_1472_power__zero__numeral,axiom,
% 0.32/0.51 ! [K: num] :
% 0.32/0.51 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 0.32/0.51 = zero_zero_rat ) ).
% 0.32/0.51
% 0.32/0.51 % power_zero_numeral
% 0.32/0.51 thf(fact_1473_power__zero__numeral,axiom,
% 0.32/0.51 ! [K: num] :
% 0.32/0.51 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % power_zero_numeral
% 0.32/0.51 thf(fact_1474_power__zero__numeral,axiom,
% 0.32/0.51 ! [K: num] :
% 0.32/0.51 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 0.32/0.51 = zero_zero_real ) ).
% 0.32/0.51
% 0.32/0.51 % power_zero_numeral
% 0.32/0.51 thf(fact_1475_power__zero__numeral,axiom,
% 0.32/0.51 ! [K: num] :
% 0.32/0.51 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % power_zero_numeral
% 0.32/0.51 thf(fact_1476_power__zero__numeral,axiom,
% 0.32/0.51 ! [K: num] :
% 0.32/0.51 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 0.32/0.51 = zero_zero_complex ) ).
% 0.32/0.51
% 0.32/0.51 % power_zero_numeral
% 0.32/0.51 thf(fact_1477_nat__power__eq__Suc__0__iff,axiom,
% 0.32/0.51 ! [X: nat,M: nat] :
% 0.32/0.51 ( ( ( power_power_nat @ X @ M )
% 0.32/0.51 = ( suc @ zero_zero_nat ) )
% 0.32/0.51 = ( ( M = zero_zero_nat )
% 0.32/0.51 | ( X
% 0.32/0.51 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nat_power_eq_Suc_0_iff
% 0.32/0.51 thf(fact_1478_power__Suc__0,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.51 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_Suc_0
% 0.32/0.51 thf(fact_1479_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.51 ! [X: nat,B: nat,W: nat] :
% 0.32/0.51 ( ( ( semiri8010041392384452111omplex @ X )
% 0.32/0.51 = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
% 0.32/0.51 = ( X
% 0.32/0.51 = ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % of_nat_power_eq_of_nat_cancel_iff
% 0.32/0.51 thf(fact_1480_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.51 ! [X: nat,B: nat,W: nat] :
% 0.32/0.51 ( ( ( semiri5074537144036343181t_real @ X )
% 0.32/0.51 = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.32/0.51 = ( X
% 0.32/0.51 = ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % of_nat_power_eq_of_nat_cancel_iff
% 0.32/0.51 thf(fact_1481_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.51 ! [X: nat,B: nat,W: nat] :
% 0.32/0.51 ( ( ( semiri681578069525770553at_rat @ X )
% 0.32/0.51 = ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.32/0.51 = ( X
% 0.32/0.51 = ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % of_nat_power_eq_of_nat_cancel_iff
% 0.32/0.51 thf(fact_1482_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.51 ! [X: nat,B: nat,W: nat] :
% 0.32/0.51 ( ( ( semiri1316708129612266289at_nat @ X )
% 0.32/0.51 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.32/0.51 = ( X
% 0.32/0.51 = ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % of_nat_power_eq_of_nat_cancel_iff
% 0.32/0.51 thf(fact_1483_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.51 ! [X: nat,B: nat,W: nat] :
% 0.32/0.51 ( ( ( semiri1314217659103216013at_int @ X )
% 0.32/0.51 = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.32/0.51 = ( X
% 0.32/0.51 = ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % of_nat_power_eq_of_nat_cancel_iff
% 0.32/0.51 thf(fact_1484_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.32/0.51 ! [B: nat,W: nat,X: nat] :
% 0.32/0.51 ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
% 0.32/0.51 = ( semiri8010041392384452111omplex @ X ) )
% 0.32/0.51 = ( ( power_power_nat @ B @ W )
% 0.32/0.51 = X ) ) ).
% 0.32/0.51
% 0.32/0.51 % of_nat_eq_of_nat_power_cancel_iff
% 0.32/0.51 thf(fact_1485_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.32/0.51 ! [B: nat,W: nat,X: nat] :
% 0.32/0.51 ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
% 0.32/0.51 = ( semiri5074537144036343181t_real @ X ) )
% 0.32/0.51 = ( ( power_power_nat @ B @ W )
% 0.32/0.51 = X ) ) ).
% 0.32/0.51
% 0.32/0.51 % of_nat_eq_of_nat_power_cancel_iff
% 0.32/0.51 thf(fact_1486_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.32/0.51 ! [B: nat,W: nat,X: nat] :
% 0.32/0.51 ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W )
% 0.32/0.51 = ( semiri681578069525770553at_rat @ X ) )
% 0.32/0.51 = ( ( power_power_nat @ B @ W )
% 0.32/0.51 = X ) ) ).
% 0.32/0.51
% 0.32/0.51 % of_nat_eq_of_nat_power_cancel_iff
% 0.32/0.51 thf(fact_1487_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.32/0.51 ! [B: nat,W: nat,X: nat] :
% 0.32/0.51 ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
% 0.32/0.51 = ( semiri1316708129612266289at_nat @ X ) )
% 0.32/0.51 = ( ( power_power_nat @ B @ W )
% 0.32/0.51 = X ) ) ).
% 0.32/0.51
% 0.32/0.51 % of_nat_eq_of_nat_power_cancel_iff
% 0.32/0.51 thf(fact_1488_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.32/0.51 ! [B: nat,W: nat,X: nat] :
% 0.32/0.51 ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
% 0.32/0.51 = ( semiri1314217659103216013at_int @ X ) )
% 0.32/0.51 = ( ( power_power_nat @ B @ W )
% 0.32/0.51 = X ) ) ).
% 0.32/0.51
% 0.32/0.51 % of_nat_eq_of_nat_power_cancel_iff
% 0.32/0.51 thf(fact_1489_of__nat__power,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
% 0.32/0.51 = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % of_nat_power
% 0.32/0.51 thf(fact_1490_of__nat__power,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
% 0.32/0.51 = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % of_nat_power
% 0.32/0.51 thf(fact_1491_of__nat__power,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
% 0.32/0.51 = ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % of_nat_power
% 0.32/0.51 thf(fact_1492_of__nat__power,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
% 0.32/0.51 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % of_nat_power
% 0.32/0.51 thf(fact_1493_of__nat__power,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
% 0.32/0.51 = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % of_nat_power
% 0.32/0.51 thf(fact_1494_power__not__zero,axiom,
% 0.32/0.51 ! [A: rat,N: nat] :
% 0.32/0.51 ( ( A != zero_zero_rat )
% 0.32/0.51 => ( ( power_power_rat @ A @ N )
% 0.32/0.51 != zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_not_zero
% 0.32/0.51 thf(fact_1495_power__not__zero,axiom,
% 0.32/0.51 ! [A: nat,N: nat] :
% 0.32/0.51 ( ( A != zero_zero_nat )
% 0.32/0.51 => ( ( power_power_nat @ A @ N )
% 0.32/0.51 != zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_not_zero
% 0.32/0.51 thf(fact_1496_power__not__zero,axiom,
% 0.32/0.51 ! [A: real,N: nat] :
% 0.32/0.51 ( ( A != zero_zero_real )
% 0.32/0.51 => ( ( power_power_real @ A @ N )
% 0.32/0.51 != zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_not_zero
% 0.32/0.51 thf(fact_1497_power__not__zero,axiom,
% 0.32/0.51 ! [A: int,N: nat] :
% 0.32/0.51 ( ( A != zero_zero_int )
% 0.32/0.51 => ( ( power_power_int @ A @ N )
% 0.32/0.51 != zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_not_zero
% 0.32/0.51 thf(fact_1498_power__not__zero,axiom,
% 0.32/0.51 ! [A: complex,N: nat] :
% 0.32/0.51 ( ( A != zero_zero_complex )
% 0.32/0.51 => ( ( power_power_complex @ A @ N )
% 0.32/0.51 != zero_zero_complex ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_not_zero
% 0.32/0.51 thf(fact_1499_power__commuting__commutes,axiom,
% 0.32/0.51 ! [X: complex,Y: complex,N: nat] :
% 0.32/0.51 ( ( ( times_times_complex @ X @ Y )
% 0.32/0.51 = ( times_times_complex @ Y @ X ) )
% 0.32/0.51 => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y )
% 0.32/0.51 = ( times_times_complex @ Y @ ( power_power_complex @ X @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_commuting_commutes
% 0.32/0.51 thf(fact_1500_power__commuting__commutes,axiom,
% 0.32/0.51 ! [X: real,Y: real,N: nat] :
% 0.32/0.51 ( ( ( times_times_real @ X @ Y )
% 0.32/0.51 = ( times_times_real @ Y @ X ) )
% 0.32/0.51 => ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y )
% 0.32/0.51 = ( times_times_real @ Y @ ( power_power_real @ X @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_commuting_commutes
% 0.32/0.51 thf(fact_1501_power__commuting__commutes,axiom,
% 0.32/0.51 ! [X: rat,Y: rat,N: nat] :
% 0.32/0.51 ( ( ( times_times_rat @ X @ Y )
% 0.32/0.51 = ( times_times_rat @ Y @ X ) )
% 0.32/0.51 => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y )
% 0.32/0.51 = ( times_times_rat @ Y @ ( power_power_rat @ X @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_commuting_commutes
% 0.32/0.51 thf(fact_1502_power__commuting__commutes,axiom,
% 0.32/0.51 ! [X: nat,Y: nat,N: nat] :
% 0.32/0.51 ( ( ( times_times_nat @ X @ Y )
% 0.32/0.51 = ( times_times_nat @ Y @ X ) )
% 0.32/0.51 => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
% 0.32/0.51 = ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_commuting_commutes
% 0.32/0.51 thf(fact_1503_power__commuting__commutes,axiom,
% 0.32/0.51 ! [X: int,Y: int,N: nat] :
% 0.32/0.51 ( ( ( times_times_int @ X @ Y )
% 0.32/0.51 = ( times_times_int @ Y @ X ) )
% 0.32/0.51 => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
% 0.32/0.51 = ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_commuting_commutes
% 0.32/0.51 thf(fact_1504_power__mult__distrib,axiom,
% 0.32/0.51 ! [A: complex,B: complex,N: nat] :
% 0.32/0.51 ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
% 0.32/0.51 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mult_distrib
% 0.32/0.51 thf(fact_1505_power__mult__distrib,axiom,
% 0.32/0.51 ! [A: real,B: real,N: nat] :
% 0.32/0.51 ( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
% 0.32/0.51 = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mult_distrib
% 0.32/0.51 thf(fact_1506_power__mult__distrib,axiom,
% 0.32/0.51 ! [A: rat,B: rat,N: nat] :
% 0.32/0.51 ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
% 0.32/0.51 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mult_distrib
% 0.32/0.51 thf(fact_1507_power__mult__distrib,axiom,
% 0.32/0.51 ! [A: nat,B: nat,N: nat] :
% 0.32/0.51 ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
% 0.32/0.51 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mult_distrib
% 0.32/0.51 thf(fact_1508_power__mult__distrib,axiom,
% 0.32/0.51 ! [A: int,B: int,N: nat] :
% 0.32/0.51 ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
% 0.32/0.51 = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mult_distrib
% 0.32/0.51 thf(fact_1509_power__commutes,axiom,
% 0.32/0.51 ! [A: complex,N: nat] :
% 0.32/0.51 ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
% 0.32/0.51 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_commutes
% 0.32/0.51 thf(fact_1510_power__commutes,axiom,
% 0.32/0.51 ! [A: real,N: nat] :
% 0.32/0.51 ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
% 0.32/0.51 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_commutes
% 0.32/0.51 thf(fact_1511_power__commutes,axiom,
% 0.32/0.51 ! [A: rat,N: nat] :
% 0.32/0.51 ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
% 0.32/0.51 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_commutes
% 0.32/0.51 thf(fact_1512_power__commutes,axiom,
% 0.32/0.51 ! [A: nat,N: nat] :
% 0.32/0.51 ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
% 0.32/0.51 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_commutes
% 0.32/0.51 thf(fact_1513_power__commutes,axiom,
% 0.32/0.51 ! [A: int,N: nat] :
% 0.32/0.51 ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
% 0.32/0.51 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_commutes
% 0.32/0.51 thf(fact_1514_power__add,axiom,
% 0.32/0.51 ! [A: complex,M: nat,N: nat] :
% 0.32/0.51 ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.51 = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_add
% 0.32/0.51 thf(fact_1515_power__add,axiom,
% 0.32/0.51 ! [A: real,M: nat,N: nat] :
% 0.32/0.51 ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.51 = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_add
% 0.32/0.51 thf(fact_1516_power__add,axiom,
% 0.32/0.51 ! [A: rat,M: nat,N: nat] :
% 0.32/0.51 ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.51 = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_add
% 0.32/0.51 thf(fact_1517_power__add,axiom,
% 0.32/0.51 ! [A: nat,M: nat,N: nat] :
% 0.32/0.51 ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.51 = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_add
% 0.32/0.51 thf(fact_1518_power__add,axiom,
% 0.32/0.51 ! [A: int,M: nat,N: nat] :
% 0.32/0.51 ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.51 = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_add
% 0.32/0.51 thf(fact_1519_power__divide,axiom,
% 0.32/0.51 ! [A: complex,B: complex,N: nat] :
% 0.32/0.51 ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
% 0.32/0.51 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_divide
% 0.32/0.51 thf(fact_1520_power__divide,axiom,
% 0.32/0.51 ! [A: real,B: real,N: nat] :
% 0.32/0.51 ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
% 0.32/0.51 = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_divide
% 0.32/0.51 thf(fact_1521_power__divide,axiom,
% 0.32/0.51 ! [A: rat,B: rat,N: nat] :
% 0.32/0.51 ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
% 0.32/0.51 = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_divide
% 0.32/0.51 thf(fact_1522_dvd__power__same,axiom,
% 0.32/0.51 ! [X: code_integer,Y: code_integer,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_Code_integer @ X @ Y )
% 0.32/0.51 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_power_same
% 0.32/0.51 thf(fact_1523_dvd__power__same,axiom,
% 0.32/0.51 ! [X: nat,Y: nat,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_nat @ X @ Y )
% 0.32/0.51 => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_power_same
% 0.32/0.51 thf(fact_1524_dvd__power__same,axiom,
% 0.32/0.51 ! [X: real,Y: real,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_real @ X @ Y )
% 0.32/0.51 => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_power_same
% 0.32/0.51 thf(fact_1525_dvd__power__same,axiom,
% 0.32/0.51 ! [X: int,Y: int,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_int @ X @ Y )
% 0.32/0.51 => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_power_same
% 0.32/0.51 thf(fact_1526_dvd__power__same,axiom,
% 0.32/0.51 ! [X: complex,Y: complex,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_complex @ X @ Y )
% 0.32/0.51 => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dvd_power_same
% 0.32/0.51 thf(fact_1527_power__mult,axiom,
% 0.32/0.51 ! [A: nat,M: nat,N: nat] :
% 0.32/0.51 ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
% 0.32/0.51 = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mult
% 0.32/0.51 thf(fact_1528_power__mult,axiom,
% 0.32/0.51 ! [A: real,M: nat,N: nat] :
% 0.32/0.51 ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
% 0.32/0.51 = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mult
% 0.32/0.51 thf(fact_1529_power__mult,axiom,
% 0.32/0.51 ! [A: int,M: nat,N: nat] :
% 0.32/0.51 ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
% 0.32/0.51 = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mult
% 0.32/0.51 thf(fact_1530_power__mult,axiom,
% 0.32/0.51 ! [A: complex,M: nat,N: nat] :
% 0.32/0.51 ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
% 0.32/0.51 = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mult
% 0.32/0.51 thf(fact_1531_left__right__inverse__power,axiom,
% 0.32/0.51 ! [X: complex,Y: complex,N: nat] :
% 0.32/0.51 ( ( ( times_times_complex @ X @ Y )
% 0.32/0.51 = one_one_complex )
% 0.32/0.51 => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
% 0.32/0.51 = one_one_complex ) ) ).
% 0.32/0.51
% 0.32/0.51 % left_right_inverse_power
% 0.32/0.51 thf(fact_1532_left__right__inverse__power,axiom,
% 0.32/0.51 ! [X: real,Y: real,N: nat] :
% 0.32/0.51 ( ( ( times_times_real @ X @ Y )
% 0.32/0.51 = one_one_real )
% 0.32/0.51 => ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
% 0.32/0.51 = one_one_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % left_right_inverse_power
% 0.32/0.51 thf(fact_1533_left__right__inverse__power,axiom,
% 0.32/0.51 ! [X: rat,Y: rat,N: nat] :
% 0.32/0.51 ( ( ( times_times_rat @ X @ Y )
% 0.32/0.51 = one_one_rat )
% 0.32/0.51 => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
% 0.32/0.51 = one_one_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % left_right_inverse_power
% 0.32/0.51 thf(fact_1534_left__right__inverse__power,axiom,
% 0.32/0.51 ! [X: nat,Y: nat,N: nat] :
% 0.32/0.51 ( ( ( times_times_nat @ X @ Y )
% 0.32/0.51 = one_one_nat )
% 0.32/0.51 => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
% 0.32/0.51 = one_one_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % left_right_inverse_power
% 0.32/0.51 thf(fact_1535_left__right__inverse__power,axiom,
% 0.32/0.51 ! [X: int,Y: int,N: nat] :
% 0.32/0.51 ( ( ( times_times_int @ X @ Y )
% 0.32/0.51 = one_one_int )
% 0.32/0.51 => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
% 0.32/0.51 = one_one_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % left_right_inverse_power
% 0.32/0.51 thf(fact_1536_power__one__over,axiom,
% 0.32/0.51 ! [A: complex,N: nat] :
% 0.32/0.51 ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
% 0.32/0.51 = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_one_over
% 0.32/0.51 thf(fact_1537_power__one__over,axiom,
% 0.32/0.51 ! [A: real,N: nat] :
% 0.32/0.51 ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
% 0.32/0.51 = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_one_over
% 0.32/0.51 thf(fact_1538_power__one__over,axiom,
% 0.32/0.51 ! [A: rat,N: nat] :
% 0.32/0.51 ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
% 0.32/0.51 = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_one_over
% 0.32/0.51 thf(fact_1539_power__Suc2,axiom,
% 0.32/0.51 ! [A: complex,N: nat] :
% 0.32/0.51 ( ( power_power_complex @ A @ ( suc @ N ) )
% 0.32/0.51 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_Suc2
% 0.32/0.51 thf(fact_1540_power__Suc2,axiom,
% 0.32/0.51 ! [A: real,N: nat] :
% 0.32/0.51 ( ( power_power_real @ A @ ( suc @ N ) )
% 0.32/0.51 = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_Suc2
% 0.32/0.51 thf(fact_1541_power__Suc2,axiom,
% 0.32/0.51 ! [A: rat,N: nat] :
% 0.32/0.51 ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.32/0.51 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_Suc2
% 0.32/0.51 thf(fact_1542_power__Suc2,axiom,
% 0.32/0.51 ! [A: nat,N: nat] :
% 0.32/0.51 ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.32/0.51 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_Suc2
% 0.32/0.51 thf(fact_1543_power__Suc2,axiom,
% 0.32/0.51 ! [A: int,N: nat] :
% 0.32/0.51 ( ( power_power_int @ A @ ( suc @ N ) )
% 0.32/0.51 = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_Suc2
% 0.32/0.51 thf(fact_1544_power__Suc,axiom,
% 0.32/0.51 ! [A: complex,N: nat] :
% 0.32/0.51 ( ( power_power_complex @ A @ ( suc @ N ) )
% 0.32/0.51 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_Suc
% 0.32/0.51 thf(fact_1545_power__Suc,axiom,
% 0.32/0.51 ! [A: real,N: nat] :
% 0.32/0.51 ( ( power_power_real @ A @ ( suc @ N ) )
% 0.32/0.51 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_Suc
% 0.32/0.51 thf(fact_1546_power__Suc,axiom,
% 0.32/0.51 ! [A: rat,N: nat] :
% 0.32/0.51 ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.32/0.51 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_Suc
% 0.32/0.51 thf(fact_1547_power__Suc,axiom,
% 0.32/0.51 ! [A: nat,N: nat] :
% 0.32/0.51 ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.32/0.51 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_Suc
% 0.32/0.51 thf(fact_1548_power__Suc,axiom,
% 0.32/0.51 ! [A: int,N: nat] :
% 0.32/0.51 ( ( power_power_int @ A @ ( suc @ N ) )
% 0.32/0.51 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_Suc
% 0.32/0.51 thf(fact_1549_power__0,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( power_power_rat @ A @ zero_zero_nat )
% 0.32/0.51 = one_one_rat ) ).
% 0.32/0.51
% 0.32/0.51 % power_0
% 0.32/0.51 thf(fact_1550_power__0,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( power_power_nat @ A @ zero_zero_nat )
% 0.32/0.51 = one_one_nat ) ).
% 0.32/0.51
% 0.32/0.51 % power_0
% 0.32/0.51 thf(fact_1551_power__0,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( power_power_real @ A @ zero_zero_nat )
% 0.32/0.51 = one_one_real ) ).
% 0.32/0.51
% 0.32/0.51 % power_0
% 0.32/0.51 thf(fact_1552_power__0,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( power_power_int @ A @ zero_zero_nat )
% 0.32/0.51 = one_one_int ) ).
% 0.32/0.51
% 0.32/0.51 % power_0
% 0.32/0.51 thf(fact_1553_power__0,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( power_power_complex @ A @ zero_zero_nat )
% 0.32/0.51 = one_one_complex ) ).
% 0.32/0.51
% 0.32/0.51 % power_0
% 0.32/0.51 thf(fact_1554_div__power,axiom,
% 0.32/0.51 ! [B: code_integer,A: code_integer,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.51 => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N )
% 0.32/0.51 = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_power
% 0.32/0.51 thf(fact_1555_div__power,axiom,
% 0.32/0.51 ! [B: nat,A: nat,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.51 => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
% 0.32/0.51 = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_power
% 0.32/0.51 thf(fact_1556_div__power,axiom,
% 0.32/0.51 ! [B: int,A: int,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_int @ B @ A )
% 0.32/0.51 => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
% 0.32/0.51 = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_power
% 0.32/0.51 thf(fact_1557_power__0__left,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ( N = zero_zero_nat )
% 0.32/0.51 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.32/0.51 = one_one_rat ) )
% 0.32/0.51 & ( ( N != zero_zero_nat )
% 0.32/0.51 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.32/0.51 = zero_zero_rat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_0_left
% 0.32/0.51 thf(fact_1558_power__0__left,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ( N = zero_zero_nat )
% 0.32/0.51 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.32/0.51 = one_one_nat ) )
% 0.32/0.51 & ( ( N != zero_zero_nat )
% 0.32/0.51 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.32/0.51 = zero_zero_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_0_left
% 0.32/0.51 thf(fact_1559_power__0__left,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ( N = zero_zero_nat )
% 0.32/0.51 => ( ( power_power_real @ zero_zero_real @ N )
% 0.32/0.51 = one_one_real ) )
% 0.32/0.51 & ( ( N != zero_zero_nat )
% 0.32/0.51 => ( ( power_power_real @ zero_zero_real @ N )
% 0.32/0.51 = zero_zero_real ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_0_left
% 0.32/0.51 thf(fact_1560_power__0__left,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ( N = zero_zero_nat )
% 0.32/0.51 => ( ( power_power_int @ zero_zero_int @ N )
% 0.32/0.51 = one_one_int ) )
% 0.32/0.51 & ( ( N != zero_zero_nat )
% 0.32/0.51 => ( ( power_power_int @ zero_zero_int @ N )
% 0.32/0.51 = zero_zero_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_0_left
% 0.32/0.51 thf(fact_1561_power__0__left,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ( N = zero_zero_nat )
% 0.32/0.51 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.32/0.51 = one_one_complex ) )
% 0.32/0.51 & ( ( N != zero_zero_nat )
% 0.32/0.51 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.32/0.51 = zero_zero_complex ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_0_left
% 0.32/0.51 thf(fact_1562_is__unit__power__iff,axiom,
% 0.32/0.51 ! [A: code_integer,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
% 0.32/0.51 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.51 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % is_unit_power_iff
% 0.32/0.51 thf(fact_1563_is__unit__power__iff,axiom,
% 0.32/0.51 ! [A: nat,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
% 0.32/0.51 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.51 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % is_unit_power_iff
% 0.32/0.51 thf(fact_1564_is__unit__power__iff,axiom,
% 0.32/0.51 ! [A: int,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
% 0.32/0.51 = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.51 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % is_unit_power_iff
% 0.32/0.51 thf(fact_1565_zero__power2,axiom,
% 0.32/0.51 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_rat ) ).
% 0.32/0.51
% 0.32/0.51 % zero_power2
% 0.32/0.51 thf(fact_1566_zero__power2,axiom,
% 0.32/0.51 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % zero_power2
% 0.32/0.51 thf(fact_1567_zero__power2,axiom,
% 0.32/0.51 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_real ) ).
% 0.32/0.51
% 0.32/0.51 % zero_power2
% 0.32/0.51 thf(fact_1568_zero__power2,axiom,
% 0.32/0.51 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % zero_power2
% 0.32/0.51 thf(fact_1569_zero__power2,axiom,
% 0.32/0.51 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_complex ) ).
% 0.32/0.51
% 0.32/0.51 % zero_power2
% 0.32/0.51 thf(fact_1570_power4__eq__xxxx,axiom,
% 0.32/0.51 ! [X: complex] :
% 0.32/0.51 ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).
% 0.32/0.51
% 0.32/0.51 % power4_eq_xxxx
% 0.32/0.51 thf(fact_1571_power4__eq__xxxx,axiom,
% 0.32/0.51 ! [X: real] :
% 0.32/0.51 ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% 0.32/0.51
% 0.32/0.51 % power4_eq_xxxx
% 0.32/0.51 thf(fact_1572_power4__eq__xxxx,axiom,
% 0.32/0.51 ! [X: rat] :
% 0.32/0.51 ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).
% 0.32/0.51
% 0.32/0.51 % power4_eq_xxxx
% 0.32/0.51 thf(fact_1573_power4__eq__xxxx,axiom,
% 0.32/0.51 ! [X: nat] :
% 0.32/0.51 ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% 0.32/0.51
% 0.32/0.51 % power4_eq_xxxx
% 0.32/0.51 thf(fact_1574_power4__eq__xxxx,axiom,
% 0.32/0.51 ! [X: int] :
% 0.32/0.51 ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% 0.32/0.51
% 0.32/0.51 % power4_eq_xxxx
% 0.32/0.51 thf(fact_1575_power2__eq__square,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( times_times_complex @ A @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_eq_square
% 0.32/0.51 thf(fact_1576_power2__eq__square,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( times_times_real @ A @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_eq_square
% 0.32/0.51 thf(fact_1577_power2__eq__square,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( times_times_rat @ A @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_eq_square
% 0.32/0.51 thf(fact_1578_power2__eq__square,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( times_times_nat @ A @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_eq_square
% 0.32/0.51 thf(fact_1579_power2__eq__square,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( times_times_int @ A @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_eq_square
% 0.32/0.51 thf(fact_1580_one__power2,axiom,
% 0.32/0.51 ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = one_one_rat ) ).
% 0.32/0.51
% 0.32/0.51 % one_power2
% 0.32/0.51 thf(fact_1581_one__power2,axiom,
% 0.32/0.51 ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = one_one_nat ) ).
% 0.32/0.51
% 0.32/0.51 % one_power2
% 0.32/0.51 thf(fact_1582_one__power2,axiom,
% 0.32/0.51 ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = one_one_real ) ).
% 0.32/0.51
% 0.32/0.51 % one_power2
% 0.32/0.51 thf(fact_1583_one__power2,axiom,
% 0.32/0.51 ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = one_one_int ) ).
% 0.32/0.51
% 0.32/0.51 % one_power2
% 0.32/0.51 thf(fact_1584_one__power2,axiom,
% 0.32/0.51 ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = one_one_complex ) ).
% 0.32/0.51
% 0.32/0.51 % one_power2
% 0.32/0.51 thf(fact_1585_power2__commute,axiom,
% 0.32/0.51 ! [X: complex,Y: complex] :
% 0.32/0.51 ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( power_power_complex @ ( minus_minus_complex @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_commute
% 0.32/0.51 thf(fact_1586_power2__commute,axiom,
% 0.32/0.51 ! [X: real,Y: real] :
% 0.32/0.51 ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( power_power_real @ ( minus_minus_real @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_commute
% 0.32/0.51 thf(fact_1587_power2__commute,axiom,
% 0.32/0.51 ! [X: rat,Y: rat] :
% 0.32/0.51 ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( power_power_rat @ ( minus_minus_rat @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_commute
% 0.32/0.51 thf(fact_1588_power2__commute,axiom,
% 0.32/0.51 ! [X: int,Y: int] :
% 0.32/0.51 ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( power_power_int @ ( minus_minus_int @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_commute
% 0.32/0.51 thf(fact_1589_power__even__eq,axiom,
% 0.32/0.51 ! [A: nat,N: nat] :
% 0.32/0.51 ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.51 = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_even_eq
% 0.32/0.51 thf(fact_1590_power__even__eq,axiom,
% 0.32/0.51 ! [A: real,N: nat] :
% 0.32/0.51 ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.51 = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_even_eq
% 0.32/0.51 thf(fact_1591_power__even__eq,axiom,
% 0.32/0.51 ! [A: int,N: nat] :
% 0.32/0.51 ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.51 = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_even_eq
% 0.32/0.51 thf(fact_1592_power__even__eq,axiom,
% 0.32/0.51 ! [A: complex,N: nat] :
% 0.32/0.51 ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.51 = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_even_eq
% 0.32/0.51 thf(fact_1593_power__eq__if,axiom,
% 0.32/0.51 ( power_power_complex
% 0.32/0.51 = ( ^ [P3: complex,M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P3 @ ( power_power_complex @ P3 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_eq_if
% 0.32/0.51 thf(fact_1594_power__eq__if,axiom,
% 0.32/0.51 ( power_power_real
% 0.32/0.51 = ( ^ [P3: real,M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P3 @ ( power_power_real @ P3 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_eq_if
% 0.32/0.51 thf(fact_1595_power__eq__if,axiom,
% 0.32/0.51 ( power_power_rat
% 0.32/0.51 = ( ^ [P3: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P3 @ ( power_power_rat @ P3 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_eq_if
% 0.32/0.51 thf(fact_1596_power__eq__if,axiom,
% 0.32/0.51 ( power_power_nat
% 0.32/0.51 = ( ^ [P3: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P3 @ ( power_power_nat @ P3 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_eq_if
% 0.32/0.51 thf(fact_1597_power__eq__if,axiom,
% 0.32/0.51 ( power_power_int
% 0.32/0.51 = ( ^ [P3: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P3 @ ( power_power_int @ P3 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_eq_if
% 0.32/0.51 thf(fact_1598_power2__sum,axiom,
% 0.32/0.51 ! [X: complex,Y: complex] :
% 0.32/0.51 ( ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_sum
% 0.32/0.51 thf(fact_1599_power2__sum,axiom,
% 0.32/0.51 ! [X: real,Y: real] :
% 0.32/0.51 ( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_sum
% 0.32/0.51 thf(fact_1600_power2__sum,axiom,
% 0.32/0.51 ! [X: rat,Y: rat] :
% 0.32/0.51 ( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_sum
% 0.32/0.51 thf(fact_1601_power2__sum,axiom,
% 0.32/0.51 ! [X: nat,Y: nat] :
% 0.32/0.51 ( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_sum
% 0.32/0.51 thf(fact_1602_power2__sum,axiom,
% 0.32/0.51 ! [X: int,Y: int] :
% 0.32/0.51 ( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_sum
% 0.32/0.51 thf(fact_1603_power__odd__eq,axiom,
% 0.32/0.51 ! [A: complex,N: nat] :
% 0.32/0.51 ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.51 = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_odd_eq
% 0.32/0.51 thf(fact_1604_power__odd__eq,axiom,
% 0.32/0.51 ! [A: real,N: nat] :
% 0.32/0.51 ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.51 = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_odd_eq
% 0.32/0.51 thf(fact_1605_power__odd__eq,axiom,
% 0.32/0.51 ! [A: rat,N: nat] :
% 0.32/0.51 ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.51 = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_odd_eq
% 0.32/0.51 thf(fact_1606_power__odd__eq,axiom,
% 0.32/0.51 ! [A: nat,N: nat] :
% 0.32/0.51 ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.51 = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_odd_eq
% 0.32/0.51 thf(fact_1607_power__odd__eq,axiom,
% 0.32/0.51 ! [A: int,N: nat] :
% 0.32/0.51 ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.51 = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_odd_eq
% 0.32/0.51 thf(fact_1608_high__def,axiom,
% 0.32/0.51 ( vEBT_VEBT_high
% 0.32/0.51 = ( ^ [X3: nat,N3: nat] : ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % high_def
% 0.32/0.51 thf(fact_1609_even__succ__div__exp,axiom,
% 0.32/0.51 ! [A: code_integer,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.51 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.51 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.51 = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % even_succ_div_exp
% 0.32/0.51 thf(fact_1610_even__succ__div__exp,axiom,
% 0.32/0.51 ! [A: nat,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.51 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.51 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.51 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % even_succ_div_exp
% 0.32/0.51 thf(fact_1611_even__succ__div__exp,axiom,
% 0.32/0.51 ! [A: int,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.51 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.51 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.51 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % even_succ_div_exp
% 0.32/0.51 thf(fact_1612_flip__bit__0,axiom,
% 0.32/0.51 ! [A: code_integer] :
% 0.32/0.51 ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 0.32/0.51 = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % flip_bit_0
% 0.32/0.51 thf(fact_1613_flip__bit__0,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 0.32/0.51 = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % flip_bit_0
% 0.32/0.51 thf(fact_1614_flip__bit__0,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 0.32/0.51 = ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % flip_bit_0
% 0.32/0.51 thf(fact_1615_add__scale__eq__noteq,axiom,
% 0.32/0.51 ! [R: real,A: real,B: real,C: real,D: real] :
% 0.32/0.51 ( ( R != zero_zero_real )
% 0.32/0.51 => ( ( ( A = B )
% 0.32/0.51 & ( C != D ) )
% 0.32/0.51 => ( ( plus_plus_real @ A @ ( times_times_real @ R @ C ) )
% 0.32/0.51 != ( plus_plus_real @ B @ ( times_times_real @ R @ D ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_scale_eq_noteq
% 0.32/0.51 thf(fact_1616_add__scale__eq__noteq,axiom,
% 0.32/0.51 ! [R: rat,A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.51 ( ( R != zero_zero_rat )
% 0.32/0.51 => ( ( ( A = B )
% 0.32/0.51 & ( C != D ) )
% 0.32/0.51 => ( ( plus_plus_rat @ A @ ( times_times_rat @ R @ C ) )
% 0.32/0.51 != ( plus_plus_rat @ B @ ( times_times_rat @ R @ D ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_scale_eq_noteq
% 0.32/0.51 thf(fact_1617_add__scale__eq__noteq,axiom,
% 0.32/0.51 ! [R: nat,A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.51 ( ( R != zero_zero_nat )
% 0.32/0.51 => ( ( ( A = B )
% 0.32/0.51 & ( C != D ) )
% 0.32/0.51 => ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
% 0.32/0.51 != ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_scale_eq_noteq
% 0.32/0.51 thf(fact_1618_add__scale__eq__noteq,axiom,
% 0.32/0.51 ! [R: int,A: int,B: int,C: int,D: int] :
% 0.32/0.51 ( ( R != zero_zero_int )
% 0.32/0.51 => ( ( ( A = B )
% 0.32/0.51 & ( C != D ) )
% 0.32/0.51 => ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
% 0.32/0.51 != ( plus_plus_int @ B @ ( times_times_int @ R @ D ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_scale_eq_noteq
% 0.32/0.51 thf(fact_1619_num_Osize__gen_I2_J,axiom,
% 0.32/0.51 ! [X2: num] :
% 0.32/0.51 ( ( size_num @ ( bit0 @ X2 ) )
% 0.32/0.51 = ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % num.size_gen(2)
% 0.32/0.51 thf(fact_1620_power__numeral,axiom,
% 0.32/0.51 ! [K: num,L: num] :
% 0.32/0.51 ( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.32/0.51 = ( numera6690914467698888265omplex @ ( pow @ K @ L ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_numeral
% 0.32/0.51 thf(fact_1621_power__numeral,axiom,
% 0.32/0.51 ! [K: num,L: num] :
% 0.32/0.51 ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.32/0.51 = ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_numeral
% 0.32/0.51 thf(fact_1622_power__numeral,axiom,
% 0.32/0.51 ! [K: num,L: num] :
% 0.32/0.51 ( ( power_power_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.32/0.51 = ( numeral_numeral_rat @ ( pow @ K @ L ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_numeral
% 0.32/0.51 thf(fact_1623_power__numeral,axiom,
% 0.32/0.51 ! [K: num,L: num] :
% 0.32/0.51 ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.32/0.51 = ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_numeral
% 0.32/0.51 thf(fact_1624_power__numeral,axiom,
% 0.32/0.51 ! [K: num,L: num] :
% 0.32/0.51 ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
% 0.32/0.51 = ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_numeral
% 0.32/0.51 thf(fact_1625_even__mask__div__iff,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.51 = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.32/0.51 = zero_z3403309356797280102nteger )
% 0.32/0.51 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % even_mask_div_iff
% 0.32/0.51 thf(fact_1626_even__mask__div__iff,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.51 = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.51 = zero_zero_nat )
% 0.32/0.51 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % even_mask_div_iff
% 0.32/0.51 thf(fact_1627_even__mask__div__iff,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.51 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.32/0.51 = zero_zero_int )
% 0.32/0.51 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % even_mask_div_iff
% 0.32/0.51 thf(fact_1628_arith__geo__mean,axiom,
% 0.32/0.51 ! [U: real,X: real,Y: real] :
% 0.32/0.51 ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( times_times_real @ X @ Y ) )
% 0.32/0.51 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.51 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.32/0.51 => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % arith_geo_mean
% 0.32/0.51 thf(fact_1629_arith__geo__mean,axiom,
% 0.32/0.51 ! [U: rat,X: rat,Y: rat] :
% 0.32/0.51 ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( times_times_rat @ X @ Y ) )
% 0.32/0.51 => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.51 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.32/0.51 => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % arith_geo_mean
% 0.32/0.51 thf(fact_1630_zero__less__power__eq__numeral,axiom,
% 0.32/0.51 ! [A: real,W: num] :
% 0.32/0.51 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.51 = ( ( ( numeral_numeral_nat @ W )
% 0.32/0.51 = zero_zero_nat )
% 0.32/0.51 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.51 & ( A != zero_zero_real ) )
% 0.32/0.51 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.51 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_less_power_eq_numeral
% 0.32/0.51 thf(fact_1631_zero__less__power__eq__numeral,axiom,
% 0.32/0.51 ! [A: rat,W: num] :
% 0.32/0.51 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.51 = ( ( ( numeral_numeral_nat @ W )
% 0.32/0.51 = zero_zero_nat )
% 0.32/0.51 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.51 & ( A != zero_zero_rat ) )
% 0.32/0.51 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.51 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_less_power_eq_numeral
% 0.32/0.51 thf(fact_1632_zero__less__power__eq__numeral,axiom,
% 0.32/0.51 ! [A: int,W: num] :
% 0.32/0.51 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.51 = ( ( ( numeral_numeral_nat @ W )
% 0.32/0.51 = zero_zero_nat )
% 0.32/0.51 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.51 & ( A != zero_zero_int ) )
% 0.32/0.51 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.51 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_less_power_eq_numeral
% 0.32/0.51 thf(fact_1633_Bernoulli__inequality__even,axiom,
% 0.32/0.51 ! [N: nat,X: real] :
% 0.32/0.51 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.51 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % Bernoulli_inequality_even
% 0.32/0.51 thf(fact_1634_semiring__norm_I78_J,axiom,
% 0.32/0.51 ! [M: num,N: num] :
% 0.32/0.51 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.32/0.51 = ( ord_less_num @ M @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % semiring_norm(78)
% 0.32/0.51 thf(fact_1635_semiring__norm_I71_J,axiom,
% 0.32/0.51 ! [M: num,N: num] :
% 0.32/0.51 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.32/0.51 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % semiring_norm(71)
% 0.32/0.51 thf(fact_1636_semiring__norm_I75_J,axiom,
% 0.32/0.51 ! [M: num] :
% 0.32/0.51 ~ ( ord_less_num @ M @ one ) ).
% 0.32/0.51
% 0.32/0.51 % semiring_norm(75)
% 0.32/0.51 thf(fact_1637_semiring__norm_I68_J,axiom,
% 0.32/0.51 ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 0.32/0.51
% 0.32/0.51 % semiring_norm(68)
% 0.32/0.51 thf(fact_1638_set__bit__negative__int__iff,axiom,
% 0.32/0.51 ! [N: nat,K: int] :
% 0.32/0.51 ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 0.32/0.51 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % set_bit_negative_int_iff
% 0.32/0.51 thf(fact_1639_flip__bit__negative__int__iff,axiom,
% 0.32/0.51 ! [N: nat,K: int] :
% 0.32/0.51 ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 0.32/0.51 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % flip_bit_negative_int_iff
% 0.32/0.51 thf(fact_1640_unset__bit__negative__int__iff,axiom,
% 0.32/0.51 ! [N: nat,K: int] :
% 0.32/0.51 ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 0.32/0.51 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % unset_bit_negative_int_iff
% 0.32/0.51 thf(fact_1641_set__bit__nonnegative__int__iff,axiom,
% 0.32/0.51 ! [N: nat,K: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
% 0.32/0.51 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.51
% 0.32/0.51 % set_bit_nonnegative_int_iff
% 0.32/0.51 thf(fact_1642_flip__bit__nonnegative__int__iff,axiom,
% 0.32/0.51 ! [N: nat,K: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
% 0.32/0.51 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.51
% 0.32/0.51 % flip_bit_nonnegative_int_iff
% 0.32/0.51 thf(fact_1643_unset__bit__nonnegative__int__iff,axiom,
% 0.32/0.51 ! [N: nat,K: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
% 0.32/0.51 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.51
% 0.32/0.51 % unset_bit_nonnegative_int_iff
% 0.32/0.51 thf(fact_1644_not__real__square__gt__zero,axiom,
% 0.32/0.51 ! [X: real] :
% 0.32/0.51 ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
% 0.32/0.51 = ( X = zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % not_real_square_gt_zero
% 0.32/0.51 thf(fact_1645_high__bound__aux,axiom,
% 0.32/0.51 ! [Ma: nat,N: nat,M: nat] :
% 0.32/0.51 ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.32/0.51 => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % high_bound_aux
% 0.32/0.51 thf(fact_1646_le__zero__eq,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.32/0.51 = ( N = zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % le_zero_eq
% 0.32/0.51 thf(fact_1647_not__gr__zero,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.32/0.51 = ( N = zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % not_gr_zero
% 0.32/0.51 thf(fact_1648_numeral__le__iff,axiom,
% 0.32/0.51 ! [M: num,N: num] :
% 0.32/0.51 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.32/0.51 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_le_iff
% 0.32/0.51 thf(fact_1649_numeral__le__iff,axiom,
% 0.32/0.51 ! [M: num,N: num] :
% 0.32/0.51 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.32/0.51 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_le_iff
% 0.32/0.51 thf(fact_1650_numeral__le__iff,axiom,
% 0.32/0.51 ! [M: num,N: num] :
% 0.32/0.51 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.51 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_le_iff
% 0.32/0.51 thf(fact_1651_numeral__le__iff,axiom,
% 0.32/0.51 ! [M: num,N: num] :
% 0.32/0.51 ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.51 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_le_iff
% 0.32/0.51 thf(fact_1652_numeral__less__iff,axiom,
% 0.32/0.51 ! [M: num,N: num] :
% 0.32/0.51 ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.32/0.51 = ( ord_less_num @ M @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_less_iff
% 0.32/0.51 thf(fact_1653_numeral__less__iff,axiom,
% 0.32/0.51 ! [M: num,N: num] :
% 0.32/0.51 ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.32/0.51 = ( ord_less_num @ M @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_less_iff
% 0.32/0.51 thf(fact_1654_numeral__less__iff,axiom,
% 0.32/0.51 ! [M: num,N: num] :
% 0.32/0.51 ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.51 = ( ord_less_num @ M @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_less_iff
% 0.32/0.51 thf(fact_1655_numeral__less__iff,axiom,
% 0.32/0.51 ! [M: num,N: num] :
% 0.32/0.51 ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.51 = ( ord_less_num @ M @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % numeral_less_iff
% 0.32/0.51 thf(fact_1656_add__le__cancel__right,axiom,
% 0.32/0.51 ! [A: real,C: real,B: real] :
% 0.32/0.51 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.32/0.51 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_le_cancel_right
% 0.32/0.51 thf(fact_1657_add__le__cancel__right,axiom,
% 0.32/0.51 ! [A: rat,C: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.51 = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_le_cancel_right
% 0.32/0.51 thf(fact_1658_add__le__cancel__right,axiom,
% 0.32/0.51 ! [A: nat,C: nat,B: nat] :
% 0.32/0.51 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.51 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_le_cancel_right
% 0.32/0.51 thf(fact_1659_add__le__cancel__right,axiom,
% 0.32/0.51 ! [A: int,C: int,B: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.32/0.51 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_le_cancel_right
% 0.32/0.51 thf(fact_1660_add__le__cancel__left,axiom,
% 0.32/0.51 ! [C: real,A: real,B: real] :
% 0.32/0.51 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.32/0.51 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_le_cancel_left
% 0.32/0.51 thf(fact_1661_add__le__cancel__left,axiom,
% 0.32/0.51 ! [C: rat,A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.32/0.51 = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_le_cancel_left
% 0.32/0.51 thf(fact_1662_add__le__cancel__left,axiom,
% 0.32/0.51 ! [C: nat,A: nat,B: nat] :
% 0.32/0.51 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.32/0.51 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_le_cancel_left
% 0.32/0.51 thf(fact_1663_add__le__cancel__left,axiom,
% 0.32/0.51 ! [C: int,A: int,B: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.32/0.51 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_le_cancel_left
% 0.32/0.51 thf(fact_1664_add__less__cancel__right,axiom,
% 0.32/0.51 ! [A: real,C: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.32/0.52 = ( ord_less_real @ A @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_cancel_right
% 0.32/0.52 thf(fact_1665_add__less__cancel__right,axiom,
% 0.32/0.52 ! [A: rat,C: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.52 = ( ord_less_rat @ A @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_cancel_right
% 0.32/0.52 thf(fact_1666_add__less__cancel__right,axiom,
% 0.32/0.52 ! [A: nat,C: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.52 = ( ord_less_nat @ A @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_cancel_right
% 0.32/0.52 thf(fact_1667_add__less__cancel__right,axiom,
% 0.32/0.52 ! [A: int,C: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.32/0.52 = ( ord_less_int @ A @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_cancel_right
% 0.32/0.52 thf(fact_1668_add__less__cancel__left,axiom,
% 0.32/0.52 ! [C: real,A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.32/0.52 = ( ord_less_real @ A @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_cancel_left
% 0.32/0.52 thf(fact_1669_add__less__cancel__left,axiom,
% 0.32/0.52 ! [C: rat,A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.32/0.52 = ( ord_less_rat @ A @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_cancel_left
% 0.32/0.52 thf(fact_1670_add__less__cancel__left,axiom,
% 0.32/0.52 ! [C: nat,A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.32/0.52 = ( ord_less_nat @ A @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_cancel_left
% 0.32/0.52 thf(fact_1671_add__less__cancel__left,axiom,
% 0.32/0.52 ! [C: int,A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.32/0.52 = ( ord_less_int @ A @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_cancel_left
% 0.32/0.52 thf(fact_1672_semiring__norm_I76_J,axiom,
% 0.32/0.52 ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % semiring_norm(76)
% 0.32/0.52 thf(fact_1673_semiring__norm_I69_J,axiom,
% 0.32/0.52 ! [M: num] :
% 0.32/0.52 ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 0.32/0.52
% 0.32/0.52 % semiring_norm(69)
% 0.32/0.52 thf(fact_1674_Suc__less__eq,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.32/0.52 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % Suc_less_eq
% 0.32/0.52 thf(fact_1675_Suc__mono,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ M @ N )
% 0.32/0.52 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % Suc_mono
% 0.32/0.52 thf(fact_1676_lessI,axiom,
% 0.32/0.52 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % lessI
% 0.32/0.52 thf(fact_1677_less__nat__zero__code,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.32/0.52
% 0.32/0.52 % less_nat_zero_code
% 0.32/0.52 thf(fact_1678_neq0__conv,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( N != zero_zero_nat )
% 0.32/0.52 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % neq0_conv
% 0.32/0.52 thf(fact_1679_bot__nat__0_Onot__eq__extremum,axiom,
% 0.32/0.52 ! [A: nat] :
% 0.32/0.52 ( ( A != zero_zero_nat )
% 0.32/0.52 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % bot_nat_0.not_eq_extremum
% 0.32/0.52 thf(fact_1680_Suc__le__mono,axiom,
% 0.32/0.52 ! [N: nat,M: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.32/0.52 = ( ord_less_eq_nat @ N @ M ) ) ).
% 0.32/0.52
% 0.32/0.52 % Suc_le_mono
% 0.32/0.52 thf(fact_1681_le0,axiom,
% 0.32/0.52 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.32/0.52
% 0.32/0.52 % le0
% 0.32/0.52 thf(fact_1682_bot__nat__0_Oextremum,axiom,
% 0.32/0.52 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.32/0.52
% 0.32/0.52 % bot_nat_0.extremum
% 0.32/0.52 thf(fact_1683_nat__add__left__cancel__less,axiom,
% 0.32/0.52 ! [K: nat,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.32/0.52 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % nat_add_left_cancel_less
% 0.32/0.52 thf(fact_1684_nat__add__left__cancel__le,axiom,
% 0.32/0.52 ! [K: nat,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.32/0.52 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % nat_add_left_cancel_le
% 0.32/0.52 thf(fact_1685_diff__diff__cancel,axiom,
% 0.32/0.52 ! [I: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ I @ N )
% 0.32/0.52 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 0.32/0.52 = I ) ) ).
% 0.32/0.52
% 0.32/0.52 % diff_diff_cancel
% 0.32/0.52 thf(fact_1686_of__bool__less__eq__iff,axiom,
% 0.32/0.52 ! [P: $o,Q: $o] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.32/0.52 = ( P
% 0.32/0.52 => Q ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_eq_iff
% 0.32/0.52 thf(fact_1687_of__bool__less__eq__iff,axiom,
% 0.32/0.52 ! [P: $o,Q: $o] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.32/0.52 = ( P
% 0.32/0.52 => Q ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_eq_iff
% 0.32/0.52 thf(fact_1688_of__bool__less__eq__iff,axiom,
% 0.32/0.52 ! [P: $o,Q: $o] :
% 0.32/0.52 ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.32/0.52 = ( P
% 0.32/0.52 => Q ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_eq_iff
% 0.32/0.52 thf(fact_1689_of__bool__less__eq__iff,axiom,
% 0.32/0.52 ! [P: $o,Q: $o] :
% 0.32/0.52 ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 0.32/0.52 = ( P
% 0.32/0.52 => Q ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_eq_iff
% 0.32/0.52 thf(fact_1690_of__bool__eq_I1_J,axiom,
% 0.32/0.52 ( ( zero_n3304061248610475627l_real @ $false )
% 0.32/0.52 = zero_zero_real ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq(1)
% 0.32/0.52 thf(fact_1691_of__bool__eq_I1_J,axiom,
% 0.32/0.52 ( ( zero_n2052037380579107095ol_rat @ $false )
% 0.32/0.52 = zero_zero_rat ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq(1)
% 0.32/0.52 thf(fact_1692_of__bool__eq_I1_J,axiom,
% 0.32/0.52 ( ( zero_n2687167440665602831ol_nat @ $false )
% 0.32/0.52 = zero_zero_nat ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq(1)
% 0.32/0.52 thf(fact_1693_of__bool__eq_I1_J,axiom,
% 0.32/0.52 ( ( zero_n2684676970156552555ol_int @ $false )
% 0.32/0.52 = zero_zero_int ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq(1)
% 0.32/0.52 thf(fact_1694_of__bool__eq_I1_J,axiom,
% 0.32/0.52 ( ( zero_n356916108424825756nteger @ $false )
% 0.32/0.52 = zero_z3403309356797280102nteger ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq(1)
% 0.32/0.52 thf(fact_1695_of__bool__eq__0__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ( zero_n3304061248610475627l_real @ P )
% 0.32/0.52 = zero_zero_real )
% 0.32/0.52 = ( ~ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq_0_iff
% 0.32/0.52 thf(fact_1696_of__bool__eq__0__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.32/0.52 = zero_zero_rat )
% 0.32/0.52 = ( ~ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq_0_iff
% 0.32/0.52 thf(fact_1697_of__bool__eq__0__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.32/0.52 = zero_zero_nat )
% 0.32/0.52 = ( ~ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq_0_iff
% 0.32/0.52 thf(fact_1698_of__bool__eq__0__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.32/0.52 = zero_zero_int )
% 0.32/0.52 = ( ~ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq_0_iff
% 0.32/0.52 thf(fact_1699_of__bool__eq__0__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ( zero_n356916108424825756nteger @ P )
% 0.32/0.52 = zero_z3403309356797280102nteger )
% 0.32/0.52 = ( ~ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq_0_iff
% 0.32/0.52 thf(fact_1700_of__bool__less__iff,axiom,
% 0.32/0.52 ! [P: $o,Q: $o] :
% 0.32/0.52 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 0.32/0.52 = ( ~ P
% 0.32/0.52 & Q ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_iff
% 0.32/0.52 thf(fact_1701_of__bool__less__iff,axiom,
% 0.32/0.52 ! [P: $o,Q: $o] :
% 0.32/0.52 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.32/0.52 = ( ~ P
% 0.32/0.52 & Q ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_iff
% 0.32/0.52 thf(fact_1702_of__bool__less__iff,axiom,
% 0.32/0.52 ! [P: $o,Q: $o] :
% 0.32/0.52 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.32/0.52 = ( ~ P
% 0.32/0.52 & Q ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_iff
% 0.32/0.52 thf(fact_1703_of__bool__less__iff,axiom,
% 0.32/0.52 ! [P: $o,Q: $o] :
% 0.32/0.52 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.32/0.52 = ( ~ P
% 0.32/0.52 & Q ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_iff
% 0.32/0.52 thf(fact_1704_of__bool__less__iff,axiom,
% 0.32/0.52 ! [P: $o,Q: $o] :
% 0.32/0.52 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 0.32/0.52 = ( ~ P
% 0.32/0.52 & Q ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_iff
% 0.32/0.52 thf(fact_1705_of__bool__eq__1__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ( zero_n1201886186963655149omplex @ P )
% 0.32/0.52 = one_one_complex )
% 0.32/0.52 = P ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq_1_iff
% 0.32/0.52 thf(fact_1706_of__bool__eq__1__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ( zero_n3304061248610475627l_real @ P )
% 0.32/0.52 = one_one_real )
% 0.32/0.52 = P ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq_1_iff
% 0.32/0.52 thf(fact_1707_of__bool__eq__1__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.32/0.52 = one_one_rat )
% 0.32/0.52 = P ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq_1_iff
% 0.32/0.52 thf(fact_1708_of__bool__eq__1__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.32/0.52 = one_one_nat )
% 0.32/0.52 = P ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq_1_iff
% 0.32/0.52 thf(fact_1709_of__bool__eq__1__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.32/0.52 = one_one_int )
% 0.32/0.52 = P ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq_1_iff
% 0.32/0.52 thf(fact_1710_of__bool__eq__1__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ( zero_n356916108424825756nteger @ P )
% 0.32/0.52 = one_one_Code_integer )
% 0.32/0.52 = P ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq_1_iff
% 0.32/0.52 thf(fact_1711_of__bool__eq_I2_J,axiom,
% 0.32/0.52 ( ( zero_n1201886186963655149omplex @ $true )
% 0.32/0.52 = one_one_complex ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq(2)
% 0.32/0.52 thf(fact_1712_of__bool__eq_I2_J,axiom,
% 0.32/0.52 ( ( zero_n3304061248610475627l_real @ $true )
% 0.32/0.52 = one_one_real ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq(2)
% 0.32/0.52 thf(fact_1713_of__bool__eq_I2_J,axiom,
% 0.32/0.52 ( ( zero_n2052037380579107095ol_rat @ $true )
% 0.32/0.52 = one_one_rat ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq(2)
% 0.32/0.52 thf(fact_1714_of__bool__eq_I2_J,axiom,
% 0.32/0.52 ( ( zero_n2687167440665602831ol_nat @ $true )
% 0.32/0.52 = one_one_nat ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq(2)
% 0.32/0.52 thf(fact_1715_of__bool__eq_I2_J,axiom,
% 0.32/0.52 ( ( zero_n2684676970156552555ol_int @ $true )
% 0.32/0.52 = one_one_int ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq(2)
% 0.32/0.52 thf(fact_1716_of__bool__eq_I2_J,axiom,
% 0.32/0.52 ( ( zero_n356916108424825756nteger @ $true )
% 0.32/0.52 = one_one_Code_integer ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq(2)
% 0.32/0.52 thf(fact_1717_zle__add1__eq__le,axiom,
% 0.32/0.52 ! [W: int,Z2: int] :
% 0.32/0.52 ( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
% 0.32/0.52 = ( ord_less_eq_int @ W @ Z2 ) ) ).
% 0.32/0.52
% 0.32/0.52 % zle_add1_eq_le
% 0.32/0.52 thf(fact_1718_div__neg__neg__trivial,axiom,
% 0.32/0.52 ! [K: int,L: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_int @ L @ K )
% 0.32/0.52 => ( ( divide_divide_int @ K @ L )
% 0.32/0.52 = zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % div_neg_neg_trivial
% 0.32/0.52 thf(fact_1719_div__pos__pos__trivial,axiom,
% 0.32/0.52 ! [K: int,L: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.52 => ( ( ord_less_int @ K @ L )
% 0.32/0.52 => ( ( divide_divide_int @ K @ L )
% 0.32/0.52 = zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % div_pos_pos_trivial
% 0.32/0.52 thf(fact_1720_of__nat__of__bool,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.32/0.52 = ( zero_n1201886186963655149omplex @ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_of_bool
% 0.32/0.52 thf(fact_1721_of__nat__of__bool,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.32/0.52 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_of_bool
% 0.32/0.52 thf(fact_1722_of__nat__of__bool,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.32/0.52 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_of_bool
% 0.32/0.52 thf(fact_1723_of__nat__of__bool,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.32/0.52 = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_of_bool
% 0.32/0.52 thf(fact_1724_of__nat__of__bool,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.32/0.52 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_of_bool
% 0.32/0.52 thf(fact_1725_of__nat__of__bool,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.32/0.52 = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_of_bool
% 0.32/0.52 thf(fact_1726_zle__diff1__eq,axiom,
% 0.32/0.52 ! [W: int,Z2: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
% 0.32/0.52 = ( ord_less_int @ W @ Z2 ) ) ).
% 0.32/0.52
% 0.32/0.52 % zle_diff1_eq
% 0.32/0.52 thf(fact_1727_push__bit__negative__int__iff,axiom,
% 0.32/0.52 ! [N: nat,K: int] :
% 0.32/0.52 ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 0.32/0.52 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.52
% 0.32/0.52 % push_bit_negative_int_iff
% 0.32/0.52 thf(fact_1728_push__bit__nonnegative__int__iff,axiom,
% 0.32/0.52 ! [N: nat,K: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
% 0.32/0.52 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.52
% 0.32/0.52 % push_bit_nonnegative_int_iff
% 0.32/0.52 thf(fact_1729_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.32/0.52 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_double_add_iff_zero_le_single_add
% 0.32/0.52 thf(fact_1730_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.32/0.52 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_double_add_iff_zero_le_single_add
% 0.32/0.52 thf(fact_1731_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.32/0.52 ! [A: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.32/0.52 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_double_add_iff_zero_le_single_add
% 0.32/0.52 thf(fact_1732_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.32/0.52 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.32/0.52
% 0.32/0.52 % double_add_le_zero_iff_single_add_le_zero
% 0.32/0.52 thf(fact_1733_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.32/0.52 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.52
% 0.32/0.52 % double_add_le_zero_iff_single_add_le_zero
% 0.32/0.52 thf(fact_1734_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.32/0.52 ! [A: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.32/0.52 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.32/0.52
% 0.32/0.52 % double_add_le_zero_iff_single_add_le_zero
% 0.32/0.52 thf(fact_1735_le__add__same__cancel2,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.32/0.52 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_add_same_cancel2
% 0.32/0.52 thf(fact_1736_le__add__same__cancel2,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.32/0.52 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_add_same_cancel2
% 0.32/0.52 thf(fact_1737_le__add__same__cancel2,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.32/0.52 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_add_same_cancel2
% 0.32/0.52 thf(fact_1738_le__add__same__cancel2,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.32/0.52 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_add_same_cancel2
% 0.32/0.52 thf(fact_1739_le__add__same__cancel1,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.32/0.52 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_add_same_cancel1
% 0.32/0.52 thf(fact_1740_le__add__same__cancel1,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.52 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_add_same_cancel1
% 0.32/0.52 thf(fact_1741_le__add__same__cancel1,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.52 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_add_same_cancel1
% 0.32/0.52 thf(fact_1742_le__add__same__cancel1,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.32/0.52 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_add_same_cancel1
% 0.32/0.52 thf(fact_1743_add__le__same__cancel2,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.32/0.52 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_le_same_cancel2
% 0.32/0.52 thf(fact_1744_add__le__same__cancel2,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.32/0.52 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_le_same_cancel2
% 0.32/0.52 thf(fact_1745_add__le__same__cancel2,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.32/0.52 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_le_same_cancel2
% 0.32/0.52 thf(fact_1746_add__le__same__cancel2,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.32/0.52 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_le_same_cancel2
% 0.32/0.52 thf(fact_1747_add__le__same__cancel1,axiom,
% 0.32/0.52 ! [B: real,A: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.32/0.52 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_le_same_cancel1
% 0.32/0.52 thf(fact_1748_add__le__same__cancel1,axiom,
% 0.32/0.52 ! [B: rat,A: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 0.32/0.52 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_le_same_cancel1
% 0.32/0.52 thf(fact_1749_add__le__same__cancel1,axiom,
% 0.32/0.52 ! [B: nat,A: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.32/0.52 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_le_same_cancel1
% 0.32/0.52 thf(fact_1750_add__le__same__cancel1,axiom,
% 0.32/0.52 ! [B: int,A: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.32/0.52 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_le_same_cancel1
% 0.32/0.52 thf(fact_1751_diff__ge__0__iff__ge,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.32/0.52 = ( ord_less_eq_real @ B @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % diff_ge_0_iff_ge
% 0.32/0.52 thf(fact_1752_diff__ge__0__iff__ge,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 0.32/0.52 = ( ord_less_eq_rat @ B @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % diff_ge_0_iff_ge
% 0.32/0.52 thf(fact_1753_diff__ge__0__iff__ge,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.32/0.52 = ( ord_less_eq_int @ B @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % diff_ge_0_iff_ge
% 0.32/0.52 thf(fact_1754_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.32/0.52 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_double_add_iff_zero_less_single_add
% 0.32/0.52 thf(fact_1755_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.32/0.52 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_double_add_iff_zero_less_single_add
% 0.32/0.52 thf(fact_1756_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.32/0.52 ! [A: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.32/0.52 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_double_add_iff_zero_less_single_add
% 0.32/0.52 thf(fact_1757_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.32/0.52 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.32/0.52
% 0.32/0.52 % double_add_less_zero_iff_single_add_less_zero
% 0.32/0.52 thf(fact_1758_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.32/0.52 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.52
% 0.32/0.52 % double_add_less_zero_iff_single_add_less_zero
% 0.32/0.52 thf(fact_1759_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.32/0.52 ! [A: int] :
% 0.32/0.52 ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.32/0.52 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.32/0.52
% 0.32/0.52 % double_add_less_zero_iff_single_add_less_zero
% 0.32/0.52 thf(fact_1760_less__add__same__cancel2,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.32/0.52 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_add_same_cancel2
% 0.32/0.52 thf(fact_1761_less__add__same__cancel2,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.32/0.52 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_add_same_cancel2
% 0.32/0.52 thf(fact_1762_less__add__same__cancel2,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.32/0.52 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_add_same_cancel2
% 0.32/0.52 thf(fact_1763_less__add__same__cancel2,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.32/0.52 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_add_same_cancel2
% 0.32/0.52 thf(fact_1764_less__add__same__cancel1,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.32/0.52 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_add_same_cancel1
% 0.32/0.52 thf(fact_1765_less__add__same__cancel1,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.52 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_add_same_cancel1
% 0.32/0.52 thf(fact_1766_less__add__same__cancel1,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.52 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_add_same_cancel1
% 0.32/0.52 thf(fact_1767_less__add__same__cancel1,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.32/0.52 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_add_same_cancel1
% 0.32/0.52 thf(fact_1768_add__less__same__cancel2,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.32/0.52 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_same_cancel2
% 0.32/0.52 thf(fact_1769_add__less__same__cancel2,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.32/0.52 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_same_cancel2
% 0.32/0.52 thf(fact_1770_add__less__same__cancel2,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.32/0.52 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_same_cancel2
% 0.32/0.52 thf(fact_1771_add__less__same__cancel2,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.32/0.52 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_same_cancel2
% 0.32/0.52 thf(fact_1772_add__less__same__cancel1,axiom,
% 0.32/0.52 ! [B: real,A: real] :
% 0.32/0.52 ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.32/0.52 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_same_cancel1
% 0.32/0.52 thf(fact_1773_add__less__same__cancel1,axiom,
% 0.32/0.52 ! [B: rat,A: rat] :
% 0.32/0.52 ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 0.32/0.52 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_same_cancel1
% 0.32/0.52 thf(fact_1774_add__less__same__cancel1,axiom,
% 0.32/0.52 ! [B: nat,A: nat] :
% 0.32/0.52 ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.32/0.52 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_same_cancel1
% 0.32/0.52 thf(fact_1775_add__less__same__cancel1,axiom,
% 0.32/0.52 ! [B: int,A: int] :
% 0.32/0.52 ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.32/0.52 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_same_cancel1
% 0.32/0.52 thf(fact_1776_diff__gt__0__iff__gt,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.32/0.52 = ( ord_less_real @ B @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % diff_gt_0_iff_gt
% 0.32/0.52 thf(fact_1777_diff__gt__0__iff__gt,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 0.32/0.52 = ( ord_less_rat @ B @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % diff_gt_0_iff_gt
% 0.32/0.52 thf(fact_1778_diff__gt__0__iff__gt,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.32/0.52 = ( ord_less_int @ B @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % diff_gt_0_iff_gt
% 0.32/0.52 thf(fact_1779_le__add__diff__inverse2,axiom,
% 0.32/0.52 ! [B: real,A: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.52 => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.32/0.52 = A ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_add_diff_inverse2
% 0.32/0.52 thf(fact_1780_le__add__diff__inverse2,axiom,
% 0.32/0.52 ! [B: rat,A: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.52 => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 0.32/0.52 = A ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_add_diff_inverse2
% 0.32/0.52 thf(fact_1781_le__add__diff__inverse2,axiom,
% 0.32/0.52 ! [B: nat,A: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.52 => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 0.32/0.52 = A ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_add_diff_inverse2
% 0.32/0.52 thf(fact_1782_le__add__diff__inverse2,axiom,
% 0.32/0.52 ! [B: int,A: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.52 => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.32/0.52 = A ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_add_diff_inverse2
% 0.32/0.52 thf(fact_1783_le__add__diff__inverse,axiom,
% 0.32/0.52 ! [B: real,A: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.52 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 0.32/0.52 = A ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_add_diff_inverse
% 0.32/0.52 thf(fact_1784_le__add__diff__inverse,axiom,
% 0.32/0.52 ! [B: rat,A: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.52 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 0.32/0.52 = A ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_add_diff_inverse
% 0.32/0.52 thf(fact_1785_le__add__diff__inverse,axiom,
% 0.32/0.52 ! [B: nat,A: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.52 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 0.32/0.52 = A ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_add_diff_inverse
% 0.32/0.52 thf(fact_1786_le__add__diff__inverse,axiom,
% 0.32/0.52 ! [B: int,A: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.52 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 0.32/0.52 = A ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_add_diff_inverse
% 0.32/0.52 thf(fact_1787_power__inject__exp,axiom,
% 0.32/0.52 ! [A: real,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.52 => ( ( ( power_power_real @ A @ M )
% 0.32/0.52 = ( power_power_real @ A @ N ) )
% 0.32/0.52 = ( M = N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_inject_exp
% 0.32/0.52 thf(fact_1788_power__inject__exp,axiom,
% 0.32/0.52 ! [A: rat,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.52 => ( ( ( power_power_rat @ A @ M )
% 0.32/0.52 = ( power_power_rat @ A @ N ) )
% 0.32/0.52 = ( M = N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_inject_exp
% 0.32/0.52 thf(fact_1789_power__inject__exp,axiom,
% 0.32/0.52 ! [A: nat,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.52 => ( ( ( power_power_nat @ A @ M )
% 0.32/0.52 = ( power_power_nat @ A @ N ) )
% 0.32/0.52 = ( M = N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_inject_exp
% 0.32/0.52 thf(fact_1790_power__inject__exp,axiom,
% 0.32/0.52 ! [A: int,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.52 => ( ( ( power_power_int @ A @ M )
% 0.32/0.52 = ( power_power_int @ A @ N ) )
% 0.32/0.52 = ( M = N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_inject_exp
% 0.32/0.52 thf(fact_1791_of__nat__less__iff,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.52 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_less_iff
% 0.32/0.52 thf(fact_1792_of__nat__less__iff,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.32/0.52 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_less_iff
% 0.32/0.52 thf(fact_1793_of__nat__less__iff,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.52 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_less_iff
% 0.32/0.52 thf(fact_1794_of__nat__less__iff,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.52 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_less_iff
% 0.32/0.52 thf(fact_1795_of__nat__le__iff,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.52 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_le_iff
% 0.32/0.52 thf(fact_1796_of__nat__le__iff,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.32/0.52 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_le_iff
% 0.32/0.52 thf(fact_1797_of__nat__le__iff,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.52 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_le_iff
% 0.32/0.52 thf(fact_1798_of__nat__le__iff,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.52 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_le_iff
% 0.32/0.52 thf(fact_1799_less__Suc0,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.32/0.52 = ( N = zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_Suc0
% 0.32/0.52 thf(fact_1800_zero__less__Suc,axiom,
% 0.32/0.52 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_Suc
% 0.32/0.52 thf(fact_1801_add__gr__0,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.52 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.52 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_gr_0
% 0.32/0.52 thf(fact_1802_zero__less__of__bool__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.32/0.52 = P ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_of_bool_iff
% 0.32/0.52 thf(fact_1803_zero__less__of__bool__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 0.32/0.52 = P ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_of_bool_iff
% 0.32/0.52 thf(fact_1804_zero__less__of__bool__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.32/0.52 = P ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_of_bool_iff
% 0.32/0.52 thf(fact_1805_zero__less__of__bool__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.32/0.52 = P ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_of_bool_iff
% 0.32/0.52 thf(fact_1806_zero__less__of__bool__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
% 0.32/0.52 = P ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_of_bool_iff
% 0.32/0.52 thf(fact_1807_zero__less__diff,axiom,
% 0.32/0.52 ! [N: nat,M: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 0.32/0.52 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_diff
% 0.32/0.52 thf(fact_1808_nat__mult__less__cancel__disj,axiom,
% 0.32/0.52 ! [K: nat,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.52 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.52 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nat_mult_less_cancel_disj
% 0.32/0.52 thf(fact_1809_nat__0__less__mult__iff,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 0.32/0.52 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.52 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nat_0_less_mult_iff
% 0.32/0.52 thf(fact_1810_mult__less__cancel2,axiom,
% 0.32/0.52 ! [M: nat,K: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.32/0.52 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.52 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_cancel2
% 0.32/0.52 thf(fact_1811_diff__is__0__eq_H,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.52 => ( ( minus_minus_nat @ M @ N )
% 0.32/0.52 = zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % diff_is_0_eq'
% 0.32/0.52 thf(fact_1812_diff__is__0__eq,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ( minus_minus_nat @ M @ N )
% 0.32/0.52 = zero_zero_nat )
% 0.32/0.52 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % diff_is_0_eq
% 0.32/0.52 thf(fact_1813_of__bool__less__one__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 0.32/0.52 = ( ~ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_one_iff
% 0.32/0.52 thf(fact_1814_of__bool__less__one__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 0.32/0.52 = ( ~ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_one_iff
% 0.32/0.52 thf(fact_1815_of__bool__less__one__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 0.32/0.52 = ( ~ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_one_iff
% 0.32/0.52 thf(fact_1816_of__bool__less__one__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 0.32/0.52 = ( ~ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_one_iff
% 0.32/0.52 thf(fact_1817_of__bool__less__one__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
% 0.32/0.52 = ( ~ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_one_iff
% 0.32/0.52 thf(fact_1818_less__one,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ N @ one_one_nat )
% 0.32/0.52 = ( N = zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_one
% 0.32/0.52 thf(fact_1819_of__bool__not__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( zero_n1201886186963655149omplex @ ~ P )
% 0.32/0.52 = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_not_iff
% 0.32/0.52 thf(fact_1820_of__bool__not__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( zero_n3304061248610475627l_real @ ~ P )
% 0.32/0.52 = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_not_iff
% 0.32/0.52 thf(fact_1821_of__bool__not__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 0.32/0.52 = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_not_iff
% 0.32/0.52 thf(fact_1822_of__bool__not__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( zero_n2684676970156552555ol_int @ ~ P )
% 0.32/0.52 = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_not_iff
% 0.32/0.52 thf(fact_1823_of__bool__not__iff,axiom,
% 0.32/0.52 ! [P: $o] :
% 0.32/0.52 ( ( zero_n356916108424825756nteger @ ~ P )
% 0.32/0.52 = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_not_iff
% 0.32/0.52 thf(fact_1824_Nat_Odiff__diff__right,axiom,
% 0.32/0.52 ! [K: nat,J: nat,I: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.52 => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.32/0.52 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % Nat.diff_diff_right
% 0.32/0.52 thf(fact_1825_Nat_Oadd__diff__assoc2,axiom,
% 0.32/0.52 ! [K: nat,J: nat,I: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.52 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.32/0.52 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % Nat.add_diff_assoc2
% 0.32/0.52 thf(fact_1826_Nat_Oadd__diff__assoc,axiom,
% 0.32/0.52 ! [K: nat,J: nat,I: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.52 => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.32/0.52 = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % Nat.add_diff_assoc
% 0.32/0.52 thf(fact_1827_div__less,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ M @ N )
% 0.32/0.52 => ( ( divide_divide_nat @ M @ N )
% 0.32/0.52 = zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % div_less
% 0.32/0.52 thf(fact_1828_nat__zero__less__power__iff,axiom,
% 0.32/0.52 ! [X: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
% 0.32/0.52 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.32/0.52 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nat_zero_less_power_iff
% 0.32/0.52 thf(fact_1829_zero__le__divide__1__iff,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.32/0.52 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_divide_1_iff
% 0.32/0.52 thf(fact_1830_zero__le__divide__1__iff,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.32/0.52 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_divide_1_iff
% 0.32/0.52 thf(fact_1831_divide__le__0__1__iff,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.32/0.52 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_le_0_1_iff
% 0.32/0.52 thf(fact_1832_divide__le__0__1__iff,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.32/0.52 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_le_0_1_iff
% 0.32/0.52 thf(fact_1833_numeral__le__one__iff,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.32/0.52 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.32/0.52
% 0.32/0.52 % numeral_le_one_iff
% 0.32/0.52 thf(fact_1834_numeral__le__one__iff,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 0.32/0.52 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.32/0.52
% 0.32/0.52 % numeral_le_one_iff
% 0.32/0.52 thf(fact_1835_numeral__le__one__iff,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.32/0.52 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.32/0.52
% 0.32/0.52 % numeral_le_one_iff
% 0.32/0.52 thf(fact_1836_numeral__le__one__iff,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.32/0.52 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.32/0.52
% 0.32/0.52 % numeral_le_one_iff
% 0.32/0.52 thf(fact_1837_zero__less__divide__1__iff,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.32/0.52 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_divide_1_iff
% 0.32/0.52 thf(fact_1838_zero__less__divide__1__iff,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.32/0.52 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_divide_1_iff
% 0.32/0.52 thf(fact_1839_less__divide__eq__1__pos,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.32/0.52 = ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_divide_eq_1_pos
% 0.32/0.52 thf(fact_1840_less__divide__eq__1__pos,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.32/0.52 = ( ord_less_rat @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_divide_eq_1_pos
% 0.32/0.52 thf(fact_1841_less__divide__eq__1__neg,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.32/0.52 = ( ord_less_real @ B @ A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_divide_eq_1_neg
% 0.32/0.52 thf(fact_1842_less__divide__eq__1__neg,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.32/0.52 = ( ord_less_rat @ B @ A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_divide_eq_1_neg
% 0.32/0.52 thf(fact_1843_divide__less__eq__1__pos,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.32/0.52 = ( ord_less_real @ B @ A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_less_eq_1_pos
% 0.32/0.52 thf(fact_1844_divide__less__eq__1__pos,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.32/0.52 = ( ord_less_rat @ B @ A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_less_eq_1_pos
% 0.32/0.52 thf(fact_1845_divide__less__eq__1__neg,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.32/0.52 = ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_less_eq_1_neg
% 0.32/0.52 thf(fact_1846_divide__less__eq__1__neg,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.32/0.52 = ( ord_less_rat @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_less_eq_1_neg
% 0.32/0.52 thf(fact_1847_divide__less__0__1__iff,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.32/0.52 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_less_0_1_iff
% 0.32/0.52 thf(fact_1848_divide__less__0__1__iff,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.32/0.52 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_less_0_1_iff
% 0.32/0.52 thf(fact_1849_le__divide__eq__numeral1_I1_J,axiom,
% 0.32/0.52 ! [A: real,B: real,W: num] :
% 0.32/0.52 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.52 = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_divide_eq_numeral1(1)
% 0.32/0.52 thf(fact_1850_le__divide__eq__numeral1_I1_J,axiom,
% 0.32/0.52 ! [A: rat,B: rat,W: num] :
% 0.32/0.52 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.32/0.52 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_divide_eq_numeral1(1)
% 0.32/0.52 thf(fact_1851_divide__le__eq__numeral1_I1_J,axiom,
% 0.32/0.52 ! [B: real,W: num,A: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 0.32/0.52 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_le_eq_numeral1(1)
% 0.32/0.52 thf(fact_1852_divide__le__eq__numeral1_I1_J,axiom,
% 0.32/0.52 ! [B: rat,W: num,A: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 0.32/0.52 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_le_eq_numeral1(1)
% 0.32/0.52 thf(fact_1853_one__less__numeral__iff,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.32/0.52 = ( ord_less_num @ one @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % one_less_numeral_iff
% 0.32/0.52 thf(fact_1854_one__less__numeral__iff,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.32/0.52 = ( ord_less_num @ one @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % one_less_numeral_iff
% 0.32/0.52 thf(fact_1855_one__less__numeral__iff,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 0.32/0.52 = ( ord_less_num @ one @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % one_less_numeral_iff
% 0.32/0.52 thf(fact_1856_one__less__numeral__iff,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.32/0.52 = ( ord_less_num @ one @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % one_less_numeral_iff
% 0.32/0.52 thf(fact_1857_less__divide__eq__numeral1_I1_J,axiom,
% 0.32/0.52 ! [A: real,B: real,W: num] :
% 0.32/0.52 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.52 = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_divide_eq_numeral1(1)
% 0.32/0.52 thf(fact_1858_less__divide__eq__numeral1_I1_J,axiom,
% 0.32/0.52 ! [A: rat,B: rat,W: num] :
% 0.32/0.52 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.32/0.52 = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_divide_eq_numeral1(1)
% 0.32/0.52 thf(fact_1859_divide__less__eq__numeral1_I1_J,axiom,
% 0.32/0.52 ! [B: real,W: num,A: real] :
% 0.32/0.52 ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 0.32/0.52 = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_less_eq_numeral1(1)
% 0.32/0.52 thf(fact_1860_divide__less__eq__numeral1_I1_J,axiom,
% 0.32/0.52 ! [B: rat,W: num,A: rat] :
% 0.32/0.52 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 0.32/0.52 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_less_eq_numeral1(1)
% 0.32/0.52 thf(fact_1861_of__nat__le__0__iff,axiom,
% 0.32/0.52 ! [M: nat] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 0.32/0.52 = ( M = zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_le_0_iff
% 0.32/0.52 thf(fact_1862_of__nat__le__0__iff,axiom,
% 0.32/0.52 ! [M: nat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 0.32/0.52 = ( M = zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_le_0_iff
% 0.32/0.52 thf(fact_1863_of__nat__le__0__iff,axiom,
% 0.32/0.52 ! [M: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 0.32/0.52 = ( M = zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_le_0_iff
% 0.32/0.52 thf(fact_1864_of__nat__le__0__iff,axiom,
% 0.32/0.52 ! [M: nat] :
% 0.32/0.52 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 0.32/0.52 = ( M = zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_le_0_iff
% 0.32/0.52 thf(fact_1865_power__strict__increasing__iff,axiom,
% 0.32/0.52 ! [B: real,X: nat,Y: nat] :
% 0.32/0.52 ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.52 => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
% 0.32/0.52 = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_strict_increasing_iff
% 0.32/0.52 thf(fact_1866_power__strict__increasing__iff,axiom,
% 0.32/0.52 ! [B: rat,X: nat,Y: nat] :
% 0.32/0.52 ( ( ord_less_rat @ one_one_rat @ B )
% 0.32/0.52 => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 0.32/0.52 = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_strict_increasing_iff
% 0.32/0.52 thf(fact_1867_power__strict__increasing__iff,axiom,
% 0.32/0.52 ! [B: nat,X: nat,Y: nat] :
% 0.32/0.52 ( ( ord_less_nat @ one_one_nat @ B )
% 0.32/0.52 => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 0.32/0.52 = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_strict_increasing_iff
% 0.32/0.52 thf(fact_1868_power__strict__increasing__iff,axiom,
% 0.32/0.52 ! [B: int,X: nat,Y: nat] :
% 0.32/0.52 ( ( ord_less_int @ one_one_int @ B )
% 0.32/0.52 => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 0.32/0.52 = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_strict_increasing_iff
% 0.32/0.52 thf(fact_1869_power__eq__0__iff,axiom,
% 0.32/0.52 ! [A: rat,N: nat] :
% 0.32/0.52 ( ( ( power_power_rat @ A @ N )
% 0.32/0.52 = zero_zero_rat )
% 0.32/0.52 = ( ( A = zero_zero_rat )
% 0.32/0.52 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_eq_0_iff
% 0.32/0.52 thf(fact_1870_power__eq__0__iff,axiom,
% 0.32/0.52 ! [A: nat,N: nat] :
% 0.32/0.52 ( ( ( power_power_nat @ A @ N )
% 0.32/0.52 = zero_zero_nat )
% 0.32/0.52 = ( ( A = zero_zero_nat )
% 0.32/0.52 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_eq_0_iff
% 0.32/0.52 thf(fact_1871_power__eq__0__iff,axiom,
% 0.32/0.52 ! [A: real,N: nat] :
% 0.32/0.52 ( ( ( power_power_real @ A @ N )
% 0.32/0.52 = zero_zero_real )
% 0.32/0.52 = ( ( A = zero_zero_real )
% 0.32/0.52 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_eq_0_iff
% 0.32/0.52 thf(fact_1872_power__eq__0__iff,axiom,
% 0.32/0.52 ! [A: int,N: nat] :
% 0.32/0.52 ( ( ( power_power_int @ A @ N )
% 0.32/0.52 = zero_zero_int )
% 0.32/0.52 = ( ( A = zero_zero_int )
% 0.32/0.52 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_eq_0_iff
% 0.32/0.52 thf(fact_1873_power__eq__0__iff,axiom,
% 0.32/0.52 ! [A: complex,N: nat] :
% 0.32/0.52 ( ( ( power_power_complex @ A @ N )
% 0.32/0.52 = zero_zero_complex )
% 0.32/0.52 = ( ( A = zero_zero_complex )
% 0.32/0.52 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_eq_0_iff
% 0.32/0.52 thf(fact_1874_pow__divides__pow__iff,axiom,
% 0.32/0.52 ! [N: nat,A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.32/0.52 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pow_divides_pow_iff
% 0.32/0.52 thf(fact_1875_pow__divides__pow__iff,axiom,
% 0.32/0.52 ! [N: nat,A: int,B: int] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.32/0.52 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pow_divides_pow_iff
% 0.32/0.52 thf(fact_1876_Suc__pred,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.32/0.52 = N ) ) ).
% 0.32/0.52
% 0.32/0.52 % Suc_pred
% 0.32/0.52 thf(fact_1877_one__le__mult__iff,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 0.32/0.52 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.32/0.52 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % one_le_mult_iff
% 0.32/0.52 thf(fact_1878_nat__mult__le__cancel__disj,axiom,
% 0.32/0.52 ! [K: nat,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.52 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.52 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nat_mult_le_cancel_disj
% 0.32/0.52 thf(fact_1879_mult__le__cancel2,axiom,
% 0.32/0.52 ! [M: nat,K: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.32/0.52 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.52 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_le_cancel2
% 0.32/0.52 thf(fact_1880_diff__Suc__diff__eq2,axiom,
% 0.32/0.52 ! [K: nat,J: nat,I: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.52 => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 0.32/0.52 = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % diff_Suc_diff_eq2
% 0.32/0.52 thf(fact_1881_diff__Suc__diff__eq1,axiom,
% 0.32/0.52 ! [K: nat,J: nat,I: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.52 => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 0.32/0.52 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % diff_Suc_diff_eq1
% 0.32/0.52 thf(fact_1882_div__mult__self1__is__m,axiom,
% 0.32/0.52 ! [N: nat,M: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 0.32/0.52 = M ) ) ).
% 0.32/0.52
% 0.32/0.52 % div_mult_self1_is_m
% 0.32/0.52 thf(fact_1883_div__mult__self__is__m,axiom,
% 0.32/0.52 ! [N: nat,M: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 0.32/0.52 = M ) ) ).
% 0.32/0.52
% 0.32/0.52 % div_mult_self_is_m
% 0.32/0.52 thf(fact_1884_numeral__less__real__of__nat__iff,axiom,
% 0.32/0.52 ! [W: num,N: nat] :
% 0.32/0.52 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.52 = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % numeral_less_real_of_nat_iff
% 0.32/0.52 thf(fact_1885_real__of__nat__less__numeral__iff,axiom,
% 0.32/0.52 ! [N: nat,W: num] :
% 0.32/0.52 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
% 0.32/0.52 = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % real_of_nat_less_numeral_iff
% 0.32/0.52 thf(fact_1886_le__divide__eq__1__pos,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.32/0.52 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_divide_eq_1_pos
% 0.32/0.52 thf(fact_1887_le__divide__eq__1__pos,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.32/0.52 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_divide_eq_1_pos
% 0.32/0.52 thf(fact_1888_le__divide__eq__1__neg,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.32/0.52 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_divide_eq_1_neg
% 0.32/0.52 thf(fact_1889_le__divide__eq__1__neg,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.32/0.52 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_divide_eq_1_neg
% 0.32/0.52 thf(fact_1890_divide__le__eq__1__pos,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.32/0.52 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_le_eq_1_pos
% 0.32/0.52 thf(fact_1891_divide__le__eq__1__pos,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.32/0.52 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_le_eq_1_pos
% 0.32/0.52 thf(fact_1892_divide__le__eq__1__neg,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.32/0.52 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_le_eq_1_neg
% 0.32/0.52 thf(fact_1893_divide__le__eq__1__neg,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.32/0.52 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_le_eq_1_neg
% 0.32/0.52 thf(fact_1894_power__strict__decreasing__iff,axiom,
% 0.32/0.52 ! [B: real,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.52 => ( ( ord_less_real @ B @ one_one_real )
% 0.32/0.52 => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 0.32/0.52 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_strict_decreasing_iff
% 0.32/0.52 thf(fact_1895_power__strict__decreasing__iff,axiom,
% 0.32/0.52 ! [B: rat,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.32/0.52 => ( ( ord_less_rat @ B @ one_one_rat )
% 0.32/0.52 => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 0.32/0.52 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_strict_decreasing_iff
% 0.32/0.52 thf(fact_1896_power__strict__decreasing__iff,axiom,
% 0.32/0.52 ! [B: nat,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.52 => ( ( ord_less_nat @ B @ one_one_nat )
% 0.32/0.52 => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 0.32/0.52 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_strict_decreasing_iff
% 0.32/0.52 thf(fact_1897_power__strict__decreasing__iff,axiom,
% 0.32/0.52 ! [B: int,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ( ord_less_int @ B @ one_one_int )
% 0.32/0.52 => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 0.32/0.52 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_strict_decreasing_iff
% 0.32/0.52 thf(fact_1898_power__increasing__iff,axiom,
% 0.32/0.52 ! [B: real,X: nat,Y: nat] :
% 0.32/0.52 ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.52 => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
% 0.32/0.52 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_increasing_iff
% 0.32/0.52 thf(fact_1899_power__increasing__iff,axiom,
% 0.32/0.52 ! [B: rat,X: nat,Y: nat] :
% 0.32/0.52 ( ( ord_less_rat @ one_one_rat @ B )
% 0.32/0.52 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 0.32/0.52 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_increasing_iff
% 0.32/0.52 thf(fact_1900_power__increasing__iff,axiom,
% 0.32/0.52 ! [B: nat,X: nat,Y: nat] :
% 0.32/0.52 ( ( ord_less_nat @ one_one_nat @ B )
% 0.32/0.52 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 0.32/0.52 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_increasing_iff
% 0.32/0.52 thf(fact_1901_power__increasing__iff,axiom,
% 0.32/0.52 ! [B: int,X: nat,Y: nat] :
% 0.32/0.52 ( ( ord_less_int @ one_one_int @ B )
% 0.32/0.52 => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 0.32/0.52 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_increasing_iff
% 0.32/0.52 thf(fact_1902_power__mono__iff,axiom,
% 0.32/0.52 ! [A: real,B: real,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 0.32/0.52 = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_mono_iff
% 0.32/0.52 thf(fact_1903_power__mono__iff,axiom,
% 0.32/0.52 ! [A: rat,B: rat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 0.32/0.52 = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_mono_iff
% 0.32/0.52 thf(fact_1904_power__mono__iff,axiom,
% 0.32/0.52 ! [A: nat,B: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.32/0.52 = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_mono_iff
% 0.32/0.52 thf(fact_1905_power__mono__iff,axiom,
% 0.32/0.52 ! [A: int,B: int,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.32/0.52 = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_mono_iff
% 0.32/0.52 thf(fact_1906_of__nat__0__less__iff,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.52 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_0_less_iff
% 0.32/0.52 thf(fact_1907_of__nat__0__less__iff,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.32/0.52 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_0_less_iff
% 0.32/0.52 thf(fact_1908_of__nat__0__less__iff,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.52 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_0_less_iff
% 0.32/0.52 thf(fact_1909_of__nat__0__less__iff,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.52 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_0_less_iff
% 0.32/0.52 thf(fact_1910_odd__of__bool__self,axiom,
% 0.32/0.52 ! [P2: $o] :
% 0.32/0.52 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P2 ) ) )
% 0.32/0.52 = P2 ) ).
% 0.32/0.52
% 0.32/0.52 % odd_of_bool_self
% 0.32/0.52 thf(fact_1911_odd__of__bool__self,axiom,
% 0.32/0.52 ! [P2: $o] :
% 0.32/0.52 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P2 ) ) )
% 0.32/0.52 = P2 ) ).
% 0.32/0.52
% 0.32/0.52 % odd_of_bool_self
% 0.32/0.52 thf(fact_1912_odd__of__bool__self,axiom,
% 0.32/0.52 ! [P2: $o] :
% 0.32/0.52 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P2 ) ) )
% 0.32/0.52 = P2 ) ).
% 0.32/0.52
% 0.32/0.52 % odd_of_bool_self
% 0.32/0.52 thf(fact_1913_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.32/0.52 ! [B: nat,W: nat,X: nat] :
% 0.32/0.52 ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.32/0.52 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_less_of_nat_power_cancel_iff
% 0.32/0.52 thf(fact_1914_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.32/0.52 ! [B: nat,W: nat,X: nat] :
% 0.32/0.52 ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.32/0.52 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_less_of_nat_power_cancel_iff
% 0.32/0.52 thf(fact_1915_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.32/0.52 ! [B: nat,W: nat,X: nat] :
% 0.32/0.52 ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.32/0.52 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_less_of_nat_power_cancel_iff
% 0.32/0.52 thf(fact_1916_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.32/0.52 ! [B: nat,W: nat,X: nat] :
% 0.32/0.52 ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.32/0.52 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_less_of_nat_power_cancel_iff
% 0.32/0.52 thf(fact_1917_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.52 ! [X: nat,B: nat,W: nat] :
% 0.32/0.52 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.32/0.52 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_power_less_of_nat_cancel_iff
% 0.32/0.52 thf(fact_1918_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.52 ! [X: nat,B: nat,W: nat] :
% 0.32/0.52 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.32/0.52 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_power_less_of_nat_cancel_iff
% 0.32/0.52 thf(fact_1919_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.52 ! [X: nat,B: nat,W: nat] :
% 0.32/0.52 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.32/0.52 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_power_less_of_nat_cancel_iff
% 0.32/0.52 thf(fact_1920_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.52 ! [X: nat,B: nat,W: nat] :
% 0.32/0.52 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.32/0.52 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_power_less_of_nat_cancel_iff
% 0.32/0.52 thf(fact_1921_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.32/0.52 ! [B: nat,W: nat,X: nat] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.32/0.52 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_le_of_nat_power_cancel_iff
% 0.32/0.52 thf(fact_1922_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.32/0.52 ! [B: nat,W: nat,X: nat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.32/0.52 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_le_of_nat_power_cancel_iff
% 0.32/0.52 thf(fact_1923_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.32/0.52 ! [B: nat,W: nat,X: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.32/0.52 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_le_of_nat_power_cancel_iff
% 0.32/0.52 thf(fact_1924_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.32/0.52 ! [B: nat,W: nat,X: nat] :
% 0.32/0.52 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.32/0.52 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_le_of_nat_power_cancel_iff
% 0.32/0.52 thf(fact_1925_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.52 ! [X: nat,B: nat,W: nat] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.32/0.52 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_power_le_of_nat_cancel_iff
% 0.32/0.52 thf(fact_1926_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.52 ! [X: nat,B: nat,W: nat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.32/0.52 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_power_le_of_nat_cancel_iff
% 0.32/0.52 thf(fact_1927_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.52 ! [X: nat,B: nat,W: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.32/0.52 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_power_le_of_nat_cancel_iff
% 0.32/0.52 thf(fact_1928_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.52 ! [X: nat,B: nat,W: nat] :
% 0.32/0.52 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.32/0.52 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_power_le_of_nat_cancel_iff
% 0.32/0.52 thf(fact_1929_Suc__diff__1,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.32/0.52 = N ) ) ).
% 0.32/0.52
% 0.32/0.52 % Suc_diff_1
% 0.32/0.52 thf(fact_1930_half__negative__int__iff,axiom,
% 0.32/0.52 ! [K: int] :
% 0.32/0.52 ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.32/0.52 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.52
% 0.32/0.52 % half_negative_int_iff
% 0.32/0.52 thf(fact_1931_half__nonnegative__int__iff,axiom,
% 0.32/0.52 ! [K: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.32/0.52 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.52
% 0.32/0.52 % half_nonnegative_int_iff
% 0.32/0.52 thf(fact_1932_numeral__le__real__of__nat__iff,axiom,
% 0.32/0.52 ! [N: num,M: nat] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 0.32/0.52 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 0.32/0.52
% 0.32/0.52 % numeral_le_real_of_nat_iff
% 0.32/0.52 thf(fact_1933_power__decreasing__iff,axiom,
% 0.32/0.52 ! [B: real,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.52 => ( ( ord_less_real @ B @ one_one_real )
% 0.32/0.52 => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 0.32/0.52 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_decreasing_iff
% 0.32/0.52 thf(fact_1934_power__decreasing__iff,axiom,
% 0.32/0.52 ! [B: rat,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.32/0.52 => ( ( ord_less_rat @ B @ one_one_rat )
% 0.32/0.52 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 0.32/0.52 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_decreasing_iff
% 0.32/0.52 thf(fact_1935_power__decreasing__iff,axiom,
% 0.32/0.52 ! [B: nat,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.52 => ( ( ord_less_nat @ B @ one_one_nat )
% 0.32/0.52 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 0.32/0.52 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_decreasing_iff
% 0.32/0.52 thf(fact_1936_power__decreasing__iff,axiom,
% 0.32/0.52 ! [B: int,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ( ord_less_int @ B @ one_one_int )
% 0.32/0.52 => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 0.32/0.52 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_decreasing_iff
% 0.32/0.52 thf(fact_1937_power2__eq__iff__nonneg,axiom,
% 0.32/0.52 ! [X: real,Y: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.32/0.52 => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.52 = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.52 = ( X = Y ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power2_eq_iff_nonneg
% 0.32/0.52 thf(fact_1938_power2__eq__iff__nonneg,axiom,
% 0.32/0.52 ! [X: rat,Y: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.32/0.52 => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.52 = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.52 = ( X = Y ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power2_eq_iff_nonneg
% 0.32/0.52 thf(fact_1939_power2__eq__iff__nonneg,axiom,
% 0.32/0.52 ! [X: nat,Y: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.32/0.52 => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.52 = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.52 = ( X = Y ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power2_eq_iff_nonneg
% 0.32/0.52 thf(fact_1940_power2__eq__iff__nonneg,axiom,
% 0.32/0.52 ! [X: int,Y: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.32/0.52 => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.52 = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.52 = ( X = Y ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power2_eq_iff_nonneg
% 0.32/0.52 thf(fact_1941_power2__less__eq__zero__iff,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 0.32/0.52 = ( A = zero_zero_real ) ) ).
% 0.32/0.52
% 0.32/0.52 % power2_less_eq_zero_iff
% 0.32/0.52 thf(fact_1942_power2__less__eq__zero__iff,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 0.32/0.52 = ( A = zero_zero_rat ) ) ).
% 0.32/0.52
% 0.32/0.52 % power2_less_eq_zero_iff
% 0.32/0.52 thf(fact_1943_power2__less__eq__zero__iff,axiom,
% 0.32/0.52 ! [A: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.32/0.52 = ( A = zero_zero_int ) ) ).
% 0.32/0.52
% 0.32/0.52 % power2_less_eq_zero_iff
% 0.32/0.52 thf(fact_1944_zero__less__power2,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.52 = ( A != zero_zero_real ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_power2
% 0.32/0.52 thf(fact_1945_zero__less__power2,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.52 = ( A != zero_zero_rat ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_power2
% 0.32/0.52 thf(fact_1946_zero__less__power2,axiom,
% 0.32/0.52 ! [A: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.52 = ( A != zero_zero_int ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_power2
% 0.32/0.52 thf(fact_1947_of__nat__zero__less__power__iff,axiom,
% 0.32/0.52 ! [X: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
% 0.32/0.52 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.32/0.52 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_zero_less_power_iff
% 0.32/0.52 thf(fact_1948_of__nat__zero__less__power__iff,axiom,
% 0.32/0.52 ! [X: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
% 0.32/0.52 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.32/0.52 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_zero_less_power_iff
% 0.32/0.52 thf(fact_1949_of__nat__zero__less__power__iff,axiom,
% 0.32/0.52 ! [X: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
% 0.32/0.52 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.32/0.52 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_zero_less_power_iff
% 0.32/0.52 thf(fact_1950_of__nat__zero__less__power__iff,axiom,
% 0.32/0.52 ! [X: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
% 0.32/0.52 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.32/0.52 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_zero_less_power_iff
% 0.32/0.52 thf(fact_1951_of__bool__half__eq__0,axiom,
% 0.32/0.52 ! [B: $o] :
% 0.32/0.52 ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.52 = zero_zero_nat ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_half_eq_0
% 0.32/0.52 thf(fact_1952_of__bool__half__eq__0,axiom,
% 0.32/0.52 ! [B: $o] :
% 0.32/0.52 ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.52 = zero_zero_int ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_half_eq_0
% 0.32/0.52 thf(fact_1953_of__bool__half__eq__0,axiom,
% 0.32/0.52 ! [B: $o] :
% 0.32/0.52 ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.52 = zero_z3403309356797280102nteger ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_half_eq_0
% 0.32/0.52 thf(fact_1954_zero__le__power__eq__numeral,axiom,
% 0.32/0.52 ! [A: real,W: num] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.52 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_power_eq_numeral
% 0.32/0.52 thf(fact_1955_zero__le__power__eq__numeral,axiom,
% 0.32/0.52 ! [A: rat,W: num] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.52 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_power_eq_numeral
% 0.32/0.52 thf(fact_1956_zero__le__power__eq__numeral,axiom,
% 0.32/0.52 ! [A: int,W: num] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.52 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_power_eq_numeral
% 0.32/0.52 thf(fact_1957_even__power,axiom,
% 0.32/0.52 ! [A: code_integer,N: nat] :
% 0.32/0.52 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.32/0.52 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.52 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % even_power
% 0.32/0.52 thf(fact_1958_even__power,axiom,
% 0.32/0.52 ! [A: nat,N: nat] :
% 0.32/0.52 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
% 0.32/0.52 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.52 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % even_power
% 0.32/0.52 thf(fact_1959_even__power,axiom,
% 0.32/0.52 ! [A: int,N: nat] :
% 0.32/0.52 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
% 0.32/0.52 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.52 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % even_power
% 0.32/0.52 thf(fact_1960_power__less__zero__eq__numeral,axiom,
% 0.32/0.52 ! [A: real,W: num] :
% 0.32/0.52 ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 0.32/0.52 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_less_zero_eq_numeral
% 0.32/0.52 thf(fact_1961_power__less__zero__eq__numeral,axiom,
% 0.32/0.52 ! [A: rat,W: num] :
% 0.32/0.52 ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 0.32/0.52 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_less_zero_eq_numeral
% 0.32/0.52 thf(fact_1962_power__less__zero__eq__numeral,axiom,
% 0.32/0.52 ! [A: int,W: num] :
% 0.32/0.52 ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 0.32/0.52 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_less_zero_eq_numeral
% 0.32/0.52 thf(fact_1963_power__less__zero__eq,axiom,
% 0.32/0.52 ! [A: real,N: nat] :
% 0.32/0.52 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 0.32/0.52 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.52 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_less_zero_eq
% 0.32/0.52 thf(fact_1964_power__less__zero__eq,axiom,
% 0.32/0.52 ! [A: rat,N: nat] :
% 0.32/0.52 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 0.32/0.52 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.52 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_less_zero_eq
% 0.32/0.52 thf(fact_1965_power__less__zero__eq,axiom,
% 0.32/0.52 ! [A: int,N: nat] :
% 0.32/0.52 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 0.32/0.52 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.52 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_less_zero_eq
% 0.32/0.52 thf(fact_1966_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.32/0.52 ! [X: nat,I: num,N: nat] :
% 0.32/0.52 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 0.32/0.52 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_less_numeral_power_cancel_iff
% 0.32/0.52 thf(fact_1967_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.32/0.52 ! [X: nat,I: num,N: nat] :
% 0.32/0.52 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 0.32/0.52 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_less_numeral_power_cancel_iff
% 0.32/0.52 thf(fact_1968_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.32/0.52 ! [X: nat,I: num,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 0.32/0.52 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_less_numeral_power_cancel_iff
% 0.32/0.52 thf(fact_1969_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.32/0.52 ! [X: nat,I: num,N: nat] :
% 0.32/0.52 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 0.32/0.52 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_less_numeral_power_cancel_iff
% 0.32/0.52 thf(fact_1970_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.52 ! [I: num,N: nat,X: nat] :
% 0.32/0.52 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.32/0.52 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.32/0.52
% 0.32/0.52 % numeral_power_less_of_nat_cancel_iff
% 0.32/0.52 thf(fact_1971_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.52 ! [I: num,N: nat,X: nat] :
% 0.32/0.52 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.32/0.52 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.32/0.52
% 0.32/0.52 % numeral_power_less_of_nat_cancel_iff
% 0.32/0.52 thf(fact_1972_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.52 ! [I: num,N: nat,X: nat] :
% 0.32/0.52 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.32/0.52 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.32/0.52
% 0.32/0.52 % numeral_power_less_of_nat_cancel_iff
% 0.32/0.52 thf(fact_1973_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.52 ! [I: num,N: nat,X: nat] :
% 0.32/0.52 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.32/0.52 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.32/0.52
% 0.32/0.52 % numeral_power_less_of_nat_cancel_iff
% 0.32/0.52 thf(fact_1974_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.32/0.52 ! [X: nat,I: num,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 0.32/0.52 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_le_numeral_power_cancel_iff
% 0.32/0.52 thf(fact_1975_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.32/0.52 ! [X: nat,I: num,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 0.32/0.52 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_le_numeral_power_cancel_iff
% 0.32/0.52 thf(fact_1976_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.32/0.52 ! [X: nat,I: num,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 0.32/0.52 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_le_numeral_power_cancel_iff
% 0.32/0.52 thf(fact_1977_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.32/0.52 ! [X: nat,I: num,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 0.32/0.52 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_nat_le_numeral_power_cancel_iff
% 0.32/0.52 thf(fact_1978_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.52 ! [I: num,N: nat,X: nat] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.32/0.52 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.32/0.52
% 0.32/0.52 % numeral_power_le_of_nat_cancel_iff
% 0.32/0.52 thf(fact_1979_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.52 ! [I: num,N: nat,X: nat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.32/0.52 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.32/0.52
% 0.32/0.52 % numeral_power_le_of_nat_cancel_iff
% 0.32/0.52 thf(fact_1980_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.52 ! [I: num,N: nat,X: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.32/0.52 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.32/0.52
% 0.32/0.52 % numeral_power_le_of_nat_cancel_iff
% 0.32/0.52 thf(fact_1981_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.52 ! [I: num,N: nat,X: nat] :
% 0.32/0.52 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.32/0.52 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.32/0.52
% 0.32/0.52 % numeral_power_le_of_nat_cancel_iff
% 0.32/0.52 thf(fact_1982_even__diff__nat,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.52 = ( ( ord_less_nat @ M @ N )
% 0.32/0.52 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % even_diff_nat
% 0.32/0.52 thf(fact_1983_one__div__2__pow__eq,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.52 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % one_div_2_pow_eq
% 0.32/0.52 thf(fact_1984_one__div__2__pow__eq,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.52 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % one_div_2_pow_eq
% 0.32/0.52 thf(fact_1985_one__div__2__pow__eq,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.52 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % one_div_2_pow_eq
% 0.32/0.52 thf(fact_1986_bits__1__div__exp,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.52 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % bits_1_div_exp
% 0.32/0.52 thf(fact_1987_bits__1__div__exp,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.52 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % bits_1_div_exp
% 0.32/0.52 thf(fact_1988_bits__1__div__exp,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.52 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % bits_1_div_exp
% 0.32/0.52 thf(fact_1989_power__le__zero__eq__numeral,axiom,
% 0.32/0.52 ! [A: real,W: num] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 0.32/0.52 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.32/0.52 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 & ( A = zero_zero_real ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_le_zero_eq_numeral
% 0.32/0.52 thf(fact_1990_power__le__zero__eq__numeral,axiom,
% 0.32/0.52 ! [A: rat,W: num] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 0.32/0.52 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.32/0.52 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_le_zero_eq_numeral
% 0.32/0.52 thf(fact_1991_power__le__zero__eq__numeral,axiom,
% 0.32/0.52 ! [A: int,W: num] :
% 0.32/0.52 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 0.32/0.52 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.32/0.52 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.52 & ( A = zero_zero_int ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_le_zero_eq_numeral
% 0.32/0.52 thf(fact_1992_zero__less__eq__of__bool,axiom,
% 0.32/0.52 ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_eq_of_bool
% 0.32/0.52 thf(fact_1993_zero__less__eq__of__bool,axiom,
% 0.32/0.52 ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_eq_of_bool
% 0.32/0.52 thf(fact_1994_zero__less__eq__of__bool,axiom,
% 0.32/0.52 ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_eq_of_bool
% 0.32/0.52 thf(fact_1995_zero__less__eq__of__bool,axiom,
% 0.32/0.52 ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_eq_of_bool
% 0.32/0.52 thf(fact_1996_zero__less__eq__of__bool,axiom,
% 0.32/0.52 ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_eq_of_bool
% 0.32/0.52 thf(fact_1997_of__bool__less__eq__one,axiom,
% 0.32/0.52 ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_eq_one
% 0.32/0.52 thf(fact_1998_of__bool__less__eq__one,axiom,
% 0.32/0.52 ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_eq_one
% 0.32/0.52 thf(fact_1999_of__bool__less__eq__one,axiom,
% 0.32/0.52 ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_eq_one
% 0.32/0.52 thf(fact_2000_of__bool__less__eq__one,axiom,
% 0.32/0.52 ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_eq_one
% 0.32/0.52 thf(fact_2001_of__bool__less__eq__one,axiom,
% 0.32/0.52 ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_less_eq_one
% 0.32/0.52 thf(fact_2002_verit__la__disequality,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( A = B )
% 0.32/0.52 | ~ ( ord_less_eq_rat @ A @ B )
% 0.32/0.52 | ~ ( ord_less_eq_rat @ B @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % verit_la_disequality
% 0.32/0.52 thf(fact_2003_verit__la__disequality,axiom,
% 0.32/0.52 ! [A: num,B: num] :
% 0.32/0.52 ( ( A = B )
% 0.32/0.52 | ~ ( ord_less_eq_num @ A @ B )
% 0.32/0.52 | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % verit_la_disequality
% 0.32/0.52 thf(fact_2004_verit__la__disequality,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( A = B )
% 0.32/0.52 | ~ ( ord_less_eq_nat @ A @ B )
% 0.32/0.52 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % verit_la_disequality
% 0.32/0.52 thf(fact_2005_verit__la__disequality,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( A = B )
% 0.32/0.52 | ~ ( ord_less_eq_int @ A @ B )
% 0.32/0.52 | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % verit_la_disequality
% 0.32/0.52 thf(fact_2006_linorder__neqE__linordered__idom,axiom,
% 0.32/0.52 ! [X: real,Y: real] :
% 0.32/0.52 ( ( X != Y )
% 0.32/0.52 => ( ~ ( ord_less_real @ X @ Y )
% 0.32/0.52 => ( ord_less_real @ Y @ X ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % linorder_neqE_linordered_idom
% 0.32/0.52 thf(fact_2007_linorder__neqE__linordered__idom,axiom,
% 0.32/0.52 ! [X: rat,Y: rat] :
% 0.32/0.52 ( ( X != Y )
% 0.32/0.52 => ( ~ ( ord_less_rat @ X @ Y )
% 0.32/0.52 => ( ord_less_rat @ Y @ X ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % linorder_neqE_linordered_idom
% 0.32/0.52 thf(fact_2008_linorder__neqE__linordered__idom,axiom,
% 0.32/0.52 ! [X: int,Y: int] :
% 0.32/0.52 ( ( X != Y )
% 0.32/0.52 => ( ~ ( ord_less_int @ X @ Y )
% 0.32/0.52 => ( ord_less_int @ Y @ X ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % linorder_neqE_linordered_idom
% 0.32/0.52 thf(fact_2009_of__bool__eq__iff,axiom,
% 0.32/0.52 ! [P2: $o,Q2: $o] :
% 0.32/0.52 ( ( ( zero_n2687167440665602831ol_nat @ P2 )
% 0.32/0.52 = ( zero_n2687167440665602831ol_nat @ Q2 ) )
% 0.32/0.52 = ( P2 = Q2 ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq_iff
% 0.32/0.52 thf(fact_2010_of__bool__eq__iff,axiom,
% 0.32/0.52 ! [P2: $o,Q2: $o] :
% 0.32/0.52 ( ( ( zero_n2684676970156552555ol_int @ P2 )
% 0.32/0.52 = ( zero_n2684676970156552555ol_int @ Q2 ) )
% 0.32/0.52 = ( P2 = Q2 ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq_iff
% 0.32/0.52 thf(fact_2011_of__bool__eq__iff,axiom,
% 0.32/0.52 ! [P2: $o,Q2: $o] :
% 0.32/0.52 ( ( ( zero_n356916108424825756nteger @ P2 )
% 0.32/0.52 = ( zero_n356916108424825756nteger @ Q2 ) )
% 0.32/0.52 = ( P2 = Q2 ) ) ).
% 0.32/0.52
% 0.32/0.52 % of_bool_eq_iff
% 0.32/0.52 thf(fact_2012_verit__comp__simplify1_I1_J,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ~ ( ord_less_real @ A @ A ) ).
% 0.32/0.52
% 0.32/0.52 % verit_comp_simplify1(1)
% 0.32/0.52 thf(fact_2013_verit__comp__simplify1_I1_J,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ~ ( ord_less_rat @ A @ A ) ).
% 0.32/0.52
% 0.32/0.52 % verit_comp_simplify1(1)
% 0.32/0.52 thf(fact_2014_verit__comp__simplify1_I1_J,axiom,
% 0.32/0.52 ! [A: num] :
% 0.32/0.52 ~ ( ord_less_num @ A @ A ) ).
% 0.32/0.52
% 0.32/0.52 % verit_comp_simplify1(1)
% 0.32/0.52 thf(fact_2015_verit__comp__simplify1_I1_J,axiom,
% 0.32/0.52 ! [A: nat] :
% 0.32/0.52 ~ ( ord_less_nat @ A @ A ) ).
% 0.32/0.52
% 0.32/0.52 % verit_comp_simplify1(1)
% 0.32/0.52 thf(fact_2016_verit__comp__simplify1_I1_J,axiom,
% 0.32/0.52 ! [A: int] :
% 0.32/0.52 ~ ( ord_less_int @ A @ A ) ).
% 0.32/0.52
% 0.32/0.52 % verit_comp_simplify1(1)
% 0.32/0.52 thf(fact_2017_verit__comp__simplify1_I2_J,axiom,
% 0.32/0.52 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% 0.32/0.52
% 0.32/0.52 % verit_comp_simplify1(2)
% 0.32/0.52 thf(fact_2018_verit__comp__simplify1_I2_J,axiom,
% 0.32/0.52 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.32/0.52
% 0.32/0.52 % verit_comp_simplify1(2)
% 0.32/0.52 thf(fact_2019_verit__comp__simplify1_I2_J,axiom,
% 0.32/0.52 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.32/0.52
% 0.32/0.52 % verit_comp_simplify1(2)
% 0.32/0.52 thf(fact_2020_verit__comp__simplify1_I2_J,axiom,
% 0.32/0.52 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.32/0.52
% 0.32/0.52 % verit_comp_simplify1(2)
% 0.32/0.52 thf(fact_2021_verit__comp__simplify1_I2_J,axiom,
% 0.32/0.52 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.32/0.52
% 0.32/0.52 % verit_comp_simplify1(2)
% 0.32/0.52 thf(fact_2022_verit__comp__simplify1_I3_J,axiom,
% 0.32/0.52 ! [B6: real,A5: real] :
% 0.32/0.52 ( ( ~ ( ord_less_eq_real @ B6 @ A5 ) )
% 0.32/0.52 = ( ord_less_real @ A5 @ B6 ) ) ).
% 0.32/0.52
% 0.32/0.52 % verit_comp_simplify1(3)
% 0.32/0.52 thf(fact_2023_verit__comp__simplify1_I3_J,axiom,
% 0.32/0.52 ! [B6: rat,A5: rat] :
% 0.32/0.52 ( ( ~ ( ord_less_eq_rat @ B6 @ A5 ) )
% 0.32/0.52 = ( ord_less_rat @ A5 @ B6 ) ) ).
% 0.32/0.52
% 0.32/0.52 % verit_comp_simplify1(3)
% 0.32/0.52 thf(fact_2024_verit__comp__simplify1_I3_J,axiom,
% 0.32/0.52 ! [B6: num,A5: num] :
% 0.32/0.52 ( ( ~ ( ord_less_eq_num @ B6 @ A5 ) )
% 0.32/0.52 = ( ord_less_num @ A5 @ B6 ) ) ).
% 0.32/0.52
% 0.32/0.52 % verit_comp_simplify1(3)
% 0.32/0.52 thf(fact_2025_verit__comp__simplify1_I3_J,axiom,
% 0.32/0.52 ! [B6: nat,A5: nat] :
% 0.32/0.52 ( ( ~ ( ord_less_eq_nat @ B6 @ A5 ) )
% 0.32/0.52 = ( ord_less_nat @ A5 @ B6 ) ) ).
% 0.32/0.52
% 0.32/0.52 % verit_comp_simplify1(3)
% 0.32/0.52 thf(fact_2026_verit__comp__simplify1_I3_J,axiom,
% 0.32/0.52 ! [B6: int,A5: int] :
% 0.32/0.52 ( ( ~ ( ord_less_eq_int @ B6 @ A5 ) )
% 0.32/0.52 = ( ord_less_int @ A5 @ B6 ) ) ).
% 0.32/0.52
% 0.32/0.52 % verit_comp_simplify1(3)
% 0.32/0.52 thf(fact_2027_less__mono__imp__le__mono,axiom,
% 0.32/0.52 ! [F: nat > nat,I: nat,J: nat] :
% 0.32/0.52 ( ! [I2: nat,J2: nat] :
% 0.32/0.52 ( ( ord_less_nat @ I2 @ J2 )
% 0.32/0.52 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.32/0.52 => ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.52 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_mono_imp_le_mono
% 0.32/0.52 thf(fact_2028_le__neq__implies__less,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.52 => ( ( M != N )
% 0.32/0.52 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_neq_implies_less
% 0.32/0.52 thf(fact_2029_Nat_Oex__has__greatest__nat,axiom,
% 0.32/0.52 ! [P: nat > $o,K: nat,B: nat] :
% 0.32/0.52 ( ( P @ K )
% 0.32/0.52 => ( ! [Y4: nat] :
% 0.32/0.52 ( ( P @ Y4 )
% 0.32/0.52 => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.32/0.52 => ? [X4: nat] :
% 0.32/0.52 ( ( P @ X4 )
% 0.32/0.52 & ! [Y5: nat] :
% 0.32/0.52 ( ( P @ Y5 )
% 0.32/0.52 => ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % Nat.ex_has_greatest_nat
% 0.32/0.52 thf(fact_2030_linorder__neqE__nat,axiom,
% 0.32/0.52 ! [X: nat,Y: nat] :
% 0.32/0.52 ( ( X != Y )
% 0.32/0.52 => ( ~ ( ord_less_nat @ X @ Y )
% 0.32/0.52 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % linorder_neqE_nat
% 0.32/0.52 thf(fact_2031_less__or__eq__imp__le,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ( ord_less_nat @ M @ N )
% 0.32/0.52 | ( M = N ) )
% 0.32/0.52 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_or_eq_imp_le
% 0.32/0.52 thf(fact_2032_le__eq__less__or__eq,axiom,
% 0.32/0.52 ( ord_less_eq_nat
% 0.32/0.52 = ( ^ [M3: nat,N3: nat] :
% 0.32/0.52 ( ( ord_less_nat @ M3 @ N3 )
% 0.32/0.52 | ( M3 = N3 ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_eq_less_or_eq
% 0.32/0.52 thf(fact_2033_infinite__descent,axiom,
% 0.32/0.52 ! [P: nat > $o,N: nat] :
% 0.32/0.52 ( ! [N2: nat] :
% 0.32/0.52 ( ~ ( P @ N2 )
% 0.32/0.52 => ? [M4: nat] :
% 0.32/0.52 ( ( ord_less_nat @ M4 @ N2 )
% 0.32/0.52 & ~ ( P @ M4 ) ) )
% 0.32/0.52 => ( P @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % infinite_descent
% 0.32/0.52 thf(fact_2034_nat__less__induct,axiom,
% 0.32/0.52 ! [P: nat > $o,N: nat] :
% 0.32/0.52 ( ! [N2: nat] :
% 0.32/0.52 ( ! [M4: nat] :
% 0.32/0.52 ( ( ord_less_nat @ M4 @ N2 )
% 0.32/0.52 => ( P @ M4 ) )
% 0.32/0.52 => ( P @ N2 ) )
% 0.32/0.52 => ( P @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % nat_less_induct
% 0.32/0.52 thf(fact_2035_less__irrefl__nat,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ~ ( ord_less_nat @ N @ N ) ).
% 0.32/0.52
% 0.32/0.52 % less_irrefl_nat
% 0.32/0.52 thf(fact_2036_less__imp__le__nat,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ M @ N )
% 0.32/0.52 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_imp_le_nat
% 0.32/0.52 thf(fact_2037_less__not__refl3,axiom,
% 0.32/0.52 ! [S: nat,T: nat] :
% 0.32/0.52 ( ( ord_less_nat @ S @ T )
% 0.32/0.52 => ( S != T ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_not_refl3
% 0.32/0.52 thf(fact_2038_less__not__refl2,axiom,
% 0.32/0.52 ! [N: nat,M: nat] :
% 0.32/0.52 ( ( ord_less_nat @ N @ M )
% 0.32/0.52 => ( M != N ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_not_refl2
% 0.32/0.52 thf(fact_2039_nat__le__linear,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.52 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.32/0.52
% 0.32/0.52 % nat_le_linear
% 0.32/0.52 thf(fact_2040_less__not__refl,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ~ ( ord_less_nat @ N @ N ) ).
% 0.32/0.52
% 0.32/0.52 % less_not_refl
% 0.32/0.52 thf(fact_2041_nat__neq__iff,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( M != N )
% 0.32/0.52 = ( ( ord_less_nat @ M @ N )
% 0.32/0.52 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nat_neq_iff
% 0.32/0.52 thf(fact_2042_nat__less__le,axiom,
% 0.32/0.52 ( ord_less_nat
% 0.32/0.52 = ( ^ [M3: nat,N3: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.32/0.52 & ( M3 != N3 ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nat_less_le
% 0.32/0.52 thf(fact_2043_le__antisym,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.52 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.52 => ( M = N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_antisym
% 0.32/0.52 thf(fact_2044_eq__imp__le,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( M = N )
% 0.32/0.52 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % eq_imp_le
% 0.32/0.52 thf(fact_2045_le__trans,axiom,
% 0.32/0.52 ! [I: nat,J: nat,K: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.52 => ( ( ord_less_eq_nat @ J @ K )
% 0.32/0.52 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_trans
% 0.32/0.52 thf(fact_2046_le__refl,axiom,
% 0.32/0.52 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.32/0.52
% 0.32/0.52 % le_refl
% 0.32/0.52 thf(fact_2047_less__eq__real__def,axiom,
% 0.32/0.52 ( ord_less_eq_real
% 0.32/0.52 = ( ^ [X3: real,Y6: real] :
% 0.32/0.52 ( ( ord_less_real @ X3 @ Y6 )
% 0.32/0.52 | ( X3 = Y6 ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_eq_real_def
% 0.32/0.52 thf(fact_2048_complete__real,axiom,
% 0.32/0.52 ! [S2: set_real] :
% 0.32/0.52 ( ? [X5: real] : ( member_real @ X5 @ S2 )
% 0.32/0.52 => ( ? [Z3: real] :
% 0.32/0.52 ! [X4: real] :
% 0.32/0.52 ( ( member_real @ X4 @ S2 )
% 0.32/0.52 => ( ord_less_eq_real @ X4 @ Z3 ) )
% 0.32/0.52 => ? [Y4: real] :
% 0.32/0.52 ( ! [X5: real] :
% 0.32/0.52 ( ( member_real @ X5 @ S2 )
% 0.32/0.52 => ( ord_less_eq_real @ X5 @ Y4 ) )
% 0.32/0.52 & ! [Z3: real] :
% 0.32/0.52 ( ! [X4: real] :
% 0.32/0.52 ( ( member_real @ X4 @ S2 )
% 0.32/0.52 => ( ord_less_eq_real @ X4 @ Z3 ) )
% 0.32/0.52 => ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % complete_real
% 0.32/0.52 thf(fact_2049_linordered__field__no__ub,axiom,
% 0.32/0.52 ! [X5: real] :
% 0.32/0.52 ? [X_12: real] : ( ord_less_real @ X5 @ X_12 ) ).
% 0.32/0.52
% 0.32/0.52 % linordered_field_no_ub
% 0.32/0.52 thf(fact_2050_linordered__field__no__ub,axiom,
% 0.32/0.52 ! [X5: rat] :
% 0.32/0.52 ? [X_12: rat] : ( ord_less_rat @ X5 @ X_12 ) ).
% 0.32/0.52
% 0.32/0.52 % linordered_field_no_ub
% 0.32/0.52 thf(fact_2051_linordered__field__no__lb,axiom,
% 0.32/0.52 ! [X5: real] :
% 0.32/0.52 ? [Y4: real] : ( ord_less_real @ Y4 @ X5 ) ).
% 0.32/0.52
% 0.32/0.52 % linordered_field_no_lb
% 0.32/0.52 thf(fact_2052_linordered__field__no__lb,axiom,
% 0.32/0.52 ! [X5: rat] :
% 0.32/0.52 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X5 ) ).
% 0.32/0.52
% 0.32/0.52 % linordered_field_no_lb
% 0.32/0.52 thf(fact_2053_pinf_I1_J,axiom,
% 0.32/0.52 ! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
% 0.32/0.52 ( ? [Z3: real] :
% 0.32/0.52 ! [X4: real] :
% 0.32/0.52 ( ( ord_less_real @ Z3 @ X4 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: real] :
% 0.32/0.52 ! [X4: real] :
% 0.32/0.52 ( ( ord_less_real @ Z3 @ X4 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: real] :
% 0.32/0.52 ! [X5: real] :
% 0.32/0.52 ( ( ord_less_real @ Z4 @ X5 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 & ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(1)
% 0.32/0.52 thf(fact_2054_pinf_I1_J,axiom,
% 0.32/0.52 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
% 0.32/0.52 ( ? [Z3: rat] :
% 0.32/0.52 ! [X4: rat] :
% 0.32/0.52 ( ( ord_less_rat @ Z3 @ X4 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: rat] :
% 0.32/0.52 ! [X4: rat] :
% 0.32/0.52 ( ( ord_less_rat @ Z3 @ X4 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: rat] :
% 0.32/0.52 ! [X5: rat] :
% 0.32/0.52 ( ( ord_less_rat @ Z4 @ X5 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 & ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(1)
% 0.32/0.52 thf(fact_2055_pinf_I1_J,axiom,
% 0.32/0.52 ! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
% 0.32/0.52 ( ? [Z3: num] :
% 0.32/0.52 ! [X4: num] :
% 0.32/0.52 ( ( ord_less_num @ Z3 @ X4 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: num] :
% 0.32/0.52 ! [X4: num] :
% 0.32/0.52 ( ( ord_less_num @ Z3 @ X4 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: num] :
% 0.32/0.52 ! [X5: num] :
% 0.32/0.52 ( ( ord_less_num @ Z4 @ X5 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 & ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(1)
% 0.32/0.52 thf(fact_2056_pinf_I1_J,axiom,
% 0.32/0.52 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
% 0.32/0.52 ( ? [Z3: nat] :
% 0.32/0.52 ! [X4: nat] :
% 0.32/0.52 ( ( ord_less_nat @ Z3 @ X4 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: nat] :
% 0.32/0.52 ! [X4: nat] :
% 0.32/0.52 ( ( ord_less_nat @ Z3 @ X4 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: nat] :
% 0.32/0.52 ! [X5: nat] :
% 0.32/0.52 ( ( ord_less_nat @ Z4 @ X5 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 & ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(1)
% 0.32/0.52 thf(fact_2057_pinf_I1_J,axiom,
% 0.32/0.52 ! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
% 0.32/0.52 ( ? [Z3: int] :
% 0.32/0.52 ! [X4: int] :
% 0.32/0.52 ( ( ord_less_int @ Z3 @ X4 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: int] :
% 0.32/0.52 ! [X4: int] :
% 0.32/0.52 ( ( ord_less_int @ Z3 @ X4 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: int] :
% 0.32/0.52 ! [X5: int] :
% 0.32/0.52 ( ( ord_less_int @ Z4 @ X5 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 & ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(1)
% 0.32/0.52 thf(fact_2058_pinf_I2_J,axiom,
% 0.32/0.52 ! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
% 0.32/0.52 ( ? [Z3: real] :
% 0.32/0.52 ! [X4: real] :
% 0.32/0.52 ( ( ord_less_real @ Z3 @ X4 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: real] :
% 0.32/0.52 ! [X4: real] :
% 0.32/0.52 ( ( ord_less_real @ Z3 @ X4 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: real] :
% 0.32/0.52 ! [X5: real] :
% 0.32/0.52 ( ( ord_less_real @ Z4 @ X5 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 | ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(2)
% 0.32/0.52 thf(fact_2059_pinf_I2_J,axiom,
% 0.32/0.52 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
% 0.32/0.52 ( ? [Z3: rat] :
% 0.32/0.52 ! [X4: rat] :
% 0.32/0.52 ( ( ord_less_rat @ Z3 @ X4 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: rat] :
% 0.32/0.52 ! [X4: rat] :
% 0.32/0.52 ( ( ord_less_rat @ Z3 @ X4 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: rat] :
% 0.32/0.52 ! [X5: rat] :
% 0.32/0.52 ( ( ord_less_rat @ Z4 @ X5 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 | ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(2)
% 0.32/0.52 thf(fact_2060_pinf_I2_J,axiom,
% 0.32/0.52 ! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
% 0.32/0.52 ( ? [Z3: num] :
% 0.32/0.52 ! [X4: num] :
% 0.32/0.52 ( ( ord_less_num @ Z3 @ X4 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: num] :
% 0.32/0.52 ! [X4: num] :
% 0.32/0.52 ( ( ord_less_num @ Z3 @ X4 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: num] :
% 0.32/0.52 ! [X5: num] :
% 0.32/0.52 ( ( ord_less_num @ Z4 @ X5 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 | ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(2)
% 0.32/0.52 thf(fact_2061_pinf_I2_J,axiom,
% 0.32/0.52 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
% 0.32/0.52 ( ? [Z3: nat] :
% 0.32/0.52 ! [X4: nat] :
% 0.32/0.52 ( ( ord_less_nat @ Z3 @ X4 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: nat] :
% 0.32/0.52 ! [X4: nat] :
% 0.32/0.52 ( ( ord_less_nat @ Z3 @ X4 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: nat] :
% 0.32/0.52 ! [X5: nat] :
% 0.32/0.52 ( ( ord_less_nat @ Z4 @ X5 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 | ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(2)
% 0.32/0.52 thf(fact_2062_pinf_I2_J,axiom,
% 0.32/0.52 ! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
% 0.32/0.52 ( ? [Z3: int] :
% 0.32/0.52 ! [X4: int] :
% 0.32/0.52 ( ( ord_less_int @ Z3 @ X4 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: int] :
% 0.32/0.52 ! [X4: int] :
% 0.32/0.52 ( ( ord_less_int @ Z3 @ X4 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: int] :
% 0.32/0.52 ! [X5: int] :
% 0.32/0.52 ( ( ord_less_int @ Z4 @ X5 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 | ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(2)
% 0.32/0.52 thf(fact_2063_pinf_I3_J,axiom,
% 0.32/0.52 ! [T: real] :
% 0.32/0.52 ? [Z4: real] :
% 0.32/0.52 ! [X5: real] :
% 0.32/0.52 ( ( ord_less_real @ Z4 @ X5 )
% 0.32/0.52 => ( X5 != T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(3)
% 0.32/0.52 thf(fact_2064_pinf_I3_J,axiom,
% 0.32/0.52 ! [T: rat] :
% 0.32/0.52 ? [Z4: rat] :
% 0.32/0.52 ! [X5: rat] :
% 0.32/0.52 ( ( ord_less_rat @ Z4 @ X5 )
% 0.32/0.52 => ( X5 != T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(3)
% 0.32/0.52 thf(fact_2065_pinf_I3_J,axiom,
% 0.32/0.52 ! [T: num] :
% 0.32/0.52 ? [Z4: num] :
% 0.32/0.52 ! [X5: num] :
% 0.32/0.52 ( ( ord_less_num @ Z4 @ X5 )
% 0.32/0.52 => ( X5 != T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(3)
% 0.32/0.52 thf(fact_2066_pinf_I3_J,axiom,
% 0.32/0.52 ! [T: nat] :
% 0.32/0.52 ? [Z4: nat] :
% 0.32/0.52 ! [X5: nat] :
% 0.32/0.52 ( ( ord_less_nat @ Z4 @ X5 )
% 0.32/0.52 => ( X5 != T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(3)
% 0.32/0.52 thf(fact_2067_pinf_I3_J,axiom,
% 0.32/0.52 ! [T: int] :
% 0.32/0.52 ? [Z4: int] :
% 0.32/0.52 ! [X5: int] :
% 0.32/0.52 ( ( ord_less_int @ Z4 @ X5 )
% 0.32/0.52 => ( X5 != T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(3)
% 0.32/0.52 thf(fact_2068_pinf_I4_J,axiom,
% 0.32/0.52 ! [T: real] :
% 0.32/0.52 ? [Z4: real] :
% 0.32/0.52 ! [X5: real] :
% 0.32/0.52 ( ( ord_less_real @ Z4 @ X5 )
% 0.32/0.52 => ( X5 != T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(4)
% 0.32/0.52 thf(fact_2069_pinf_I4_J,axiom,
% 0.32/0.52 ! [T: rat] :
% 0.32/0.52 ? [Z4: rat] :
% 0.32/0.52 ! [X5: rat] :
% 0.32/0.52 ( ( ord_less_rat @ Z4 @ X5 )
% 0.32/0.52 => ( X5 != T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(4)
% 0.32/0.52 thf(fact_2070_pinf_I4_J,axiom,
% 0.32/0.52 ! [T: num] :
% 0.32/0.52 ? [Z4: num] :
% 0.32/0.52 ! [X5: num] :
% 0.32/0.52 ( ( ord_less_num @ Z4 @ X5 )
% 0.32/0.52 => ( X5 != T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(4)
% 0.32/0.52 thf(fact_2071_pinf_I4_J,axiom,
% 0.32/0.52 ! [T: nat] :
% 0.32/0.52 ? [Z4: nat] :
% 0.32/0.52 ! [X5: nat] :
% 0.32/0.52 ( ( ord_less_nat @ Z4 @ X5 )
% 0.32/0.52 => ( X5 != T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(4)
% 0.32/0.52 thf(fact_2072_pinf_I4_J,axiom,
% 0.32/0.52 ! [T: int] :
% 0.32/0.52 ? [Z4: int] :
% 0.32/0.52 ! [X5: int] :
% 0.32/0.52 ( ( ord_less_int @ Z4 @ X5 )
% 0.32/0.52 => ( X5 != T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(4)
% 0.32/0.52 thf(fact_2073_pinf_I5_J,axiom,
% 0.32/0.52 ! [T: real] :
% 0.32/0.52 ? [Z4: real] :
% 0.32/0.52 ! [X5: real] :
% 0.32/0.52 ( ( ord_less_real @ Z4 @ X5 )
% 0.32/0.52 => ~ ( ord_less_real @ X5 @ T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(5)
% 0.32/0.52 thf(fact_2074_pinf_I5_J,axiom,
% 0.32/0.52 ! [T: rat] :
% 0.32/0.52 ? [Z4: rat] :
% 0.32/0.52 ! [X5: rat] :
% 0.32/0.52 ( ( ord_less_rat @ Z4 @ X5 )
% 0.32/0.52 => ~ ( ord_less_rat @ X5 @ T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(5)
% 0.32/0.52 thf(fact_2075_pinf_I5_J,axiom,
% 0.32/0.52 ! [T: num] :
% 0.32/0.52 ? [Z4: num] :
% 0.32/0.52 ! [X5: num] :
% 0.32/0.52 ( ( ord_less_num @ Z4 @ X5 )
% 0.32/0.52 => ~ ( ord_less_num @ X5 @ T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(5)
% 0.32/0.52 thf(fact_2076_pinf_I5_J,axiom,
% 0.32/0.52 ! [T: nat] :
% 0.32/0.52 ? [Z4: nat] :
% 0.32/0.52 ! [X5: nat] :
% 0.32/0.52 ( ( ord_less_nat @ Z4 @ X5 )
% 0.32/0.52 => ~ ( ord_less_nat @ X5 @ T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(5)
% 0.32/0.52 thf(fact_2077_pinf_I5_J,axiom,
% 0.32/0.52 ! [T: int] :
% 0.32/0.52 ? [Z4: int] :
% 0.32/0.52 ! [X5: int] :
% 0.32/0.52 ( ( ord_less_int @ Z4 @ X5 )
% 0.32/0.52 => ~ ( ord_less_int @ X5 @ T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(5)
% 0.32/0.52 thf(fact_2078_pinf_I6_J,axiom,
% 0.32/0.52 ! [T: real] :
% 0.32/0.52 ? [Z4: real] :
% 0.32/0.52 ! [X5: real] :
% 0.32/0.52 ( ( ord_less_real @ Z4 @ X5 )
% 0.32/0.52 => ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(6)
% 0.32/0.52 thf(fact_2079_pinf_I6_J,axiom,
% 0.32/0.52 ! [T: rat] :
% 0.32/0.52 ? [Z4: rat] :
% 0.32/0.52 ! [X5: rat] :
% 0.32/0.52 ( ( ord_less_rat @ Z4 @ X5 )
% 0.32/0.52 => ~ ( ord_less_eq_rat @ X5 @ T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(6)
% 0.32/0.52 thf(fact_2080_pinf_I6_J,axiom,
% 0.32/0.52 ! [T: num] :
% 0.32/0.52 ? [Z4: num] :
% 0.32/0.52 ! [X5: num] :
% 0.32/0.52 ( ( ord_less_num @ Z4 @ X5 )
% 0.32/0.52 => ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(6)
% 0.32/0.52 thf(fact_2081_pinf_I6_J,axiom,
% 0.32/0.52 ! [T: nat] :
% 0.32/0.52 ? [Z4: nat] :
% 0.32/0.52 ! [X5: nat] :
% 0.32/0.52 ( ( ord_less_nat @ Z4 @ X5 )
% 0.32/0.52 => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(6)
% 0.32/0.52 thf(fact_2082_pinf_I6_J,axiom,
% 0.32/0.52 ! [T: int] :
% 0.32/0.52 ? [Z4: int] :
% 0.32/0.52 ! [X5: int] :
% 0.32/0.52 ( ( ord_less_int @ Z4 @ X5 )
% 0.32/0.52 => ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(6)
% 0.32/0.52 thf(fact_2083_pinf_I7_J,axiom,
% 0.32/0.52 ! [T: real] :
% 0.32/0.52 ? [Z4: real] :
% 0.32/0.52 ! [X5: real] :
% 0.32/0.52 ( ( ord_less_real @ Z4 @ X5 )
% 0.32/0.52 => ( ord_less_real @ T @ X5 ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(7)
% 0.32/0.52 thf(fact_2084_pinf_I7_J,axiom,
% 0.32/0.52 ! [T: rat] :
% 0.32/0.52 ? [Z4: rat] :
% 0.32/0.52 ! [X5: rat] :
% 0.32/0.52 ( ( ord_less_rat @ Z4 @ X5 )
% 0.32/0.52 => ( ord_less_rat @ T @ X5 ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(7)
% 0.32/0.52 thf(fact_2085_pinf_I7_J,axiom,
% 0.32/0.52 ! [T: num] :
% 0.32/0.52 ? [Z4: num] :
% 0.32/0.52 ! [X5: num] :
% 0.32/0.52 ( ( ord_less_num @ Z4 @ X5 )
% 0.32/0.52 => ( ord_less_num @ T @ X5 ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(7)
% 0.32/0.52 thf(fact_2086_pinf_I7_J,axiom,
% 0.32/0.52 ! [T: nat] :
% 0.32/0.52 ? [Z4: nat] :
% 0.32/0.52 ! [X5: nat] :
% 0.32/0.52 ( ( ord_less_nat @ Z4 @ X5 )
% 0.32/0.52 => ( ord_less_nat @ T @ X5 ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(7)
% 0.32/0.52 thf(fact_2087_pinf_I7_J,axiom,
% 0.32/0.52 ! [T: int] :
% 0.32/0.52 ? [Z4: int] :
% 0.32/0.52 ! [X5: int] :
% 0.32/0.52 ( ( ord_less_int @ Z4 @ X5 )
% 0.32/0.52 => ( ord_less_int @ T @ X5 ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(7)
% 0.32/0.52 thf(fact_2088_pinf_I8_J,axiom,
% 0.32/0.52 ! [T: real] :
% 0.32/0.52 ? [Z4: real] :
% 0.32/0.52 ! [X5: real] :
% 0.32/0.52 ( ( ord_less_real @ Z4 @ X5 )
% 0.32/0.52 => ( ord_less_eq_real @ T @ X5 ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(8)
% 0.32/0.52 thf(fact_2089_pinf_I8_J,axiom,
% 0.32/0.52 ! [T: rat] :
% 0.32/0.52 ? [Z4: rat] :
% 0.32/0.52 ! [X5: rat] :
% 0.32/0.52 ( ( ord_less_rat @ Z4 @ X5 )
% 0.32/0.52 => ( ord_less_eq_rat @ T @ X5 ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(8)
% 0.32/0.52 thf(fact_2090_pinf_I8_J,axiom,
% 0.32/0.52 ! [T: num] :
% 0.32/0.52 ? [Z4: num] :
% 0.32/0.52 ! [X5: num] :
% 0.32/0.52 ( ( ord_less_num @ Z4 @ X5 )
% 0.32/0.52 => ( ord_less_eq_num @ T @ X5 ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(8)
% 0.32/0.52 thf(fact_2091_pinf_I8_J,axiom,
% 0.32/0.52 ! [T: nat] :
% 0.32/0.52 ? [Z4: nat] :
% 0.32/0.52 ! [X5: nat] :
% 0.32/0.52 ( ( ord_less_nat @ Z4 @ X5 )
% 0.32/0.52 => ( ord_less_eq_nat @ T @ X5 ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(8)
% 0.32/0.52 thf(fact_2092_pinf_I8_J,axiom,
% 0.32/0.52 ! [T: int] :
% 0.32/0.52 ? [Z4: int] :
% 0.32/0.52 ! [X5: int] :
% 0.32/0.52 ( ( ord_less_int @ Z4 @ X5 )
% 0.32/0.52 => ( ord_less_eq_int @ T @ X5 ) ) ).
% 0.32/0.52
% 0.32/0.52 % pinf(8)
% 0.32/0.52 thf(fact_2093_minf_I1_J,axiom,
% 0.32/0.52 ! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
% 0.32/0.52 ( ? [Z3: real] :
% 0.32/0.52 ! [X4: real] :
% 0.32/0.52 ( ( ord_less_real @ X4 @ Z3 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: real] :
% 0.32/0.52 ! [X4: real] :
% 0.32/0.52 ( ( ord_less_real @ X4 @ Z3 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: real] :
% 0.32/0.52 ! [X5: real] :
% 0.32/0.52 ( ( ord_less_real @ X5 @ Z4 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 & ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % minf(1)
% 0.32/0.52 thf(fact_2094_minf_I1_J,axiom,
% 0.32/0.52 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
% 0.32/0.52 ( ? [Z3: rat] :
% 0.32/0.52 ! [X4: rat] :
% 0.32/0.52 ( ( ord_less_rat @ X4 @ Z3 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: rat] :
% 0.32/0.52 ! [X4: rat] :
% 0.32/0.52 ( ( ord_less_rat @ X4 @ Z3 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: rat] :
% 0.32/0.52 ! [X5: rat] :
% 0.32/0.52 ( ( ord_less_rat @ X5 @ Z4 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 & ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % minf(1)
% 0.32/0.52 thf(fact_2095_minf_I1_J,axiom,
% 0.32/0.52 ! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
% 0.32/0.52 ( ? [Z3: num] :
% 0.32/0.52 ! [X4: num] :
% 0.32/0.52 ( ( ord_less_num @ X4 @ Z3 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: num] :
% 0.32/0.52 ! [X4: num] :
% 0.32/0.52 ( ( ord_less_num @ X4 @ Z3 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: num] :
% 0.32/0.52 ! [X5: num] :
% 0.32/0.52 ( ( ord_less_num @ X5 @ Z4 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 & ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % minf(1)
% 0.32/0.52 thf(fact_2096_minf_I1_J,axiom,
% 0.32/0.52 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
% 0.32/0.52 ( ? [Z3: nat] :
% 0.32/0.52 ! [X4: nat] :
% 0.32/0.52 ( ( ord_less_nat @ X4 @ Z3 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: nat] :
% 0.32/0.52 ! [X4: nat] :
% 0.32/0.52 ( ( ord_less_nat @ X4 @ Z3 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: nat] :
% 0.32/0.52 ! [X5: nat] :
% 0.32/0.52 ( ( ord_less_nat @ X5 @ Z4 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 & ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % minf(1)
% 0.32/0.52 thf(fact_2097_minf_I1_J,axiom,
% 0.32/0.52 ! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
% 0.32/0.52 ( ? [Z3: int] :
% 0.32/0.52 ! [X4: int] :
% 0.32/0.52 ( ( ord_less_int @ X4 @ Z3 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: int] :
% 0.32/0.52 ! [X4: int] :
% 0.32/0.52 ( ( ord_less_int @ X4 @ Z3 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: int] :
% 0.32/0.52 ! [X5: int] :
% 0.32/0.52 ( ( ord_less_int @ X5 @ Z4 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 & ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % minf(1)
% 0.32/0.52 thf(fact_2098_minf_I2_J,axiom,
% 0.32/0.52 ! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
% 0.32/0.52 ( ? [Z3: real] :
% 0.32/0.52 ! [X4: real] :
% 0.32/0.52 ( ( ord_less_real @ X4 @ Z3 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: real] :
% 0.32/0.52 ! [X4: real] :
% 0.32/0.52 ( ( ord_less_real @ X4 @ Z3 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: real] :
% 0.32/0.52 ! [X5: real] :
% 0.32/0.52 ( ( ord_less_real @ X5 @ Z4 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 | ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % minf(2)
% 0.32/0.52 thf(fact_2099_minf_I2_J,axiom,
% 0.32/0.52 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
% 0.32/0.52 ( ? [Z3: rat] :
% 0.32/0.52 ! [X4: rat] :
% 0.32/0.52 ( ( ord_less_rat @ X4 @ Z3 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: rat] :
% 0.32/0.52 ! [X4: rat] :
% 0.32/0.52 ( ( ord_less_rat @ X4 @ Z3 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: rat] :
% 0.32/0.52 ! [X5: rat] :
% 0.32/0.52 ( ( ord_less_rat @ X5 @ Z4 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 | ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % minf(2)
% 0.32/0.52 thf(fact_2100_minf_I2_J,axiom,
% 0.32/0.52 ! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
% 0.32/0.52 ( ? [Z3: num] :
% 0.32/0.52 ! [X4: num] :
% 0.32/0.52 ( ( ord_less_num @ X4 @ Z3 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: num] :
% 0.32/0.52 ! [X4: num] :
% 0.32/0.52 ( ( ord_less_num @ X4 @ Z3 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: num] :
% 0.32/0.52 ! [X5: num] :
% 0.32/0.52 ( ( ord_less_num @ X5 @ Z4 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 | ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % minf(2)
% 0.32/0.52 thf(fact_2101_minf_I2_J,axiom,
% 0.32/0.52 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
% 0.32/0.52 ( ? [Z3: nat] :
% 0.32/0.52 ! [X4: nat] :
% 0.32/0.52 ( ( ord_less_nat @ X4 @ Z3 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: nat] :
% 0.32/0.52 ! [X4: nat] :
% 0.32/0.52 ( ( ord_less_nat @ X4 @ Z3 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: nat] :
% 0.32/0.52 ! [X5: nat] :
% 0.32/0.52 ( ( ord_less_nat @ X5 @ Z4 )
% 0.32/0.52 => ( ( ( P @ X5 )
% 0.32/0.52 | ( Q @ X5 ) )
% 0.32/0.52 = ( ( P4 @ X5 )
% 0.32/0.52 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % minf(2)
% 0.32/0.52 thf(fact_2102_minf_I2_J,axiom,
% 0.32/0.52 ! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
% 0.32/0.52 ( ? [Z3: int] :
% 0.32/0.52 ! [X4: int] :
% 0.32/0.52 ( ( ord_less_int @ X4 @ Z3 )
% 0.32/0.52 => ( ( P @ X4 )
% 0.32/0.52 = ( P4 @ X4 ) ) )
% 0.32/0.52 => ( ? [Z3: int] :
% 0.32/0.52 ! [X4: int] :
% 0.32/0.52 ( ( ord_less_int @ X4 @ Z3 )
% 0.32/0.52 => ( ( Q @ X4 )
% 0.32/0.52 = ( Q3 @ X4 ) ) )
% 0.32/0.52 => ? [Z4: int] :
% 0.32/0.53 ! [X5: int] :
% 0.32/0.53 ( ( ord_less_int @ X5 @ Z4 )
% 0.32/0.53 => ( ( ( P @ X5 )
% 0.32/0.53 | ( Q @ X5 ) )
% 0.32/0.53 = ( ( P4 @ X5 )
% 0.32/0.53 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(2)
% 0.32/0.53 thf(fact_2103_minf_I3_J,axiom,
% 0.32/0.53 ! [T: real] :
% 0.32/0.53 ? [Z4: real] :
% 0.32/0.53 ! [X5: real] :
% 0.32/0.53 ( ( ord_less_real @ X5 @ Z4 )
% 0.32/0.53 => ( X5 != T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(3)
% 0.32/0.53 thf(fact_2104_minf_I3_J,axiom,
% 0.32/0.53 ! [T: rat] :
% 0.32/0.53 ? [Z4: rat] :
% 0.32/0.53 ! [X5: rat] :
% 0.32/0.53 ( ( ord_less_rat @ X5 @ Z4 )
% 0.32/0.53 => ( X5 != T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(3)
% 0.32/0.53 thf(fact_2105_minf_I3_J,axiom,
% 0.32/0.53 ! [T: num] :
% 0.32/0.53 ? [Z4: num] :
% 0.32/0.53 ! [X5: num] :
% 0.32/0.53 ( ( ord_less_num @ X5 @ Z4 )
% 0.32/0.53 => ( X5 != T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(3)
% 0.32/0.53 thf(fact_2106_minf_I3_J,axiom,
% 0.32/0.53 ! [T: nat] :
% 0.32/0.53 ? [Z4: nat] :
% 0.32/0.53 ! [X5: nat] :
% 0.32/0.53 ( ( ord_less_nat @ X5 @ Z4 )
% 0.32/0.53 => ( X5 != T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(3)
% 0.32/0.53 thf(fact_2107_minf_I3_J,axiom,
% 0.32/0.53 ! [T: int] :
% 0.32/0.53 ? [Z4: int] :
% 0.32/0.53 ! [X5: int] :
% 0.32/0.53 ( ( ord_less_int @ X5 @ Z4 )
% 0.32/0.53 => ( X5 != T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(3)
% 0.32/0.53 thf(fact_2108_minf_I4_J,axiom,
% 0.32/0.53 ! [T: real] :
% 0.32/0.53 ? [Z4: real] :
% 0.32/0.53 ! [X5: real] :
% 0.32/0.53 ( ( ord_less_real @ X5 @ Z4 )
% 0.32/0.53 => ( X5 != T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(4)
% 0.32/0.53 thf(fact_2109_minf_I4_J,axiom,
% 0.32/0.53 ! [T: rat] :
% 0.32/0.53 ? [Z4: rat] :
% 0.32/0.53 ! [X5: rat] :
% 0.32/0.53 ( ( ord_less_rat @ X5 @ Z4 )
% 0.32/0.53 => ( X5 != T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(4)
% 0.32/0.53 thf(fact_2110_minf_I4_J,axiom,
% 0.32/0.53 ! [T: num] :
% 0.32/0.53 ? [Z4: num] :
% 0.32/0.53 ! [X5: num] :
% 0.32/0.53 ( ( ord_less_num @ X5 @ Z4 )
% 0.32/0.53 => ( X5 != T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(4)
% 0.32/0.53 thf(fact_2111_minf_I4_J,axiom,
% 0.32/0.53 ! [T: nat] :
% 0.32/0.53 ? [Z4: nat] :
% 0.32/0.53 ! [X5: nat] :
% 0.32/0.53 ( ( ord_less_nat @ X5 @ Z4 )
% 0.32/0.53 => ( X5 != T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(4)
% 0.32/0.53 thf(fact_2112_minf_I4_J,axiom,
% 0.32/0.53 ! [T: int] :
% 0.32/0.53 ? [Z4: int] :
% 0.32/0.53 ! [X5: int] :
% 0.32/0.53 ( ( ord_less_int @ X5 @ Z4 )
% 0.32/0.53 => ( X5 != T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(4)
% 0.32/0.53 thf(fact_2113_minf_I5_J,axiom,
% 0.32/0.53 ! [T: real] :
% 0.32/0.53 ? [Z4: real] :
% 0.32/0.53 ! [X5: real] :
% 0.32/0.53 ( ( ord_less_real @ X5 @ Z4 )
% 0.32/0.53 => ( ord_less_real @ X5 @ T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(5)
% 0.32/0.53 thf(fact_2114_minf_I5_J,axiom,
% 0.32/0.53 ! [T: rat] :
% 0.32/0.53 ? [Z4: rat] :
% 0.32/0.53 ! [X5: rat] :
% 0.32/0.53 ( ( ord_less_rat @ X5 @ Z4 )
% 0.32/0.53 => ( ord_less_rat @ X5 @ T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(5)
% 0.32/0.53 thf(fact_2115_minf_I5_J,axiom,
% 0.32/0.53 ! [T: num] :
% 0.32/0.53 ? [Z4: num] :
% 0.32/0.53 ! [X5: num] :
% 0.32/0.53 ( ( ord_less_num @ X5 @ Z4 )
% 0.32/0.53 => ( ord_less_num @ X5 @ T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(5)
% 0.32/0.53 thf(fact_2116_minf_I5_J,axiom,
% 0.32/0.53 ! [T: nat] :
% 0.32/0.53 ? [Z4: nat] :
% 0.32/0.53 ! [X5: nat] :
% 0.32/0.53 ( ( ord_less_nat @ X5 @ Z4 )
% 0.32/0.53 => ( ord_less_nat @ X5 @ T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(5)
% 0.32/0.53 thf(fact_2117_minf_I5_J,axiom,
% 0.32/0.53 ! [T: int] :
% 0.32/0.53 ? [Z4: int] :
% 0.32/0.53 ! [X5: int] :
% 0.32/0.53 ( ( ord_less_int @ X5 @ Z4 )
% 0.32/0.53 => ( ord_less_int @ X5 @ T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(5)
% 0.32/0.53 thf(fact_2118_minf_I6_J,axiom,
% 0.32/0.53 ! [T: real] :
% 0.32/0.53 ? [Z4: real] :
% 0.32/0.53 ! [X5: real] :
% 0.32/0.53 ( ( ord_less_real @ X5 @ Z4 )
% 0.32/0.53 => ( ord_less_eq_real @ X5 @ T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(6)
% 0.32/0.53 thf(fact_2119_minf_I6_J,axiom,
% 0.32/0.53 ! [T: rat] :
% 0.32/0.53 ? [Z4: rat] :
% 0.32/0.53 ! [X5: rat] :
% 0.32/0.53 ( ( ord_less_rat @ X5 @ Z4 )
% 0.32/0.53 => ( ord_less_eq_rat @ X5 @ T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(6)
% 0.32/0.53 thf(fact_2120_minf_I6_J,axiom,
% 0.32/0.53 ! [T: num] :
% 0.32/0.53 ? [Z4: num] :
% 0.32/0.53 ! [X5: num] :
% 0.32/0.53 ( ( ord_less_num @ X5 @ Z4 )
% 0.32/0.53 => ( ord_less_eq_num @ X5 @ T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(6)
% 0.32/0.53 thf(fact_2121_minf_I6_J,axiom,
% 0.32/0.53 ! [T: nat] :
% 0.32/0.53 ? [Z4: nat] :
% 0.32/0.53 ! [X5: nat] :
% 0.32/0.53 ( ( ord_less_nat @ X5 @ Z4 )
% 0.32/0.53 => ( ord_less_eq_nat @ X5 @ T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(6)
% 0.32/0.53 thf(fact_2122_minf_I6_J,axiom,
% 0.32/0.53 ! [T: int] :
% 0.32/0.53 ? [Z4: int] :
% 0.32/0.53 ! [X5: int] :
% 0.32/0.53 ( ( ord_less_int @ X5 @ Z4 )
% 0.32/0.53 => ( ord_less_eq_int @ X5 @ T ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(6)
% 0.32/0.53 thf(fact_2123_minf_I7_J,axiom,
% 0.32/0.53 ! [T: real] :
% 0.32/0.53 ? [Z4: real] :
% 0.32/0.53 ! [X5: real] :
% 0.32/0.53 ( ( ord_less_real @ X5 @ Z4 )
% 0.32/0.53 => ~ ( ord_less_real @ T @ X5 ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(7)
% 0.32/0.53 thf(fact_2124_minf_I7_J,axiom,
% 0.32/0.53 ! [T: rat] :
% 0.32/0.53 ? [Z4: rat] :
% 0.32/0.53 ! [X5: rat] :
% 0.32/0.53 ( ( ord_less_rat @ X5 @ Z4 )
% 0.32/0.53 => ~ ( ord_less_rat @ T @ X5 ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(7)
% 0.32/0.53 thf(fact_2125_minf_I7_J,axiom,
% 0.32/0.53 ! [T: num] :
% 0.32/0.53 ? [Z4: num] :
% 0.32/0.53 ! [X5: num] :
% 0.32/0.53 ( ( ord_less_num @ X5 @ Z4 )
% 0.32/0.53 => ~ ( ord_less_num @ T @ X5 ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(7)
% 0.32/0.53 thf(fact_2126_minf_I7_J,axiom,
% 0.32/0.53 ! [T: nat] :
% 0.32/0.53 ? [Z4: nat] :
% 0.32/0.53 ! [X5: nat] :
% 0.32/0.53 ( ( ord_less_nat @ X5 @ Z4 )
% 0.32/0.53 => ~ ( ord_less_nat @ T @ X5 ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(7)
% 0.32/0.53 thf(fact_2127_minf_I7_J,axiom,
% 0.32/0.53 ! [T: int] :
% 0.32/0.53 ? [Z4: int] :
% 0.32/0.53 ! [X5: int] :
% 0.32/0.53 ( ( ord_less_int @ X5 @ Z4 )
% 0.32/0.53 => ~ ( ord_less_int @ T @ X5 ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(7)
% 0.32/0.53 thf(fact_2128_minf_I8_J,axiom,
% 0.32/0.53 ! [T: real] :
% 0.32/0.53 ? [Z4: real] :
% 0.32/0.53 ! [X5: real] :
% 0.32/0.53 ( ( ord_less_real @ X5 @ Z4 )
% 0.32/0.53 => ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(8)
% 0.32/0.53 thf(fact_2129_minf_I8_J,axiom,
% 0.32/0.53 ! [T: rat] :
% 0.32/0.53 ? [Z4: rat] :
% 0.32/0.53 ! [X5: rat] :
% 0.32/0.53 ( ( ord_less_rat @ X5 @ Z4 )
% 0.32/0.53 => ~ ( ord_less_eq_rat @ T @ X5 ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(8)
% 0.32/0.53 thf(fact_2130_minf_I8_J,axiom,
% 0.32/0.53 ! [T: num] :
% 0.32/0.53 ? [Z4: num] :
% 0.32/0.53 ! [X5: num] :
% 0.32/0.53 ( ( ord_less_num @ X5 @ Z4 )
% 0.32/0.53 => ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(8)
% 0.32/0.53 thf(fact_2131_minf_I8_J,axiom,
% 0.32/0.53 ! [T: nat] :
% 0.32/0.53 ? [Z4: nat] :
% 0.32/0.53 ! [X5: nat] :
% 0.32/0.53 ( ( ord_less_nat @ X5 @ Z4 )
% 0.32/0.53 => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(8)
% 0.32/0.53 thf(fact_2132_minf_I8_J,axiom,
% 0.32/0.53 ! [T: int] :
% 0.32/0.53 ? [Z4: int] :
% 0.32/0.53 ! [X5: int] :
% 0.32/0.53 ( ( ord_less_int @ X5 @ Z4 )
% 0.32/0.53 => ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% 0.32/0.53
% 0.32/0.53 % minf(8)
% 0.32/0.53 thf(fact_2133_add__less__le__mono,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.53 ( ( ord_less_real @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_real @ C @ D )
% 0.32/0.53 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_less_le_mono
% 0.32/0.53 thf(fact_2134_add__less__le__mono,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.53 ( ( ord_less_rat @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_rat @ C @ D )
% 0.32/0.53 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_less_le_mono
% 0.32/0.53 thf(fact_2135_add__less__le__mono,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.53 ( ( ord_less_nat @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_nat @ C @ D )
% 0.32/0.53 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_less_le_mono
% 0.32/0.53 thf(fact_2136_add__less__le__mono,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.53 ( ( ord_less_int @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_int @ C @ D )
% 0.32/0.53 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_less_le_mono
% 0.32/0.53 thf(fact_2137_add__le__less__mono,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.53 => ( ( ord_less_real @ C @ D )
% 0.32/0.53 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_le_less_mono
% 0.32/0.53 thf(fact_2138_add__le__less__mono,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.53 => ( ( ord_less_rat @ C @ D )
% 0.32/0.53 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_le_less_mono
% 0.32/0.53 thf(fact_2139_add__le__less__mono,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.53 => ( ( ord_less_nat @ C @ D )
% 0.32/0.53 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_le_less_mono
% 0.32/0.53 thf(fact_2140_add__le__less__mono,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.53 => ( ( ord_less_int @ C @ D )
% 0.32/0.53 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_le_less_mono
% 0.32/0.53 thf(fact_2141_add__mono__thms__linordered__field_I3_J,axiom,
% 0.32/0.53 ! [I: real,J: real,K: real,L: real] :
% 0.32/0.53 ( ( ( ord_less_real @ I @ J )
% 0.32/0.53 & ( ord_less_eq_real @ K @ L ) )
% 0.32/0.53 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(3)
% 0.32/0.53 thf(fact_2142_add__mono__thms__linordered__field_I3_J,axiom,
% 0.32/0.53 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.32/0.53 ( ( ( ord_less_rat @ I @ J )
% 0.32/0.53 & ( ord_less_eq_rat @ K @ L ) )
% 0.32/0.53 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(3)
% 0.32/0.53 thf(fact_2143_add__mono__thms__linordered__field_I3_J,axiom,
% 0.32/0.53 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.53 ( ( ( ord_less_nat @ I @ J )
% 0.32/0.53 & ( ord_less_eq_nat @ K @ L ) )
% 0.32/0.53 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(3)
% 0.32/0.53 thf(fact_2144_add__mono__thms__linordered__field_I3_J,axiom,
% 0.32/0.53 ! [I: int,J: int,K: int,L: int] :
% 0.32/0.53 ( ( ( ord_less_int @ I @ J )
% 0.32/0.53 & ( ord_less_eq_int @ K @ L ) )
% 0.32/0.53 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(3)
% 0.32/0.53 thf(fact_2145_add__mono__thms__linordered__field_I4_J,axiom,
% 0.32/0.53 ! [I: real,J: real,K: real,L: real] :
% 0.32/0.53 ( ( ( ord_less_eq_real @ I @ J )
% 0.32/0.53 & ( ord_less_real @ K @ L ) )
% 0.32/0.53 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(4)
% 0.32/0.53 thf(fact_2146_add__mono__thms__linordered__field_I4_J,axiom,
% 0.32/0.53 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.32/0.53 ( ( ( ord_less_eq_rat @ I @ J )
% 0.32/0.53 & ( ord_less_rat @ K @ L ) )
% 0.32/0.53 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(4)
% 0.32/0.53 thf(fact_2147_add__mono__thms__linordered__field_I4_J,axiom,
% 0.32/0.53 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.53 ( ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.53 & ( ord_less_nat @ K @ L ) )
% 0.32/0.53 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(4)
% 0.32/0.53 thf(fact_2148_add__mono__thms__linordered__field_I4_J,axiom,
% 0.32/0.53 ! [I: int,J: int,K: int,L: int] :
% 0.32/0.53 ( ( ( ord_less_eq_int @ I @ J )
% 0.32/0.53 & ( ord_less_int @ K @ L ) )
% 0.32/0.53 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(4)
% 0.32/0.53 thf(fact_2149_lift__Suc__mono__less,axiom,
% 0.32/0.53 ! [F: nat > real,N: nat,N4: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.53 => ( ( ord_less_nat @ N @ N4 )
% 0.32/0.53 => ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_mono_less
% 0.32/0.53 thf(fact_2150_lift__Suc__mono__less,axiom,
% 0.32/0.53 ! [F: nat > rat,N: nat,N4: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.53 => ( ( ord_less_nat @ N @ N4 )
% 0.32/0.53 => ( ord_less_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_mono_less
% 0.32/0.53 thf(fact_2151_lift__Suc__mono__less,axiom,
% 0.32/0.53 ! [F: nat > num,N: nat,N4: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.53 => ( ( ord_less_nat @ N @ N4 )
% 0.32/0.53 => ( ord_less_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_mono_less
% 0.32/0.53 thf(fact_2152_lift__Suc__mono__less,axiom,
% 0.32/0.53 ! [F: nat > nat,N: nat,N4: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.53 => ( ( ord_less_nat @ N @ N4 )
% 0.32/0.53 => ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_mono_less
% 0.32/0.53 thf(fact_2153_lift__Suc__mono__less,axiom,
% 0.32/0.53 ! [F: nat > int,N: nat,N4: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.53 => ( ( ord_less_nat @ N @ N4 )
% 0.32/0.53 => ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_mono_less
% 0.32/0.53 thf(fact_2154_lift__Suc__mono__less__iff,axiom,
% 0.32/0.53 ! [F: nat > real,N: nat,M: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.53 => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 0.32/0.53 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_mono_less_iff
% 0.32/0.53 thf(fact_2155_lift__Suc__mono__less__iff,axiom,
% 0.32/0.53 ! [F: nat > rat,N: nat,M: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.53 => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 0.32/0.53 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_mono_less_iff
% 0.32/0.53 thf(fact_2156_lift__Suc__mono__less__iff,axiom,
% 0.32/0.53 ! [F: nat > num,N: nat,M: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.53 => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 0.32/0.53 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_mono_less_iff
% 0.32/0.53 thf(fact_2157_lift__Suc__mono__less__iff,axiom,
% 0.32/0.53 ! [F: nat > nat,N: nat,M: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.53 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 0.32/0.53 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_mono_less_iff
% 0.32/0.53 thf(fact_2158_lift__Suc__mono__less__iff,axiom,
% 0.32/0.53 ! [F: nat > int,N: nat,M: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.53 => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 0.32/0.53 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_mono_less_iff
% 0.32/0.53 thf(fact_2159_lift__Suc__mono__le,axiom,
% 0.32/0.53 ! [F: nat > set_nat,N: nat,N4: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/0.53 => ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_mono_le
% 0.32/0.53 thf(fact_2160_lift__Suc__mono__le,axiom,
% 0.32/0.53 ! [F: nat > rat,N: nat,N4: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/0.53 => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_mono_le
% 0.32/0.53 thf(fact_2161_lift__Suc__mono__le,axiom,
% 0.32/0.53 ! [F: nat > num,N: nat,N4: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/0.53 => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_mono_le
% 0.32/0.53 thf(fact_2162_lift__Suc__mono__le,axiom,
% 0.32/0.53 ! [F: nat > nat,N: nat,N4: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/0.53 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_mono_le
% 0.32/0.53 thf(fact_2163_lift__Suc__mono__le,axiom,
% 0.32/0.53 ! [F: nat > int,N: nat,N4: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/0.53 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_mono_le
% 0.32/0.53 thf(fact_2164_lift__Suc__antimono__le,axiom,
% 0.32/0.53 ! [F: nat > set_nat,N: nat,N4: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/0.53 => ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_antimono_le
% 0.32/0.53 thf(fact_2165_lift__Suc__antimono__le,axiom,
% 0.32/0.53 ! [F: nat > rat,N: nat,N4: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/0.53 => ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_antimono_le
% 0.32/0.53 thf(fact_2166_lift__Suc__antimono__le,axiom,
% 0.32/0.53 ! [F: nat > num,N: nat,N4: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/0.53 => ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_antimono_le
% 0.32/0.53 thf(fact_2167_lift__Suc__antimono__le,axiom,
% 0.32/0.53 ! [F: nat > nat,N: nat,N4: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/0.53 => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_antimono_le
% 0.32/0.53 thf(fact_2168_lift__Suc__antimono__le,axiom,
% 0.32/0.53 ! [F: nat > int,N: nat,N4: nat] :
% 0.32/0.53 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.32/0.53 => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % lift_Suc_antimono_le
% 0.32/0.53 thf(fact_2169_Suc__leI,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ M @ N )
% 0.32/0.53 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % Suc_leI
% 0.32/0.53 thf(fact_2170_Suc__le__eq,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.32/0.53 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % Suc_le_eq
% 0.32/0.53 thf(fact_2171_dec__induct,axiom,
% 0.32/0.53 ! [I: nat,J: nat,P: nat > $o] :
% 0.32/0.53 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.53 => ( ( P @ I )
% 0.32/0.53 => ( ! [N2: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ I @ N2 )
% 0.32/0.53 => ( ( ord_less_nat @ N2 @ J )
% 0.32/0.53 => ( ( P @ N2 )
% 0.32/0.53 => ( P @ ( suc @ N2 ) ) ) ) )
% 0.32/0.53 => ( P @ J ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % dec_induct
% 0.32/0.53 thf(fact_2172_inc__induct,axiom,
% 0.32/0.53 ! [I: nat,J: nat,P: nat > $o] :
% 0.32/0.53 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.53 => ( ( P @ J )
% 0.32/0.53 => ( ! [N2: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ I @ N2 )
% 0.32/0.53 => ( ( ord_less_nat @ N2 @ J )
% 0.32/0.53 => ( ( P @ ( suc @ N2 ) )
% 0.32/0.53 => ( P @ N2 ) ) ) )
% 0.32/0.53 => ( P @ I ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % inc_induct
% 0.32/0.53 thf(fact_2173_Suc__le__lessD,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.32/0.53 => ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % Suc_le_lessD
% 0.32/0.53 thf(fact_2174_le__less__Suc__eq,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.53 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.32/0.53 = ( N = M ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_less_Suc_eq
% 0.32/0.53 thf(fact_2175_less__Suc__eq__le,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.32/0.53 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_Suc_eq_le
% 0.32/0.53 thf(fact_2176_less__eq__Suc__le,axiom,
% 0.32/0.53 ( ord_less_nat
% 0.32/0.53 = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_eq_Suc_le
% 0.32/0.53 thf(fact_2177_le__imp__less__Suc,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.53 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_imp_less_Suc
% 0.32/0.53 thf(fact_2178_ex__least__nat__le,axiom,
% 0.32/0.53 ! [P: nat > $o,N: nat] :
% 0.32/0.53 ( ( P @ N )
% 0.32/0.53 => ( ~ ( P @ zero_zero_nat )
% 0.32/0.53 => ? [K3: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ K3 @ N )
% 0.32/0.53 & ! [I3: nat] :
% 0.32/0.53 ( ( ord_less_nat @ I3 @ K3 )
% 0.32/0.53 => ~ ( P @ I3 ) )
% 0.32/0.53 & ( P @ K3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % ex_least_nat_le
% 0.32/0.53 thf(fact_2179_of__nat__less__imp__less,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.53 => ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % of_nat_less_imp_less
% 0.32/0.53 thf(fact_2180_of__nat__less__imp__less,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.32/0.53 => ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % of_nat_less_imp_less
% 0.32/0.53 thf(fact_2181_of__nat__less__imp__less,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.53 => ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % of_nat_less_imp_less
% 0.32/0.53 thf(fact_2182_of__nat__less__imp__less,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.53 => ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % of_nat_less_imp_less
% 0.32/0.53 thf(fact_2183_less__imp__of__nat__less,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ M @ N )
% 0.32/0.53 => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_imp_of_nat_less
% 0.32/0.53 thf(fact_2184_less__imp__of__nat__less,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ M @ N )
% 0.32/0.53 => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_imp_of_nat_less
% 0.32/0.53 thf(fact_2185_less__imp__of__nat__less,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ M @ N )
% 0.32/0.53 => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_imp_of_nat_less
% 0.32/0.53 thf(fact_2186_less__imp__of__nat__less,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ M @ N )
% 0.32/0.53 => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_imp_of_nat_less
% 0.32/0.53 thf(fact_2187_of__nat__mono,axiom,
% 0.32/0.53 ! [I: nat,J: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.53 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % of_nat_mono
% 0.32/0.53 thf(fact_2188_of__nat__mono,axiom,
% 0.32/0.53 ! [I: nat,J: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.53 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % of_nat_mono
% 0.32/0.53 thf(fact_2189_of__nat__mono,axiom,
% 0.32/0.53 ! [I: nat,J: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.53 => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % of_nat_mono
% 0.32/0.53 thf(fact_2190_of__nat__mono,axiom,
% 0.32/0.53 ! [I: nat,J: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.53 => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % of_nat_mono
% 0.32/0.53 thf(fact_2191_mono__nat__linear__lb,axiom,
% 0.32/0.53 ! [F: nat > nat,M: nat,K: nat] :
% 0.32/0.53 ( ! [M2: nat,N2: nat] :
% 0.32/0.53 ( ( ord_less_nat @ M2 @ N2 )
% 0.32/0.53 => ( ord_less_nat @ ( F @ M2 ) @ ( F @ N2 ) ) )
% 0.32/0.53 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mono_nat_linear_lb
% 0.32/0.53 thf(fact_2192_diff__less__mono,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat] :
% 0.32/0.53 ( ( ord_less_nat @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_nat @ C @ A )
% 0.32/0.53 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_less_mono
% 0.32/0.53 thf(fact_2193_less__diff__iff,axiom,
% 0.32/0.53 ! [K: nat,M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ K @ M )
% 0.32/0.53 => ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.53 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.32/0.53 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_diff_iff
% 0.32/0.53 thf(fact_2194_nat__int__comparison_I2_J,axiom,
% 0.32/0.53 ( ord_less_nat
% 0.32/0.53 = ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % nat_int_comparison(2)
% 0.32/0.53 thf(fact_2195_zle__int,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.53 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % zle_int
% 0.32/0.53 thf(fact_2196_nat__int__comparison_I3_J,axiom,
% 0.32/0.53 ( ord_less_eq_nat
% 0.32/0.53 = ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % nat_int_comparison(3)
% 0.32/0.53 thf(fact_2197_power__le__imp__le__exp,axiom,
% 0.32/0.53 ! [A: real,M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.53 => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 0.32/0.53 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_le_imp_le_exp
% 0.32/0.53 thf(fact_2198_power__le__imp__le__exp,axiom,
% 0.32/0.53 ! [A: rat,M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.53 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 0.32/0.53 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_le_imp_le_exp
% 0.32/0.53 thf(fact_2199_power__le__imp__le__exp,axiom,
% 0.32/0.53 ! [A: nat,M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.53 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.32/0.53 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_le_imp_le_exp
% 0.32/0.53 thf(fact_2200_power__le__imp__le__exp,axiom,
% 0.32/0.53 ! [A: int,M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.53 => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.32/0.53 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_le_imp_le_exp
% 0.32/0.53 thf(fact_2201_subset__decode__imp__le,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 0.32/0.53 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % subset_decode_imp_le
% 0.32/0.53 thf(fact_2202_power__strict__mono,axiom,
% 0.32/0.53 ! [A: real,B: real,N: nat] :
% 0.32/0.53 ( ( ord_less_real @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_strict_mono
% 0.32/0.53 thf(fact_2203_power__strict__mono,axiom,
% 0.32/0.53 ! [A: rat,B: rat,N: nat] :
% 0.32/0.53 ( ( ord_less_rat @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_strict_mono
% 0.32/0.53 thf(fact_2204_power__strict__mono,axiom,
% 0.32/0.53 ! [A: nat,B: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_strict_mono
% 0.32/0.53 thf(fact_2205_power__strict__mono,axiom,
% 0.32/0.53 ! [A: int,B: int,N: nat] :
% 0.32/0.53 ( ( ord_less_int @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_strict_mono
% 0.32/0.53 thf(fact_2206_mult__le__cancel__left,axiom,
% 0.32/0.53 ! [C: real,A: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_eq_real @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left
% 0.32/0.53 thf(fact_2207_mult__le__cancel__left,axiom,
% 0.32/0.53 ! [C: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left
% 0.32/0.53 thf(fact_2208_mult__le__cancel__left,axiom,
% 0.32/0.53 ! [C: int,A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_eq_int @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.53 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left
% 0.32/0.53 thf(fact_2209_mult__le__cancel__right,axiom,
% 0.32/0.53 ! [A: real,C: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_eq_real @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_right
% 0.32/0.53 thf(fact_2210_mult__le__cancel__right,axiom,
% 0.32/0.53 ! [A: rat,C: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_right
% 0.32/0.53 thf(fact_2211_mult__le__cancel__right,axiom,
% 0.32/0.53 ! [A: int,C: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_eq_int @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.53 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_right
% 0.32/0.53 thf(fact_2212_mult__left__less__imp__less,axiom,
% 0.32/0.53 ! [C: real,A: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_less_imp_less
% 0.32/0.53 thf(fact_2213_mult__left__less__imp__less,axiom,
% 0.32/0.53 ! [C: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_less_imp_less
% 0.32/0.53 thf(fact_2214_mult__left__less__imp__less,axiom,
% 0.32/0.53 ! [C: nat,A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.53 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_less_imp_less
% 0.32/0.53 thf(fact_2215_mult__left__less__imp__less,axiom,
% 0.32/0.53 ! [C: int,A: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_less_imp_less
% 0.32/0.53 thf(fact_2216_mult__strict__mono,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.53 ( ( ord_less_real @ A @ B )
% 0.32/0.53 => ( ( ord_less_real @ C @ D )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_strict_mono
% 0.32/0.53 thf(fact_2217_mult__strict__mono,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.53 ( ( ord_less_rat @ A @ B )
% 0.32/0.53 => ( ( ord_less_rat @ C @ D )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_strict_mono
% 0.32/0.53 thf(fact_2218_mult__strict__mono,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.53 ( ( ord_less_nat @ A @ B )
% 0.32/0.53 => ( ( ord_less_nat @ C @ D )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.53 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_strict_mono
% 0.32/0.53 thf(fact_2219_mult__strict__mono,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.53 ( ( ord_less_int @ A @ B )
% 0.32/0.53 => ( ( ord_less_int @ C @ D )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_strict_mono
% 0.32/0.53 thf(fact_2220_mult__less__cancel__left,axiom,
% 0.32/0.53 ! [C: real,A: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_left
% 0.32/0.53 thf(fact_2221_mult__less__cancel__left,axiom,
% 0.32/0.53 ! [C: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_left
% 0.32/0.53 thf(fact_2222_mult__less__cancel__left,axiom,
% 0.32/0.53 ! [C: int,A: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.53 => ( ord_less_int @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_left
% 0.32/0.53 thf(fact_2223_mult__right__less__imp__less,axiom,
% 0.32/0.53 ! [A: real,C: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_right_less_imp_less
% 0.32/0.53 thf(fact_2224_mult__right__less__imp__less,axiom,
% 0.32/0.53 ! [A: rat,C: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_right_less_imp_less
% 0.32/0.53 thf(fact_2225_mult__right__less__imp__less,axiom,
% 0.32/0.53 ! [A: nat,C: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.53 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_right_less_imp_less
% 0.32/0.53 thf(fact_2226_mult__right__less__imp__less,axiom,
% 0.32/0.53 ! [A: int,C: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_right_less_imp_less
% 0.32/0.53 thf(fact_2227_mult__strict__mono_H,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.53 ( ( ord_less_real @ A @ B )
% 0.32/0.53 => ( ( ord_less_real @ C @ D )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_strict_mono'
% 0.32/0.53 thf(fact_2228_mult__strict__mono_H,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.53 ( ( ord_less_rat @ A @ B )
% 0.32/0.53 => ( ( ord_less_rat @ C @ D )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_strict_mono'
% 0.32/0.53 thf(fact_2229_mult__strict__mono_H,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.53 ( ( ord_less_nat @ A @ B )
% 0.32/0.53 => ( ( ord_less_nat @ C @ D )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.53 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_strict_mono'
% 0.32/0.53 thf(fact_2230_mult__strict__mono_H,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.53 ( ( ord_less_int @ A @ B )
% 0.32/0.53 => ( ( ord_less_int @ C @ D )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_strict_mono'
% 0.32/0.53 thf(fact_2231_mult__less__cancel__right,axiom,
% 0.32/0.53 ! [A: real,C: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_right
% 0.32/0.53 thf(fact_2232_mult__less__cancel__right,axiom,
% 0.32/0.53 ! [A: rat,C: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_right
% 0.32/0.53 thf(fact_2233_mult__less__cancel__right,axiom,
% 0.32/0.53 ! [A: int,C: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.53 => ( ord_less_int @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_right
% 0.32/0.53 thf(fact_2234_mult__le__cancel__left__neg,axiom,
% 0.32/0.53 ! [C: real,A: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.53 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left_neg
% 0.32/0.53 thf(fact_2235_mult__le__cancel__left__neg,axiom,
% 0.32/0.53 ! [C: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.53 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left_neg
% 0.32/0.53 thf(fact_2236_mult__le__cancel__left__neg,axiom,
% 0.32/0.53 ! [C: int,A: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.53 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.53 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left_neg
% 0.32/0.53 thf(fact_2237_mult__le__cancel__left__pos,axiom,
% 0.32/0.53 ! [C: real,A: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.53 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left_pos
% 0.32/0.53 thf(fact_2238_mult__le__cancel__left__pos,axiom,
% 0.32/0.53 ! [C: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.53 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left_pos
% 0.32/0.53 thf(fact_2239_mult__le__cancel__left__pos,axiom,
% 0.32/0.53 ! [C: int,A: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.53 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left_pos
% 0.32/0.53 thf(fact_2240_mult__left__le__imp__le,axiom,
% 0.32/0.53 ! [C: real,A: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_le_imp_le
% 0.32/0.53 thf(fact_2241_mult__left__le__imp__le,axiom,
% 0.32/0.53 ! [C: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_le_imp_le
% 0.32/0.53 thf(fact_2242_mult__left__le__imp__le,axiom,
% 0.32/0.53 ! [C: nat,A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.32/0.53 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_le_imp_le
% 0.32/0.53 thf(fact_2243_mult__left__le__imp__le,axiom,
% 0.32/0.53 ! [C: int,A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_le_imp_le
% 0.32/0.53 thf(fact_2244_mult__right__le__imp__le,axiom,
% 0.32/0.53 ! [A: real,C: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_right_le_imp_le
% 0.32/0.53 thf(fact_2245_mult__right__le__imp__le,axiom,
% 0.32/0.53 ! [A: rat,C: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_right_le_imp_le
% 0.32/0.53 thf(fact_2246_mult__right__le__imp__le,axiom,
% 0.32/0.53 ! [A: nat,C: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.32/0.53 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_right_le_imp_le
% 0.32/0.53 thf(fact_2247_mult__right__le__imp__le,axiom,
% 0.32/0.53 ! [A: int,C: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_right_le_imp_le
% 0.32/0.53 thf(fact_2248_mult__le__less__imp__less,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.53 => ( ( ord_less_real @ C @ D )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_less_imp_less
% 0.32/0.53 thf(fact_2249_mult__le__less__imp__less,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.53 => ( ( ord_less_rat @ C @ D )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_less_imp_less
% 0.32/0.53 thf(fact_2250_mult__le__less__imp__less,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.53 => ( ( ord_less_nat @ C @ D )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.53 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_less_imp_less
% 0.32/0.53 thf(fact_2251_mult__le__less__imp__less,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.53 => ( ( ord_less_int @ C @ D )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_less_imp_less
% 0.32/0.53 thf(fact_2252_mult__less__le__imp__less,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.53 ( ( ord_less_real @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_real @ C @ D )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_le_imp_less
% 0.32/0.53 thf(fact_2253_mult__less__le__imp__less,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.53 ( ( ord_less_rat @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_rat @ C @ D )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_le_imp_less
% 0.32/0.53 thf(fact_2254_mult__less__le__imp__less,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.53 ( ( ord_less_nat @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_nat @ C @ D )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.32/0.53 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_le_imp_less
% 0.32/0.53 thf(fact_2255_mult__less__le__imp__less,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.53 ( ( ord_less_int @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_int @ C @ D )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_le_imp_less
% 0.32/0.53 thf(fact_2256_add__strict__increasing2,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ( ord_less_real @ B @ C )
% 0.32/0.53 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_increasing2
% 0.32/0.53 thf(fact_2257_add__strict__increasing2,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ( ord_less_rat @ B @ C )
% 0.32/0.53 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_increasing2
% 0.32/0.53 thf(fact_2258_add__strict__increasing2,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ B @ C )
% 0.32/0.53 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_increasing2
% 0.32/0.53 thf(fact_2259_add__strict__increasing2,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_int @ B @ C )
% 0.32/0.53 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_increasing2
% 0.32/0.53 thf(fact_2260_add__strict__increasing,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ( ord_less_eq_real @ B @ C )
% 0.32/0.53 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_increasing
% 0.32/0.53 thf(fact_2261_add__strict__increasing,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.53 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_increasing
% 0.32/0.53 thf(fact_2262_add__strict__increasing,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.53 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_increasing
% 0.32/0.53 thf(fact_2263_add__strict__increasing,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int] :
% 0.32/0.53 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_eq_int @ B @ C )
% 0.32/0.53 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_increasing
% 0.32/0.53 thf(fact_2264_add__pos__nonneg,axiom,
% 0.32/0.53 ! [A: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.53 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_pos_nonneg
% 0.32/0.53 thf(fact_2265_add__pos__nonneg,axiom,
% 0.32/0.53 ! [A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.53 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_pos_nonneg
% 0.32/0.53 thf(fact_2266_add__pos__nonneg,axiom,
% 0.32/0.53 ! [A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.53 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_pos_nonneg
% 0.32/0.53 thf(fact_2267_add__pos__nonneg,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_pos_nonneg
% 0.32/0.53 thf(fact_2268_add__nonpos__neg,axiom,
% 0.32/0.53 ! [A: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.53 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_nonpos_neg
% 0.32/0.53 thf(fact_2269_add__nonpos__neg,axiom,
% 0.32/0.53 ! [A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_nonpos_neg
% 0.32/0.53 thf(fact_2270_add__nonpos__neg,axiom,
% 0.32/0.53 ! [A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/0.53 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.32/0.53 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_nonpos_neg
% 0.32/0.53 thf(fact_2271_add__nonpos__neg,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.53 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.53 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_nonpos_neg
% 0.32/0.53 thf(fact_2272_add__nonneg__pos,axiom,
% 0.32/0.53 ! [A: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.53 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_nonneg_pos
% 0.32/0.53 thf(fact_2273_add__nonneg__pos,axiom,
% 0.32/0.53 ! [A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.32/0.53 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_nonneg_pos
% 0.32/0.53 thf(fact_2274_add__nonneg__pos,axiom,
% 0.32/0.53 ! [A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.53 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_nonneg_pos
% 0.32/0.53 thf(fact_2275_add__nonneg__pos,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_nonneg_pos
% 0.32/0.53 thf(fact_2276_add__neg__nonpos,axiom,
% 0.32/0.53 ! [A: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.53 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_neg_nonpos
% 0.32/0.53 thf(fact_2277_add__neg__nonpos,axiom,
% 0.32/0.53 ! [A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_neg_nonpos
% 0.32/0.53 thf(fact_2278_add__neg__nonpos,axiom,
% 0.32/0.53 ! [A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.32/0.53 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.32/0.53 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_neg_nonpos
% 0.32/0.53 thf(fact_2279_add__neg__nonpos,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.53 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.32/0.53 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_neg_nonpos
% 0.32/0.53 thf(fact_2280_field__le__epsilon,axiom,
% 0.32/0.53 ! [X: real,Y: real] :
% 0.32/0.53 ( ! [E2: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.32/0.53 => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
% 0.32/0.53 => ( ord_less_eq_real @ X @ Y ) ) ).
% 0.32/0.53
% 0.32/0.53 % field_le_epsilon
% 0.32/0.53 thf(fact_2281_field__le__epsilon,axiom,
% 0.32/0.53 ! [X: rat,Y: rat] :
% 0.32/0.53 ( ! [E2: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.32/0.53 => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E2 ) ) )
% 0.32/0.53 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.32/0.53
% 0.32/0.53 % field_le_epsilon
% 0.32/0.53 thf(fact_2282_divide__nonpos__pos,axiom,
% 0.32/0.53 ! [X: real,Y: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.53 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_nonpos_pos
% 0.32/0.53 thf(fact_2283_divide__nonpos__pos,axiom,
% 0.32/0.53 ! [X: rat,Y: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.32/0.53 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_nonpos_pos
% 0.32/0.53 thf(fact_2284_divide__nonpos__neg,axiom,
% 0.32/0.53 ! [X: real,Y: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.53 => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_nonpos_neg
% 0.32/0.53 thf(fact_2285_divide__nonpos__neg,axiom,
% 0.32/0.53 ! [X: rat,Y: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_nonpos_neg
% 0.32/0.53 thf(fact_2286_divide__nonneg__pos,axiom,
% 0.32/0.53 ! [X: real,Y: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.53 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_nonneg_pos
% 0.32/0.53 thf(fact_2287_divide__nonneg__pos,axiom,
% 0.32/0.53 ! [X: rat,Y: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.32/0.53 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_nonneg_pos
% 0.32/0.53 thf(fact_2288_divide__nonneg__neg,axiom,
% 0.32/0.53 ! [X: real,Y: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.53 => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_nonneg_neg
% 0.32/0.53 thf(fact_2289_divide__nonneg__neg,axiom,
% 0.32/0.53 ! [X: rat,Y: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.53 => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_nonneg_neg
% 0.32/0.53 thf(fact_2290_divide__le__cancel,axiom,
% 0.32/0.53 ! [A: real,C: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_eq_real @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_le_cancel
% 0.32/0.53 thf(fact_2291_divide__le__cancel,axiom,
% 0.32/0.53 ! [A: rat,C: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_le_cancel
% 0.32/0.53 thf(fact_2292_frac__less2,axiom,
% 0.32/0.53 ! [X: real,Y: real,W: real,Z2: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.53 => ( ( ord_less_eq_real @ X @ Y )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.32/0.53 => ( ( ord_less_real @ W @ Z2 )
% 0.32/0.53 => ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % frac_less2
% 0.32/0.53 thf(fact_2293_frac__less2,axiom,
% 0.32/0.53 ! [X: rat,Y: rat,W: rat,Z2: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.32/0.53 => ( ( ord_less_eq_rat @ X @ Y )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.32/0.53 => ( ( ord_less_rat @ W @ Z2 )
% 0.32/0.53 => ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % frac_less2
% 0.32/0.53 thf(fact_2294_frac__less,axiom,
% 0.32/0.53 ! [X: real,Y: real,W: real,Z2: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.53 => ( ( ord_less_real @ X @ Y )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.32/0.53 => ( ( ord_less_eq_real @ W @ Z2 )
% 0.32/0.53 => ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % frac_less
% 0.32/0.53 thf(fact_2295_frac__less,axiom,
% 0.32/0.53 ! [X: rat,Y: rat,W: rat,Z2: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.53 => ( ( ord_less_rat @ X @ Y )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.32/0.53 => ( ( ord_less_eq_rat @ W @ Z2 )
% 0.32/0.53 => ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % frac_less
% 0.32/0.53 thf(fact_2296_frac__le,axiom,
% 0.32/0.53 ! [Y: real,X: real,W: real,Z2: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.32/0.53 => ( ( ord_less_eq_real @ X @ Y )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.32/0.53 => ( ( ord_less_eq_real @ W @ Z2 )
% 0.32/0.53 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % frac_le
% 0.32/0.53 thf(fact_2297_frac__le,axiom,
% 0.32/0.53 ! [Y: rat,X: rat,W: rat,Z2: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.32/0.53 => ( ( ord_less_eq_rat @ X @ Y )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.32/0.53 => ( ( ord_less_eq_rat @ W @ Z2 )
% 0.32/0.53 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % frac_le
% 0.32/0.53 thf(fact_2298_div__positive,axiom,
% 0.32/0.53 ! [B: nat,A: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.53 => ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.53 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_positive
% 0.32/0.53 thf(fact_2299_div__positive,axiom,
% 0.32/0.53 ! [B: int,A: int] :
% 0.32/0.53 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ( ord_less_eq_int @ B @ A )
% 0.32/0.53 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_positive
% 0.32/0.53 thf(fact_2300_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.32/0.53 ! [A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ A @ B )
% 0.32/0.53 => ( ( divide_divide_nat @ A @ B )
% 0.32/0.53 = zero_zero_nat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % unique_euclidean_semiring_numeral_class.div_less
% 0.32/0.53 thf(fact_2301_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_int @ A @ B )
% 0.32/0.53 => ( ( divide_divide_int @ A @ B )
% 0.32/0.53 = zero_zero_int ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % unique_euclidean_semiring_numeral_class.div_less
% 0.32/0.53 thf(fact_2302_power__less__imp__less__base,axiom,
% 0.32/0.53 ! [A: real,N: nat,B: real] :
% 0.32/0.53 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.53 => ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_less_imp_less_base
% 0.32/0.53 thf(fact_2303_power__less__imp__less__base,axiom,
% 0.32/0.53 ! [A: rat,N: nat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.53 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_less_imp_less_base
% 0.32/0.53 thf(fact_2304_power__less__imp__less__base,axiom,
% 0.32/0.53 ! [A: nat,N: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.53 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_less_imp_less_base
% 0.32/0.53 thf(fact_2305_power__less__imp__less__base,axiom,
% 0.32/0.53 ! [A: int,N: nat,B: int] :
% 0.32/0.53 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ord_less_int @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_less_imp_less_base
% 0.32/0.53 thf(fact_2306_discrete,axiom,
% 0.32/0.53 ( ord_less_nat
% 0.32/0.53 = ( ^ [A2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % discrete
% 0.32/0.53 thf(fact_2307_discrete,axiom,
% 0.32/0.53 ( ord_less_int
% 0.32/0.53 = ( ^ [A2: int] : ( ord_less_eq_int @ ( plus_plus_int @ A2 @ one_one_int ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % discrete
% 0.32/0.53 thf(fact_2308_power__strict__increasing,axiom,
% 0.32/0.53 ! [N: nat,N5: nat,A: real] :
% 0.32/0.53 ( ( ord_less_nat @ N @ N5 )
% 0.32/0.53 => ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.53 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_strict_increasing
% 0.32/0.53 thf(fact_2309_power__strict__increasing,axiom,
% 0.32/0.53 ! [N: nat,N5: nat,A: rat] :
% 0.32/0.53 ( ( ord_less_nat @ N @ N5 )
% 0.32/0.53 => ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.53 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_strict_increasing
% 0.32/0.53 thf(fact_2310_power__strict__increasing,axiom,
% 0.32/0.53 ! [N: nat,N5: nat,A: nat] :
% 0.32/0.53 ( ( ord_less_nat @ N @ N5 )
% 0.32/0.53 => ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.53 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_strict_increasing
% 0.32/0.53 thf(fact_2311_power__strict__increasing,axiom,
% 0.32/0.53 ! [N: nat,N5: nat,A: int] :
% 0.32/0.53 ( ( ord_less_nat @ N @ N5 )
% 0.32/0.53 => ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.53 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_strict_increasing
% 0.32/0.53 thf(fact_2312_power__less__imp__less__exp,axiom,
% 0.32/0.53 ! [A: real,M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.53 => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 0.32/0.53 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_less_imp_less_exp
% 0.32/0.53 thf(fact_2313_power__less__imp__less__exp,axiom,
% 0.32/0.53 ! [A: rat,M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.53 => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 0.32/0.53 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_less_imp_less_exp
% 0.32/0.53 thf(fact_2314_power__less__imp__less__exp,axiom,
% 0.32/0.53 ! [A: nat,M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.32/0.53 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_less_imp_less_exp
% 0.32/0.53 thf(fact_2315_power__less__imp__less__exp,axiom,
% 0.32/0.53 ! [A: int,M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.53 => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.32/0.53 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_less_imp_less_exp
% 0.32/0.53 thf(fact_2316_power__increasing,axiom,
% 0.32/0.53 ! [N: nat,N5: nat,A: real] :
% 0.32/0.53 ( ( ord_less_eq_nat @ N @ N5 )
% 0.32/0.53 => ( ( ord_less_eq_real @ one_one_real @ A )
% 0.32/0.53 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_increasing
% 0.32/0.53 thf(fact_2317_power__increasing,axiom,
% 0.32/0.53 ! [N: nat,N5: nat,A: rat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ N @ N5 )
% 0.32/0.53 => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.32/0.53 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_increasing
% 0.32/0.53 thf(fact_2318_power__increasing,axiom,
% 0.32/0.53 ! [N: nat,N5: nat,A: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ N @ N5 )
% 0.32/0.53 => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.32/0.53 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_increasing
% 0.32/0.53 thf(fact_2319_power__increasing,axiom,
% 0.32/0.53 ! [N: nat,N5: nat,A: int] :
% 0.32/0.53 ( ( ord_less_eq_nat @ N @ N5 )
% 0.32/0.53 => ( ( ord_less_eq_int @ one_one_int @ A )
% 0.32/0.53 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_increasing
% 0.32/0.53 thf(fact_2320_ex__least__nat__less,axiom,
% 0.32/0.53 ! [P: nat > $o,N: nat] :
% 0.32/0.53 ( ( P @ N )
% 0.32/0.53 => ( ~ ( P @ zero_zero_nat )
% 0.32/0.53 => ? [K3: nat] :
% 0.32/0.53 ( ( ord_less_nat @ K3 @ N )
% 0.32/0.53 & ! [I3: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ I3 @ K3 )
% 0.32/0.53 => ~ ( P @ I3 ) )
% 0.32/0.53 & ( P @ ( suc @ K3 ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % ex_least_nat_less
% 0.32/0.53 thf(fact_2321_nat__mult__le__cancel1,axiom,
% 0.32/0.53 ! [K: nat,M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.53 => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.53 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % nat_mult_le_cancel1
% 0.32/0.53 thf(fact_2322_realpow__pos__nth__unique,axiom,
% 0.32/0.53 ! [N: nat,A: real] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.53 => ? [X4: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.32/0.53 & ( ( power_power_real @ X4 @ N )
% 0.32/0.53 = A )
% 0.32/0.53 & ! [Y5: real] :
% 0.32/0.53 ( ( ( ord_less_real @ zero_zero_real @ Y5 )
% 0.32/0.53 & ( ( power_power_real @ Y5 @ N )
% 0.32/0.53 = A ) )
% 0.32/0.53 => ( Y5 = X4 ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % realpow_pos_nth_unique
% 0.32/0.53 thf(fact_2323_realpow__pos__nth,axiom,
% 0.32/0.53 ! [N: nat,A: real] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.53 => ? [R2: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.32/0.53 & ( ( power_power_real @ R2 @ N )
% 0.32/0.53 = A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % realpow_pos_nth
% 0.32/0.53 thf(fact_2324_int__one__le__iff__zero__less,axiom,
% 0.32/0.53 ! [Z2: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ one_one_int @ Z2 )
% 0.32/0.53 = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 0.32/0.53
% 0.32/0.53 % int_one_le_iff_zero_less
% 0.32/0.53 thf(fact_2325_dvd__imp__le,axiom,
% 0.32/0.53 ! [K: nat,N: nat] :
% 0.32/0.53 ( ( dvd_dvd_nat @ K @ N )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % dvd_imp_le
% 0.32/0.53 thf(fact_2326_less__diff__conv2,axiom,
% 0.32/0.53 ! [K: nat,J: nat,I: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.53 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.32/0.53 = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_diff_conv2
% 0.32/0.53 thf(fact_2327_div__greater__zero__iff,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 0.32/0.53 = ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.53 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_greater_zero_iff
% 0.32/0.53 thf(fact_2328_div__le__mono2,axiom,
% 0.32/0.53 ! [M: nat,N: nat,K: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.53 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.53 => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_le_mono2
% 0.32/0.53 thf(fact_2329_add1__zle__eq,axiom,
% 0.32/0.53 ! [W: int,Z2: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
% 0.32/0.53 = ( ord_less_int @ W @ Z2 ) ) ).
% 0.32/0.53
% 0.32/0.53 % add1_zle_eq
% 0.32/0.53 thf(fact_2330_zless__imp__add1__zle,axiom,
% 0.32/0.53 ! [W: int,Z2: int] :
% 0.32/0.53 ( ( ord_less_int @ W @ Z2 )
% 0.32/0.53 => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% 0.32/0.53
% 0.32/0.53 % zless_imp_add1_zle
% 0.32/0.53 thf(fact_2331_zdiv__mono1,axiom,
% 0.32/0.53 ! [A: int,A5: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ A @ A5 )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % zdiv_mono1
% 0.32/0.53 thf(fact_2332_zdiv__mono2,axiom,
% 0.32/0.53 ! [A: int,B6: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.32/0.53 => ( ( ord_less_eq_int @ B6 @ B )
% 0.32/0.53 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B6 ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % zdiv_mono2
% 0.32/0.53 thf(fact_2333_zdiv__eq__0__iff,axiom,
% 0.32/0.53 ! [I: int,K: int] :
% 0.32/0.53 ( ( ( divide_divide_int @ I @ K )
% 0.32/0.53 = zero_zero_int )
% 0.32/0.53 = ( ( K = zero_zero_int )
% 0.32/0.53 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.32/0.53 & ( ord_less_int @ I @ K ) )
% 0.32/0.53 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.32/0.53 & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % zdiv_eq_0_iff
% 0.32/0.53 thf(fact_2334_zdiv__mono1__neg,axiom,
% 0.32/0.53 ! [A: int,A5: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ A @ A5 )
% 0.32/0.53 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.53 => ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % zdiv_mono1_neg
% 0.32/0.53 thf(fact_2335_zdiv__mono2__neg,axiom,
% 0.32/0.53 ! [A: int,B6: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.32/0.53 => ( ( ord_less_eq_int @ B6 @ B )
% 0.32/0.53 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B6 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % zdiv_mono2_neg
% 0.32/0.53 thf(fact_2336_div__int__pos__iff,axiom,
% 0.32/0.53 ! [K: int,L: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
% 0.32/0.53 = ( ( K = zero_zero_int )
% 0.32/0.53 | ( L = zero_zero_int )
% 0.32/0.53 | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.53 & ( ord_less_eq_int @ zero_zero_int @ L ) )
% 0.32/0.53 | ( ( ord_less_int @ K @ zero_zero_int )
% 0.32/0.53 & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_int_pos_iff
% 0.32/0.53 thf(fact_2337_div__positive__int,axiom,
% 0.32/0.53 ! [L: int,K: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ L @ K )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ L )
% 0.32/0.53 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_positive_int
% 0.32/0.53 thf(fact_2338_div__nonneg__neg__le0,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.53 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_nonneg_neg_le0
% 0.32/0.53 thf(fact_2339_div__nonpos__pos__le0,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_nonpos_pos_le0
% 0.32/0.53 thf(fact_2340_pos__imp__zdiv__pos__iff,axiom,
% 0.32/0.53 ! [K: int,I: int] :
% 0.32/0.53 ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 0.32/0.53 = ( ord_less_eq_int @ K @ I ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % pos_imp_zdiv_pos_iff
% 0.32/0.53 thf(fact_2341_neg__imp__zdiv__nonneg__iff,axiom,
% 0.32/0.53 ! [B: int,A: int] :
% 0.32/0.53 ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.32/0.53 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % neg_imp_zdiv_nonneg_iff
% 0.32/0.53 thf(fact_2342_pos__imp__zdiv__nonneg__iff,axiom,
% 0.32/0.53 ! [B: int,A: int] :
% 0.32/0.53 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.32/0.53 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % pos_imp_zdiv_nonneg_iff
% 0.32/0.53 thf(fact_2343_nonneg1__imp__zdiv__pos__iff,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.32/0.53 = ( ( ord_less_eq_int @ B @ A )
% 0.32/0.53 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % nonneg1_imp_zdiv_pos_iff
% 0.32/0.53 thf(fact_2344_zdvd__imp__le,axiom,
% 0.32/0.53 ! [Z2: int,N: int] :
% 0.32/0.53 ( ( dvd_dvd_int @ Z2 @ N )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.32/0.53 => ( ord_less_eq_int @ Z2 @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % zdvd_imp_le
% 0.32/0.53 thf(fact_2345_nat__le__real__less,axiom,
% 0.32/0.53 ( ord_less_eq_nat
% 0.32/0.53 = ( ^ [N3: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % nat_le_real_less
% 0.32/0.53 thf(fact_2346_nat__less__real__le,axiom,
% 0.32/0.53 ( ord_less_nat
% 0.32/0.53 = ( ^ [N3: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % nat_less_real_le
% 0.32/0.53 thf(fact_2347_of__bool__conj,axiom,
% 0.32/0.53 ! [P: $o,Q: $o] :
% 0.32/0.53 ( ( zero_n3304061248610475627l_real
% 0.32/0.53 @ ( P
% 0.32/0.53 & Q ) )
% 0.32/0.53 = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % of_bool_conj
% 0.32/0.53 thf(fact_2348_of__bool__conj,axiom,
% 0.32/0.53 ! [P: $o,Q: $o] :
% 0.32/0.53 ( ( zero_n2052037380579107095ol_rat
% 0.32/0.53 @ ( P
% 0.32/0.53 & Q ) )
% 0.32/0.53 = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % of_bool_conj
% 0.32/0.53 thf(fact_2349_of__bool__conj,axiom,
% 0.32/0.53 ! [P: $o,Q: $o] :
% 0.32/0.53 ( ( zero_n2687167440665602831ol_nat
% 0.32/0.53 @ ( P
% 0.32/0.53 & Q ) )
% 0.32/0.53 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % of_bool_conj
% 0.32/0.53 thf(fact_2350_of__bool__conj,axiom,
% 0.32/0.53 ! [P: $o,Q: $o] :
% 0.32/0.53 ( ( zero_n2684676970156552555ol_int
% 0.32/0.53 @ ( P
% 0.32/0.53 & Q ) )
% 0.32/0.53 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % of_bool_conj
% 0.32/0.53 thf(fact_2351_of__bool__conj,axiom,
% 0.32/0.53 ! [P: $o,Q: $o] :
% 0.32/0.53 ( ( zero_n356916108424825756nteger
% 0.32/0.53 @ ( P
% 0.32/0.53 & Q ) )
% 0.32/0.53 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % of_bool_conj
% 0.32/0.53 thf(fact_2352_gr__zeroI,axiom,
% 0.32/0.53 ! [N: nat] :
% 0.32/0.53 ( ( N != zero_zero_nat )
% 0.32/0.53 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % gr_zeroI
% 0.32/0.53 thf(fact_2353_not__less__zero,axiom,
% 0.32/0.53 ! [N: nat] :
% 0.32/0.53 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.32/0.53
% 0.32/0.53 % not_less_zero
% 0.32/0.53 thf(fact_2354_gr__implies__not__zero,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ M @ N )
% 0.32/0.53 => ( N != zero_zero_nat ) ) ).
% 0.32/0.53
% 0.32/0.53 % gr_implies_not_zero
% 0.32/0.53 thf(fact_2355_zero__less__iff__neq__zero,axiom,
% 0.32/0.53 ! [N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 = ( N != zero_zero_nat ) ) ).
% 0.32/0.53
% 0.32/0.53 % zero_less_iff_neq_zero
% 0.32/0.53 thf(fact_2356_field__lbound__gt__zero,axiom,
% 0.32/0.53 ! [D1: real,D22: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ D1 )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ D22 )
% 0.32/0.53 => ? [E2: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.32/0.53 & ( ord_less_real @ E2 @ D1 )
% 0.32/0.53 & ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % field_lbound_gt_zero
% 0.32/0.53 thf(fact_2357_field__lbound__gt__zero,axiom,
% 0.32/0.53 ! [D1: rat,D22: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ D22 )
% 0.32/0.53 => ? [E2: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.32/0.53 & ( ord_less_rat @ E2 @ D1 )
% 0.32/0.53 & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % field_lbound_gt_zero
% 0.32/0.53 thf(fact_2358_less__numeral__extra_I3_J,axiom,
% 0.32/0.53 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 0.32/0.53
% 0.32/0.53 % less_numeral_extra(3)
% 0.32/0.53 thf(fact_2359_less__numeral__extra_I3_J,axiom,
% 0.32/0.53 ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 0.32/0.53
% 0.32/0.53 % less_numeral_extra(3)
% 0.32/0.53 thf(fact_2360_less__numeral__extra_I3_J,axiom,
% 0.32/0.53 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.32/0.53
% 0.32/0.53 % less_numeral_extra(3)
% 0.32/0.53 thf(fact_2361_less__numeral__extra_I3_J,axiom,
% 0.32/0.53 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.32/0.53
% 0.32/0.53 % less_numeral_extra(3)
% 0.32/0.53 thf(fact_2362_zero__le,axiom,
% 0.32/0.53 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.32/0.53
% 0.32/0.53 % zero_le
% 0.32/0.53 thf(fact_2363_le__numeral__extra_I3_J,axiom,
% 0.32/0.53 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.32/0.53
% 0.32/0.53 % le_numeral_extra(3)
% 0.32/0.53 thf(fact_2364_le__numeral__extra_I3_J,axiom,
% 0.32/0.53 ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 0.32/0.53
% 0.32/0.53 % le_numeral_extra(3)
% 0.32/0.53 thf(fact_2365_le__numeral__extra_I3_J,axiom,
% 0.32/0.53 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.32/0.53
% 0.32/0.53 % le_numeral_extra(3)
% 0.32/0.53 thf(fact_2366_le__numeral__extra_I3_J,axiom,
% 0.32/0.53 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.32/0.53
% 0.32/0.53 % le_numeral_extra(3)
% 0.32/0.53 thf(fact_2367_less__numeral__extra_I4_J,axiom,
% 0.32/0.53 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 0.32/0.53
% 0.32/0.53 % less_numeral_extra(4)
% 0.32/0.53 thf(fact_2368_less__numeral__extra_I4_J,axiom,
% 0.32/0.53 ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 0.32/0.53
% 0.32/0.53 % less_numeral_extra(4)
% 0.32/0.53 thf(fact_2369_less__numeral__extra_I4_J,axiom,
% 0.32/0.53 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 0.32/0.53
% 0.32/0.53 % less_numeral_extra(4)
% 0.32/0.53 thf(fact_2370_less__numeral__extra_I4_J,axiom,
% 0.32/0.53 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 0.32/0.53
% 0.32/0.53 % less_numeral_extra(4)
% 0.32/0.53 thf(fact_2371_add__less__imp__less__right,axiom,
% 0.32/0.53 ! [A: real,C: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.32/0.53 => ( ord_less_real @ A @ B ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_less_imp_less_right
% 0.32/0.53 thf(fact_2372_add__less__imp__less__right,axiom,
% 0.32/0.53 ! [A: rat,C: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.53 => ( ord_less_rat @ A @ B ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_less_imp_less_right
% 0.32/0.53 thf(fact_2373_add__less__imp__less__right,axiom,
% 0.32/0.53 ! [A: nat,C: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.53 => ( ord_less_nat @ A @ B ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_less_imp_less_right
% 0.32/0.53 thf(fact_2374_add__less__imp__less__right,axiom,
% 0.32/0.53 ! [A: int,C: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.32/0.53 => ( ord_less_int @ A @ B ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_less_imp_less_right
% 0.32/0.53 thf(fact_2375_add__less__imp__less__left,axiom,
% 0.32/0.53 ! [C: real,A: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.32/0.53 => ( ord_less_real @ A @ B ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_less_imp_less_left
% 0.32/0.53 thf(fact_2376_add__less__imp__less__left,axiom,
% 0.32/0.53 ! [C: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.32/0.53 => ( ord_less_rat @ A @ B ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_less_imp_less_left
% 0.32/0.53 thf(fact_2377_add__less__imp__less__left,axiom,
% 0.32/0.53 ! [C: nat,A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.32/0.53 => ( ord_less_nat @ A @ B ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_less_imp_less_left
% 0.32/0.53 thf(fact_2378_add__less__imp__less__left,axiom,
% 0.32/0.53 ! [C: int,A: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.32/0.53 => ( ord_less_int @ A @ B ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_less_imp_less_left
% 0.32/0.53 thf(fact_2379_add__strict__right__mono,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real] :
% 0.32/0.53 ( ( ord_less_real @ A @ B )
% 0.32/0.53 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_right_mono
% 0.32/0.53 thf(fact_2380_add__strict__right__mono,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_rat @ A @ B )
% 0.32/0.53 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_right_mono
% 0.32/0.53 thf(fact_2381_add__strict__right__mono,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat] :
% 0.32/0.53 ( ( ord_less_nat @ A @ B )
% 0.32/0.53 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_right_mono
% 0.32/0.53 thf(fact_2382_add__strict__right__mono,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int] :
% 0.32/0.53 ( ( ord_less_int @ A @ B )
% 0.32/0.53 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_right_mono
% 0.32/0.53 thf(fact_2383_add__strict__left__mono,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real] :
% 0.32/0.53 ( ( ord_less_real @ A @ B )
% 0.32/0.53 => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_left_mono
% 0.32/0.53 thf(fact_2384_add__strict__left__mono,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_rat @ A @ B )
% 0.32/0.53 => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_left_mono
% 0.32/0.53 thf(fact_2385_add__strict__left__mono,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat] :
% 0.32/0.53 ( ( ord_less_nat @ A @ B )
% 0.32/0.53 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_left_mono
% 0.32/0.53 thf(fact_2386_add__strict__left__mono,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int] :
% 0.32/0.53 ( ( ord_less_int @ A @ B )
% 0.32/0.53 => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_left_mono
% 0.32/0.53 thf(fact_2387_add__strict__mono,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.53 ( ( ord_less_real @ A @ B )
% 0.32/0.53 => ( ( ord_less_real @ C @ D )
% 0.32/0.53 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_mono
% 0.32/0.53 thf(fact_2388_add__strict__mono,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.53 ( ( ord_less_rat @ A @ B )
% 0.32/0.53 => ( ( ord_less_rat @ C @ D )
% 0.32/0.53 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_mono
% 0.32/0.53 thf(fact_2389_add__strict__mono,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.53 ( ( ord_less_nat @ A @ B )
% 0.32/0.53 => ( ( ord_less_nat @ C @ D )
% 0.32/0.53 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_mono
% 0.32/0.53 thf(fact_2390_add__strict__mono,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.53 ( ( ord_less_int @ A @ B )
% 0.32/0.53 => ( ( ord_less_int @ C @ D )
% 0.32/0.53 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_mono
% 0.32/0.53 thf(fact_2391_add__mono__thms__linordered__field_I1_J,axiom,
% 0.32/0.53 ! [I: real,J: real,K: real,L: real] :
% 0.32/0.53 ( ( ( ord_less_real @ I @ J )
% 0.32/0.53 & ( K = L ) )
% 0.32/0.53 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(1)
% 0.32/0.53 thf(fact_2392_add__mono__thms__linordered__field_I1_J,axiom,
% 0.32/0.53 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.32/0.53 ( ( ( ord_less_rat @ I @ J )
% 0.32/0.53 & ( K = L ) )
% 0.32/0.53 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(1)
% 0.32/0.53 thf(fact_2393_add__mono__thms__linordered__field_I1_J,axiom,
% 0.32/0.53 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.53 ( ( ( ord_less_nat @ I @ J )
% 0.32/0.53 & ( K = L ) )
% 0.32/0.53 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(1)
% 0.32/0.53 thf(fact_2394_add__mono__thms__linordered__field_I1_J,axiom,
% 0.32/0.53 ! [I: int,J: int,K: int,L: int] :
% 0.32/0.53 ( ( ( ord_less_int @ I @ J )
% 0.32/0.53 & ( K = L ) )
% 0.32/0.53 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(1)
% 0.32/0.53 thf(fact_2395_add__mono__thms__linordered__field_I2_J,axiom,
% 0.32/0.53 ! [I: real,J: real,K: real,L: real] :
% 0.32/0.53 ( ( ( I = J )
% 0.32/0.53 & ( ord_less_real @ K @ L ) )
% 0.32/0.53 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(2)
% 0.32/0.53 thf(fact_2396_add__mono__thms__linordered__field_I2_J,axiom,
% 0.32/0.53 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.32/0.53 ( ( ( I = J )
% 0.32/0.53 & ( ord_less_rat @ K @ L ) )
% 0.32/0.53 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(2)
% 0.32/0.53 thf(fact_2397_add__mono__thms__linordered__field_I2_J,axiom,
% 0.32/0.53 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.53 ( ( ( I = J )
% 0.32/0.53 & ( ord_less_nat @ K @ L ) )
% 0.32/0.53 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(2)
% 0.32/0.53 thf(fact_2398_add__mono__thms__linordered__field_I2_J,axiom,
% 0.32/0.53 ! [I: int,J: int,K: int,L: int] :
% 0.32/0.53 ( ( ( I = J )
% 0.32/0.53 & ( ord_less_int @ K @ L ) )
% 0.32/0.53 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(2)
% 0.32/0.53 thf(fact_2399_add__mono__thms__linordered__field_I5_J,axiom,
% 0.32/0.53 ! [I: real,J: real,K: real,L: real] :
% 0.32/0.53 ( ( ( ord_less_real @ I @ J )
% 0.32/0.53 & ( ord_less_real @ K @ L ) )
% 0.32/0.53 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(5)
% 0.32/0.53 thf(fact_2400_add__mono__thms__linordered__field_I5_J,axiom,
% 0.32/0.53 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.32/0.53 ( ( ( ord_less_rat @ I @ J )
% 0.32/0.53 & ( ord_less_rat @ K @ L ) )
% 0.32/0.53 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(5)
% 0.32/0.53 thf(fact_2401_add__mono__thms__linordered__field_I5_J,axiom,
% 0.32/0.53 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.53 ( ( ( ord_less_nat @ I @ J )
% 0.32/0.53 & ( ord_less_nat @ K @ L ) )
% 0.32/0.53 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(5)
% 0.32/0.53 thf(fact_2402_add__mono__thms__linordered__field_I5_J,axiom,
% 0.32/0.53 ! [I: int,J: int,K: int,L: int] :
% 0.32/0.53 ( ( ( ord_less_int @ I @ J )
% 0.32/0.53 & ( ord_less_int @ K @ L ) )
% 0.32/0.53 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_field(5)
% 0.32/0.53 thf(fact_2403_diff__strict__mono,axiom,
% 0.32/0.53 ! [A: real,B: real,D: real,C: real] :
% 0.32/0.53 ( ( ord_less_real @ A @ B )
% 0.32/0.53 => ( ( ord_less_real @ D @ C )
% 0.32/0.53 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_strict_mono
% 0.32/0.53 thf(fact_2404_diff__strict__mono,axiom,
% 0.32/0.53 ! [A: rat,B: rat,D: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_rat @ A @ B )
% 0.32/0.53 => ( ( ord_less_rat @ D @ C )
% 0.32/0.53 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_strict_mono
% 0.32/0.53 thf(fact_2405_diff__strict__mono,axiom,
% 0.32/0.53 ! [A: int,B: int,D: int,C: int] :
% 0.32/0.53 ( ( ord_less_int @ A @ B )
% 0.32/0.53 => ( ( ord_less_int @ D @ C )
% 0.32/0.53 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_strict_mono
% 0.32/0.53 thf(fact_2406_diff__eq__diff__less,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.53 ( ( ( minus_minus_real @ A @ B )
% 0.32/0.53 = ( minus_minus_real @ C @ D ) )
% 0.32/0.53 => ( ( ord_less_real @ A @ B )
% 0.32/0.53 = ( ord_less_real @ C @ D ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_eq_diff_less
% 0.32/0.53 thf(fact_2407_diff__eq__diff__less,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.53 ( ( ( minus_minus_rat @ A @ B )
% 0.32/0.53 = ( minus_minus_rat @ C @ D ) )
% 0.32/0.53 => ( ( ord_less_rat @ A @ B )
% 0.32/0.53 = ( ord_less_rat @ C @ D ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_eq_diff_less
% 0.32/0.53 thf(fact_2408_diff__eq__diff__less,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.53 ( ( ( minus_minus_int @ A @ B )
% 0.32/0.53 = ( minus_minus_int @ C @ D ) )
% 0.32/0.53 => ( ( ord_less_int @ A @ B )
% 0.32/0.53 = ( ord_less_int @ C @ D ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_eq_diff_less
% 0.32/0.53 thf(fact_2409_diff__strict__left__mono,axiom,
% 0.32/0.53 ! [B: real,A: real,C: real] :
% 0.32/0.53 ( ( ord_less_real @ B @ A )
% 0.32/0.53 => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_strict_left_mono
% 0.32/0.53 thf(fact_2410_diff__strict__left__mono,axiom,
% 0.32/0.53 ! [B: rat,A: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_rat @ B @ A )
% 0.32/0.53 => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_strict_left_mono
% 0.32/0.53 thf(fact_2411_diff__strict__left__mono,axiom,
% 0.32/0.53 ! [B: int,A: int,C: int] :
% 0.32/0.53 ( ( ord_less_int @ B @ A )
% 0.32/0.53 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_strict_left_mono
% 0.32/0.53 thf(fact_2412_diff__strict__right__mono,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real] :
% 0.32/0.53 ( ( ord_less_real @ A @ B )
% 0.32/0.53 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_strict_right_mono
% 0.32/0.53 thf(fact_2413_diff__strict__right__mono,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_rat @ A @ B )
% 0.32/0.53 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_strict_right_mono
% 0.32/0.53 thf(fact_2414_diff__strict__right__mono,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int] :
% 0.32/0.53 ( ( ord_less_int @ A @ B )
% 0.32/0.53 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_strict_right_mono
% 0.32/0.53 thf(fact_2415_le__numeral__extra_I4_J,axiom,
% 0.32/0.53 ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.32/0.53
% 0.32/0.53 % le_numeral_extra(4)
% 0.32/0.53 thf(fact_2416_le__numeral__extra_I4_J,axiom,
% 0.32/0.53 ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 0.32/0.53
% 0.32/0.53 % le_numeral_extra(4)
% 0.32/0.53 thf(fact_2417_le__numeral__extra_I4_J,axiom,
% 0.32/0.53 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.32/0.53
% 0.32/0.53 % le_numeral_extra(4)
% 0.32/0.53 thf(fact_2418_le__numeral__extra_I4_J,axiom,
% 0.32/0.53 ord_less_eq_int @ one_one_int @ one_one_int ).
% 0.32/0.53
% 0.32/0.53 % le_numeral_extra(4)
% 0.32/0.53 thf(fact_2419_add__le__imp__le__right,axiom,
% 0.32/0.53 ! [A: real,C: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.32/0.53 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_le_imp_le_right
% 0.32/0.53 thf(fact_2420_add__le__imp__le__right,axiom,
% 0.32/0.53 ! [A: rat,C: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.53 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_le_imp_le_right
% 0.32/0.53 thf(fact_2421_add__le__imp__le__right,axiom,
% 0.32/0.53 ! [A: nat,C: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.53 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_le_imp_le_right
% 0.32/0.53 thf(fact_2422_add__le__imp__le__right,axiom,
% 0.32/0.53 ! [A: int,C: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.32/0.53 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_le_imp_le_right
% 0.32/0.53 thf(fact_2423_add__le__imp__le__left,axiom,
% 0.32/0.53 ! [C: real,A: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.32/0.53 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_le_imp_le_left
% 0.32/0.53 thf(fact_2424_add__le__imp__le__left,axiom,
% 0.32/0.53 ! [C: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.32/0.53 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_le_imp_le_left
% 0.32/0.53 thf(fact_2425_add__le__imp__le__left,axiom,
% 0.32/0.53 ! [C: nat,A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.32/0.53 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_le_imp_le_left
% 0.32/0.53 thf(fact_2426_add__le__imp__le__left,axiom,
% 0.32/0.53 ! [C: int,A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.32/0.53 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_le_imp_le_left
% 0.32/0.53 thf(fact_2427_le__iff__add,axiom,
% 0.32/0.53 ( ord_less_eq_nat
% 0.32/0.53 = ( ^ [A2: nat,B2: nat] :
% 0.32/0.53 ? [C4: nat] :
% 0.32/0.53 ( B2
% 0.32/0.53 = ( plus_plus_nat @ A2 @ C4 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_iff_add
% 0.32/0.53 thf(fact_2428_add__right__mono,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.53 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_right_mono
% 0.32/0.53 thf(fact_2429_add__right__mono,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.53 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_right_mono
% 0.32/0.53 thf(fact_2430_add__right__mono,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.53 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_right_mono
% 0.32/0.53 thf(fact_2431_add__right__mono,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.53 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_right_mono
% 0.32/0.53 thf(fact_2432_less__eqE,axiom,
% 0.32/0.53 ! [A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.53 => ~ ! [C3: nat] :
% 0.32/0.53 ( B
% 0.32/0.53 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_eqE
% 0.32/0.53 thf(fact_2433_add__left__mono,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.53 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_left_mono
% 0.32/0.53 thf(fact_2434_add__left__mono,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.53 => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_left_mono
% 0.32/0.53 thf(fact_2435_add__left__mono,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.53 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_left_mono
% 0.32/0.53 thf(fact_2436_add__left__mono,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.53 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_left_mono
% 0.32/0.53 thf(fact_2437_add__mono,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_real @ C @ D )
% 0.32/0.53 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono
% 0.32/0.53 thf(fact_2438_add__mono,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_rat @ C @ D )
% 0.32/0.53 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono
% 0.32/0.53 thf(fact_2439_add__mono,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_nat @ C @ D )
% 0.32/0.53 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono
% 0.32/0.53 thf(fact_2440_add__mono,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_int @ C @ D )
% 0.32/0.53 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono
% 0.32/0.53 thf(fact_2441_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.32/0.53 ! [I: real,J: real,K: real,L: real] :
% 0.32/0.53 ( ( ( ord_less_eq_real @ I @ J )
% 0.32/0.53 & ( ord_less_eq_real @ K @ L ) )
% 0.32/0.53 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_semiring(1)
% 0.32/0.53 thf(fact_2442_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.32/0.53 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.32/0.53 ( ( ( ord_less_eq_rat @ I @ J )
% 0.32/0.53 & ( ord_less_eq_rat @ K @ L ) )
% 0.32/0.53 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_semiring(1)
% 0.32/0.53 thf(fact_2443_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.32/0.53 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.53 ( ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.53 & ( ord_less_eq_nat @ K @ L ) )
% 0.32/0.53 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_semiring(1)
% 0.32/0.53 thf(fact_2444_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.32/0.53 ! [I: int,J: int,K: int,L: int] :
% 0.32/0.53 ( ( ( ord_less_eq_int @ I @ J )
% 0.32/0.53 & ( ord_less_eq_int @ K @ L ) )
% 0.32/0.53 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_semiring(1)
% 0.32/0.53 thf(fact_2445_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.32/0.53 ! [I: real,J: real,K: real,L: real] :
% 0.32/0.53 ( ( ( I = J )
% 0.32/0.53 & ( ord_less_eq_real @ K @ L ) )
% 0.32/0.53 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_semiring(2)
% 0.32/0.53 thf(fact_2446_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.32/0.53 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.32/0.53 ( ( ( I = J )
% 0.32/0.53 & ( ord_less_eq_rat @ K @ L ) )
% 0.32/0.53 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_semiring(2)
% 0.32/0.53 thf(fact_2447_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.32/0.53 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.53 ( ( ( I = J )
% 0.32/0.53 & ( ord_less_eq_nat @ K @ L ) )
% 0.32/0.53 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_semiring(2)
% 0.32/0.53 thf(fact_2448_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.32/0.53 ! [I: int,J: int,K: int,L: int] :
% 0.32/0.53 ( ( ( I = J )
% 0.32/0.53 & ( ord_less_eq_int @ K @ L ) )
% 0.32/0.53 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_semiring(2)
% 0.32/0.53 thf(fact_2449_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.32/0.53 ! [I: real,J: real,K: real,L: real] :
% 0.32/0.53 ( ( ( ord_less_eq_real @ I @ J )
% 0.32/0.53 & ( K = L ) )
% 0.32/0.53 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_semiring(3)
% 0.32/0.53 thf(fact_2450_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.32/0.53 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.32/0.53 ( ( ( ord_less_eq_rat @ I @ J )
% 0.32/0.53 & ( K = L ) )
% 0.32/0.53 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_semiring(3)
% 0.32/0.53 thf(fact_2451_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.32/0.53 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.53 ( ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.53 & ( K = L ) )
% 0.32/0.53 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_semiring(3)
% 0.32/0.53 thf(fact_2452_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.32/0.53 ! [I: int,J: int,K: int,L: int] :
% 0.32/0.53 ( ( ( ord_less_eq_int @ I @ J )
% 0.32/0.53 & ( K = L ) )
% 0.32/0.53 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_mono_thms_linordered_semiring(3)
% 0.32/0.53 thf(fact_2453_le__num__One__iff,axiom,
% 0.32/0.53 ! [X: num] :
% 0.32/0.53 ( ( ord_less_eq_num @ X @ one )
% 0.32/0.53 = ( X = one ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_num_One_iff
% 0.32/0.53 thf(fact_2454_diff__mono,axiom,
% 0.32/0.53 ! [A: real,B: real,D: real,C: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_real @ D @ C )
% 0.32/0.53 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_mono
% 0.32/0.53 thf(fact_2455_diff__mono,axiom,
% 0.32/0.53 ! [A: rat,B: rat,D: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_rat @ D @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_mono
% 0.32/0.53 thf(fact_2456_diff__mono,axiom,
% 0.32/0.53 ! [A: int,B: int,D: int,C: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_int @ D @ C )
% 0.32/0.53 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_mono
% 0.32/0.53 thf(fact_2457_diff__left__mono,axiom,
% 0.32/0.53 ! [B: real,A: real,C: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.53 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_left_mono
% 0.32/0.53 thf(fact_2458_diff__left__mono,axiom,
% 0.32/0.53 ! [B: rat,A: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.53 => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_left_mono
% 0.32/0.53 thf(fact_2459_diff__left__mono,axiom,
% 0.32/0.53 ! [B: int,A: int,C: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.53 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_left_mono
% 0.32/0.53 thf(fact_2460_diff__right__mono,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.53 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_right_mono
% 0.32/0.53 thf(fact_2461_diff__right__mono,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.53 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_right_mono
% 0.32/0.53 thf(fact_2462_diff__right__mono,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.53 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_right_mono
% 0.32/0.53 thf(fact_2463_diff__eq__diff__less__eq,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.53 ( ( ( minus_minus_real @ A @ B )
% 0.32/0.53 = ( minus_minus_real @ C @ D ) )
% 0.32/0.53 => ( ( ord_less_eq_real @ A @ B )
% 0.32/0.53 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_eq_diff_less_eq
% 0.32/0.53 thf(fact_2464_diff__eq__diff__less__eq,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.53 ( ( ( minus_minus_rat @ A @ B )
% 0.32/0.53 = ( minus_minus_rat @ C @ D ) )
% 0.32/0.53 => ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.53 = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_eq_diff_less_eq
% 0.32/0.53 thf(fact_2465_diff__eq__diff__less__eq,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.53 ( ( ( minus_minus_int @ A @ B )
% 0.32/0.53 = ( minus_minus_int @ C @ D ) )
% 0.32/0.53 => ( ( ord_less_eq_int @ A @ B )
% 0.32/0.53 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_eq_diff_less_eq
% 0.32/0.53 thf(fact_2466_reals__Archimedean2,axiom,
% 0.32/0.53 ! [X: real] :
% 0.32/0.53 ? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 0.32/0.53
% 0.32/0.53 % reals_Archimedean2
% 0.32/0.53 thf(fact_2467_reals__Archimedean2,axiom,
% 0.32/0.53 ! [X: rat] :
% 0.32/0.53 ? [N2: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 0.32/0.53
% 0.32/0.53 % reals_Archimedean2
% 0.32/0.53 thf(fact_2468_not__less__less__Suc__eq,axiom,
% 0.32/0.53 ! [N: nat,M: nat] :
% 0.32/0.53 ( ~ ( ord_less_nat @ N @ M )
% 0.32/0.53 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.32/0.53 = ( N = M ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % not_less_less_Suc_eq
% 0.32/0.53 thf(fact_2469_strict__inc__induct,axiom,
% 0.32/0.53 ! [I: nat,J: nat,P: nat > $o] :
% 0.32/0.53 ( ( ord_less_nat @ I @ J )
% 0.32/0.53 => ( ! [I2: nat] :
% 0.32/0.53 ( ( J
% 0.32/0.53 = ( suc @ I2 ) )
% 0.32/0.53 => ( P @ I2 ) )
% 0.32/0.53 => ( ! [I2: nat] :
% 0.32/0.53 ( ( ord_less_nat @ I2 @ J )
% 0.32/0.53 => ( ( P @ ( suc @ I2 ) )
% 0.32/0.53 => ( P @ I2 ) ) )
% 0.32/0.53 => ( P @ I ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % strict_inc_induct
% 0.32/0.53 thf(fact_2470_less__Suc__induct,axiom,
% 0.32/0.53 ! [I: nat,J: nat,P: nat > nat > $o] :
% 0.32/0.53 ( ( ord_less_nat @ I @ J )
% 0.32/0.53 => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 0.32/0.53 => ( ! [I2: nat,J2: nat,K3: nat] :
% 0.32/0.53 ( ( ord_less_nat @ I2 @ J2 )
% 0.32/0.53 => ( ( ord_less_nat @ J2 @ K3 )
% 0.32/0.53 => ( ( P @ I2 @ J2 )
% 0.32/0.53 => ( ( P @ J2 @ K3 )
% 0.32/0.53 => ( P @ I2 @ K3 ) ) ) ) )
% 0.32/0.53 => ( P @ I @ J ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_Suc_induct
% 0.32/0.53 thf(fact_2471_less__trans__Suc,axiom,
% 0.32/0.53 ! [I: nat,J: nat,K: nat] :
% 0.32/0.53 ( ( ord_less_nat @ I @ J )
% 0.32/0.53 => ( ( ord_less_nat @ J @ K )
% 0.32/0.53 => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_trans_Suc
% 0.32/0.53 thf(fact_2472_Suc__less__SucD,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.32/0.53 => ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % Suc_less_SucD
% 0.32/0.53 thf(fact_2473_less__antisym,axiom,
% 0.32/0.53 ! [N: nat,M: nat] :
% 0.32/0.53 ( ~ ( ord_less_nat @ N @ M )
% 0.32/0.53 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.32/0.53 => ( M = N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_antisym
% 0.32/0.53 thf(fact_2474_Suc__less__eq2,axiom,
% 0.32/0.53 ! [N: nat,M: nat] :
% 0.32/0.53 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.53 = ( ? [M5: nat] :
% 0.32/0.53 ( ( M
% 0.32/0.53 = ( suc @ M5 ) )
% 0.32/0.53 & ( ord_less_nat @ N @ M5 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % Suc_less_eq2
% 0.32/0.53 thf(fact_2475_All__less__Suc,axiom,
% 0.32/0.53 ! [N: nat,P: nat > $o] :
% 0.32/0.53 ( ( ! [I4: nat] :
% 0.32/0.53 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.32/0.53 => ( P @ I4 ) ) )
% 0.32/0.53 = ( ( P @ N )
% 0.32/0.53 & ! [I4: nat] :
% 0.32/0.53 ( ( ord_less_nat @ I4 @ N )
% 0.32/0.53 => ( P @ I4 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % All_less_Suc
% 0.32/0.53 thf(fact_2476_not__less__eq,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ~ ( ord_less_nat @ M @ N ) )
% 0.32/0.53 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % not_less_eq
% 0.32/0.53 thf(fact_2477_less__Suc__eq,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.32/0.53 = ( ( ord_less_nat @ M @ N )
% 0.32/0.53 | ( M = N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_Suc_eq
% 0.32/0.53 thf(fact_2478_Ex__less__Suc,axiom,
% 0.32/0.53 ! [N: nat,P: nat > $o] :
% 0.32/0.53 ( ( ? [I4: nat] :
% 0.32/0.53 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.32/0.53 & ( P @ I4 ) ) )
% 0.32/0.53 = ( ( P @ N )
% 0.32/0.53 | ? [I4: nat] :
% 0.32/0.53 ( ( ord_less_nat @ I4 @ N )
% 0.32/0.53 & ( P @ I4 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % Ex_less_Suc
% 0.32/0.53 thf(fact_2479_less__SucI,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ M @ N )
% 0.32/0.53 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_SucI
% 0.32/0.53 thf(fact_2480_less__SucE,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.32/0.53 => ( ~ ( ord_less_nat @ M @ N )
% 0.32/0.53 => ( M = N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_SucE
% 0.32/0.53 thf(fact_2481_Suc__lessI,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ M @ N )
% 0.32/0.53 => ( ( ( suc @ M )
% 0.32/0.53 != N )
% 0.32/0.53 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % Suc_lessI
% 0.32/0.53 thf(fact_2482_Suc__lessE,axiom,
% 0.32/0.53 ! [I: nat,K: nat] :
% 0.32/0.53 ( ( ord_less_nat @ ( suc @ I ) @ K )
% 0.32/0.53 => ~ ! [J2: nat] :
% 0.32/0.53 ( ( ord_less_nat @ I @ J2 )
% 0.32/0.53 => ( K
% 0.32/0.53 != ( suc @ J2 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % Suc_lessE
% 0.32/0.53 thf(fact_2483_Suc__lessD,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 0.32/0.53 => ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % Suc_lessD
% 0.32/0.53 thf(fact_2484_Nat_OlessE,axiom,
% 0.32/0.53 ! [I: nat,K: nat] :
% 0.32/0.53 ( ( ord_less_nat @ I @ K )
% 0.32/0.53 => ( ( K
% 0.32/0.53 != ( suc @ I ) )
% 0.32/0.53 => ~ ! [J2: nat] :
% 0.32/0.53 ( ( ord_less_nat @ I @ J2 )
% 0.32/0.53 => ( K
% 0.32/0.53 != ( suc @ J2 ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % Nat.lessE
% 0.32/0.53 thf(fact_2485_infinite__descent0,axiom,
% 0.32/0.53 ! [P: nat > $o,N: nat] :
% 0.32/0.53 ( ( P @ zero_zero_nat )
% 0.32/0.53 => ( ! [N2: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.53 => ( ~ ( P @ N2 )
% 0.32/0.53 => ? [M4: nat] :
% 0.32/0.53 ( ( ord_less_nat @ M4 @ N2 )
% 0.32/0.53 & ~ ( P @ M4 ) ) ) )
% 0.32/0.53 => ( P @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % infinite_descent0
% 0.32/0.53 thf(fact_2486_gr__implies__not0,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ M @ N )
% 0.32/0.53 => ( N != zero_zero_nat ) ) ).
% 0.32/0.53
% 0.32/0.53 % gr_implies_not0
% 0.32/0.53 thf(fact_2487_less__zeroE,axiom,
% 0.32/0.53 ! [N: nat] :
% 0.32/0.53 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.32/0.53
% 0.32/0.53 % less_zeroE
% 0.32/0.53 thf(fact_2488_not__less0,axiom,
% 0.32/0.53 ! [N: nat] :
% 0.32/0.53 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.32/0.53
% 0.32/0.53 % not_less0
% 0.32/0.53 thf(fact_2489_not__gr0,axiom,
% 0.32/0.53 ! [N: nat] :
% 0.32/0.53 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.32/0.53 = ( N = zero_zero_nat ) ) ).
% 0.32/0.53
% 0.32/0.53 % not_gr0
% 0.32/0.53 thf(fact_2490_gr0I,axiom,
% 0.32/0.53 ! [N: nat] :
% 0.32/0.53 ( ( N != zero_zero_nat )
% 0.32/0.53 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % gr0I
% 0.32/0.53 thf(fact_2491_bot__nat__0_Oextremum__strict,axiom,
% 0.32/0.53 ! [A: nat] :
% 0.32/0.53 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.32/0.53
% 0.32/0.53 % bot_nat_0.extremum_strict
% 0.32/0.53 thf(fact_2492_transitive__stepwise__le,axiom,
% 0.32/0.53 ! [M: nat,N: nat,R3: nat > nat > $o] :
% 0.32/0.53 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.53 => ( ! [X4: nat] : ( R3 @ X4 @ X4 )
% 0.32/0.53 => ( ! [X4: nat,Y4: nat,Z4: nat] :
% 0.32/0.53 ( ( R3 @ X4 @ Y4 )
% 0.32/0.53 => ( ( R3 @ Y4 @ Z4 )
% 0.32/0.53 => ( R3 @ X4 @ Z4 ) ) )
% 0.32/0.53 => ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
% 0.32/0.53 => ( R3 @ M @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % transitive_stepwise_le
% 0.32/0.53 thf(fact_2493_nat__induct__at__least,axiom,
% 0.32/0.53 ! [M: nat,N: nat,P: nat > $o] :
% 0.32/0.53 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.53 => ( ( P @ M )
% 0.32/0.53 => ( ! [N2: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.53 => ( ( P @ N2 )
% 0.32/0.53 => ( P @ ( suc @ N2 ) ) ) )
% 0.32/0.53 => ( P @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % nat_induct_at_least
% 0.32/0.53 thf(fact_2494_full__nat__induct,axiom,
% 0.32/0.53 ! [P: nat > $o,N: nat] :
% 0.32/0.53 ( ! [N2: nat] :
% 0.32/0.53 ( ! [M4: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
% 0.32/0.53 => ( P @ M4 ) )
% 0.32/0.53 => ( P @ N2 ) )
% 0.32/0.53 => ( P @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % full_nat_induct
% 0.32/0.53 thf(fact_2495_not__less__eq__eq,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 0.32/0.53 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 0.32/0.53
% 0.32/0.53 % not_less_eq_eq
% 0.32/0.53 thf(fact_2496_Suc__n__not__le__n,axiom,
% 0.32/0.53 ! [N: nat] :
% 0.32/0.53 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 0.32/0.53
% 0.32/0.53 % Suc_n_not_le_n
% 0.32/0.53 thf(fact_2497_le__Suc__eq,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.32/0.53 = ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.53 | ( M
% 0.32/0.53 = ( suc @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_Suc_eq
% 0.32/0.53 thf(fact_2498_Suc__le__D,axiom,
% 0.32/0.53 ! [N: nat,M6: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
% 0.32/0.53 => ? [M2: nat] :
% 0.32/0.53 ( M6
% 0.32/0.53 = ( suc @ M2 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % Suc_le_D
% 0.32/0.53 thf(fact_2499_le__SucI,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.53 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_SucI
% 0.32/0.53 thf(fact_2500_le__SucE,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.32/0.53 => ( ~ ( ord_less_eq_nat @ M @ N )
% 0.32/0.53 => ( M
% 0.32/0.53 = ( suc @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_SucE
% 0.32/0.53 thf(fact_2501_Suc__leD,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.32/0.53 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % Suc_leD
% 0.32/0.53 thf(fact_2502_le__0__eq,axiom,
% 0.32/0.53 ! [N: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.32/0.53 = ( N = zero_zero_nat ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_0_eq
% 0.32/0.53 thf(fact_2503_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.32/0.53 ! [A: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/0.53 => ( A = zero_zero_nat ) ) ).
% 0.32/0.53
% 0.32/0.53 % bot_nat_0.extremum_uniqueI
% 0.32/0.53 thf(fact_2504_bot__nat__0_Oextremum__unique,axiom,
% 0.32/0.53 ! [A: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/0.53 = ( A = zero_zero_nat ) ) ).
% 0.32/0.53
% 0.32/0.53 % bot_nat_0.extremum_unique
% 0.32/0.53 thf(fact_2505_less__eq__nat_Osimps_I1_J,axiom,
% 0.32/0.53 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.32/0.53
% 0.32/0.53 % less_eq_nat.simps(1)
% 0.32/0.53 thf(fact_2506_real__arch__simple,axiom,
% 0.32/0.53 ! [X: real] :
% 0.32/0.53 ? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 0.32/0.53
% 0.32/0.53 % real_arch_simple
% 0.32/0.53 thf(fact_2507_real__arch__simple,axiom,
% 0.32/0.53 ! [X: rat] :
% 0.32/0.53 ? [N2: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 0.32/0.53
% 0.32/0.53 % real_arch_simple
% 0.32/0.53 thf(fact_2508_less__int__code_I1_J,axiom,
% 0.32/0.53 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.32/0.53
% 0.32/0.53 % less_int_code(1)
% 0.32/0.53 thf(fact_2509_imp__le__cong,axiom,
% 0.32/0.53 ! [X: int,X6: int,P: $o,P4: $o] :
% 0.32/0.53 ( ( X = X6 )
% 0.32/0.53 => ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
% 0.32/0.53 => ( P = P4 ) )
% 0.32/0.53 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.53 => P )
% 0.32/0.53 = ( ( ord_less_eq_int @ zero_zero_int @ X6 )
% 0.32/0.53 => P4 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % imp_le_cong
% 0.32/0.53 thf(fact_2510_conj__le__cong,axiom,
% 0.32/0.53 ! [X: int,X6: int,P: $o,P4: $o] :
% 0.32/0.53 ( ( X = X6 )
% 0.32/0.53 => ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
% 0.32/0.53 => ( P = P4 ) )
% 0.32/0.53 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.53 & P )
% 0.32/0.53 = ( ( ord_less_eq_int @ zero_zero_int @ X6 )
% 0.32/0.53 & P4 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % conj_le_cong
% 0.32/0.53 thf(fact_2511_less__eq__int__code_I1_J,axiom,
% 0.32/0.53 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.32/0.53
% 0.32/0.53 % less_eq_int_code(1)
% 0.32/0.53 thf(fact_2512_add__lessD1,axiom,
% 0.32/0.53 ! [I: nat,J: nat,K: nat] :
% 0.32/0.53 ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.32/0.53 => ( ord_less_nat @ I @ K ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_lessD1
% 0.32/0.53 thf(fact_2513_add__less__mono,axiom,
% 0.32/0.53 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.53 ( ( ord_less_nat @ I @ J )
% 0.32/0.53 => ( ( ord_less_nat @ K @ L )
% 0.32/0.53 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_less_mono
% 0.32/0.53 thf(fact_2514_not__add__less1,axiom,
% 0.32/0.53 ! [I: nat,J: nat] :
% 0.32/0.53 ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 0.32/0.53
% 0.32/0.53 % not_add_less1
% 0.32/0.53 thf(fact_2515_not__add__less2,axiom,
% 0.32/0.53 ! [J: nat,I: nat] :
% 0.32/0.53 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 0.32/0.53
% 0.32/0.53 % not_add_less2
% 0.32/0.53 thf(fact_2516_add__less__mono1,axiom,
% 0.32/0.53 ! [I: nat,J: nat,K: nat] :
% 0.32/0.53 ( ( ord_less_nat @ I @ J )
% 0.32/0.53 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_less_mono1
% 0.32/0.53 thf(fact_2517_trans__less__add1,axiom,
% 0.32/0.53 ! [I: nat,J: nat,M: nat] :
% 0.32/0.53 ( ( ord_less_nat @ I @ J )
% 0.32/0.53 => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % trans_less_add1
% 0.32/0.53 thf(fact_2518_trans__less__add2,axiom,
% 0.32/0.53 ! [I: nat,J: nat,M: nat] :
% 0.32/0.53 ( ( ord_less_nat @ I @ J )
% 0.32/0.53 => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % trans_less_add2
% 0.32/0.53 thf(fact_2519_less__add__eq__less,axiom,
% 0.32/0.53 ! [K: nat,L: nat,M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ K @ L )
% 0.32/0.53 => ( ( ( plus_plus_nat @ M @ L )
% 0.32/0.53 = ( plus_plus_nat @ K @ N ) )
% 0.32/0.53 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_add_eq_less
% 0.32/0.53 thf(fact_2520_add__leE,axiom,
% 0.32/0.53 ! [M: nat,K: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.32/0.53 => ~ ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.53 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_leE
% 0.32/0.53 thf(fact_2521_le__add1,axiom,
% 0.32/0.53 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_add1
% 0.32/0.53 thf(fact_2522_le__add2,axiom,
% 0.32/0.53 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_add2
% 0.32/0.53 thf(fact_2523_add__leD1,axiom,
% 0.32/0.53 ! [M: nat,K: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.32/0.53 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_leD1
% 0.32/0.53 thf(fact_2524_add__leD2,axiom,
% 0.32/0.53 ! [M: nat,K: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.32/0.53 => ( ord_less_eq_nat @ K @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_leD2
% 0.32/0.53 thf(fact_2525_le__Suc__ex,axiom,
% 0.32/0.53 ! [K: nat,L: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ K @ L )
% 0.32/0.53 => ? [N2: nat] :
% 0.32/0.53 ( L
% 0.32/0.53 = ( plus_plus_nat @ K @ N2 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_Suc_ex
% 0.32/0.53 thf(fact_2526_add__le__mono,axiom,
% 0.32/0.53 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.53 => ( ( ord_less_eq_nat @ K @ L )
% 0.32/0.53 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_le_mono
% 0.32/0.53 thf(fact_2527_add__le__mono1,axiom,
% 0.32/0.53 ! [I: nat,J: nat,K: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.53 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_le_mono1
% 0.32/0.53 thf(fact_2528_trans__le__add1,axiom,
% 0.32/0.53 ! [I: nat,J: nat,M: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.53 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % trans_le_add1
% 0.32/0.53 thf(fact_2529_trans__le__add2,axiom,
% 0.32/0.53 ! [I: nat,J: nat,M: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.53 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % trans_le_add2
% 0.32/0.53 thf(fact_2530_nat__le__iff__add,axiom,
% 0.32/0.53 ( ord_less_eq_nat
% 0.32/0.53 = ( ^ [M3: nat,N3: nat] :
% 0.32/0.53 ? [K2: nat] :
% 0.32/0.53 ( N3
% 0.32/0.53 = ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % nat_le_iff_add
% 0.32/0.53 thf(fact_2531_less__imp__diff__less,axiom,
% 0.32/0.53 ! [J: nat,K: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ J @ K )
% 0.32/0.53 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_imp_diff_less
% 0.32/0.54 thf(fact_2532_diff__less__mono2,axiom,
% 0.32/0.54 ! [M: nat,N: nat,L: nat] :
% 0.32/0.54 ( ( ord_less_nat @ M @ N )
% 0.32/0.54 => ( ( ord_less_nat @ M @ L )
% 0.32/0.54 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % diff_less_mono2
% 0.32/0.54 thf(fact_2533_diff__le__mono2,axiom,
% 0.32/0.54 ! [M: nat,N: nat,L: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.54 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % diff_le_mono2
% 0.32/0.54 thf(fact_2534_le__diff__iff_H,axiom,
% 0.32/0.54 ! [A: nat,C: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ A @ C )
% 0.32/0.54 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.54 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.32/0.54 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_diff_iff'
% 0.32/0.54 thf(fact_2535_diff__le__self,axiom,
% 0.32/0.54 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.32/0.54
% 0.32/0.54 % diff_le_self
% 0.32/0.54 thf(fact_2536_diff__le__mono,axiom,
% 0.32/0.54 ! [M: nat,N: nat,L: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.54 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % diff_le_mono
% 0.32/0.54 thf(fact_2537_Nat_Odiff__diff__eq,axiom,
% 0.32/0.54 ! [K: nat,M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ K @ M )
% 0.32/0.54 => ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.54 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.32/0.54 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % Nat.diff_diff_eq
% 0.32/0.54 thf(fact_2538_le__diff__iff,axiom,
% 0.32/0.54 ! [K: nat,M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ K @ M )
% 0.32/0.54 => ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.54 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.32/0.54 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_diff_iff
% 0.32/0.54 thf(fact_2539_eq__diff__iff,axiom,
% 0.32/0.54 ! [K: nat,M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ K @ M )
% 0.32/0.54 => ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.54 => ( ( ( minus_minus_nat @ M @ K )
% 0.32/0.54 = ( minus_minus_nat @ N @ K ) )
% 0.32/0.54 = ( M = N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % eq_diff_iff
% 0.32/0.54 thf(fact_2540_le__cube,axiom,
% 0.32/0.54 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_cube
% 0.32/0.54 thf(fact_2541_le__square,axiom,
% 0.32/0.54 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_square
% 0.32/0.54 thf(fact_2542_mult__le__mono,axiom,
% 0.32/0.54 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.54 => ( ( ord_less_eq_nat @ K @ L )
% 0.32/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_mono
% 0.32/0.54 thf(fact_2543_mult__le__mono1,axiom,
% 0.32/0.54 ! [I: nat,J: nat,K: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_mono1
% 0.32/0.54 thf(fact_2544_mult__le__mono2,axiom,
% 0.32/0.54 ! [I: nat,J: nat,K: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_mono2
% 0.32/0.54 thf(fact_2545_real__arch__pow,axiom,
% 0.32/0.54 ! [X: real,Y: real] :
% 0.32/0.54 ( ( ord_less_real @ one_one_real @ X )
% 0.32/0.54 => ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % real_arch_pow
% 0.32/0.54 thf(fact_2546_real__arch__pow__inv,axiom,
% 0.32/0.54 ! [Y: real,X: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.54 => ( ( ord_less_real @ X @ one_one_real )
% 0.32/0.54 => ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % real_arch_pow_inv
% 0.32/0.54 thf(fact_2547_div__le__dividend,axiom,
% 0.32/0.54 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 0.32/0.54
% 0.32/0.54 % div_le_dividend
% 0.32/0.54 thf(fact_2548_div__le__mono,axiom,
% 0.32/0.54 ! [M: nat,N: nat,K: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.54 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % div_le_mono
% 0.32/0.54 thf(fact_2549_mult__le__cancel__left1,axiom,
% 0.32/0.54 ! [C: real,B: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_eq_real @ one_one_real @ B ) )
% 0.32/0.54 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_cancel_left1
% 0.32/0.54 thf(fact_2550_mult__le__cancel__left1,axiom,
% 0.32/0.54 ! [C: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.32/0.54 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_cancel_left1
% 0.32/0.54 thf(fact_2551_mult__le__cancel__left1,axiom,
% 0.32/0.54 ! [C: int,B: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 0.32/0.54 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.32/0.54 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.54 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_cancel_left1
% 0.32/0.54 thf(fact_2552_mult__le__cancel__left2,axiom,
% 0.32/0.54 ! [C: real,A: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.32/0.54 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_cancel_left2
% 0.32/0.54 thf(fact_2553_mult__le__cancel__left2,axiom,
% 0.32/0.54 ! [C: rat,A: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.32/0.54 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_cancel_left2
% 0.32/0.54 thf(fact_2554_mult__le__cancel__left2,axiom,
% 0.32/0.54 ! [C: int,A: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 0.32/0.54 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.32/0.54 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.54 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_cancel_left2
% 0.32/0.54 thf(fact_2555_mult__le__cancel__right1,axiom,
% 0.32/0.54 ! [C: real,B: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_eq_real @ one_one_real @ B ) )
% 0.32/0.54 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_cancel_right1
% 0.32/0.54 thf(fact_2556_mult__le__cancel__right1,axiom,
% 0.32/0.54 ! [C: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.32/0.54 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_cancel_right1
% 0.32/0.54 thf(fact_2557_mult__le__cancel__right1,axiom,
% 0.32/0.54 ! [C: int,B: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 0.32/0.54 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.32/0.54 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.54 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_cancel_right1
% 0.32/0.54 thf(fact_2558_mult__le__cancel__right2,axiom,
% 0.32/0.54 ! [A: real,C: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.32/0.54 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_cancel_right2
% 0.32/0.54 thf(fact_2559_mult__le__cancel__right2,axiom,
% 0.32/0.54 ! [A: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.32/0.54 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_cancel_right2
% 0.32/0.54 thf(fact_2560_mult__le__cancel__right2,axiom,
% 0.32/0.54 ! [A: int,C: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 0.32/0.54 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.32/0.54 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.54 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_cancel_right2
% 0.32/0.54 thf(fact_2561_mult__less__cancel__left1,axiom,
% 0.32/0.54 ! [C: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
% 0.32/0.54 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_real @ one_one_real @ B ) )
% 0.32/0.54 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_left1
% 0.32/0.54 thf(fact_2562_mult__less__cancel__left1,axiom,
% 0.32/0.54 ! [C: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.32/0.54 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_rat @ one_one_rat @ B ) )
% 0.32/0.54 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_left1
% 0.32/0.54 thf(fact_2563_mult__less__cancel__left1,axiom,
% 0.32/0.54 ! [C: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 0.32/0.54 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ord_less_int @ one_one_int @ B ) )
% 0.32/0.54 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.54 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_left1
% 0.32/0.54 thf(fact_2564_mult__less__cancel__left2,axiom,
% 0.32/0.54 ! [C: real,A: real] :
% 0.32/0.54 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 0.32/0.54 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_real @ A @ one_one_real ) )
% 0.32/0.54 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_left2
% 0.32/0.54 thf(fact_2565_mult__less__cancel__left2,axiom,
% 0.32/0.54 ! [C: rat,A: rat] :
% 0.32/0.54 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.32/0.54 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_rat @ A @ one_one_rat ) )
% 0.32/0.54 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_left2
% 0.32/0.54 thf(fact_2566_mult__less__cancel__left2,axiom,
% 0.32/0.54 ! [C: int,A: int] :
% 0.32/0.54 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 0.32/0.54 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ord_less_int @ A @ one_one_int ) )
% 0.32/0.54 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.54 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_left2
% 0.32/0.54 thf(fact_2567_mult__less__cancel__right1,axiom,
% 0.32/0.54 ! [C: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
% 0.32/0.54 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_real @ one_one_real @ B ) )
% 0.32/0.54 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_right1
% 0.32/0.54 thf(fact_2568_mult__less__cancel__right1,axiom,
% 0.32/0.54 ! [C: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.32/0.54 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_rat @ one_one_rat @ B ) )
% 0.32/0.54 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_right1
% 0.32/0.54 thf(fact_2569_mult__less__cancel__right1,axiom,
% 0.32/0.54 ! [C: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 0.32/0.54 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ord_less_int @ one_one_int @ B ) )
% 0.32/0.54 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.54 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_right1
% 0.32/0.54 thf(fact_2570_mult__less__cancel__right2,axiom,
% 0.32/0.54 ! [A: real,C: real] :
% 0.32/0.54 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 0.32/0.54 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_real @ A @ one_one_real ) )
% 0.32/0.54 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_right2
% 0.32/0.54 thf(fact_2571_mult__less__cancel__right2,axiom,
% 0.32/0.54 ! [A: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.32/0.54 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_rat @ A @ one_one_rat ) )
% 0.32/0.54 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_right2
% 0.32/0.54 thf(fact_2572_mult__less__cancel__right2,axiom,
% 0.32/0.54 ! [A: int,C: int] :
% 0.32/0.54 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 0.32/0.54 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ord_less_int @ A @ one_one_int ) )
% 0.32/0.54 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.54 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_right2
% 0.32/0.54 thf(fact_2573_field__le__mult__one__interval,axiom,
% 0.32/0.54 ! [X: real,Y: real] :
% 0.32/0.54 ( ! [Z4: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ Z4 )
% 0.32/0.54 => ( ( ord_less_real @ Z4 @ one_one_real )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ Z4 @ X ) @ Y ) ) )
% 0.32/0.54 => ( ord_less_eq_real @ X @ Y ) ) ).
% 0.32/0.54
% 0.32/0.54 % field_le_mult_one_interval
% 0.32/0.54 thf(fact_2574_field__le__mult__one__interval,axiom,
% 0.32/0.54 ! [X: rat,Y: rat] :
% 0.32/0.54 ( ! [Z4: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ Z4 )
% 0.32/0.54 => ( ( ord_less_rat @ Z4 @ one_one_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ Z4 @ X ) @ Y ) ) )
% 0.32/0.54 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.32/0.54
% 0.32/0.54 % field_le_mult_one_interval
% 0.32/0.54 thf(fact_2575_divide__left__mono__neg,axiom,
% 0.32/0.54 ! [A: real,B: real,C: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.54 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_left_mono_neg
% 0.32/0.54 thf(fact_2576_divide__left__mono__neg,axiom,
% 0.32/0.54 ! [A: rat,B: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.54 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_left_mono_neg
% 0.32/0.54 thf(fact_2577_mult__imp__le__div__pos,axiom,
% 0.32/0.54 ! [Y: real,Z2: real,X: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.54 => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ Y ) @ X )
% 0.32/0.54 => ( ord_less_eq_real @ Z2 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_imp_le_div_pos
% 0.32/0.54 thf(fact_2578_mult__imp__le__div__pos,axiom,
% 0.32/0.54 ! [Y: rat,Z2: rat,X: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.32/0.54 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ Y ) @ X )
% 0.32/0.54 => ( ord_less_eq_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_imp_le_div_pos
% 0.32/0.54 thf(fact_2579_mult__imp__div__pos__le,axiom,
% 0.32/0.54 ! [Y: real,X: real,Z2: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.54 => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z2 @ Y ) )
% 0.32/0.54 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z2 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_imp_div_pos_le
% 0.32/0.54 thf(fact_2580_mult__imp__div__pos__le,axiom,
% 0.32/0.54 ! [Y: rat,X: rat,Z2: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.32/0.54 => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z2 @ Y ) )
% 0.32/0.54 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_imp_div_pos_le
% 0.32/0.54 thf(fact_2581_pos__le__divide__eq,axiom,
% 0.32/0.54 ! [C: real,A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.54 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pos_le_divide_eq
% 0.32/0.54 thf(fact_2582_pos__le__divide__eq,axiom,
% 0.32/0.54 ! [C: rat,A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.54 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pos_le_divide_eq
% 0.32/0.54 thf(fact_2583_pos__divide__le__eq,axiom,
% 0.32/0.54 ! [C: real,B: real,A: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.54 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pos_divide_le_eq
% 0.32/0.54 thf(fact_2584_pos__divide__le__eq,axiom,
% 0.32/0.54 ! [C: rat,B: rat,A: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.54 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pos_divide_le_eq
% 0.32/0.54 thf(fact_2585_neg__le__divide__eq,axiom,
% 0.32/0.54 ! [C: real,A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.54 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % neg_le_divide_eq
% 0.32/0.54 thf(fact_2586_neg__le__divide__eq,axiom,
% 0.32/0.54 ! [C: rat,A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.54 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % neg_le_divide_eq
% 0.32/0.54 thf(fact_2587_neg__divide__le__eq,axiom,
% 0.32/0.54 ! [C: real,B: real,A: real] :
% 0.32/0.54 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.54 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % neg_divide_le_eq
% 0.32/0.54 thf(fact_2588_neg__divide__le__eq,axiom,
% 0.32/0.54 ! [C: rat,B: rat,A: rat] :
% 0.32/0.54 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.54 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % neg_divide_le_eq
% 0.32/0.54 thf(fact_2589_divide__left__mono,axiom,
% 0.32/0.54 ! [B: real,A: real,C: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.54 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_left_mono
% 0.32/0.54 thf(fact_2590_divide__left__mono,axiom,
% 0.32/0.54 ! [B: rat,A: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.54 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_left_mono
% 0.32/0.54 thf(fact_2591_le__divide__eq,axiom,
% 0.32/0.54 ! [A: real,B: real,C: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.32/0.54 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.32/0.54 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_divide_eq
% 0.32/0.54 thf(fact_2592_le__divide__eq,axiom,
% 0.32/0.54 ! [A: rat,B: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.32/0.54 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.32/0.54 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_divide_eq
% 0.32/0.54 thf(fact_2593_divide__le__eq,axiom,
% 0.32/0.54 ! [B: real,C: real,A: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.32/0.54 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.32/0.54 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_le_eq
% 0.32/0.54 thf(fact_2594_divide__le__eq,axiom,
% 0.32/0.54 ! [B: rat,C: rat,A: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.32/0.54 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.32/0.54 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_le_eq
% 0.32/0.54 thf(fact_2595_le__divide__eq__1,axiom,
% 0.32/0.54 ! [B: real,A: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 & ( ord_less_eq_real @ A @ B ) )
% 0.32/0.54 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.54 & ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_divide_eq_1
% 0.32/0.54 thf(fact_2596_le__divide__eq__1,axiom,
% 0.32/0.54 ! [B: rat,A: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.54 & ( ord_less_eq_rat @ A @ B ) )
% 0.32/0.54 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.54 & ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_divide_eq_1
% 0.32/0.54 thf(fact_2597_divide__le__eq__1,axiom,
% 0.32/0.54 ! [B: real,A: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 & ( ord_less_eq_real @ B @ A ) )
% 0.32/0.54 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.54 & ( ord_less_eq_real @ A @ B ) )
% 0.32/0.54 | ( A = zero_zero_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_le_eq_1
% 0.32/0.54 thf(fact_2598_divide__le__eq__1,axiom,
% 0.32/0.54 ! [B: rat,A: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.54 & ( ord_less_eq_rat @ B @ A ) )
% 0.32/0.54 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.54 & ( ord_less_eq_rat @ A @ B ) )
% 0.32/0.54 | ( A = zero_zero_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_le_eq_1
% 0.32/0.54 thf(fact_2599_power__strict__decreasing,axiom,
% 0.32/0.54 ! [N: nat,N5: nat,A: real] :
% 0.32/0.54 ( ( ord_less_nat @ N @ N5 )
% 0.32/0.54 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_real @ A @ one_one_real )
% 0.32/0.54 => ( ord_less_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_strict_decreasing
% 0.32/0.54 thf(fact_2600_power__strict__decreasing,axiom,
% 0.32/0.54 ! [N: nat,N5: nat,A: rat] :
% 0.32/0.54 ( ( ord_less_nat @ N @ N5 )
% 0.32/0.54 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.32/0.54 => ( ord_less_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_strict_decreasing
% 0.32/0.54 thf(fact_2601_power__strict__decreasing,axiom,
% 0.32/0.54 ! [N: nat,N5: nat,A: nat] :
% 0.32/0.54 ( ( ord_less_nat @ N @ N5 )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.32/0.54 => ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_strict_decreasing
% 0.32/0.54 thf(fact_2602_power__strict__decreasing,axiom,
% 0.32/0.54 ! [N: nat,N5: nat,A: int] :
% 0.32/0.54 ( ( ord_less_nat @ N @ N5 )
% 0.32/0.54 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_int @ A @ one_one_int )
% 0.32/0.54 => ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_strict_decreasing
% 0.32/0.54 thf(fact_2603_power__decreasing,axiom,
% 0.32/0.54 ! [N: nat,N5: nat,A: real] :
% 0.32/0.54 ( ( ord_less_eq_nat @ N @ N5 )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.32/0.54 => ( ord_less_eq_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_decreasing
% 0.32/0.54 thf(fact_2604_power__decreasing,axiom,
% 0.32/0.54 ! [N: nat,N5: nat,A: rat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ N @ N5 )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_decreasing
% 0.32/0.54 thf(fact_2605_power__decreasing,axiom,
% 0.32/0.54 ! [N: nat,N5: nat,A: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ N @ N5 )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.32/0.54 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_decreasing
% 0.32/0.54 thf(fact_2606_power__decreasing,axiom,
% 0.32/0.54 ! [N: nat,N5: nat,A: int] :
% 0.32/0.54 ( ( ord_less_eq_nat @ N @ N5 )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.32/0.54 => ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_decreasing
% 0.32/0.54 thf(fact_2607_power__eq__imp__eq__base,axiom,
% 0.32/0.54 ! [A: real,N: nat,B: real] :
% 0.32/0.54 ( ( ( power_power_real @ A @ N )
% 0.32/0.54 = ( power_power_real @ B @ N ) )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( A = B ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_eq_imp_eq_base
% 0.32/0.54 thf(fact_2608_power__eq__imp__eq__base,axiom,
% 0.32/0.54 ! [A: rat,N: nat,B: rat] :
% 0.32/0.54 ( ( ( power_power_rat @ A @ N )
% 0.32/0.54 = ( power_power_rat @ B @ N ) )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( A = B ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_eq_imp_eq_base
% 0.32/0.54 thf(fact_2609_power__eq__imp__eq__base,axiom,
% 0.32/0.54 ! [A: nat,N: nat,B: nat] :
% 0.32/0.54 ( ( ( power_power_nat @ A @ N )
% 0.32/0.54 = ( power_power_nat @ B @ N ) )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( A = B ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_eq_imp_eq_base
% 0.32/0.54 thf(fact_2610_power__eq__imp__eq__base,axiom,
% 0.32/0.54 ! [A: int,N: nat,B: int] :
% 0.32/0.54 ( ( ( power_power_int @ A @ N )
% 0.32/0.54 = ( power_power_int @ B @ N ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( A = B ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_eq_imp_eq_base
% 0.32/0.54 thf(fact_2611_power__eq__iff__eq__base,axiom,
% 0.32/0.54 ! [N: nat,A: real,B: real] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.54 => ( ( ( power_power_real @ A @ N )
% 0.32/0.54 = ( power_power_real @ B @ N ) )
% 0.32/0.54 = ( A = B ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_eq_iff_eq_base
% 0.32/0.54 thf(fact_2612_power__eq__iff__eq__base,axiom,
% 0.32/0.54 ! [N: nat,A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.54 => ( ( ( power_power_rat @ A @ N )
% 0.32/0.54 = ( power_power_rat @ B @ N ) )
% 0.32/0.54 = ( A = B ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_eq_iff_eq_base
% 0.32/0.54 thf(fact_2613_power__eq__iff__eq__base,axiom,
% 0.32/0.54 ! [N: nat,A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.54 => ( ( ( power_power_nat @ A @ N )
% 0.32/0.54 = ( power_power_nat @ B @ N ) )
% 0.32/0.54 = ( A = B ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_eq_iff_eq_base
% 0.32/0.54 thf(fact_2614_power__eq__iff__eq__base,axiom,
% 0.32/0.54 ! [N: nat,A: int,B: int] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.54 => ( ( ( power_power_int @ A @ N )
% 0.32/0.54 = ( power_power_int @ B @ N ) )
% 0.32/0.54 = ( A = B ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_eq_iff_eq_base
% 0.32/0.54 thf(fact_2615_one__less__power,axiom,
% 0.32/0.54 ! [A: real,N: nat] :
% 0.32/0.54 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % one_less_power
% 0.32/0.54 thf(fact_2616_one__less__power,axiom,
% 0.32/0.54 ! [A: rat,N: nat] :
% 0.32/0.54 ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % one_less_power
% 0.32/0.54 thf(fact_2617_one__less__power,axiom,
% 0.32/0.54 ! [A: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % one_less_power
% 0.32/0.54 thf(fact_2618_one__less__power,axiom,
% 0.32/0.54 ! [A: int,N: nat] :
% 0.32/0.54 ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % one_less_power
% 0.32/0.54 thf(fact_2619_self__le__power,axiom,
% 0.32/0.54 ! [A: real,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_real @ one_one_real @ A )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % self_le_power
% 0.32/0.54 thf(fact_2620_self__le__power,axiom,
% 0.32/0.54 ! [A: rat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % self_le_power
% 0.32/0.54 thf(fact_2621_self__le__power,axiom,
% 0.32/0.54 ! [A: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % self_le_power
% 0.32/0.54 thf(fact_2622_self__le__power,axiom,
% 0.32/0.54 ! [A: int,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_int @ one_one_int @ A )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % self_le_power
% 0.32/0.54 thf(fact_2623_zero__less__imp__eq__int,axiom,
% 0.32/0.54 ! [K: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.54 => ? [N2: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.54 & ( K
% 0.32/0.54 = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_imp_eq_int
% 0.32/0.54 thf(fact_2624_pos__int__cases,axiom,
% 0.32/0.54 ! [K: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.54 => ~ ! [N2: nat] :
% 0.32/0.54 ( ( K
% 0.32/0.54 = ( semiri1314217659103216013at_int @ N2 ) )
% 0.32/0.54 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pos_int_cases
% 0.32/0.54 thf(fact_2625_nat__less__add__iff2,axiom,
% 0.32/0.54 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.54 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.32/0.54 = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % nat_less_add_iff2
% 0.32/0.54 thf(fact_2626_nat__less__add__iff1,axiom,
% 0.32/0.54 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ J @ I )
% 0.32/0.54 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.32/0.54 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % nat_less_add_iff1
% 0.32/0.54 thf(fact_2627_zmult__zless__mono2__lemma,axiom,
% 0.32/0.54 ! [I: int,J: int,K: nat] :
% 0.32/0.54 ( ( ord_less_int @ I @ J )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.54 => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zmult_zless_mono2_lemma
% 0.32/0.54 thf(fact_2628_le__imp__0__less,axiom,
% 0.32/0.54 ! [Z2: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.32/0.54 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_imp_0_less
% 0.32/0.54 thf(fact_2629_div__nat__eqI,axiom,
% 0.32/0.54 ! [N: nat,Q2: nat,M: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
% 0.32/0.54 => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
% 0.32/0.54 => ( ( divide_divide_nat @ M @ N )
% 0.32/0.54 = Q2 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % div_nat_eqI
% 0.32/0.54 thf(fact_2630_less__eq__div__iff__mult__less__eq,axiom,
% 0.32/0.54 ! [Q2: nat,M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 0.32/0.54 => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
% 0.32/0.54 = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_eq_div_iff_mult_less_eq
% 0.32/0.54 thf(fact_2631_unique__quotient__lemma__neg,axiom,
% 0.32/0.54 ! [B: int,Q4: int,R4: int,Q2: int,R: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_int @ B @ R )
% 0.32/0.54 => ( ( ord_less_int @ B @ R4 )
% 0.32/0.54 => ( ord_less_eq_int @ Q2 @ Q4 ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % unique_quotient_lemma_neg
% 0.32/0.54 thf(fact_2632_unique__quotient__lemma,axiom,
% 0.32/0.54 ! [B: int,Q4: int,R4: int,Q2: int,R: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.32/0.54 => ( ( ord_less_int @ R4 @ B )
% 0.32/0.54 => ( ( ord_less_int @ R @ B )
% 0.32/0.54 => ( ord_less_eq_int @ Q4 @ Q2 ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % unique_quotient_lemma
% 0.32/0.54 thf(fact_2633_zdiv__mono2__neg__lemma,axiom,
% 0.32/0.54 ! [B: int,Q2: int,R: int,B6: int,Q4: int,R4: int] :
% 0.32/0.54 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R )
% 0.32/0.54 = ( plus_plus_int @ ( times_times_int @ B6 @ Q4 ) @ R4 ) )
% 0.32/0.54 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q4 ) @ R4 ) @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_int @ R @ B )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.32/0.54 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.32/0.54 => ( ( ord_less_eq_int @ B6 @ B )
% 0.32/0.54 => ( ord_less_eq_int @ Q4 @ Q2 ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zdiv_mono2_neg_lemma
% 0.32/0.54 thf(fact_2634_incr__mult__lemma,axiom,
% 0.32/0.54 ! [D: int,P: int > $o,K: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ D )
% 0.32/0.54 => ( ! [X4: int] :
% 0.32/0.54 ( ( P @ X4 )
% 0.32/0.54 => ( P @ ( plus_plus_int @ X4 @ D ) ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.54 => ! [X5: int] :
% 0.32/0.54 ( ( P @ X5 )
% 0.32/0.54 => ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % incr_mult_lemma
% 0.32/0.54 thf(fact_2635_zdiv__mono2__lemma,axiom,
% 0.32/0.54 ! [B: int,Q2: int,R: int,B6: int,Q4: int,R4: int] :
% 0.32/0.54 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R )
% 0.32/0.54 = ( plus_plus_int @ ( times_times_int @ B6 @ Q4 ) @ R4 ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q4 ) @ R4 ) )
% 0.32/0.54 => ( ( ord_less_int @ R4 @ B6 )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.32/0.54 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.32/0.54 => ( ( ord_less_eq_int @ B6 @ B )
% 0.32/0.54 => ( ord_less_eq_int @ Q2 @ Q4 ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zdiv_mono2_lemma
% 0.32/0.54 thf(fact_2636_q__pos__lemma,axiom,
% 0.32/0.54 ! [B6: int,Q4: int,R4: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q4 ) @ R4 ) )
% 0.32/0.54 => ( ( ord_less_int @ R4 @ B6 )
% 0.32/0.54 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.32/0.54 => ( ord_less_eq_int @ zero_zero_int @ Q4 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % q_pos_lemma
% 0.32/0.54 thf(fact_2637_real__archimedian__rdiv__eq__0,axiom,
% 0.32/0.54 ! [X: real,C: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ! [M2: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ M2 )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ X ) @ C ) )
% 0.32/0.54 => ( X = zero_zero_real ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % real_archimedian_rdiv_eq_0
% 0.32/0.54 thf(fact_2638_decr__mult__lemma,axiom,
% 0.32/0.54 ! [D: int,P: int > $o,K: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ D )
% 0.32/0.54 => ( ! [X4: int] :
% 0.32/0.54 ( ( P @ X4 )
% 0.32/0.54 => ( P @ ( minus_minus_int @ X4 @ D ) ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.54 => ! [X5: int] :
% 0.32/0.54 ( ( P @ X5 )
% 0.32/0.54 => ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % decr_mult_lemma
% 0.32/0.54 thf(fact_2639_power__dvd__imp__le,axiom,
% 0.32/0.54 ! [I: nat,M: nat,N: nat] :
% 0.32/0.54 ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.32/0.54 => ( ( ord_less_nat @ one_one_nat @ I )
% 0.32/0.54 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_dvd_imp_le
% 0.32/0.54 thf(fact_2640_convex__bound__lt,axiom,
% 0.32/0.54 ! [X: real,A: real,Y: real,U: real,V: real] :
% 0.32/0.54 ( ( ord_less_real @ X @ A )
% 0.32/0.54 => ( ( ord_less_real @ Y @ A )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.32/0.54 => ( ( ( plus_plus_real @ U @ V )
% 0.32/0.54 = one_one_real )
% 0.32/0.54 => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % convex_bound_lt
% 0.32/0.54 thf(fact_2641_convex__bound__lt,axiom,
% 0.32/0.54 ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
% 0.32/0.54 ( ( ord_less_rat @ X @ A )
% 0.32/0.54 => ( ( ord_less_rat @ Y @ A )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.32/0.54 => ( ( ( plus_plus_rat @ U @ V )
% 0.32/0.54 = one_one_rat )
% 0.32/0.54 => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % convex_bound_lt
% 0.32/0.54 thf(fact_2642_convex__bound__lt,axiom,
% 0.32/0.54 ! [X: int,A: int,Y: int,U: int,V: int] :
% 0.32/0.54 ( ( ord_less_int @ X @ A )
% 0.32/0.54 => ( ( ord_less_int @ Y @ A )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.32/0.54 => ( ( ( plus_plus_int @ U @ V )
% 0.32/0.54 = one_one_int )
% 0.32/0.54 => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % convex_bound_lt
% 0.32/0.54 thf(fact_2643_divide__le__eq__numeral_I1_J,axiom,
% 0.32/0.54 ! [B: real,C: real,W: num] :
% 0.32/0.54 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.32/0.54 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.32/0.54 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_le_eq_numeral(1)
% 0.32/0.54 thf(fact_2644_divide__le__eq__numeral_I1_J,axiom,
% 0.32/0.54 ! [B: rat,C: rat,W: num] :
% 0.32/0.54 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.32/0.54 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.32/0.54 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_le_eq_numeral(1)
% 0.32/0.54 thf(fact_2645_le__divide__eq__numeral_I1_J,axiom,
% 0.32/0.54 ! [W: num,B: real,C: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.32/0.54 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.32/0.54 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_divide_eq_numeral(1)
% 0.32/0.54 thf(fact_2646_le__divide__eq__numeral_I1_J,axiom,
% 0.32/0.54 ! [W: num,B: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.32/0.54 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.32/0.54 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_divide_eq_numeral(1)
% 0.32/0.54 thf(fact_2647_of__bool__def,axiom,
% 0.32/0.54 ( zero_n1201886186963655149omplex
% 0.32/0.54 = ( ^ [P3: $o] : ( if_complex @ P3 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % of_bool_def
% 0.32/0.54 thf(fact_2648_of__bool__def,axiom,
% 0.32/0.54 ( zero_n3304061248610475627l_real
% 0.32/0.54 = ( ^ [P3: $o] : ( if_real @ P3 @ one_one_real @ zero_zero_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % of_bool_def
% 0.32/0.54 thf(fact_2649_of__bool__def,axiom,
% 0.32/0.54 ( zero_n2052037380579107095ol_rat
% 0.32/0.54 = ( ^ [P3: $o] : ( if_rat @ P3 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % of_bool_def
% 0.32/0.54 thf(fact_2650_of__bool__def,axiom,
% 0.32/0.54 ( zero_n2687167440665602831ol_nat
% 0.32/0.54 = ( ^ [P3: $o] : ( if_nat @ P3 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % of_bool_def
% 0.32/0.54 thf(fact_2651_of__bool__def,axiom,
% 0.32/0.54 ( zero_n2684676970156552555ol_int
% 0.32/0.54 = ( ^ [P3: $o] : ( if_int @ P3 @ one_one_int @ zero_zero_int ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % of_bool_def
% 0.32/0.54 thf(fact_2652_of__bool__def,axiom,
% 0.32/0.54 ( zero_n356916108424825756nteger
% 0.32/0.54 = ( ^ [P3: $o] : ( if_Code_integer @ P3 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % of_bool_def
% 0.32/0.54 thf(fact_2653_split__of__bool,axiom,
% 0.32/0.54 ! [P: complex > $o,P2: $o] :
% 0.32/0.54 ( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
% 0.32/0.54 = ( ( P2
% 0.32/0.54 => ( P @ one_one_complex ) )
% 0.32/0.54 & ( ~ P2
% 0.32/0.54 => ( P @ zero_zero_complex ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_of_bool
% 0.32/0.54 thf(fact_2654_split__of__bool,axiom,
% 0.32/0.54 ! [P: real > $o,P2: $o] :
% 0.32/0.54 ( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
% 0.32/0.54 = ( ( P2
% 0.32/0.54 => ( P @ one_one_real ) )
% 0.32/0.54 & ( ~ P2
% 0.32/0.54 => ( P @ zero_zero_real ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_of_bool
% 0.32/0.54 thf(fact_2655_split__of__bool,axiom,
% 0.32/0.54 ! [P: rat > $o,P2: $o] :
% 0.32/0.54 ( ( P @ ( zero_n2052037380579107095ol_rat @ P2 ) )
% 0.32/0.54 = ( ( P2
% 0.32/0.54 => ( P @ one_one_rat ) )
% 0.32/0.54 & ( ~ P2
% 0.32/0.54 => ( P @ zero_zero_rat ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_of_bool
% 0.32/0.54 thf(fact_2656_split__of__bool,axiom,
% 0.32/0.54 ! [P: nat > $o,P2: $o] :
% 0.32/0.54 ( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
% 0.32/0.54 = ( ( P2
% 0.32/0.54 => ( P @ one_one_nat ) )
% 0.32/0.54 & ( ~ P2
% 0.32/0.54 => ( P @ zero_zero_nat ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_of_bool
% 0.32/0.54 thf(fact_2657_split__of__bool,axiom,
% 0.32/0.54 ! [P: int > $o,P2: $o] :
% 0.32/0.54 ( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
% 0.32/0.54 = ( ( P2
% 0.32/0.54 => ( P @ one_one_int ) )
% 0.32/0.54 & ( ~ P2
% 0.32/0.54 => ( P @ zero_zero_int ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_of_bool
% 0.32/0.54 thf(fact_2658_split__of__bool,axiom,
% 0.32/0.54 ! [P: code_integer > $o,P2: $o] :
% 0.32/0.54 ( ( P @ ( zero_n356916108424825756nteger @ P2 ) )
% 0.32/0.54 = ( ( P2
% 0.32/0.54 => ( P @ one_one_Code_integer ) )
% 0.32/0.54 & ( ~ P2
% 0.32/0.54 => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_of_bool
% 0.32/0.54 thf(fact_2659_split__of__bool__asm,axiom,
% 0.32/0.54 ! [P: complex > $o,P2: $o] :
% 0.32/0.54 ( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
% 0.32/0.54 = ( ~ ( ( P2
% 0.32/0.54 & ~ ( P @ one_one_complex ) )
% 0.32/0.54 | ( ~ P2
% 0.32/0.54 & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_of_bool_asm
% 0.32/0.54 thf(fact_2660_split__of__bool__asm,axiom,
% 0.32/0.54 ! [P: real > $o,P2: $o] :
% 0.32/0.54 ( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
% 0.32/0.54 = ( ~ ( ( P2
% 0.32/0.54 & ~ ( P @ one_one_real ) )
% 0.32/0.54 | ( ~ P2
% 0.32/0.54 & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_of_bool_asm
% 0.32/0.54 thf(fact_2661_split__of__bool__asm,axiom,
% 0.32/0.54 ! [P: rat > $o,P2: $o] :
% 0.32/0.54 ( ( P @ ( zero_n2052037380579107095ol_rat @ P2 ) )
% 0.32/0.54 = ( ~ ( ( P2
% 0.32/0.54 & ~ ( P @ one_one_rat ) )
% 0.32/0.54 | ( ~ P2
% 0.32/0.54 & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_of_bool_asm
% 0.32/0.54 thf(fact_2662_split__of__bool__asm,axiom,
% 0.32/0.54 ! [P: nat > $o,P2: $o] :
% 0.32/0.54 ( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
% 0.32/0.54 = ( ~ ( ( P2
% 0.32/0.54 & ~ ( P @ one_one_nat ) )
% 0.32/0.54 | ( ~ P2
% 0.32/0.54 & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_of_bool_asm
% 0.32/0.54 thf(fact_2663_split__of__bool__asm,axiom,
% 0.32/0.54 ! [P: int > $o,P2: $o] :
% 0.32/0.54 ( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
% 0.32/0.54 = ( ~ ( ( P2
% 0.32/0.54 & ~ ( P @ one_one_int ) )
% 0.32/0.54 | ( ~ P2
% 0.32/0.54 & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_of_bool_asm
% 0.32/0.54 thf(fact_2664_split__of__bool__asm,axiom,
% 0.32/0.54 ! [P: code_integer > $o,P2: $o] :
% 0.32/0.54 ( ( P @ ( zero_n356916108424825756nteger @ P2 ) )
% 0.32/0.54 = ( ~ ( ( P2
% 0.32/0.54 & ~ ( P @ one_one_Code_integer ) )
% 0.32/0.54 | ( ~ P2
% 0.32/0.54 & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_of_bool_asm
% 0.32/0.54 thf(fact_2665_inverse__of__nat__le,axiom,
% 0.32/0.54 ! [N: nat,M: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.54 => ( ( N != zero_zero_nat )
% 0.32/0.54 => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % inverse_of_nat_le
% 0.32/0.54 thf(fact_2666_inverse__of__nat__le,axiom,
% 0.32/0.54 ! [N: nat,M: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.54 => ( ( N != zero_zero_nat )
% 0.32/0.54 => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % inverse_of_nat_le
% 0.32/0.54 thf(fact_2667_le__div__geq,axiom,
% 0.32/0.54 ! [N: nat,M: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.54 => ( ( divide_divide_nat @ M @ N )
% 0.32/0.54 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_div_geq
% 0.32/0.54 thf(fact_2668_split__div_H,axiom,
% 0.32/0.54 ! [P: nat > $o,M: nat,N: nat] :
% 0.32/0.54 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.32/0.54 = ( ( ( N = zero_zero_nat )
% 0.32/0.54 & ( P @ zero_zero_nat ) )
% 0.32/0.54 | ? [Q5: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q5 ) @ M )
% 0.32/0.54 & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q5 ) ) )
% 0.32/0.54 & ( P @ Q5 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_div'
% 0.32/0.54 thf(fact_2669_zero__less__numeral,axiom,
% 0.32/0.54 ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_numeral
% 0.32/0.54 thf(fact_2670_zero__less__numeral,axiom,
% 0.32/0.54 ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_numeral
% 0.32/0.54 thf(fact_2671_zero__less__numeral,axiom,
% 0.32/0.54 ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_numeral
% 0.32/0.54 thf(fact_2672_zero__less__numeral,axiom,
% 0.32/0.54 ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_numeral
% 0.32/0.54 thf(fact_2673_not__numeral__less__zero,axiom,
% 0.32/0.54 ! [N: num] :
% 0.32/0.54 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.32/0.54
% 0.32/0.54 % not_numeral_less_zero
% 0.32/0.54 thf(fact_2674_not__numeral__less__zero,axiom,
% 0.32/0.54 ! [N: num] :
% 0.32/0.54 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.32/0.54
% 0.32/0.54 % not_numeral_less_zero
% 0.32/0.54 thf(fact_2675_not__numeral__less__zero,axiom,
% 0.32/0.54 ! [N: num] :
% 0.32/0.54 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.32/0.54
% 0.32/0.54 % not_numeral_less_zero
% 0.32/0.54 thf(fact_2676_not__numeral__less__zero,axiom,
% 0.32/0.54 ! [N: num] :
% 0.32/0.54 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.32/0.54
% 0.32/0.54 % not_numeral_less_zero
% 0.32/0.54 thf(fact_2677_mult__neg__neg,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.54 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.32/0.54 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_neg_neg
% 0.32/0.54 thf(fact_2678_mult__neg__neg,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.54 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_neg_neg
% 0.32/0.54 thf(fact_2679_mult__neg__neg,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.54 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_neg_neg
% 0.32/0.54 thf(fact_2680_not__square__less__zero,axiom,
% 0.32/0.54 ! [A: real] :
% 0.32/0.54 ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 0.32/0.54
% 0.32/0.54 % not_square_less_zero
% 0.32/0.54 thf(fact_2681_not__square__less__zero,axiom,
% 0.32/0.54 ! [A: rat] :
% 0.32/0.54 ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 0.32/0.54
% 0.32/0.54 % not_square_less_zero
% 0.32/0.54 thf(fact_2682_not__square__less__zero,axiom,
% 0.32/0.54 ! [A: int] :
% 0.32/0.54 ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 0.32/0.54
% 0.32/0.54 % not_square_less_zero
% 0.32/0.54 thf(fact_2683_mult__less__0__iff,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 & ( ord_less_real @ B @ zero_zero_real ) )
% 0.32/0.54 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.54 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_0_iff
% 0.32/0.54 thf(fact_2684_mult__less__0__iff,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.54 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.32/0.54 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.54 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_0_iff
% 0.32/0.54 thf(fact_2685_mult__less__0__iff,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 0.32/0.54 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.54 & ( ord_less_int @ B @ zero_zero_int ) )
% 0.32/0.54 | ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.54 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_0_iff
% 0.32/0.54 thf(fact_2686_mult__neg__pos,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.54 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.54 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_neg_pos
% 0.32/0.54 thf(fact_2687_mult__neg__pos,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.54 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.32/0.54 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_neg_pos
% 0.32/0.54 thf(fact_2688_mult__neg__pos,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.54 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_neg_pos
% 0.32/0.54 thf(fact_2689_mult__neg__pos,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.54 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_neg_pos
% 0.32/0.54 thf(fact_2690_mult__pos__neg,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.32/0.54 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_pos_neg
% 0.32/0.54 thf(fact_2691_mult__pos__neg,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_pos_neg
% 0.32/0.54 thf(fact_2692_mult__pos__neg,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.32/0.54 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_pos_neg
% 0.32/0.54 thf(fact_2693_mult__pos__neg,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.54 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_pos_neg
% 0.32/0.54 thf(fact_2694_mult__pos__pos,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.54 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_pos_pos
% 0.32/0.54 thf(fact_2695_mult__pos__pos,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.32/0.54 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_pos_pos
% 0.32/0.54 thf(fact_2696_mult__pos__pos,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.54 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_pos_pos
% 0.32/0.54 thf(fact_2697_mult__pos__pos,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.54 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_pos_pos
% 0.32/0.54 thf(fact_2698_mult__pos__neg2,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.32/0.54 => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_pos_neg2
% 0.32/0.54 thf(fact_2699_mult__pos__neg2,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_pos_neg2
% 0.32/0.54 thf(fact_2700_mult__pos__neg2,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.32/0.54 => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_pos_neg2
% 0.32/0.54 thf(fact_2701_mult__pos__neg2,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.54 => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_pos_neg2
% 0.32/0.54 thf(fact_2702_zero__less__mult__iff,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 & ( ord_less_real @ zero_zero_real @ B ) )
% 0.32/0.54 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.54 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_mult_iff
% 0.32/0.54 thf(fact_2703_zero__less__mult__iff,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.54 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.32/0.54 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.54 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_mult_iff
% 0.32/0.54 thf(fact_2704_zero__less__mult__iff,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.32/0.54 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.54 & ( ord_less_int @ zero_zero_int @ B ) )
% 0.32/0.54 | ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.54 & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_mult_iff
% 0.32/0.54 thf(fact_2705_zero__less__mult__pos,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.54 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_mult_pos
% 0.32/0.54 thf(fact_2706_zero__less__mult__pos,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.54 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_mult_pos
% 0.32/0.54 thf(fact_2707_zero__less__mult__pos,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_mult_pos
% 0.32/0.54 thf(fact_2708_zero__less__mult__pos,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.32/0.54 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_mult_pos
% 0.32/0.54 thf(fact_2709_zero__less__mult__pos2,axiom,
% 0.32/0.54 ! [B: real,A: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
% 0.32/0.54 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_mult_pos2
% 0.32/0.54 thf(fact_2710_zero__less__mult__pos2,axiom,
% 0.32/0.54 ! [B: rat,A: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
% 0.32/0.54 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_mult_pos2
% 0.32/0.54 thf(fact_2711_zero__less__mult__pos2,axiom,
% 0.32/0.54 ! [B: nat,A: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_mult_pos2
% 0.32/0.54 thf(fact_2712_zero__less__mult__pos2,axiom,
% 0.32/0.54 ! [B: int,A: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
% 0.32/0.54 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_mult_pos2
% 0.32/0.54 thf(fact_2713_mult__less__cancel__left__neg,axiom,
% 0.32/0.54 ! [C: real,A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.54 = ( ord_less_real @ B @ A ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_left_neg
% 0.32/0.54 thf(fact_2714_mult__less__cancel__left__neg,axiom,
% 0.32/0.54 ! [C: rat,A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.54 = ( ord_less_rat @ B @ A ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_left_neg
% 0.32/0.54 thf(fact_2715_mult__less__cancel__left__neg,axiom,
% 0.32/0.54 ! [C: int,A: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.54 = ( ord_less_int @ B @ A ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_left_neg
% 0.32/0.54 thf(fact_2716_mult__less__cancel__left__pos,axiom,
% 0.32/0.54 ! [C: real,A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.54 = ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_left_pos
% 0.32/0.54 thf(fact_2717_mult__less__cancel__left__pos,axiom,
% 0.32/0.54 ! [C: rat,A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.54 = ( ord_less_rat @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_left_pos
% 0.32/0.54 thf(fact_2718_mult__less__cancel__left__pos,axiom,
% 0.32/0.54 ! [C: int,A: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.54 = ( ord_less_int @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_left_pos
% 0.32/0.54 thf(fact_2719_mult__strict__left__mono__neg,axiom,
% 0.32/0.54 ! [B: real,A: real,C: real] :
% 0.32/0.54 ( ( ord_less_real @ B @ A )
% 0.32/0.54 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_strict_left_mono_neg
% 0.32/0.54 thf(fact_2720_mult__strict__left__mono__neg,axiom,
% 0.32/0.54 ! [B: rat,A: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_rat @ B @ A )
% 0.32/0.54 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_strict_left_mono_neg
% 0.32/0.54 thf(fact_2721_mult__strict__left__mono__neg,axiom,
% 0.32/0.54 ! [B: int,A: int,C: int] :
% 0.32/0.54 ( ( ord_less_int @ B @ A )
% 0.32/0.54 => ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.54 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_strict_left_mono_neg
% 0.32/0.54 thf(fact_2722_mult__strict__left__mono,axiom,
% 0.32/0.54 ! [A: real,B: real,C: real] :
% 0.32/0.54 ( ( ord_less_real @ A @ B )
% 0.32/0.54 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_strict_left_mono
% 0.32/0.54 thf(fact_2723_mult__strict__left__mono,axiom,
% 0.32/0.54 ! [A: rat,B: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_rat @ A @ B )
% 0.32/0.54 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_strict_left_mono
% 0.32/0.54 thf(fact_2724_mult__strict__left__mono,axiom,
% 0.32/0.54 ! [A: nat,B: nat,C: nat] :
% 0.32/0.54 ( ( ord_less_nat @ A @ B )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.32/0.54 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_strict_left_mono
% 0.32/0.54 thf(fact_2725_mult__strict__left__mono,axiom,
% 0.32/0.54 ! [A: int,B: int,C: int] :
% 0.32/0.54 ( ( ord_less_int @ A @ B )
% 0.32/0.54 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_strict_left_mono
% 0.32/0.54 thf(fact_2726_mult__less__cancel__left__disj,axiom,
% 0.32/0.54 ! [C: real,A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 & ( ord_less_real @ A @ B ) )
% 0.32/0.54 | ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_left_disj
% 0.32/0.54 thf(fact_2727_mult__less__cancel__left__disj,axiom,
% 0.32/0.54 ! [C: rat,A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 & ( ord_less_rat @ A @ B ) )
% 0.32/0.54 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_left_disj
% 0.32/0.54 thf(fact_2728_mult__less__cancel__left__disj,axiom,
% 0.32/0.54 ! [C: int,A: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.54 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.54 & ( ord_less_int @ A @ B ) )
% 0.32/0.54 | ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.54 & ( ord_less_int @ B @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_left_disj
% 0.32/0.54 thf(fact_2729_mult__strict__right__mono__neg,axiom,
% 0.32/0.54 ! [B: real,A: real,C: real] :
% 0.32/0.54 ( ( ord_less_real @ B @ A )
% 0.32/0.54 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_strict_right_mono_neg
% 0.32/0.54 thf(fact_2730_mult__strict__right__mono__neg,axiom,
% 0.32/0.54 ! [B: rat,A: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_rat @ B @ A )
% 0.32/0.54 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_strict_right_mono_neg
% 0.32/0.54 thf(fact_2731_mult__strict__right__mono__neg,axiom,
% 0.32/0.54 ! [B: int,A: int,C: int] :
% 0.32/0.54 ( ( ord_less_int @ B @ A )
% 0.32/0.54 => ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.54 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_strict_right_mono_neg
% 0.32/0.54 thf(fact_2732_mult__strict__right__mono,axiom,
% 0.32/0.54 ! [A: real,B: real,C: real] :
% 0.32/0.54 ( ( ord_less_real @ A @ B )
% 0.32/0.54 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_strict_right_mono
% 0.32/0.54 thf(fact_2733_mult__strict__right__mono,axiom,
% 0.32/0.54 ! [A: rat,B: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_rat @ A @ B )
% 0.32/0.54 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_strict_right_mono
% 0.32/0.54 thf(fact_2734_mult__strict__right__mono,axiom,
% 0.32/0.54 ! [A: nat,B: nat,C: nat] :
% 0.32/0.54 ( ( ord_less_nat @ A @ B )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.32/0.54 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_strict_right_mono
% 0.32/0.54 thf(fact_2735_mult__strict__right__mono,axiom,
% 0.32/0.54 ! [A: int,B: int,C: int] :
% 0.32/0.54 ( ( ord_less_int @ A @ B )
% 0.32/0.54 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_strict_right_mono
% 0.32/0.54 thf(fact_2736_mult__less__cancel__right__disj,axiom,
% 0.32/0.54 ! [A: real,C: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 & ( ord_less_real @ A @ B ) )
% 0.32/0.54 | ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_right_disj
% 0.32/0.54 thf(fact_2737_mult__less__cancel__right__disj,axiom,
% 0.32/0.54 ! [A: rat,C: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 & ( ord_less_rat @ A @ B ) )
% 0.32/0.54 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_right_disj
% 0.32/0.54 thf(fact_2738_mult__less__cancel__right__disj,axiom,
% 0.32/0.54 ! [A: int,C: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.54 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.54 & ( ord_less_int @ A @ B ) )
% 0.32/0.54 | ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.54 & ( ord_less_int @ B @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_less_cancel_right_disj
% 0.32/0.54 thf(fact_2739_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.32/0.54 ! [A: real,B: real,C: real] :
% 0.32/0.54 ( ( ord_less_real @ A @ B )
% 0.32/0.54 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.32/0.54 thf(fact_2740_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.32/0.54 ! [A: rat,B: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_rat @ A @ B )
% 0.32/0.54 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.32/0.54 thf(fact_2741_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.32/0.54 ! [A: nat,B: nat,C: nat] :
% 0.32/0.54 ( ( ord_less_nat @ A @ B )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.32/0.54 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.32/0.54 thf(fact_2742_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.32/0.54 ! [A: int,B: int,C: int] :
% 0.32/0.54 ( ( ord_less_int @ A @ B )
% 0.32/0.54 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.32/0.54 thf(fact_2743_zdiff__int__split,axiom,
% 0.32/0.54 ! [P: int > $o,X: nat,Y: nat] :
% 0.32/0.54 ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
% 0.32/0.54 = ( ( ( ord_less_eq_nat @ Y @ X )
% 0.32/0.54 => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
% 0.32/0.54 & ( ( ord_less_nat @ X @ Y )
% 0.32/0.54 => ( P @ zero_zero_int ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zdiff_int_split
% 0.32/0.54 thf(fact_2744_int__div__pos__eq,axiom,
% 0.32/0.54 ! [A: int,B: int,Q2: int,R: int] :
% 0.32/0.54 ( ( A
% 0.32/0.54 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.32/0.54 => ( ( ord_less_int @ R @ B )
% 0.32/0.54 => ( ( divide_divide_int @ A @ B )
% 0.32/0.54 = Q2 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % int_div_pos_eq
% 0.32/0.54 thf(fact_2745_int__div__neg__eq,axiom,
% 0.32/0.54 ! [A: int,B: int,Q2: int,R: int] :
% 0.32/0.54 ( ( A
% 0.32/0.54 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_int @ B @ R )
% 0.32/0.54 => ( ( divide_divide_int @ A @ B )
% 0.32/0.54 = Q2 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % int_div_neg_eq
% 0.32/0.54 thf(fact_2746_split__zdiv,axiom,
% 0.32/0.54 ! [P: int > $o,N: int,K: int] :
% 0.32/0.54 ( ( P @ ( divide_divide_int @ N @ K ) )
% 0.32/0.54 = ( ( ( K = zero_zero_int )
% 0.32/0.54 => ( P @ zero_zero_int ) )
% 0.32/0.54 & ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.54 => ! [I4: int,J3: int] :
% 0.32/0.54 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.32/0.54 & ( ord_less_int @ J3 @ K )
% 0.32/0.54 & ( N
% 0.32/0.54 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.32/0.54 => ( P @ I4 ) ) )
% 0.32/0.54 & ( ( ord_less_int @ K @ zero_zero_int )
% 0.32/0.54 => ! [I4: int,J3: int] :
% 0.32/0.54 ( ( ( ord_less_int @ K @ J3 )
% 0.32/0.54 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.32/0.54 & ( N
% 0.32/0.54 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.32/0.54 => ( P @ I4 ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_zdiv
% 0.32/0.54 thf(fact_2747_zero__less__one,axiom,
% 0.32/0.54 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.32/0.54
% 0.32/0.54 % zero_less_one
% 0.32/0.54 thf(fact_2748_zero__less__one,axiom,
% 0.32/0.54 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.32/0.54
% 0.32/0.54 % zero_less_one
% 0.32/0.54 thf(fact_2749_zero__less__one,axiom,
% 0.32/0.54 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.32/0.54
% 0.32/0.54 % zero_less_one
% 0.32/0.54 thf(fact_2750_zero__less__one,axiom,
% 0.32/0.54 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.32/0.54
% 0.32/0.54 % zero_less_one
% 0.32/0.54 thf(fact_2751_not__one__less__zero,axiom,
% 0.32/0.54 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 0.32/0.54
% 0.32/0.54 % not_one_less_zero
% 0.32/0.54 thf(fact_2752_not__one__less__zero,axiom,
% 0.32/0.54 ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.32/0.54
% 0.32/0.54 % not_one_less_zero
% 0.32/0.54 thf(fact_2753_not__one__less__zero,axiom,
% 0.32/0.54 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.32/0.54
% 0.32/0.54 % not_one_less_zero
% 0.32/0.54 thf(fact_2754_not__one__less__zero,axiom,
% 0.32/0.54 ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 0.32/0.54
% 0.32/0.54 % not_one_less_zero
% 0.32/0.54 thf(fact_2755_less__numeral__extra_I1_J,axiom,
% 0.32/0.54 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.32/0.54
% 0.32/0.54 % less_numeral_extra(1)
% 0.32/0.54 thf(fact_2756_less__numeral__extra_I1_J,axiom,
% 0.32/0.54 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.32/0.54
% 0.32/0.54 % less_numeral_extra(1)
% 0.32/0.54 thf(fact_2757_less__numeral__extra_I1_J,axiom,
% 0.32/0.54 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.32/0.54
% 0.32/0.54 % less_numeral_extra(1)
% 0.32/0.54 thf(fact_2758_less__numeral__extra_I1_J,axiom,
% 0.32/0.54 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.32/0.54
% 0.32/0.54 % less_numeral_extra(1)
% 0.32/0.54 thf(fact_2759_pos__add__strict,axiom,
% 0.32/0.54 ! [A: real,B: real,C: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_real @ B @ C )
% 0.32/0.54 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pos_add_strict
% 0.32/0.54 thf(fact_2760_pos__add__strict,axiom,
% 0.32/0.54 ! [A: rat,B: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_rat @ B @ C )
% 0.32/0.54 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pos_add_strict
% 0.32/0.54 thf(fact_2761_pos__add__strict,axiom,
% 0.32/0.54 ! [A: nat,B: nat,C: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_nat @ B @ C )
% 0.32/0.54 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pos_add_strict
% 0.32/0.54 thf(fact_2762_pos__add__strict,axiom,
% 0.32/0.54 ! [A: int,B: int,C: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_int @ B @ C )
% 0.32/0.54 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pos_add_strict
% 0.32/0.54 thf(fact_2763_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_nat @ A @ B )
% 0.32/0.54 => ~ ! [C3: nat] :
% 0.32/0.54 ( ( B
% 0.32/0.54 = ( plus_plus_nat @ A @ C3 ) )
% 0.32/0.54 => ( C3 = zero_zero_nat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % canonically_ordered_monoid_add_class.lessE
% 0.32/0.54 thf(fact_2764_add__pos__pos,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.54 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_pos_pos
% 0.32/0.54 thf(fact_2765_add__pos__pos,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.32/0.54 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_pos_pos
% 0.32/0.54 thf(fact_2766_add__pos__pos,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.54 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_pos_pos
% 0.32/0.54 thf(fact_2767_add__pos__pos,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.54 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_pos_pos
% 0.32/0.54 thf(fact_2768_add__neg__neg,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.54 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.32/0.54 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_neg_neg
% 0.32/0.54 thf(fact_2769_add__neg__neg,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.54 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_neg_neg
% 0.32/0.54 thf(fact_2770_add__neg__neg,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.32/0.54 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.32/0.54 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_neg_neg
% 0.32/0.54 thf(fact_2771_add__neg__neg,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.54 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_neg_neg
% 0.32/0.54 thf(fact_2772_add__less__zeroD,axiom,
% 0.32/0.54 ! [X: real,Y: real] :
% 0.32/0.54 ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 0.32/0.54 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.32/0.54 | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_less_zeroD
% 0.32/0.54 thf(fact_2773_add__less__zeroD,axiom,
% 0.32/0.54 ! [X: rat,Y: rat] :
% 0.32/0.54 ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
% 0.32/0.54 => ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.32/0.54 | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_less_zeroD
% 0.32/0.54 thf(fact_2774_add__less__zeroD,axiom,
% 0.32/0.54 ! [X: int,Y: int] :
% 0.32/0.54 ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_int @ X @ zero_zero_int )
% 0.32/0.54 | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_less_zeroD
% 0.32/0.54 thf(fact_2775_less__iff__diff__less__0,axiom,
% 0.32/0.54 ( ord_less_real
% 0.32/0.54 = ( ^ [A2: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_iff_diff_less_0
% 0.32/0.54 thf(fact_2776_less__iff__diff__less__0,axiom,
% 0.32/0.54 ( ord_less_rat
% 0.32/0.54 = ( ^ [A2: rat,B2: rat] : ( ord_less_rat @ ( minus_minus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_iff_diff_less_0
% 0.32/0.54 thf(fact_2777_less__iff__diff__less__0,axiom,
% 0.32/0.54 ( ord_less_int
% 0.32/0.54 = ( ^ [A2: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_iff_diff_less_0
% 0.32/0.54 thf(fact_2778_divide__strict__right__mono__neg,axiom,
% 0.32/0.54 ! [B: real,A: real,C: real] :
% 0.32/0.54 ( ( ord_less_real @ B @ A )
% 0.32/0.54 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_strict_right_mono_neg
% 0.32/0.54 thf(fact_2779_divide__strict__right__mono__neg,axiom,
% 0.32/0.54 ! [B: rat,A: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_rat @ B @ A )
% 0.32/0.54 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_strict_right_mono_neg
% 0.32/0.54 thf(fact_2780_divide__strict__right__mono,axiom,
% 0.32/0.54 ! [A: real,B: real,C: real] :
% 0.32/0.54 ( ( ord_less_real @ A @ B )
% 0.32/0.54 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_strict_right_mono
% 0.32/0.54 thf(fact_2781_divide__strict__right__mono,axiom,
% 0.32/0.54 ! [A: rat,B: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_rat @ A @ B )
% 0.32/0.54 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_strict_right_mono
% 0.32/0.54 thf(fact_2782_zero__less__divide__iff,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 & ( ord_less_real @ zero_zero_real @ B ) )
% 0.32/0.54 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.54 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_divide_iff
% 0.32/0.54 thf(fact_2783_zero__less__divide__iff,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.54 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.32/0.54 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.54 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_divide_iff
% 0.32/0.54 thf(fact_2784_divide__less__cancel,axiom,
% 0.32/0.54 ! [A: real,C: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_real @ A @ B ) )
% 0.32/0.54 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_real @ B @ A ) )
% 0.32/0.54 & ( C != zero_zero_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_less_cancel
% 0.32/0.54 thf(fact_2785_divide__less__cancel,axiom,
% 0.32/0.54 ! [A: rat,C: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_rat @ A @ B ) )
% 0.32/0.54 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_rat @ B @ A ) )
% 0.32/0.54 & ( C != zero_zero_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_less_cancel
% 0.32/0.54 thf(fact_2786_divide__less__0__iff,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.32/0.54 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 & ( ord_less_real @ B @ zero_zero_real ) )
% 0.32/0.54 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.54 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_less_0_iff
% 0.32/0.54 thf(fact_2787_divide__less__0__iff,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 0.32/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.54 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.32/0.54 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.54 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_less_0_iff
% 0.32/0.54 thf(fact_2788_divide__pos__pos,axiom,
% 0.32/0.54 ! [X: real,Y: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.54 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.54 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_pos_pos
% 0.32/0.54 thf(fact_2789_divide__pos__pos,axiom,
% 0.32/0.54 ! [X: rat,Y: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.32/0.54 => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.32/0.54 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_pos_pos
% 0.32/0.54 thf(fact_2790_divide__pos__neg,axiom,
% 0.32/0.54 ! [X: real,Y: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.54 => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.32/0.54 => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_pos_neg
% 0.32/0.54 thf(fact_2791_divide__pos__neg,axiom,
% 0.32/0.54 ! [X: rat,Y: rat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.32/0.54 => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_pos_neg
% 0.32/0.54 thf(fact_2792_divide__neg__pos,axiom,
% 0.32/0.54 ! [X: real,Y: real] :
% 0.32/0.54 ( ( ord_less_real @ X @ zero_zero_real )
% 0.32/0.54 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.54 => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_neg_pos
% 0.32/0.54 thf(fact_2793_divide__neg__pos,axiom,
% 0.32/0.54 ! [X: rat,Y: rat] :
% 0.32/0.54 ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.32/0.54 => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.32/0.54 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_neg_pos
% 0.32/0.54 thf(fact_2794_divide__neg__neg,axiom,
% 0.32/0.54 ! [X: real,Y: real] :
% 0.32/0.54 ( ( ord_less_real @ X @ zero_zero_real )
% 0.32/0.54 => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.32/0.54 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_neg_neg
% 0.32/0.54 thf(fact_2795_divide__neg__neg,axiom,
% 0.32/0.54 ! [X: rat,Y: rat] :
% 0.32/0.54 ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.32/0.54 => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divide_neg_neg
% 0.32/0.54 thf(fact_2796_not__numeral__less__one,axiom,
% 0.32/0.54 ! [N: num] :
% 0.32/0.54 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 0.32/0.54
% 0.32/0.54 % not_numeral_less_one
% 0.32/0.54 thf(fact_2797_not__numeral__less__one,axiom,
% 0.32/0.54 ! [N: num] :
% 0.32/0.54 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 0.32/0.54
% 0.32/0.54 % not_numeral_less_one
% 0.32/0.54 thf(fact_2798_not__numeral__less__one,axiom,
% 0.32/0.54 ! [N: num] :
% 0.32/0.54 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 0.32/0.54
% 0.32/0.54 % not_numeral_less_one
% 0.32/0.54 thf(fact_2799_not__numeral__less__one,axiom,
% 0.32/0.54 ! [N: num] :
% 0.32/0.54 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 0.32/0.54
% 0.32/0.54 % not_numeral_less_one
% 0.32/0.54 thf(fact_2800_less__1__mult,axiom,
% 0.32/0.54 ! [M: real,N: real] :
% 0.32/0.54 ( ( ord_less_real @ one_one_real @ M )
% 0.32/0.54 => ( ( ord_less_real @ one_one_real @ N )
% 0.32/0.54 => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_1_mult
% 0.32/0.54 thf(fact_2801_less__1__mult,axiom,
% 0.32/0.54 ! [M: rat,N: rat] :
% 0.32/0.54 ( ( ord_less_rat @ one_one_rat @ M )
% 0.32/0.54 => ( ( ord_less_rat @ one_one_rat @ N )
% 0.32/0.54 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_1_mult
% 0.32/0.54 thf(fact_2802_less__1__mult,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ one_one_nat @ M )
% 0.32/0.54 => ( ( ord_less_nat @ one_one_nat @ N )
% 0.32/0.54 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_1_mult
% 0.32/0.54 thf(fact_2803_less__1__mult,axiom,
% 0.32/0.54 ! [M: int,N: int] :
% 0.32/0.54 ( ( ord_less_int @ one_one_int @ M )
% 0.32/0.54 => ( ( ord_less_int @ one_one_int @ N )
% 0.32/0.54 => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_1_mult
% 0.32/0.54 thf(fact_2804_zero__le__numeral,axiom,
% 0.32/0.54 ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_le_numeral
% 0.32/0.54 thf(fact_2805_zero__le__numeral,axiom,
% 0.32/0.54 ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_le_numeral
% 0.32/0.54 thf(fact_2806_zero__le__numeral,axiom,
% 0.32/0.54 ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_le_numeral
% 0.32/0.54 thf(fact_2807_zero__le__numeral,axiom,
% 0.32/0.54 ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_le_numeral
% 0.32/0.54 thf(fact_2808_not__numeral__le__zero,axiom,
% 0.32/0.54 ! [N: num] :
% 0.32/0.54 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.32/0.54
% 0.32/0.54 % not_numeral_le_zero
% 0.32/0.54 thf(fact_2809_not__numeral__le__zero,axiom,
% 0.32/0.54 ! [N: num] :
% 0.32/0.54 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.32/0.54
% 0.32/0.54 % not_numeral_le_zero
% 0.32/0.54 thf(fact_2810_not__numeral__le__zero,axiom,
% 0.32/0.54 ! [N: num] :
% 0.32/0.54 ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.32/0.54
% 0.32/0.54 % not_numeral_le_zero
% 0.32/0.54 thf(fact_2811_not__numeral__le__zero,axiom,
% 0.32/0.54 ! [N: num] :
% 0.32/0.54 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.32/0.54
% 0.32/0.54 % not_numeral_le_zero
% 0.32/0.54 thf(fact_2812_zero__less__power,axiom,
% 0.32/0.54 ! [A: real,N: nat] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_power
% 0.32/0.54 thf(fact_2813_zero__less__power,axiom,
% 0.32/0.54 ! [A: rat,N: nat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_power
% 0.32/0.54 thf(fact_2814_zero__less__power,axiom,
% 0.32/0.54 ! [A: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_power
% 0.32/0.54 thf(fact_2815_zero__less__power,axiom,
% 0.32/0.54 ! [A: int,N: nat] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_power
% 0.32/0.54 thf(fact_2816_mult__mono,axiom,
% 0.32/0.54 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_real @ C @ D )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_mono
% 0.32/0.54 thf(fact_2817_mult__mono,axiom,
% 0.32/0.54 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_rat @ C @ D )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_mono
% 0.32/0.54 thf(fact_2818_mult__mono,axiom,
% 0.32/0.54 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_nat @ C @ D )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_mono
% 0.32/0.54 thf(fact_2819_mult__mono,axiom,
% 0.32/0.54 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_int @ C @ D )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_mono
% 0.32/0.54 thf(fact_2820_mult__mono_H,axiom,
% 0.32/0.54 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_real @ C @ D )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_mono'
% 0.32/0.54 thf(fact_2821_mult__mono_H,axiom,
% 0.32/0.54 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_rat @ C @ D )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_mono'
% 0.32/0.54 thf(fact_2822_mult__mono_H,axiom,
% 0.32/0.54 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_nat @ C @ D )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_mono'
% 0.32/0.54 thf(fact_2823_mult__mono_H,axiom,
% 0.32/0.54 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_int @ C @ D )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_mono'
% 0.32/0.54 thf(fact_2824_zero__le__square,axiom,
% 0.32/0.54 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_le_square
% 0.32/0.54 thf(fact_2825_zero__le__square,axiom,
% 0.32/0.54 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_le_square
% 0.32/0.54 thf(fact_2826_zero__le__square,axiom,
% 0.32/0.54 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_le_square
% 0.32/0.54 thf(fact_2827_split__mult__pos__le,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.54 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.32/0.54 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.54 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.32/0.54 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_mult_pos_le
% 0.32/0.54 thf(fact_2828_split__mult__pos__le,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.54 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.32/0.54 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.54 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.32/0.54 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_mult_pos_le
% 0.32/0.54 thf(fact_2829_split__mult__pos__le,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.54 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 0.32/0.54 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.54 & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.32/0.54 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_mult_pos_le
% 0.32/0.54 thf(fact_2830_mult__left__mono__neg,axiom,
% 0.32/0.54 ! [B: real,A: real,C: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.54 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_left_mono_neg
% 0.32/0.54 thf(fact_2831_mult__left__mono__neg,axiom,
% 0.32/0.54 ! [B: rat,A: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.54 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_left_mono_neg
% 0.32/0.54 thf(fact_2832_mult__left__mono__neg,axiom,
% 0.32/0.54 ! [B: int,A: int,C: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.54 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.54 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_left_mono_neg
% 0.32/0.54 thf(fact_2833_mult__nonpos__nonpos,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.54 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonpos_nonpos
% 0.32/0.54 thf(fact_2834_mult__nonpos__nonpos,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.54 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonpos_nonpos
% 0.32/0.54 thf(fact_2835_mult__nonpos__nonpos,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.32/0.54 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonpos_nonpos
% 0.32/0.54 thf(fact_2836_mult__left__mono,axiom,
% 0.32/0.54 ! [A: real,B: real,C: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_left_mono
% 0.32/0.54 thf(fact_2837_mult__left__mono,axiom,
% 0.32/0.54 ! [A: rat,B: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_left_mono
% 0.32/0.54 thf(fact_2838_mult__left__mono,axiom,
% 0.32/0.54 ! [A: nat,B: nat,C: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_left_mono
% 0.32/0.54 thf(fact_2839_mult__left__mono,axiom,
% 0.32/0.54 ! [A: int,B: int,C: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_left_mono
% 0.32/0.54 thf(fact_2840_mult__right__mono__neg,axiom,
% 0.32/0.54 ! [B: real,A: real,C: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.54 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_right_mono_neg
% 0.32/0.54 thf(fact_2841_mult__right__mono__neg,axiom,
% 0.32/0.54 ! [B: rat,A: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.54 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_right_mono_neg
% 0.32/0.54 thf(fact_2842_mult__right__mono__neg,axiom,
% 0.32/0.54 ! [B: int,A: int,C: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.54 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.54 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_right_mono_neg
% 0.32/0.54 thf(fact_2843_mult__right__mono,axiom,
% 0.32/0.54 ! [A: real,B: real,C: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_right_mono
% 0.32/0.54 thf(fact_2844_mult__right__mono,axiom,
% 0.32/0.54 ! [A: rat,B: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_right_mono
% 0.32/0.54 thf(fact_2845_mult__right__mono,axiom,
% 0.32/0.54 ! [A: nat,B: nat,C: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_right_mono
% 0.32/0.54 thf(fact_2846_mult__right__mono,axiom,
% 0.32/0.54 ! [A: int,B: int,C: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_right_mono
% 0.32/0.54 thf(fact_2847_mult__le__0__iff,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.32/0.54 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.54 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.32/0.54 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.54 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_0_iff
% 0.32/0.54 thf(fact_2848_mult__le__0__iff,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.32/0.54 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.54 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.32/0.54 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.54 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_0_iff
% 0.32/0.54 thf(fact_2849_mult__le__0__iff,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 0.32/0.54 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.54 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.32/0.54 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.54 & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_le_0_iff
% 0.32/0.54 thf(fact_2850_split__mult__neg__le,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.54 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.32/0.54 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.54 & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_mult_neg_le
% 0.32/0.54 thf(fact_2851_split__mult__neg__le,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.54 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.32/0.54 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.54 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 0.32/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_mult_neg_le
% 0.32/0.54 thf(fact_2852_split__mult__neg__le,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.54 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 0.32/0.54 | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/0.54 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 0.32/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_mult_neg_le
% 0.32/0.54 thf(fact_2853_split__mult__neg__le,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.54 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.32/0.54 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.54 & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 0.32/0.54 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_mult_neg_le
% 0.32/0.54 thf(fact_2854_mult__nonneg__nonneg,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.54 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonneg_nonneg
% 0.32/0.54 thf(fact_2855_mult__nonneg__nonneg,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.54 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonneg_nonneg
% 0.32/0.54 thf(fact_2856_mult__nonneg__nonneg,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.54 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonneg_nonneg
% 0.32/0.54 thf(fact_2857_mult__nonneg__nonneg,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.54 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonneg_nonneg
% 0.32/0.54 thf(fact_2858_mult__nonneg__nonpos,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonneg_nonpos
% 0.32/0.54 thf(fact_2859_mult__nonneg__nonpos,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonneg_nonpos
% 0.32/0.54 thf(fact_2860_mult__nonneg__nonpos,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.32/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonneg_nonpos
% 0.32/0.54 thf(fact_2861_mult__nonneg__nonpos,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.32/0.54 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonneg_nonpos
% 0.32/0.54 thf(fact_2862_mult__nonpos__nonneg,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonpos_nonneg
% 0.32/0.54 thf(fact_2863_mult__nonpos__nonneg,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonpos_nonneg
% 0.32/0.54 thf(fact_2864_mult__nonpos__nonneg,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonpos_nonneg
% 0.32/0.54 thf(fact_2865_mult__nonpos__nonneg,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.54 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonpos_nonneg
% 0.32/0.54 thf(fact_2866_mult__nonneg__nonpos2,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonneg_nonpos2
% 0.32/0.54 thf(fact_2867_mult__nonneg__nonpos2,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonneg_nonpos2
% 0.32/0.54 thf(fact_2868_mult__nonneg__nonpos2,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.32/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonneg_nonpos2
% 0.32/0.54 thf(fact_2869_mult__nonneg__nonpos2,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.32/0.54 => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_nonneg_nonpos2
% 0.32/0.54 thf(fact_2870_zero__le__mult__iff,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.54 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.54 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.32/0.54 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.54 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_le_mult_iff
% 0.32/0.54 thf(fact_2871_zero__le__mult__iff,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.54 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.54 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.32/0.54 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.54 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_le_mult_iff
% 0.32/0.54 thf(fact_2872_zero__le__mult__iff,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.32/0.54 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.54 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 0.32/0.54 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.54 & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_le_mult_iff
% 0.32/0.54 thf(fact_2873_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.32/0.54 ! [A: real,B: real,C: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.32/0.54 thf(fact_2874_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.32/0.54 ! [A: rat,B: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.32/0.54 thf(fact_2875_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.32/0.54 ! [A: nat,B: nat,C: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.32/0.54 thf(fact_2876_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.32/0.54 ! [A: int,B: int,C: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.32/0.54 thf(fact_2877_add__mono1,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ A @ B )
% 0.32/0.54 => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_mono1
% 0.32/0.54 thf(fact_2878_add__mono1,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ A @ B )
% 0.32/0.54 => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_mono1
% 0.32/0.54 thf(fact_2879_add__mono1,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_nat @ A @ B )
% 0.32/0.54 => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_mono1
% 0.32/0.54 thf(fact_2880_add__mono1,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_int @ A @ B )
% 0.32/0.54 => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_mono1
% 0.32/0.54 thf(fact_2881_less__add__one,axiom,
% 0.32/0.54 ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_add_one
% 0.32/0.54 thf(fact_2882_less__add__one,axiom,
% 0.32/0.54 ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_add_one
% 0.32/0.54 thf(fact_2883_less__add__one,axiom,
% 0.32/0.54 ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_add_one
% 0.32/0.54 thf(fact_2884_less__add__one,axiom,
% 0.32/0.54 ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_add_one
% 0.32/0.54 thf(fact_2885_zero__less__one__class_Ozero__le__one,axiom,
% 0.32/0.54 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.32/0.54
% 0.32/0.54 % zero_less_one_class.zero_le_one
% 0.32/0.54 thf(fact_2886_zero__less__one__class_Ozero__le__one,axiom,
% 0.32/0.54 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.32/0.54
% 0.32/0.54 % zero_less_one_class.zero_le_one
% 0.32/0.54 thf(fact_2887_zero__less__one__class_Ozero__le__one,axiom,
% 0.32/0.54 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.32/0.54
% 0.32/0.54 % zero_less_one_class.zero_le_one
% 0.32/0.54 thf(fact_2888_zero__less__one__class_Ozero__le__one,axiom,
% 0.32/0.54 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.32/0.54
% 0.32/0.54 % zero_less_one_class.zero_le_one
% 0.32/0.54 thf(fact_2889_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.32/0.54 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.32/0.54
% 0.32/0.54 % linordered_nonzero_semiring_class.zero_le_one
% 0.32/0.54 thf(fact_2890_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.32/0.54 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.32/0.54
% 0.32/0.54 % linordered_nonzero_semiring_class.zero_le_one
% 0.32/0.54 thf(fact_2891_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.32/0.54 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.32/0.54
% 0.32/0.54 % linordered_nonzero_semiring_class.zero_le_one
% 0.32/0.54 thf(fact_2892_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.32/0.54 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.32/0.54
% 0.32/0.54 % linordered_nonzero_semiring_class.zero_le_one
% 0.32/0.54 thf(fact_2893_not__one__le__zero,axiom,
% 0.32/0.54 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 0.32/0.54
% 0.32/0.54 % not_one_le_zero
% 0.32/0.54 thf(fact_2894_not__one__le__zero,axiom,
% 0.32/0.54 ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.32/0.54
% 0.32/0.54 % not_one_le_zero
% 0.32/0.54 thf(fact_2895_not__one__le__zero,axiom,
% 0.32/0.54 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.32/0.54
% 0.32/0.54 % not_one_le_zero
% 0.32/0.54 thf(fact_2896_not__one__le__zero,axiom,
% 0.32/0.54 ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 0.32/0.54
% 0.32/0.54 % not_one_le_zero
% 0.32/0.54 thf(fact_2897_add__nonpos__eq__0__iff,axiom,
% 0.32/0.54 ! [X: real,Y: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.54 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.32/0.54 => ( ( ( plus_plus_real @ X @ Y )
% 0.32/0.54 = zero_zero_real )
% 0.32/0.54 = ( ( X = zero_zero_real )
% 0.32/0.54 & ( Y = zero_zero_real ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_nonpos_eq_0_iff
% 0.32/0.54 thf(fact_2898_add__nonpos__eq__0__iff,axiom,
% 0.32/0.54 ! [X: rat,Y: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.32/0.54 => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 0.32/0.54 => ( ( ( plus_plus_rat @ X @ Y )
% 0.32/0.54 = zero_zero_rat )
% 0.32/0.54 = ( ( X = zero_zero_rat )
% 0.32/0.54 & ( Y = zero_zero_rat ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_nonpos_eq_0_iff
% 0.32/0.54 thf(fact_2899_add__nonpos__eq__0__iff,axiom,
% 0.32/0.54 ! [X: nat,Y: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 0.32/0.54 => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
% 0.32/0.54 => ( ( ( plus_plus_nat @ X @ Y )
% 0.32/0.54 = zero_zero_nat )
% 0.32/0.54 = ( ( X = zero_zero_nat )
% 0.32/0.54 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_nonpos_eq_0_iff
% 0.32/0.54 thf(fact_2900_add__nonpos__eq__0__iff,axiom,
% 0.32/0.54 ! [X: int,Y: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 0.32/0.54 => ( ( ( plus_plus_int @ X @ Y )
% 0.32/0.54 = zero_zero_int )
% 0.32/0.54 = ( ( X = zero_zero_int )
% 0.32/0.54 & ( Y = zero_zero_int ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_nonpos_eq_0_iff
% 0.32/0.54 thf(fact_2901_add__nonneg__eq__0__iff,axiom,
% 0.32/0.54 ! [X: real,Y: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.32/0.54 => ( ( ( plus_plus_real @ X @ Y )
% 0.32/0.54 = zero_zero_real )
% 0.32/0.54 = ( ( X = zero_zero_real )
% 0.32/0.54 & ( Y = zero_zero_real ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_nonneg_eq_0_iff
% 0.32/0.54 thf(fact_2902_add__nonneg__eq__0__iff,axiom,
% 0.32/0.54 ! [X: rat,Y: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.32/0.54 => ( ( ( plus_plus_rat @ X @ Y )
% 0.32/0.54 = zero_zero_rat )
% 0.32/0.54 = ( ( X = zero_zero_rat )
% 0.32/0.54 & ( Y = zero_zero_rat ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_nonneg_eq_0_iff
% 0.32/0.54 thf(fact_2903_add__nonneg__eq__0__iff,axiom,
% 0.32/0.54 ! [X: nat,Y: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.32/0.54 => ( ( ( plus_plus_nat @ X @ Y )
% 0.32/0.54 = zero_zero_nat )
% 0.32/0.54 = ( ( X = zero_zero_nat )
% 0.32/0.54 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_nonneg_eq_0_iff
% 0.32/0.54 thf(fact_2904_add__nonneg__eq__0__iff,axiom,
% 0.32/0.54 ! [X: int,Y: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.32/0.54 => ( ( ( plus_plus_int @ X @ Y )
% 0.32/0.54 = zero_zero_int )
% 0.32/0.54 = ( ( X = zero_zero_int )
% 0.32/0.54 & ( Y = zero_zero_int ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_nonneg_eq_0_iff
% 0.32/0.54 thf(fact_2905_add__nonpos__nonpos,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.54 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.32/0.54 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_nonpos_nonpos
% 0.32/0.54 thf(fact_2906_add__nonpos__nonpos,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.54 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_nonpos_nonpos
% 0.32/0.54 thf(fact_2907_add__nonpos__nonpos,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/0.54 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.32/0.54 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_nonpos_nonpos
% 0.32/0.54 thf(fact_2908_add__nonpos__nonpos,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.32/0.54 => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_nonpos_nonpos
% 0.32/0.54 thf(fact_2909_add__nonneg__nonneg,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.54 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_nonneg_nonneg
% 0.32/0.54 thf(fact_2910_add__nonneg__nonneg,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.54 => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_nonneg_nonneg
% 0.32/0.54 thf(fact_2911_add__nonneg__nonneg,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.54 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_nonneg_nonneg
% 0.32/0.54 thf(fact_2912_add__nonneg__nonneg,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.54 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_nonneg_nonneg
% 0.32/0.54 thf(fact_2913_add__increasing2,axiom,
% 0.32/0.54 ! [C: real,B: real,A: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.54 => ( ( ord_less_eq_real @ B @ A )
% 0.32/0.54 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_increasing2
% 0.32/0.54 thf(fact_2914_add__increasing2,axiom,
% 0.32/0.54 ! [C: rat,B: rat,A: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.54 => ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.54 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_increasing2
% 0.32/0.54 thf(fact_2915_add__increasing2,axiom,
% 0.32/0.54 ! [C: nat,B: nat,A: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.54 => ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.54 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_increasing2
% 0.32/0.54 thf(fact_2916_add__increasing2,axiom,
% 0.32/0.54 ! [C: int,B: int,A: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.54 => ( ( ord_less_eq_int @ B @ A )
% 0.32/0.54 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_increasing2
% 0.32/0.54 thf(fact_2917_add__decreasing2,axiom,
% 0.32/0.54 ! [C: real,A: real,B: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.54 => ( ( ord_less_eq_real @ A @ B )
% 0.32/0.54 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_decreasing2
% 0.32/0.54 thf(fact_2918_add__decreasing2,axiom,
% 0.32/0.54 ! [C: rat,A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.54 => ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.54 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_decreasing2
% 0.32/0.54 thf(fact_2919_add__decreasing2,axiom,
% 0.32/0.54 ! [C: nat,A: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 0.32/0.54 => ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.54 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_decreasing2
% 0.32/0.54 thf(fact_2920_add__decreasing2,axiom,
% 0.32/0.54 ! [C: int,A: int,B: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_eq_int @ A @ B )
% 0.32/0.54 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_decreasing2
% 0.32/0.54 thf(fact_2921_add__increasing,axiom,
% 0.32/0.54 ! [A: real,B: real,C: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_eq_real @ B @ C )
% 0.32/0.54 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_increasing
% 0.32/0.54 thf(fact_2922_add__increasing,axiom,
% 0.32/0.54 ! [A: rat,B: rat,C: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.54 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_increasing
% 0.32/0.54 thf(fact_2923_add__increasing,axiom,
% 0.32/0.54 ! [A: nat,B: nat,C: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.54 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_increasing
% 0.32/0.54 thf(fact_2924_add__increasing,axiom,
% 0.32/0.54 ! [A: int,B: int,C: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_eq_int @ B @ C )
% 0.32/0.54 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_increasing
% 0.32/0.54 thf(fact_2925_add__decreasing,axiom,
% 0.32/0.54 ! [A: real,C: real,B: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.54 => ( ( ord_less_eq_real @ C @ B )
% 0.32/0.54 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_decreasing
% 0.32/0.54 thf(fact_2926_add__decreasing,axiom,
% 0.32/0.54 ! [A: rat,C: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.54 => ( ( ord_less_eq_rat @ C @ B )
% 0.32/0.54 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_decreasing
% 0.32/0.54 thf(fact_2927_add__decreasing,axiom,
% 0.32/0.54 ! [A: nat,C: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/0.54 => ( ( ord_less_eq_nat @ C @ B )
% 0.32/0.54 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_decreasing
% 0.32/0.54 thf(fact_2928_add__decreasing,axiom,
% 0.32/0.54 ! [A: int,C: int,B: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_eq_int @ C @ B )
% 0.32/0.54 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_decreasing
% 0.32/0.54 thf(fact_2929_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ~ ( ord_less_real @ A @ B )
% 0.32/0.54 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 0.32/0.54 = A ) ) ).
% 0.32/0.54
% 0.32/0.54 % linordered_semidom_class.add_diff_inverse
% 0.32/0.54 thf(fact_2930_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ~ ( ord_less_rat @ A @ B )
% 0.32/0.54 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 0.32/0.54 = A ) ) ).
% 0.32/0.54
% 0.32/0.54 % linordered_semidom_class.add_diff_inverse
% 0.32/0.55 thf(fact_2931_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.32/0.55 ! [A: nat,B: nat] :
% 0.32/0.55 ( ~ ( ord_less_nat @ A @ B )
% 0.32/0.55 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 0.32/0.55 = A ) ) ).
% 0.32/0.55
% 0.32/0.55 % linordered_semidom_class.add_diff_inverse
% 0.32/0.55 thf(fact_2932_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.32/0.55 ! [A: int,B: int] :
% 0.32/0.55 ( ~ ( ord_less_int @ A @ B )
% 0.32/0.55 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 0.32/0.55 = A ) ) ).
% 0.32/0.55
% 0.32/0.55 % linordered_semidom_class.add_diff_inverse
% 0.32/0.55 thf(fact_2933_diff__less__eq,axiom,
% 0.32/0.55 ! [A: real,B: real,C: real] :
% 0.32/0.55 ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.32/0.55 = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_less_eq
% 0.32/0.55 thf(fact_2934_diff__less__eq,axiom,
% 0.32/0.55 ! [A: rat,B: rat,C: rat] :
% 0.32/0.55 ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.32/0.55 = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_less_eq
% 0.32/0.55 thf(fact_2935_diff__less__eq,axiom,
% 0.32/0.55 ! [A: int,B: int,C: int] :
% 0.32/0.55 ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.32/0.55 = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_less_eq
% 0.32/0.55 thf(fact_2936_less__diff__eq,axiom,
% 0.32/0.55 ! [A: real,C: real,B: real] :
% 0.32/0.55 ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.32/0.55 = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_diff_eq
% 0.32/0.55 thf(fact_2937_less__diff__eq,axiom,
% 0.32/0.55 ! [A: rat,C: rat,B: rat] :
% 0.32/0.55 ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 0.32/0.55 = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_diff_eq
% 0.32/0.55 thf(fact_2938_less__diff__eq,axiom,
% 0.32/0.55 ! [A: int,C: int,B: int] :
% 0.32/0.55 ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.32/0.55 = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_diff_eq
% 0.32/0.55 thf(fact_2939_le__iff__diff__le__0,axiom,
% 0.32/0.55 ( ord_less_eq_real
% 0.32/0.55 = ( ^ [A2: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % le_iff_diff_le_0
% 0.32/0.55 thf(fact_2940_le__iff__diff__le__0,axiom,
% 0.32/0.55 ( ord_less_eq_rat
% 0.32/0.55 = ( ^ [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % le_iff_diff_le_0
% 0.32/0.55 thf(fact_2941_le__iff__diff__le__0,axiom,
% 0.32/0.55 ( ord_less_eq_int
% 0.32/0.55 = ( ^ [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % le_iff_diff_le_0
% 0.32/0.55 thf(fact_2942_of__nat__less__0__iff,axiom,
% 0.32/0.55 ! [M: nat] :
% 0.32/0.55 ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 0.32/0.55
% 0.32/0.55 % of_nat_less_0_iff
% 0.32/0.55 thf(fact_2943_of__nat__less__0__iff,axiom,
% 0.32/0.55 ! [M: nat] :
% 0.32/0.55 ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 0.32/0.55
% 0.32/0.55 % of_nat_less_0_iff
% 0.32/0.55 thf(fact_2944_of__nat__less__0__iff,axiom,
% 0.32/0.55 ! [M: nat] :
% 0.32/0.55 ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 0.32/0.55
% 0.32/0.55 % of_nat_less_0_iff
% 0.32/0.55 thf(fact_2945_of__nat__less__0__iff,axiom,
% 0.32/0.55 ! [M: nat] :
% 0.32/0.55 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 0.32/0.55
% 0.32/0.55 % of_nat_less_0_iff
% 0.32/0.55 thf(fact_2946_divide__right__mono__neg,axiom,
% 0.32/0.55 ! [A: real,B: real,C: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.55 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_right_mono_neg
% 0.32/0.55 thf(fact_2947_divide__right__mono__neg,axiom,
% 0.32/0.55 ! [A: rat,B: rat,C: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.55 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_right_mono_neg
% 0.32/0.55 thf(fact_2948_divide__nonpos__nonpos,axiom,
% 0.32/0.55 ! [X: real,Y: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.55 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.32/0.55 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_nonpos_nonpos
% 0.32/0.55 thf(fact_2949_divide__nonpos__nonpos,axiom,
% 0.32/0.55 ! [X: rat,Y: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.32/0.55 => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 0.32/0.55 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_nonpos_nonpos
% 0.32/0.55 thf(fact_2950_divide__nonpos__nonneg,axiom,
% 0.32/0.55 ! [X: real,Y: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.32/0.55 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_nonpos_nonneg
% 0.32/0.55 thf(fact_2951_divide__nonpos__nonneg,axiom,
% 0.32/0.55 ! [X: rat,Y: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.32/0.55 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_nonpos_nonneg
% 0.32/0.55 thf(fact_2952_divide__nonneg__nonpos,axiom,
% 0.32/0.55 ! [X: real,Y: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.55 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.32/0.55 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_nonneg_nonpos
% 0.32/0.55 thf(fact_2953_divide__nonneg__nonpos,axiom,
% 0.32/0.55 ! [X: rat,Y: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.55 => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 0.32/0.55 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_nonneg_nonpos
% 0.32/0.55 thf(fact_2954_divide__nonneg__nonneg,axiom,
% 0.32/0.55 ! [X: real,Y: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.32/0.55 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_nonneg_nonneg
% 0.32/0.55 thf(fact_2955_divide__nonneg__nonneg,axiom,
% 0.32/0.55 ! [X: rat,Y: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.32/0.55 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_nonneg_nonneg
% 0.32/0.55 thf(fact_2956_zero__le__divide__iff,axiom,
% 0.32/0.55 ! [A: real,B: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.32/0.55 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.55 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.32/0.55 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.55 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zero_le_divide_iff
% 0.32/0.55 thf(fact_2957_zero__le__divide__iff,axiom,
% 0.32/0.55 ! [A: rat,B: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 0.32/0.55 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.55 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.32/0.55 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.55 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zero_le_divide_iff
% 0.32/0.55 thf(fact_2958_divide__right__mono,axiom,
% 0.32/0.55 ! [A: real,B: real,C: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_right_mono
% 0.32/0.55 thf(fact_2959_divide__right__mono,axiom,
% 0.32/0.55 ! [A: rat,B: rat,C: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_right_mono
% 0.32/0.55 thf(fact_2960_divide__le__0__iff,axiom,
% 0.32/0.55 ! [A: real,B: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.32/0.55 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.55 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.32/0.55 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.55 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_le_0_iff
% 0.32/0.55 thf(fact_2961_divide__le__0__iff,axiom,
% 0.32/0.55 ! [A: rat,B: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 0.32/0.55 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.55 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.32/0.55 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.55 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_le_0_iff
% 0.32/0.55 thf(fact_2962_one__le__numeral,axiom,
% 0.32/0.55 ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % one_le_numeral
% 0.32/0.55 thf(fact_2963_one__le__numeral,axiom,
% 0.32/0.55 ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % one_le_numeral
% 0.32/0.55 thf(fact_2964_one__le__numeral,axiom,
% 0.32/0.55 ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % one_le_numeral
% 0.32/0.55 thf(fact_2965_one__le__numeral,axiom,
% 0.32/0.55 ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % one_le_numeral
% 0.32/0.55 thf(fact_2966_zero__le__power,axiom,
% 0.32/0.55 ! [A: real,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.55 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zero_le_power
% 0.32/0.55 thf(fact_2967_zero__le__power,axiom,
% 0.32/0.55 ! [A: rat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.55 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zero_le_power
% 0.32/0.55 thf(fact_2968_zero__le__power,axiom,
% 0.32/0.55 ! [A: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.55 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zero_le_power
% 0.32/0.55 thf(fact_2969_zero__le__power,axiom,
% 0.32/0.55 ! [A: int,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.55 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zero_le_power
% 0.32/0.55 thf(fact_2970_power__mono,axiom,
% 0.32/0.55 ! [A: real,B: real,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.55 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_mono
% 0.32/0.55 thf(fact_2971_power__mono,axiom,
% 0.32/0.55 ! [A: rat,B: rat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.55 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_mono
% 0.32/0.55 thf(fact_2972_power__mono,axiom,
% 0.32/0.55 ! [A: nat,B: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.55 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_mono
% 0.32/0.55 thf(fact_2973_power__mono,axiom,
% 0.32/0.55 ! [A: int,B: int,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.55 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_mono
% 0.32/0.55 thf(fact_2974_pinf_I9_J,axiom,
% 0.32/0.55 ! [D: code_integer,S: code_integer] :
% 0.32/0.55 ? [Z4: code_integer] :
% 0.32/0.55 ! [X5: code_integer] :
% 0.32/0.55 ( ( ord_le6747313008572928689nteger @ Z4 @ X5 )
% 0.32/0.55 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) )
% 0.32/0.55 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(9)
% 0.32/0.55 thf(fact_2975_pinf_I9_J,axiom,
% 0.32/0.55 ! [D: real,S: real] :
% 0.32/0.55 ? [Z4: real] :
% 0.32/0.55 ! [X5: real] :
% 0.32/0.55 ( ( ord_less_real @ Z4 @ X5 )
% 0.32/0.55 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
% 0.32/0.55 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(9)
% 0.32/0.55 thf(fact_2976_pinf_I9_J,axiom,
% 0.32/0.55 ! [D: rat,S: rat] :
% 0.32/0.55 ? [Z4: rat] :
% 0.32/0.55 ! [X5: rat] :
% 0.32/0.55 ( ( ord_less_rat @ Z4 @ X5 )
% 0.32/0.55 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
% 0.32/0.55 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(9)
% 0.32/0.55 thf(fact_2977_pinf_I9_J,axiom,
% 0.32/0.55 ! [D: nat,S: nat] :
% 0.32/0.55 ? [Z4: nat] :
% 0.32/0.55 ! [X5: nat] :
% 0.32/0.55 ( ( ord_less_nat @ Z4 @ X5 )
% 0.32/0.55 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
% 0.32/0.55 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(9)
% 0.32/0.55 thf(fact_2978_pinf_I9_J,axiom,
% 0.32/0.55 ! [D: int,S: int] :
% 0.32/0.55 ? [Z4: int] :
% 0.32/0.55 ! [X5: int] :
% 0.32/0.55 ( ( ord_less_int @ Z4 @ X5 )
% 0.32/0.55 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
% 0.32/0.55 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(9)
% 0.32/0.55 thf(fact_2979_pinf_I10_J,axiom,
% 0.32/0.55 ! [D: code_integer,S: code_integer] :
% 0.32/0.55 ? [Z4: code_integer] :
% 0.32/0.55 ! [X5: code_integer] :
% 0.32/0.55 ( ( ord_le6747313008572928689nteger @ Z4 @ X5 )
% 0.32/0.55 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) )
% 0.32/0.55 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(10)
% 0.32/0.55 thf(fact_2980_pinf_I10_J,axiom,
% 0.32/0.55 ! [D: real,S: real] :
% 0.32/0.55 ? [Z4: real] :
% 0.32/0.55 ! [X5: real] :
% 0.32/0.55 ( ( ord_less_real @ Z4 @ X5 )
% 0.32/0.55 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
% 0.32/0.55 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(10)
% 0.32/0.55 thf(fact_2981_pinf_I10_J,axiom,
% 0.32/0.55 ! [D: rat,S: rat] :
% 0.32/0.55 ? [Z4: rat] :
% 0.32/0.55 ! [X5: rat] :
% 0.32/0.55 ( ( ord_less_rat @ Z4 @ X5 )
% 0.32/0.55 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
% 0.32/0.55 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(10)
% 0.32/0.55 thf(fact_2982_pinf_I10_J,axiom,
% 0.32/0.55 ! [D: nat,S: nat] :
% 0.32/0.55 ? [Z4: nat] :
% 0.32/0.55 ! [X5: nat] :
% 0.32/0.55 ( ( ord_less_nat @ Z4 @ X5 )
% 0.32/0.55 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
% 0.32/0.55 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(10)
% 0.32/0.55 thf(fact_2983_pinf_I10_J,axiom,
% 0.32/0.55 ! [D: int,S: int] :
% 0.32/0.55 ? [Z4: int] :
% 0.32/0.55 ! [X5: int] :
% 0.32/0.55 ( ( ord_less_int @ Z4 @ X5 )
% 0.32/0.55 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
% 0.32/0.55 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(10)
% 0.32/0.55 thf(fact_2984_minf_I9_J,axiom,
% 0.32/0.55 ! [D: code_integer,S: code_integer] :
% 0.32/0.55 ? [Z4: code_integer] :
% 0.32/0.55 ! [X5: code_integer] :
% 0.32/0.55 ( ( ord_le6747313008572928689nteger @ X5 @ Z4 )
% 0.32/0.55 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) )
% 0.32/0.55 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(9)
% 0.32/0.55 thf(fact_2985_minf_I9_J,axiom,
% 0.32/0.55 ! [D: real,S: real] :
% 0.32/0.55 ? [Z4: real] :
% 0.32/0.55 ! [X5: real] :
% 0.32/0.55 ( ( ord_less_real @ X5 @ Z4 )
% 0.32/0.55 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
% 0.32/0.55 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(9)
% 0.32/0.55 thf(fact_2986_minf_I9_J,axiom,
% 0.32/0.55 ! [D: rat,S: rat] :
% 0.32/0.55 ? [Z4: rat] :
% 0.32/0.55 ! [X5: rat] :
% 0.32/0.55 ( ( ord_less_rat @ X5 @ Z4 )
% 0.32/0.55 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
% 0.32/0.55 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(9)
% 0.32/0.55 thf(fact_2987_minf_I9_J,axiom,
% 0.32/0.55 ! [D: nat,S: nat] :
% 0.32/0.55 ? [Z4: nat] :
% 0.32/0.55 ! [X5: nat] :
% 0.32/0.55 ( ( ord_less_nat @ X5 @ Z4 )
% 0.32/0.55 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
% 0.32/0.55 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(9)
% 0.32/0.55 thf(fact_2988_minf_I9_J,axiom,
% 0.32/0.55 ! [D: int,S: int] :
% 0.32/0.55 ? [Z4: int] :
% 0.32/0.55 ! [X5: int] :
% 0.32/0.55 ( ( ord_less_int @ X5 @ Z4 )
% 0.32/0.55 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
% 0.32/0.55 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(9)
% 0.32/0.55 thf(fact_2989_minf_I10_J,axiom,
% 0.32/0.55 ! [D: code_integer,S: code_integer] :
% 0.32/0.55 ? [Z4: code_integer] :
% 0.32/0.55 ! [X5: code_integer] :
% 0.32/0.55 ( ( ord_le6747313008572928689nteger @ X5 @ Z4 )
% 0.32/0.55 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) )
% 0.32/0.55 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(10)
% 0.32/0.55 thf(fact_2990_minf_I10_J,axiom,
% 0.32/0.55 ! [D: real,S: real] :
% 0.32/0.55 ? [Z4: real] :
% 0.32/0.55 ! [X5: real] :
% 0.32/0.55 ( ( ord_less_real @ X5 @ Z4 )
% 0.32/0.55 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
% 0.32/0.55 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(10)
% 0.32/0.55 thf(fact_2991_minf_I10_J,axiom,
% 0.32/0.55 ! [D: rat,S: rat] :
% 0.32/0.55 ? [Z4: rat] :
% 0.32/0.55 ! [X5: rat] :
% 0.32/0.55 ( ( ord_less_rat @ X5 @ Z4 )
% 0.32/0.55 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
% 0.32/0.55 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(10)
% 0.32/0.55 thf(fact_2992_minf_I10_J,axiom,
% 0.32/0.55 ! [D: nat,S: nat] :
% 0.32/0.55 ? [Z4: nat] :
% 0.32/0.55 ! [X5: nat] :
% 0.32/0.55 ( ( ord_less_nat @ X5 @ Z4 )
% 0.32/0.55 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
% 0.32/0.55 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(10)
% 0.32/0.55 thf(fact_2993_minf_I10_J,axiom,
% 0.32/0.55 ! [D: int,S: int] :
% 0.32/0.55 ? [Z4: int] :
% 0.32/0.55 ! [X5: int] :
% 0.32/0.55 ( ( ord_less_int @ X5 @ Z4 )
% 0.32/0.55 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
% 0.32/0.55 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(10)
% 0.32/0.55 thf(fact_2994_add__le__add__imp__diff__le,axiom,
% 0.32/0.55 ! [I: real,K: real,N: real,J: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.32/0.55 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.32/0.55 => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.32/0.55 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.32/0.55 => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_le_add_imp_diff_le
% 0.32/0.55 thf(fact_2995_add__le__add__imp__diff__le,axiom,
% 0.32/0.55 ! [I: rat,K: rat,N: rat,J: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.32/0.55 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.32/0.55 => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.32/0.55 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.32/0.55 => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_le_add_imp_diff_le
% 0.32/0.55 thf(fact_2996_add__le__add__imp__diff__le,axiom,
% 0.32/0.55 ! [I: nat,K: nat,N: nat,J: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.32/0.55 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.32/0.55 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.32/0.55 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.32/0.55 => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_le_add_imp_diff_le
% 0.32/0.55 thf(fact_2997_add__le__add__imp__diff__le,axiom,
% 0.32/0.55 ! [I: int,K: int,N: int,J: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.32/0.55 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.32/0.55 => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.32/0.55 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.32/0.55 => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_le_add_imp_diff_le
% 0.32/0.55 thf(fact_2998_add__le__imp__le__diff,axiom,
% 0.32/0.55 ! [I: real,K: real,N: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.32/0.55 => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_le_imp_le_diff
% 0.32/0.55 thf(fact_2999_add__le__imp__le__diff,axiom,
% 0.32/0.55 ! [I: rat,K: rat,N: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.32/0.55 => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_le_imp_le_diff
% 0.32/0.55 thf(fact_3000_add__le__imp__le__diff,axiom,
% 0.32/0.55 ! [I: nat,K: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.32/0.55 => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_le_imp_le_diff
% 0.32/0.55 thf(fact_3001_add__le__imp__le__diff,axiom,
% 0.32/0.55 ! [I: int,K: int,N: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.32/0.55 => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_le_imp_le_diff
% 0.32/0.55 thf(fact_3002_diff__le__eq,axiom,
% 0.32/0.55 ! [A: real,B: real,C: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.32/0.55 = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_le_eq
% 0.32/0.55 thf(fact_3003_diff__le__eq,axiom,
% 0.32/0.55 ! [A: rat,B: rat,C: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.32/0.55 = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_le_eq
% 0.32/0.55 thf(fact_3004_diff__le__eq,axiom,
% 0.32/0.55 ! [A: int,B: int,C: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.32/0.55 = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_le_eq
% 0.32/0.55 thf(fact_3005_le__diff__eq,axiom,
% 0.32/0.55 ! [A: real,C: real,B: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.32/0.55 = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.32/0.55
% 0.32/0.55 % le_diff_eq
% 0.32/0.55 thf(fact_3006_le__diff__eq,axiom,
% 0.32/0.55 ! [A: rat,C: rat,B: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 0.32/0.55 = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.32/0.55
% 0.32/0.55 % le_diff_eq
% 0.32/0.55 thf(fact_3007_le__diff__eq,axiom,
% 0.32/0.55 ! [A: int,C: int,B: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.32/0.55 = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.32/0.55
% 0.32/0.55 % le_diff_eq
% 0.32/0.55 thf(fact_3008_diff__add,axiom,
% 0.32/0.55 ! [A: nat,B: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.55 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 0.32/0.55 = B ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_add
% 0.32/0.55 thf(fact_3009_le__add__diff,axiom,
% 0.32/0.55 ! [A: nat,B: nat,C: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.55 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % le_add_diff
% 0.32/0.55 thf(fact_3010_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.32/0.55 ! [A: nat,B: nat,C: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.55 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.32/0.55 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.32/0.55 thf(fact_3011_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.32/0.55 ! [A: nat,B: nat,C: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.55 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.32/0.55 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.32/0.55 thf(fact_3012_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.32/0.55 ! [A: nat,B: nat,C: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.55 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 0.32/0.55 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.32/0.55 thf(fact_3013_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.32/0.55 ! [A: nat,B: nat,C: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.55 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 0.32/0.55 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.32/0.55 thf(fact_3014_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.32/0.55 ! [A: nat,B: nat,C: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.55 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 0.32/0.55 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.32/0.55 thf(fact_3015_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.32/0.55 ! [A: nat,B: nat,C: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.55 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.32/0.55 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.32/0.55 thf(fact_3016_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.32/0.55 ! [A: nat,B: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.55 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 0.32/0.55 = B ) ) ).
% 0.32/0.55
% 0.32/0.55 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.32/0.55 thf(fact_3017_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.32/0.55 ! [A: nat,B: nat,C: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.55 => ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.55 => ( ( ( minus_minus_nat @ B @ A )
% 0.32/0.55 = C )
% 0.32/0.55 = ( B
% 0.32/0.55 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.32/0.55 thf(fact_3018_of__nat__0__le__iff,axiom,
% 0.32/0.55 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % of_nat_0_le_iff
% 0.32/0.55 thf(fact_3019_of__nat__0__le__iff,axiom,
% 0.32/0.55 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % of_nat_0_le_iff
% 0.32/0.55 thf(fact_3020_of__nat__0__le__iff,axiom,
% 0.32/0.55 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % of_nat_0_le_iff
% 0.32/0.55 thf(fact_3021_of__nat__0__le__iff,axiom,
% 0.32/0.55 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % of_nat_0_le_iff
% 0.32/0.55 thf(fact_3022_one__le__power,axiom,
% 0.32/0.55 ! [A: real,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_real @ one_one_real @ A )
% 0.32/0.55 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % one_le_power
% 0.32/0.55 thf(fact_3023_one__le__power,axiom,
% 0.32/0.55 ! [A: rat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.32/0.55 => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % one_le_power
% 0.32/0.55 thf(fact_3024_one__le__power,axiom,
% 0.32/0.55 ! [A: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.32/0.55 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % one_le_power
% 0.32/0.55 thf(fact_3025_one__le__power,axiom,
% 0.32/0.55 ! [A: int,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_int @ one_one_int @ A )
% 0.32/0.55 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % one_le_power
% 0.32/0.55 thf(fact_3026_Ex__less__Suc2,axiom,
% 0.32/0.55 ! [N: nat,P: nat > $o] :
% 0.32/0.55 ( ( ? [I4: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.32/0.55 & ( P @ I4 ) ) )
% 0.32/0.55 = ( ( P @ zero_zero_nat )
% 0.32/0.55 | ? [I4: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I4 @ N )
% 0.32/0.55 & ( P @ ( suc @ I4 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Ex_less_Suc2
% 0.32/0.55 thf(fact_3027_gr0__conv__Suc,axiom,
% 0.32/0.55 ! [N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 = ( ? [M3: nat] :
% 0.32/0.55 ( N
% 0.32/0.55 = ( suc @ M3 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % gr0_conv_Suc
% 0.32/0.55 thf(fact_3028_All__less__Suc2,axiom,
% 0.32/0.55 ! [N: nat,P: nat > $o] :
% 0.32/0.55 ( ( ! [I4: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.32/0.55 => ( P @ I4 ) ) )
% 0.32/0.55 = ( ( P @ zero_zero_nat )
% 0.32/0.55 & ! [I4: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I4 @ N )
% 0.32/0.55 => ( P @ ( suc @ I4 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % All_less_Suc2
% 0.32/0.55 thf(fact_3029_gr0__implies__Suc,axiom,
% 0.32/0.55 ! [N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ? [M2: nat] :
% 0.32/0.55 ( N
% 0.32/0.55 = ( suc @ M2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % gr0_implies_Suc
% 0.32/0.55 thf(fact_3030_less__Suc__eq__0__disj,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.32/0.55 = ( ( M = zero_zero_nat )
% 0.32/0.55 | ? [J3: nat] :
% 0.32/0.55 ( ( M
% 0.32/0.55 = ( suc @ J3 ) )
% 0.32/0.55 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_Suc_eq_0_disj
% 0.32/0.55 thf(fact_3031_le__imp__power__dvd,axiom,
% 0.32/0.55 ! [M: nat,N: nat,A: code_integer] :
% 0.32/0.55 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.55 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % le_imp_power_dvd
% 0.32/0.55 thf(fact_3032_le__imp__power__dvd,axiom,
% 0.32/0.55 ! [M: nat,N: nat,A: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.55 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % le_imp_power_dvd
% 0.32/0.55 thf(fact_3033_le__imp__power__dvd,axiom,
% 0.32/0.55 ! [M: nat,N: nat,A: real] :
% 0.32/0.55 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.55 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % le_imp_power_dvd
% 0.32/0.55 thf(fact_3034_le__imp__power__dvd,axiom,
% 0.32/0.55 ! [M: nat,N: nat,A: int] :
% 0.32/0.55 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.55 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % le_imp_power_dvd
% 0.32/0.55 thf(fact_3035_le__imp__power__dvd,axiom,
% 0.32/0.55 ! [M: nat,N: nat,A: complex] :
% 0.32/0.55 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.55 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % le_imp_power_dvd
% 0.32/0.55 thf(fact_3036_power__le__dvd,axiom,
% 0.32/0.55 ! [A: code_integer,N: nat,B: code_integer,M: nat] :
% 0.32/0.55 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
% 0.32/0.55 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.55 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_dvd
% 0.32/0.55 thf(fact_3037_power__le__dvd,axiom,
% 0.32/0.55 ! [A: nat,N: nat,B: nat,M: nat] :
% 0.32/0.55 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
% 0.32/0.55 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.55 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_dvd
% 0.32/0.55 thf(fact_3038_power__le__dvd,axiom,
% 0.32/0.55 ! [A: real,N: nat,B: real,M: nat] :
% 0.32/0.55 ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
% 0.32/0.55 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.55 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_dvd
% 0.32/0.55 thf(fact_3039_power__le__dvd,axiom,
% 0.32/0.55 ! [A: int,N: nat,B: int,M: nat] :
% 0.32/0.55 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
% 0.32/0.55 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.55 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_dvd
% 0.32/0.55 thf(fact_3040_power__le__dvd,axiom,
% 0.32/0.55 ! [A: complex,N: nat,B: complex,M: nat] :
% 0.32/0.55 ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
% 0.32/0.55 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.55 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_dvd
% 0.32/0.55 thf(fact_3041_dvd__power__le,axiom,
% 0.32/0.55 ! [X: code_integer,Y: code_integer,N: nat,M: nat] :
% 0.32/0.55 ( ( dvd_dvd_Code_integer @ X @ Y )
% 0.32/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_power_le
% 0.32/0.55 thf(fact_3042_dvd__power__le,axiom,
% 0.32/0.55 ! [X: nat,Y: nat,N: nat,M: nat] :
% 0.32/0.55 ( ( dvd_dvd_nat @ X @ Y )
% 0.32/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_power_le
% 0.32/0.55 thf(fact_3043_dvd__power__le,axiom,
% 0.32/0.55 ! [X: real,Y: real,N: nat,M: nat] :
% 0.32/0.55 ( ( dvd_dvd_real @ X @ Y )
% 0.32/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ M ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_power_le
% 0.32/0.55 thf(fact_3044_dvd__power__le,axiom,
% 0.32/0.55 ! [X: int,Y: int,N: nat,M: nat] :
% 0.32/0.55 ( ( dvd_dvd_int @ X @ Y )
% 0.32/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_power_le
% 0.32/0.55 thf(fact_3045_dvd__power__le,axiom,
% 0.32/0.55 ! [X: complex,Y: complex,N: nat,M: nat] :
% 0.32/0.55 ( ( dvd_dvd_complex @ X @ Y )
% 0.32/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ M ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_power_le
% 0.32/0.55 thf(fact_3046_less__natE,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ M @ N )
% 0.32/0.55 => ~ ! [Q6: nat] :
% 0.32/0.55 ( N
% 0.32/0.55 != ( suc @ ( plus_plus_nat @ M @ Q6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_natE
% 0.32/0.55 thf(fact_3047_less__add__Suc1,axiom,
% 0.32/0.55 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_add_Suc1
% 0.32/0.55 thf(fact_3048_less__add__Suc2,axiom,
% 0.32/0.55 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_add_Suc2
% 0.32/0.55 thf(fact_3049_less__iff__Suc__add,axiom,
% 0.32/0.55 ( ord_less_nat
% 0.32/0.55 = ( ^ [M3: nat,N3: nat] :
% 0.32/0.55 ? [K2: nat] :
% 0.32/0.55 ( N3
% 0.32/0.55 = ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_iff_Suc_add
% 0.32/0.55 thf(fact_3050_less__imp__Suc__add,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ M @ N )
% 0.32/0.55 => ? [K3: nat] :
% 0.32/0.55 ( N
% 0.32/0.55 = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_imp_Suc_add
% 0.32/0.55 thf(fact_3051_less__imp__add__positive,axiom,
% 0.32/0.55 ! [I: nat,J: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I @ J )
% 0.32/0.55 => ? [K3: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.32/0.55 & ( ( plus_plus_nat @ I @ K3 )
% 0.32/0.55 = J ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_imp_add_positive
% 0.32/0.55 thf(fact_3052_Suc__diff__Suc,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_nat @ N @ M )
% 0.32/0.55 => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 0.32/0.55 = ( minus_minus_nat @ M @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Suc_diff_Suc
% 0.32/0.55 thf(fact_3053_diff__less__Suc,axiom,
% 0.32/0.55 ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_less_Suc
% 0.32/0.55 thf(fact_3054_diff__less,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.55 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_less
% 0.32/0.55 thf(fact_3055_Suc__mult__less__cancel1,axiom,
% 0.32/0.55 ! [K: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.32/0.55 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % Suc_mult_less_cancel1
% 0.32/0.55 thf(fact_3056_nat__mult__less__cancel1,axiom,
% 0.32/0.55 ! [K: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.55 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.55 = ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_mult_less_cancel1
% 0.32/0.55 thf(fact_3057_nat__mult__eq__cancel1,axiom,
% 0.32/0.55 ! [K: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.55 => ( ( ( times_times_nat @ K @ M )
% 0.32/0.55 = ( times_times_nat @ K @ N ) )
% 0.32/0.55 = ( M = N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_mult_eq_cancel1
% 0.32/0.55 thf(fact_3058_mult__less__mono2,axiom,
% 0.32/0.55 ! [I: nat,J: nat,K: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I @ J )
% 0.32/0.55 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.55 => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_less_mono2
% 0.32/0.55 thf(fact_3059_mult__less__mono1,axiom,
% 0.32/0.55 ! [I: nat,J: nat,K: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I @ J )
% 0.32/0.55 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.55 => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_less_mono1
% 0.32/0.55 thf(fact_3060_Suc__diff__le,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.32/0.55 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Suc_diff_le
% 0.32/0.55 thf(fact_3061_Suc__mult__le__cancel1,axiom,
% 0.32/0.55 ! [K: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.32/0.55 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % Suc_mult_le_cancel1
% 0.32/0.55 thf(fact_3062_realpow__pos__nth2,axiom,
% 0.32/0.55 ! [A: real,N: nat] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.55 => ? [R2: real] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.32/0.55 & ( ( power_power_real @ R2 @ ( suc @ N ) )
% 0.32/0.55 = A ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % realpow_pos_nth2
% 0.32/0.55 thf(fact_3063_nat__dvd__not__less,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.55 => ( ( ord_less_nat @ M @ N )
% 0.32/0.55 => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_dvd_not_less
% 0.32/0.55 thf(fact_3064_dvd__pos__nat,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ( dvd_dvd_nat @ M @ N )
% 0.32/0.55 => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_pos_nat
% 0.32/0.55 thf(fact_3065_add__diff__inverse__nat,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ~ ( ord_less_nat @ M @ N )
% 0.32/0.55 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.55 = M ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_diff_inverse_nat
% 0.32/0.55 thf(fact_3066_less__diff__conv,axiom,
% 0.32/0.55 ! [I: nat,J: nat,K: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.32/0.55 = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_diff_conv
% 0.32/0.55 thf(fact_3067_zmult__zless__mono2,axiom,
% 0.32/0.55 ! [I: int,J: int,K: int] :
% 0.32/0.55 ( ( ord_less_int @ I @ J )
% 0.32/0.55 => ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.55 => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zmult_zless_mono2
% 0.32/0.55 thf(fact_3068_Nat_Ole__imp__diff__is__add,axiom,
% 0.32/0.55 ! [I: nat,J: nat,K: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.55 => ( ( ( minus_minus_nat @ J @ I )
% 0.32/0.55 = K )
% 0.32/0.55 = ( J
% 0.32/0.55 = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Nat.le_imp_diff_is_add
% 0.32/0.55 thf(fact_3069_Nat_Odiff__add__assoc2,axiom,
% 0.32/0.55 ! [K: nat,J: nat,I: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.55 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 0.32/0.55 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Nat.diff_add_assoc2
% 0.32/0.55 thf(fact_3070_Nat_Odiff__add__assoc,axiom,
% 0.32/0.55 ! [K: nat,J: nat,I: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.55 => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.32/0.55 = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Nat.diff_add_assoc
% 0.32/0.55 thf(fact_3071_Nat_Ole__diff__conv2,axiom,
% 0.32/0.55 ! [K: nat,J: nat,I: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.55 => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.32/0.55 = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Nat.le_diff_conv2
% 0.32/0.55 thf(fact_3072_le__diff__conv,axiom,
% 0.32/0.55 ! [J: nat,K: nat,I: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.32/0.55 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % le_diff_conv
% 0.32/0.55 thf(fact_3073_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ( ( divide_divide_nat @ M @ N )
% 0.32/0.55 = zero_zero_nat )
% 0.32/0.55 = ( ( ord_less_nat @ M @ N )
% 0.32/0.55 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Euclidean_Division.div_eq_0_iff
% 0.32/0.55 thf(fact_3074_Suc__div__le__mono,axiom,
% 0.32/0.55 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % Suc_div_le_mono
% 0.32/0.55 thf(fact_3075_nat__power__less__imp__less,axiom,
% 0.32/0.55 ! [I: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ I )
% 0.32/0.55 => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.32/0.55 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_power_less_imp_less
% 0.32/0.55 thf(fact_3076_int__gr__induct,axiom,
% 0.32/0.55 ! [K: int,I: int,P: int > $o] :
% 0.32/0.55 ( ( ord_less_int @ K @ I )
% 0.32/0.55 => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 0.32/0.55 => ( ! [I2: int] :
% 0.32/0.55 ( ( ord_less_int @ K @ I2 )
% 0.32/0.55 => ( ( P @ I2 )
% 0.32/0.55 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 0.32/0.55 => ( P @ I ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % int_gr_induct
% 0.32/0.55 thf(fact_3077_zless__add1__eq,axiom,
% 0.32/0.55 ! [W: int,Z2: int] :
% 0.32/0.55 ( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
% 0.32/0.55 = ( ( ord_less_int @ W @ Z2 )
% 0.32/0.55 | ( W = Z2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zless_add1_eq
% 0.32/0.55 thf(fact_3078_zero__le__imp__eq__int,axiom,
% 0.32/0.55 ! [K: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.55 => ? [N2: nat] :
% 0.32/0.55 ( K
% 0.32/0.55 = ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zero_le_imp_eq_int
% 0.32/0.55 thf(fact_3079_nonneg__int__cases,axiom,
% 0.32/0.55 ! [K: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.55 => ~ ! [N2: nat] :
% 0.32/0.55 ( K
% 0.32/0.55 != ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nonneg_int_cases
% 0.32/0.55 thf(fact_3080_dvd__minus__self,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 0.32/0.55 = ( ( ord_less_nat @ N @ M )
% 0.32/0.55 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_minus_self
% 0.32/0.55 thf(fact_3081_pos__imp__zdiv__neg__iff,axiom,
% 0.32/0.55 ! [B: int,A: int] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.55 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 0.32/0.55 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pos_imp_zdiv_neg_iff
% 0.32/0.55 thf(fact_3082_neg__imp__zdiv__neg__iff,axiom,
% 0.32/0.55 ! [B: int,A: int] :
% 0.32/0.55 ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.55 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 0.32/0.55 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % neg_imp_zdiv_neg_iff
% 0.32/0.55 thf(fact_3083_div__neg__pos__less0,axiom,
% 0.32/0.55 ! [A: int,B: int] :
% 0.32/0.55 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.55 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.55 => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % div_neg_pos_less0
% 0.32/0.55 thf(fact_3084_dvd__diffD,axiom,
% 0.32/0.55 ! [K: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.55 => ( ( dvd_dvd_nat @ K @ N )
% 0.32/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_diffD
% 0.32/0.55 thf(fact_3085_dvd__diffD1,axiom,
% 0.32/0.55 ! [K: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.55 => ( ( dvd_dvd_nat @ K @ M )
% 0.32/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_diffD1
% 0.32/0.55 thf(fact_3086_less__eq__dvd__minus,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.55 => ( ( dvd_dvd_nat @ M @ N )
% 0.32/0.55 = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_eq_dvd_minus
% 0.32/0.55 thf(fact_3087_zdvd__not__zless,axiom,
% 0.32/0.55 ! [M: int,N: int] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ M )
% 0.32/0.55 => ( ( ord_less_int @ M @ N )
% 0.32/0.55 => ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zdvd_not_zless
% 0.32/0.55 thf(fact_3088_int__ge__induct,axiom,
% 0.32/0.55 ! [K: int,I: int,P: int > $o] :
% 0.32/0.55 ( ( ord_less_eq_int @ K @ I )
% 0.32/0.55 => ( ( P @ K )
% 0.32/0.55 => ( ! [I2: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ K @ I2 )
% 0.32/0.55 => ( ( P @ I2 )
% 0.32/0.55 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 0.32/0.55 => ( P @ I ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % int_ge_induct
% 0.32/0.55 thf(fact_3089_int__less__induct,axiom,
% 0.32/0.55 ! [I: int,K: int,P: int > $o] :
% 0.32/0.55 ( ( ord_less_int @ I @ K )
% 0.32/0.55 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 0.32/0.55 => ( ! [I2: int] :
% 0.32/0.55 ( ( ord_less_int @ I2 @ K )
% 0.32/0.55 => ( ( P @ I2 )
% 0.32/0.55 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 0.32/0.55 => ( P @ I ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % int_less_induct
% 0.32/0.55 thf(fact_3090_less__mult__imp__div__less,axiom,
% 0.32/0.55 ! [M: nat,I: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 0.32/0.55 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_mult_imp_div_less
% 0.32/0.55 thf(fact_3091_div__times__less__eq__dividend,axiom,
% 0.32/0.55 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 0.32/0.55
% 0.32/0.55 % div_times_less_eq_dividend
% 0.32/0.55 thf(fact_3092_times__div__less__eq__dividend,axiom,
% 0.32/0.55 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 0.32/0.55
% 0.32/0.55 % times_div_less_eq_dividend
% 0.32/0.55 thf(fact_3093_zdvd__antisym__nonneg,axiom,
% 0.32/0.55 ! [M: int,N: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ N )
% 0.32/0.55 => ( ( dvd_dvd_int @ M @ N )
% 0.32/0.55 => ( ( dvd_dvd_int @ N @ M )
% 0.32/0.55 => ( M = N ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zdvd_antisym_nonneg
% 0.32/0.55 thf(fact_3094_reals__Archimedean3,axiom,
% 0.32/0.55 ! [X: real] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.55 => ! [Y5: real] :
% 0.32/0.55 ? [N2: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % reals_Archimedean3
% 0.32/0.55 thf(fact_3095_zle__iff__zadd,axiom,
% 0.32/0.55 ( ord_less_eq_int
% 0.32/0.55 = ( ^ [W2: int,Z5: int] :
% 0.32/0.55 ? [N3: nat] :
% 0.32/0.55 ( Z5
% 0.32/0.55 = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zle_iff_zadd
% 0.32/0.55 thf(fact_3096_int__le__induct,axiom,
% 0.32/0.55 ! [I: int,K: int,P: int > $o] :
% 0.32/0.55 ( ( ord_less_eq_int @ I @ K )
% 0.32/0.55 => ( ( P @ K )
% 0.32/0.55 => ( ! [I2: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ I2 @ K )
% 0.32/0.55 => ( ( P @ I2 )
% 0.32/0.55 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 0.32/0.55 => ( P @ I ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % int_le_induct
% 0.32/0.55 thf(fact_3097_power2__less__imp__less,axiom,
% 0.32/0.55 ! [X: real,Y: real] :
% 0.32/0.55 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.32/0.55 => ( ord_less_real @ X @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power2_less_imp_less
% 0.32/0.55 thf(fact_3098_power2__less__imp__less,axiom,
% 0.32/0.55 ! [X: rat,Y: rat] :
% 0.32/0.55 ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.32/0.55 => ( ord_less_rat @ X @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power2_less_imp_less
% 0.32/0.55 thf(fact_3099_power2__less__imp__less,axiom,
% 0.32/0.55 ! [X: nat,Y: nat] :
% 0.32/0.55 ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.32/0.55 => ( ord_less_nat @ X @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power2_less_imp_less
% 0.32/0.55 thf(fact_3100_power2__less__imp__less,axiom,
% 0.32/0.55 ! [X: int,Y: int] :
% 0.32/0.55 ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.32/0.55 => ( ord_less_int @ X @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power2_less_imp_less
% 0.32/0.55 thf(fact_3101_pow_Osimps_I1_J,axiom,
% 0.32/0.55 ! [X: num] :
% 0.32/0.55 ( ( pow @ X @ one )
% 0.32/0.55 = X ) ).
% 0.32/0.55
% 0.32/0.55 % pow.simps(1)
% 0.32/0.55 thf(fact_3102_div__pos__geq,axiom,
% 0.32/0.55 ! [L: int,K: int] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ L )
% 0.32/0.55 => ( ( ord_less_eq_int @ L @ K )
% 0.32/0.55 => ( ( divide_divide_int @ K @ L )
% 0.32/0.55 = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % div_pos_geq
% 0.32/0.55 thf(fact_3103_not__sum__squares__lt__zero,axiom,
% 0.32/0.55 ! [X: real,Y: real] :
% 0.32/0.55 ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% 0.32/0.55
% 0.32/0.55 % not_sum_squares_lt_zero
% 0.32/0.55 thf(fact_3104_not__sum__squares__lt__zero,axiom,
% 0.32/0.55 ! [X: rat,Y: rat] :
% 0.32/0.55 ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).
% 0.32/0.55
% 0.32/0.55 % not_sum_squares_lt_zero
% 0.32/0.55 thf(fact_3105_not__sum__squares__lt__zero,axiom,
% 0.32/0.55 ! [X: int,Y: int] :
% 0.32/0.55 ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% 0.32/0.55
% 0.32/0.55 % not_sum_squares_lt_zero
% 0.32/0.55 thf(fact_3106_sum__squares__gt__zero__iff,axiom,
% 0.32/0.55 ! [X: real,Y: real] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
% 0.32/0.55 = ( ( X != zero_zero_real )
% 0.32/0.55 | ( Y != zero_zero_real ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % sum_squares_gt_zero_iff
% 0.32/0.55 thf(fact_3107_sum__squares__gt__zero__iff,axiom,
% 0.32/0.55 ! [X: rat,Y: rat] :
% 0.32/0.55 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
% 0.32/0.55 = ( ( X != zero_zero_rat )
% 0.32/0.55 | ( Y != zero_zero_rat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % sum_squares_gt_zero_iff
% 0.32/0.55 thf(fact_3108_sum__squares__gt__zero__iff,axiom,
% 0.32/0.55 ! [X: int,Y: int] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
% 0.32/0.55 = ( ( X != zero_zero_int )
% 0.32/0.55 | ( Y != zero_zero_int ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % sum_squares_gt_zero_iff
% 0.32/0.55 thf(fact_3109_zero__less__two,axiom,
% 0.32/0.55 ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 0.32/0.55
% 0.32/0.55 % zero_less_two
% 0.32/0.55 thf(fact_3110_zero__less__two,axiom,
% 0.32/0.55 ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 0.32/0.55
% 0.32/0.55 % zero_less_two
% 0.32/0.55 thf(fact_3111_zero__less__two,axiom,
% 0.32/0.55 ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 0.32/0.55
% 0.32/0.55 % zero_less_two
% 0.32/0.55 thf(fact_3112_zero__less__two,axiom,
% 0.32/0.55 ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 0.32/0.55
% 0.32/0.55 % zero_less_two
% 0.32/0.55 thf(fact_3113_ex__power__ivl2,axiom,
% 0.32/0.55 ! [B: nat,K: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.32/0.55 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.32/0.55 => ? [N2: nat] :
% 0.32/0.55 ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 0.32/0.55 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % ex_power_ivl2
% 0.32/0.55 thf(fact_3114_ex__power__ivl1,axiom,
% 0.32/0.55 ! [B: nat,K: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.32/0.55 => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 0.32/0.55 => ? [N2: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 0.32/0.55 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % ex_power_ivl1
% 0.32/0.55 thf(fact_3115_divide__strict__left__mono__neg,axiom,
% 0.32/0.55 ! [A: real,B: real,C: real] :
% 0.32/0.55 ( ( ord_less_real @ A @ B )
% 0.32/0.55 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.55 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_strict_left_mono_neg
% 0.32/0.55 thf(fact_3116_divide__strict__left__mono__neg,axiom,
% 0.32/0.55 ! [A: rat,B: rat,C: rat] :
% 0.32/0.55 ( ( ord_less_rat @ A @ B )
% 0.32/0.55 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.55 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_strict_left_mono_neg
% 0.32/0.55 thf(fact_3117_divide__strict__left__mono,axiom,
% 0.32/0.55 ! [B: real,A: real,C: real] :
% 0.32/0.55 ( ( ord_less_real @ B @ A )
% 0.32/0.55 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.55 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_strict_left_mono
% 0.32/0.55 thf(fact_3118_divide__strict__left__mono,axiom,
% 0.32/0.55 ! [B: rat,A: rat,C: rat] :
% 0.32/0.55 ( ( ord_less_rat @ B @ A )
% 0.32/0.55 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.55 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_strict_left_mono
% 0.32/0.55 thf(fact_3119_mult__imp__less__div__pos,axiom,
% 0.32/0.55 ! [Y: real,Z2: real,X: real] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.55 => ( ( ord_less_real @ ( times_times_real @ Z2 @ Y ) @ X )
% 0.32/0.55 => ( ord_less_real @ Z2 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_imp_less_div_pos
% 0.32/0.55 thf(fact_3120_mult__imp__less__div__pos,axiom,
% 0.32/0.55 ! [Y: rat,Z2: rat,X: rat] :
% 0.32/0.55 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.32/0.55 => ( ( ord_less_rat @ ( times_times_rat @ Z2 @ Y ) @ X )
% 0.32/0.55 => ( ord_less_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_imp_less_div_pos
% 0.32/0.55 thf(fact_3121_mult__imp__div__pos__less,axiom,
% 0.32/0.55 ! [Y: real,X: real,Z2: real] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.55 => ( ( ord_less_real @ X @ ( times_times_real @ Z2 @ Y ) )
% 0.32/0.55 => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_imp_div_pos_less
% 0.32/0.55 thf(fact_3122_mult__imp__div__pos__less,axiom,
% 0.32/0.55 ! [Y: rat,X: rat,Z2: rat] :
% 0.32/0.55 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.32/0.55 => ( ( ord_less_rat @ X @ ( times_times_rat @ Z2 @ Y ) )
% 0.32/0.55 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_imp_div_pos_less
% 0.32/0.55 thf(fact_3123_pos__less__divide__eq,axiom,
% 0.32/0.55 ! [C: real,A: real,B: real] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.55 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pos_less_divide_eq
% 0.32/0.55 thf(fact_3124_pos__less__divide__eq,axiom,
% 0.32/0.55 ! [C: rat,A: rat,B: rat] :
% 0.32/0.55 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.55 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pos_less_divide_eq
% 0.32/0.55 thf(fact_3125_pos__divide__less__eq,axiom,
% 0.32/0.55 ! [C: real,B: real,A: real] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.55 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pos_divide_less_eq
% 0.32/0.55 thf(fact_3126_pos__divide__less__eq,axiom,
% 0.32/0.55 ! [C: rat,B: rat,A: rat] :
% 0.32/0.55 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.55 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pos_divide_less_eq
% 0.32/0.55 thf(fact_3127_neg__less__divide__eq,axiom,
% 0.32/0.55 ! [C: real,A: real,B: real] :
% 0.32/0.55 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.55 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % neg_less_divide_eq
% 0.32/0.55 thf(fact_3128_neg__less__divide__eq,axiom,
% 0.32/0.55 ! [C: rat,A: rat,B: rat] :
% 0.32/0.55 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.55 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % neg_less_divide_eq
% 0.32/0.55 thf(fact_3129_neg__divide__less__eq,axiom,
% 0.32/0.55 ! [C: real,B: real,A: real] :
% 0.32/0.55 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.55 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % neg_divide_less_eq
% 0.32/0.55 thf(fact_3130_neg__divide__less__eq,axiom,
% 0.32/0.55 ! [C: rat,B: rat,A: rat] :
% 0.32/0.55 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.55 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % neg_divide_less_eq
% 0.32/0.55 thf(fact_3131_less__divide__eq,axiom,
% 0.32/0.55 ! [A: real,B: real,C: real] :
% 0.32/0.55 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.55 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.32/0.55 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.32/0.55 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_divide_eq
% 0.32/0.55 thf(fact_3132_less__divide__eq,axiom,
% 0.32/0.55 ! [A: rat,B: rat,C: rat] :
% 0.32/0.55 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.32/0.55 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.32/0.55 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_divide_eq
% 0.32/0.55 thf(fact_3133_divide__less__eq,axiom,
% 0.32/0.55 ! [B: real,C: real,A: real] :
% 0.32/0.55 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.55 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.32/0.55 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.32/0.55 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_less_eq
% 0.32/0.55 thf(fact_3134_divide__less__eq,axiom,
% 0.32/0.55 ! [B: rat,C: rat,A: rat] :
% 0.32/0.55 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.32/0.55 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.32/0.55 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_less_eq
% 0.32/0.55 thf(fact_3135_less__divide__eq__1,axiom,
% 0.32/0.55 ! [B: real,A: real] :
% 0.32/0.55 ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.32/0.55 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.55 & ( ord_less_real @ A @ B ) )
% 0.32/0.55 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.55 & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_divide_eq_1
% 0.32/0.55 thf(fact_3136_less__divide__eq__1,axiom,
% 0.32/0.55 ! [B: rat,A: rat] :
% 0.32/0.55 ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.32/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.55 & ( ord_less_rat @ A @ B ) )
% 0.32/0.55 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.55 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_divide_eq_1
% 0.32/0.55 thf(fact_3137_divide__less__eq__1,axiom,
% 0.32/0.55 ! [B: real,A: real] :
% 0.32/0.55 ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.32/0.55 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.55 & ( ord_less_real @ B @ A ) )
% 0.32/0.55 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.55 & ( ord_less_real @ A @ B ) )
% 0.32/0.55 | ( A = zero_zero_real ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_less_eq_1
% 0.32/0.55 thf(fact_3138_divide__less__eq__1,axiom,
% 0.32/0.55 ! [B: rat,A: rat] :
% 0.32/0.55 ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.32/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.55 & ( ord_less_rat @ B @ A ) )
% 0.32/0.55 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.55 & ( ord_less_rat @ A @ B ) )
% 0.32/0.55 | ( A = zero_zero_rat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_less_eq_1
% 0.32/0.55 thf(fact_3139_less__add__iff1,axiom,
% 0.32/0.55 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.32/0.55 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.32/0.55 = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_add_iff1
% 0.32/0.55 thf(fact_3140_less__add__iff1,axiom,
% 0.32/0.55 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.32/0.55 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.32/0.55 = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_add_iff1
% 0.32/0.55 thf(fact_3141_less__add__iff1,axiom,
% 0.32/0.55 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.32/0.55 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.32/0.55 = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_add_iff1
% 0.32/0.55 thf(fact_3142_less__add__iff2,axiom,
% 0.32/0.55 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.32/0.55 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.32/0.55 = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_add_iff2
% 0.32/0.55 thf(fact_3143_less__add__iff2,axiom,
% 0.32/0.55 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.32/0.55 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.32/0.55 = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_add_iff2
% 0.32/0.55 thf(fact_3144_less__add__iff2,axiom,
% 0.32/0.55 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.32/0.55 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.32/0.55 = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_add_iff2
% 0.32/0.55 thf(fact_3145_mult__left__le__one__le,axiom,
% 0.32/0.55 ! [X: real,Y: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.32/0.55 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.32/0.55 => ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_left_le_one_le
% 0.32/0.55 thf(fact_3146_mult__left__le__one__le,axiom,
% 0.32/0.55 ! [X: rat,Y: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.32/0.55 => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 0.32/0.55 => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_left_le_one_le
% 0.32/0.55 thf(fact_3147_mult__left__le__one__le,axiom,
% 0.32/0.55 ! [X: int,Y: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.32/0.55 => ( ( ord_less_eq_int @ Y @ one_one_int )
% 0.32/0.55 => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_left_le_one_le
% 0.32/0.55 thf(fact_3148_mult__right__le__one__le,axiom,
% 0.32/0.55 ! [X: real,Y: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.32/0.55 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.32/0.55 => ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_right_le_one_le
% 0.32/0.55 thf(fact_3149_mult__right__le__one__le,axiom,
% 0.32/0.55 ! [X: rat,Y: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.32/0.55 => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 0.32/0.55 => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_right_le_one_le
% 0.32/0.55 thf(fact_3150_mult__right__le__one__le,axiom,
% 0.32/0.55 ! [X: int,Y: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.32/0.55 => ( ( ord_less_eq_int @ Y @ one_one_int )
% 0.32/0.55 => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_right_le_one_le
% 0.32/0.55 thf(fact_3151_mult__le__one,axiom,
% 0.32/0.55 ! [A: real,B: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ A @ one_one_real )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.55 => ( ( ord_less_eq_real @ B @ one_one_real )
% 0.32/0.55 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_le_one
% 0.32/0.55 thf(fact_3152_mult__le__one,axiom,
% 0.32/0.55 ! [A: rat,B: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.55 => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 0.32/0.55 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_le_one
% 0.32/0.55 thf(fact_3153_mult__le__one,axiom,
% 0.32/0.55 ! [A: nat,B: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.32/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.55 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 0.32/0.55 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_le_one
% 0.32/0.55 thf(fact_3154_mult__le__one,axiom,
% 0.32/0.55 ! [A: int,B: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ A @ one_one_int )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.55 => ( ( ord_less_eq_int @ B @ one_one_int )
% 0.32/0.55 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_le_one
% 0.32/0.55 thf(fact_3155_mult__left__le,axiom,
% 0.32/0.55 ! [C: real,A: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ C @ one_one_real )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.55 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_left_le
% 0.32/0.55 thf(fact_3156_mult__left__le,axiom,
% 0.32/0.55 ! [C: rat,A: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ C @ one_one_rat )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.55 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_left_le
% 0.32/0.55 thf(fact_3157_mult__left__le,axiom,
% 0.32/0.55 ! [C: nat,A: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 0.32/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.55 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_left_le
% 0.32/0.55 thf(fact_3158_mult__left__le,axiom,
% 0.32/0.55 ! [C: int,A: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ C @ one_one_int )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.55 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_left_le
% 0.32/0.55 thf(fact_3159_sum__squares__ge__zero,axiom,
% 0.32/0.55 ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % sum_squares_ge_zero
% 0.32/0.55 thf(fact_3160_sum__squares__ge__zero,axiom,
% 0.32/0.55 ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % sum_squares_ge_zero
% 0.32/0.55 thf(fact_3161_sum__squares__ge__zero,axiom,
% 0.32/0.55 ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % sum_squares_ge_zero
% 0.32/0.55 thf(fact_3162_sum__squares__le__zero__iff,axiom,
% 0.32/0.55 ! [X: real,Y: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
% 0.32/0.55 = ( ( X = zero_zero_real )
% 0.32/0.55 & ( Y = zero_zero_real ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % sum_squares_le_zero_iff
% 0.32/0.55 thf(fact_3163_sum__squares__le__zero__iff,axiom,
% 0.32/0.55 ! [X: rat,Y: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
% 0.32/0.55 = ( ( X = zero_zero_rat )
% 0.32/0.55 & ( Y = zero_zero_rat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % sum_squares_le_zero_iff
% 0.32/0.55 thf(fact_3164_sum__squares__le__zero__iff,axiom,
% 0.32/0.55 ! [X: int,Y: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
% 0.32/0.55 = ( ( X = zero_zero_int )
% 0.32/0.55 & ( Y = zero_zero_int ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % sum_squares_le_zero_iff
% 0.32/0.55 thf(fact_3165_ex__less__of__nat__mult,axiom,
% 0.32/0.55 ! [X: real,Y: real] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.55 => ? [N2: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % ex_less_of_nat_mult
% 0.32/0.55 thf(fact_3166_ex__less__of__nat__mult,axiom,
% 0.32/0.55 ! [X: rat,Y: rat] :
% 0.32/0.55 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.32/0.55 => ? [N2: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ X ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % ex_less_of_nat_mult
% 0.32/0.55 thf(fact_3167_less__half__sum,axiom,
% 0.32/0.55 ! [A: real,B: real] :
% 0.32/0.55 ( ( ord_less_real @ A @ B )
% 0.32/0.55 => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_half_sum
% 0.32/0.55 thf(fact_3168_less__half__sum,axiom,
% 0.32/0.55 ! [A: rat,B: rat] :
% 0.32/0.55 ( ( ord_less_rat @ A @ B )
% 0.32/0.55 => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_half_sum
% 0.32/0.55 thf(fact_3169_gt__half__sum,axiom,
% 0.32/0.55 ! [A: real,B: real] :
% 0.32/0.55 ( ( ord_less_real @ A @ B )
% 0.32/0.55 => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% 0.32/0.55
% 0.32/0.55 % gt_half_sum
% 0.32/0.55 thf(fact_3170_gt__half__sum,axiom,
% 0.32/0.55 ! [A: rat,B: rat] :
% 0.32/0.55 ( ( ord_less_rat @ A @ B )
% 0.32/0.55 => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% 0.32/0.55
% 0.32/0.55 % gt_half_sum
% 0.32/0.55 thf(fact_3171_power__gt1__lemma,axiom,
% 0.32/0.55 ! [A: real,N: nat] :
% 0.32/0.55 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.55 => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_gt1_lemma
% 0.32/0.55 thf(fact_3172_power__gt1__lemma,axiom,
% 0.32/0.55 ! [A: rat,N: nat] :
% 0.32/0.55 ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.55 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_gt1_lemma
% 0.32/0.55 thf(fact_3173_power__gt1__lemma,axiom,
% 0.32/0.55 ! [A: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.55 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_gt1_lemma
% 0.32/0.55 thf(fact_3174_power__gt1__lemma,axiom,
% 0.32/0.55 ! [A: int,N: nat] :
% 0.32/0.55 ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.55 => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_gt1_lemma
% 0.32/0.55 thf(fact_3175_power__less__power__Suc,axiom,
% 0.32/0.55 ! [A: real,N: nat] :
% 0.32/0.55 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.55 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_less_power_Suc
% 0.32/0.55 thf(fact_3176_power__less__power__Suc,axiom,
% 0.32/0.55 ! [A: rat,N: nat] :
% 0.32/0.55 ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.55 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_less_power_Suc
% 0.32/0.55 thf(fact_3177_power__less__power__Suc,axiom,
% 0.32/0.55 ! [A: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.55 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_less_power_Suc
% 0.32/0.55 thf(fact_3178_power__less__power__Suc,axiom,
% 0.32/0.55 ! [A: int,N: nat] :
% 0.32/0.55 ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.55 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_less_power_Suc
% 0.32/0.55 thf(fact_3179_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.32/0.55 ! [C: nat,A: nat,B: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.55 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.55 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.32/0.55 thf(fact_3180_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.32/0.55 ! [C: int,A: int,B: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.55 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.55 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.32/0.55 thf(fact_3181_power__le__one,axiom,
% 0.32/0.55 ! [A: real,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.55 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.32/0.55 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_one
% 0.32/0.55 thf(fact_3182_power__le__one,axiom,
% 0.32/0.55 ! [A: rat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.55 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.32/0.55 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_one
% 0.32/0.55 thf(fact_3183_power__le__one,axiom,
% 0.32/0.55 ! [A: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.55 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.32/0.55 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_one
% 0.32/0.55 thf(fact_3184_power__le__one,axiom,
% 0.32/0.55 ! [A: int,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.55 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.32/0.55 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_one
% 0.32/0.55 thf(fact_3185_power__gt1,axiom,
% 0.32/0.55 ! [A: real,N: nat] :
% 0.32/0.55 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.55 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_gt1
% 0.32/0.55 thf(fact_3186_power__gt1,axiom,
% 0.32/0.55 ! [A: rat,N: nat] :
% 0.32/0.55 ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.55 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_gt1
% 0.32/0.55 thf(fact_3187_power__gt1,axiom,
% 0.32/0.55 ! [A: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.55 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_gt1
% 0.32/0.55 thf(fact_3188_power__gt1,axiom,
% 0.32/0.55 ! [A: int,N: nat] :
% 0.32/0.55 ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.55 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_gt1
% 0.32/0.55 thf(fact_3189_ordered__ring__class_Ole__add__iff1,axiom,
% 0.32/0.55 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.32/0.55 = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.32/0.55
% 0.32/0.55 % ordered_ring_class.le_add_iff1
% 0.32/0.55 thf(fact_3190_ordered__ring__class_Ole__add__iff1,axiom,
% 0.32/0.55 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.32/0.55 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.32/0.55
% 0.32/0.55 % ordered_ring_class.le_add_iff1
% 0.32/0.55 thf(fact_3191_ordered__ring__class_Ole__add__iff1,axiom,
% 0.32/0.55 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.32/0.55 = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.32/0.55
% 0.32/0.55 % ordered_ring_class.le_add_iff1
% 0.32/0.55 thf(fact_3192_ordered__ring__class_Ole__add__iff2,axiom,
% 0.32/0.55 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.32/0.55 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % ordered_ring_class.le_add_iff2
% 0.32/0.55 thf(fact_3193_ordered__ring__class_Ole__add__iff2,axiom,
% 0.32/0.55 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.32/0.55 = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % ordered_ring_class.le_add_iff2
% 0.32/0.55 thf(fact_3194_ordered__ring__class_Ole__add__iff2,axiom,
% 0.32/0.55 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.32/0.55 = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % ordered_ring_class.le_add_iff2
% 0.32/0.55 thf(fact_3195_int__power__div__base,axiom,
% 0.32/0.55 ! [M: nat,K: int] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.55 => ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.55 => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 0.32/0.55 = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % int_power_div_base
% 0.32/0.55 thf(fact_3196_power__inject__base,axiom,
% 0.32/0.55 ! [A: real,N: nat,B: real] :
% 0.32/0.55 ( ( ( power_power_real @ A @ ( suc @ N ) )
% 0.32/0.55 = ( power_power_real @ B @ ( suc @ N ) ) )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.55 => ( A = B ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_inject_base
% 0.32/0.55 thf(fact_3197_power__inject__base,axiom,
% 0.32/0.55 ! [A: rat,N: nat,B: rat] :
% 0.32/0.55 ( ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.32/0.55 = ( power_power_rat @ B @ ( suc @ N ) ) )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.55 => ( A = B ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_inject_base
% 0.32/0.55 thf(fact_3198_power__inject__base,axiom,
% 0.32/0.55 ! [A: nat,N: nat,B: nat] :
% 0.32/0.55 ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.32/0.55 = ( power_power_nat @ B @ ( suc @ N ) ) )
% 0.32/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.55 => ( A = B ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_inject_base
% 0.32/0.55 thf(fact_3199_power__inject__base,axiom,
% 0.32/0.55 ! [A: int,N: nat,B: int] :
% 0.32/0.55 ( ( ( power_power_int @ A @ ( suc @ N ) )
% 0.32/0.55 = ( power_power_int @ B @ ( suc @ N ) ) )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.55 => ( A = B ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_inject_base
% 0.32/0.55 thf(fact_3200_power__le__imp__le__base,axiom,
% 0.32/0.55 ! [A: real,N: nat,B: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.55 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_imp_le_base
% 0.32/0.55 thf(fact_3201_power__le__imp__le__base,axiom,
% 0.32/0.55 ! [A: rat,N: nat,B: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.55 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_imp_le_base
% 0.32/0.55 thf(fact_3202_power__le__imp__le__base,axiom,
% 0.32/0.55 ! [A: nat,N: nat,B: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
% 0.32/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.55 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_imp_le_base
% 0.32/0.55 thf(fact_3203_power__le__imp__le__base,axiom,
% 0.32/0.55 ! [A: int,N: nat,B: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.55 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_imp_le_base
% 0.32/0.55 thf(fact_3204_zero__power,axiom,
% 0.32/0.55 ! [N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.32/0.55 = zero_zero_rat ) ) ).
% 0.32/0.55
% 0.32/0.55 % zero_power
% 0.32/0.55 thf(fact_3205_zero__power,axiom,
% 0.32/0.55 ! [N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.32/0.55 = zero_zero_nat ) ) ).
% 0.32/0.55
% 0.32/0.55 % zero_power
% 0.32/0.55 thf(fact_3206_zero__power,axiom,
% 0.32/0.55 ! [N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ( power_power_real @ zero_zero_real @ N )
% 0.32/0.55 = zero_zero_real ) ) ).
% 0.32/0.55
% 0.32/0.55 % zero_power
% 0.32/0.55 thf(fact_3207_zero__power,axiom,
% 0.32/0.55 ! [N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ( power_power_int @ zero_zero_int @ N )
% 0.32/0.55 = zero_zero_int ) ) ).
% 0.32/0.55
% 0.32/0.55 % zero_power
% 0.32/0.55 thf(fact_3208_zero__power,axiom,
% 0.32/0.55 ! [N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.32/0.55 = zero_zero_complex ) ) ).
% 0.32/0.55
% 0.32/0.55 % zero_power
% 0.32/0.55 thf(fact_3209_diff__Suc__less,axiom,
% 0.32/0.55 ! [N: nat,I: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_Suc_less
% 0.32/0.55 thf(fact_3210_n__less__n__mult__m,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.32/0.55 => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % n_less_n_mult_m
% 0.32/0.55 thf(fact_3211_n__less__m__mult__n,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.32/0.55 => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % n_less_m_mult_n
% 0.32/0.55 thf(fact_3212_one__less__mult,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.55 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.32/0.55 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % one_less_mult
% 0.32/0.55 thf(fact_3213_of__nat__diff,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.55 = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % of_nat_diff
% 0.32/0.55 thf(fact_3214_of__nat__diff,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.55 = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % of_nat_diff
% 0.32/0.55 thf(fact_3215_of__nat__diff,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.55 = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % of_nat_diff
% 0.32/0.55 thf(fact_3216_of__nat__diff,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.55 = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % of_nat_diff
% 0.32/0.55 thf(fact_3217_of__nat__diff,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.55 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % of_nat_diff
% 0.32/0.55 thf(fact_3218_nat__induct__non__zero,axiom,
% 0.32/0.55 ! [N: nat,P: nat > $o] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ( P @ one_one_nat )
% 0.32/0.55 => ( ! [N2: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.55 => ( ( P @ N2 )
% 0.32/0.55 => ( P @ ( suc @ N2 ) ) ) )
% 0.32/0.55 => ( P @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_induct_non_zero
% 0.32/0.55 thf(fact_3219_nat__diff__split,axiom,
% 0.32/0.55 ! [P: nat > $o,A: nat,B: nat] :
% 0.32/0.55 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 0.32/0.55 = ( ( ( ord_less_nat @ A @ B )
% 0.32/0.55 => ( P @ zero_zero_nat ) )
% 0.32/0.55 & ! [D4: nat] :
% 0.32/0.55 ( ( A
% 0.32/0.55 = ( plus_plus_nat @ B @ D4 ) )
% 0.32/0.55 => ( P @ D4 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_diff_split
% 0.32/0.55 thf(fact_3220_nat__diff__split__asm,axiom,
% 0.32/0.55 ! [P: nat > $o,A: nat,B: nat] :
% 0.32/0.55 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 0.32/0.55 = ( ~ ( ( ( ord_less_nat @ A @ B )
% 0.32/0.55 & ~ ( P @ zero_zero_nat ) )
% 0.32/0.55 | ? [D4: nat] :
% 0.32/0.55 ( ( A
% 0.32/0.55 = ( plus_plus_nat @ B @ D4 ) )
% 0.32/0.55 & ~ ( P @ D4 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_diff_split_asm
% 0.32/0.55 thf(fact_3221_power__gt__expt,axiom,
% 0.32/0.55 ! [N: nat,K: nat] :
% 0.32/0.55 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.55 => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_gt_expt
% 0.32/0.55 thf(fact_3222_zless__iff__Suc__zadd,axiom,
% 0.32/0.55 ( ord_less_int
% 0.32/0.55 = ( ^ [W2: int,Z5: int] :
% 0.32/0.55 ? [N3: nat] :
% 0.32/0.55 ( Z5
% 0.32/0.55 = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zless_iff_Suc_zadd
% 0.32/0.55 thf(fact_3223_nat__one__le__power,axiom,
% 0.32/0.55 ! [I: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 0.32/0.55 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_one_le_power
% 0.32/0.55 thf(fact_3224_pos__zmult__eq__1__iff,axiom,
% 0.32/0.55 ! [M: int,N: int] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ M )
% 0.32/0.55 => ( ( ( times_times_int @ M @ N )
% 0.32/0.55 = one_one_int )
% 0.32/0.55 = ( ( M = one_one_int )
% 0.32/0.55 & ( N = one_one_int ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pos_zmult_eq_1_iff
% 0.32/0.55 thf(fact_3225_nat__mult__dvd__cancel1,axiom,
% 0.32/0.55 ! [K: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.55 => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.55 = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_mult_dvd_cancel1
% 0.32/0.55 thf(fact_3226_dvd__mult__cancel,axiom,
% 0.32/0.55 ! [K: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.55 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.55 => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_mult_cancel
% 0.32/0.55 thf(fact_3227_odd__less__0__iff,axiom,
% 0.32/0.55 ! [Z2: int] :
% 0.32/0.55 ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
% 0.32/0.55 = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 0.32/0.55
% 0.32/0.55 % odd_less_0_iff
% 0.32/0.55 thf(fact_3228_div__less__iff__less__mult,axiom,
% 0.32/0.55 ! [Q2: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 0.32/0.55 => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
% 0.32/0.55 = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % div_less_iff_less_mult
% 0.32/0.55 thf(fact_3229_nat__mult__div__cancel1,axiom,
% 0.32/0.55 ! [K: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.55 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.55 = ( divide_divide_nat @ M @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_mult_div_cancel1
% 0.32/0.55 thf(fact_3230_nat__eq__add__iff1,axiom,
% 0.32/0.55 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ J @ I )
% 0.32/0.55 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.32/0.55 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.32/0.55 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 0.32/0.55 = N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_eq_add_iff1
% 0.32/0.55 thf(fact_3231_nat__eq__add__iff2,axiom,
% 0.32/0.55 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.55 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.32/0.55 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.32/0.55 = ( M
% 0.32/0.55 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_eq_add_iff2
% 0.32/0.55 thf(fact_3232_nat__le__add__iff1,axiom,
% 0.32/0.55 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ J @ I )
% 0.32/0.55 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.32/0.55 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_le_add_iff1
% 0.32/0.55 thf(fact_3233_nat__le__add__iff2,axiom,
% 0.32/0.55 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.55 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.32/0.55 = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_le_add_iff2
% 0.32/0.55 thf(fact_3234_nat__diff__add__eq1,axiom,
% 0.32/0.55 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ J @ I )
% 0.32/0.55 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.32/0.55 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_diff_add_eq1
% 0.32/0.55 thf(fact_3235_nat__diff__add__eq2,axiom,
% 0.32/0.55 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.55 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.32/0.55 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_diff_add_eq2
% 0.32/0.55 thf(fact_3236_div__eq__dividend__iff,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.55 => ( ( ( divide_divide_nat @ M @ N )
% 0.32/0.55 = M )
% 0.32/0.55 = ( N = one_one_nat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % div_eq_dividend_iff
% 0.32/0.55 thf(fact_3237_div__less__dividend,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_nat @ one_one_nat @ N )
% 0.32/0.55 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.55 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % div_less_dividend
% 0.32/0.55 thf(fact_3238_int__div__less__self,axiom,
% 0.32/0.55 ! [X: int,K: int] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ X )
% 0.32/0.55 => ( ( ord_less_int @ one_one_int @ K )
% 0.32/0.55 => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % int_div_less_self
% 0.32/0.55 thf(fact_3239_plusinfinity,axiom,
% 0.32/0.55 ! [D: int,P4: int > $o,P: int > $o] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ D )
% 0.32/0.55 => ( ! [X4: int,K3: int] :
% 0.32/0.55 ( ( P4 @ X4 )
% 0.32/0.55 = ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) )
% 0.32/0.55 => ( ? [Z3: int] :
% 0.32/0.55 ! [X4: int] :
% 0.32/0.55 ( ( ord_less_int @ Z3 @ X4 )
% 0.32/0.55 => ( ( P @ X4 )
% 0.32/0.55 = ( P4 @ X4 ) ) )
% 0.32/0.55 => ( ? [X_1: int] : ( P4 @ X_1 )
% 0.32/0.55 => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % plusinfinity
% 0.32/0.55 thf(fact_3240_minusinfinity,axiom,
% 0.32/0.55 ! [D: int,P1: int > $o,P: int > $o] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ D )
% 0.32/0.55 => ( ! [X4: int,K3: int] :
% 0.32/0.55 ( ( P1 @ X4 )
% 0.32/0.55 = ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) )
% 0.32/0.55 => ( ? [Z3: int] :
% 0.32/0.55 ! [X4: int] :
% 0.32/0.55 ( ( ord_less_int @ X4 @ Z3 )
% 0.32/0.55 => ( ( P @ X4 )
% 0.32/0.55 = ( P1 @ X4 ) ) )
% 0.32/0.55 => ( ? [X_1: int] : ( P1 @ X_1 )
% 0.32/0.55 => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minusinfinity
% 0.32/0.55 thf(fact_3241_zdiv__zmult2__eq,axiom,
% 0.32/0.55 ! [C: int,A: int,B: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.55 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.55 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zdiv_zmult2_eq
% 0.32/0.55 thf(fact_3242_int__induct,axiom,
% 0.32/0.55 ! [P: int > $o,K: int,I: int] :
% 0.32/0.55 ( ( P @ K )
% 0.32/0.55 => ( ! [I2: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ K @ I2 )
% 0.32/0.55 => ( ( P @ I2 )
% 0.32/0.55 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 0.32/0.55 => ( ! [I2: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ I2 @ K )
% 0.32/0.55 => ( ( P @ I2 )
% 0.32/0.55 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 0.32/0.55 => ( P @ I ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % int_induct
% 0.32/0.55 thf(fact_3243_real__of__nat__div4,axiom,
% 0.32/0.55 ! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % real_of_nat_div4
% 0.32/0.55 thf(fact_3244_power__le__zero__eq,axiom,
% 0.32/0.55 ! [A: real,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 0.32/0.55 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.55 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.32/0.55 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.55 & ( A = zero_zero_real ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_zero_eq
% 0.32/0.55 thf(fact_3245_power__le__zero__eq,axiom,
% 0.32/0.55 ! [A: rat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 0.32/0.55 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.55 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.32/0.55 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.55 & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_zero_eq
% 0.32/0.55 thf(fact_3246_power__le__zero__eq,axiom,
% 0.32/0.55 ! [A: int,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 0.32/0.55 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.55 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.32/0.55 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.55 & ( A = zero_zero_int ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_zero_eq
% 0.32/0.55 thf(fact_3247_exp__div__exp__eq,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.55 = ( times_times_nat
% 0.32/0.55 @ ( zero_n2687167440665602831ol_nat
% 0.32/0.55 @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.32/0.55 != zero_zero_nat )
% 0.32/0.55 & ( ord_less_eq_nat @ N @ M ) ) )
% 0.32/0.55 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % exp_div_exp_eq
% 0.32/0.55 thf(fact_3248_exp__div__exp__eq,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.55 = ( times_times_int
% 0.32/0.55 @ ( zero_n2684676970156552555ol_int
% 0.32/0.55 @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.32/0.55 != zero_zero_int )
% 0.32/0.55 & ( ord_less_eq_nat @ N @ M ) ) )
% 0.32/0.55 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % exp_div_exp_eq
% 0.32/0.55 thf(fact_3249_exp__div__exp__eq,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.55 = ( times_3573771949741848930nteger
% 0.32/0.55 @ ( zero_n356916108424825756nteger
% 0.32/0.55 @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 0.32/0.55 != zero_z3403309356797280102nteger )
% 0.32/0.55 & ( ord_less_eq_nat @ N @ M ) ) )
% 0.32/0.55 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % exp_div_exp_eq
% 0.32/0.55 thf(fact_3250_less__divide__eq__numeral_I1_J,axiom,
% 0.32/0.55 ! [W: num,B: real,C: real] :
% 0.32/0.55 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 0.32/0.55 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.32/0.55 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.32/0.55 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_divide_eq_numeral(1)
% 0.32/0.55 thf(fact_3251_less__divide__eq__numeral_I1_J,axiom,
% 0.32/0.55 ! [W: num,B: rat,C: rat] :
% 0.32/0.55 ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.32/0.55 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.32/0.55 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_divide_eq_numeral(1)
% 0.32/0.55 thf(fact_3252_divide__less__eq__numeral_I1_J,axiom,
% 0.32/0.55 ! [B: real,C: real,W: num] :
% 0.32/0.55 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 0.32/0.55 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.32/0.55 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.32/0.55 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_less_eq_numeral(1)
% 0.32/0.55 thf(fact_3253_divide__less__eq__numeral_I1_J,axiom,
% 0.32/0.55 ! [B: rat,C: rat,W: num] :
% 0.32/0.55 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 0.32/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.32/0.55 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.32/0.55 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_less_eq_numeral(1)
% 0.32/0.55 thf(fact_3254_frac__less__eq,axiom,
% 0.32/0.55 ! [Y: real,Z2: real,X: real,W: real] :
% 0.32/0.55 ( ( Y != zero_zero_real )
% 0.32/0.55 => ( ( Z2 != zero_zero_real )
% 0.32/0.55 => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
% 0.32/0.55 = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % frac_less_eq
% 0.32/0.55 thf(fact_3255_frac__less__eq,axiom,
% 0.32/0.55 ! [Y: rat,Z2: rat,X: rat,W: rat] :
% 0.32/0.55 ( ( Y != zero_zero_rat )
% 0.32/0.55 => ( ( Z2 != zero_zero_rat )
% 0.32/0.55 => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z2 ) )
% 0.32/0.55 = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % frac_less_eq
% 0.32/0.55 thf(fact_3256_power__Suc__less,axiom,
% 0.32/0.55 ! [A: real,N: nat] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.55 => ( ( ord_less_real @ A @ one_one_real )
% 0.32/0.55 => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_Suc_less
% 0.32/0.55 thf(fact_3257_power__Suc__less,axiom,
% 0.32/0.55 ! [A: rat,N: nat] :
% 0.32/0.55 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.55 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.32/0.55 => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_Suc_less
% 0.32/0.55 thf(fact_3258_power__Suc__less,axiom,
% 0.32/0.55 ! [A: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.55 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.32/0.55 => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_Suc_less
% 0.32/0.55 thf(fact_3259_power__Suc__less,axiom,
% 0.32/0.55 ! [A: int,N: nat] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.55 => ( ( ord_less_int @ A @ one_one_int )
% 0.32/0.55 => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_Suc_less
% 0.32/0.55 thf(fact_3260_power__Suc__less__one,axiom,
% 0.32/0.55 ! [A: real,N: nat] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.55 => ( ( ord_less_real @ A @ one_one_real )
% 0.32/0.55 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_Suc_less_one
% 0.32/0.55 thf(fact_3261_power__Suc__less__one,axiom,
% 0.32/0.55 ! [A: rat,N: nat] :
% 0.32/0.55 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.55 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.32/0.55 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_Suc_less_one
% 0.32/0.55 thf(fact_3262_power__Suc__less__one,axiom,
% 0.32/0.55 ! [A: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.55 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.32/0.55 => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_Suc_less_one
% 0.32/0.55 thf(fact_3263_power__Suc__less__one,axiom,
% 0.32/0.55 ! [A: int,N: nat] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.55 => ( ( ord_less_int @ A @ one_one_int )
% 0.32/0.55 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_Suc_less_one
% 0.32/0.55 thf(fact_3264_convex__bound__le,axiom,
% 0.32/0.55 ! [X: real,A: real,Y: real,U: real,V: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ X @ A )
% 0.32/0.55 => ( ( ord_less_eq_real @ Y @ A )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.32/0.55 => ( ( ( plus_plus_real @ U @ V )
% 0.32/0.55 = one_one_real )
% 0.32/0.55 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % convex_bound_le
% 0.32/0.55 thf(fact_3265_convex__bound__le,axiom,
% 0.32/0.55 ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ X @ A )
% 0.32/0.55 => ( ( ord_less_eq_rat @ Y @ A )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.32/0.55 => ( ( ( plus_plus_rat @ U @ V )
% 0.32/0.55 = one_one_rat )
% 0.32/0.55 => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % convex_bound_le
% 0.32/0.55 thf(fact_3266_convex__bound__le,axiom,
% 0.32/0.55 ! [X: int,A: int,Y: int,U: int,V: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ X @ A )
% 0.32/0.55 => ( ( ord_less_eq_int @ Y @ A )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.32/0.55 => ( ( ( plus_plus_int @ U @ V )
% 0.32/0.55 = one_one_int )
% 0.32/0.55 => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % convex_bound_le
% 0.32/0.55 thf(fact_3267_frac__le__eq,axiom,
% 0.32/0.55 ! [Y: real,Z2: real,X: real,W: real] :
% 0.32/0.55 ( ( Y != zero_zero_real )
% 0.32/0.55 => ( ( Z2 != zero_zero_real )
% 0.32/0.55 => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
% 0.32/0.55 = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % frac_le_eq
% 0.32/0.55 thf(fact_3268_frac__le__eq,axiom,
% 0.32/0.55 ! [Y: rat,Z2: rat,X: rat,W: rat] :
% 0.32/0.55 ( ( Y != zero_zero_rat )
% 0.32/0.55 => ( ( Z2 != zero_zero_rat )
% 0.32/0.55 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z2 ) )
% 0.32/0.55 = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % frac_le_eq
% 0.32/0.55 thf(fact_3269_power__Suc__le__self,axiom,
% 0.32/0.55 ! [A: real,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.55 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.32/0.55 => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_Suc_le_self
% 0.32/0.55 thf(fact_3270_power__Suc__le__self,axiom,
% 0.32/0.55 ! [A: rat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.55 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.32/0.55 => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_Suc_le_self
% 0.32/0.55 thf(fact_3271_power__Suc__le__self,axiom,
% 0.32/0.55 ! [A: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.55 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.32/0.55 => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_Suc_le_self
% 0.32/0.55 thf(fact_3272_power__Suc__le__self,axiom,
% 0.32/0.55 ! [A: int,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.55 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.32/0.55 => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_Suc_le_self
% 0.32/0.55 thf(fact_3273_dvd__power__iff,axiom,
% 0.32/0.55 ! [X: code_integer,M: nat,N: nat] :
% 0.32/0.55 ( ( X != zero_z3403309356797280102nteger )
% 0.32/0.55 => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
% 0.32/0.55 = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
% 0.32/0.55 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_power_iff
% 0.32/0.55 thf(fact_3274_dvd__power__iff,axiom,
% 0.32/0.55 ! [X: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( X != zero_zero_nat )
% 0.32/0.55 => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
% 0.32/0.55 = ( ( dvd_dvd_nat @ X @ one_one_nat )
% 0.32/0.55 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_power_iff
% 0.32/0.55 thf(fact_3275_dvd__power__iff,axiom,
% 0.32/0.55 ! [X: int,M: nat,N: nat] :
% 0.32/0.55 ( ( X != zero_zero_int )
% 0.32/0.55 => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
% 0.32/0.55 = ( ( dvd_dvd_int @ X @ one_one_int )
% 0.32/0.55 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_power_iff
% 0.32/0.55 thf(fact_3276_pos2,axiom,
% 0.32/0.55 ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 0.32/0.55
% 0.32/0.55 % pos2
% 0.32/0.55 thf(fact_3277_dvd__power,axiom,
% 0.32/0.55 ! [N: nat,X: code_integer] :
% 0.32/0.55 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 | ( X = one_one_Code_integer ) )
% 0.32/0.55 => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_power
% 0.32/0.55 thf(fact_3278_dvd__power,axiom,
% 0.32/0.55 ! [N: nat,X: rat] :
% 0.32/0.55 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 | ( X = one_one_rat ) )
% 0.32/0.55 => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_power
% 0.32/0.55 thf(fact_3279_dvd__power,axiom,
% 0.32/0.55 ! [N: nat,X: nat] :
% 0.32/0.55 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 | ( X = one_one_nat ) )
% 0.32/0.55 => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_power
% 0.32/0.55 thf(fact_3280_dvd__power,axiom,
% 0.32/0.55 ! [N: nat,X: real] :
% 0.32/0.55 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 | ( X = one_one_real ) )
% 0.32/0.55 => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_power
% 0.32/0.55 thf(fact_3281_dvd__power,axiom,
% 0.32/0.55 ! [N: nat,X: int] :
% 0.32/0.55 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 | ( X = one_one_int ) )
% 0.32/0.55 => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_power
% 0.32/0.55 thf(fact_3282_dvd__power,axiom,
% 0.32/0.55 ! [N: nat,X: complex] :
% 0.32/0.55 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 | ( X = one_one_complex ) )
% 0.32/0.55 => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_power
% 0.32/0.55 thf(fact_3283_power__diff,axiom,
% 0.32/0.55 ! [A: complex,N: nat,M: nat] :
% 0.32/0.55 ( ( A != zero_zero_complex )
% 0.32/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.55 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_diff
% 0.32/0.55 thf(fact_3284_power__diff,axiom,
% 0.32/0.55 ! [A: real,N: nat,M: nat] :
% 0.32/0.55 ( ( A != zero_zero_real )
% 0.32/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.55 = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_diff
% 0.32/0.55 thf(fact_3285_power__diff,axiom,
% 0.32/0.55 ! [A: rat,N: nat,M: nat] :
% 0.32/0.55 ( ( A != zero_zero_rat )
% 0.32/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.55 = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_diff
% 0.32/0.55 thf(fact_3286_power__diff,axiom,
% 0.32/0.55 ! [A: nat,N: nat,M: nat] :
% 0.32/0.55 ( ( A != zero_zero_nat )
% 0.32/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.55 = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_diff
% 0.32/0.55 thf(fact_3287_power__diff,axiom,
% 0.32/0.55 ! [A: int,N: nat,M: nat] :
% 0.32/0.55 ( ( A != zero_zero_int )
% 0.32/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.55 = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_diff
% 0.32/0.55 thf(fact_3288_less__exp,axiom,
% 0.32/0.55 ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_exp
% 0.32/0.55 thf(fact_3289_power2__nat__le__imp__le,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 0.32/0.55 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % power2_nat_le_imp_le
% 0.32/0.55 thf(fact_3290_power2__nat__le__eq__le,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.55 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % power2_nat_le_eq_le
% 0.32/0.55 thf(fact_3291_self__le__ge2__pow,axiom,
% 0.32/0.55 ! [K: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.32/0.55 => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % self_le_ge2_pow
% 0.32/0.55 thf(fact_3292_two__realpow__ge__one,axiom,
% 0.32/0.55 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % two_realpow_ge_one
% 0.32/0.55 thf(fact_3293_Suc__diff__eq__diff__pred,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.32/0.55 = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Suc_diff_eq_diff_pred
% 0.32/0.55 thf(fact_3294_Suc__pred_H,axiom,
% 0.32/0.55 ! [N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( N
% 0.32/0.55 = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Suc_pred'
% 0.32/0.55 thf(fact_3295_div__if,axiom,
% 0.32/0.55 ( divide_divide_nat
% 0.32/0.55 = ( ^ [M3: nat,N3: nat] :
% 0.32/0.55 ( if_nat
% 0.32/0.55 @ ( ( ord_less_nat @ M3 @ N3 )
% 0.32/0.55 | ( N3 = zero_zero_nat ) )
% 0.32/0.55 @ zero_zero_nat
% 0.32/0.55 @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % div_if
% 0.32/0.55 thf(fact_3296_div__geq,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ~ ( ord_less_nat @ M @ N )
% 0.32/0.55 => ( ( divide_divide_nat @ M @ N )
% 0.32/0.55 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % div_geq
% 0.32/0.55 thf(fact_3297_dividend__less__times__div,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dividend_less_times_div
% 0.32/0.55 thf(fact_3298_dividend__less__div__times,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dividend_less_div_times
% 0.32/0.55 thf(fact_3299_split__div,axiom,
% 0.32/0.55 ! [P: nat > $o,M: nat,N: nat] :
% 0.32/0.55 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.32/0.55 = ( ( ( N = zero_zero_nat )
% 0.32/0.55 => ( P @ zero_zero_nat ) )
% 0.32/0.55 & ( ( N != zero_zero_nat )
% 0.32/0.55 => ! [I4: nat,J3: nat] :
% 0.32/0.55 ( ( ord_less_nat @ J3 @ N )
% 0.32/0.55 => ( ( M
% 0.32/0.55 = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 0.32/0.55 => ( P @ I4 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % split_div
% 0.32/0.55 thf(fact_3300_dvd__mult__cancel2,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.55 => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 0.32/0.55 = ( N = one_one_nat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_mult_cancel2
% 0.32/0.55 thf(fact_3301_dvd__mult__cancel1,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.55 => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 0.32/0.55 = ( N = one_one_nat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_mult_cancel1
% 0.32/0.55 thf(fact_3302_int__ops_I6_J,axiom,
% 0.32/0.55 ! [A: nat,B: nat] :
% 0.32/0.55 ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 0.32/0.55 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 0.32/0.55 = zero_zero_int ) )
% 0.32/0.55 & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 0.32/0.55 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 0.32/0.55 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % int_ops(6)
% 0.32/0.55 thf(fact_3303_dvd__minus__add,axiom,
% 0.32/0.55 ! [Q2: nat,N: nat,R: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ Q2 @ N )
% 0.32/0.55 => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R @ M ) )
% 0.32/0.55 => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
% 0.32/0.55 = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % dvd_minus_add
% 0.32/0.55 thf(fact_3304_even__mult__exp__div__exp__iff,axiom,
% 0.32/0.55 ! [A: code_integer,M: nat,N: nat] :
% 0.32/0.55 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.55 = ( ( ord_less_nat @ N @ M )
% 0.32/0.55 | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.32/0.55 = zero_z3403309356797280102nteger )
% 0.32/0.55 | ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.55 & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % even_mult_exp_div_exp_iff
% 0.32/0.55 thf(fact_3305_even__mult__exp__div__exp__iff,axiom,
% 0.32/0.55 ! [A: nat,M: nat,N: nat] :
% 0.32/0.55 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.55 = ( ( ord_less_nat @ N @ M )
% 0.32/0.55 | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.55 = zero_zero_nat )
% 0.32/0.55 | ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.55 & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % even_mult_exp_div_exp_iff
% 0.32/0.55 thf(fact_3306_even__mult__exp__div__exp__iff,axiom,
% 0.32/0.55 ! [A: int,M: nat,N: nat] :
% 0.32/0.55 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.55 = ( ( ord_less_nat @ N @ M )
% 0.32/0.55 | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.32/0.55 = zero_zero_int )
% 0.32/0.55 | ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.55 & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % even_mult_exp_div_exp_iff
% 0.32/0.55 thf(fact_3307_real__of__nat__div2,axiom,
% 0.32/0.55 ! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % real_of_nat_div2
% 0.32/0.55 thf(fact_3308_real__of__nat__div3,axiom,
% 0.32/0.55 ! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).
% 0.32/0.55
% 0.32/0.55 % real_of_nat_div3
% 0.32/0.55 thf(fact_3309_half__gt__zero__iff,axiom,
% 0.32/0.55 ! [A: real] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.55 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.32/0.55
% 0.32/0.55 % half_gt_zero_iff
% 0.32/0.55 thf(fact_3310_half__gt__zero__iff,axiom,
% 0.32/0.55 ! [A: rat] :
% 0.32/0.55 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.32/0.55 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.55
% 0.32/0.55 % half_gt_zero_iff
% 0.32/0.55 thf(fact_3311_half__gt__zero,axiom,
% 0.32/0.55 ! [A: real] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.55 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % half_gt_zero
% 0.32/0.55 thf(fact_3312_half__gt__zero,axiom,
% 0.32/0.55 ! [A: rat] :
% 0.32/0.55 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.55 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % half_gt_zero
% 0.32/0.55 thf(fact_3313_field__less__half__sum,axiom,
% 0.32/0.55 ! [X: real,Y: real] :
% 0.32/0.55 ( ( ord_less_real @ X @ Y )
% 0.32/0.55 => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % field_less_half_sum
% 0.32/0.55 thf(fact_3314_field__less__half__sum,axiom,
% 0.32/0.55 ! [X: rat,Y: rat] :
% 0.32/0.55 ( ( ord_less_rat @ X @ Y )
% 0.32/0.55 => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % field_less_half_sum
% 0.32/0.55 thf(fact_3315_power2__less__0,axiom,
% 0.32/0.55 ! [A: real] :
% 0.32/0.55 ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 0.32/0.55
% 0.32/0.55 % power2_less_0
% 0.32/0.55 thf(fact_3316_power2__less__0,axiom,
% 0.32/0.55 ! [A: rat] :
% 0.32/0.55 ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 0.32/0.55
% 0.32/0.55 % power2_less_0
% 0.32/0.55 thf(fact_3317_power2__less__0,axiom,
% 0.32/0.55 ! [A: int] :
% 0.32/0.55 ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 0.32/0.55
% 0.32/0.55 % power2_less_0
% 0.32/0.55 thf(fact_3318_nat__approx__posE,axiom,
% 0.32/0.55 ! [E: real] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ E )
% 0.32/0.55 => ~ ! [N2: nat] :
% 0.32/0.55 ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ E ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_approx_posE
% 0.32/0.55 thf(fact_3319_nat__approx__posE,axiom,
% 0.32/0.55 ! [E: rat] :
% 0.32/0.55 ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.32/0.55 => ~ ! [N2: nat] :
% 0.32/0.55 ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ E ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_approx_posE
% 0.32/0.55 thf(fact_3320_scaling__mono,axiom,
% 0.32/0.55 ! [U: real,V: real,R: real,S: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ U @ V )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ R )
% 0.32/0.55 => ( ( ord_less_eq_real @ R @ S )
% 0.32/0.55 => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % scaling_mono
% 0.32/0.55 thf(fact_3321_scaling__mono,axiom,
% 0.32/0.55 ! [U: rat,V: rat,R: rat,S: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ U @ V )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ R )
% 0.32/0.55 => ( ( ord_less_eq_rat @ R @ S )
% 0.32/0.55 => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % scaling_mono
% 0.32/0.55 thf(fact_3322_of__nat__less__two__power,axiom,
% 0.32/0.55 ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % of_nat_less_two_power
% 0.32/0.55 thf(fact_3323_of__nat__less__two__power,axiom,
% 0.32/0.55 ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % of_nat_less_two_power
% 0.32/0.55 thf(fact_3324_of__nat__less__two__power,axiom,
% 0.32/0.55 ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % of_nat_less_two_power
% 0.32/0.55 thf(fact_3325_power2__le__imp__le,axiom,
% 0.32/0.55 ! [X: real,Y: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.32/0.55 => ( ord_less_eq_real @ X @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power2_le_imp_le
% 0.32/0.55 thf(fact_3326_power2__le__imp__le,axiom,
% 0.32/0.55 ! [X: rat,Y: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.32/0.55 => ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power2_le_imp_le
% 0.32/0.55 thf(fact_3327_power2__le__imp__le,axiom,
% 0.32/0.55 ! [X: nat,Y: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.32/0.55 => ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power2_le_imp_le
% 0.32/0.55 thf(fact_3328_power2__le__imp__le,axiom,
% 0.32/0.55 ! [X: int,Y: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.32/0.55 => ( ord_less_eq_int @ X @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power2_le_imp_le
% 0.32/0.55 thf(fact_3329_power2__eq__imp__eq,axiom,
% 0.32/0.55 ! [X: real,Y: real] :
% 0.32/0.55 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.55 = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.32/0.55 => ( X = Y ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power2_eq_imp_eq
% 0.32/0.55 thf(fact_3330_power2__eq__imp__eq,axiom,
% 0.32/0.55 ! [X: rat,Y: rat] :
% 0.32/0.55 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.55 = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.32/0.55 => ( X = Y ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power2_eq_imp_eq
% 0.32/0.55 thf(fact_3331_power2__eq__imp__eq,axiom,
% 0.32/0.55 ! [X: nat,Y: nat] :
% 0.32/0.55 ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.55 = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.32/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.32/0.55 => ( X = Y ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power2_eq_imp_eq
% 0.32/0.55 thf(fact_3332_power2__eq__imp__eq,axiom,
% 0.32/0.55 ! [X: int,Y: int] :
% 0.32/0.55 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.55 = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.32/0.55 => ( X = Y ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power2_eq_imp_eq
% 0.32/0.55 thf(fact_3333_zero__le__power2,axiom,
% 0.32/0.55 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zero_le_power2
% 0.32/0.55 thf(fact_3334_zero__le__power2,axiom,
% 0.32/0.55 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zero_le_power2
% 0.32/0.55 thf(fact_3335_zero__le__power2,axiom,
% 0.32/0.55 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % zero_le_power2
% 0.32/0.55 thf(fact_3336_num_Osize__gen_I1_J,axiom,
% 0.32/0.55 ( ( size_num @ one )
% 0.32/0.55 = zero_zero_nat ) ).
% 0.32/0.55
% 0.32/0.55 % num.size_gen(1)
% 0.32/0.55 thf(fact_3337_less__2__cases__iff,axiom,
% 0.32/0.55 ! [N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.55 = ( ( N = zero_zero_nat )
% 0.32/0.55 | ( N
% 0.32/0.55 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_2_cases_iff
% 0.32/0.55 thf(fact_3338_less__2__cases,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 => ( ( N = zero_zero_nat )
% 0.32/0.56 | ( N
% 0.32/0.56 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % less_2_cases
% 0.32/0.56 thf(fact_3339_power__diff__power__eq,axiom,
% 0.32/0.56 ! [A: nat,N: nat,M: nat] :
% 0.32/0.56 ( ( A != zero_zero_nat )
% 0.32/0.56 => ( ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.56 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.32/0.56 = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.32/0.56 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.32/0.56 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.32/0.56 = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_diff_power_eq
% 0.32/0.56 thf(fact_3340_power__diff__power__eq,axiom,
% 0.32/0.56 ! [A: int,N: nat,M: nat] :
% 0.32/0.56 ( ( A != zero_zero_int )
% 0.32/0.56 => ( ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.56 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.32/0.56 = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.32/0.56 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.32/0.56 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.32/0.56 = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_diff_power_eq
% 0.32/0.56 thf(fact_3341_power__mono__odd,axiom,
% 0.32/0.56 ! [N: nat,A: real,B: real] :
% 0.32/0.56 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 => ( ( ord_less_eq_real @ A @ B )
% 0.32/0.56 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_mono_odd
% 0.32/0.56 thf(fact_3342_power__mono__odd,axiom,
% 0.32/0.56 ! [N: nat,A: rat,B: rat] :
% 0.32/0.56 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 => ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.56 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_mono_odd
% 0.32/0.56 thf(fact_3343_power__mono__odd,axiom,
% 0.32/0.56 ! [N: nat,A: int,B: int] :
% 0.32/0.56 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 => ( ( ord_less_eq_int @ A @ B )
% 0.32/0.56 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_mono_odd
% 0.32/0.56 thf(fact_3344_odd__pos,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.56
% 0.32/0.56 % odd_pos
% 0.32/0.56 thf(fact_3345_power__minus__mult,axiom,
% 0.32/0.56 ! [N: nat,A: complex] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.32/0.56 = ( power_power_complex @ A @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_minus_mult
% 0.32/0.56 thf(fact_3346_power__minus__mult,axiom,
% 0.32/0.56 ! [N: nat,A: real] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.32/0.56 = ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_minus_mult
% 0.32/0.56 thf(fact_3347_power__minus__mult,axiom,
% 0.32/0.56 ! [N: nat,A: rat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.32/0.56 = ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_minus_mult
% 0.32/0.56 thf(fact_3348_power__minus__mult,axiom,
% 0.32/0.56 ! [N: nat,A: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.32/0.56 = ( power_power_nat @ A @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_minus_mult
% 0.32/0.56 thf(fact_3349_power__minus__mult,axiom,
% 0.32/0.56 ! [N: nat,A: int] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.32/0.56 = ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_minus_mult
% 0.32/0.56 thf(fact_3350_diff__le__diff__pow,axiom,
% 0.32/0.56 ! [K: nat,M: nat,N: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.32/0.56 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % diff_le_diff_pow
% 0.32/0.56 thf(fact_3351_dvd__power__iff__le,axiom,
% 0.32/0.56 ! [K: nat,M: nat,N: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.32/0.56 => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 0.32/0.56 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_power_iff_le
% 0.32/0.56 thf(fact_3352_not__exp__less__eq__0__int,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% 0.32/0.56
% 0.32/0.56 % not_exp_less_eq_0_int
% 0.32/0.56 thf(fact_3353_bits__induct,axiom,
% 0.32/0.56 ! [P: nat > $o,A: nat] :
% 0.32/0.56 ( ! [A4: nat] :
% 0.32/0.56 ( ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = A4 )
% 0.32/0.56 => ( P @ A4 ) )
% 0.32/0.56 => ( ! [A4: nat,B4: $o] :
% 0.32/0.56 ( ( P @ A4 )
% 0.32/0.56 => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = A4 )
% 0.32/0.56 => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 0.32/0.56 => ( P @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % bits_induct
% 0.32/0.56 thf(fact_3354_bits__induct,axiom,
% 0.32/0.56 ! [P: int > $o,A: int] :
% 0.32/0.56 ( ! [A4: int] :
% 0.32/0.56 ( ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.56 = A4 )
% 0.32/0.56 => ( P @ A4 ) )
% 0.32/0.56 => ( ! [A4: int,B4: $o] :
% 0.32/0.56 ( ( P @ A4 )
% 0.32/0.56 => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.56 = A4 )
% 0.32/0.56 => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 0.32/0.56 => ( P @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % bits_induct
% 0.32/0.56 thf(fact_3355_bits__induct,axiom,
% 0.32/0.56 ! [P: code_integer > $o,A: code_integer] :
% 0.32/0.56 ( ! [A4: code_integer] :
% 0.32/0.56 ( ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.56 = A4 )
% 0.32/0.56 => ( P @ A4 ) )
% 0.32/0.56 => ( ! [A4: code_integer,B4: $o] :
% 0.32/0.56 ( ( P @ A4 )
% 0.32/0.56 => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.56 = A4 )
% 0.32/0.56 => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 0.32/0.56 => ( P @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % bits_induct
% 0.32/0.56 thf(fact_3356_sum__power2__gt__zero__iff,axiom,
% 0.32/0.56 ! [X: real,Y: real] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.56 = ( ( X != zero_zero_real )
% 0.32/0.56 | ( Y != zero_zero_real ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_gt_zero_iff
% 0.32/0.56 thf(fact_3357_sum__power2__gt__zero__iff,axiom,
% 0.32/0.56 ! [X: rat,Y: rat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.56 = ( ( X != zero_zero_rat )
% 0.32/0.56 | ( Y != zero_zero_rat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_gt_zero_iff
% 0.32/0.56 thf(fact_3358_sum__power2__gt__zero__iff,axiom,
% 0.32/0.56 ! [X: int,Y: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.56 = ( ( X != zero_zero_int )
% 0.32/0.56 | ( Y != zero_zero_int ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_gt_zero_iff
% 0.32/0.56 thf(fact_3359_not__sum__power2__lt__zero,axiom,
% 0.32/0.56 ! [X: real,Y: real] :
% 0.32/0.56 ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 0.32/0.56
% 0.32/0.56 % not_sum_power2_lt_zero
% 0.32/0.56 thf(fact_3360_not__sum__power2__lt__zero,axiom,
% 0.32/0.56 ! [X: rat,Y: rat] :
% 0.32/0.56 ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 0.32/0.56
% 0.32/0.56 % not_sum_power2_lt_zero
% 0.32/0.56 thf(fact_3361_not__sum__power2__lt__zero,axiom,
% 0.32/0.56 ! [X: int,Y: int] :
% 0.32/0.56 ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 0.32/0.56
% 0.32/0.56 % not_sum_power2_lt_zero
% 0.32/0.56 thf(fact_3362_sum__power2__le__zero__iff,axiom,
% 0.32/0.56 ! [X: real,Y: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 0.32/0.56 = ( ( X = zero_zero_real )
% 0.32/0.56 & ( Y = zero_zero_real ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_le_zero_iff
% 0.32/0.56 thf(fact_3363_sum__power2__le__zero__iff,axiom,
% 0.32/0.56 ! [X: rat,Y: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 0.32/0.56 = ( ( X = zero_zero_rat )
% 0.32/0.56 & ( Y = zero_zero_rat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_le_zero_iff
% 0.32/0.56 thf(fact_3364_sum__power2__le__zero__iff,axiom,
% 0.32/0.56 ! [X: int,Y: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 0.32/0.56 = ( ( X = zero_zero_int )
% 0.32/0.56 & ( Y = zero_zero_int ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_le_zero_iff
% 0.32/0.56 thf(fact_3365_sum__power2__ge__zero,axiom,
% 0.32/0.56 ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_ge_zero
% 0.32/0.56 thf(fact_3366_sum__power2__ge__zero,axiom,
% 0.32/0.56 ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_ge_zero
% 0.32/0.56 thf(fact_3367_sum__power2__ge__zero,axiom,
% 0.32/0.56 ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_ge_zero
% 0.32/0.56 thf(fact_3368_zero__le__even__power_H,axiom,
% 0.32/0.56 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_even_power'
% 0.32/0.56 thf(fact_3369_zero__le__even__power_H,axiom,
% 0.32/0.56 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_even_power'
% 0.32/0.56 thf(fact_3370_zero__le__even__power_H,axiom,
% 0.32/0.56 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_even_power'
% 0.32/0.56 thf(fact_3371_zero__le__even__power,axiom,
% 0.32/0.56 ! [N: nat,A: real] :
% 0.32/0.56 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_even_power
% 0.32/0.56 thf(fact_3372_zero__le__even__power,axiom,
% 0.32/0.56 ! [N: nat,A: rat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_even_power
% 0.32/0.56 thf(fact_3373_zero__le__even__power,axiom,
% 0.32/0.56 ! [N: nat,A: int] :
% 0.32/0.56 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_even_power
% 0.32/0.56 thf(fact_3374_zero__le__odd__power,axiom,
% 0.32/0.56 ! [N: nat,A: real] :
% 0.32/0.56 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.32/0.56 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_odd_power
% 0.32/0.56 thf(fact_3375_zero__le__odd__power,axiom,
% 0.32/0.56 ! [N: nat,A: rat] :
% 0.32/0.56 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.32/0.56 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_odd_power
% 0.32/0.56 thf(fact_3376_zero__le__odd__power,axiom,
% 0.32/0.56 ! [N: nat,A: int] :
% 0.32/0.56 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.32/0.56 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_odd_power
% 0.32/0.56 thf(fact_3377_zero__le__power__eq,axiom,
% 0.32/0.56 ! [A: real,N: nat] :
% 0.32/0.56 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.32/0.56 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_power_eq
% 0.32/0.56 thf(fact_3378_zero__le__power__eq,axiom,
% 0.32/0.56 ! [A: rat,N: nat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.32/0.56 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_power_eq
% 0.32/0.56 thf(fact_3379_zero__le__power__eq,axiom,
% 0.32/0.56 ! [A: int,N: nat] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.32/0.56 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_power_eq
% 0.32/0.56 thf(fact_3380_nat__bit__induct,axiom,
% 0.32/0.56 ! [P: nat > $o,N: nat] :
% 0.32/0.56 ( ( P @ zero_zero_nat )
% 0.32/0.56 => ( ! [N2: nat] :
% 0.32/0.56 ( ( P @ N2 )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.32/0.56 => ( ! [N2: nat] :
% 0.32/0.56 ( ( P @ N2 )
% 0.32/0.56 => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.32/0.56 => ( P @ N ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nat_bit_induct
% 0.32/0.56 thf(fact_3381_Suc__n__div__2__gt__zero,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % Suc_n_div_2_gt_zero
% 0.32/0.56 thf(fact_3382_div__2__gt__zero,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_2_gt_zero
% 0.32/0.56 thf(fact_3383_add__0__iff,axiom,
% 0.32/0.56 ! [B: real,A: real] :
% 0.32/0.56 ( ( B
% 0.32/0.56 = ( plus_plus_real @ B @ A ) )
% 0.32/0.56 = ( A = zero_zero_real ) ) ).
% 0.32/0.56
% 0.32/0.56 % add_0_iff
% 0.32/0.56 thf(fact_3384_add__0__iff,axiom,
% 0.32/0.56 ! [B: rat,A: rat] :
% 0.32/0.56 ( ( B
% 0.32/0.56 = ( plus_plus_rat @ B @ A ) )
% 0.32/0.56 = ( A = zero_zero_rat ) ) ).
% 0.32/0.56
% 0.32/0.56 % add_0_iff
% 0.32/0.56 thf(fact_3385_add__0__iff,axiom,
% 0.32/0.56 ! [B: nat,A: nat] :
% 0.32/0.56 ( ( B
% 0.32/0.56 = ( plus_plus_nat @ B @ A ) )
% 0.32/0.56 = ( A = zero_zero_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % add_0_iff
% 0.32/0.56 thf(fact_3386_add__0__iff,axiom,
% 0.32/0.56 ! [B: int,A: int] :
% 0.32/0.56 ( ( B
% 0.32/0.56 = ( plus_plus_int @ B @ A ) )
% 0.32/0.56 = ( A = zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % add_0_iff
% 0.32/0.56 thf(fact_3387_crossproduct__eq,axiom,
% 0.32/0.56 ! [W: real,Y: real,X: real,Z2: real] :
% 0.32/0.56 ( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z2 ) )
% 0.32/0.56 = ( plus_plus_real @ ( times_times_real @ W @ Z2 ) @ ( times_times_real @ X @ Y ) ) )
% 0.32/0.56 = ( ( W = X )
% 0.32/0.56 | ( Y = Z2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % crossproduct_eq
% 0.32/0.56 thf(fact_3388_crossproduct__eq,axiom,
% 0.32/0.56 ! [W: rat,Y: rat,X: rat,Z2: rat] :
% 0.32/0.56 ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z2 ) )
% 0.32/0.56 = ( plus_plus_rat @ ( times_times_rat @ W @ Z2 ) @ ( times_times_rat @ X @ Y ) ) )
% 0.32/0.56 = ( ( W = X )
% 0.32/0.56 | ( Y = Z2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % crossproduct_eq
% 0.32/0.56 thf(fact_3389_crossproduct__eq,axiom,
% 0.32/0.56 ! [W: nat,Y: nat,X: nat,Z2: nat] :
% 0.32/0.56 ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z2 ) )
% 0.32/0.56 = ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X @ Y ) ) )
% 0.32/0.56 = ( ( W = X )
% 0.32/0.56 | ( Y = Z2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % crossproduct_eq
% 0.32/0.56 thf(fact_3390_crossproduct__eq,axiom,
% 0.32/0.56 ! [W: int,Y: int,X: int,Z2: int] :
% 0.32/0.56 ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z2 ) )
% 0.32/0.56 = ( plus_plus_int @ ( times_times_int @ W @ Z2 ) @ ( times_times_int @ X @ Y ) ) )
% 0.32/0.56 = ( ( W = X )
% 0.32/0.56 | ( Y = Z2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % crossproduct_eq
% 0.32/0.56 thf(fact_3391_crossproduct__noteq,axiom,
% 0.32/0.56 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.56 ( ( ( A != B )
% 0.32/0.56 & ( C != D ) )
% 0.32/0.56 = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
% 0.32/0.56 != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % crossproduct_noteq
% 0.32/0.56 thf(fact_3392_crossproduct__noteq,axiom,
% 0.32/0.56 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.56 ( ( ( A != B )
% 0.32/0.56 & ( C != D ) )
% 0.32/0.56 = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
% 0.32/0.56 != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % crossproduct_noteq
% 0.32/0.56 thf(fact_3393_crossproduct__noteq,axiom,
% 0.32/0.56 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.56 ( ( ( A != B )
% 0.32/0.56 & ( C != D ) )
% 0.32/0.56 = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
% 0.32/0.56 != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % crossproduct_noteq
% 0.32/0.56 thf(fact_3394_crossproduct__noteq,axiom,
% 0.32/0.56 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.56 ( ( ( A != B )
% 0.32/0.56 & ( C != D ) )
% 0.32/0.56 = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
% 0.32/0.56 != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % crossproduct_noteq
% 0.32/0.56 thf(fact_3395_odd__power__less__zero,axiom,
% 0.32/0.56 ! [A: real,N: nat] :
% 0.32/0.56 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.56 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 0.32/0.56
% 0.32/0.56 % odd_power_less_zero
% 0.32/0.56 thf(fact_3396_odd__power__less__zero,axiom,
% 0.32/0.56 ! [A: rat,N: nat] :
% 0.32/0.56 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 0.32/0.56
% 0.32/0.56 % odd_power_less_zero
% 0.32/0.56 thf(fact_3397_odd__power__less__zero,axiom,
% 0.32/0.56 ! [A: int,N: nat] :
% 0.32/0.56 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.56 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % odd_power_less_zero
% 0.32/0.56 thf(fact_3398_sum__squares__bound,axiom,
% 0.32/0.56 ! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_squares_bound
% 0.32/0.56 thf(fact_3399_sum__squares__bound,axiom,
% 0.32/0.56 ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_squares_bound
% 0.32/0.56 thf(fact_3400_zero__less__power__eq,axiom,
% 0.32/0.56 ! [A: real,N: nat] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.32/0.56 = ( ( N = zero_zero_nat )
% 0.32/0.56 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 & ( A != zero_zero_real ) )
% 0.32/0.56 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_less_power_eq
% 0.32/0.56 thf(fact_3401_zero__less__power__eq,axiom,
% 0.32/0.56 ! [A: rat,N: nat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.32/0.56 = ( ( N = zero_zero_nat )
% 0.32/0.56 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 & ( A != zero_zero_rat ) )
% 0.32/0.56 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_less_power_eq
% 0.32/0.56 thf(fact_3402_zero__less__power__eq,axiom,
% 0.32/0.56 ! [A: int,N: nat] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.32/0.56 = ( ( N = zero_zero_nat )
% 0.32/0.56 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 & ( A != zero_zero_int ) )
% 0.32/0.56 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.56 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_less_power_eq
% 0.32/0.56 thf(fact_3403_odd__0__le__power__imp__0__le,axiom,
% 0.32/0.56 ! [A: real,N: nat] :
% 0.32/0.56 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.32/0.56 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % odd_0_le_power_imp_0_le
% 0.32/0.56 thf(fact_3404_odd__0__le__power__imp__0__le,axiom,
% 0.32/0.56 ! [A: rat,N: nat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.32/0.56 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % odd_0_le_power_imp_0_le
% 0.32/0.56 thf(fact_3405_odd__0__le__power__imp__0__le,axiom,
% 0.32/0.56 ! [A: int,N: nat] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.32/0.56 => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % odd_0_le_power_imp_0_le
% 0.32/0.56 thf(fact_3406_L2__set__mult__ineq__lemma,axiom,
% 0.32/0.56 ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % L2_set_mult_ineq_lemma
% 0.32/0.56 thf(fact_3407_even__mask__div__iff_H,axiom,
% 0.32/0.56 ! [M: nat,N: nat] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.56 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_mask_div_iff'
% 0.32/0.56 thf(fact_3408_even__mask__div__iff_H,axiom,
% 0.32/0.56 ! [M: nat,N: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.56 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_mask_div_iff'
% 0.32/0.56 thf(fact_3409_even__mask__div__iff_H,axiom,
% 0.32/0.56 ! [M: nat,N: nat] :
% 0.32/0.56 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.56 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_mask_div_iff'
% 0.32/0.56 thf(fact_3410_pos__zdiv__mult__2,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.56 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.56 = ( divide_divide_int @ B @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % pos_zdiv_mult_2
% 0.32/0.56 thf(fact_3411_neg__zdiv__mult__2,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.56 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.56 = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % neg_zdiv_mult_2
% 0.32/0.56 thf(fact_3412_linear__plus__1__le__power,axiom,
% 0.32/0.56 ! [X: real,N: nat] :
% 0.32/0.56 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.56 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % linear_plus_1_le_power
% 0.32/0.56 thf(fact_3413_enat__ord__number_I1_J,axiom,
% 0.32/0.56 ! [M: num,N: num] :
% 0.32/0.56 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.32/0.56 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % enat_ord_number(1)
% 0.32/0.56 thf(fact_3414_enat__ord__number_I2_J,axiom,
% 0.32/0.56 ! [M: num,N: num] :
% 0.32/0.56 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.32/0.56 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % enat_ord_number(2)
% 0.32/0.56 thf(fact_3415_mult__le__cancel__iff1,axiom,
% 0.32/0.56 ! [Z2: real,X: real,Y: real] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ Z2 )
% 0.32/0.56 => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
% 0.32/0.56 = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_iff1
% 0.32/0.56 thf(fact_3416_mult__le__cancel__iff1,axiom,
% 0.32/0.56 ! [Z2: rat,X: rat,Y: rat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 0.32/0.56 => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
% 0.32/0.56 = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_iff1
% 0.32/0.56 thf(fact_3417_mult__le__cancel__iff1,axiom,
% 0.32/0.56 ! [Z2: int,X: int,Y: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.32/0.56 => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
% 0.32/0.56 = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_iff1
% 0.32/0.56 thf(fact_3418_mult__le__cancel__iff2,axiom,
% 0.32/0.56 ! [Z2: real,X: real,Y: real] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ Z2 )
% 0.32/0.56 => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ ( times_times_real @ Z2 @ Y ) )
% 0.32/0.56 = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_iff2
% 0.32/0.56 thf(fact_3419_mult__le__cancel__iff2,axiom,
% 0.32/0.56 ! [Z2: rat,X: rat,Y: rat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 0.32/0.56 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ ( times_times_rat @ Z2 @ Y ) )
% 0.32/0.56 = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_iff2
% 0.32/0.56 thf(fact_3420_mult__le__cancel__iff2,axiom,
% 0.32/0.56 ! [Z2: int,X: int,Y: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.32/0.56 => ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X ) @ ( times_times_int @ Z2 @ Y ) )
% 0.32/0.56 = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_iff2
% 0.32/0.56 thf(fact_3421_lemma__termdiff3,axiom,
% 0.32/0.56 ! [H: real,Z2: real,K5: real,N: nat] :
% 0.32/0.56 ( ( H != zero_zero_real )
% 0.32/0.56 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ K5 )
% 0.32/0.56 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z2 @ H ) ) @ K5 )
% 0.32/0.56 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H ) @ N ) @ ( power_power_real @ Z2 @ N ) ) @ H ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % lemma_termdiff3
% 0.32/0.56 thf(fact_3422_lemma__termdiff3,axiom,
% 0.32/0.56 ! [H: complex,Z2: complex,K5: real,N: nat] :
% 0.32/0.56 ( ( H != zero_zero_complex )
% 0.32/0.56 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ K5 )
% 0.32/0.56 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z2 @ H ) ) @ K5 )
% 0.32/0.56 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H ) @ N ) @ ( power_power_complex @ Z2 @ N ) ) @ H ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % lemma_termdiff3
% 0.32/0.56 thf(fact_3423_even__succ__mod__exp,axiom,
% 0.32/0.56 ! [A: nat,N: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.56 = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_succ_mod_exp
% 0.32/0.56 thf(fact_3424_even__succ__mod__exp,axiom,
% 0.32/0.56 ! [A: int,N: nat] :
% 0.32/0.56 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.56 = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_succ_mod_exp
% 0.32/0.56 thf(fact_3425_even__succ__mod__exp,axiom,
% 0.32/0.56 ! [A: code_integer,N: nat] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.56 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_succ_mod_exp
% 0.32/0.56 thf(fact_3426_even__succ__mod__exp,axiom,
% 0.32/0.56 ! [A: code_natural,N: nat] :
% 0.32/0.56 ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.56 = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_succ_mod_exp
% 0.32/0.56 thf(fact_3427_mod__less,axiom,
% 0.32/0.56 ! [M: nat,N: nat] :
% 0.32/0.56 ( ( ord_less_nat @ M @ N )
% 0.32/0.56 => ( ( modulo_modulo_nat @ M @ N )
% 0.32/0.56 = M ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_less
% 0.32/0.56 thf(fact_3428_mod__mod__trivial,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.32/0.56 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mod_trivial
% 0.32/0.56 thf(fact_3429_mod__mod__trivial,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.32/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mod_trivial
% 0.32/0.56 thf(fact_3430_mod__mod__trivial,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.32/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mod_trivial
% 0.32/0.56 thf(fact_3431_mod__mod__trivial,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( modulo8411746178871703098atural @ A @ B ) @ B )
% 0.32/0.56 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mod_trivial
% 0.32/0.56 thf(fact_3432_bits__mod__0,axiom,
% 0.32/0.56 ! [A: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.32/0.56 = zero_zero_nat ) ).
% 0.32/0.56
% 0.32/0.56 % bits_mod_0
% 0.32/0.56 thf(fact_3433_bits__mod__0,axiom,
% 0.32/0.56 ! [A: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.32/0.56 = zero_zero_int ) ).
% 0.32/0.56
% 0.32/0.56 % bits_mod_0
% 0.32/0.56 thf(fact_3434_bits__mod__0,axiom,
% 0.32/0.56 ! [A: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.56 = zero_z3403309356797280102nteger ) ).
% 0.32/0.56
% 0.32/0.56 % bits_mod_0
% 0.32/0.56 thf(fact_3435_bits__mod__0,axiom,
% 0.32/0.56 ! [A: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ zero_z2226904508553997617atural @ A )
% 0.32/0.56 = zero_z2226904508553997617atural ) ).
% 0.32/0.56
% 0.32/0.56 % bits_mod_0
% 0.32/0.56 thf(fact_3436_mod__self,axiom,
% 0.32/0.56 ! [A: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ A @ A )
% 0.32/0.56 = zero_zero_nat ) ).
% 0.32/0.56
% 0.32/0.56 % mod_self
% 0.32/0.56 thf(fact_3437_mod__self,axiom,
% 0.32/0.56 ! [A: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ A @ A )
% 0.32/0.56 = zero_zero_int ) ).
% 0.32/0.56
% 0.32/0.56 % mod_self
% 0.32/0.56 thf(fact_3438_mod__self,axiom,
% 0.32/0.56 ! [A: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ A @ A )
% 0.32/0.56 = zero_z3403309356797280102nteger ) ).
% 0.32/0.56
% 0.32/0.56 % mod_self
% 0.32/0.56 thf(fact_3439_mod__self,axiom,
% 0.32/0.56 ! [A: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ A @ A )
% 0.32/0.56 = zero_z2226904508553997617atural ) ).
% 0.32/0.56
% 0.32/0.56 % mod_self
% 0.32/0.56 thf(fact_3440_mod__by__0,axiom,
% 0.32/0.56 ! [A: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % mod_by_0
% 0.32/0.56 thf(fact_3441_mod__by__0,axiom,
% 0.32/0.56 ! [A: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ A @ zero_zero_int )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % mod_by_0
% 0.32/0.56 thf(fact_3442_mod__by__0,axiom,
% 0.32/0.56 ! [A: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % mod_by_0
% 0.32/0.56 thf(fact_3443_mod__by__0,axiom,
% 0.32/0.56 ! [A: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ A @ zero_z2226904508553997617atural )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % mod_by_0
% 0.32/0.56 thf(fact_3444_mod__0,axiom,
% 0.32/0.56 ! [A: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.32/0.56 = zero_zero_nat ) ).
% 0.32/0.56
% 0.32/0.56 % mod_0
% 0.32/0.56 thf(fact_3445_mod__0,axiom,
% 0.32/0.56 ! [A: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.32/0.56 = zero_zero_int ) ).
% 0.32/0.56
% 0.32/0.56 % mod_0
% 0.32/0.56 thf(fact_3446_mod__0,axiom,
% 0.32/0.56 ! [A: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.56 = zero_z3403309356797280102nteger ) ).
% 0.32/0.56
% 0.32/0.56 % mod_0
% 0.32/0.56 thf(fact_3447_mod__0,axiom,
% 0.32/0.56 ! [A: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ zero_z2226904508553997617atural @ A )
% 0.32/0.56 = zero_z2226904508553997617atural ) ).
% 0.32/0.56
% 0.32/0.56 % mod_0
% 0.32/0.56 thf(fact_3448_mod__by__Suc__0,axiom,
% 0.32/0.56 ! [M: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.32/0.56 = zero_zero_nat ) ).
% 0.32/0.56
% 0.32/0.56 % mod_by_Suc_0
% 0.32/0.56 thf(fact_3449_mod__add__self1,axiom,
% 0.32/0.56 ! [B: nat,A: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.32/0.56 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_self1
% 0.32/0.56 thf(fact_3450_mod__add__self1,axiom,
% 0.32/0.56 ! [B: int,A: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.32/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_self1
% 0.32/0.56 thf(fact_3451_mod__add__self1,axiom,
% 0.32/0.56 ! [B: code_integer,A: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 0.32/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_self1
% 0.32/0.56 thf(fact_3452_mod__add__self1,axiom,
% 0.32/0.56 ! [B: code_natural,A: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ B @ A ) @ B )
% 0.32/0.56 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_self1
% 0.32/0.56 thf(fact_3453_mod__add__self2,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.32/0.56 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_self2
% 0.32/0.56 thf(fact_3454_mod__add__self2,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.32/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_self2
% 0.32/0.56 thf(fact_3455_mod__add__self2,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 0.32/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_self2
% 0.32/0.56 thf(fact_3456_mod__add__self2,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ B )
% 0.32/0.56 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_self2
% 0.32/0.56 thf(fact_3457_minus__mod__self2,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.32/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % minus_mod_self2
% 0.32/0.56 thf(fact_3458_minus__mod__self2,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
% 0.32/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % minus_mod_self2
% 0.32/0.56 thf(fact_3459_mod__mult__self1__is__0,axiom,
% 0.32/0.56 ! [B: nat,A: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
% 0.32/0.56 = zero_zero_nat ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self1_is_0
% 0.32/0.56 thf(fact_3460_mod__mult__self1__is__0,axiom,
% 0.32/0.56 ! [B: int,A: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
% 0.32/0.56 = zero_zero_int ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self1_is_0
% 0.32/0.56 thf(fact_3461_mod__mult__self1__is__0,axiom,
% 0.32/0.56 ! [B: code_integer,A: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
% 0.32/0.56 = zero_z3403309356797280102nteger ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self1_is_0
% 0.32/0.56 thf(fact_3462_mod__mult__self1__is__0,axiom,
% 0.32/0.56 ! [B: code_natural,A: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ B @ A ) @ B )
% 0.32/0.56 = zero_z2226904508553997617atural ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self1_is_0
% 0.32/0.56 thf(fact_3463_mod__mult__self2__is__0,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.32/0.56 = zero_zero_nat ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self2_is_0
% 0.32/0.56 thf(fact_3464_mod__mult__self2__is__0,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
% 0.32/0.56 = zero_zero_int ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self2_is_0
% 0.32/0.56 thf(fact_3465_mod__mult__self2__is__0,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 0.32/0.56 = zero_z3403309356797280102nteger ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self2_is_0
% 0.32/0.56 thf(fact_3466_mod__mult__self2__is__0,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ B )
% 0.32/0.56 = zero_z2226904508553997617atural ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self2_is_0
% 0.32/0.56 thf(fact_3467_bits__mod__by__1,axiom,
% 0.32/0.56 ! [A: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.32/0.56 = zero_zero_nat ) ).
% 0.32/0.56
% 0.32/0.56 % bits_mod_by_1
% 0.32/0.56 thf(fact_3468_bits__mod__by__1,axiom,
% 0.32/0.56 ! [A: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ A @ one_one_int )
% 0.32/0.56 = zero_zero_int ) ).
% 0.32/0.56
% 0.32/0.56 % bits_mod_by_1
% 0.32/0.56 thf(fact_3469_bits__mod__by__1,axiom,
% 0.32/0.56 ! [A: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.32/0.56 = zero_z3403309356797280102nteger ) ).
% 0.32/0.56
% 0.32/0.56 % bits_mod_by_1
% 0.32/0.56 thf(fact_3470_bits__mod__by__1,axiom,
% 0.32/0.56 ! [A: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ A @ one_one_Code_natural )
% 0.32/0.56 = zero_z2226904508553997617atural ) ).
% 0.32/0.56
% 0.32/0.56 % bits_mod_by_1
% 0.32/0.56 thf(fact_3471_mod__by__1,axiom,
% 0.32/0.56 ! [A: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.32/0.56 = zero_zero_nat ) ).
% 0.32/0.56
% 0.32/0.56 % mod_by_1
% 0.32/0.56 thf(fact_3472_mod__by__1,axiom,
% 0.32/0.56 ! [A: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ A @ one_one_int )
% 0.32/0.56 = zero_zero_int ) ).
% 0.32/0.56
% 0.32/0.56 % mod_by_1
% 0.32/0.56 thf(fact_3473_mod__by__1,axiom,
% 0.32/0.56 ! [A: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.32/0.56 = zero_z3403309356797280102nteger ) ).
% 0.32/0.56
% 0.32/0.56 % mod_by_1
% 0.32/0.56 thf(fact_3474_mod__by__1,axiom,
% 0.32/0.56 ! [A: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ A @ one_one_Code_natural )
% 0.32/0.56 = zero_z2226904508553997617atural ) ).
% 0.32/0.56
% 0.32/0.56 % mod_by_1
% 0.32/0.56 thf(fact_3475_bits__mod__div__trivial,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.32/0.56 = zero_zero_nat ) ).
% 0.32/0.56
% 0.32/0.56 % bits_mod_div_trivial
% 0.32/0.56 thf(fact_3476_bits__mod__div__trivial,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.32/0.56 = zero_zero_int ) ).
% 0.32/0.56
% 0.32/0.56 % bits_mod_div_trivial
% 0.32/0.56 thf(fact_3477_bits__mod__div__trivial,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.32/0.56 = zero_z3403309356797280102nteger ) ).
% 0.32/0.56
% 0.32/0.56 % bits_mod_div_trivial
% 0.32/0.56 thf(fact_3478_bits__mod__div__trivial,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural] :
% 0.32/0.56 ( ( divide5121882707175180666atural @ ( modulo8411746178871703098atural @ A @ B ) @ B )
% 0.32/0.56 = zero_z2226904508553997617atural ) ).
% 0.32/0.56
% 0.32/0.56 % bits_mod_div_trivial
% 0.32/0.56 thf(fact_3479_mod__div__trivial,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.32/0.56 = zero_zero_nat ) ).
% 0.32/0.56
% 0.32/0.56 % mod_div_trivial
% 0.32/0.56 thf(fact_3480_mod__div__trivial,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.32/0.56 = zero_zero_int ) ).
% 0.32/0.56
% 0.32/0.56 % mod_div_trivial
% 0.32/0.56 thf(fact_3481_mod__div__trivial,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.32/0.56 = zero_z3403309356797280102nteger ) ).
% 0.32/0.56
% 0.32/0.56 % mod_div_trivial
% 0.32/0.56 thf(fact_3482_mod__div__trivial,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural] :
% 0.32/0.56 ( ( divide5121882707175180666atural @ ( modulo8411746178871703098atural @ A @ B ) @ B )
% 0.32/0.56 = zero_z2226904508553997617atural ) ).
% 0.32/0.56
% 0.32/0.56 % mod_div_trivial
% 0.32/0.56 thf(fact_3483_mod__mult__self1,axiom,
% 0.32/0.56 ! [A: nat,C: nat,B: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 0.32/0.56 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self1
% 0.32/0.56 thf(fact_3484_mod__mult__self1,axiom,
% 0.32/0.56 ! [A: int,C: int,B: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 0.32/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self1
% 0.32/0.56 thf(fact_3485_mod__mult__self1,axiom,
% 0.32/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 0.32/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self1
% 0.32/0.56 thf(fact_3486_mod__mult__self1,axiom,
% 0.32/0.56 ! [A: code_natural,C: code_natural,B: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ C @ B ) ) @ B )
% 0.32/0.56 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self1
% 0.32/0.56 thf(fact_3487_mod__mult__self2,axiom,
% 0.32/0.56 ! [A: nat,B: nat,C: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 0.32/0.56 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self2
% 0.32/0.56 thf(fact_3488_mod__mult__self2,axiom,
% 0.32/0.56 ! [A: int,B: int,C: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 0.32/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self2
% 0.32/0.56 thf(fact_3489_mod__mult__self2,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 0.32/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self2
% 0.32/0.56 thf(fact_3490_mod__mult__self2,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural,C: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ B @ C ) ) @ B )
% 0.32/0.56 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self2
% 0.32/0.56 thf(fact_3491_mod__mult__self3,axiom,
% 0.32/0.56 ! [C: nat,B: nat,A: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 0.32/0.56 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self3
% 0.32/0.56 thf(fact_3492_mod__mult__self3,axiom,
% 0.32/0.56 ! [C: int,B: int,A: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 0.32/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self3
% 0.32/0.56 thf(fact_3493_mod__mult__self3,axiom,
% 0.32/0.56 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 0.32/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self3
% 0.32/0.56 thf(fact_3494_mod__mult__self3,axiom,
% 0.32/0.56 ! [C: code_natural,B: code_natural,A: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C @ B ) @ A ) @ B )
% 0.32/0.56 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self3
% 0.32/0.56 thf(fact_3495_mod__mult__self4,axiom,
% 0.32/0.56 ! [B: nat,C: nat,A: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 0.32/0.56 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self4
% 0.32/0.56 thf(fact_3496_mod__mult__self4,axiom,
% 0.32/0.56 ! [B: int,C: int,A: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 0.32/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self4
% 0.32/0.56 thf(fact_3497_mod__mult__self4,axiom,
% 0.32/0.56 ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 0.32/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self4
% 0.32/0.56 thf(fact_3498_mod__mult__self4,axiom,
% 0.32/0.56 ! [B: code_natural,C: code_natural,A: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ C ) @ A ) @ B )
% 0.32/0.56 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_self4
% 0.32/0.56 thf(fact_3499_dvd__imp__mod__0,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.56 => ( ( modulo_modulo_nat @ B @ A )
% 0.32/0.56 = zero_zero_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_imp_mod_0
% 0.32/0.56 thf(fact_3500_dvd__imp__mod__0,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.56 => ( ( modulo_modulo_int @ B @ A )
% 0.32/0.56 = zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_imp_mod_0
% 0.32/0.56 thf(fact_3501_dvd__imp__mod__0,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.56 => ( ( modulo364778990260209775nteger @ B @ A )
% 0.32/0.56 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_imp_mod_0
% 0.32/0.56 thf(fact_3502_dvd__imp__mod__0,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural] :
% 0.32/0.56 ( ( dvd_dvd_Code_natural @ A @ B )
% 0.32/0.56 => ( ( modulo8411746178871703098atural @ B @ A )
% 0.32/0.56 = zero_z2226904508553997617atural ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_imp_mod_0
% 0.32/0.56 thf(fact_3503_Suc__mod__mult__self1,axiom,
% 0.32/0.56 ! [M: nat,K: nat,N: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
% 0.32/0.56 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.32/0.56
% 0.32/0.56 % Suc_mod_mult_self1
% 0.32/0.56 thf(fact_3504_Suc__mod__mult__self2,axiom,
% 0.32/0.56 ! [M: nat,N: nat,K: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
% 0.32/0.56 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.32/0.56
% 0.32/0.56 % Suc_mod_mult_self2
% 0.32/0.56 thf(fact_3505_Suc__mod__mult__self3,axiom,
% 0.32/0.56 ! [K: nat,N: nat,M: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
% 0.32/0.56 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.32/0.56
% 0.32/0.56 % Suc_mod_mult_self3
% 0.32/0.56 thf(fact_3506_Suc__mod__mult__self4,axiom,
% 0.32/0.56 ! [N: nat,K: nat,M: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
% 0.32/0.56 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.32/0.56
% 0.32/0.56 % Suc_mod_mult_self4
% 0.32/0.56 thf(fact_3507_mod__pos__pos__trivial,axiom,
% 0.32/0.56 ! [K: int,L: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.56 => ( ( ord_less_int @ K @ L )
% 0.32/0.56 => ( ( modulo_modulo_int @ K @ L )
% 0.32/0.56 = K ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_pos_pos_trivial
% 0.32/0.56 thf(fact_3508_mod__neg__neg__trivial,axiom,
% 0.32/0.56 ! [K: int,L: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.32/0.56 => ( ( ord_less_int @ L @ K )
% 0.32/0.56 => ( ( modulo_modulo_int @ K @ L )
% 0.32/0.56 = K ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_neg_neg_trivial
% 0.32/0.56 thf(fact_3509_Suc__0__mod__eq,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.56 = ( zero_n2687167440665602831ol_nat
% 0.32/0.56 @ ( N
% 0.32/0.56 != ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % Suc_0_mod_eq
% 0.32/0.56 thf(fact_3510_mod2__Suc__Suc,axiom,
% 0.32/0.56 ! [M: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod2_Suc_Suc
% 0.32/0.56 thf(fact_3511_zmod__numeral__Bit0,axiom,
% 0.32/0.56 ! [V: num,W: num] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.32/0.56 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zmod_numeral_Bit0
% 0.32/0.56 thf(fact_3512_Suc__times__numeral__mod__eq,axiom,
% 0.32/0.56 ! [K: num,N: nat] :
% 0.32/0.56 ( ( ( numeral_numeral_nat @ K )
% 0.32/0.56 != one_one_nat )
% 0.32/0.56 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
% 0.32/0.56 = one_one_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % Suc_times_numeral_mod_eq
% 0.32/0.56 thf(fact_3513_bits__one__mod__two__eq__one,axiom,
% 0.32/0.56 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = one_one_nat ) ).
% 0.32/0.56
% 0.32/0.56 % bits_one_mod_two_eq_one
% 0.32/0.56 thf(fact_3514_bits__one__mod__two__eq__one,axiom,
% 0.32/0.56 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.56 = one_one_int ) ).
% 0.32/0.56
% 0.32/0.56 % bits_one_mod_two_eq_one
% 0.32/0.56 thf(fact_3515_bits__one__mod__two__eq__one,axiom,
% 0.32/0.56 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.56 = one_one_Code_integer ) ).
% 0.32/0.56
% 0.32/0.56 % bits_one_mod_two_eq_one
% 0.32/0.56 thf(fact_3516_bits__one__mod__two__eq__one,axiom,
% 0.32/0.56 ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 0.32/0.56 = one_one_Code_natural ) ).
% 0.32/0.56
% 0.32/0.56 % bits_one_mod_two_eq_one
% 0.32/0.56 thf(fact_3517_one__mod__two__eq__one,axiom,
% 0.32/0.56 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = one_one_nat ) ).
% 0.32/0.56
% 0.32/0.56 % one_mod_two_eq_one
% 0.32/0.56 thf(fact_3518_one__mod__two__eq__one,axiom,
% 0.32/0.56 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.56 = one_one_int ) ).
% 0.32/0.56
% 0.32/0.56 % one_mod_two_eq_one
% 0.32/0.56 thf(fact_3519_one__mod__two__eq__one,axiom,
% 0.32/0.56 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.56 = one_one_Code_integer ) ).
% 0.32/0.56
% 0.32/0.56 % one_mod_two_eq_one
% 0.32/0.56 thf(fact_3520_one__mod__two__eq__one,axiom,
% 0.32/0.56 ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 0.32/0.56 = one_one_Code_natural ) ).
% 0.32/0.56
% 0.32/0.56 % one_mod_two_eq_one
% 0.32/0.56 thf(fact_3521_not__mod2__eq__Suc__0__eq__0,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 != ( suc @ zero_zero_nat ) )
% 0.32/0.56 = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = zero_zero_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % not_mod2_eq_Suc_0_eq_0
% 0.32/0.56 thf(fact_3522_even__mod__2__iff,axiom,
% 0.32/0.56 ! [A: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.56 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_mod_2_iff
% 0.32/0.56 thf(fact_3523_even__mod__2__iff,axiom,
% 0.32/0.56 ! [A: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.32/0.56 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_mod_2_iff
% 0.32/0.56 thf(fact_3524_even__mod__2__iff,axiom,
% 0.32/0.56 ! [A: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.32/0.56 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_mod_2_iff
% 0.32/0.56 thf(fact_3525_even__mod__2__iff,axiom,
% 0.32/0.56 ! [A: code_natural] :
% 0.32/0.56 ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
% 0.32/0.56 = ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_mod_2_iff
% 0.32/0.56 thf(fact_3526_add__self__mod__2,axiom,
% 0.32/0.56 ! [M: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = zero_zero_nat ) ).
% 0.32/0.56
% 0.32/0.56 % add_self_mod_2
% 0.32/0.56 thf(fact_3527_not__mod__2__eq__0__eq__1,axiom,
% 0.32/0.56 ! [A: nat] :
% 0.32/0.56 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 != zero_zero_nat )
% 0.32/0.56 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = one_one_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % not_mod_2_eq_0_eq_1
% 0.32/0.56 thf(fact_3528_not__mod__2__eq__0__eq__1,axiom,
% 0.32/0.56 ! [A: int] :
% 0.32/0.56 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.56 != zero_zero_int )
% 0.32/0.56 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.56 = one_one_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % not_mod_2_eq_0_eq_1
% 0.32/0.56 thf(fact_3529_not__mod__2__eq__0__eq__1,axiom,
% 0.32/0.56 ! [A: code_integer] :
% 0.32/0.56 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.56 != zero_z3403309356797280102nteger )
% 0.32/0.56 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.56 = one_one_Code_integer ) ) ).
% 0.32/0.56
% 0.32/0.56 % not_mod_2_eq_0_eq_1
% 0.32/0.56 thf(fact_3530_not__mod__2__eq__0__eq__1,axiom,
% 0.32/0.56 ! [A: code_natural] :
% 0.32/0.56 ( ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 0.32/0.56 != zero_z2226904508553997617atural )
% 0.32/0.56 = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 0.32/0.56 = one_one_Code_natural ) ) ).
% 0.32/0.56
% 0.32/0.56 % not_mod_2_eq_0_eq_1
% 0.32/0.56 thf(fact_3531_not__mod__2__eq__1__eq__0,axiom,
% 0.32/0.56 ! [A: nat] :
% 0.32/0.56 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 != one_one_nat )
% 0.32/0.56 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = zero_zero_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % not_mod_2_eq_1_eq_0
% 0.32/0.56 thf(fact_3532_not__mod__2__eq__1__eq__0,axiom,
% 0.32/0.56 ! [A: int] :
% 0.32/0.56 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.56 != one_one_int )
% 0.32/0.56 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.56 = zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % not_mod_2_eq_1_eq_0
% 0.32/0.56 thf(fact_3533_not__mod__2__eq__1__eq__0,axiom,
% 0.32/0.56 ! [A: code_integer] :
% 0.32/0.56 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.56 != one_one_Code_integer )
% 0.32/0.56 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.56 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.56
% 0.32/0.56 % not_mod_2_eq_1_eq_0
% 0.32/0.56 thf(fact_3534_not__mod__2__eq__1__eq__0,axiom,
% 0.32/0.56 ! [A: code_natural] :
% 0.32/0.56 ( ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 0.32/0.56 != one_one_Code_natural )
% 0.32/0.56 = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 0.32/0.56 = zero_z2226904508553997617atural ) ) ).
% 0.32/0.56
% 0.32/0.56 % not_mod_2_eq_1_eq_0
% 0.32/0.56 thf(fact_3535_mod2__gr__0,axiom,
% 0.32/0.56 ! [M: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.56 = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = one_one_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod2_gr_0
% 0.32/0.56 thf(fact_3536_one__mod__2__pow__eq,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.56 = ( zero_n8403883297036319079atural @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % one_mod_2_pow_eq
% 0.32/0.56 thf(fact_3537_one__mod__2__pow__eq,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.56 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % one_mod_2_pow_eq
% 0.32/0.56 thf(fact_3538_one__mod__2__pow__eq,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.56 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % one_mod_2_pow_eq
% 0.32/0.56 thf(fact_3539_one__mod__2__pow__eq,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.56 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % one_mod_2_pow_eq
% 0.32/0.56 thf(fact_3540_add__diff__assoc__enat,axiom,
% 0.32/0.56 ! [Z2: extended_enat,Y: extended_enat,X: extended_enat] :
% 0.32/0.56 ( ( ord_le2932123472753598470d_enat @ Z2 @ Y )
% 0.32/0.56 => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z2 ) )
% 0.32/0.56 = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % add_diff_assoc_enat
% 0.32/0.56 thf(fact_3541_verit__la__generic,axiom,
% 0.32/0.56 ! [A: int,X: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ A @ X )
% 0.32/0.56 | ( A = X )
% 0.32/0.56 | ( ord_less_eq_int @ X @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % verit_la_generic
% 0.32/0.56 thf(fact_3542_unset__bit__less__eq,axiom,
% 0.32/0.56 ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% 0.32/0.56
% 0.32/0.56 % unset_bit_less_eq
% 0.32/0.56 thf(fact_3543_set__bit__greater__eq,axiom,
% 0.32/0.56 ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% 0.32/0.56
% 0.32/0.56 % set_bit_greater_eq
% 0.32/0.56 thf(fact_3544_enat__0__less__mult__iff,axiom,
% 0.32/0.56 ! [M: extended_enat,N: extended_enat] :
% 0.32/0.56 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 0.32/0.56 = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 0.32/0.56 & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % enat_0_less_mult_iff
% 0.32/0.56 thf(fact_3545_of__nat__mod,axiom,
% 0.32/0.56 ! [M: nat,N: nat] :
% 0.32/0.56 ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % of_nat_mod
% 0.32/0.56 thf(fact_3546_of__nat__mod,axiom,
% 0.32/0.56 ! [M: nat,N: nat] :
% 0.32/0.56 ( ( semiri3763490453095760265atural @ ( modulo_modulo_nat @ M @ N ) )
% 0.32/0.56 = ( modulo8411746178871703098atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % of_nat_mod
% 0.32/0.56 thf(fact_3547_of__nat__mod,axiom,
% 0.32/0.56 ! [M: nat,N: nat] :
% 0.32/0.56 ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
% 0.32/0.56 = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % of_nat_mod
% 0.32/0.56 thf(fact_3548_of__nat__mod,axiom,
% 0.32/0.56 ! [M: nat,N: nat] :
% 0.32/0.56 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
% 0.32/0.56 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % of_nat_mod
% 0.32/0.56 thf(fact_3549_zmod__int,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
% 0.32/0.56 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zmod_int
% 0.32/0.56 thf(fact_3550_mod__mult__eq,axiom,
% 0.32/0.56 ! [A: nat,C: nat,B: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_eq
% 0.32/0.56 thf(fact_3551_mod__mult__eq,axiom,
% 0.32/0.56 ! [A: int,C: int,B: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_eq
% 0.32/0.56 thf(fact_3552_mod__mult__eq,axiom,
% 0.32/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_eq
% 0.32/0.56 thf(fact_3553_mod__mult__eq,axiom,
% 0.32/0.56 ! [A: code_natural,C: code_natural,B: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ C ) @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_eq
% 0.32/0.56 thf(fact_3554_mod__mult__cong,axiom,
% 0.32/0.56 ! [A: nat,C: nat,A5: nat,B: nat,B6: nat] :
% 0.32/0.56 ( ( ( modulo_modulo_nat @ A @ C )
% 0.32/0.56 = ( modulo_modulo_nat @ A5 @ C ) )
% 0.32/0.56 => ( ( ( modulo_modulo_nat @ B @ C )
% 0.32/0.56 = ( modulo_modulo_nat @ B6 @ C ) )
% 0.32/0.56 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.56 = ( modulo_modulo_nat @ ( times_times_nat @ A5 @ B6 ) @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_cong
% 0.32/0.56 thf(fact_3555_mod__mult__cong,axiom,
% 0.32/0.56 ! [A: int,C: int,A5: int,B: int,B6: int] :
% 0.32/0.56 ( ( ( modulo_modulo_int @ A @ C )
% 0.32/0.56 = ( modulo_modulo_int @ A5 @ C ) )
% 0.32/0.56 => ( ( ( modulo_modulo_int @ B @ C )
% 0.32/0.56 = ( modulo_modulo_int @ B6 @ C ) )
% 0.32/0.56 => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.56 = ( modulo_modulo_int @ ( times_times_int @ A5 @ B6 ) @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_cong
% 0.32/0.56 thf(fact_3556_mod__mult__cong,axiom,
% 0.32/0.56 ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B6: code_integer] :
% 0.32/0.56 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ A5 @ C ) )
% 0.32/0.56 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 0.32/0.56 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A5 @ B6 ) @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_cong
% 0.32/0.56 thf(fact_3557_mod__mult__cong,axiom,
% 0.32/0.56 ! [A: code_natural,C: code_natural,A5: code_natural,B: code_natural,B6: code_natural] :
% 0.32/0.56 ( ( ( modulo8411746178871703098atural @ A @ C )
% 0.32/0.56 = ( modulo8411746178871703098atural @ A5 @ C ) )
% 0.32/0.56 => ( ( ( modulo8411746178871703098atural @ B @ C )
% 0.32/0.56 = ( modulo8411746178871703098atural @ B6 @ C ) )
% 0.32/0.56 => ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C )
% 0.32/0.56 = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A5 @ B6 ) @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_cong
% 0.32/0.56 thf(fact_3558_mod__mult__mult2,axiom,
% 0.32/0.56 ! [A: nat,C: nat,B: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.32/0.56 = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_mult2
% 0.32/0.56 thf(fact_3559_mod__mult__mult2,axiom,
% 0.32/0.56 ! [A: int,C: int,B: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.56 = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_mult2
% 0.32/0.56 thf(fact_3560_mod__mult__mult2,axiom,
% 0.32/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.56 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_mult2
% 0.32/0.56 thf(fact_3561_mod__mult__mult2,axiom,
% 0.32/0.56 ! [A: code_natural,C: code_natural,B: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) )
% 0.32/0.56 = ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_mult2
% 0.32/0.56 thf(fact_3562_mult__mod__right,axiom,
% 0.32/0.56 ! [C: nat,A: nat,B: nat] :
% 0.32/0.56 ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 0.32/0.56 = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_mod_right
% 0.32/0.56 thf(fact_3563_mult__mod__right,axiom,
% 0.32/0.56 ! [C: int,A: int,B: int] :
% 0.32/0.56 ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 0.32/0.56 = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_mod_right
% 0.32/0.56 thf(fact_3564_mult__mod__right,axiom,
% 0.32/0.56 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_mod_right
% 0.32/0.56 thf(fact_3565_mult__mod__right,axiom,
% 0.32/0.56 ! [C: code_natural,A: code_natural,B: code_natural] :
% 0.32/0.56 ( ( times_2397367101498566445atural @ C @ ( modulo8411746178871703098atural @ A @ B ) )
% 0.32/0.56 = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_mod_right
% 0.32/0.56 thf(fact_3566_mod__mult__left__eq,axiom,
% 0.32/0.56 ! [A: nat,C: nat,B: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 0.32/0.56 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_left_eq
% 0.32/0.56 thf(fact_3567_mod__mult__left__eq,axiom,
% 0.32/0.56 ! [A: int,C: int,B: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.32/0.56 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_left_eq
% 0.32/0.56 thf(fact_3568_mod__mult__left__eq,axiom,
% 0.32/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_left_eq
% 0.32/0.56 thf(fact_3569_mod__mult__left__eq,axiom,
% 0.32/0.56 ! [A: code_natural,C: code_natural,B: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ C ) @ B ) @ C )
% 0.32/0.56 = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_left_eq
% 0.32/0.56 thf(fact_3570_mod__mult__right__eq,axiom,
% 0.32/0.56 ! [A: nat,B: nat,C: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_right_eq
% 0.32/0.56 thf(fact_3571_mod__mult__right__eq,axiom,
% 0.32/0.56 ! [A: int,B: int,C: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_right_eq
% 0.32/0.56 thf(fact_3572_mod__mult__right__eq,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_right_eq
% 0.32/0.56 thf(fact_3573_mod__mult__right__eq,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural,C: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_right_eq
% 0.32/0.56 thf(fact_3574_mod__add__eq,axiom,
% 0.32/0.56 ! [A: nat,C: nat,B: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_eq
% 0.32/0.56 thf(fact_3575_mod__add__eq,axiom,
% 0.32/0.56 ! [A: int,C: int,B: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_eq
% 0.32/0.56 thf(fact_3576_mod__add__eq,axiom,
% 0.32/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_eq
% 0.32/0.56 thf(fact_3577_mod__add__eq,axiom,
% 0.32/0.56 ! [A: code_natural,C: code_natural,B: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C ) @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_eq
% 0.32/0.56 thf(fact_3578_mod__add__cong,axiom,
% 0.32/0.56 ! [A: nat,C: nat,A5: nat,B: nat,B6: nat] :
% 0.32/0.56 ( ( ( modulo_modulo_nat @ A @ C )
% 0.32/0.56 = ( modulo_modulo_nat @ A5 @ C ) )
% 0.32/0.56 => ( ( ( modulo_modulo_nat @ B @ C )
% 0.32/0.56 = ( modulo_modulo_nat @ B6 @ C ) )
% 0.32/0.56 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.56 = ( modulo_modulo_nat @ ( plus_plus_nat @ A5 @ B6 ) @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_cong
% 0.32/0.56 thf(fact_3579_mod__add__cong,axiom,
% 0.32/0.56 ! [A: int,C: int,A5: int,B: int,B6: int] :
% 0.32/0.56 ( ( ( modulo_modulo_int @ A @ C )
% 0.32/0.56 = ( modulo_modulo_int @ A5 @ C ) )
% 0.32/0.56 => ( ( ( modulo_modulo_int @ B @ C )
% 0.32/0.56 = ( modulo_modulo_int @ B6 @ C ) )
% 0.32/0.56 => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.56 = ( modulo_modulo_int @ ( plus_plus_int @ A5 @ B6 ) @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_cong
% 0.32/0.56 thf(fact_3580_mod__add__cong,axiom,
% 0.32/0.56 ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B6: code_integer] :
% 0.32/0.56 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ A5 @ C ) )
% 0.32/0.56 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 0.32/0.56 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A5 @ B6 ) @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_cong
% 0.32/0.56 thf(fact_3581_mod__add__cong,axiom,
% 0.32/0.56 ! [A: code_natural,C: code_natural,A5: code_natural,B: code_natural,B6: code_natural] :
% 0.32/0.56 ( ( ( modulo8411746178871703098atural @ A @ C )
% 0.32/0.56 = ( modulo8411746178871703098atural @ A5 @ C ) )
% 0.32/0.56 => ( ( ( modulo8411746178871703098atural @ B @ C )
% 0.32/0.56 = ( modulo8411746178871703098atural @ B6 @ C ) )
% 0.32/0.56 => ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
% 0.32/0.56 = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A5 @ B6 ) @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_cong
% 0.32/0.56 thf(fact_3582_mod__add__left__eq,axiom,
% 0.32/0.56 ! [A: nat,C: nat,B: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 0.32/0.56 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_left_eq
% 0.32/0.56 thf(fact_3583_mod__add__left__eq,axiom,
% 0.32/0.56 ! [A: int,C: int,B: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.32/0.56 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_left_eq
% 0.32/0.56 thf(fact_3584_mod__add__left__eq,axiom,
% 0.32/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_left_eq
% 0.32/0.56 thf(fact_3585_mod__add__left__eq,axiom,
% 0.32/0.56 ! [A: code_natural,C: code_natural,B: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C ) @ B ) @ C )
% 0.32/0.56 = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_left_eq
% 0.32/0.56 thf(fact_3586_mod__add__right__eq,axiom,
% 0.32/0.56 ! [A: nat,B: nat,C: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_right_eq
% 0.32/0.56 thf(fact_3587_mod__add__right__eq,axiom,
% 0.32/0.56 ! [A: int,B: int,C: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_right_eq
% 0.32/0.56 thf(fact_3588_mod__add__right__eq,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_right_eq
% 0.32/0.56 thf(fact_3589_mod__add__right__eq,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural,C: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_add_right_eq
% 0.32/0.56 thf(fact_3590_mod__diff__eq,axiom,
% 0.32/0.56 ! [A: int,C: int,B: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_diff_eq
% 0.32/0.56 thf(fact_3591_mod__diff__eq,axiom,
% 0.32/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_diff_eq
% 0.32/0.56 thf(fact_3592_mod__diff__cong,axiom,
% 0.32/0.56 ! [A: int,C: int,A5: int,B: int,B6: int] :
% 0.32/0.56 ( ( ( modulo_modulo_int @ A @ C )
% 0.32/0.56 = ( modulo_modulo_int @ A5 @ C ) )
% 0.32/0.56 => ( ( ( modulo_modulo_int @ B @ C )
% 0.32/0.56 = ( modulo_modulo_int @ B6 @ C ) )
% 0.32/0.56 => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.32/0.56 = ( modulo_modulo_int @ ( minus_minus_int @ A5 @ B6 ) @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_diff_cong
% 0.32/0.56 thf(fact_3593_mod__diff__cong,axiom,
% 0.32/0.56 ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B6: code_integer] :
% 0.32/0.56 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ A5 @ C ) )
% 0.32/0.56 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 0.32/0.56 => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A5 @ B6 ) @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_diff_cong
% 0.32/0.56 thf(fact_3594_mod__diff__left__eq,axiom,
% 0.32/0.56 ! [A: int,C: int,B: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.32/0.56 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_diff_left_eq
% 0.32/0.56 thf(fact_3595_mod__diff__left__eq,axiom,
% 0.32/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_diff_left_eq
% 0.32/0.56 thf(fact_3596_mod__diff__right__eq,axiom,
% 0.32/0.56 ! [A: int,B: int,C: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_diff_right_eq
% 0.32/0.56 thf(fact_3597_mod__diff__right__eq,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_diff_right_eq
% 0.32/0.56 thf(fact_3598_power__mod,axiom,
% 0.32/0.56 ! [A: nat,B: nat,N: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
% 0.32/0.56 = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_mod
% 0.32/0.56 thf(fact_3599_power__mod,axiom,
% 0.32/0.56 ! [A: int,B: int,N: nat] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
% 0.32/0.56 = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_mod
% 0.32/0.56 thf(fact_3600_power__mod,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer,N: nat] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_mod
% 0.32/0.56 thf(fact_3601_power__mod,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural,N: nat] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ ( power_7079662738309270450atural @ ( modulo8411746178871703098atural @ A @ B ) @ N ) @ B )
% 0.32/0.56 = ( modulo8411746178871703098atural @ ( power_7079662738309270450atural @ A @ N ) @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_mod
% 0.32/0.56 thf(fact_3602_dvd__mod__imp__dvd,axiom,
% 0.32/0.56 ! [C: nat,A: nat,B: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 0.32/0.56 => ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.56 => ( dvd_dvd_nat @ C @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mod_imp_dvd
% 0.32/0.56 thf(fact_3603_dvd__mod__imp__dvd,axiom,
% 0.32/0.56 ! [C: int,A: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 0.32/0.56 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.56 => ( dvd_dvd_int @ C @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mod_imp_dvd
% 0.32/0.56 thf(fact_3604_dvd__mod__imp__dvd,axiom,
% 0.32/0.56 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.32/0.56 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.56 => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mod_imp_dvd
% 0.32/0.56 thf(fact_3605_dvd__mod__imp__dvd,axiom,
% 0.32/0.56 ! [C: code_natural,A: code_natural,B: code_natural] :
% 0.32/0.56 ( ( dvd_dvd_Code_natural @ C @ ( modulo8411746178871703098atural @ A @ B ) )
% 0.32/0.56 => ( ( dvd_dvd_Code_natural @ C @ B )
% 0.32/0.56 => ( dvd_dvd_Code_natural @ C @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mod_imp_dvd
% 0.32/0.56 thf(fact_3606_dvd__mod__iff,axiom,
% 0.32/0.56 ! [C: nat,B: nat,A: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.56 => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 0.32/0.56 = ( dvd_dvd_nat @ C @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mod_iff
% 0.32/0.56 thf(fact_3607_dvd__mod__iff,axiom,
% 0.32/0.56 ! [C: int,B: int,A: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.56 => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 0.32/0.56 = ( dvd_dvd_int @ C @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mod_iff
% 0.32/0.56 thf(fact_3608_dvd__mod__iff,axiom,
% 0.32/0.56 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.56 => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.32/0.56 = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mod_iff
% 0.32/0.56 thf(fact_3609_dvd__mod__iff,axiom,
% 0.32/0.56 ! [C: code_natural,B: code_natural,A: code_natural] :
% 0.32/0.56 ( ( dvd_dvd_Code_natural @ C @ B )
% 0.32/0.56 => ( ( dvd_dvd_Code_natural @ C @ ( modulo8411746178871703098atural @ A @ B ) )
% 0.32/0.56 = ( dvd_dvd_Code_natural @ C @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mod_iff
% 0.32/0.56 thf(fact_3610_mod__mod__cancel,axiom,
% 0.32/0.56 ! [C: nat,B: nat,A: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.56 => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
% 0.32/0.56 = ( modulo_modulo_nat @ A @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mod_cancel
% 0.32/0.56 thf(fact_3611_mod__mod__cancel,axiom,
% 0.32/0.56 ! [C: int,B: int,A: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.56 => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
% 0.32/0.56 = ( modulo_modulo_int @ A @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mod_cancel
% 0.32/0.56 thf(fact_3612_mod__mod__cancel,axiom,
% 0.32/0.56 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.56 => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ A @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mod_cancel
% 0.32/0.56 thf(fact_3613_mod__mod__cancel,axiom,
% 0.32/0.56 ! [C: code_natural,B: code_natural,A: code_natural] :
% 0.32/0.56 ( ( dvd_dvd_Code_natural @ C @ B )
% 0.32/0.56 => ( ( modulo8411746178871703098atural @ ( modulo8411746178871703098atural @ A @ B ) @ C )
% 0.32/0.56 = ( modulo8411746178871703098atural @ A @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mod_cancel
% 0.32/0.56 thf(fact_3614_dvd__mod,axiom,
% 0.32/0.56 ! [K: nat,M: nat,N: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ K @ M )
% 0.32/0.56 => ( ( dvd_dvd_nat @ K @ N )
% 0.32/0.56 => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mod
% 0.32/0.56 thf(fact_3615_dvd__mod,axiom,
% 0.32/0.56 ! [K: int,M: int,N: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ K @ M )
% 0.32/0.56 => ( ( dvd_dvd_int @ K @ N )
% 0.32/0.56 => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mod
% 0.32/0.56 thf(fact_3616_dvd__mod,axiom,
% 0.32/0.56 ! [K: code_integer,M: code_integer,N: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ K @ M )
% 0.32/0.56 => ( ( dvd_dvd_Code_integer @ K @ N )
% 0.32/0.56 => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mod
% 0.32/0.56 thf(fact_3617_dvd__mod,axiom,
% 0.32/0.56 ! [K: code_natural,M: code_natural,N: code_natural] :
% 0.32/0.56 ( ( dvd_dvd_Code_natural @ K @ M )
% 0.32/0.56 => ( ( dvd_dvd_Code_natural @ K @ N )
% 0.32/0.56 => ( dvd_dvd_Code_natural @ K @ ( modulo8411746178871703098atural @ M @ N ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mod
% 0.32/0.56 thf(fact_3618_mod__Suc__eq,axiom,
% 0.32/0.56 ! [M: nat,N: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
% 0.32/0.56 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_Suc_eq
% 0.32/0.56 thf(fact_3619_mod__Suc__Suc__eq,axiom,
% 0.32/0.56 ! [M: nat,N: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
% 0.32/0.56 = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_Suc_Suc_eq
% 0.32/0.56 thf(fact_3620_zmod__le__nonneg__dividend,axiom,
% 0.32/0.56 ! [M: int,K: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 0.32/0.56 => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% 0.32/0.56
% 0.32/0.56 % zmod_le_nonneg_dividend
% 0.32/0.56 thf(fact_3621_neg__mod__bound,axiom,
% 0.32/0.56 ! [L: int,K: int] :
% 0.32/0.56 ( ( ord_less_int @ L @ zero_zero_int )
% 0.32/0.56 => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % neg_mod_bound
% 0.32/0.56 thf(fact_3622_Euclidean__Division_Opos__mod__bound,axiom,
% 0.32/0.56 ! [L: int,K: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ L )
% 0.32/0.56 => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% 0.32/0.56
% 0.32/0.56 % Euclidean_Division.pos_mod_bound
% 0.32/0.56 thf(fact_3623_neg__mod__conj,axiom,
% 0.32/0.56 ! [B: int,A: int] :
% 0.32/0.56 ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.56 => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
% 0.32/0.56 & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % neg_mod_conj
% 0.32/0.56 thf(fact_3624_pos__mod__conj,axiom,
% 0.32/0.56 ! [B: int,A: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
% 0.32/0.56 & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % pos_mod_conj
% 0.32/0.56 thf(fact_3625_zmod__trivial__iff,axiom,
% 0.32/0.56 ! [I: int,K: int] :
% 0.32/0.56 ( ( ( modulo_modulo_int @ I @ K )
% 0.32/0.56 = I )
% 0.32/0.56 = ( ( K = zero_zero_int )
% 0.32/0.56 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.32/0.56 & ( ord_less_int @ I @ K ) )
% 0.32/0.56 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.32/0.56 & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zmod_trivial_iff
% 0.32/0.56 thf(fact_3626_Euclidean__Division_Opos__mod__sign,axiom,
% 0.32/0.56 ! [L: int,K: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ L )
% 0.32/0.56 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % Euclidean_Division.pos_mod_sign
% 0.32/0.56 thf(fact_3627_neg__mod__sign,axiom,
% 0.32/0.56 ! [L: int,K: int] :
% 0.32/0.56 ( ( ord_less_int @ L @ zero_zero_int )
% 0.32/0.56 => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % neg_mod_sign
% 0.32/0.56 thf(fact_3628_mod__less__eq__dividend,axiom,
% 0.32/0.56 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 0.32/0.56
% 0.32/0.56 % mod_less_eq_dividend
% 0.32/0.56 thf(fact_3629_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.56 => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.32/0.56 thf(fact_3630_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.56 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.32/0.56 thf(fact_3631_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.56 => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.32/0.56 thf(fact_3632_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.32/0.56 ! [B: nat,A: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.56 => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.32/0.56 thf(fact_3633_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.32/0.56 ! [B: int,A: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.56 => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.32/0.56 thf(fact_3634_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.32/0.56 ! [B: code_integer,A: code_integer] :
% 0.32/0.56 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.32/0.56 => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.32/0.56 thf(fact_3635_mod__eq__self__iff__div__eq__0,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( ( modulo_modulo_nat @ A @ B )
% 0.32/0.56 = A )
% 0.32/0.56 = ( ( divide_divide_nat @ A @ B )
% 0.32/0.56 = zero_zero_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_eq_self_iff_div_eq_0
% 0.32/0.56 thf(fact_3636_mod__eq__self__iff__div__eq__0,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( ( modulo_modulo_int @ A @ B )
% 0.32/0.56 = A )
% 0.32/0.56 = ( ( divide_divide_int @ A @ B )
% 0.32/0.56 = zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_eq_self_iff_div_eq_0
% 0.32/0.56 thf(fact_3637_mod__eq__self__iff__div__eq__0,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.32/0.56 = A )
% 0.32/0.56 = ( ( divide6298287555418463151nteger @ A @ B )
% 0.32/0.56 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_eq_self_iff_div_eq_0
% 0.32/0.56 thf(fact_3638_mod__eq__self__iff__div__eq__0,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural] :
% 0.32/0.56 ( ( ( modulo8411746178871703098atural @ A @ B )
% 0.32/0.56 = A )
% 0.32/0.56 = ( ( divide5121882707175180666atural @ A @ B )
% 0.32/0.56 = zero_z2226904508553997617atural ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_eq_self_iff_div_eq_0
% 0.32/0.56 thf(fact_3639_cong__exp__iff__simps_I9_J,axiom,
% 0.32/0.56 ! [M: num,Q2: num,N: num] :
% 0.32/0.56 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.32/0.56 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.32/0.56 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.32/0.56 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(9)
% 0.32/0.56 thf(fact_3640_cong__exp__iff__simps_I9_J,axiom,
% 0.32/0.56 ! [M: num,Q2: num,N: num] :
% 0.32/0.56 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.32/0.56 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.32/0.56 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.32/0.56 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(9)
% 0.32/0.56 thf(fact_3641_cong__exp__iff__simps_I9_J,axiom,
% 0.32/0.56 ! [M: num,Q2: num,N: num] :
% 0.32/0.56 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.32/0.56 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(9)
% 0.32/0.56 thf(fact_3642_mod__eqE,axiom,
% 0.32/0.56 ! [A: int,C: int,B: int] :
% 0.32/0.56 ( ( ( modulo_modulo_int @ A @ C )
% 0.32/0.56 = ( modulo_modulo_int @ B @ C ) )
% 0.32/0.56 => ~ ! [D2: int] :
% 0.32/0.56 ( B
% 0.32/0.56 != ( plus_plus_int @ A @ ( times_times_int @ C @ D2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_eqE
% 0.32/0.56 thf(fact_3643_mod__eqE,axiom,
% 0.32/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.56 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ B @ C ) )
% 0.32/0.56 => ~ ! [D2: code_integer] :
% 0.32/0.56 ( B
% 0.32/0.56 != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_eqE
% 0.32/0.56 thf(fact_3644_cong__exp__iff__simps_I4_J,axiom,
% 0.32/0.56 ! [M: num,N: num] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
% 0.32/0.56 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(4)
% 0.32/0.56 thf(fact_3645_cong__exp__iff__simps_I4_J,axiom,
% 0.32/0.56 ! [M: num,N: num] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
% 0.32/0.56 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(4)
% 0.32/0.56 thf(fact_3646_cong__exp__iff__simps_I4_J,axiom,
% 0.32/0.56 ! [M: num,N: num] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
% 0.32/0.56 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(4)
% 0.32/0.56 thf(fact_3647_mod__0__imp__dvd,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( ( modulo_modulo_nat @ A @ B )
% 0.32/0.56 = zero_zero_nat )
% 0.32/0.56 => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_0_imp_dvd
% 0.32/0.56 thf(fact_3648_mod__0__imp__dvd,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( ( modulo_modulo_int @ A @ B )
% 0.32/0.56 = zero_zero_int )
% 0.32/0.56 => ( dvd_dvd_int @ B @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_0_imp_dvd
% 0.32/0.56 thf(fact_3649_mod__0__imp__dvd,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.32/0.56 = zero_z3403309356797280102nteger )
% 0.32/0.56 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_0_imp_dvd
% 0.32/0.56 thf(fact_3650_mod__0__imp__dvd,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural] :
% 0.32/0.56 ( ( ( modulo8411746178871703098atural @ A @ B )
% 0.32/0.56 = zero_z2226904508553997617atural )
% 0.32/0.56 => ( dvd_dvd_Code_natural @ B @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_0_imp_dvd
% 0.32/0.56 thf(fact_3651_dvd__eq__mod__eq__0,axiom,
% 0.32/0.56 ( dvd_dvd_nat
% 0.32/0.56 = ( ^ [A2: nat,B2: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ B2 @ A2 )
% 0.32/0.56 = zero_zero_nat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_eq_mod_eq_0
% 0.32/0.56 thf(fact_3652_dvd__eq__mod__eq__0,axiom,
% 0.32/0.56 ( dvd_dvd_int
% 0.32/0.56 = ( ^ [A2: int,B2: int] :
% 0.32/0.56 ( ( modulo_modulo_int @ B2 @ A2 )
% 0.32/0.56 = zero_zero_int ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_eq_mod_eq_0
% 0.32/0.56 thf(fact_3653_dvd__eq__mod__eq__0,axiom,
% 0.32/0.56 ( dvd_dvd_Code_integer
% 0.32/0.56 = ( ^ [A2: code_integer,B2: code_integer] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ B2 @ A2 )
% 0.32/0.56 = zero_z3403309356797280102nteger ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_eq_mod_eq_0
% 0.32/0.56 thf(fact_3654_dvd__eq__mod__eq__0,axiom,
% 0.32/0.56 ( dvd_dvd_Code_natural
% 0.32/0.56 = ( ^ [A2: code_natural,B2: code_natural] :
% 0.32/0.56 ( ( modulo8411746178871703098atural @ B2 @ A2 )
% 0.32/0.56 = zero_z2226904508553997617atural ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_eq_mod_eq_0
% 0.32/0.56 thf(fact_3655_mod__eq__0__iff__dvd,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( ( modulo_modulo_nat @ A @ B )
% 0.32/0.56 = zero_zero_nat )
% 0.32/0.56 = ( dvd_dvd_nat @ B @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_eq_0_iff_dvd
% 0.32/0.56 thf(fact_3656_mod__eq__0__iff__dvd,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( ( modulo_modulo_int @ A @ B )
% 0.32/0.56 = zero_zero_int )
% 0.32/0.56 = ( dvd_dvd_int @ B @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_eq_0_iff_dvd
% 0.32/0.56 thf(fact_3657_mod__eq__0__iff__dvd,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.32/0.56 = zero_z3403309356797280102nteger )
% 0.32/0.56 = ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_eq_0_iff_dvd
% 0.32/0.56 thf(fact_3658_mod__eq__0__iff__dvd,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural] :
% 0.32/0.56 ( ( ( modulo8411746178871703098atural @ A @ B )
% 0.32/0.56 = zero_z2226904508553997617atural )
% 0.32/0.56 = ( dvd_dvd_Code_natural @ B @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_eq_0_iff_dvd
% 0.32/0.56 thf(fact_3659_div__add1__eq,axiom,
% 0.32/0.56 ! [A: nat,B: nat,C: nat] :
% 0.32/0.56 ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.56 = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_add1_eq
% 0.32/0.56 thf(fact_3660_div__add1__eq,axiom,
% 0.32/0.56 ! [A: int,B: int,C: int] :
% 0.32/0.56 ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.56 = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_add1_eq
% 0.32/0.56 thf(fact_3661_div__add1__eq,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.56 ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.32/0.56 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_add1_eq
% 0.32/0.56 thf(fact_3662_div__add1__eq,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural,C: code_natural] :
% 0.32/0.56 ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
% 0.32/0.56 = ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ C ) @ ( divide5121882707175180666atural @ B @ C ) ) @ ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C ) @ ( modulo8411746178871703098atural @ B @ C ) ) @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_add1_eq
% 0.32/0.56 thf(fact_3663_dvd__minus__mod,axiom,
% 0.32/0.56 ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_minus_mod
% 0.32/0.56 thf(fact_3664_dvd__minus__mod,axiom,
% 0.32/0.56 ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_minus_mod
% 0.32/0.56 thf(fact_3665_dvd__minus__mod,axiom,
% 0.32/0.56 ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_minus_mod
% 0.32/0.56 thf(fact_3666_dvd__minus__mod,axiom,
% 0.32/0.56 ! [B: code_natural,A: code_natural] : ( dvd_dvd_Code_natural @ B @ ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_minus_mod
% 0.32/0.56 thf(fact_3667_mod__eq__dvd__iff,axiom,
% 0.32/0.56 ! [A: int,C: int,B: int] :
% 0.32/0.56 ( ( ( modulo_modulo_int @ A @ C )
% 0.32/0.56 = ( modulo_modulo_int @ B @ C ) )
% 0.32/0.56 = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_eq_dvd_iff
% 0.32/0.56 thf(fact_3668_mod__eq__dvd__iff,axiom,
% 0.32/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.56 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.32/0.56 = ( modulo364778990260209775nteger @ B @ C ) )
% 0.32/0.56 = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_eq_dvd_iff
% 0.32/0.56 thf(fact_3669_mod__Suc,axiom,
% 0.32/0.56 ! [M: nat,N: nat] :
% 0.32/0.56 ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.32/0.56 = N )
% 0.32/0.56 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.32/0.56 = zero_zero_nat ) )
% 0.32/0.56 & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.32/0.56 != N )
% 0.32/0.56 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.32/0.56 = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_Suc
% 0.32/0.56 thf(fact_3670_mod__induct,axiom,
% 0.32/0.56 ! [P: nat > $o,N: nat,P2: nat,M: nat] :
% 0.32/0.56 ( ( P @ N )
% 0.32/0.56 => ( ( ord_less_nat @ N @ P2 )
% 0.32/0.56 => ( ( ord_less_nat @ M @ P2 )
% 0.32/0.56 => ( ! [N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ N2 @ P2 )
% 0.32/0.56 => ( ( P @ N2 )
% 0.32/0.56 => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P2 ) ) ) )
% 0.32/0.56 => ( P @ M ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_induct
% 0.32/0.56 thf(fact_3671_gcd__nat__induct,axiom,
% 0.32/0.56 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.32/0.56 ( ! [M2: nat] : ( P @ M2 @ zero_zero_nat )
% 0.32/0.56 => ( ! [M2: nat,N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ( P @ N2 @ ( modulo_modulo_nat @ M2 @ N2 ) )
% 0.32/0.56 => ( P @ M2 @ N2 ) ) )
% 0.32/0.56 => ( P @ M @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % gcd_nat_induct
% 0.32/0.56 thf(fact_3672_mod__less__divisor,axiom,
% 0.32/0.56 ! [N: nat,M: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_less_divisor
% 0.32/0.56 thf(fact_3673_mod__Suc__le__divisor,axiom,
% 0.32/0.56 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 0.32/0.56
% 0.32/0.56 % mod_Suc_le_divisor
% 0.32/0.56 thf(fact_3674_mod__eq__0D,axiom,
% 0.32/0.56 ! [M: nat,D: nat] :
% 0.32/0.56 ( ( ( modulo_modulo_nat @ M @ D )
% 0.32/0.56 = zero_zero_nat )
% 0.32/0.56 => ? [Q6: nat] :
% 0.32/0.56 ( M
% 0.32/0.56 = ( times_times_nat @ D @ Q6 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_eq_0D
% 0.32/0.56 thf(fact_3675_mod__if,axiom,
% 0.32/0.56 ( modulo_modulo_nat
% 0.32/0.56 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N3 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_if
% 0.32/0.56 thf(fact_3676_mod__geq,axiom,
% 0.32/0.56 ! [M: nat,N: nat] :
% 0.32/0.56 ( ~ ( ord_less_nat @ M @ N )
% 0.32/0.56 => ( ( modulo_modulo_nat @ M @ N )
% 0.32/0.56 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_geq
% 0.32/0.56 thf(fact_3677_le__mod__geq,axiom,
% 0.32/0.56 ! [N: nat,M: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.56 => ( ( modulo_modulo_nat @ M @ N )
% 0.32/0.56 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % le_mod_geq
% 0.32/0.56 thf(fact_3678_zmod__eq__0__iff,axiom,
% 0.32/0.56 ! [M: int,D: int] :
% 0.32/0.56 ( ( ( modulo_modulo_int @ M @ D )
% 0.32/0.56 = zero_zero_int )
% 0.32/0.56 = ( ? [Q5: int] :
% 0.32/0.56 ( M
% 0.32/0.56 = ( times_times_int @ D @ Q5 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zmod_eq_0_iff
% 0.32/0.56 thf(fact_3679_zmod__eq__0D,axiom,
% 0.32/0.56 ! [M: int,D: int] :
% 0.32/0.56 ( ( ( modulo_modulo_int @ M @ D )
% 0.32/0.56 = zero_zero_int )
% 0.32/0.56 => ? [Q6: int] :
% 0.32/0.56 ( M
% 0.32/0.56 = ( times_times_int @ D @ Q6 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zmod_eq_0D
% 0.32/0.56 thf(fact_3680_nat__mod__eq__iff,axiom,
% 0.32/0.56 ! [X: nat,N: nat,Y: nat] :
% 0.32/0.56 ( ( ( modulo_modulo_nat @ X @ N )
% 0.32/0.56 = ( modulo_modulo_nat @ Y @ N ) )
% 0.32/0.56 = ( ? [Q1: nat,Q22: nat] :
% 0.32/0.56 ( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
% 0.32/0.56 = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nat_mod_eq_iff
% 0.32/0.56 thf(fact_3681_mod__pos__neg__trivial,axiom,
% 0.32/0.56 ! [K: int,L: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.56 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 0.32/0.56 => ( ( modulo_modulo_int @ K @ L )
% 0.32/0.56 = ( plus_plus_int @ K @ L ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_pos_neg_trivial
% 0.32/0.56 thf(fact_3682_mod__pos__geq,axiom,
% 0.32/0.56 ! [L: int,K: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ L )
% 0.32/0.56 => ( ( ord_less_eq_int @ L @ K )
% 0.32/0.56 => ( ( modulo_modulo_int @ K @ L )
% 0.32/0.56 = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_pos_geq
% 0.32/0.56 thf(fact_3683_zdiv__mono__strict,axiom,
% 0.32/0.56 ! [A3: int,B5: int,N: int] :
% 0.32/0.56 ( ( ord_less_int @ A3 @ B5 )
% 0.32/0.56 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.32/0.56 => ( ( ( modulo_modulo_int @ A3 @ N )
% 0.32/0.56 = zero_zero_int )
% 0.32/0.56 => ( ( ( modulo_modulo_int @ B5 @ N )
% 0.32/0.56 = zero_zero_int )
% 0.32/0.56 => ( ord_less_int @ ( divide_divide_int @ A3 @ N ) @ ( divide_divide_int @ B5 @ N ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zdiv_mono_strict
% 0.32/0.56 thf(fact_3684_mod__int__pos__iff,axiom,
% 0.32/0.56 ! [K: int,L: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
% 0.32/0.56 = ( ( dvd_dvd_int @ L @ K )
% 0.32/0.56 | ( ( L = zero_zero_int )
% 0.32/0.56 & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 0.32/0.56 | ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_int_pos_iff
% 0.32/0.56 thf(fact_3685_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.56 => ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.32/0.56 => ( ( modulo364778990260209775nteger @ A @ B )
% 0.32/0.56 = A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unique_euclidean_semiring_numeral_class.mod_less
% 0.32/0.56 thf(fact_3686_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.56 => ( ( ord_less_nat @ A @ B )
% 0.32/0.56 => ( ( modulo_modulo_nat @ A @ B )
% 0.32/0.56 = A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unique_euclidean_semiring_numeral_class.mod_less
% 0.32/0.56 thf(fact_3687_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.56 => ( ( ord_less_int @ A @ B )
% 0.32/0.56 => ( ( modulo_modulo_int @ A @ B )
% 0.32/0.56 = A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unique_euclidean_semiring_numeral_class.mod_less
% 0.32/0.56 thf(fact_3688_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.32/0.56 ! [B: code_integer,A: code_integer] :
% 0.32/0.56 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.32/0.56 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.32/0.56 thf(fact_3689_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.32/0.56 ! [B: nat,A: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.56 => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.32/0.56 thf(fact_3690_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.32/0.56 ! [B: int,A: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.56 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.32/0.56 thf(fact_3691_cong__exp__iff__simps_I2_J,axiom,
% 0.32/0.56 ! [N: num,Q2: num] :
% 0.32/0.56 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.32/0.56 = zero_zero_nat )
% 0.32/0.56 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.32/0.56 = zero_zero_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(2)
% 0.32/0.56 thf(fact_3692_cong__exp__iff__simps_I2_J,axiom,
% 0.32/0.56 ! [N: num,Q2: num] :
% 0.32/0.56 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.32/0.56 = zero_zero_int )
% 0.32/0.56 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 0.32/0.56 = zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(2)
% 0.32/0.56 thf(fact_3693_cong__exp__iff__simps_I2_J,axiom,
% 0.32/0.56 ! [N: num,Q2: num] :
% 0.32/0.56 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.32/0.56 = zero_z3403309356797280102nteger )
% 0.32/0.56 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.32/0.56 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(2)
% 0.32/0.56 thf(fact_3694_cong__exp__iff__simps_I1_J,axiom,
% 0.32/0.56 ! [N: num] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 0.32/0.56 = zero_zero_nat ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(1)
% 0.32/0.56 thf(fact_3695_cong__exp__iff__simps_I1_J,axiom,
% 0.32/0.56 ! [N: num] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 0.32/0.56 = zero_zero_int ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(1)
% 0.32/0.56 thf(fact_3696_cong__exp__iff__simps_I1_J,axiom,
% 0.32/0.56 ! [N: num] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 0.32/0.56 = zero_z3403309356797280102nteger ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(1)
% 0.32/0.56 thf(fact_3697_cong__exp__iff__simps_I8_J,axiom,
% 0.32/0.56 ! [M: num,Q2: num] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.32/0.56 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(8)
% 0.32/0.56 thf(fact_3698_cong__exp__iff__simps_I8_J,axiom,
% 0.32/0.56 ! [M: num,Q2: num] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.32/0.56 != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(8)
% 0.32/0.56 thf(fact_3699_cong__exp__iff__simps_I8_J,axiom,
% 0.32/0.56 ! [M: num,Q2: num] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.32/0.56 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(8)
% 0.32/0.56 thf(fact_3700_cong__exp__iff__simps_I6_J,axiom,
% 0.32/0.56 ! [Q2: num,N: num] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.32/0.56 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(6)
% 0.32/0.56 thf(fact_3701_cong__exp__iff__simps_I6_J,axiom,
% 0.32/0.56 ! [Q2: num,N: num] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.32/0.56 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(6)
% 0.32/0.56 thf(fact_3702_cong__exp__iff__simps_I6_J,axiom,
% 0.32/0.56 ! [Q2: num,N: num] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.32/0.56 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % cong_exp_iff_simps(6)
% 0.32/0.56 thf(fact_3703_cancel__div__mod__rules_I2_J,axiom,
% 0.32/0.56 ! [B: nat,A: nat,C: nat] :
% 0.32/0.56 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 0.32/0.56 = ( plus_plus_nat @ A @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % cancel_div_mod_rules(2)
% 0.32/0.56 thf(fact_3704_cancel__div__mod__rules_I2_J,axiom,
% 0.32/0.56 ! [B: int,A: int,C: int] :
% 0.32/0.56 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 0.32/0.56 = ( plus_plus_int @ A @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % cancel_div_mod_rules(2)
% 0.32/0.56 thf(fact_3705_cancel__div__mod__rules_I2_J,axiom,
% 0.32/0.56 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.56 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 0.32/0.56 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % cancel_div_mod_rules(2)
% 0.32/0.56 thf(fact_3706_cancel__div__mod__rules_I2_J,axiom,
% 0.32/0.56 ! [B: code_natural,A: code_natural,C: code_natural] :
% 0.32/0.56 ( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C )
% 0.32/0.56 = ( plus_p4538020629002901425atural @ A @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % cancel_div_mod_rules(2)
% 0.32/0.56 thf(fact_3707_cancel__div__mod__rules_I1_J,axiom,
% 0.32/0.56 ! [A: nat,B: nat,C: nat] :
% 0.32/0.56 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 0.32/0.56 = ( plus_plus_nat @ A @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % cancel_div_mod_rules(1)
% 0.32/0.56 thf(fact_3708_cancel__div__mod__rules_I1_J,axiom,
% 0.32/0.56 ! [A: int,B: int,C: int] :
% 0.32/0.56 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 0.32/0.56 = ( plus_plus_int @ A @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % cancel_div_mod_rules(1)
% 0.32/0.56 thf(fact_3709_cancel__div__mod__rules_I1_J,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.56 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 0.32/0.56 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % cancel_div_mod_rules(1)
% 0.32/0.56 thf(fact_3710_cancel__div__mod__rules_I1_J,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural,C: code_natural] :
% 0.32/0.56 ( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C )
% 0.32/0.56 = ( plus_p4538020629002901425atural @ A @ C ) ) ).
% 0.32/0.56
% 0.32/0.56 % cancel_div_mod_rules(1)
% 0.32/0.56 thf(fact_3711_mod__div__decomp,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( A
% 0.32/0.56 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_div_decomp
% 0.32/0.56 thf(fact_3712_mod__div__decomp,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( A
% 0.32/0.56 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_div_decomp
% 0.32/0.56 thf(fact_3713_mod__div__decomp,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( A
% 0.32/0.56 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_div_decomp
% 0.32/0.56 thf(fact_3714_mod__div__decomp,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural] :
% 0.32/0.56 ( A
% 0.32/0.56 = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_div_decomp
% 0.32/0.56 thf(fact_3715_div__mult__mod__eq,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % div_mult_mod_eq
% 0.32/0.56 thf(fact_3716_div__mult__mod__eq,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % div_mult_mod_eq
% 0.32/0.56 thf(fact_3717_div__mult__mod__eq,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % div_mult_mod_eq
% 0.32/0.56 thf(fact_3718_div__mult__mod__eq,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural] :
% 0.32/0.56 ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % div_mult_mod_eq
% 0.32/0.56 thf(fact_3719_mod__div__mult__eq,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % mod_div_mult_eq
% 0.32/0.56 thf(fact_3720_mod__div__mult__eq,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % mod_div_mult_eq
% 0.32/0.56 thf(fact_3721_mod__div__mult__eq,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % mod_div_mult_eq
% 0.32/0.56 thf(fact_3722_mod__div__mult__eq,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural] :
% 0.32/0.56 ( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ B ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % mod_div_mult_eq
% 0.32/0.56 thf(fact_3723_mod__mult__div__eq,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_div_eq
% 0.32/0.56 thf(fact_3724_mod__mult__div__eq,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_div_eq
% 0.32/0.56 thf(fact_3725_mod__mult__div__eq,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_div_eq
% 0.32/0.56 thf(fact_3726_mod__mult__div__eq,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural] :
% 0.32/0.56 ( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ B ) @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % mod_mult_div_eq
% 0.32/0.56 thf(fact_3727_mult__div__mod__eq,axiom,
% 0.32/0.56 ! [B: nat,A: nat] :
% 0.32/0.56 ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % mult_div_mod_eq
% 0.32/0.56 thf(fact_3728_mult__div__mod__eq,axiom,
% 0.32/0.56 ! [B: int,A: int] :
% 0.32/0.56 ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % mult_div_mod_eq
% 0.32/0.56 thf(fact_3729_mult__div__mod__eq,axiom,
% 0.32/0.56 ! [B: code_integer,A: code_integer] :
% 0.32/0.56 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % mult_div_mod_eq
% 0.32/0.56 thf(fact_3730_mult__div__mod__eq,axiom,
% 0.32/0.56 ! [B: code_natural,A: code_natural] :
% 0.32/0.56 ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % mult_div_mod_eq
% 0.32/0.56 thf(fact_3731_div__mult1__eq,axiom,
% 0.32/0.56 ! [A: nat,B: nat,C: nat] :
% 0.32/0.56 ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.56 = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_mult1_eq
% 0.32/0.56 thf(fact_3732_div__mult1__eq,axiom,
% 0.32/0.56 ! [A: int,B: int,C: int] :
% 0.32/0.56 ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.56 = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_mult1_eq
% 0.32/0.56 thf(fact_3733_div__mult1__eq,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.56 ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.32/0.56 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_mult1_eq
% 0.32/0.56 thf(fact_3734_div__mult1__eq,axiom,
% 0.32/0.56 ! [A: code_natural,B: code_natural,C: code_natural] :
% 0.32/0.57 ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C )
% 0.32/0.57 = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C ) ) @ ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_mult1_eq
% 0.32/0.57 thf(fact_3735_unit__imp__mod__eq__0,axiom,
% 0.32/0.57 ! [B: nat,A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.57 => ( ( modulo_modulo_nat @ A @ B )
% 0.32/0.57 = zero_zero_nat ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_imp_mod_eq_0
% 0.32/0.57 thf(fact_3736_unit__imp__mod__eq__0,axiom,
% 0.32/0.57 ! [B: int,A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.57 => ( ( modulo_modulo_int @ A @ B )
% 0.32/0.57 = zero_zero_int ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_imp_mod_eq_0
% 0.32/0.57 thf(fact_3737_unit__imp__mod__eq__0,axiom,
% 0.32/0.57 ! [B: code_integer,A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.57 => ( ( modulo364778990260209775nteger @ A @ B )
% 0.32/0.57 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_imp_mod_eq_0
% 0.32/0.57 thf(fact_3738_unit__imp__mod__eq__0,axiom,
% 0.32/0.57 ! [B: code_natural,A: code_natural] :
% 0.32/0.57 ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
% 0.32/0.57 => ( ( modulo8411746178871703098atural @ A @ B )
% 0.32/0.57 = zero_z2226904508553997617atural ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_imp_mod_eq_0
% 0.32/0.57 thf(fact_3739_minus__mult__div__eq__mod,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 0.32/0.57 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % minus_mult_div_eq_mod
% 0.32/0.57 thf(fact_3740_minus__mult__div__eq__mod,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 0.32/0.57 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % minus_mult_div_eq_mod
% 0.32/0.57 thf(fact_3741_minus__mult__div__eq__mod,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer] :
% 0.32/0.57 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 0.32/0.57 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % minus_mult_div_eq_mod
% 0.32/0.57 thf(fact_3742_minus__mult__div__eq__mod,axiom,
% 0.32/0.57 ! [A: code_natural,B: code_natural] :
% 0.32/0.57 ( ( minus_7197305767214868737atural @ A @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) )
% 0.32/0.57 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % minus_mult_div_eq_mod
% 0.32/0.57 thf(fact_3743_minus__mod__eq__mult__div,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 0.32/0.57 = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minus_mod_eq_mult_div
% 0.32/0.57 thf(fact_3744_minus__mod__eq__mult__div,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 0.32/0.57 = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minus_mod_eq_mult_div
% 0.32/0.57 thf(fact_3745_minus__mod__eq__mult__div,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer] :
% 0.32/0.57 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.32/0.57 = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % minus_mod_eq_mult_div
% 0.33/0.57 thf(fact_3746_minus__mod__eq__mult__div,axiom,
% 0.33/0.57 ! [A: code_natural,B: code_natural] :
% 0.33/0.57 ( ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) )
% 0.33/0.57 = ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % minus_mod_eq_mult_div
% 0.33/0.57 thf(fact_3747_minus__mod__eq__div__mult,axiom,
% 0.33/0.57 ! [A: nat,B: nat] :
% 0.33/0.57 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 0.33/0.57 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% 0.33/0.57
% 0.33/0.57 % minus_mod_eq_div_mult
% 0.33/0.57 thf(fact_3748_minus__mod__eq__div__mult,axiom,
% 0.33/0.57 ! [A: int,B: int] :
% 0.33/0.57 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 0.33/0.57 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% 0.33/0.57
% 0.33/0.57 % minus_mod_eq_div_mult
% 0.33/0.57 thf(fact_3749_minus__mod__eq__div__mult,axiom,
% 0.33/0.57 ! [A: code_integer,B: code_integer] :
% 0.33/0.57 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.33/0.57 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% 0.33/0.57
% 0.33/0.57 % minus_mod_eq_div_mult
% 0.33/0.57 thf(fact_3750_minus__mod__eq__div__mult,axiom,
% 0.33/0.57 ! [A: code_natural,B: code_natural] :
% 0.33/0.57 ( ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) )
% 0.33/0.57 = ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) ) ).
% 0.33/0.57
% 0.33/0.57 % minus_mod_eq_div_mult
% 0.33/0.57 thf(fact_3751_minus__div__mult__eq__mod,axiom,
% 0.33/0.57 ! [A: nat,B: nat] :
% 0.33/0.57 ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 0.33/0.57 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.33/0.57
% 0.33/0.57 % minus_div_mult_eq_mod
% 0.33/0.57 thf(fact_3752_minus__div__mult__eq__mod,axiom,
% 0.33/0.57 ! [A: int,B: int] :
% 0.33/0.57 ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 0.33/0.57 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.33/0.57
% 0.33/0.57 % minus_div_mult_eq_mod
% 0.33/0.57 thf(fact_3753_minus__div__mult__eq__mod,axiom,
% 0.33/0.57 ! [A: code_integer,B: code_integer] :
% 0.33/0.57 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 0.33/0.57 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.33/0.57
% 0.33/0.57 % minus_div_mult_eq_mod
% 0.33/0.57 thf(fact_3754_minus__div__mult__eq__mod,axiom,
% 0.33/0.57 ! [A: code_natural,B: code_natural] :
% 0.33/0.57 ( ( minus_7197305767214868737atural @ A @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) )
% 0.33/0.57 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 0.33/0.57
% 0.33/0.57 % minus_div_mult_eq_mod
% 0.33/0.57 thf(fact_3755_mod__le__divisor,axiom,
% 0.33/0.57 ! [N: nat,M: nat] :
% 0.33/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.57 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod_le_divisor
% 0.33/0.57 thf(fact_3756_mod__greater__zero__iff__not__dvd,axiom,
% 0.33/0.57 ! [M: nat,N: nat] :
% 0.33/0.57 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 0.33/0.57 = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod_greater_zero_iff_not_dvd
% 0.33/0.57 thf(fact_3757_div__less__mono,axiom,
% 0.33/0.57 ! [A3: nat,B5: nat,N: nat] :
% 0.33/0.57 ( ( ord_less_nat @ A3 @ B5 )
% 0.33/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.57 => ( ( ( modulo_modulo_nat @ A3 @ N )
% 0.33/0.57 = zero_zero_nat )
% 0.33/0.57 => ( ( ( modulo_modulo_nat @ B5 @ N )
% 0.33/0.57 = zero_zero_nat )
% 0.33/0.57 => ( ord_less_nat @ ( divide_divide_nat @ A3 @ N ) @ ( divide_divide_nat @ B5 @ N ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % div_less_mono
% 0.33/0.57 thf(fact_3758_nat__mod__eq__lemma,axiom,
% 0.33/0.57 ! [X: nat,N: nat,Y: nat] :
% 0.33/0.57 ( ( ( modulo_modulo_nat @ X @ N )
% 0.33/0.57 = ( modulo_modulo_nat @ Y @ N ) )
% 0.33/0.57 => ( ( ord_less_eq_nat @ Y @ X )
% 0.33/0.57 => ? [Q6: nat] :
% 0.33/0.57 ( X
% 0.33/0.57 = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q6 ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % nat_mod_eq_lemma
% 0.33/0.57 thf(fact_3759_mod__eq__nat2E,axiom,
% 0.33/0.57 ! [M: nat,Q2: nat,N: nat] :
% 0.33/0.57 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.33/0.57 = ( modulo_modulo_nat @ N @ Q2 ) )
% 0.33/0.57 => ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.57 => ~ ! [S3: nat] :
% 0.33/0.57 ( N
% 0.33/0.57 != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod_eq_nat2E
% 0.33/0.57 thf(fact_3760_mod__eq__nat1E,axiom,
% 0.33/0.57 ! [M: nat,Q2: nat,N: nat] :
% 0.33/0.57 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.33/0.57 = ( modulo_modulo_nat @ N @ Q2 ) )
% 0.33/0.57 => ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.57 => ~ ! [S3: nat] :
% 0.33/0.57 ( M
% 0.33/0.57 != ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod_eq_nat1E
% 0.33/0.57 thf(fact_3761_mod__eq__dvd__iff__nat,axiom,
% 0.33/0.57 ! [N: nat,M: nat,Q2: nat] :
% 0.33/0.57 ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.57 => ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.33/0.57 = ( modulo_modulo_nat @ N @ Q2 ) )
% 0.33/0.57 = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod_eq_dvd_iff_nat
% 0.33/0.57 thf(fact_3762_split__zmod,axiom,
% 0.33/0.57 ! [P: int > $o,N: int,K: int] :
% 0.33/0.57 ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 0.33/0.57 = ( ( ( K = zero_zero_int )
% 0.33/0.57 => ( P @ N ) )
% 0.33/0.57 & ( ( ord_less_int @ zero_zero_int @ K )
% 0.33/0.57 => ! [I4: int,J3: int] :
% 0.33/0.57 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.33/0.57 & ( ord_less_int @ J3 @ K )
% 0.33/0.57 & ( N
% 0.33/0.57 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.33/0.57 => ( P @ J3 ) ) )
% 0.33/0.57 & ( ( ord_less_int @ K @ zero_zero_int )
% 0.33/0.57 => ! [I4: int,J3: int] :
% 0.33/0.57 ( ( ( ord_less_int @ K @ J3 )
% 0.33/0.57 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.33/0.57 & ( N
% 0.33/0.57 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.33/0.57 => ( P @ J3 ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % split_zmod
% 0.33/0.57 thf(fact_3763_int__mod__neg__eq,axiom,
% 0.33/0.57 ! [A: int,B: int,Q2: int,R: int] :
% 0.33/0.57 ( ( A
% 0.33/0.57 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.33/0.57 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 0.33/0.57 => ( ( ord_less_int @ B @ R )
% 0.33/0.57 => ( ( modulo_modulo_int @ A @ B )
% 0.33/0.57 = R ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % int_mod_neg_eq
% 0.33/0.57 thf(fact_3764_int__mod__pos__eq,axiom,
% 0.33/0.57 ! [A: int,B: int,Q2: int,R: int] :
% 0.33/0.57 ( ( A
% 0.33/0.57 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.33/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.33/0.57 => ( ( ord_less_int @ R @ B )
% 0.33/0.57 => ( ( modulo_modulo_int @ A @ B )
% 0.33/0.57 = R ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % int_mod_pos_eq
% 0.33/0.57 thf(fact_3765_div__mod__decomp,axiom,
% 0.33/0.57 ! [A3: nat,N: nat] :
% 0.33/0.57 ( A3
% 0.33/0.57 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A3 @ N ) @ N ) @ ( modulo_modulo_nat @ A3 @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % div_mod_decomp
% 0.33/0.57 thf(fact_3766_mod__mult2__eq,axiom,
% 0.33/0.57 ! [M: nat,N: nat,Q2: nat] :
% 0.33/0.57 ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 0.33/0.57 = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod_mult2_eq
% 0.33/0.57 thf(fact_3767_modulo__nat__def,axiom,
% 0.33/0.57 ( modulo_modulo_nat
% 0.33/0.57 = ( ^ [M3: nat,N3: nat] : ( minus_minus_nat @ M3 @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N3 ) @ N3 ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % modulo_nat_def
% 0.33/0.57 thf(fact_3768_div__mod__decomp__int,axiom,
% 0.33/0.57 ! [A3: int,N: int] :
% 0.33/0.57 ( A3
% 0.33/0.57 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A3 @ N ) @ N ) @ ( modulo_modulo_int @ A3 @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % div_mod_decomp_int
% 0.33/0.57 thf(fact_3769_mod__mult2__eq_H,axiom,
% 0.33/0.57 ! [A: code_integer,M: nat,N: nat] :
% 0.33/0.57 ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
% 0.33/0.57 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod_mult2_eq'
% 0.33/0.57 thf(fact_3770_mod__mult2__eq_H,axiom,
% 0.33/0.57 ! [A: code_natural,M: nat,N: nat] :
% 0.33/0.57 ( ( modulo8411746178871703098atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) )
% 0.33/0.57 = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( modulo8411746178871703098atural @ ( divide5121882707175180666atural @ A @ ( semiri3763490453095760265atural @ M ) ) @ ( semiri3763490453095760265atural @ N ) ) ) @ ( modulo8411746178871703098atural @ A @ ( semiri3763490453095760265atural @ M ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod_mult2_eq'
% 0.33/0.57 thf(fact_3771_mod__mult2__eq_H,axiom,
% 0.33/0.57 ! [A: nat,M: nat,N: nat] :
% 0.33/0.57 ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 0.33/0.57 = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod_mult2_eq'
% 0.33/0.57 thf(fact_3772_mod__mult2__eq_H,axiom,
% 0.33/0.57 ! [A: int,M: nat,N: nat] :
% 0.33/0.57 ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.33/0.57 = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod_mult2_eq'
% 0.33/0.57 thf(fact_3773_even__even__mod__4__iff,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.57 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % even_even_mod_4_iff
% 0.33/0.57 thf(fact_3774_field__char__0__class_Oof__nat__div,axiom,
% 0.33/0.57 ! [M: nat,N: nat] :
% 0.33/0.57 ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
% 0.33/0.57 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % field_char_0_class.of_nat_div
% 0.33/0.57 thf(fact_3775_field__char__0__class_Oof__nat__div,axiom,
% 0.33/0.57 ! [M: nat,N: nat] :
% 0.33/0.57 ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
% 0.33/0.57 = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % field_char_0_class.of_nat_div
% 0.33/0.57 thf(fact_3776_field__char__0__class_Oof__nat__div,axiom,
% 0.33/0.57 ! [M: nat,N: nat] :
% 0.33/0.57 ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
% 0.33/0.57 = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % field_char_0_class.of_nat_div
% 0.33/0.57 thf(fact_3777_split__mod,axiom,
% 0.33/0.57 ! [P: nat > $o,M: nat,N: nat] :
% 0.33/0.57 ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 0.33/0.57 = ( ( ( N = zero_zero_nat )
% 0.33/0.57 => ( P @ M ) )
% 0.33/0.57 & ( ( N != zero_zero_nat )
% 0.33/0.57 => ! [I4: nat,J3: nat] :
% 0.33/0.57 ( ( ord_less_nat @ J3 @ N )
% 0.33/0.57 => ( ( M
% 0.33/0.57 = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 0.33/0.57 => ( P @ J3 ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % split_mod
% 0.33/0.57 thf(fact_3778_mod__nat__eqI,axiom,
% 0.33/0.57 ! [R: nat,N: nat,M: nat] :
% 0.33/0.57 ( ( ord_less_nat @ R @ N )
% 0.33/0.57 => ( ( ord_less_eq_nat @ R @ M )
% 0.33/0.57 => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R ) )
% 0.33/0.57 => ( ( modulo_modulo_nat @ M @ N )
% 0.33/0.57 = R ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod_nat_eqI
% 0.33/0.57 thf(fact_3779_verit__le__mono__div__int,axiom,
% 0.33/0.57 ! [A3: int,B5: int,N: int] :
% 0.33/0.57 ( ( ord_less_int @ A3 @ B5 )
% 0.33/0.57 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.33/0.57 => ( ord_less_eq_int
% 0.33/0.57 @ ( plus_plus_int @ ( divide_divide_int @ A3 @ N )
% 0.33/0.57 @ ( if_int
% 0.33/0.57 @ ( ( modulo_modulo_int @ B5 @ N )
% 0.33/0.57 = zero_zero_int )
% 0.33/0.57 @ one_one_int
% 0.33/0.57 @ zero_zero_int ) )
% 0.33/0.57 @ ( divide_divide_int @ B5 @ N ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % verit_le_mono_div_int
% 0.33/0.57 thf(fact_3780_zmod__zmult2__eq,axiom,
% 0.33/0.57 ! [C: int,A: int,B: int] :
% 0.33/0.57 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.33/0.57 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 0.33/0.57 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % zmod_zmult2_eq
% 0.33/0.57 thf(fact_3781_split__neg__lemma,axiom,
% 0.33/0.57 ! [K: int,P: int > int > $o,N: int] :
% 0.33/0.57 ( ( ord_less_int @ K @ zero_zero_int )
% 0.33/0.57 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.33/0.57 = ( ! [I4: int,J3: int] :
% 0.33/0.57 ( ( ( ord_less_int @ K @ J3 )
% 0.33/0.57 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.33/0.57 & ( N
% 0.33/0.57 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.33/0.57 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % split_neg_lemma
% 0.33/0.57 thf(fact_3782_split__pos__lemma,axiom,
% 0.33/0.57 ! [K: int,P: int > int > $o,N: int] :
% 0.33/0.57 ( ( ord_less_int @ zero_zero_int @ K )
% 0.33/0.57 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.33/0.57 = ( ! [I4: int,J3: int] :
% 0.33/0.57 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.33/0.57 & ( ord_less_int @ J3 @ K )
% 0.33/0.57 & ( N
% 0.33/0.57 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.33/0.57 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % split_pos_lemma
% 0.33/0.57 thf(fact_3783_real__of__nat__div__aux,axiom,
% 0.33/0.57 ! [X: nat,D: nat] :
% 0.33/0.57 ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
% 0.33/0.57 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % real_of_nat_div_aux
% 0.33/0.57 thf(fact_3784_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.33/0.57 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.33/0.57 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.33/0.57 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.33/0.57 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.33/0.57 thf(fact_3785_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.33/0.57 ! [C: nat,A: nat,B: nat] :
% 0.33/0.57 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.33/0.57 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.33/0.57 = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.33/0.57 thf(fact_3786_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.33/0.57 ! [C: int,A: int,B: int] :
% 0.33/0.57 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.33/0.57 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 0.33/0.57 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.33/0.57 thf(fact_3787_even__iff__mod__2__eq__zero,axiom,
% 0.33/0.57 ! [A: nat] :
% 0.33/0.57 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.57 = zero_zero_nat ) ) ).
% 0.33/0.57
% 0.33/0.57 % even_iff_mod_2_eq_zero
% 0.33/0.57 thf(fact_3788_even__iff__mod__2__eq__zero,axiom,
% 0.33/0.57 ! [A: int] :
% 0.33/0.57 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.57 = zero_zero_int ) ) ).
% 0.33/0.57
% 0.33/0.57 % even_iff_mod_2_eq_zero
% 0.33/0.57 thf(fact_3789_even__iff__mod__2__eq__zero,axiom,
% 0.33/0.57 ! [A: code_integer] :
% 0.33/0.57 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.57 = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.57
% 0.33/0.57 % even_iff_mod_2_eq_zero
% 0.33/0.57 thf(fact_3790_even__iff__mod__2__eq__zero,axiom,
% 0.33/0.57 ! [A: code_natural] :
% 0.33/0.57 ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 0.33/0.57 = zero_z2226904508553997617atural ) ) ).
% 0.33/0.57
% 0.33/0.57 % even_iff_mod_2_eq_zero
% 0.33/0.57 thf(fact_3791_odd__iff__mod__2__eq__one,axiom,
% 0.33/0.57 ! [A: nat] :
% 0.33/0.57 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.33/0.57 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.57 = one_one_nat ) ) ).
% 0.33/0.57
% 0.33/0.57 % odd_iff_mod_2_eq_one
% 0.33/0.57 thf(fact_3792_odd__iff__mod__2__eq__one,axiom,
% 0.33/0.57 ! [A: int] :
% 0.33/0.57 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.33/0.57 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.57 = one_one_int ) ) ).
% 0.33/0.57
% 0.33/0.57 % odd_iff_mod_2_eq_one
% 0.33/0.57 thf(fact_3793_odd__iff__mod__2__eq__one,axiom,
% 0.33/0.57 ! [A: code_integer] :
% 0.33/0.57 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.33/0.57 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.57 = one_one_Code_integer ) ) ).
% 0.33/0.57
% 0.33/0.57 % odd_iff_mod_2_eq_one
% 0.33/0.57 thf(fact_3794_odd__iff__mod__2__eq__one,axiom,
% 0.33/0.57 ! [A: code_natural] :
% 0.33/0.57 ( ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) )
% 0.33/0.57 = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 0.33/0.57 = one_one_Code_natural ) ) ).
% 0.33/0.57
% 0.33/0.57 % odd_iff_mod_2_eq_one
% 0.33/0.57 thf(fact_3795_of__bool__odd__eq__mod__2,axiom,
% 0.33/0.57 ! [A: code_natural] :
% 0.33/0.57 ( ( zero_n8403883297036319079atural
% 0.33/0.57 @ ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) )
% 0.33/0.57 = ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % of_bool_odd_eq_mod_2
% 0.33/0.57 thf(fact_3796_of__bool__odd__eq__mod__2,axiom,
% 0.33/0.57 ! [A: nat] :
% 0.33/0.57 ( ( zero_n2687167440665602831ol_nat
% 0.33/0.57 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.33/0.57 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % of_bool_odd_eq_mod_2
% 0.33/0.57 thf(fact_3797_of__bool__odd__eq__mod__2,axiom,
% 0.33/0.57 ! [A: int] :
% 0.33/0.57 ( ( zero_n2684676970156552555ol_int
% 0.33/0.57 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.33/0.57 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % of_bool_odd_eq_mod_2
% 0.33/0.57 thf(fact_3798_of__bool__odd__eq__mod__2,axiom,
% 0.33/0.57 ! [A: code_integer] :
% 0.33/0.57 ( ( zero_n356916108424825756nteger
% 0.33/0.57 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.33/0.57 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % of_bool_odd_eq_mod_2
% 0.33/0.57 thf(fact_3799_Suc__times__mod__eq,axiom,
% 0.33/0.57 ! [M: nat,N: nat] :
% 0.33/0.57 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.33/0.57 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 0.33/0.57 = one_one_nat ) ) ).
% 0.33/0.57
% 0.33/0.57 % Suc_times_mod_eq
% 0.33/0.57 thf(fact_3800_divmod__digit__0_I2_J,axiom,
% 0.33/0.57 ! [B: nat,A: nat] :
% 0.33/0.57 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.33/0.57 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.33/0.57 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
% 0.33/0.57 = ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % divmod_digit_0(2)
% 0.33/0.57 thf(fact_3801_divmod__digit__0_I2_J,axiom,
% 0.33/0.57 ! [B: int,A: int] :
% 0.33/0.57 ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.57 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.33/0.57 => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
% 0.33/0.57 = ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % divmod_digit_0(2)
% 0.33/0.57 thf(fact_3802_divmod__digit__0_I2_J,axiom,
% 0.33/0.57 ! [B: code_integer,A: code_integer] :
% 0.33/0.57 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.33/0.57 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.33/0.57 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
% 0.33/0.57 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % divmod_digit_0(2)
% 0.33/0.57 thf(fact_3803_bits__stable__imp__add__self,axiom,
% 0.33/0.57 ! [A: nat] :
% 0.33/0.57 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.57 = A )
% 0.33/0.57 => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.57 = zero_zero_nat ) ) ).
% 0.33/0.57
% 0.33/0.57 % bits_stable_imp_add_self
% 0.33/0.57 thf(fact_3804_bits__stable__imp__add__self,axiom,
% 0.33/0.57 ! [A: int] :
% 0.33/0.57 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.57 = A )
% 0.33/0.57 => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.33/0.57 = zero_zero_int ) ) ).
% 0.33/0.57
% 0.33/0.57 % bits_stable_imp_add_self
% 0.33/0.57 thf(fact_3805_bits__stable__imp__add__self,axiom,
% 0.33/0.57 ! [A: code_integer] :
% 0.33/0.57 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.57 = A )
% 0.33/0.57 => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.33/0.57 = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.57
% 0.33/0.57 % bits_stable_imp_add_self
% 0.33/0.57 thf(fact_3806_bits__stable__imp__add__self,axiom,
% 0.33/0.57 ! [A: code_natural] :
% 0.33/0.57 ( ( ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 0.33/0.57 = A )
% 0.33/0.57 => ( ( plus_p4538020629002901425atural @ A @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
% 0.33/0.57 = zero_z2226904508553997617atural ) ) ).
% 0.33/0.57
% 0.33/0.57 % bits_stable_imp_add_self
% 0.33/0.57 thf(fact_3807_mod2__eq__if,axiom,
% 0.33/0.57 ! [A: nat] :
% 0.33/0.57 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.57 = zero_zero_nat ) )
% 0.33/0.57 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.57 = one_one_nat ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod2_eq_if
% 0.33/0.57 thf(fact_3808_mod2__eq__if,axiom,
% 0.33/0.57 ! [A: int] :
% 0.33/0.57 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.57 = zero_zero_int ) )
% 0.33/0.57 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.57 = one_one_int ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod2_eq_if
% 0.33/0.57 thf(fact_3809_mod2__eq__if,axiom,
% 0.33/0.57 ! [A: code_integer] :
% 0.33/0.57 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.57 = zero_z3403309356797280102nteger ) )
% 0.33/0.57 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.57 = one_one_Code_integer ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod2_eq_if
% 0.33/0.57 thf(fact_3810_mod2__eq__if,axiom,
% 0.33/0.57 ! [A: code_natural] :
% 0.33/0.57 ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 0.33/0.57 = zero_z2226904508553997617atural ) )
% 0.33/0.57 & ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 0.33/0.57 = one_one_Code_natural ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod2_eq_if
% 0.33/0.57 thf(fact_3811_parity__cases,axiom,
% 0.33/0.57 ! [A: nat] :
% 0.33/0.57 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.57 != zero_zero_nat ) )
% 0.33/0.57 => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.57 != one_one_nat ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % parity_cases
% 0.33/0.57 thf(fact_3812_parity__cases,axiom,
% 0.33/0.57 ! [A: int] :
% 0.33/0.57 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.57 != zero_zero_int ) )
% 0.33/0.57 => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.57 != one_one_int ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % parity_cases
% 0.33/0.57 thf(fact_3813_parity__cases,axiom,
% 0.33/0.57 ! [A: code_integer] :
% 0.33/0.57 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.57 != zero_z3403309356797280102nteger ) )
% 0.33/0.57 => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.57 != one_one_Code_integer ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % parity_cases
% 0.33/0.57 thf(fact_3814_parity__cases,axiom,
% 0.33/0.57 ! [A: code_natural] :
% 0.33/0.57 ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 0.33/0.57 != zero_z2226904508553997617atural ) )
% 0.33/0.57 => ~ ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
% 0.33/0.57 => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 0.33/0.57 != one_one_Code_natural ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % parity_cases
% 0.33/0.57 thf(fact_3815_div__exp__mod__exp__eq,axiom,
% 0.33/0.57 ! [A: nat,N: nat,M: nat] :
% 0.33/0.57 ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.57 = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % div_exp_mod_exp_eq
% 0.33/0.57 thf(fact_3816_div__exp__mod__exp__eq,axiom,
% 0.33/0.57 ! [A: int,N: nat,M: nat] :
% 0.33/0.57 ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.57 = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % div_exp_mod_exp_eq
% 0.33/0.57 thf(fact_3817_div__exp__mod__exp__eq,axiom,
% 0.33/0.57 ! [A: code_integer,N: nat,M: nat] :
% 0.33/0.57 ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.57 = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % div_exp_mod_exp_eq
% 0.33/0.57 thf(fact_3818_div__exp__mod__exp__eq,axiom,
% 0.33/0.57 ! [A: code_natural,N: nat,M: nat] :
% 0.33/0.57 ( ( modulo8411746178871703098atural @ ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.57 = ( divide5121882707175180666atural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % div_exp_mod_exp_eq
% 0.33/0.57 thf(fact_3819_verit__le__mono__div,axiom,
% 0.33/0.57 ! [A3: nat,B5: nat,N: nat] :
% 0.33/0.57 ( ( ord_less_nat @ A3 @ B5 )
% 0.33/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.57 => ( ord_less_eq_nat
% 0.33/0.57 @ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ N )
% 0.33/0.57 @ ( if_nat
% 0.33/0.57 @ ( ( modulo_modulo_nat @ B5 @ N )
% 0.33/0.57 = zero_zero_nat )
% 0.33/0.57 @ one_one_nat
% 0.33/0.57 @ zero_zero_nat ) )
% 0.33/0.57 @ ( divide_divide_nat @ B5 @ N ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % verit_le_mono_div
% 0.33/0.57 thf(fact_3820_divmod__digit__0_I1_J,axiom,
% 0.33/0.57 ! [B: nat,A: nat] :
% 0.33/0.57 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.33/0.57 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.33/0.57 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.33/0.57 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % divmod_digit_0(1)
% 0.33/0.57 thf(fact_3821_divmod__digit__0_I1_J,axiom,
% 0.33/0.57 ! [B: int,A: int] :
% 0.33/0.57 ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.57 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.33/0.57 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.33/0.57 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % divmod_digit_0(1)
% 0.33/0.57 thf(fact_3822_divmod__digit__0_I1_J,axiom,
% 0.33/0.57 ! [B: code_integer,A: code_integer] :
% 0.33/0.57 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.33/0.57 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.33/0.57 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.33/0.57 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % divmod_digit_0(1)
% 0.33/0.57 thf(fact_3823_mult__exp__mod__exp__eq,axiom,
% 0.33/0.57 ! [M: nat,N: nat,A: nat] :
% 0.33/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.57 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.57 = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mult_exp_mod_exp_eq
% 0.33/0.57 thf(fact_3824_mult__exp__mod__exp__eq,axiom,
% 0.33/0.57 ! [M: nat,N: nat,A: int] :
% 0.33/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.57 => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.57 = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mult_exp_mod_exp_eq
% 0.33/0.57 thf(fact_3825_mult__exp__mod__exp__eq,axiom,
% 0.33/0.57 ! [M: nat,N: nat,A: code_integer] :
% 0.33/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.57 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.57 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mult_exp_mod_exp_eq
% 0.33/0.57 thf(fact_3826_mult__exp__mod__exp__eq,axiom,
% 0.33/0.57 ! [M: nat,N: nat,A: code_natural] :
% 0.33/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.57 => ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.57 = ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mult_exp_mod_exp_eq
% 0.33/0.57 thf(fact_3827_exp__mod__exp,axiom,
% 0.33/0.57 ! [M: nat,N: nat] :
% 0.33/0.57 ( ( modulo8411746178871703098atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.57 = ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_nat @ M @ N ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % exp_mod_exp
% 0.33/0.57 thf(fact_3828_exp__mod__exp,axiom,
% 0.33/0.57 ! [M: nat,N: nat] :
% 0.33/0.57 ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.57 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % exp_mod_exp
% 0.33/0.57 thf(fact_3829_exp__mod__exp,axiom,
% 0.33/0.57 ! [M: nat,N: nat] :
% 0.33/0.57 ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.57 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % exp_mod_exp
% 0.33/0.57 thf(fact_3830_exp__mod__exp,axiom,
% 0.33/0.57 ! [M: nat,N: nat] :
% 0.33/0.57 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.57 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % exp_mod_exp
% 0.33/0.57 thf(fact_3831_pos__zmod__mult__2,axiom,
% 0.33/0.57 ! [A: int,B: int] :
% 0.33/0.57 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.57 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.33/0.57 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pos_zmod_mult_2
% 0.33/0.57 thf(fact_3832_Bolzano,axiom,
% 0.33/0.57 ! [A: real,B: real,P: real > real > $o] :
% 0.33/0.57 ( ( ord_less_eq_real @ A @ B )
% 0.33/0.57 => ( ! [A4: real,B4: real,C3: real] :
% 0.33/0.57 ( ( P @ A4 @ B4 )
% 0.33/0.57 => ( ( P @ B4 @ C3 )
% 0.33/0.57 => ( ( ord_less_eq_real @ A4 @ B4 )
% 0.33/0.57 => ( ( ord_less_eq_real @ B4 @ C3 )
% 0.33/0.57 => ( P @ A4 @ C3 ) ) ) ) )
% 0.33/0.57 => ( ! [X4: real] :
% 0.33/0.57 ( ( ord_less_eq_real @ A @ X4 )
% 0.33/0.57 => ( ( ord_less_eq_real @ X4 @ B )
% 0.33/0.57 => ? [D5: real] :
% 0.33/0.57 ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.33/0.57 & ! [A4: real,B4: real] :
% 0.33/0.57 ( ( ( ord_less_eq_real @ A4 @ X4 )
% 0.33/0.57 & ( ord_less_eq_real @ X4 @ B4 )
% 0.33/0.57 & ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D5 ) )
% 0.33/0.57 => ( P @ A4 @ B4 ) ) ) ) )
% 0.33/0.57 => ( P @ A @ B ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % Bolzano
% 0.33/0.57 thf(fact_3833_mod__double__modulus,axiom,
% 0.33/0.57 ! [M: code_integer,X: code_integer] :
% 0.33/0.57 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 0.33/0.57 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 0.33/0.57 => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.57 = ( modulo364778990260209775nteger @ X @ M ) )
% 0.33/0.57 | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.57 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod_double_modulus
% 0.33/0.57 thf(fact_3834_mod__double__modulus,axiom,
% 0.33/0.57 ! [M: nat,X: nat] :
% 0.33/0.57 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.57 => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.33/0.57 => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.57 = ( modulo_modulo_nat @ X @ M ) )
% 0.33/0.57 | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.57 = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod_double_modulus
% 0.33/0.57 thf(fact_3835_mod__double__modulus,axiom,
% 0.33/0.57 ! [M: int,X: int] :
% 0.33/0.57 ( ( ord_less_int @ zero_zero_int @ M )
% 0.33/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.57 => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.57 = ( modulo_modulo_int @ X @ M ) )
% 0.33/0.57 | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.57 = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mod_double_modulus
% 0.33/0.57 thf(fact_3836_divmod__digit__1_I2_J,axiom,
% 0.33/0.57 ! [A: code_integer,B: code_integer] :
% 0.33/0.57 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.33/0.57 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.33/0.57 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.33/0.57 => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.33/0.57 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % divmod_digit_1(2)
% 0.33/0.57 thf(fact_3837_divmod__digit__1_I2_J,axiom,
% 0.33/0.57 ! [A: nat,B: nat] :
% 0.33/0.57 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.57 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.33/0.57 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.33/0.57 => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.33/0.57 = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % divmod_digit_1(2)
% 0.33/0.57 thf(fact_3838_divmod__digit__1_I2_J,axiom,
% 0.33/0.57 ! [A: int,B: int] :
% 0.33/0.57 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.57 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.57 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.33/0.57 => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.33/0.57 = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % divmod_digit_1(2)
% 0.33/0.57 thf(fact_3839_unset__bit__Suc,axiom,
% 0.33/0.57 ! [N: nat,A: code_integer] :
% 0.33/0.57 ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
% 0.33/0.57 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % unset_bit_Suc
% 0.33/0.57 thf(fact_3840_unset__bit__Suc,axiom,
% 0.33/0.57 ! [N: nat,A: code_natural] :
% 0.33/0.57 ( ( bit_se7083795435491715335atural @ ( suc @ N ) @ A )
% 0.33/0.57 = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se7083795435491715335atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % unset_bit_Suc
% 0.33/0.57 thf(fact_3841_unset__bit__Suc,axiom,
% 0.33/0.57 ! [N: nat,A: int] :
% 0.33/0.57 ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
% 0.33/0.57 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % unset_bit_Suc
% 0.33/0.57 thf(fact_3842_unset__bit__Suc,axiom,
% 0.33/0.57 ! [N: nat,A: nat] :
% 0.33/0.57 ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
% 0.33/0.57 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % unset_bit_Suc
% 0.33/0.57 thf(fact_3843_set__bit__Suc,axiom,
% 0.33/0.57 ! [N: nat,A: code_integer] :
% 0.33/0.57 ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
% 0.33/0.57 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % set_bit_Suc
% 0.33/0.57 thf(fact_3844_set__bit__Suc,axiom,
% 0.33/0.57 ! [N: nat,A: code_natural] :
% 0.33/0.57 ( ( bit_se1617098188084679374atural @ ( suc @ N ) @ A )
% 0.33/0.57 = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se1617098188084679374atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % set_bit_Suc
% 0.33/0.57 thf(fact_3845_set__bit__Suc,axiom,
% 0.33/0.57 ! [N: nat,A: int] :
% 0.33/0.57 ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
% 0.33/0.57 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % set_bit_Suc
% 0.33/0.57 thf(fact_3846_set__bit__Suc,axiom,
% 0.33/0.57 ! [N: nat,A: nat] :
% 0.33/0.57 ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
% 0.33/0.57 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % set_bit_Suc
% 0.33/0.57 thf(fact_3847_flip__bit__Suc,axiom,
% 0.33/0.57 ! [N: nat,A: code_integer] :
% 0.33/0.57 ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
% 0.33/0.57 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % flip_bit_Suc
% 0.33/0.57 thf(fact_3848_flip__bit__Suc,axiom,
% 0.33/0.57 ! [N: nat,A: code_natural] :
% 0.33/0.57 ( ( bit_se168947363167071951atural @ ( suc @ N ) @ A )
% 0.33/0.57 = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se168947363167071951atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % flip_bit_Suc
% 0.33/0.57 thf(fact_3849_flip__bit__Suc,axiom,
% 0.33/0.57 ! [N: nat,A: int] :
% 0.33/0.57 ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
% 0.33/0.57 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % flip_bit_Suc
% 0.33/0.57 thf(fact_3850_flip__bit__Suc,axiom,
% 0.33/0.57 ! [N: nat,A: nat] :
% 0.33/0.57 ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
% 0.33/0.57 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % flip_bit_Suc
% 0.33/0.57 thf(fact_3851_even__mod__4__div__2,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.57 = ( suc @ zero_zero_nat ) )
% 0.33/0.57 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % even_mod_4_div_2
% 0.33/0.57 thf(fact_3852_neg__zmod__mult__2,axiom,
% 0.33/0.57 ! [A: int,B: int] :
% 0.33/0.57 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.33/0.57 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.33/0.57 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % neg_zmod_mult_2
% 0.33/0.57 thf(fact_3853_divmod__digit__1_I1_J,axiom,
% 0.33/0.57 ! [A: code_integer,B: code_integer] :
% 0.33/0.57 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.33/0.57 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.33/0.57 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.33/0.57 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
% 0.33/0.57 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % divmod_digit_1(1)
% 0.33/0.57 thf(fact_3854_divmod__digit__1_I1_J,axiom,
% 0.33/0.57 ! [A: nat,B: nat] :
% 0.33/0.57 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.57 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.33/0.57 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.33/0.57 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
% 0.33/0.57 = ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % divmod_digit_1(1)
% 0.33/0.57 thf(fact_3855_divmod__digit__1_I1_J,axiom,
% 0.33/0.57 ! [A: int,B: int] :
% 0.33/0.57 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.57 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.57 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.33/0.57 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
% 0.33/0.57 = ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % divmod_digit_1(1)
% 0.33/0.57 thf(fact_3856_mult__less__iff1,axiom,
% 0.33/0.57 ! [Z2: real,X: real,Y: real] :
% 0.33/0.57 ( ( ord_less_real @ zero_zero_real @ Z2 )
% 0.33/0.57 => ( ( ord_less_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
% 0.33/0.57 = ( ord_less_real @ X @ Y ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mult_less_iff1
% 0.33/0.57 thf(fact_3857_mult__less__iff1,axiom,
% 0.33/0.57 ! [Z2: rat,X: rat,Y: rat] :
% 0.33/0.57 ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 0.33/0.57 => ( ( ord_less_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
% 0.33/0.57 = ( ord_less_rat @ X @ Y ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mult_less_iff1
% 0.33/0.57 thf(fact_3858_mult__less__iff1,axiom,
% 0.33/0.57 ! [Z2: int,X: int,Y: int] :
% 0.33/0.57 ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.33/0.57 => ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
% 0.33/0.57 = ( ord_less_int @ X @ Y ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % mult_less_iff1
% 0.33/0.57 thf(fact_3859_norm__divide__numeral,axiom,
% 0.33/0.57 ! [A: real,W: num] :
% 0.33/0.57 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.57 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_divide_numeral
% 0.33/0.57 thf(fact_3860_norm__divide__numeral,axiom,
% 0.33/0.57 ! [A: complex,W: num] :
% 0.33/0.57 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 0.33/0.57 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_divide_numeral
% 0.33/0.57 thf(fact_3861_norm__mult__numeral2,axiom,
% 0.33/0.57 ! [A: real,W: num] :
% 0.33/0.57 ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.57 = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_mult_numeral2
% 0.33/0.57 thf(fact_3862_norm__mult__numeral2,axiom,
% 0.33/0.57 ! [A: complex,W: num] :
% 0.33/0.57 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 0.33/0.57 = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_mult_numeral2
% 0.33/0.57 thf(fact_3863_norm__mult__numeral1,axiom,
% 0.33/0.57 ! [W: num,A: real] :
% 0.33/0.57 ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 0.33/0.57 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_mult_numeral1
% 0.33/0.57 thf(fact_3864_norm__mult__numeral1,axiom,
% 0.33/0.57 ! [W: num,A: complex] :
% 0.33/0.57 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 0.33/0.57 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_mult_numeral1
% 0.33/0.57 thf(fact_3865_norm__le__zero__iff,axiom,
% 0.33/0.57 ! [X: real] :
% 0.33/0.57 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
% 0.33/0.57 = ( X = zero_zero_real ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_le_zero_iff
% 0.33/0.57 thf(fact_3866_norm__le__zero__iff,axiom,
% 0.33/0.57 ! [X: complex] :
% 0.33/0.57 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
% 0.33/0.57 = ( X = zero_zero_complex ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_le_zero_iff
% 0.33/0.57 thf(fact_3867_zero__less__norm__iff,axiom,
% 0.33/0.57 ! [X: real] :
% 0.33/0.57 ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
% 0.33/0.57 = ( X != zero_zero_real ) ) ).
% 0.33/0.57
% 0.33/0.57 % zero_less_norm_iff
% 0.33/0.57 thf(fact_3868_zero__less__norm__iff,axiom,
% 0.33/0.57 ! [X: complex] :
% 0.33/0.57 ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
% 0.33/0.57 = ( X != zero_zero_complex ) ) ).
% 0.33/0.57
% 0.33/0.57 % zero_less_norm_iff
% 0.33/0.57 thf(fact_3869_norm__of__nat,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ( real_V7735802525324610683m_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.33/0.57 = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_of_nat
% 0.33/0.57 thf(fact_3870_norm__of__nat,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ( real_V1022390504157884413omplex @ ( semiri8010041392384452111omplex @ N ) )
% 0.33/0.57 = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_of_nat
% 0.33/0.57 thf(fact_3871_norm__numeral,axiom,
% 0.33/0.57 ! [W: num] :
% 0.33/0.57 ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
% 0.33/0.57 = ( numeral_numeral_real @ W ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_numeral
% 0.33/0.57 thf(fact_3872_norm__numeral,axiom,
% 0.33/0.57 ! [W: num] :
% 0.33/0.57 ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.33/0.57 = ( numeral_numeral_real @ W ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_numeral
% 0.33/0.57 thf(fact_3873_norm__one,axiom,
% 0.33/0.57 ( ( real_V7735802525324610683m_real @ one_one_real )
% 0.33/0.57 = one_one_real ) ).
% 0.33/0.57
% 0.33/0.57 % norm_one
% 0.33/0.57 thf(fact_3874_norm__one,axiom,
% 0.33/0.57 ( ( real_V1022390504157884413omplex @ one_one_complex )
% 0.33/0.57 = one_one_real ) ).
% 0.33/0.57
% 0.33/0.57 % norm_one
% 0.33/0.57 thf(fact_3875_idiff__0,axiom,
% 0.33/0.57 ! [N: extended_enat] :
% 0.33/0.57 ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.33/0.57 = zero_z5237406670263579293d_enat ) ).
% 0.33/0.57
% 0.33/0.57 % idiff_0
% 0.33/0.57 thf(fact_3876_idiff__0__right,axiom,
% 0.33/0.57 ! [N: extended_enat] :
% 0.33/0.57 ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
% 0.33/0.57 = N ) ).
% 0.33/0.57
% 0.33/0.57 % idiff_0_right
% 0.33/0.57 thf(fact_3877_norm__zero,axiom,
% 0.33/0.57 ( ( real_V7735802525324610683m_real @ zero_zero_real )
% 0.33/0.57 = zero_zero_real ) ).
% 0.33/0.57
% 0.33/0.57 % norm_zero
% 0.33/0.57 thf(fact_3878_norm__zero,axiom,
% 0.33/0.57 ( ( real_V1022390504157884413omplex @ zero_zero_complex )
% 0.33/0.57 = zero_zero_real ) ).
% 0.33/0.57
% 0.33/0.57 % norm_zero
% 0.33/0.57 thf(fact_3879_norm__eq__zero,axiom,
% 0.33/0.57 ! [X: real] :
% 0.33/0.57 ( ( ( real_V7735802525324610683m_real @ X )
% 0.33/0.57 = zero_zero_real )
% 0.33/0.57 = ( X = zero_zero_real ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_eq_zero
% 0.33/0.57 thf(fact_3880_norm__eq__zero,axiom,
% 0.33/0.57 ! [X: complex] :
% 0.33/0.57 ( ( ( real_V1022390504157884413omplex @ X )
% 0.33/0.57 = zero_zero_real )
% 0.33/0.57 = ( X = zero_zero_complex ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_eq_zero
% 0.33/0.57 thf(fact_3881_zero__one__enat__neq_I1_J,axiom,
% 0.33/0.57 zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% 0.33/0.57
% 0.33/0.57 % zero_one_enat_neq(1)
% 0.33/0.57 thf(fact_3882_imult__is__0,axiom,
% 0.33/0.57 ! [M: extended_enat,N: extended_enat] :
% 0.33/0.57 ( ( ( times_7803423173614009249d_enat @ M @ N )
% 0.33/0.57 = zero_z5237406670263579293d_enat )
% 0.33/0.57 = ( ( M = zero_z5237406670263579293d_enat )
% 0.33/0.57 | ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % imult_is_0
% 0.33/0.57 thf(fact_3883_iadd__is__0,axiom,
% 0.33/0.57 ! [M: extended_enat,N: extended_enat] :
% 0.33/0.57 ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 0.33/0.57 = zero_z5237406670263579293d_enat )
% 0.33/0.57 = ( ( M = zero_z5237406670263579293d_enat )
% 0.33/0.57 & ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % iadd_is_0
% 0.33/0.57 thf(fact_3884_norm__minus__commute,axiom,
% 0.33/0.57 ! [A: real,B: real] :
% 0.33/0.57 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
% 0.33/0.57 = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_minus_commute
% 0.33/0.57 thf(fact_3885_norm__minus__commute,axiom,
% 0.33/0.57 ! [A: complex,B: complex] :
% 0.33/0.57 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
% 0.33/0.57 = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_minus_commute
% 0.33/0.57 thf(fact_3886_norm__mult,axiom,
% 0.33/0.57 ! [X: real,Y: real] :
% 0.33/0.57 ( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) )
% 0.33/0.57 = ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_mult
% 0.33/0.57 thf(fact_3887_norm__mult,axiom,
% 0.33/0.57 ! [X: complex,Y: complex] :
% 0.33/0.57 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) )
% 0.33/0.57 = ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_mult
% 0.33/0.57 thf(fact_3888_norm__divide,axiom,
% 0.33/0.57 ! [A: real,B: real] :
% 0.33/0.57 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.57 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_divide
% 0.33/0.57 thf(fact_3889_norm__divide,axiom,
% 0.33/0.57 ! [A: complex,B: complex] :
% 0.33/0.57 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.57 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_divide
% 0.33/0.57 thf(fact_3890_norm__power,axiom,
% 0.33/0.57 ! [X: real,N: nat] :
% 0.33/0.57 ( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) )
% 0.33/0.57 = ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_power
% 0.33/0.57 thf(fact_3891_norm__power,axiom,
% 0.33/0.57 ! [X: complex,N: nat] :
% 0.33/0.57 ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) )
% 0.33/0.57 = ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_power
% 0.33/0.57 thf(fact_3892_nonzero__norm__divide,axiom,
% 0.33/0.57 ! [B: real,A: real] :
% 0.33/0.57 ( ( B != zero_zero_real )
% 0.33/0.57 => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.57 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % nonzero_norm_divide
% 0.33/0.57 thf(fact_3893_nonzero__norm__divide,axiom,
% 0.33/0.57 ! [B: complex,A: complex] :
% 0.33/0.57 ( ( B != zero_zero_complex )
% 0.33/0.57 => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.57 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % nonzero_norm_divide
% 0.33/0.57 thf(fact_3894_power__eq__imp__eq__norm,axiom,
% 0.33/0.57 ! [W: real,N: nat,Z2: real] :
% 0.33/0.57 ( ( ( power_power_real @ W @ N )
% 0.33/0.57 = ( power_power_real @ Z2 @ N ) )
% 0.33/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.57 => ( ( real_V7735802525324610683m_real @ W )
% 0.33/0.57 = ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % power_eq_imp_eq_norm
% 0.33/0.57 thf(fact_3895_power__eq__imp__eq__norm,axiom,
% 0.33/0.57 ! [W: complex,N: nat,Z2: complex] :
% 0.33/0.57 ( ( ( power_power_complex @ W @ N )
% 0.33/0.57 = ( power_power_complex @ Z2 @ N ) )
% 0.33/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.57 => ( ( real_V1022390504157884413omplex @ W )
% 0.33/0.57 = ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % power_eq_imp_eq_norm
% 0.33/0.57 thf(fact_3896_norm__mult__less,axiom,
% 0.33/0.57 ! [X: real,R: real,Y: real,S: real] :
% 0.33/0.57 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R )
% 0.33/0.57 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
% 0.33/0.57 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R @ S ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_mult_less
% 0.33/0.57 thf(fact_3897_norm__mult__less,axiom,
% 0.33/0.57 ! [X: complex,R: real,Y: complex,S: real] :
% 0.33/0.57 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R )
% 0.33/0.57 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
% 0.33/0.57 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R @ S ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_mult_less
% 0.33/0.57 thf(fact_3898_norm__mult__ineq,axiom,
% 0.33/0.57 ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_mult_ineq
% 0.33/0.57 thf(fact_3899_norm__mult__ineq,axiom,
% 0.33/0.57 ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_mult_ineq
% 0.33/0.57 thf(fact_3900_norm__triangle__lt,axiom,
% 0.33/0.57 ! [X: real,Y: real,E: real] :
% 0.33/0.57 ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
% 0.33/0.57 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_triangle_lt
% 0.33/0.57 thf(fact_3901_norm__triangle__lt,axiom,
% 0.33/0.57 ! [X: complex,Y: complex,E: real] :
% 0.33/0.57 ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
% 0.33/0.57 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_triangle_lt
% 0.33/0.57 thf(fact_3902_norm__add__less,axiom,
% 0.33/0.57 ! [X: real,R: real,Y: real,S: real] :
% 0.33/0.57 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R )
% 0.33/0.57 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
% 0.33/0.57 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_add_less
% 0.33/0.57 thf(fact_3903_norm__add__less,axiom,
% 0.33/0.57 ! [X: complex,R: real,Y: complex,S: real] :
% 0.33/0.57 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R )
% 0.33/0.57 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
% 0.33/0.57 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_add_less
% 0.33/0.57 thf(fact_3904_norm__triangle__mono,axiom,
% 0.33/0.57 ! [A: real,R: real,B: real,S: real] :
% 0.33/0.57 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R )
% 0.33/0.57 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
% 0.33/0.57 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_triangle_mono
% 0.33/0.57 thf(fact_3905_norm__triangle__mono,axiom,
% 0.33/0.57 ! [A: complex,R: real,B: complex,S: real] :
% 0.33/0.57 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R )
% 0.33/0.57 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
% 0.33/0.57 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_triangle_mono
% 0.33/0.57 thf(fact_3906_norm__triangle__ineq,axiom,
% 0.33/0.57 ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_triangle_ineq
% 0.33/0.57 thf(fact_3907_norm__triangle__ineq,axiom,
% 0.33/0.57 ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_triangle_ineq
% 0.33/0.57 thf(fact_3908_norm__triangle__le,axiom,
% 0.33/0.57 ! [X: real,Y: real,E: real] :
% 0.33/0.57 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
% 0.33/0.57 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_triangle_le
% 0.33/0.57 thf(fact_3909_norm__triangle__le,axiom,
% 0.33/0.57 ! [X: complex,Y: complex,E: real] :
% 0.33/0.57 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
% 0.33/0.57 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_triangle_le
% 0.33/0.57 thf(fact_3910_norm__add__leD,axiom,
% 0.33/0.57 ! [A: real,B: real,C: real] :
% 0.33/0.57 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
% 0.33/0.57 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_add_leD
% 0.33/0.57 thf(fact_3911_norm__add__leD,axiom,
% 0.33/0.57 ! [A: complex,B: complex,C: real] :
% 0.33/0.57 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
% 0.33/0.57 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_add_leD
% 0.33/0.57 thf(fact_3912_norm__diff__triangle__less,axiom,
% 0.33/0.57 ! [X: real,Y: real,E1: real,Z2: real,E22: real] :
% 0.33/0.57 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
% 0.33/0.57 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z2 ) ) @ E22 )
% 0.33/0.57 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_diff_triangle_less
% 0.33/0.57 thf(fact_3913_norm__diff__triangle__less,axiom,
% 0.33/0.57 ! [X: complex,Y: complex,E1: real,Z2: complex,E22: real] :
% 0.33/0.57 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
% 0.33/0.57 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z2 ) ) @ E22 )
% 0.33/0.57 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_diff_triangle_less
% 0.33/0.57 thf(fact_3914_norm__power__ineq,axiom,
% 0.33/0.57 ! [X: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_power_ineq
% 0.33/0.57 thf(fact_3915_norm__power__ineq,axiom,
% 0.33/0.57 ! [X: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_power_ineq
% 0.33/0.57 thf(fact_3916_norm__triangle__le__diff,axiom,
% 0.33/0.57 ! [X: real,Y: real,E: real] :
% 0.33/0.57 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
% 0.33/0.57 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_triangle_le_diff
% 0.33/0.57 thf(fact_3917_norm__triangle__le__diff,axiom,
% 0.33/0.57 ! [X: complex,Y: complex,E: real] :
% 0.33/0.57 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
% 0.33/0.57 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_triangle_le_diff
% 0.33/0.57 thf(fact_3918_norm__diff__triangle__le,axiom,
% 0.33/0.57 ! [X: real,Y: real,E1: real,Z2: real,E22: real] :
% 0.33/0.57 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
% 0.33/0.57 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z2 ) ) @ E22 )
% 0.33/0.57 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_diff_triangle_le
% 0.33/0.57 thf(fact_3919_norm__diff__triangle__le,axiom,
% 0.33/0.57 ! [X: complex,Y: complex,E1: real,Z2: complex,E22: real] :
% 0.33/0.57 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
% 0.33/0.57 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z2 ) ) @ E22 )
% 0.33/0.57 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_diff_triangle_le
% 0.33/0.57 thf(fact_3920_norm__triangle__ineq4,axiom,
% 0.33/0.57 ! [A: real,B: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_triangle_ineq4
% 0.33/0.57 thf(fact_3921_norm__triangle__ineq4,axiom,
% 0.33/0.57 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_triangle_ineq4
% 0.33/0.57 thf(fact_3922_norm__triangle__sub,axiom,
% 0.33/0.57 ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_triangle_sub
% 0.33/0.57 thf(fact_3923_norm__triangle__sub,axiom,
% 0.33/0.57 ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_triangle_sub
% 0.33/0.57 thf(fact_3924_norm__diff__ineq,axiom,
% 0.33/0.57 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_diff_ineq
% 0.33/0.57 thf(fact_3925_norm__diff__ineq,axiom,
% 0.33/0.57 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_diff_ineq
% 0.33/0.57 thf(fact_3926_norm__triangle__ineq2,axiom,
% 0.33/0.57 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_triangle_ineq2
% 0.33/0.57 thf(fact_3927_norm__triangle__ineq2,axiom,
% 0.33/0.57 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_triangle_ineq2
% 0.33/0.57 thf(fact_3928_power__eq__1__iff,axiom,
% 0.33/0.57 ! [W: real,N: nat] :
% 0.33/0.57 ( ( ( power_power_real @ W @ N )
% 0.33/0.57 = one_one_real )
% 0.33/0.57 => ( ( ( real_V7735802525324610683m_real @ W )
% 0.33/0.57 = one_one_real )
% 0.33/0.57 | ( N = zero_zero_nat ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % power_eq_1_iff
% 0.33/0.57 thf(fact_3929_power__eq__1__iff,axiom,
% 0.33/0.57 ! [W: complex,N: nat] :
% 0.33/0.57 ( ( ( power_power_complex @ W @ N )
% 0.33/0.57 = one_one_complex )
% 0.33/0.57 => ( ( ( real_V1022390504157884413omplex @ W )
% 0.33/0.57 = one_one_real )
% 0.33/0.57 | ( N = zero_zero_nat ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % power_eq_1_iff
% 0.33/0.57 thf(fact_3930_norm__diff__triangle__ineq,axiom,
% 0.33/0.57 ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_diff_triangle_ineq
% 0.33/0.57 thf(fact_3931_norm__diff__triangle__ineq,axiom,
% 0.33/0.57 ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_diff_triangle_ineq
% 0.33/0.57 thf(fact_3932_square__norm__one,axiom,
% 0.33/0.57 ! [X: real] :
% 0.33/0.57 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.57 = one_one_real )
% 0.33/0.57 => ( ( real_V7735802525324610683m_real @ X )
% 0.33/0.57 = one_one_real ) ) ).
% 0.33/0.57
% 0.33/0.57 % square_norm_one
% 0.33/0.57 thf(fact_3933_square__norm__one,axiom,
% 0.33/0.57 ! [X: complex] :
% 0.33/0.57 ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.57 = one_one_complex )
% 0.33/0.57 => ( ( real_V1022390504157884413omplex @ X )
% 0.33/0.57 = one_one_real ) ) ).
% 0.33/0.57
% 0.33/0.57 % square_norm_one
% 0.33/0.57 thf(fact_3934_norm__power__diff,axiom,
% 0.33/0.57 ! [Z2: real,W: real,M: nat] :
% 0.33/0.57 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ one_one_real )
% 0.33/0.57 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
% 0.33/0.57 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z2 @ M ) @ ( power_power_real @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z2 @ W ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_power_diff
% 0.33/0.57 thf(fact_3935_norm__power__diff,axiom,
% 0.33/0.57 ! [Z2: complex,W: complex,M: nat] :
% 0.33/0.57 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
% 0.33/0.57 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
% 0.33/0.57 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z2 @ M ) @ ( power_power_complex @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z2 @ W ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % norm_power_diff
% 0.33/0.57 thf(fact_3936_arcosh__1,axiom,
% 0.33/0.57 ( ( arcosh_real @ one_one_real )
% 0.33/0.57 = zero_zero_real ) ).
% 0.33/0.57
% 0.33/0.57 % arcosh_1
% 0.33/0.57 thf(fact_3937_low__def,axiom,
% 0.33/0.57 ( vEBT_VEBT_low
% 0.33/0.57 = ( ^ [X3: nat,N3: nat] : ( modulo_modulo_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % low_def
% 0.33/0.57 thf(fact_3938_arsinh__0,axiom,
% 0.33/0.57 ( ( arsinh_real @ zero_zero_real )
% 0.33/0.57 = zero_zero_real ) ).
% 0.33/0.57
% 0.33/0.57 % arsinh_0
% 0.33/0.57 thf(fact_3939_artanh__0,axiom,
% 0.33/0.57 ( ( artanh_real @ zero_zero_real )
% 0.33/0.57 = zero_zero_real ) ).
% 0.33/0.57
% 0.33/0.57 % artanh_0
% 0.33/0.57 thf(fact_3940_pochhammer__double,axiom,
% 0.33/0.57 ! [Z2: complex,N: nat] :
% 0.33/0.57 ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.57 = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s2602460028002588243omplex @ Z2 @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_double
% 0.33/0.57 thf(fact_3941_pochhammer__double,axiom,
% 0.33/0.57 ! [Z2: real,N: nat] :
% 0.33/0.57 ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.57 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_double
% 0.33/0.57 thf(fact_3942_pochhammer__double,axiom,
% 0.33/0.57 ! [Z2: rat,N: nat] :
% 0.33/0.57 ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.57 = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_double
% 0.33/0.57 thf(fact_3943_central__binomial__lower__bound,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.57 => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % central_binomial_lower_bound
% 0.33/0.57 thf(fact_3944_pochhammer__0,axiom,
% 0.33/0.57 ! [A: complex] :
% 0.33/0.57 ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 0.33/0.57 = one_one_complex ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_0
% 0.33/0.57 thf(fact_3945_pochhammer__0,axiom,
% 0.33/0.57 ! [A: real] :
% 0.33/0.57 ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 0.33/0.57 = one_one_real ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_0
% 0.33/0.57 thf(fact_3946_pochhammer__0,axiom,
% 0.33/0.57 ! [A: rat] :
% 0.33/0.57 ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 0.33/0.57 = one_one_rat ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_0
% 0.33/0.57 thf(fact_3947_pochhammer__0,axiom,
% 0.33/0.57 ! [A: nat] :
% 0.33/0.57 ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 0.33/0.57 = one_one_nat ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_0
% 0.33/0.57 thf(fact_3948_pochhammer__0,axiom,
% 0.33/0.57 ! [A: int] :
% 0.33/0.57 ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 0.33/0.57 = one_one_int ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_0
% 0.33/0.57 thf(fact_3949_pochhammer__of__nat,axiom,
% 0.33/0.57 ! [X: nat,N: nat] :
% 0.33/0.57 ( ( comm_s2602460028002588243omplex @ ( semiri8010041392384452111omplex @ X ) @ N )
% 0.33/0.57 = ( semiri8010041392384452111omplex @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_of_nat
% 0.33/0.57 thf(fact_3950_pochhammer__of__nat,axiom,
% 0.33/0.57 ! [X: nat,N: nat] :
% 0.33/0.57 ( ( comm_s7457072308508201937r_real @ ( semiri5074537144036343181t_real @ X ) @ N )
% 0.33/0.57 = ( semiri5074537144036343181t_real @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_of_nat
% 0.33/0.57 thf(fact_3951_pochhammer__of__nat,axiom,
% 0.33/0.57 ! [X: nat,N: nat] :
% 0.33/0.57 ( ( comm_s4028243227959126397er_rat @ ( semiri681578069525770553at_rat @ X ) @ N )
% 0.33/0.57 = ( semiri681578069525770553at_rat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_of_nat
% 0.33/0.57 thf(fact_3952_pochhammer__of__nat,axiom,
% 0.33/0.57 ! [X: nat,N: nat] :
% 0.33/0.57 ( ( comm_s4663373288045622133er_nat @ ( semiri1316708129612266289at_nat @ X ) @ N )
% 0.33/0.57 = ( semiri1316708129612266289at_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_of_nat
% 0.33/0.57 thf(fact_3953_pochhammer__of__nat,axiom,
% 0.33/0.57 ! [X: nat,N: nat] :
% 0.33/0.57 ( ( comm_s4660882817536571857er_int @ ( semiri1314217659103216013at_int @ X ) @ N )
% 0.33/0.57 = ( semiri1314217659103216013at_int @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_of_nat
% 0.33/0.57 thf(fact_3954_pochhammer__pos,axiom,
% 0.33/0.57 ! [X: real,N: nat] :
% 0.33/0.57 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.57 => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_pos
% 0.33/0.57 thf(fact_3955_pochhammer__pos,axiom,
% 0.33/0.57 ! [X: rat,N: nat] :
% 0.33/0.57 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.33/0.57 => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_pos
% 0.33/0.57 thf(fact_3956_pochhammer__pos,axiom,
% 0.33/0.57 ! [X: nat,N: nat] :
% 0.33/0.57 ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.33/0.57 => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_pos
% 0.33/0.57 thf(fact_3957_pochhammer__pos,axiom,
% 0.33/0.57 ! [X: int,N: nat] :
% 0.33/0.57 ( ( ord_less_int @ zero_zero_int @ X )
% 0.33/0.57 => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_pos
% 0.33/0.57 thf(fact_3958_pochhammer__eq__0__mono,axiom,
% 0.33/0.57 ! [A: real,N: nat,M: nat] :
% 0.33/0.57 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.33/0.57 = zero_zero_real )
% 0.33/0.57 => ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.57 => ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.33/0.57 = zero_zero_real ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_eq_0_mono
% 0.33/0.57 thf(fact_3959_pochhammer__eq__0__mono,axiom,
% 0.33/0.57 ! [A: rat,N: nat,M: nat] :
% 0.33/0.57 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.33/0.57 = zero_zero_rat )
% 0.33/0.57 => ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.57 => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.33/0.57 = zero_zero_rat ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_eq_0_mono
% 0.33/0.57 thf(fact_3960_pochhammer__neq__0__mono,axiom,
% 0.33/0.57 ! [A: real,M: nat,N: nat] :
% 0.33/0.57 ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.33/0.57 != zero_zero_real )
% 0.33/0.57 => ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.57 => ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.33/0.57 != zero_zero_real ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_neq_0_mono
% 0.33/0.57 thf(fact_3961_pochhammer__neq__0__mono,axiom,
% 0.33/0.57 ! [A: rat,M: nat,N: nat] :
% 0.33/0.57 ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.33/0.57 != zero_zero_rat )
% 0.33/0.57 => ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.57 => ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.33/0.57 != zero_zero_rat ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_neq_0_mono
% 0.33/0.57 thf(fact_3962_pochhammer__nonneg,axiom,
% 0.33/0.57 ! [X: real,N: nat] :
% 0.33/0.57 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.57 => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_nonneg
% 0.33/0.57 thf(fact_3963_pochhammer__nonneg,axiom,
% 0.33/0.57 ! [X: rat,N: nat] :
% 0.33/0.57 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.33/0.57 => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_nonneg
% 0.33/0.57 thf(fact_3964_pochhammer__nonneg,axiom,
% 0.33/0.57 ! [X: nat,N: nat] :
% 0.33/0.57 ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.33/0.57 => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_nonneg
% 0.33/0.57 thf(fact_3965_pochhammer__nonneg,axiom,
% 0.33/0.57 ! [X: int,N: nat] :
% 0.33/0.57 ( ( ord_less_int @ zero_zero_int @ X )
% 0.33/0.57 => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_nonneg
% 0.33/0.57 thf(fact_3966_pochhammer__0__left,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ( ( N = zero_zero_nat )
% 0.33/0.57 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 0.33/0.57 = one_one_complex ) )
% 0.33/0.57 & ( ( N != zero_zero_nat )
% 0.33/0.57 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 0.33/0.57 = zero_zero_complex ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_0_left
% 0.33/0.57 thf(fact_3967_pochhammer__0__left,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ( ( N = zero_zero_nat )
% 0.33/0.57 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 0.33/0.57 = one_one_real ) )
% 0.33/0.57 & ( ( N != zero_zero_nat )
% 0.33/0.57 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 0.33/0.57 = zero_zero_real ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_0_left
% 0.33/0.57 thf(fact_3968_pochhammer__0__left,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ( ( N = zero_zero_nat )
% 0.33/0.57 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 0.33/0.57 = one_one_rat ) )
% 0.33/0.57 & ( ( N != zero_zero_nat )
% 0.33/0.57 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 0.33/0.57 = zero_zero_rat ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_0_left
% 0.33/0.57 thf(fact_3969_pochhammer__0__left,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ( ( N = zero_zero_nat )
% 0.33/0.57 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 0.33/0.57 = one_one_nat ) )
% 0.33/0.57 & ( ( N != zero_zero_nat )
% 0.33/0.57 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 0.33/0.57 = zero_zero_nat ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_0_left
% 0.33/0.57 thf(fact_3970_pochhammer__0__left,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ( ( N = zero_zero_nat )
% 0.33/0.57 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 0.33/0.57 = one_one_int ) )
% 0.33/0.57 & ( ( N != zero_zero_nat )
% 0.33/0.57 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 0.33/0.57 = zero_zero_int ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_0_left
% 0.33/0.57 thf(fact_3971_pochhammer__rec,axiom,
% 0.33/0.57 ! [A: complex,N: nat] :
% 0.33/0.57 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 0.33/0.57 = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_rec
% 0.33/0.57 thf(fact_3972_pochhammer__rec,axiom,
% 0.33/0.57 ! [A: real,N: nat] :
% 0.33/0.57 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.33/0.57 = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_rec
% 0.33/0.57 thf(fact_3973_pochhammer__rec,axiom,
% 0.33/0.57 ! [A: rat,N: nat] :
% 0.33/0.57 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.33/0.57 = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_rec
% 0.33/0.57 thf(fact_3974_pochhammer__rec,axiom,
% 0.33/0.57 ! [A: nat,N: nat] :
% 0.33/0.57 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.33/0.57 = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_rec
% 0.33/0.57 thf(fact_3975_pochhammer__rec,axiom,
% 0.33/0.57 ! [A: int,N: nat] :
% 0.33/0.57 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.33/0.57 = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_rec
% 0.33/0.57 thf(fact_3976_pochhammer__rec_H,axiom,
% 0.33/0.57 ! [Z2: complex,N: nat] :
% 0.33/0.57 ( ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N ) )
% 0.33/0.57 = ( times_times_complex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ N ) ) @ ( comm_s2602460028002588243omplex @ Z2 @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_rec'
% 0.33/0.57 thf(fact_3977_pochhammer__rec_H,axiom,
% 0.33/0.57 ! [Z2: real,N: nat] :
% 0.33/0.57 ( ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N ) )
% 0.33/0.57 = ( times_times_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_rec'
% 0.33/0.57 thf(fact_3978_pochhammer__rec_H,axiom,
% 0.33/0.57 ! [Z2: rat,N: nat] :
% 0.33/0.57 ( ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N ) )
% 0.33/0.57 = ( times_times_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_rec'
% 0.33/0.57 thf(fact_3979_pochhammer__rec_H,axiom,
% 0.33/0.57 ! [Z2: nat,N: nat] :
% 0.33/0.57 ( ( comm_s4663373288045622133er_nat @ Z2 @ ( suc @ N ) )
% 0.33/0.57 = ( times_times_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z2 @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_rec'
% 0.33/0.57 thf(fact_3980_pochhammer__rec_H,axiom,
% 0.33/0.57 ! [Z2: int,N: nat] :
% 0.33/0.57 ( ( comm_s4660882817536571857er_int @ Z2 @ ( suc @ N ) )
% 0.33/0.57 = ( times_times_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z2 @ N ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_rec'
% 0.33/0.57 thf(fact_3981_pochhammer__Suc,axiom,
% 0.33/0.57 ! [A: complex,N: nat] :
% 0.33/0.57 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 0.33/0.57 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_Suc
% 0.33/0.57 thf(fact_3982_pochhammer__Suc,axiom,
% 0.33/0.57 ! [A: real,N: nat] :
% 0.33/0.57 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.33/0.57 = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_Suc
% 0.33/0.57 thf(fact_3983_pochhammer__Suc,axiom,
% 0.33/0.57 ! [A: rat,N: nat] :
% 0.33/0.57 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.33/0.57 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_Suc
% 0.33/0.57 thf(fact_3984_pochhammer__Suc,axiom,
% 0.33/0.57 ! [A: nat,N: nat] :
% 0.33/0.57 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.33/0.57 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_Suc
% 0.33/0.57 thf(fact_3985_pochhammer__Suc,axiom,
% 0.33/0.57 ! [A: int,N: nat] :
% 0.33/0.57 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.33/0.57 = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_Suc
% 0.33/0.57 thf(fact_3986_pochhammer__product_H,axiom,
% 0.33/0.57 ! [Z2: complex,N: nat,M: nat] :
% 0.33/0.57 ( ( comm_s2602460028002588243omplex @ Z2 @ ( plus_plus_nat @ N @ M ) )
% 0.33/0.57 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ N ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ N ) ) @ M ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_product'
% 0.33/0.57 thf(fact_3987_pochhammer__product_H,axiom,
% 0.33/0.57 ! [Z2: real,N: nat,M: nat] :
% 0.33/0.57 ( ( comm_s7457072308508201937r_real @ Z2 @ ( plus_plus_nat @ N @ M ) )
% 0.33/0.57 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_product'
% 0.33/0.57 thf(fact_3988_pochhammer__product_H,axiom,
% 0.33/0.57 ! [Z2: rat,N: nat,M: nat] :
% 0.33/0.57 ( ( comm_s4028243227959126397er_rat @ Z2 @ ( plus_plus_nat @ N @ M ) )
% 0.33/0.57 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_product'
% 0.33/0.57 thf(fact_3989_pochhammer__product_H,axiom,
% 0.33/0.57 ! [Z2: nat,N: nat,M: nat] :
% 0.33/0.57 ( ( comm_s4663373288045622133er_nat @ Z2 @ ( plus_plus_nat @ N @ M ) )
% 0.33/0.57 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_product'
% 0.33/0.57 thf(fact_3990_pochhammer__product_H,axiom,
% 0.33/0.57 ! [Z2: int,N: nat,M: nat] :
% 0.33/0.57 ( ( comm_s4660882817536571857er_int @ Z2 @ ( plus_plus_nat @ N @ M ) )
% 0.33/0.57 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_product'
% 0.33/0.57 thf(fact_3991_binomial__mono,axiom,
% 0.33/0.57 ! [K: nat,K6: nat,N: nat] :
% 0.33/0.57 ( ( ord_less_eq_nat @ K @ K6 )
% 0.33/0.57 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 0.33/0.57 => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % binomial_mono
% 0.33/0.57 thf(fact_3992_binomial__maximum_H,axiom,
% 0.33/0.57 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 0.33/0.57
% 0.33/0.57 % binomial_maximum'
% 0.33/0.57 thf(fact_3993_binomial__maximum,axiom,
% 0.33/0.57 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % binomial_maximum
% 0.33/0.57 thf(fact_3994_binomial__antimono,axiom,
% 0.33/0.57 ! [K: nat,K6: nat,N: nat] :
% 0.33/0.57 ( ( ord_less_eq_nat @ K @ K6 )
% 0.33/0.57 => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 0.33/0.57 => ( ( ord_less_eq_nat @ K6 @ N )
% 0.33/0.57 => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % binomial_antimono
% 0.33/0.57 thf(fact_3995_pochhammer__product,axiom,
% 0.33/0.57 ! [M: nat,N: nat,Z2: complex] :
% 0.33/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.57 => ( ( comm_s2602460028002588243omplex @ Z2 @ N )
% 0.33/0.57 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_product
% 0.33/0.57 thf(fact_3996_pochhammer__product,axiom,
% 0.33/0.57 ! [M: nat,N: nat,Z2: real] :
% 0.33/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.57 => ( ( comm_s7457072308508201937r_real @ Z2 @ N )
% 0.33/0.57 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_product
% 0.33/0.57 thf(fact_3997_pochhammer__product,axiom,
% 0.33/0.57 ! [M: nat,N: nat,Z2: rat] :
% 0.33/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.57 => ( ( comm_s4028243227959126397er_rat @ Z2 @ N )
% 0.33/0.57 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_product
% 0.33/0.57 thf(fact_3998_pochhammer__product,axiom,
% 0.33/0.57 ! [M: nat,N: nat,Z2: nat] :
% 0.33/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.57 => ( ( comm_s4663373288045622133er_nat @ Z2 @ N )
% 0.33/0.57 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_product
% 0.33/0.57 thf(fact_3999_pochhammer__product,axiom,
% 0.33/0.57 ! [M: nat,N: nat,Z2: int] :
% 0.33/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.57 => ( ( comm_s4660882817536571857er_int @ Z2 @ N )
% 0.33/0.57 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % pochhammer_product
% 0.33/0.57 thf(fact_4000_binomial__strict__mono,axiom,
% 0.33/0.57 ! [K: nat,K6: nat,N: nat] :
% 0.33/0.57 ( ( ord_less_nat @ K @ K6 )
% 0.33/0.57 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 0.33/0.57 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % binomial_strict_mono
% 0.33/0.57 thf(fact_4001_binomial__strict__antimono,axiom,
% 0.33/0.57 ! [K: nat,K6: nat,N: nat] :
% 0.33/0.57 ( ( ord_less_nat @ K @ K6 )
% 0.33/0.57 => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 0.33/0.57 => ( ( ord_less_eq_nat @ K6 @ N )
% 0.33/0.57 => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % binomial_strict_antimono
% 0.33/0.57 thf(fact_4002_binomial__less__binomial__Suc,axiom,
% 0.33/0.57 ! [K: nat,N: nat] :
% 0.33/0.57 ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.57 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % binomial_less_binomial_Suc
% 0.33/0.57 thf(fact_4003_central__binomial__odd,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.57 => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.57 = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % central_binomial_odd
% 0.33/0.57 thf(fact_4004_zero__less__binomial__iff,axiom,
% 0.33/0.57 ! [N: nat,K: nat] :
% 0.33/0.57 ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 0.33/0.57 = ( ord_less_eq_nat @ K @ N ) ) ).
% 0.33/0.57
% 0.33/0.57 % zero_less_binomial_iff
% 0.33/0.57 thf(fact_4005_choose__two,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.57 = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % choose_two
% 0.33/0.57 thf(fact_4006_binomial__n__0,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ( binomial @ N @ zero_zero_nat )
% 0.33/0.57 = one_one_nat ) ).
% 0.33/0.57
% 0.33/0.57 % binomial_n_0
% 0.33/0.57 thf(fact_4007_binomial__Suc__Suc,axiom,
% 0.33/0.57 ! [N: nat,K: nat] :
% 0.33/0.57 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.33/0.57 = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % binomial_Suc_Suc
% 0.33/0.57 thf(fact_4008_binomial__eq__0__iff,axiom,
% 0.33/0.57 ! [N: nat,K: nat] :
% 0.33/0.57 ( ( ( binomial @ N @ K )
% 0.33/0.57 = zero_zero_nat )
% 0.33/0.57 = ( ord_less_nat @ N @ K ) ) ).
% 0.33/0.57
% 0.33/0.57 % binomial_eq_0_iff
% 0.33/0.57 thf(fact_4009_binomial__0__Suc,axiom,
% 0.33/0.57 ! [K: nat] :
% 0.33/0.57 ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 0.33/0.57 = zero_zero_nat ) ).
% 0.33/0.57
% 0.33/0.57 % binomial_0_Suc
% 0.33/0.57 thf(fact_4010_binomial__1,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
% 0.33/0.57 = N ) ).
% 0.33/0.57
% 0.33/0.57 % binomial_1
% 0.33/0.57 thf(fact_4011_binomial__addition__formula,axiom,
% 0.33/0.57 ! [N: nat,K: nat] :
% 0.33/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.57 => ( ( binomial @ N @ ( suc @ K ) )
% 0.33/0.57 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % binomial_addition_formula
% 0.33/0.57 thf(fact_4012_binomial__Suc__n,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ( binomial @ ( suc @ N ) @ N )
% 0.33/0.57 = ( suc @ N ) ) ).
% 0.33/0.57
% 0.33/0.57 % binomial_Suc_n
% 0.33/0.57 thf(fact_4013_binomial__n__n,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ( binomial @ N @ N )
% 0.33/0.57 = one_one_nat ) ).
% 0.33/0.57
% 0.33/0.57 % binomial_n_n
% 0.33/0.57 thf(fact_4014_choose__one,axiom,
% 0.33/0.57 ! [N: nat] :
% 0.33/0.57 ( ( binomial @ N @ one_one_nat )
% 0.33/0.57 = N ) ).
% 0.33/0.57
% 0.33/0.57 % choose_one
% 0.33/0.57 thf(fact_4015_binomial__eq__0,axiom,
% 0.33/0.57 ! [N: nat,K: nat] :
% 0.33/0.57 ( ( ord_less_nat @ N @ K )
% 0.33/0.57 => ( ( binomial @ N @ K )
% 0.33/0.57 = zero_zero_nat ) ) ).
% 0.33/0.57
% 0.33/0.57 % binomial_eq_0
% 0.33/0.57 thf(fact_4016_Suc__times__binomial,axiom,
% 0.33/0.57 ! [K: nat,N: nat] :
% 0.33/0.57 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
% 0.33/0.57 = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % Suc_times_binomial
% 0.33/0.57 thf(fact_4017_Suc__times__binomial__eq,axiom,
% 0.33/0.57 ! [N: nat,K: nat] :
% 0.33/0.57 ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
% 0.33/0.57 = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 0.33/0.57
% 0.33/0.57 % Suc_times_binomial_eq
% 0.33/0.57 thf(fact_4018_binomial__symmetric,axiom,
% 0.33/0.57 ! [K: nat,N: nat] :
% 0.33/0.57 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.57 => ( ( binomial @ N @ K )
% 0.33/0.57 = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % binomial_symmetric
% 0.33/0.58 thf(fact_4019_choose__mult__lemma,axiom,
% 0.33/0.58 ! [M: nat,R: nat,K: nat] :
% 0.33/0.58 ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
% 0.33/0.58 = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R ) @ M ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % choose_mult_lemma
% 0.33/0.58 thf(fact_4020_binomial__le__pow,axiom,
% 0.33/0.58 ! [R: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ R @ N )
% 0.33/0.58 => ( ord_less_eq_nat @ ( binomial @ N @ R ) @ ( power_power_nat @ N @ R ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % binomial_le_pow
% 0.33/0.58 thf(fact_4021_zero__less__binomial,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.58 => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % zero_less_binomial
% 0.33/0.58 thf(fact_4022_Suc__times__binomial__add,axiom,
% 0.33/0.58 ! [A: nat,B: nat] :
% 0.33/0.58 ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
% 0.33/0.58 = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % Suc_times_binomial_add
% 0.33/0.58 thf(fact_4023_choose__mult,axiom,
% 0.33/0.58 ! [K: nat,M: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ M )
% 0.33/0.58 => ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.58 => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 0.33/0.58 = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % choose_mult
% 0.33/0.58 thf(fact_4024_binomial__Suc__Suc__eq__times,axiom,
% 0.33/0.58 ! [N: nat,K: nat] :
% 0.33/0.58 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.33/0.58 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % binomial_Suc_Suc_eq_times
% 0.33/0.58 thf(fact_4025_binomial__absorb__comp,axiom,
% 0.33/0.58 ! [N: nat,K: nat] :
% 0.33/0.58 ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
% 0.33/0.58 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % binomial_absorb_comp
% 0.33/0.58 thf(fact_4026_binomial__absorption,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 0.33/0.58 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % binomial_absorption
% 0.33/0.58 thf(fact_4027_binomial__ge__n__over__k__pow__k,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.58 => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % binomial_ge_n_over_k_pow_k
% 0.33/0.58 thf(fact_4028_binomial__ge__n__over__k__pow__k,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.58 => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % binomial_ge_n_over_k_pow_k
% 0.33/0.58 thf(fact_4029_binomial__le__pow2,axiom,
% 0.33/0.58 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % binomial_le_pow2
% 0.33/0.58 thf(fact_4030_choose__reduce__nat,axiom,
% 0.33/0.58 ! [N: nat,K: nat] :
% 0.33/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.58 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.58 => ( ( binomial @ N @ K )
% 0.33/0.58 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % choose_reduce_nat
% 0.33/0.58 thf(fact_4031_times__binomial__minus1__eq,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.58 => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 0.33/0.58 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % times_binomial_minus1_eq
% 0.33/0.58 thf(fact_4032_artanh__def,axiom,
% 0.33/0.58 ( artanh_real
% 0.33/0.58 = ( ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % artanh_def
% 0.33/0.58 thf(fact_4033_concat__bit__Suc,axiom,
% 0.33/0.58 ! [N: nat,K: int,L: int] :
% 0.33/0.58 ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
% 0.33/0.58 = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % concat_bit_Suc
% 0.33/0.58 thf(fact_4034_signed__take__bit__Suc,axiom,
% 0.33/0.58 ! [N: nat,A: code_integer] :
% 0.33/0.58 ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
% 0.33/0.58 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % signed_take_bit_Suc
% 0.33/0.58 thf(fact_4035_signed__take__bit__Suc,axiom,
% 0.33/0.58 ! [N: nat,A: int] :
% 0.33/0.58 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
% 0.33/0.58 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % signed_take_bit_Suc
% 0.33/0.58 thf(fact_4036_fact__double,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.58 = ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_double
% 0.33/0.58 thf(fact_4037_fact__double,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.58 = ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_double
% 0.33/0.58 thf(fact_4038_fact__double,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.58 = ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_double
% 0.33/0.58 thf(fact_4039_modulo__int__unfold,axiom,
% 0.33/0.58 ! [L: int,K: int,N: nat,M: nat] :
% 0.33/0.58 ( ( ( ( ( sgn_sgn_int @ L )
% 0.33/0.58 = zero_zero_int )
% 0.33/0.58 | ( ( sgn_sgn_int @ K )
% 0.33/0.58 = zero_zero_int )
% 0.33/0.58 | ( N = zero_zero_nat ) )
% 0.33/0.58 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.33/0.58 = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 0.33/0.58 & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.33/0.58 = zero_zero_int )
% 0.33/0.58 | ( ( sgn_sgn_int @ K )
% 0.33/0.58 = zero_zero_int )
% 0.33/0.58 | ( N = zero_zero_nat ) )
% 0.33/0.58 => ( ( ( ( sgn_sgn_int @ K )
% 0.33/0.58 = ( sgn_sgn_int @ L ) )
% 0.33/0.58 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.33/0.58 = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 0.33/0.58 & ( ( ( sgn_sgn_int @ K )
% 0.33/0.58 != ( sgn_sgn_int @ L ) )
% 0.33/0.58 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.33/0.58 = ( times_times_int @ ( sgn_sgn_int @ L )
% 0.33/0.58 @ ( minus_minus_int
% 0.33/0.58 @ ( semiri1314217659103216013at_int
% 0.33/0.58 @ ( times_times_nat @ N
% 0.33/0.58 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.58 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 0.33/0.58 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % modulo_int_unfold
% 0.33/0.58 thf(fact_4040_take__bit__rec,axiom,
% 0.33/0.58 ( bit_se1745604003318907178nteger
% 0.33/0.58 = ( ^ [N3: nat,A2: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_rec
% 0.33/0.58 thf(fact_4041_take__bit__rec,axiom,
% 0.33/0.58 ( bit_se569199155075624693atural
% 0.33/0.58 = ( ^ [N3: nat,A2: code_natural] : ( if_Code_natural @ ( N3 = zero_zero_nat ) @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_rec
% 0.33/0.58 thf(fact_4042_take__bit__rec,axiom,
% 0.33/0.58 ( bit_se2923211474154528505it_int
% 0.33/0.58 = ( ^ [N3: nat,A2: int] : ( if_int @ ( N3 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_rec
% 0.33/0.58 thf(fact_4043_take__bit__rec,axiom,
% 0.33/0.58 ( bit_se2925701944663578781it_nat
% 0.33/0.58 = ( ^ [N3: nat,A2: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_rec
% 0.33/0.58 thf(fact_4044_log2__of__power__le,axiom,
% 0.33/0.58 ! [M: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.58 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.58 => ( ord_less_eq_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log2_of_power_le
% 0.33/0.58 thf(fact_4045_sgn__sgn,axiom,
% 0.33/0.58 ! [A: int] :
% 0.33/0.58 ( ( sgn_sgn_int @ ( sgn_sgn_int @ A ) )
% 0.33/0.58 = ( sgn_sgn_int @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_sgn
% 0.33/0.58 thf(fact_4046_sgn__sgn,axiom,
% 0.33/0.58 ! [A: real] :
% 0.33/0.58 ( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
% 0.33/0.58 = ( sgn_sgn_real @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_sgn
% 0.33/0.58 thf(fact_4047_sgn__sgn,axiom,
% 0.33/0.58 ! [A: complex] :
% 0.33/0.58 ( ( sgn_sgn_complex @ ( sgn_sgn_complex @ A ) )
% 0.33/0.58 = ( sgn_sgn_complex @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_sgn
% 0.33/0.58 thf(fact_4048_sgn__sgn,axiom,
% 0.33/0.58 ! [A: rat] :
% 0.33/0.58 ( ( sgn_sgn_rat @ ( sgn_sgn_rat @ A ) )
% 0.33/0.58 = ( sgn_sgn_rat @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_sgn
% 0.33/0.58 thf(fact_4049_sgn__sgn,axiom,
% 0.33/0.58 ! [A: code_integer] :
% 0.33/0.58 ( ( sgn_sgn_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.33/0.58 = ( sgn_sgn_Code_integer @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_sgn
% 0.33/0.58 thf(fact_4050_take__bit__of__0,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
% 0.33/0.58 = zero_zero_int ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_of_0
% 0.33/0.58 thf(fact_4051_take__bit__of__0,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
% 0.33/0.58 = zero_zero_nat ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_of_0
% 0.33/0.58 thf(fact_4052_sgn__0,axiom,
% 0.33/0.58 ( ( sgn_sgn_complex @ zero_zero_complex )
% 0.33/0.58 = zero_zero_complex ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_0
% 0.33/0.58 thf(fact_4053_sgn__0,axiom,
% 0.33/0.58 ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
% 0.33/0.58 = zero_z3403309356797280102nteger ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_0
% 0.33/0.58 thf(fact_4054_sgn__0,axiom,
% 0.33/0.58 ( ( sgn_sgn_real @ zero_zero_real )
% 0.33/0.58 = zero_zero_real ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_0
% 0.33/0.58 thf(fact_4055_sgn__0,axiom,
% 0.33/0.58 ( ( sgn_sgn_rat @ zero_zero_rat )
% 0.33/0.58 = zero_zero_rat ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_0
% 0.33/0.58 thf(fact_4056_sgn__0,axiom,
% 0.33/0.58 ( ( sgn_sgn_int @ zero_zero_int )
% 0.33/0.58 = zero_zero_int ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_0
% 0.33/0.58 thf(fact_4057_sgn__zero,axiom,
% 0.33/0.58 ( ( sgn_sgn_complex @ zero_zero_complex )
% 0.33/0.58 = zero_zero_complex ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_zero
% 0.33/0.58 thf(fact_4058_sgn__zero,axiom,
% 0.33/0.58 ( ( sgn_sgn_real @ zero_zero_real )
% 0.33/0.58 = zero_zero_real ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_zero
% 0.33/0.58 thf(fact_4059_sgn__1,axiom,
% 0.33/0.58 ( ( sgn_sgn_int @ one_one_int )
% 0.33/0.58 = one_one_int ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_1
% 0.33/0.58 thf(fact_4060_sgn__1,axiom,
% 0.33/0.58 ( ( sgn_sgn_real @ one_one_real )
% 0.33/0.58 = one_one_real ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_1
% 0.33/0.58 thf(fact_4061_sgn__1,axiom,
% 0.33/0.58 ( ( sgn_sgn_complex @ one_one_complex )
% 0.33/0.58 = one_one_complex ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_1
% 0.33/0.58 thf(fact_4062_sgn__1,axiom,
% 0.33/0.58 ( ( sgn_sgn_rat @ one_one_rat )
% 0.33/0.58 = one_one_rat ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_1
% 0.33/0.58 thf(fact_4063_sgn__1,axiom,
% 0.33/0.58 ( ( sgn_sgn_Code_integer @ one_one_Code_integer )
% 0.33/0.58 = one_one_Code_integer ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_1
% 0.33/0.58 thf(fact_4064_sgn__one,axiom,
% 0.33/0.58 ( ( sgn_sgn_real @ one_one_real )
% 0.33/0.58 = one_one_real ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_one
% 0.33/0.58 thf(fact_4065_sgn__one,axiom,
% 0.33/0.58 ( ( sgn_sgn_complex @ one_one_complex )
% 0.33/0.58 = one_one_complex ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_one
% 0.33/0.58 thf(fact_4066_sgn__divide,axiom,
% 0.33/0.58 ! [A: complex,B: complex] :
% 0.33/0.58 ( ( sgn_sgn_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.58 = ( divide1717551699836669952omplex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_divide
% 0.33/0.58 thf(fact_4067_sgn__divide,axiom,
% 0.33/0.58 ! [A: real,B: real] :
% 0.33/0.58 ( ( sgn_sgn_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.58 = ( divide_divide_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_divide
% 0.33/0.58 thf(fact_4068_sgn__divide,axiom,
% 0.33/0.58 ! [A: rat,B: rat] :
% 0.33/0.58 ( ( sgn_sgn_rat @ ( divide_divide_rat @ A @ B ) )
% 0.33/0.58 = ( divide_divide_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_divide
% 0.33/0.58 thf(fact_4069_power__sgn,axiom,
% 0.33/0.58 ! [A: rat,N: nat] :
% 0.33/0.58 ( ( sgn_sgn_rat @ ( power_power_rat @ A @ N ) )
% 0.33/0.58 = ( power_power_rat @ ( sgn_sgn_rat @ A ) @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % power_sgn
% 0.33/0.58 thf(fact_4070_power__sgn,axiom,
% 0.33/0.58 ! [A: code_integer,N: nat] :
% 0.33/0.58 ( ( sgn_sgn_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.33/0.58 = ( power_8256067586552552935nteger @ ( sgn_sgn_Code_integer @ A ) @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % power_sgn
% 0.33/0.58 thf(fact_4071_power__sgn,axiom,
% 0.33/0.58 ! [A: real,N: nat] :
% 0.33/0.58 ( ( sgn_sgn_real @ ( power_power_real @ A @ N ) )
% 0.33/0.58 = ( power_power_real @ ( sgn_sgn_real @ A ) @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % power_sgn
% 0.33/0.58 thf(fact_4072_power__sgn,axiom,
% 0.33/0.58 ! [A: int,N: nat] :
% 0.33/0.58 ( ( sgn_sgn_int @ ( power_power_int @ A @ N ) )
% 0.33/0.58 = ( power_power_int @ ( sgn_sgn_int @ A ) @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % power_sgn
% 0.33/0.58 thf(fact_4073_of__nat__fact,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri8010041392384452111omplex @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.33/0.58 = ( semiri5044797733671781792omplex @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % of_nat_fact
% 0.33/0.58 thf(fact_4074_of__nat__fact,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri681578069525770553at_rat @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.33/0.58 = ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % of_nat_fact
% 0.33/0.58 thf(fact_4075_of__nat__fact,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri1314217659103216013at_int @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.33/0.58 = ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % of_nat_fact
% 0.33/0.58 thf(fact_4076_of__nat__fact,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri1316708129612266289at_nat @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.33/0.58 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % of_nat_fact
% 0.33/0.58 thf(fact_4077_of__nat__fact,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri5074537144036343181t_real @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.33/0.58 = ( semiri2265585572941072030t_real @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % of_nat_fact
% 0.33/0.58 thf(fact_4078_signed__take__bit__of__0,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 0.33/0.58 = zero_zero_int ) ).
% 0.33/0.58
% 0.33/0.58 % signed_take_bit_of_0
% 0.33/0.58 thf(fact_4079_concat__bit__0,axiom,
% 0.33/0.58 ! [K: int,L: int] :
% 0.33/0.58 ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
% 0.33/0.58 = L ) ).
% 0.33/0.58
% 0.33/0.58 % concat_bit_0
% 0.33/0.58 thf(fact_4080_concat__bit__of__zero__2,axiom,
% 0.33/0.58 ! [N: nat,K: int] :
% 0.33/0.58 ( ( bit_concat_bit @ N @ K @ zero_zero_int )
% 0.33/0.58 = ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 0.33/0.58
% 0.33/0.58 % concat_bit_of_zero_2
% 0.33/0.58 thf(fact_4081_sgn__greater,axiom,
% 0.33/0.58 ! [A: code_integer] :
% 0.33/0.58 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
% 0.33/0.58 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_greater
% 0.33/0.58 thf(fact_4082_sgn__greater,axiom,
% 0.33/0.58 ! [A: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
% 0.33/0.58 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_greater
% 0.33/0.58 thf(fact_4083_sgn__greater,axiom,
% 0.33/0.58 ! [A: rat] :
% 0.33/0.58 ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A ) )
% 0.33/0.58 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_greater
% 0.33/0.58 thf(fact_4084_sgn__greater,axiom,
% 0.33/0.58 ! [A: int] :
% 0.33/0.58 ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
% 0.33/0.58 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_greater
% 0.33/0.58 thf(fact_4085_sgn__less,axiom,
% 0.33/0.58 ! [A: code_integer] :
% 0.33/0.58 ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 0.33/0.58 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_less
% 0.33/0.58 thf(fact_4086_sgn__less,axiom,
% 0.33/0.58 ! [A: real] :
% 0.33/0.58 ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
% 0.33/0.58 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_less
% 0.33/0.58 thf(fact_4087_sgn__less,axiom,
% 0.33/0.58 ! [A: rat] :
% 0.33/0.58 ( ( ord_less_rat @ ( sgn_sgn_rat @ A ) @ zero_zero_rat )
% 0.33/0.58 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_less
% 0.33/0.58 thf(fact_4088_sgn__less,axiom,
% 0.33/0.58 ! [A: int] :
% 0.33/0.58 ( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
% 0.33/0.58 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_less
% 0.33/0.58 thf(fact_4089_take__bit__0,axiom,
% 0.33/0.58 ! [A: int] :
% 0.33/0.58 ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
% 0.33/0.58 = zero_zero_int ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_0
% 0.33/0.58 thf(fact_4090_take__bit__0,axiom,
% 0.33/0.58 ! [A: nat] :
% 0.33/0.58 ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
% 0.33/0.58 = zero_zero_nat ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_0
% 0.33/0.58 thf(fact_4091_ln__less__zero__iff,axiom,
% 0.33/0.58 ! [X: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 0.33/0.58 = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_less_zero_iff
% 0.33/0.58 thf(fact_4092_ln__gt__zero__iff,axiom,
% 0.33/0.58 ! [X: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.33/0.58 = ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_gt_zero_iff
% 0.33/0.58 thf(fact_4093_ln__eq__zero__iff,axiom,
% 0.33/0.58 ! [X: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ( ln_ln_real @ X )
% 0.33/0.58 = zero_zero_real )
% 0.33/0.58 = ( X = one_one_real ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_eq_zero_iff
% 0.33/0.58 thf(fact_4094_divide__sgn,axiom,
% 0.33/0.58 ! [A: real,B: real] :
% 0.33/0.58 ( ( divide_divide_real @ A @ ( sgn_sgn_real @ B ) )
% 0.33/0.58 = ( times_times_real @ A @ ( sgn_sgn_real @ B ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % divide_sgn
% 0.33/0.58 thf(fact_4095_divide__sgn,axiom,
% 0.33/0.58 ! [A: rat,B: rat] :
% 0.33/0.58 ( ( divide_divide_rat @ A @ ( sgn_sgn_rat @ B ) )
% 0.33/0.58 = ( times_times_rat @ A @ ( sgn_sgn_rat @ B ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % divide_sgn
% 0.33/0.58 thf(fact_4096_take__bit__Suc__1,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
% 0.33/0.58 = one_one_int ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_Suc_1
% 0.33/0.58 thf(fact_4097_take__bit__Suc__1,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
% 0.33/0.58 = one_one_nat ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_Suc_1
% 0.33/0.58 thf(fact_4098_take__bit__numeral__1,axiom,
% 0.33/0.58 ! [L: num] :
% 0.33/0.58 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
% 0.33/0.58 = one_one_int ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_numeral_1
% 0.33/0.58 thf(fact_4099_take__bit__numeral__1,axiom,
% 0.33/0.58 ! [L: num] :
% 0.33/0.58 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
% 0.33/0.58 = one_one_nat ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_numeral_1
% 0.33/0.58 thf(fact_4100_ln__one,axiom,
% 0.33/0.58 ( ( ln_ln_real @ one_one_real )
% 0.33/0.58 = zero_zero_real ) ).
% 0.33/0.58
% 0.33/0.58 % ln_one
% 0.33/0.58 thf(fact_4101_fact__0,axiom,
% 0.33/0.58 ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 0.33/0.58 = one_one_complex ) ).
% 0.33/0.58
% 0.33/0.58 % fact_0
% 0.33/0.58 thf(fact_4102_fact__0,axiom,
% 0.33/0.58 ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 0.33/0.58 = one_one_rat ) ).
% 0.33/0.58
% 0.33/0.58 % fact_0
% 0.33/0.58 thf(fact_4103_fact__0,axiom,
% 0.33/0.58 ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 0.33/0.58 = one_one_int ) ).
% 0.33/0.58
% 0.33/0.58 % fact_0
% 0.33/0.58 thf(fact_4104_fact__0,axiom,
% 0.33/0.58 ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 0.33/0.58 = one_one_nat ) ).
% 0.33/0.58
% 0.33/0.58 % fact_0
% 0.33/0.58 thf(fact_4105_fact__0,axiom,
% 0.33/0.58 ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 0.33/0.58 = one_one_real ) ).
% 0.33/0.58
% 0.33/0.58 % fact_0
% 0.33/0.58 thf(fact_4106_fact__1,axiom,
% 0.33/0.58 ( ( semiri5044797733671781792omplex @ one_one_nat )
% 0.33/0.58 = one_one_complex ) ).
% 0.33/0.58
% 0.33/0.58 % fact_1
% 0.33/0.58 thf(fact_4107_fact__1,axiom,
% 0.33/0.58 ( ( semiri773545260158071498ct_rat @ one_one_nat )
% 0.33/0.58 = one_one_rat ) ).
% 0.33/0.58
% 0.33/0.58 % fact_1
% 0.33/0.58 thf(fact_4108_fact__1,axiom,
% 0.33/0.58 ( ( semiri1406184849735516958ct_int @ one_one_nat )
% 0.33/0.58 = one_one_int ) ).
% 0.33/0.58
% 0.33/0.58 % fact_1
% 0.33/0.58 thf(fact_4109_fact__1,axiom,
% 0.33/0.58 ( ( semiri1408675320244567234ct_nat @ one_one_nat )
% 0.33/0.58 = one_one_nat ) ).
% 0.33/0.58
% 0.33/0.58 % fact_1
% 0.33/0.58 thf(fact_4110_fact__1,axiom,
% 0.33/0.58 ( ( semiri2265585572941072030t_real @ one_one_nat )
% 0.33/0.58 = one_one_real ) ).
% 0.33/0.58
% 0.33/0.58 % fact_1
% 0.33/0.58 thf(fact_4111_signed__take__bit__Suc__1,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
% 0.33/0.58 = one_one_int ) ).
% 0.33/0.58
% 0.33/0.58 % signed_take_bit_Suc_1
% 0.33/0.58 thf(fact_4112_signed__take__bit__numeral__of__1,axiom,
% 0.33/0.58 ! [K: num] :
% 0.33/0.58 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
% 0.33/0.58 = one_one_int ) ).
% 0.33/0.58
% 0.33/0.58 % signed_take_bit_numeral_of_1
% 0.33/0.58 thf(fact_4113_log__one,axiom,
% 0.33/0.58 ! [A: real] :
% 0.33/0.58 ( ( log2 @ A @ one_one_real )
% 0.33/0.58 = zero_zero_real ) ).
% 0.33/0.58
% 0.33/0.58 % log_one
% 0.33/0.58 thf(fact_4114_concat__bit__nonnegative__iff,axiom,
% 0.33/0.58 ! [N: nat,K: int,L: int] :
% 0.33/0.58 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
% 0.33/0.58 = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).
% 0.33/0.58
% 0.33/0.58 % concat_bit_nonnegative_iff
% 0.33/0.58 thf(fact_4115_concat__bit__negative__iff,axiom,
% 0.33/0.58 ! [N: nat,K: int,L: int] :
% 0.33/0.58 ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
% 0.33/0.58 = ( ord_less_int @ L @ zero_zero_int ) ) ).
% 0.33/0.58
% 0.33/0.58 % concat_bit_negative_iff
% 0.33/0.58 thf(fact_4116_concat__bit__of__zero__1,axiom,
% 0.33/0.58 ! [N: nat,L: int] :
% 0.33/0.58 ( ( bit_concat_bit @ N @ zero_zero_int @ L )
% 0.33/0.58 = ( bit_se545348938243370406it_int @ N @ L ) ) ).
% 0.33/0.58
% 0.33/0.58 % concat_bit_of_zero_1
% 0.33/0.58 thf(fact_4117_sgn__pos,axiom,
% 0.33/0.58 ! [A: code_integer] :
% 0.33/0.58 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 0.33/0.58 => ( ( sgn_sgn_Code_integer @ A )
% 0.33/0.58 = one_one_Code_integer ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_pos
% 0.33/0.58 thf(fact_4118_sgn__pos,axiom,
% 0.33/0.58 ! [A: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58 => ( ( sgn_sgn_real @ A )
% 0.33/0.58 = one_one_real ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_pos
% 0.33/0.58 thf(fact_4119_sgn__pos,axiom,
% 0.33/0.58 ! [A: rat] :
% 0.33/0.58 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.58 => ( ( sgn_sgn_rat @ A )
% 0.33/0.58 = one_one_rat ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_pos
% 0.33/0.58 thf(fact_4120_sgn__pos,axiom,
% 0.33/0.58 ! [A: int] :
% 0.33/0.58 ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.58 => ( ( sgn_sgn_int @ A )
% 0.33/0.58 = one_one_int ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_pos
% 0.33/0.58 thf(fact_4121_take__bit__of__1__eq__0__iff,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 0.33/0.58 = zero_zero_int )
% 0.33/0.58 = ( N = zero_zero_nat ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_of_1_eq_0_iff
% 0.33/0.58 thf(fact_4122_take__bit__of__1__eq__0__iff,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 0.33/0.58 = zero_zero_nat )
% 0.33/0.58 = ( N = zero_zero_nat ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_of_1_eq_0_iff
% 0.33/0.58 thf(fact_4123_ln__le__zero__iff,axiom,
% 0.33/0.58 ! [X: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 0.33/0.58 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_le_zero_iff
% 0.33/0.58 thf(fact_4124_ln__ge__zero__iff,axiom,
% 0.33/0.58 ! [X: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.33/0.58 = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_ge_zero_iff
% 0.33/0.58 thf(fact_4125_fact__Suc__0,axiom,
% 0.33/0.58 ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 0.33/0.58 = one_one_complex ) ).
% 0.33/0.58
% 0.33/0.58 % fact_Suc_0
% 0.33/0.58 thf(fact_4126_fact__Suc__0,axiom,
% 0.33/0.58 ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 0.33/0.58 = one_one_rat ) ).
% 0.33/0.58
% 0.33/0.58 % fact_Suc_0
% 0.33/0.58 thf(fact_4127_fact__Suc__0,axiom,
% 0.33/0.58 ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 0.33/0.58 = one_one_int ) ).
% 0.33/0.58
% 0.33/0.58 % fact_Suc_0
% 0.33/0.58 thf(fact_4128_fact__Suc__0,axiom,
% 0.33/0.58 ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 0.33/0.58 = one_one_nat ) ).
% 0.33/0.58
% 0.33/0.58 % fact_Suc_0
% 0.33/0.58 thf(fact_4129_fact__Suc__0,axiom,
% 0.33/0.58 ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 0.33/0.58 = one_one_real ) ).
% 0.33/0.58
% 0.33/0.58 % fact_Suc_0
% 0.33/0.58 thf(fact_4130_fact__Suc,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri5044797733671781792omplex @ ( suc @ N ) )
% 0.33/0.58 = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_Suc
% 0.33/0.58 thf(fact_4131_fact__Suc,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
% 0.33/0.58 = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_Suc
% 0.33/0.58 thf(fact_4132_fact__Suc,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
% 0.33/0.58 = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_Suc
% 0.33/0.58 thf(fact_4133_fact__Suc,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
% 0.33/0.58 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_Suc
% 0.33/0.58 thf(fact_4134_fact__Suc,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
% 0.33/0.58 = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_Suc
% 0.33/0.58 thf(fact_4135_sgn__mult__self__eq,axiom,
% 0.33/0.58 ! [A: real] :
% 0.33/0.58 ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ A ) )
% 0.33/0.58 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_mult_self_eq
% 0.33/0.58 thf(fact_4136_sgn__mult__self__eq,axiom,
% 0.33/0.58 ! [A: rat] :
% 0.33/0.58 ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ A ) )
% 0.33/0.58 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_mult_self_eq
% 0.33/0.58 thf(fact_4137_sgn__mult__self__eq,axiom,
% 0.33/0.58 ! [A: int] :
% 0.33/0.58 ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
% 0.33/0.58 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_mult_self_eq
% 0.33/0.58 thf(fact_4138_sgn__mult__self__eq,axiom,
% 0.33/0.58 ! [A: code_integer] :
% 0.33/0.58 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
% 0.33/0.58 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_mult_self_eq
% 0.33/0.58 thf(fact_4139_zero__less__log__cancel__iff,axiom,
% 0.33/0.58 ! [A: real,X: real] :
% 0.33/0.58 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ ( log2 @ A @ X ) )
% 0.33/0.58 = ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % zero_less_log_cancel_iff
% 0.33/0.58 thf(fact_4140_log__less__zero__cancel__iff,axiom,
% 0.33/0.58 ! [A: real,X: real] :
% 0.33/0.58 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_real @ ( log2 @ A @ X ) @ zero_zero_real )
% 0.33/0.58 = ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_less_zero_cancel_iff
% 0.33/0.58 thf(fact_4141_one__less__log__cancel__iff,axiom,
% 0.33/0.58 ! [A: real,X: real] :
% 0.33/0.58 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_real @ one_one_real @ ( log2 @ A @ X ) )
% 0.33/0.58 = ( ord_less_real @ A @ X ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % one_less_log_cancel_iff
% 0.33/0.58 thf(fact_4142_log__less__one__cancel__iff,axiom,
% 0.33/0.58 ! [A: real,X: real] :
% 0.33/0.58 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_real @ ( log2 @ A @ X ) @ one_one_real )
% 0.33/0.58 = ( ord_less_real @ X @ A ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_less_one_cancel_iff
% 0.33/0.58 thf(fact_4143_log__less__cancel__iff,axiom,
% 0.33/0.58 ! [A: real,X: real,Y: real] :
% 0.33/0.58 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.58 => ( ( ord_less_real @ ( log2 @ A @ X ) @ ( log2 @ A @ Y ) )
% 0.33/0.58 = ( ord_less_real @ X @ Y ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_less_cancel_iff
% 0.33/0.58 thf(fact_4144_log__eq__one,axiom,
% 0.33/0.58 ! [A: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58 => ( ( A != one_one_real )
% 0.33/0.58 => ( ( log2 @ A @ A )
% 0.33/0.58 = one_one_real ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_eq_one
% 0.33/0.58 thf(fact_4145_take__bit__of__Suc__0,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.33/0.58 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_of_Suc_0
% 0.33/0.58 thf(fact_4146_sgn__mult__dvd__iff,axiom,
% 0.33/0.58 ! [R: int,L: int,K: int] :
% 0.33/0.58 ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R ) @ L ) @ K )
% 0.33/0.58 = ( ( dvd_dvd_int @ L @ K )
% 0.33/0.58 & ( ( R = zero_zero_int )
% 0.33/0.58 => ( K = zero_zero_int ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_mult_dvd_iff
% 0.33/0.58 thf(fact_4147_mult__sgn__dvd__iff,axiom,
% 0.33/0.58 ! [L: int,R: int,K: int] :
% 0.33/0.58 ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R ) ) @ K )
% 0.33/0.58 = ( ( dvd_dvd_int @ L @ K )
% 0.33/0.58 & ( ( R = zero_zero_int )
% 0.33/0.58 => ( K = zero_zero_int ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % mult_sgn_dvd_iff
% 0.33/0.58 thf(fact_4148_dvd__sgn__mult__iff,axiom,
% 0.33/0.58 ! [L: int,R: int,K: int] :
% 0.33/0.58 ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R ) @ K ) )
% 0.33/0.58 = ( ( dvd_dvd_int @ L @ K )
% 0.33/0.58 | ( R = zero_zero_int ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % dvd_sgn_mult_iff
% 0.33/0.58 thf(fact_4149_dvd__mult__sgn__iff,axiom,
% 0.33/0.58 ! [L: int,K: int,R: int] :
% 0.33/0.58 ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R ) ) )
% 0.33/0.58 = ( ( dvd_dvd_int @ L @ K )
% 0.33/0.58 | ( R = zero_zero_int ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % dvd_mult_sgn_iff
% 0.33/0.58 thf(fact_4150_fact__2,axiom,
% 0.33/0.58 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.58 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_2
% 0.33/0.58 thf(fact_4151_fact__2,axiom,
% 0.33/0.58 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.58 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_2
% 0.33/0.58 thf(fact_4152_fact__2,axiom,
% 0.33/0.58 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.58 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_2
% 0.33/0.58 thf(fact_4153_fact__2,axiom,
% 0.33/0.58 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.58 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_2
% 0.33/0.58 thf(fact_4154_fact__2,axiom,
% 0.33/0.58 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.58 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_2
% 0.33/0.58 thf(fact_4155_signed__take__bit__Suc__bit0,axiom,
% 0.33/0.58 ! [N: nat,K: num] :
% 0.33/0.58 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.33/0.58 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % signed_take_bit_Suc_bit0
% 0.33/0.58 thf(fact_4156_take__bit__of__1,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
% 0.33/0.58 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_of_1
% 0.33/0.58 thf(fact_4157_take__bit__of__1,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 0.33/0.58 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_of_1
% 0.33/0.58 thf(fact_4158_take__bit__of__1,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 0.33/0.58 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_of_1
% 0.33/0.58 thf(fact_4159_zero__le__log__cancel__iff,axiom,
% 0.33/0.58 ! [A: real,X: real] :
% 0.33/0.58 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_eq_real @ zero_zero_real @ ( log2 @ A @ X ) )
% 0.33/0.58 = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % zero_le_log_cancel_iff
% 0.33/0.58 thf(fact_4160_log__le__zero__cancel__iff,axiom,
% 0.33/0.58 ! [A: real,X: real] :
% 0.33/0.58 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_eq_real @ ( log2 @ A @ X ) @ zero_zero_real )
% 0.33/0.58 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_le_zero_cancel_iff
% 0.33/0.58 thf(fact_4161_one__le__log__cancel__iff,axiom,
% 0.33/0.58 ! [A: real,X: real] :
% 0.33/0.58 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_eq_real @ one_one_real @ ( log2 @ A @ X ) )
% 0.33/0.58 = ( ord_less_eq_real @ A @ X ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % one_le_log_cancel_iff
% 0.33/0.58 thf(fact_4162_log__le__one__cancel__iff,axiom,
% 0.33/0.58 ! [A: real,X: real] :
% 0.33/0.58 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_eq_real @ ( log2 @ A @ X ) @ one_one_real )
% 0.33/0.58 = ( ord_less_eq_real @ X @ A ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_le_one_cancel_iff
% 0.33/0.58 thf(fact_4163_log__le__cancel__iff,axiom,
% 0.33/0.58 ! [A: real,X: real,Y: real] :
% 0.33/0.58 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.58 => ( ( ord_less_eq_real @ ( log2 @ A @ X ) @ ( log2 @ A @ Y ) )
% 0.33/0.58 = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_le_cancel_iff
% 0.33/0.58 thf(fact_4164_sgn__of__nat,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( sgn_sgn_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.33/0.58 = ( zero_n3304061248610475627l_real @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_of_nat
% 0.33/0.58 thf(fact_4165_sgn__of__nat,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( sgn_sgn_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.33/0.58 = ( zero_n2052037380579107095ol_rat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_of_nat
% 0.33/0.58 thf(fact_4166_sgn__of__nat,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.33/0.58 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_of_nat
% 0.33/0.58 thf(fact_4167_sgn__of__nat,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( sgn_sgn_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.33/0.58 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_of_nat
% 0.33/0.58 thf(fact_4168_even__take__bit__eq,axiom,
% 0.33/0.58 ! [N: nat,A: code_integer] :
% 0.33/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
% 0.33/0.58 = ( ( N = zero_zero_nat )
% 0.33/0.58 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % even_take_bit_eq
% 0.33/0.58 thf(fact_4169_even__take__bit__eq,axiom,
% 0.33/0.58 ! [N: nat,A: int] :
% 0.33/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.33/0.58 = ( ( N = zero_zero_nat )
% 0.33/0.58 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % even_take_bit_eq
% 0.33/0.58 thf(fact_4170_even__take__bit__eq,axiom,
% 0.33/0.58 ! [N: nat,A: nat] :
% 0.33/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 0.33/0.58 = ( ( N = zero_zero_nat )
% 0.33/0.58 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % even_take_bit_eq
% 0.33/0.58 thf(fact_4171_log__pow__cancel,axiom,
% 0.33/0.58 ! [A: real,B: nat] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58 => ( ( A != one_one_real )
% 0.33/0.58 => ( ( log2 @ A @ ( power_power_real @ A @ B ) )
% 0.33/0.58 = ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_pow_cancel
% 0.33/0.58 thf(fact_4172_take__bit__Suc__0,axiom,
% 0.33/0.58 ! [A: code_integer] :
% 0.33/0.58 ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
% 0.33/0.58 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_Suc_0
% 0.33/0.58 thf(fact_4173_take__bit__Suc__0,axiom,
% 0.33/0.58 ! [A: code_natural] :
% 0.33/0.58 ( ( bit_se569199155075624693atural @ ( suc @ zero_zero_nat ) @ A )
% 0.33/0.58 = ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_Suc_0
% 0.33/0.58 thf(fact_4174_take__bit__Suc__0,axiom,
% 0.33/0.58 ! [A: int] :
% 0.33/0.58 ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
% 0.33/0.58 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_Suc_0
% 0.33/0.58 thf(fact_4175_take__bit__Suc__0,axiom,
% 0.33/0.58 ! [A: nat] :
% 0.33/0.58 ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
% 0.33/0.58 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_Suc_0
% 0.33/0.58 thf(fact_4176_take__bit__of__exp,axiom,
% 0.33/0.58 ! [M: nat,N: nat] :
% 0.33/0.58 ( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.58 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_of_exp
% 0.33/0.58 thf(fact_4177_take__bit__of__exp,axiom,
% 0.33/0.58 ! [M: nat,N: nat] :
% 0.33/0.58 ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.58 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_of_exp
% 0.33/0.58 thf(fact_4178_take__bit__of__exp,axiom,
% 0.33/0.58 ! [M: nat,N: nat] :
% 0.33/0.58 ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.58 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_of_exp
% 0.33/0.58 thf(fact_4179_take__bit__of__2,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.58 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_of_2
% 0.33/0.58 thf(fact_4180_take__bit__of__2,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.58 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_of_2
% 0.33/0.58 thf(fact_4181_take__bit__of__2,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.58 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_of_2
% 0.33/0.58 thf(fact_4182_take__bit__of__nat,axiom,
% 0.33/0.58 ! [N: nat,M: nat] :
% 0.33/0.58 ( ( bit_se2923211474154528505it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 0.33/0.58 = ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_of_nat
% 0.33/0.58 thf(fact_4183_take__bit__of__nat,axiom,
% 0.33/0.58 ! [N: nat,M: nat] :
% 0.33/0.58 ( ( bit_se2925701944663578781it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 0.33/0.58 = ( semiri1316708129612266289at_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_of_nat
% 0.33/0.58 thf(fact_4184_concat__bit__eq__iff,axiom,
% 0.33/0.58 ! [N: nat,K: int,L: int,R: int,S: int] :
% 0.33/0.58 ( ( ( bit_concat_bit @ N @ K @ L )
% 0.33/0.58 = ( bit_concat_bit @ N @ R @ S ) )
% 0.33/0.58 = ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.33/0.58 = ( bit_se2923211474154528505it_int @ N @ R ) )
% 0.33/0.58 & ( L = S ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % concat_bit_eq_iff
% 0.33/0.58 thf(fact_4185_concat__bit__take__bit__eq,axiom,
% 0.33/0.58 ! [N: nat,B: int] :
% 0.33/0.58 ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B ) )
% 0.33/0.58 = ( bit_concat_bit @ N @ B ) ) ).
% 0.33/0.58
% 0.33/0.58 % concat_bit_take_bit_eq
% 0.33/0.58 thf(fact_4186_signed__take__bit__take__bit,axiom,
% 0.33/0.58 ! [M: nat,N: nat,A: int] :
% 0.33/0.58 ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.33/0.58 = ( if_int_int @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % signed_take_bit_take_bit
% 0.33/0.58 thf(fact_4187_signed__take__bit__eq__iff__take__bit__eq,axiom,
% 0.33/0.58 ! [N: nat,A: int,B: int] :
% 0.33/0.58 ( ( ( bit_ri631733984087533419it_int @ N @ A )
% 0.33/0.58 = ( bit_ri631733984087533419it_int @ N @ B ) )
% 0.33/0.58 = ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 0.33/0.58 = ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % signed_take_bit_eq_iff_take_bit_eq
% 0.33/0.58 thf(fact_4188_log__def,axiom,
% 0.33/0.58 ( log2
% 0.33/0.58 = ( ^ [A2: real,X3: real] : ( divide_divide_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ A2 ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_def
% 0.33/0.58 thf(fact_4189_take__bit__add,axiom,
% 0.33/0.58 ! [N: nat,A: int,B: int] :
% 0.33/0.58 ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) )
% 0.33/0.58 = ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_add
% 0.33/0.58 thf(fact_4190_take__bit__add,axiom,
% 0.33/0.58 ! [N: nat,A: nat,B: nat] :
% 0.33/0.58 ( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) )
% 0.33/0.58 = ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_add
% 0.33/0.58 thf(fact_4191_take__bit__tightened,axiom,
% 0.33/0.58 ! [N: nat,A: int,B: int,M: nat] :
% 0.33/0.58 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.33/0.58 = ( bit_se2923211474154528505it_int @ N @ B ) )
% 0.33/0.58 => ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.58 => ( ( bit_se2923211474154528505it_int @ M @ A )
% 0.33/0.58 = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_tightened
% 0.33/0.58 thf(fact_4192_take__bit__tightened,axiom,
% 0.33/0.58 ! [N: nat,A: nat,B: nat,M: nat] :
% 0.33/0.58 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.33/0.58 = ( bit_se2925701944663578781it_nat @ N @ B ) )
% 0.33/0.58 => ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.58 => ( ( bit_se2925701944663578781it_nat @ M @ A )
% 0.33/0.58 = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_tightened
% 0.33/0.58 thf(fact_4193_take__bit__signed__take__bit,axiom,
% 0.33/0.58 ! [M: nat,N: nat,A: int] :
% 0.33/0.58 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.33/0.58 => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A ) )
% 0.33/0.58 = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_signed_take_bit
% 0.33/0.58 thf(fact_4194_take__bit__nat__less__eq__self,axiom,
% 0.33/0.58 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_nat_less_eq_self
% 0.33/0.58 thf(fact_4195_take__bit__tightened__less__eq__nat,axiom,
% 0.33/0.58 ! [M: nat,N: nat,Q2: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.58 => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_tightened_less_eq_nat
% 0.33/0.58 thf(fact_4196_sgn__eq__0__iff,axiom,
% 0.33/0.58 ! [A: complex] :
% 0.33/0.58 ( ( ( sgn_sgn_complex @ A )
% 0.33/0.58 = zero_zero_complex )
% 0.33/0.58 = ( A = zero_zero_complex ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_eq_0_iff
% 0.33/0.58 thf(fact_4197_sgn__eq__0__iff,axiom,
% 0.33/0.58 ! [A: code_integer] :
% 0.33/0.58 ( ( ( sgn_sgn_Code_integer @ A )
% 0.33/0.58 = zero_z3403309356797280102nteger )
% 0.33/0.58 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_eq_0_iff
% 0.33/0.58 thf(fact_4198_sgn__eq__0__iff,axiom,
% 0.33/0.58 ! [A: real] :
% 0.33/0.58 ( ( ( sgn_sgn_real @ A )
% 0.33/0.58 = zero_zero_real )
% 0.33/0.58 = ( A = zero_zero_real ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_eq_0_iff
% 0.33/0.58 thf(fact_4199_sgn__eq__0__iff,axiom,
% 0.33/0.58 ! [A: rat] :
% 0.33/0.58 ( ( ( sgn_sgn_rat @ A )
% 0.33/0.58 = zero_zero_rat )
% 0.33/0.58 = ( A = zero_zero_rat ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_eq_0_iff
% 0.33/0.58 thf(fact_4200_sgn__eq__0__iff,axiom,
% 0.33/0.58 ! [A: int] :
% 0.33/0.58 ( ( ( sgn_sgn_int @ A )
% 0.33/0.58 = zero_zero_int )
% 0.33/0.58 = ( A = zero_zero_int ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_eq_0_iff
% 0.33/0.58 thf(fact_4201_sgn__0__0,axiom,
% 0.33/0.58 ! [A: code_integer] :
% 0.33/0.58 ( ( ( sgn_sgn_Code_integer @ A )
% 0.33/0.58 = zero_z3403309356797280102nteger )
% 0.33/0.58 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_0_0
% 0.33/0.58 thf(fact_4202_sgn__0__0,axiom,
% 0.33/0.58 ! [A: real] :
% 0.33/0.58 ( ( ( sgn_sgn_real @ A )
% 0.33/0.58 = zero_zero_real )
% 0.33/0.58 = ( A = zero_zero_real ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_0_0
% 0.33/0.58 thf(fact_4203_sgn__0__0,axiom,
% 0.33/0.58 ! [A: rat] :
% 0.33/0.58 ( ( ( sgn_sgn_rat @ A )
% 0.33/0.58 = zero_zero_rat )
% 0.33/0.58 = ( A = zero_zero_rat ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_0_0
% 0.33/0.58 thf(fact_4204_sgn__0__0,axiom,
% 0.33/0.58 ! [A: int] :
% 0.33/0.58 ( ( ( sgn_sgn_int @ A )
% 0.33/0.58 = zero_zero_int )
% 0.33/0.58 = ( A = zero_zero_int ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_0_0
% 0.33/0.58 thf(fact_4205_sgn__zero__iff,axiom,
% 0.33/0.58 ! [X: complex] :
% 0.33/0.58 ( ( ( sgn_sgn_complex @ X )
% 0.33/0.58 = zero_zero_complex )
% 0.33/0.58 = ( X = zero_zero_complex ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_zero_iff
% 0.33/0.58 thf(fact_4206_sgn__zero__iff,axiom,
% 0.33/0.58 ! [X: real] :
% 0.33/0.58 ( ( ( sgn_sgn_real @ X )
% 0.33/0.58 = zero_zero_real )
% 0.33/0.58 = ( X = zero_zero_real ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_zero_iff
% 0.33/0.58 thf(fact_4207_sgn__mult,axiom,
% 0.33/0.58 ! [A: complex,B: complex] :
% 0.33/0.58 ( ( sgn_sgn_complex @ ( times_times_complex @ A @ B ) )
% 0.33/0.58 = ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_mult
% 0.33/0.58 thf(fact_4208_sgn__mult,axiom,
% 0.33/0.58 ! [A: code_integer,B: code_integer] :
% 0.33/0.58 ( ( sgn_sgn_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.33/0.58 = ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ B ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_mult
% 0.33/0.58 thf(fact_4209_sgn__mult,axiom,
% 0.33/0.58 ! [A: real,B: real] :
% 0.33/0.58 ( ( sgn_sgn_real @ ( times_times_real @ A @ B ) )
% 0.33/0.58 = ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_mult
% 0.33/0.58 thf(fact_4210_sgn__mult,axiom,
% 0.33/0.58 ! [A: rat,B: rat] :
% 0.33/0.58 ( ( sgn_sgn_rat @ ( times_times_rat @ A @ B ) )
% 0.33/0.58 = ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_mult
% 0.33/0.58 thf(fact_4211_sgn__mult,axiom,
% 0.33/0.58 ! [A: int,B: int] :
% 0.33/0.58 ( ( sgn_sgn_int @ ( times_times_int @ A @ B ) )
% 0.33/0.58 = ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_mult
% 0.33/0.58 thf(fact_4212_Real__Vector__Spaces_Osgn__mult,axiom,
% 0.33/0.58 ! [X: complex,Y: complex] :
% 0.33/0.58 ( ( sgn_sgn_complex @ ( times_times_complex @ X @ Y ) )
% 0.33/0.58 = ( times_times_complex @ ( sgn_sgn_complex @ X ) @ ( sgn_sgn_complex @ Y ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % Real_Vector_Spaces.sgn_mult
% 0.33/0.58 thf(fact_4213_Real__Vector__Spaces_Osgn__mult,axiom,
% 0.33/0.58 ! [X: real,Y: real] :
% 0.33/0.58 ( ( sgn_sgn_real @ ( times_times_real @ X @ Y ) )
% 0.33/0.58 = ( times_times_real @ ( sgn_sgn_real @ X ) @ ( sgn_sgn_real @ Y ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % Real_Vector_Spaces.sgn_mult
% 0.33/0.58 thf(fact_4214_same__sgn__sgn__add,axiom,
% 0.33/0.58 ! [B: code_integer,A: code_integer] :
% 0.33/0.58 ( ( ( sgn_sgn_Code_integer @ B )
% 0.33/0.58 = ( sgn_sgn_Code_integer @ A ) )
% 0.33/0.58 => ( ( sgn_sgn_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.33/0.58 = ( sgn_sgn_Code_integer @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % same_sgn_sgn_add
% 0.33/0.58 thf(fact_4215_same__sgn__sgn__add,axiom,
% 0.33/0.58 ! [B: real,A: real] :
% 0.33/0.58 ( ( ( sgn_sgn_real @ B )
% 0.33/0.58 = ( sgn_sgn_real @ A ) )
% 0.33/0.58 => ( ( sgn_sgn_real @ ( plus_plus_real @ A @ B ) )
% 0.33/0.58 = ( sgn_sgn_real @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % same_sgn_sgn_add
% 0.33/0.58 thf(fact_4216_same__sgn__sgn__add,axiom,
% 0.33/0.58 ! [B: rat,A: rat] :
% 0.33/0.58 ( ( ( sgn_sgn_rat @ B )
% 0.33/0.58 = ( sgn_sgn_rat @ A ) )
% 0.33/0.58 => ( ( sgn_sgn_rat @ ( plus_plus_rat @ A @ B ) )
% 0.33/0.58 = ( sgn_sgn_rat @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % same_sgn_sgn_add
% 0.33/0.58 thf(fact_4217_same__sgn__sgn__add,axiom,
% 0.33/0.58 ! [B: int,A: int] :
% 0.33/0.58 ( ( ( sgn_sgn_int @ B )
% 0.33/0.58 = ( sgn_sgn_int @ A ) )
% 0.33/0.58 => ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B ) )
% 0.33/0.58 = ( sgn_sgn_int @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % same_sgn_sgn_add
% 0.33/0.58 thf(fact_4218_fact__nonzero,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri773545260158071498ct_rat @ N )
% 0.33/0.58 != zero_zero_rat ) ).
% 0.33/0.58
% 0.33/0.58 % fact_nonzero
% 0.33/0.58 thf(fact_4219_fact__nonzero,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri1406184849735516958ct_int @ N )
% 0.33/0.58 != zero_zero_int ) ).
% 0.33/0.58
% 0.33/0.58 % fact_nonzero
% 0.33/0.58 thf(fact_4220_fact__nonzero,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri1408675320244567234ct_nat @ N )
% 0.33/0.58 != zero_zero_nat ) ).
% 0.33/0.58
% 0.33/0.58 % fact_nonzero
% 0.33/0.58 thf(fact_4221_fact__nonzero,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( semiri2265585572941072030t_real @ N )
% 0.33/0.58 != zero_zero_real ) ).
% 0.33/0.58
% 0.33/0.58 % fact_nonzero
% 0.33/0.58 thf(fact_4222_take__bit__mult,axiom,
% 0.33/0.58 ! [N: nat,K: int,L: int] :
% 0.33/0.58 ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
% 0.33/0.58 = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_mult
% 0.33/0.58 thf(fact_4223_take__bit__diff,axiom,
% 0.33/0.58 ! [N: nat,K: int,L: int] :
% 0.33/0.58 ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
% 0.33/0.58 = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_diff
% 0.33/0.58 thf(fact_4224_signed__take__bit__mult,axiom,
% 0.33/0.58 ! [N: nat,K: int,L: int] :
% 0.33/0.58 ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 0.33/0.58 = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % signed_take_bit_mult
% 0.33/0.58 thf(fact_4225_signed__take__bit__add,axiom,
% 0.33/0.58 ! [N: nat,K: int,L: int] :
% 0.33/0.58 ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 0.33/0.58 = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % signed_take_bit_add
% 0.33/0.58 thf(fact_4226_signed__take__bit__diff,axiom,
% 0.33/0.58 ! [N: nat,K: int,L: int] :
% 0.33/0.58 ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 0.33/0.58 = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % signed_take_bit_diff
% 0.33/0.58 thf(fact_4227_concat__bit__eq,axiom,
% 0.33/0.58 ( bit_concat_bit
% 0.33/0.58 = ( ^ [N3: nat,K2: int,L2: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N3 @ K2 ) @ ( bit_se545348938243370406it_int @ N3 @ L2 ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % concat_bit_eq
% 0.33/0.58 thf(fact_4228_concat__bit__assoc,axiom,
% 0.33/0.58 ! [N: nat,K: int,M: nat,L: int,R: int] :
% 0.33/0.58 ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R ) )
% 0.33/0.58 = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R ) ) ).
% 0.33/0.58
% 0.33/0.58 % concat_bit_assoc
% 0.33/0.58 thf(fact_4229_fact__less__mono__nat,axiom,
% 0.33/0.58 ! [M: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.58 => ( ( ord_less_nat @ M @ N )
% 0.33/0.58 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_less_mono_nat
% 0.33/0.58 thf(fact_4230_take__bit__tightened__less__eq__int,axiom,
% 0.33/0.58 ! [M: nat,N: nat,K: int] :
% 0.33/0.58 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.58 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_tightened_less_eq_int
% 0.33/0.58 thf(fact_4231_take__bit__nonnegative,axiom,
% 0.33/0.58 ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_nonnegative
% 0.33/0.58 thf(fact_4232_take__bit__int__less__eq__self__iff,axiom,
% 0.33/0.58 ! [N: nat,K: int] :
% 0.33/0.58 ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.33/0.58 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_int_less_eq_self_iff
% 0.33/0.58 thf(fact_4233_not__take__bit__negative,axiom,
% 0.33/0.58 ! [N: nat,K: int] :
% 0.33/0.58 ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 0.33/0.58
% 0.33/0.58 % not_take_bit_negative
% 0.33/0.58 thf(fact_4234_take__bit__int__greater__self__iff,axiom,
% 0.33/0.58 ! [K: int,N: nat] :
% 0.33/0.58 ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.33/0.58 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_int_greater_self_iff
% 0.33/0.58 thf(fact_4235_push__bit__take__bit,axiom,
% 0.33/0.58 ! [M: nat,N: nat,A: code_integer] :
% 0.33/0.58 ( ( bit_se7788150548672797655nteger @ M @ ( bit_se1745604003318907178nteger @ N @ A ) )
% 0.33/0.58 = ( bit_se1745604003318907178nteger @ ( plus_plus_nat @ M @ N ) @ ( bit_se7788150548672797655nteger @ M @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % push_bit_take_bit
% 0.33/0.58 thf(fact_4236_push__bit__take__bit,axiom,
% 0.33/0.58 ! [M: nat,N: nat,A: code_natural] :
% 0.33/0.58 ( ( bit_se6611745700429515170atural @ M @ ( bit_se569199155075624693atural @ N @ A ) )
% 0.33/0.58 = ( bit_se569199155075624693atural @ ( plus_plus_nat @ M @ N ) @ ( bit_se6611745700429515170atural @ M @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % push_bit_take_bit
% 0.33/0.58 thf(fact_4237_push__bit__take__bit,axiom,
% 0.33/0.58 ! [M: nat,N: nat,A: int] :
% 0.33/0.58 ( ( bit_se545348938243370406it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.33/0.58 = ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ M @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % push_bit_take_bit
% 0.33/0.58 thf(fact_4238_push__bit__take__bit,axiom,
% 0.33/0.58 ! [M: nat,N: nat,A: nat] :
% 0.33/0.58 ( ( bit_se547839408752420682it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 0.33/0.58 = ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N ) @ ( bit_se547839408752420682it_nat @ M @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % push_bit_take_bit
% 0.33/0.58 thf(fact_4239_fact__ge__zero,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_ge_zero
% 0.33/0.58 thf(fact_4240_fact__ge__zero,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_ge_zero
% 0.33/0.58 thf(fact_4241_fact__ge__zero,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_ge_zero
% 0.33/0.58 thf(fact_4242_fact__ge__zero,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_ge_zero
% 0.33/0.58 thf(fact_4243_take__bit__push__bit,axiom,
% 0.33/0.58 ! [M: nat,N: nat,A: code_integer] :
% 0.33/0.58 ( ( bit_se1745604003318907178nteger @ M @ ( bit_se7788150548672797655nteger @ N @ A ) )
% 0.33/0.58 = ( bit_se7788150548672797655nteger @ N @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_push_bit
% 0.33/0.58 thf(fact_4244_take__bit__push__bit,axiom,
% 0.33/0.58 ! [M: nat,N: nat,A: code_natural] :
% 0.33/0.58 ( ( bit_se569199155075624693atural @ M @ ( bit_se6611745700429515170atural @ N @ A ) )
% 0.33/0.58 = ( bit_se6611745700429515170atural @ N @ ( bit_se569199155075624693atural @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_push_bit
% 0.33/0.58 thf(fact_4245_take__bit__push__bit,axiom,
% 0.33/0.58 ! [M: nat,N: nat,A: int] :
% 0.33/0.58 ( ( bit_se2923211474154528505it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
% 0.33/0.58 = ( bit_se545348938243370406it_int @ N @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_push_bit
% 0.33/0.58 thf(fact_4246_take__bit__push__bit,axiom,
% 0.33/0.58 ! [M: nat,N: nat,A: nat] :
% 0.33/0.58 ( ( bit_se2925701944663578781it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 0.33/0.58 = ( bit_se547839408752420682it_nat @ N @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_push_bit
% 0.33/0.58 thf(fact_4247_fact__gt__zero,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_gt_zero
% 0.33/0.58 thf(fact_4248_fact__gt__zero,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_gt_zero
% 0.33/0.58 thf(fact_4249_fact__gt__zero,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_gt_zero
% 0.33/0.58 thf(fact_4250_fact__gt__zero,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_gt_zero
% 0.33/0.58 thf(fact_4251_fact__not__neg,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% 0.33/0.58
% 0.33/0.58 % fact_not_neg
% 0.33/0.58 thf(fact_4252_fact__not__neg,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% 0.33/0.58
% 0.33/0.58 % fact_not_neg
% 0.33/0.58 thf(fact_4253_fact__not__neg,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% 0.33/0.58
% 0.33/0.58 % fact_not_neg
% 0.33/0.58 thf(fact_4254_fact__not__neg,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% 0.33/0.58
% 0.33/0.58 % fact_not_neg
% 0.33/0.58 thf(fact_4255_int__sgnE,axiom,
% 0.33/0.58 ! [K: int] :
% 0.33/0.58 ~ ! [N2: nat,L3: int] :
% 0.33/0.58 ( K
% 0.33/0.58 != ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % int_sgnE
% 0.33/0.58 thf(fact_4256_fact__ge__1,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_ge_1
% 0.33/0.58 thf(fact_4257_fact__ge__1,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_ge_1
% 0.33/0.58 thf(fact_4258_fact__ge__1,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_ge_1
% 0.33/0.58 thf(fact_4259_fact__ge__1,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_ge_1
% 0.33/0.58 thf(fact_4260_take__bit__unset__bit__eq,axiom,
% 0.33/0.58 ! [N: nat,M: nat,A: int] :
% 0.33/0.58 ( ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.58 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.33/0.58 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.33/0.58 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.33/0.58 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.33/0.58 = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_unset_bit_eq
% 0.33/0.58 thf(fact_4261_take__bit__unset__bit__eq,axiom,
% 0.33/0.58 ! [N: nat,M: nat,A: nat] :
% 0.33/0.58 ( ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.58 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.33/0.58 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.33/0.58 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.33/0.58 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.33/0.58 = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_unset_bit_eq
% 0.33/0.58 thf(fact_4262_take__bit__set__bit__eq,axiom,
% 0.33/0.58 ! [N: nat,M: nat,A: int] :
% 0.33/0.58 ( ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.58 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.33/0.58 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.33/0.58 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.33/0.58 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.33/0.58 = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_set_bit_eq
% 0.33/0.58 thf(fact_4263_take__bit__set__bit__eq,axiom,
% 0.33/0.58 ! [N: nat,M: nat,A: nat] :
% 0.33/0.58 ( ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.58 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.33/0.58 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.33/0.58 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.33/0.58 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.33/0.58 = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_set_bit_eq
% 0.33/0.58 thf(fact_4264_take__bit__flip__bit__eq,axiom,
% 0.33/0.58 ! [N: nat,M: nat,A: int] :
% 0.33/0.58 ( ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.58 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.33/0.58 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.33/0.58 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.33/0.58 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.33/0.58 = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_flip_bit_eq
% 0.33/0.58 thf(fact_4265_take__bit__flip__bit__eq,axiom,
% 0.33/0.58 ! [N: nat,M: nat,A: nat] :
% 0.33/0.58 ( ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.58 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.33/0.58 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.33/0.58 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.33/0.58 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.33/0.58 = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_flip_bit_eq
% 0.33/0.58 thf(fact_4266_fact__dvd,axiom,
% 0.33/0.58 ! [N: nat,M: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.58 => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_dvd
% 0.33/0.58 thf(fact_4267_fact__dvd,axiom,
% 0.33/0.58 ! [N: nat,M: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.58 => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_dvd
% 0.33/0.58 thf(fact_4268_fact__dvd,axiom,
% 0.33/0.58 ! [N: nat,M: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.58 => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_dvd
% 0.33/0.58 thf(fact_4269_fact__dvd,axiom,
% 0.33/0.58 ! [N: nat,M: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.58 => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_dvd
% 0.33/0.58 thf(fact_4270_pochhammer__fact,axiom,
% 0.33/0.58 ( semiri5044797733671781792omplex
% 0.33/0.58 = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).
% 0.33/0.58
% 0.33/0.58 % pochhammer_fact
% 0.33/0.58 thf(fact_4271_pochhammer__fact,axiom,
% 0.33/0.58 ( semiri773545260158071498ct_rat
% 0.33/0.58 = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% 0.33/0.58
% 0.33/0.58 % pochhammer_fact
% 0.33/0.58 thf(fact_4272_pochhammer__fact,axiom,
% 0.33/0.58 ( semiri1406184849735516958ct_int
% 0.33/0.58 = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% 0.33/0.58
% 0.33/0.58 % pochhammer_fact
% 0.33/0.58 thf(fact_4273_pochhammer__fact,axiom,
% 0.33/0.58 ( semiri1408675320244567234ct_nat
% 0.33/0.58 = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% 0.33/0.58
% 0.33/0.58 % pochhammer_fact
% 0.33/0.58 thf(fact_4274_pochhammer__fact,axiom,
% 0.33/0.58 ( semiri2265585572941072030t_real
% 0.33/0.58 = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).
% 0.33/0.58
% 0.33/0.58 % pochhammer_fact
% 0.33/0.58 thf(fact_4275_log__eq__div__ln__mult__log,axiom,
% 0.33/0.58 ! [A: real,B: real,X: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58 => ( ( A != one_one_real )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.58 => ( ( B != one_one_real )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( log2 @ A @ X )
% 0.33/0.58 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log2 @ B @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_eq_div_ln_mult_log
% 0.33/0.58 thf(fact_4276_ln__gt__zero__imp__gt__one,axiom,
% 0.33/0.58 ! [X: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_gt_zero_imp_gt_one
% 0.33/0.58 thf(fact_4277_ln__less__zero,axiom,
% 0.33/0.58 ! [X: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.58 => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_less_zero
% 0.33/0.58 thf(fact_4278_ln__gt__zero,axiom,
% 0.33/0.58 ! [X: real] :
% 0.33/0.58 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.58 => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_gt_zero
% 0.33/0.58 thf(fact_4279_ln__ge__zero,axiom,
% 0.33/0.58 ! [X: real] :
% 0.33/0.58 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.33/0.58 => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_ge_zero
% 0.33/0.58 thf(fact_4280_fact__ge__Suc__0__nat,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_ge_Suc_0_nat
% 0.33/0.58 thf(fact_4281_sgn__1__pos,axiom,
% 0.33/0.58 ! [A: code_integer] :
% 0.33/0.58 ( ( ( sgn_sgn_Code_integer @ A )
% 0.33/0.58 = one_one_Code_integer )
% 0.33/0.58 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_1_pos
% 0.33/0.58 thf(fact_4282_sgn__1__pos,axiom,
% 0.33/0.58 ! [A: real] :
% 0.33/0.58 ( ( ( sgn_sgn_real @ A )
% 0.33/0.58 = one_one_real )
% 0.33/0.58 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_1_pos
% 0.33/0.58 thf(fact_4283_sgn__1__pos,axiom,
% 0.33/0.58 ! [A: rat] :
% 0.33/0.58 ( ( ( sgn_sgn_rat @ A )
% 0.33/0.58 = one_one_rat )
% 0.33/0.58 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_1_pos
% 0.33/0.58 thf(fact_4284_sgn__1__pos,axiom,
% 0.33/0.58 ! [A: int] :
% 0.33/0.58 ( ( ( sgn_sgn_int @ A )
% 0.33/0.58 = one_one_int )
% 0.33/0.58 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_1_pos
% 0.33/0.58 thf(fact_4285_dvd__fact,axiom,
% 0.33/0.58 ! [M: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.33/0.58 => ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.58 => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % dvd_fact
% 0.33/0.58 thf(fact_4286_take__bit__decr__eq,axiom,
% 0.33/0.58 ! [N: nat,K: int] :
% 0.33/0.58 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.33/0.58 != zero_zero_int )
% 0.33/0.58 => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
% 0.33/0.58 = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_decr_eq
% 0.33/0.58 thf(fact_4287_fact__less__mono,axiom,
% 0.33/0.58 ! [M: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.58 => ( ( ord_less_nat @ M @ N )
% 0.33/0.58 => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_less_mono
% 0.33/0.58 thf(fact_4288_fact__less__mono,axiom,
% 0.33/0.58 ! [M: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.58 => ( ( ord_less_nat @ M @ N )
% 0.33/0.58 => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_less_mono
% 0.33/0.58 thf(fact_4289_fact__less__mono,axiom,
% 0.33/0.58 ! [M: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.58 => ( ( ord_less_nat @ M @ N )
% 0.33/0.58 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_less_mono
% 0.33/0.58 thf(fact_4290_fact__less__mono,axiom,
% 0.33/0.58 ! [M: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.58 => ( ( ord_less_nat @ M @ N )
% 0.33/0.58 => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_less_mono
% 0.33/0.58 thf(fact_4291_sgn__mod,axiom,
% 0.33/0.58 ! [L: int,K: int] :
% 0.33/0.58 ( ( L != zero_zero_int )
% 0.33/0.58 => ( ~ ( dvd_dvd_int @ L @ K )
% 0.33/0.58 => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
% 0.33/0.58 = ( sgn_sgn_int @ L ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % sgn_mod
% 0.33/0.58 thf(fact_4292_fact__fact__dvd__fact,axiom,
% 0.33/0.58 ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_fact_dvd_fact
% 0.33/0.58 thf(fact_4293_fact__fact__dvd__fact,axiom,
% 0.33/0.58 ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_fact_dvd_fact
% 0.33/0.58 thf(fact_4294_fact__fact__dvd__fact,axiom,
% 0.33/0.58 ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_fact_dvd_fact
% 0.33/0.58 thf(fact_4295_fact__fact__dvd__fact,axiom,
% 0.33/0.58 ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_fact_dvd_fact
% 0.33/0.58 thf(fact_4296_fact__fact__dvd__fact,axiom,
% 0.33/0.58 ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_fact_dvd_fact
% 0.33/0.58 thf(fact_4297_fact__mod,axiom,
% 0.33/0.58 ! [M: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.58 => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
% 0.33/0.58 = zero_zero_int ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_mod
% 0.33/0.58 thf(fact_4298_fact__mod,axiom,
% 0.33/0.58 ! [M: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.58 => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
% 0.33/0.58 = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_mod
% 0.33/0.58 thf(fact_4299_fact__mod,axiom,
% 0.33/0.58 ! [M: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.58 => ( ( modulo8411746178871703098atural @ ( semiri2447717529341329178atural @ N ) @ ( semiri2447717529341329178atural @ M ) )
% 0.33/0.58 = zero_z2226904508553997617atural ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_mod
% 0.33/0.58 thf(fact_4300_fact__mod,axiom,
% 0.33/0.58 ! [M: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.58 => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 0.33/0.58 = zero_zero_nat ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_mod
% 0.33/0.58 thf(fact_4301_fact__le__power,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_le_power
% 0.33/0.58 thf(fact_4302_fact__le__power,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_le_power
% 0.33/0.58 thf(fact_4303_fact__le__power,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_le_power
% 0.33/0.58 thf(fact_4304_fact__le__power,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_le_power
% 0.33/0.58 thf(fact_4305_signed__take__bit__eq__take__bit__shift,axiom,
% 0.33/0.58 ( bit_ri631733984087533419it_int
% 0.33/0.58 = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % signed_take_bit_eq_take_bit_shift
% 0.33/0.58 thf(fact_4306_ln__ge__zero__imp__ge__one,axiom,
% 0.33/0.58 ! [X: real] :
% 0.33/0.58 ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_ge_zero_imp_ge_one
% 0.33/0.58 thf(fact_4307_ln__add__one__self__le__self,axiom,
% 0.33/0.58 ! [X: real] :
% 0.33/0.58 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_add_one_self_le_self
% 0.33/0.58 thf(fact_4308_ln__mult,axiom,
% 0.33/0.58 ! [X: real,Y: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.58 => ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
% 0.33/0.58 = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_mult
% 0.33/0.58 thf(fact_4309_ln__eq__minus__one,axiom,
% 0.33/0.58 ! [X: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ( ln_ln_real @ X )
% 0.33/0.58 = ( minus_minus_real @ X @ one_one_real ) )
% 0.33/0.58 => ( X = one_one_real ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_eq_minus_one
% 0.33/0.58 thf(fact_4310_ln__div,axiom,
% 0.33/0.58 ! [X: real,Y: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.58 => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
% 0.33/0.58 = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_div
% 0.33/0.58 thf(fact_4311_log__base__change,axiom,
% 0.33/0.58 ! [A: real,B: real,X: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58 => ( ( A != one_one_real )
% 0.33/0.58 => ( ( log2 @ B @ X )
% 0.33/0.58 = ( divide_divide_real @ ( log2 @ A @ X ) @ ( log2 @ A @ B ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_base_change
% 0.33/0.58 thf(fact_4312_less__log__of__power,axiom,
% 0.33/0.58 ! [B: real,N: nat,M: real] :
% 0.33/0.58 ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
% 0.33/0.58 => ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.58 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ B @ M ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % less_log_of_power
% 0.33/0.58 thf(fact_4313_log__of__power__eq,axiom,
% 0.33/0.58 ! [M: nat,B: real,N: nat] :
% 0.33/0.58 ( ( ( semiri5074537144036343181t_real @ M )
% 0.33/0.58 = ( power_power_real @ B @ N ) )
% 0.33/0.58 => ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.58 => ( ( semiri5074537144036343181t_real @ N )
% 0.33/0.58 = ( log2 @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_of_power_eq
% 0.33/0.58 thf(fact_4314_fact__div__fact__le__pow,axiom,
% 0.33/0.58 ! [R: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ R @ N )
% 0.33/0.58 => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R ) ) ) @ ( power_power_nat @ N @ R ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_div_fact_le_pow
% 0.33/0.58 thf(fact_4315_binomial__fact__lemma,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.58 => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 0.33/0.58 = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % binomial_fact_lemma
% 0.33/0.58 thf(fact_4316_norm__sgn,axiom,
% 0.33/0.58 ! [X: real] :
% 0.33/0.58 ( ( ( X = zero_zero_real )
% 0.33/0.58 => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
% 0.33/0.58 = zero_zero_real ) )
% 0.33/0.58 & ( ( X != zero_zero_real )
% 0.33/0.58 => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
% 0.33/0.58 = one_one_real ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % norm_sgn
% 0.33/0.58 thf(fact_4317_norm__sgn,axiom,
% 0.33/0.58 ! [X: complex] :
% 0.33/0.58 ( ( ( X = zero_zero_complex )
% 0.33/0.58 => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
% 0.33/0.58 = zero_zero_real ) )
% 0.33/0.58 & ( ( X != zero_zero_complex )
% 0.33/0.58 => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
% 0.33/0.58 = one_one_real ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % norm_sgn
% 0.33/0.58 thf(fact_4318_choose__dvd,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.58 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % choose_dvd
% 0.33/0.58 thf(fact_4319_choose__dvd,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.58 => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % choose_dvd
% 0.33/0.58 thf(fact_4320_choose__dvd,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.58 => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % choose_dvd
% 0.33/0.58 thf(fact_4321_choose__dvd,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.58 => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % choose_dvd
% 0.33/0.58 thf(fact_4322_choose__dvd,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.58 => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % choose_dvd
% 0.33/0.58 thf(fact_4323_take__bit__Suc__bit0,axiom,
% 0.33/0.58 ! [N: nat,K: num] :
% 0.33/0.58 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.33/0.58 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_Suc_bit0
% 0.33/0.58 thf(fact_4324_take__bit__Suc__bit0,axiom,
% 0.33/0.58 ! [N: nat,K: num] :
% 0.33/0.58 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.33/0.58 = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_Suc_bit0
% 0.33/0.58 thf(fact_4325_take__bit__eq__mod,axiom,
% 0.33/0.58 ( bit_se1745604003318907178nteger
% 0.33/0.58 = ( ^ [N3: nat,A2: code_integer] : ( modulo364778990260209775nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_eq_mod
% 0.33/0.58 thf(fact_4326_take__bit__eq__mod,axiom,
% 0.33/0.58 ( bit_se569199155075624693atural
% 0.33/0.58 = ( ^ [N3: nat,A2: code_natural] : ( modulo8411746178871703098atural @ A2 @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_eq_mod
% 0.33/0.58 thf(fact_4327_take__bit__eq__mod,axiom,
% 0.33/0.58 ( bit_se2923211474154528505it_int
% 0.33/0.58 = ( ^ [N3: nat,A2: int] : ( modulo_modulo_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_eq_mod
% 0.33/0.58 thf(fact_4328_take__bit__eq__mod,axiom,
% 0.33/0.58 ( bit_se2925701944663578781it_nat
% 0.33/0.58 = ( ^ [N3: nat,A2: nat] : ( modulo_modulo_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_eq_mod
% 0.33/0.58 thf(fact_4329_take__bit__nat__eq__self,axiom,
% 0.33/0.58 ! [M: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.58 => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.33/0.58 = M ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_nat_eq_self
% 0.33/0.58 thf(fact_4330_take__bit__nat__less__exp,axiom,
% 0.33/0.58 ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_nat_less_exp
% 0.33/0.58 thf(fact_4331_take__bit__nat__eq__self__iff,axiom,
% 0.33/0.58 ! [N: nat,M: nat] :
% 0.33/0.58 ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.33/0.58 = M )
% 0.33/0.58 = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_nat_eq_self_iff
% 0.33/0.58 thf(fact_4332_ln__le__minus__one,axiom,
% 0.33/0.58 ! [X: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_le_minus_one
% 0.33/0.58 thf(fact_4333_ln__diff__le,axiom,
% 0.33/0.58 ! [X: real,Y: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.58 => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_diff_le
% 0.33/0.58 thf(fact_4334_ln__realpow,axiom,
% 0.33/0.58 ! [X: real,N: nat] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
% 0.33/0.58 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % ln_realpow
% 0.33/0.58 thf(fact_4335_take__bit__nat__def,axiom,
% 0.33/0.58 ( bit_se2925701944663578781it_nat
% 0.33/0.58 = ( ^ [N3: nat,M3: nat] : ( modulo_modulo_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_nat_def
% 0.33/0.58 thf(fact_4336_log__mult,axiom,
% 0.33/0.58 ! [A: real,X: real,Y: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58 => ( ( A != one_one_real )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.58 => ( ( log2 @ A @ ( times_times_real @ X @ Y ) )
% 0.33/0.58 = ( plus_plus_real @ ( log2 @ A @ X ) @ ( log2 @ A @ Y ) ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_mult
% 0.33/0.58 thf(fact_4337_log__divide,axiom,
% 0.33/0.58 ! [A: real,X: real,Y: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58 => ( ( A != one_one_real )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.58 => ( ( log2 @ A @ ( divide_divide_real @ X @ Y ) )
% 0.33/0.58 = ( minus_minus_real @ ( log2 @ A @ X ) @ ( log2 @ A @ Y ) ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_divide
% 0.33/0.58 thf(fact_4338_le__log__of__power,axiom,
% 0.33/0.58 ! [B: real,N: nat,M: real] :
% 0.33/0.58 ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
% 0.33/0.58 => ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.58 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ B @ M ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % le_log_of_power
% 0.33/0.58 thf(fact_4339_log__base__pow,axiom,
% 0.33/0.58 ! [A: real,N: nat,X: real] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58 => ( ( log2 @ ( power_power_real @ A @ N ) @ X )
% 0.33/0.58 = ( divide_divide_real @ ( log2 @ A @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_base_pow
% 0.33/0.58 thf(fact_4340_log__nat__power,axiom,
% 0.33/0.58 ! [X: real,B: real,N: nat] :
% 0.33/0.58 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58 => ( ( log2 @ B @ ( power_power_real @ X @ N ) )
% 0.33/0.58 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ B @ X ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_nat_power
% 0.33/0.58 thf(fact_4341_take__bit__int__less__exp,axiom,
% 0.33/0.58 ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_int_less_exp
% 0.33/0.58 thf(fact_4342_binomial__altdef__nat,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.58 => ( ( binomial @ N @ K )
% 0.33/0.58 = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % binomial_altdef_nat
% 0.33/0.58 thf(fact_4343_take__bit__int__def,axiom,
% 0.33/0.58 ( bit_se2923211474154528505it_int
% 0.33/0.58 = ( ^ [N3: nat,K2: int] : ( modulo_modulo_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_int_def
% 0.33/0.58 thf(fact_4344_signed__take__bit__int__less__exp,axiom,
% 0.33/0.58 ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.33/0.58
% 0.33/0.58 % signed_take_bit_int_less_exp
% 0.33/0.58 thf(fact_4345_even__signed__take__bit__iff,axiom,
% 0.33/0.58 ! [M: nat,A: code_integer] :
% 0.33/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
% 0.33/0.58 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % even_signed_take_bit_iff
% 0.33/0.58 thf(fact_4346_even__signed__take__bit__iff,axiom,
% 0.33/0.58 ! [M: nat,A: int] :
% 0.33/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
% 0.33/0.58 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % even_signed_take_bit_iff
% 0.33/0.58 thf(fact_4347_take__bit__eq__0__iff,axiom,
% 0.33/0.58 ! [N: nat,A: code_integer] :
% 0.33/0.58 ( ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.33/0.58 = zero_z3403309356797280102nteger )
% 0.33/0.58 = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_eq_0_iff
% 0.33/0.58 thf(fact_4348_take__bit__eq__0__iff,axiom,
% 0.33/0.58 ! [N: nat,A: int] :
% 0.33/0.58 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.33/0.58 = zero_zero_int )
% 0.33/0.58 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_eq_0_iff
% 0.33/0.58 thf(fact_4349_take__bit__eq__0__iff,axiom,
% 0.33/0.58 ! [N: nat,A: nat] :
% 0.33/0.58 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.33/0.58 = zero_zero_nat )
% 0.33/0.58 = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_eq_0_iff
% 0.33/0.58 thf(fact_4350_take__bit__nat__less__self__iff,axiom,
% 0.33/0.58 ! [N: nat,M: nat] :
% 0.33/0.58 ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 0.33/0.58 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_nat_less_self_iff
% 0.33/0.58 thf(fact_4351_square__fact__le__2__fact,axiom,
% 0.33/0.58 ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % square_fact_le_2_fact
% 0.33/0.58 thf(fact_4352_log__of__power__less,axiom,
% 0.33/0.58 ! [M: nat,B: real,N: nat] :
% 0.33/0.58 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 0.33/0.58 => ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.58 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.58 => ( ord_less_real @ ( log2 @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log_of_power_less
% 0.33/0.58 thf(fact_4353_log2__of__power__eq,axiom,
% 0.33/0.58 ! [M: nat,N: nat] :
% 0.33/0.58 ( ( M
% 0.33/0.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.58 => ( ( semiri5074537144036343181t_real @ N )
% 0.33/0.58 = ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % log2_of_power_eq
% 0.33/0.58 thf(fact_4354_take__bit__int__less__self__iff,axiom,
% 0.33/0.58 ! [N: nat,K: int] :
% 0.33/0.58 ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.33/0.58 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_int_less_self_iff
% 0.33/0.58 thf(fact_4355_take__bit__int__greater__eq__self__iff,axiom,
% 0.33/0.58 ! [K: int,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.33/0.58 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % take_bit_int_greater_eq_self_iff
% 0.33/0.58 thf(fact_4356_fact__num__eq__if,axiom,
% 0.33/0.58 ( semiri5044797733671781792omplex
% 0.33/0.58 = ( ^ [M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M3 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_num_eq_if
% 0.33/0.58 thf(fact_4357_fact__num__eq__if,axiom,
% 0.33/0.58 ( semiri773545260158071498ct_rat
% 0.33/0.58 = ( ^ [M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M3 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_num_eq_if
% 0.33/0.58 thf(fact_4358_fact__num__eq__if,axiom,
% 0.33/0.58 ( semiri1406184849735516958ct_int
% 0.33/0.58 = ( ^ [M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_num_eq_if
% 0.33/0.58 thf(fact_4359_fact__num__eq__if,axiom,
% 0.33/0.58 ( semiri1408675320244567234ct_nat
% 0.33/0.58 = ( ^ [M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_num_eq_if
% 0.33/0.58 thf(fact_4360_fact__num__eq__if,axiom,
% 0.33/0.58 ( semiri2265585572941072030t_real
% 0.33/0.58 = ( ^ [M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_num_eq_if
% 0.33/0.58 thf(fact_4361_fact__reduce,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.58 => ( ( semiri5044797733671781792omplex @ N )
% 0.33/0.58 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_reduce
% 0.33/0.58 thf(fact_4362_fact__reduce,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.58 => ( ( semiri773545260158071498ct_rat @ N )
% 0.33/0.58 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_reduce
% 0.33/0.58 thf(fact_4363_fact__reduce,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.58 => ( ( semiri1406184849735516958ct_int @ N )
% 0.33/0.58 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_reduce
% 0.33/0.58 thf(fact_4364_fact__reduce,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.58 => ( ( semiri1408675320244567234ct_nat @ N )
% 0.33/0.58 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_reduce
% 0.33/0.58 thf(fact_4365_fact__reduce,axiom,
% 0.33/0.58 ! [N: nat] :
% 0.33/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.58 => ( ( semiri2265585572941072030t_real @ N )
% 0.33/0.58 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_reduce
% 0.33/0.58 thf(fact_4366_signed__take__bit__int__less__self__iff,axiom,
% 0.33/0.58 ! [N: nat,K: int] :
% 0.33/0.58 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.33/0.58 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.33/0.58
% 0.33/0.58 % signed_take_bit_int_less_self_iff
% 0.33/0.58 thf(fact_4367_signed__take__bit__int__greater__eq__self__iff,axiom,
% 0.33/0.58 ! [K: int,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.33/0.58 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % signed_take_bit_int_greater_eq_self_iff
% 0.33/0.58 thf(fact_4368_fact__binomial,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.58 => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
% 0.33/0.58 = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_binomial
% 0.33/0.58 thf(fact_4369_fact__binomial,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.58 => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
% 0.33/0.58 = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_binomial
% 0.33/0.58 thf(fact_4370_fact__binomial,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.58 => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
% 0.33/0.58 = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % fact_binomial
% 0.33/0.58 thf(fact_4371_binomial__fact,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.58 => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 0.33/0.58 = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % binomial_fact
% 0.33/0.58 thf(fact_4372_binomial__fact,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.58 => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 0.33/0.58 = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.33/0.58
% 0.33/0.58 % binomial_fact
% 0.33/0.58 thf(fact_4373_binomial__fact,axiom,
% 0.33/0.58 ! [K: nat,N: nat] :
% 0.33/0.58 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.58 => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 0.33/0.58 = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % binomial_fact
% 0.33/0.59 thf(fact_4374_log__of__power__le,axiom,
% 0.33/0.59 ! [M: nat,B: real,N: nat] :
% 0.33/0.59 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 0.33/0.59 => ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.59 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.59 => ( ord_less_eq_real @ ( log2 @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % log_of_power_le
% 0.33/0.59 thf(fact_4375_take__bit__int__eq__self,axiom,
% 0.33/0.59 ! [K: int,N: nat] :
% 0.33/0.59 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.59 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.59 => ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.33/0.59 = K ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % take_bit_int_eq_self
% 0.33/0.59 thf(fact_4376_take__bit__int__eq__self__iff,axiom,
% 0.33/0.59 ! [N: nat,K: int] :
% 0.33/0.59 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.33/0.59 = K )
% 0.33/0.59 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.59 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % take_bit_int_eq_self_iff
% 0.33/0.59 thf(fact_4377_take__bit__incr__eq,axiom,
% 0.33/0.59 ! [N: nat,K: int] :
% 0.33/0.59 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.33/0.59 != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 0.33/0.59 => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 0.33/0.59 = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % take_bit_incr_eq
% 0.33/0.59 thf(fact_4378_signed__take__bit__int__less__eq,axiom,
% 0.33/0.59 ! [N: nat,K: int] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.33/0.59 => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % signed_take_bit_int_less_eq
% 0.33/0.59 thf(fact_4379_take__bit__Suc,axiom,
% 0.33/0.59 ! [N: nat,A: code_integer] :
% 0.33/0.59 ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
% 0.33/0.59 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % take_bit_Suc
% 0.33/0.59 thf(fact_4380_take__bit__Suc,axiom,
% 0.33/0.59 ! [N: nat,A: code_natural] :
% 0.33/0.59 ( ( bit_se569199155075624693atural @ ( suc @ N ) @ A )
% 0.33/0.59 = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % take_bit_Suc
% 0.33/0.59 thf(fact_4381_take__bit__Suc,axiom,
% 0.33/0.59 ! [N: nat,A: int] :
% 0.33/0.59 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 0.33/0.59 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % take_bit_Suc
% 0.33/0.59 thf(fact_4382_take__bit__Suc,axiom,
% 0.33/0.59 ! [N: nat,A: nat] :
% 0.33/0.59 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
% 0.33/0.59 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % take_bit_Suc
% 0.33/0.59 thf(fact_4383_less__log2__of__power,axiom,
% 0.33/0.59 ! [N: nat,M: nat] :
% 0.33/0.59 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.33/0.59 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % less_log2_of_power
% 0.33/0.59 thf(fact_4384_le__log2__of__power,axiom,
% 0.33/0.59 ! [N: nat,M: nat] :
% 0.33/0.59 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.33/0.59 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % le_log2_of_power
% 0.33/0.59 thf(fact_4385_take__bit__int__less__eq,axiom,
% 0.33/0.59 ! [N: nat,K: int] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.33/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.59 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % take_bit_int_less_eq
% 0.33/0.59 thf(fact_4386_take__bit__int__greater__eq,axiom,
% 0.33/0.59 ! [K: int,N: nat] :
% 0.33/0.59 ( ( ord_less_int @ K @ zero_zero_int )
% 0.33/0.59 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % take_bit_int_greater_eq
% 0.33/0.59 thf(fact_4387_stable__imp__take__bit__eq,axiom,
% 0.33/0.59 ! [A: code_integer,N: nat] :
% 0.33/0.59 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.59 = A )
% 0.33/0.59 => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.33/0.59 => ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.33/0.59 = zero_z3403309356797280102nteger ) )
% 0.33/0.59 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.33/0.59 => ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.33/0.59 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % stable_imp_take_bit_eq
% 0.33/0.59 thf(fact_4388_stable__imp__take__bit__eq,axiom,
% 0.33/0.59 ! [A: int,N: nat] :
% 0.33/0.59 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.59 = A )
% 0.33/0.59 => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.33/0.59 => ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.33/0.59 = zero_zero_int ) )
% 0.33/0.59 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.33/0.59 => ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.33/0.59 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % stable_imp_take_bit_eq
% 0.33/0.59 thf(fact_4389_stable__imp__take__bit__eq,axiom,
% 0.33/0.59 ! [A: nat,N: nat] :
% 0.33/0.59 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.59 = A )
% 0.33/0.59 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.33/0.59 => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.33/0.59 = zero_zero_nat ) )
% 0.33/0.59 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.33/0.59 => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.33/0.59 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % stable_imp_take_bit_eq
% 0.33/0.59 thf(fact_4390_ln__one__plus__pos__lower__bound,axiom,
% 0.33/0.59 ! [X: real] :
% 0.33/0.59 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.59 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.59 => ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ln_one_plus_pos_lower_bound
% 0.33/0.59 thf(fact_4391_log2__of__power__less,axiom,
% 0.33/0.59 ! [M: nat,N: nat] :
% 0.33/0.59 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.59 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.59 => ( ord_less_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % log2_of_power_less
% 0.33/0.59 thf(fact_4392_ln__2__less__1,axiom,
% 0.33/0.59 ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 0.33/0.59
% 0.33/0.59 % ln_2_less_1
% 0.33/0.59 thf(fact_4393_binomial__code,axiom,
% 0.33/0.59 ( binomial
% 0.33/0.59 = ( ^ [N3: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N3 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N3 @ ( minus_minus_nat @ N3 @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N3 @ K2 ) @ one_one_nat ) @ N3 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % binomial_code
% 0.33/0.59 thf(fact_4394_fact__diff__Suc,axiom,
% 0.33/0.59 ! [N: nat,M: nat] :
% 0.33/0.59 ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.33/0.59 => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 0.33/0.59 = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % fact_diff_Suc
% 0.33/0.59 thf(fact_4395_ceiling__log__nat__eq__powr__iff,axiom,
% 0.33/0.59 ! [B: nat,K: nat,N: nat] :
% 0.33/0.59 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.33/0.59 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.59 => ( ( ( archim7802044766580827645g_real @ ( log2 @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.33/0.59 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 0.33/0.59 = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.33/0.59 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_log_nat_eq_powr_iff
% 0.33/0.59 thf(fact_4396_ceiling__log__nat__eq__if,axiom,
% 0.33/0.59 ! [B: nat,N: nat,K: nat] :
% 0.33/0.59 ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.33/0.59 => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.33/0.59 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.33/0.59 => ( ( archim7802044766580827645g_real @ ( log2 @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.33/0.59 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_log_nat_eq_if
% 0.33/0.59 thf(fact_4397_ceiling__log2__div2,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( archim7802044766580827645g_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.33/0.59 = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_log2_div2
% 0.33/0.59 thf(fact_4398_tanh__ln__real,axiom,
% 0.33/0.59 ! [X: real] :
% 0.33/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.59 => ( ( tanh_real @ ( ln_ln_real @ X ) )
% 0.33/0.59 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % tanh_ln_real
% 0.33/0.59 thf(fact_4399_ln__one__minus__pos__lower__bound,axiom,
% 0.33/0.59 ! [X: real] :
% 0.33/0.59 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.59 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.59 => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ln_one_minus_pos_lower_bound
% 0.33/0.59 thf(fact_4400_verit__minus__simplify_I4_J,axiom,
% 0.33/0.59 ! [B: real] :
% 0.33/0.59 ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
% 0.33/0.59 = B ) ).
% 0.33/0.59
% 0.33/0.59 % verit_minus_simplify(4)
% 0.33/0.59 thf(fact_4401_verit__minus__simplify_I4_J,axiom,
% 0.33/0.59 ! [B: int] :
% 0.33/0.59 ( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
% 0.33/0.59 = B ) ).
% 0.33/0.59
% 0.33/0.59 % verit_minus_simplify(4)
% 0.33/0.59 thf(fact_4402_verit__minus__simplify_I4_J,axiom,
% 0.33/0.59 ! [B: complex] :
% 0.33/0.59 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B ) )
% 0.33/0.59 = B ) ).
% 0.33/0.59
% 0.33/0.59 % verit_minus_simplify(4)
% 0.33/0.59 thf(fact_4403_verit__minus__simplify_I4_J,axiom,
% 0.33/0.59 ! [B: rat] :
% 0.33/0.59 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B ) )
% 0.33/0.59 = B ) ).
% 0.33/0.59
% 0.33/0.59 % verit_minus_simplify(4)
% 0.33/0.59 thf(fact_4404_verit__minus__simplify_I4_J,axiom,
% 0.33/0.59 ! [B: code_integer] :
% 0.33/0.59 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.59 = B ) ).
% 0.33/0.59
% 0.33/0.59 % verit_minus_simplify(4)
% 0.33/0.59 thf(fact_4405_neg__equal__iff__equal,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( ( uminus_uminus_real @ A )
% 0.33/0.59 = ( uminus_uminus_real @ B ) )
% 0.33/0.59 = ( A = B ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_equal_iff_equal
% 0.33/0.59 thf(fact_4406_neg__equal__iff__equal,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( ( uminus_uminus_int @ A )
% 0.33/0.59 = ( uminus_uminus_int @ B ) )
% 0.33/0.59 = ( A = B ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_equal_iff_equal
% 0.33/0.59 thf(fact_4407_neg__equal__iff__equal,axiom,
% 0.33/0.59 ! [A: complex,B: complex] :
% 0.33/0.59 ( ( ( uminus1482373934393186551omplex @ A )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ B ) )
% 0.33/0.59 = ( A = B ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_equal_iff_equal
% 0.33/0.59 thf(fact_4408_neg__equal__iff__equal,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( ( uminus_uminus_rat @ A )
% 0.33/0.59 = ( uminus_uminus_rat @ B ) )
% 0.33/0.59 = ( A = B ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_equal_iff_equal
% 0.33/0.59 thf(fact_4409_neg__equal__iff__equal,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( ( uminus1351360451143612070nteger @ A )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.59 = ( A = B ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_equal_iff_equal
% 0.33/0.59 thf(fact_4410_add_Oinverse__inverse,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 0.33/0.59 = A ) ).
% 0.33/0.59
% 0.33/0.59 % add.inverse_inverse
% 0.33/0.59 thf(fact_4411_add_Oinverse__inverse,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 0.33/0.59 = A ) ).
% 0.33/0.59
% 0.33/0.59 % add.inverse_inverse
% 0.33/0.59 thf(fact_4412_add_Oinverse__inverse,axiom,
% 0.33/0.59 ! [A: complex] :
% 0.33/0.59 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
% 0.33/0.59 = A ) ).
% 0.33/0.59
% 0.33/0.59 % add.inverse_inverse
% 0.33/0.59 thf(fact_4413_add_Oinverse__inverse,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 0.33/0.59 = A ) ).
% 0.33/0.59
% 0.33/0.59 % add.inverse_inverse
% 0.33/0.59 thf(fact_4414_add_Oinverse__inverse,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.33/0.59 = A ) ).
% 0.33/0.59
% 0.33/0.59 % add.inverse_inverse
% 0.33/0.59 thf(fact_4415_neg__le__iff__le,axiom,
% 0.33/0.59 ! [B: real,A: real] :
% 0.33/0.59 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.33/0.59 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_le_iff_le
% 0.33/0.59 thf(fact_4416_neg__le__iff__le,axiom,
% 0.33/0.59 ! [B: code_integer,A: code_integer] :
% 0.33/0.59 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.33/0.59 = ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_le_iff_le
% 0.33/0.59 thf(fact_4417_neg__le__iff__le,axiom,
% 0.33/0.59 ! [B: rat,A: rat] :
% 0.33/0.59 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 0.33/0.59 = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_le_iff_le
% 0.33/0.59 thf(fact_4418_neg__le__iff__le,axiom,
% 0.33/0.59 ! [B: int,A: int] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.33/0.59 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_le_iff_le
% 0.33/0.59 thf(fact_4419_add_Oinverse__neutral,axiom,
% 0.33/0.59 ( ( uminus_uminus_real @ zero_zero_real )
% 0.33/0.59 = zero_zero_real ) ).
% 0.33/0.59
% 0.33/0.59 % add.inverse_neutral
% 0.33/0.59 thf(fact_4420_add_Oinverse__neutral,axiom,
% 0.33/0.59 ( ( uminus_uminus_int @ zero_zero_int )
% 0.33/0.59 = zero_zero_int ) ).
% 0.33/0.59
% 0.33/0.59 % add.inverse_neutral
% 0.33/0.59 thf(fact_4421_add_Oinverse__neutral,axiom,
% 0.33/0.59 ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 0.33/0.59 = zero_zero_complex ) ).
% 0.33/0.59
% 0.33/0.59 % add.inverse_neutral
% 0.33/0.59 thf(fact_4422_add_Oinverse__neutral,axiom,
% 0.33/0.59 ( ( uminus_uminus_rat @ zero_zero_rat )
% 0.33/0.59 = zero_zero_rat ) ).
% 0.33/0.59
% 0.33/0.59 % add.inverse_neutral
% 0.33/0.59 thf(fact_4423_add_Oinverse__neutral,axiom,
% 0.33/0.59 ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 0.33/0.59 = zero_z3403309356797280102nteger ) ).
% 0.33/0.59
% 0.33/0.59 % add.inverse_neutral
% 0.33/0.59 thf(fact_4424_neg__0__equal__iff__equal,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( zero_zero_real
% 0.33/0.59 = ( uminus_uminus_real @ A ) )
% 0.33/0.59 = ( zero_zero_real = A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_0_equal_iff_equal
% 0.33/0.59 thf(fact_4425_neg__0__equal__iff__equal,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( zero_zero_int
% 0.33/0.59 = ( uminus_uminus_int @ A ) )
% 0.33/0.59 = ( zero_zero_int = A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_0_equal_iff_equal
% 0.33/0.59 thf(fact_4426_neg__0__equal__iff__equal,axiom,
% 0.33/0.59 ! [A: complex] :
% 0.33/0.59 ( ( zero_zero_complex
% 0.33/0.59 = ( uminus1482373934393186551omplex @ A ) )
% 0.33/0.59 = ( zero_zero_complex = A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_0_equal_iff_equal
% 0.33/0.59 thf(fact_4427_neg__0__equal__iff__equal,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( zero_zero_rat
% 0.33/0.59 = ( uminus_uminus_rat @ A ) )
% 0.33/0.59 = ( zero_zero_rat = A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_0_equal_iff_equal
% 0.33/0.59 thf(fact_4428_neg__0__equal__iff__equal,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( zero_z3403309356797280102nteger
% 0.33/0.59 = ( uminus1351360451143612070nteger @ A ) )
% 0.33/0.59 = ( zero_z3403309356797280102nteger = A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_0_equal_iff_equal
% 0.33/0.59 thf(fact_4429_neg__equal__0__iff__equal,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( ( uminus_uminus_real @ A )
% 0.33/0.59 = zero_zero_real )
% 0.33/0.59 = ( A = zero_zero_real ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_equal_0_iff_equal
% 0.33/0.59 thf(fact_4430_neg__equal__0__iff__equal,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( ( uminus_uminus_int @ A )
% 0.33/0.59 = zero_zero_int )
% 0.33/0.59 = ( A = zero_zero_int ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_equal_0_iff_equal
% 0.33/0.59 thf(fact_4431_neg__equal__0__iff__equal,axiom,
% 0.33/0.59 ! [A: complex] :
% 0.33/0.59 ( ( ( uminus1482373934393186551omplex @ A )
% 0.33/0.59 = zero_zero_complex )
% 0.33/0.59 = ( A = zero_zero_complex ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_equal_0_iff_equal
% 0.33/0.59 thf(fact_4432_neg__equal__0__iff__equal,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( ( uminus_uminus_rat @ A )
% 0.33/0.59 = zero_zero_rat )
% 0.33/0.59 = ( A = zero_zero_rat ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_equal_0_iff_equal
% 0.33/0.59 thf(fact_4433_neg__equal__0__iff__equal,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( ( uminus1351360451143612070nteger @ A )
% 0.33/0.59 = zero_z3403309356797280102nteger )
% 0.33/0.59 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_equal_0_iff_equal
% 0.33/0.59 thf(fact_4434_equal__neg__zero,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( A
% 0.33/0.59 = ( uminus_uminus_real @ A ) )
% 0.33/0.59 = ( A = zero_zero_real ) ) ).
% 0.33/0.59
% 0.33/0.59 % equal_neg_zero
% 0.33/0.59 thf(fact_4435_equal__neg__zero,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( A
% 0.33/0.59 = ( uminus_uminus_int @ A ) )
% 0.33/0.59 = ( A = zero_zero_int ) ) ).
% 0.33/0.59
% 0.33/0.59 % equal_neg_zero
% 0.33/0.59 thf(fact_4436_equal__neg__zero,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( A
% 0.33/0.59 = ( uminus_uminus_rat @ A ) )
% 0.33/0.59 = ( A = zero_zero_rat ) ) ).
% 0.33/0.59
% 0.33/0.59 % equal_neg_zero
% 0.33/0.59 thf(fact_4437_equal__neg__zero,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( A
% 0.33/0.59 = ( uminus1351360451143612070nteger @ A ) )
% 0.33/0.59 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.59
% 0.33/0.59 % equal_neg_zero
% 0.33/0.59 thf(fact_4438_neg__equal__zero,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( ( uminus_uminus_real @ A )
% 0.33/0.59 = A )
% 0.33/0.59 = ( A = zero_zero_real ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_equal_zero
% 0.33/0.59 thf(fact_4439_neg__equal__zero,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( ( uminus_uminus_int @ A )
% 0.33/0.59 = A )
% 0.33/0.59 = ( A = zero_zero_int ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_equal_zero
% 0.33/0.59 thf(fact_4440_neg__equal__zero,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( ( uminus_uminus_rat @ A )
% 0.33/0.59 = A )
% 0.33/0.59 = ( A = zero_zero_rat ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_equal_zero
% 0.33/0.59 thf(fact_4441_neg__equal__zero,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( ( uminus1351360451143612070nteger @ A )
% 0.33/0.59 = A )
% 0.33/0.59 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_equal_zero
% 0.33/0.59 thf(fact_4442_neg__less__iff__less,axiom,
% 0.33/0.59 ! [B: real,A: real] :
% 0.33/0.59 ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.33/0.59 = ( ord_less_real @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_less_iff_less
% 0.33/0.59 thf(fact_4443_neg__less__iff__less,axiom,
% 0.33/0.59 ! [B: int,A: int] :
% 0.33/0.59 ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.33/0.59 = ( ord_less_int @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_less_iff_less
% 0.33/0.59 thf(fact_4444_neg__less__iff__less,axiom,
% 0.33/0.59 ! [B: rat,A: rat] :
% 0.33/0.59 ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 0.33/0.59 = ( ord_less_rat @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_less_iff_less
% 0.33/0.59 thf(fact_4445_neg__less__iff__less,axiom,
% 0.33/0.59 ! [B: code_integer,A: code_integer] :
% 0.33/0.59 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.33/0.59 = ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_less_iff_less
% 0.33/0.59 thf(fact_4446_neg__numeral__eq__iff,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 0.33/0.59 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.33/0.59 = ( M = N ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_eq_iff
% 0.33/0.59 thf(fact_4447_neg__numeral__eq__iff,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 0.33/0.59 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.59 = ( M = N ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_eq_iff
% 0.33/0.59 thf(fact_4448_neg__numeral__eq__iff,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.33/0.59 = ( M = N ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_eq_iff
% 0.33/0.59 thf(fact_4449_neg__numeral__eq__iff,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 0.33/0.59 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.33/0.59 = ( M = N ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_eq_iff
% 0.33/0.59 thf(fact_4450_neg__numeral__eq__iff,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.33/0.59 = ( M = N ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_eq_iff
% 0.33/0.59 thf(fact_4451_mult__minus__right,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
% 0.33/0.59 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus_right
% 0.33/0.59 thf(fact_4452_mult__minus__right,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
% 0.33/0.59 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus_right
% 0.33/0.59 thf(fact_4453_mult__minus__right,axiom,
% 0.33/0.59 ! [A: complex,B: complex] :
% 0.33/0.59 ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus_right
% 0.33/0.59 thf(fact_4454_mult__minus__right,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.33/0.59 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus_right
% 0.33/0.59 thf(fact_4455_mult__minus__right,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus_right
% 0.33/0.59 thf(fact_4456_minus__mult__minus,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.33/0.59 = ( times_times_real @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_mult_minus
% 0.33/0.59 thf(fact_4457_minus__mult__minus,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.33/0.59 = ( times_times_int @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_mult_minus
% 0.33/0.59 thf(fact_4458_minus__mult__minus,axiom,
% 0.33/0.59 ! [A: complex,B: complex] :
% 0.33/0.59 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.33/0.59 = ( times_times_complex @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_mult_minus
% 0.33/0.59 thf(fact_4459_minus__mult__minus,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.33/0.59 = ( times_times_rat @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_mult_minus
% 0.33/0.59 thf(fact_4460_minus__mult__minus,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.59 = ( times_3573771949741848930nteger @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_mult_minus
% 0.33/0.59 thf(fact_4461_mult__minus__left,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 0.33/0.59 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus_left
% 0.33/0.59 thf(fact_4462_mult__minus__left,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.59 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus_left
% 0.33/0.59 thf(fact_4463_mult__minus__left,axiom,
% 0.33/0.59 ! [A: complex,B: complex] :
% 0.33/0.59 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus_left
% 0.33/0.59 thf(fact_4464_mult__minus__left,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.33/0.59 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus_left
% 0.33/0.59 thf(fact_4465_mult__minus__left,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus_left
% 0.33/0.59 thf(fact_4466_minus__add__distrib,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.33/0.59 = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_add_distrib
% 0.33/0.59 thf(fact_4467_minus__add__distrib,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.33/0.59 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_add_distrib
% 0.33/0.59 thf(fact_4468_minus__add__distrib,axiom,
% 0.33/0.59 ! [A: complex,B: complex] :
% 0.33/0.59 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.33/0.59 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_add_distrib
% 0.33/0.59 thf(fact_4469_minus__add__distrib,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.33/0.59 = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_add_distrib
% 0.33/0.59 thf(fact_4470_minus__add__distrib,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.33/0.59 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_add_distrib
% 0.33/0.59 thf(fact_4471_minus__add__cancel,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 0.33/0.59 = B ) ).
% 0.33/0.59
% 0.33/0.59 % minus_add_cancel
% 0.33/0.59 thf(fact_4472_minus__add__cancel,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 0.33/0.59 = B ) ).
% 0.33/0.59
% 0.33/0.59 % minus_add_cancel
% 0.33/0.59 thf(fact_4473_minus__add__cancel,axiom,
% 0.33/0.59 ! [A: complex,B: complex] :
% 0.33/0.59 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
% 0.33/0.59 = B ) ).
% 0.33/0.59
% 0.33/0.59 % minus_add_cancel
% 0.33/0.59 thf(fact_4474_minus__add__cancel,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
% 0.33/0.59 = B ) ).
% 0.33/0.59
% 0.33/0.59 % minus_add_cancel
% 0.33/0.59 thf(fact_4475_minus__add__cancel,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.33/0.59 = B ) ).
% 0.33/0.59
% 0.33/0.59 % minus_add_cancel
% 0.33/0.59 thf(fact_4476_add__minus__cancel,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 0.33/0.59 = B ) ).
% 0.33/0.59
% 0.33/0.59 % add_minus_cancel
% 0.33/0.59 thf(fact_4477_add__minus__cancel,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 0.33/0.59 = B ) ).
% 0.33/0.59
% 0.33/0.59 % add_minus_cancel
% 0.33/0.59 thf(fact_4478_add__minus__cancel,axiom,
% 0.33/0.59 ! [A: complex,B: complex] :
% 0.33/0.59 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
% 0.33/0.59 = B ) ).
% 0.33/0.59
% 0.33/0.59 % add_minus_cancel
% 0.33/0.59 thf(fact_4479_add__minus__cancel,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
% 0.33/0.59 = B ) ).
% 0.33/0.59
% 0.33/0.59 % add_minus_cancel
% 0.33/0.59 thf(fact_4480_add__minus__cancel,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
% 0.33/0.59 = B ) ).
% 0.33/0.59
% 0.33/0.59 % add_minus_cancel
% 0.33/0.59 thf(fact_4481_minus__diff__eq,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 0.33/0.59 = ( minus_minus_real @ B @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_diff_eq
% 0.33/0.59 thf(fact_4482_minus__diff__eq,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 0.33/0.59 = ( minus_minus_int @ B @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_diff_eq
% 0.33/0.59 thf(fact_4483_minus__diff__eq,axiom,
% 0.33/0.59 ! [A: complex,B: complex] :
% 0.33/0.59 ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
% 0.33/0.59 = ( minus_minus_complex @ B @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_diff_eq
% 0.33/0.59 thf(fact_4484_minus__diff__eq,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
% 0.33/0.59 = ( minus_minus_rat @ B @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_diff_eq
% 0.33/0.59 thf(fact_4485_minus__diff__eq,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.33/0.59 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_diff_eq
% 0.33/0.59 thf(fact_4486_div__minus__minus,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.33/0.59 = ( divide_divide_int @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % div_minus_minus
% 0.33/0.59 thf(fact_4487_div__minus__minus,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.59 = ( divide6298287555418463151nteger @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % div_minus_minus
% 0.33/0.59 thf(fact_4488_dvd__minus__iff,axiom,
% 0.33/0.59 ! [X: real,Y: real] :
% 0.33/0.59 ( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y ) )
% 0.33/0.59 = ( dvd_dvd_real @ X @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % dvd_minus_iff
% 0.33/0.59 thf(fact_4489_dvd__minus__iff,axiom,
% 0.33/0.59 ! [X: int,Y: int] :
% 0.33/0.59 ( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y ) )
% 0.33/0.59 = ( dvd_dvd_int @ X @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % dvd_minus_iff
% 0.33/0.59 thf(fact_4490_dvd__minus__iff,axiom,
% 0.33/0.59 ! [X: complex,Y: complex] :
% 0.33/0.59 ( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y ) )
% 0.33/0.59 = ( dvd_dvd_complex @ X @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % dvd_minus_iff
% 0.33/0.59 thf(fact_4491_dvd__minus__iff,axiom,
% 0.33/0.59 ! [X: rat,Y: rat] :
% 0.33/0.59 ( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y ) )
% 0.33/0.59 = ( dvd_dvd_rat @ X @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % dvd_minus_iff
% 0.33/0.59 thf(fact_4492_dvd__minus__iff,axiom,
% 0.33/0.59 ! [X: code_integer,Y: code_integer] :
% 0.33/0.59 ( ( dvd_dvd_Code_integer @ X @ ( uminus1351360451143612070nteger @ Y ) )
% 0.33/0.59 = ( dvd_dvd_Code_integer @ X @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % dvd_minus_iff
% 0.33/0.59 thf(fact_4493_minus__dvd__iff,axiom,
% 0.33/0.59 ! [X: real,Y: real] :
% 0.33/0.59 ( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y )
% 0.33/0.59 = ( dvd_dvd_real @ X @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_dvd_iff
% 0.33/0.59 thf(fact_4494_minus__dvd__iff,axiom,
% 0.33/0.59 ! [X: int,Y: int] :
% 0.33/0.59 ( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y )
% 0.33/0.59 = ( dvd_dvd_int @ X @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_dvd_iff
% 0.33/0.59 thf(fact_4495_minus__dvd__iff,axiom,
% 0.33/0.59 ! [X: complex,Y: complex] :
% 0.33/0.59 ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y )
% 0.33/0.59 = ( dvd_dvd_complex @ X @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_dvd_iff
% 0.33/0.59 thf(fact_4496_minus__dvd__iff,axiom,
% 0.33/0.59 ! [X: rat,Y: rat] :
% 0.33/0.59 ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y )
% 0.33/0.59 = ( dvd_dvd_rat @ X @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_dvd_iff
% 0.33/0.59 thf(fact_4497_minus__dvd__iff,axiom,
% 0.33/0.59 ! [X: code_integer,Y: code_integer] :
% 0.33/0.59 ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ Y )
% 0.33/0.59 = ( dvd_dvd_Code_integer @ X @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_dvd_iff
% 0.33/0.59 thf(fact_4498_mod__minus__minus,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.33/0.59 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mod_minus_minus
% 0.33/0.59 thf(fact_4499_mod__minus__minus,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mod_minus_minus
% 0.33/0.59 thf(fact_4500_idom__abs__sgn__class_Osgn__minus,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( sgn_sgn_real @ ( uminus_uminus_real @ A ) )
% 0.33/0.59 = ( uminus_uminus_real @ ( sgn_sgn_real @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % idom_abs_sgn_class.sgn_minus
% 0.33/0.59 thf(fact_4501_idom__abs__sgn__class_Osgn__minus,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( sgn_sgn_int @ ( uminus_uminus_int @ A ) )
% 0.33/0.59 = ( uminus_uminus_int @ ( sgn_sgn_int @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % idom_abs_sgn_class.sgn_minus
% 0.33/0.59 thf(fact_4502_idom__abs__sgn__class_Osgn__minus,axiom,
% 0.33/0.59 ! [A: complex] :
% 0.33/0.59 ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ A ) )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ ( sgn_sgn_complex @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % idom_abs_sgn_class.sgn_minus
% 0.33/0.59 thf(fact_4503_idom__abs__sgn__class_Osgn__minus,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ A ) )
% 0.33/0.59 = ( uminus_uminus_rat @ ( sgn_sgn_rat @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % idom_abs_sgn_class.sgn_minus
% 0.33/0.59 thf(fact_4504_idom__abs__sgn__class_Osgn__minus,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % idom_abs_sgn_class.sgn_minus
% 0.33/0.59 thf(fact_4505_tanh__0,axiom,
% 0.33/0.59 ( ( tanh_real @ zero_zero_real )
% 0.33/0.59 = zero_zero_real ) ).
% 0.33/0.59
% 0.33/0.59 % tanh_0
% 0.33/0.59 thf(fact_4506_neg__less__eq__nonneg,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.33/0.59 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_less_eq_nonneg
% 0.33/0.59 thf(fact_4507_neg__less__eq__nonneg,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.33/0.59 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_less_eq_nonneg
% 0.33/0.59 thf(fact_4508_neg__less__eq__nonneg,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.33/0.59 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_less_eq_nonneg
% 0.33/0.59 thf(fact_4509_neg__less__eq__nonneg,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 0.33/0.59 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_less_eq_nonneg
% 0.33/0.59 thf(fact_4510_less__eq__neg__nonpos,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.33/0.59 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.33/0.59
% 0.33/0.59 % less_eq_neg_nonpos
% 0.33/0.59 thf(fact_4511_less__eq__neg__nonpos,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.33/0.59 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.33/0.59
% 0.33/0.59 % less_eq_neg_nonpos
% 0.33/0.59 thf(fact_4512_less__eq__neg__nonpos,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.33/0.59 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.33/0.59
% 0.33/0.59 % less_eq_neg_nonpos
% 0.33/0.59 thf(fact_4513_less__eq__neg__nonpos,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 0.33/0.59 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.33/0.59
% 0.33/0.59 % less_eq_neg_nonpos
% 0.33/0.59 thf(fact_4514_neg__le__0__iff__le,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.33/0.59 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_le_0_iff_le
% 0.33/0.59 thf(fact_4515_neg__le__0__iff__le,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.33/0.59 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_le_0_iff_le
% 0.33/0.59 thf(fact_4516_neg__le__0__iff__le,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.33/0.59 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_le_0_iff_le
% 0.33/0.59 thf(fact_4517_neg__le__0__iff__le,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.33/0.59 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_le_0_iff_le
% 0.33/0.59 thf(fact_4518_neg__0__le__iff__le,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.33/0.59 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_0_le_iff_le
% 0.33/0.59 thf(fact_4519_neg__0__le__iff__le,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.33/0.59 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_0_le_iff_le
% 0.33/0.59 thf(fact_4520_neg__0__le__iff__le,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.33/0.59 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_0_le_iff_le
% 0.33/0.59 thf(fact_4521_neg__0__le__iff__le,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.33/0.59 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_0_le_iff_le
% 0.33/0.59 thf(fact_4522_neg__less__0__iff__less,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.33/0.59 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_less_0_iff_less
% 0.33/0.59 thf(fact_4523_neg__less__0__iff__less,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.33/0.59 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_less_0_iff_less
% 0.33/0.59 thf(fact_4524_neg__less__0__iff__less,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.33/0.59 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_less_0_iff_less
% 0.33/0.59 thf(fact_4525_neg__less__0__iff__less,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.33/0.59 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_less_0_iff_less
% 0.33/0.59 thf(fact_4526_neg__0__less__iff__less,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.33/0.59 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_0_less_iff_less
% 0.33/0.59 thf(fact_4527_neg__0__less__iff__less,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.33/0.59 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_0_less_iff_less
% 0.33/0.59 thf(fact_4528_neg__0__less__iff__less,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.33/0.59 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_0_less_iff_less
% 0.33/0.59 thf(fact_4529_neg__0__less__iff__less,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.33/0.59 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_0_less_iff_less
% 0.33/0.59 thf(fact_4530_neg__less__pos,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 0.33/0.59 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_less_pos
% 0.33/0.59 thf(fact_4531_neg__less__pos,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 0.33/0.59 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_less_pos
% 0.33/0.59 thf(fact_4532_neg__less__pos,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.33/0.59 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_less_pos
% 0.33/0.59 thf(fact_4533_neg__less__pos,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.33/0.59 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_less_pos
% 0.33/0.59 thf(fact_4534_less__neg__neg,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 0.33/0.59 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.33/0.59
% 0.33/0.59 % less_neg_neg
% 0.33/0.59 thf(fact_4535_less__neg__neg,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 0.33/0.59 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.33/0.59
% 0.33/0.59 % less_neg_neg
% 0.33/0.59 thf(fact_4536_less__neg__neg,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.33/0.59 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.33/0.59
% 0.33/0.59 % less_neg_neg
% 0.33/0.59 thf(fact_4537_less__neg__neg,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.33/0.59 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.33/0.59
% 0.33/0.59 % less_neg_neg
% 0.33/0.59 thf(fact_4538_ab__left__minus,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.33/0.59 = zero_zero_real ) ).
% 0.33/0.59
% 0.33/0.59 % ab_left_minus
% 0.33/0.59 thf(fact_4539_ab__left__minus,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.33/0.59 = zero_zero_int ) ).
% 0.33/0.59
% 0.33/0.59 % ab_left_minus
% 0.33/0.59 thf(fact_4540_ab__left__minus,axiom,
% 0.33/0.59 ! [A: complex] :
% 0.33/0.59 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.33/0.59 = zero_zero_complex ) ).
% 0.33/0.59
% 0.33/0.59 % ab_left_minus
% 0.33/0.59 thf(fact_4541_ab__left__minus,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.33/0.59 = zero_zero_rat ) ).
% 0.33/0.59
% 0.33/0.59 % ab_left_minus
% 0.33/0.59 thf(fact_4542_ab__left__minus,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.33/0.59 = zero_z3403309356797280102nteger ) ).
% 0.33/0.59
% 0.33/0.59 % ab_left_minus
% 0.33/0.59 thf(fact_4543_add_Oright__inverse,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 0.33/0.59 = zero_zero_real ) ).
% 0.33/0.59
% 0.33/0.59 % add.right_inverse
% 0.33/0.59 thf(fact_4544_add_Oright__inverse,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 0.33/0.59 = zero_zero_int ) ).
% 0.33/0.59
% 0.33/0.59 % add.right_inverse
% 0.33/0.59 thf(fact_4545_add_Oright__inverse,axiom,
% 0.33/0.59 ! [A: complex] :
% 0.33/0.59 ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 0.33/0.59 = zero_zero_complex ) ).
% 0.33/0.59
% 0.33/0.59 % add.right_inverse
% 0.33/0.59 thf(fact_4546_add_Oright__inverse,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.33/0.59 = zero_zero_rat ) ).
% 0.33/0.59
% 0.33/0.59 % add.right_inverse
% 0.33/0.59 thf(fact_4547_add_Oright__inverse,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.33/0.59 = zero_z3403309356797280102nteger ) ).
% 0.33/0.59
% 0.33/0.59 % add.right_inverse
% 0.33/0.59 thf(fact_4548_diff__0,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( minus_minus_real @ zero_zero_real @ A )
% 0.33/0.59 = ( uminus_uminus_real @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_0
% 0.33/0.59 thf(fact_4549_diff__0,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( minus_minus_int @ zero_zero_int @ A )
% 0.33/0.59 = ( uminus_uminus_int @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_0
% 0.33/0.59 thf(fact_4550_diff__0,axiom,
% 0.33/0.59 ! [A: complex] :
% 0.33/0.59 ( ( minus_minus_complex @ zero_zero_complex @ A )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_0
% 0.33/0.59 thf(fact_4551_diff__0,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( minus_minus_rat @ zero_zero_rat @ A )
% 0.33/0.59 = ( uminus_uminus_rat @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_0
% 0.33/0.59 thf(fact_4552_diff__0,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_0
% 0.33/0.59 thf(fact_4553_verit__minus__simplify_I3_J,axiom,
% 0.33/0.59 ! [B: real] :
% 0.33/0.59 ( ( minus_minus_real @ zero_zero_real @ B )
% 0.33/0.59 = ( uminus_uminus_real @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % verit_minus_simplify(3)
% 0.33/0.59 thf(fact_4554_verit__minus__simplify_I3_J,axiom,
% 0.33/0.59 ! [B: int] :
% 0.33/0.59 ( ( minus_minus_int @ zero_zero_int @ B )
% 0.33/0.59 = ( uminus_uminus_int @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % verit_minus_simplify(3)
% 0.33/0.59 thf(fact_4555_verit__minus__simplify_I3_J,axiom,
% 0.33/0.59 ! [B: complex] :
% 0.33/0.59 ( ( minus_minus_complex @ zero_zero_complex @ B )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % verit_minus_simplify(3)
% 0.33/0.59 thf(fact_4556_verit__minus__simplify_I3_J,axiom,
% 0.33/0.59 ! [B: rat] :
% 0.33/0.59 ( ( minus_minus_rat @ zero_zero_rat @ B )
% 0.33/0.59 = ( uminus_uminus_rat @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % verit_minus_simplify(3)
% 0.33/0.59 thf(fact_4557_verit__minus__simplify_I3_J,axiom,
% 0.33/0.59 ! [B: code_integer] :
% 0.33/0.59 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % verit_minus_simplify(3)
% 0.33/0.59 thf(fact_4558_add__neg__numeral__simps_I3_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.33/0.59 = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_simps(3)
% 0.33/0.59 thf(fact_4559_add__neg__numeral__simps_I3_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.59 = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_simps(3)
% 0.33/0.59 thf(fact_4560_add__neg__numeral__simps_I3_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_simps(3)
% 0.33/0.59 thf(fact_4561_add__neg__numeral__simps_I3_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.33/0.59 = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_simps(3)
% 0.33/0.59 thf(fact_4562_add__neg__numeral__simps_I3_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_simps(3)
% 0.33/0.59 thf(fact_4563_mult__minus1__right,axiom,
% 0.33/0.59 ! [Z2: real] :
% 0.33/0.59 ( ( times_times_real @ Z2 @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.59 = ( uminus_uminus_real @ Z2 ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus1_right
% 0.33/0.59 thf(fact_4564_mult__minus1__right,axiom,
% 0.33/0.59 ! [Z2: int] :
% 0.33/0.59 ( ( times_times_int @ Z2 @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.59 = ( uminus_uminus_int @ Z2 ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus1_right
% 0.33/0.59 thf(fact_4565_mult__minus1__right,axiom,
% 0.33/0.59 ! [Z2: complex] :
% 0.33/0.59 ( ( times_times_complex @ Z2 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ Z2 ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus1_right
% 0.33/0.59 thf(fact_4566_mult__minus1__right,axiom,
% 0.33/0.59 ! [Z2: rat] :
% 0.33/0.59 ( ( times_times_rat @ Z2 @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.59 = ( uminus_uminus_rat @ Z2 ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus1_right
% 0.33/0.59 thf(fact_4567_mult__minus1__right,axiom,
% 0.33/0.59 ! [Z2: code_integer] :
% 0.33/0.59 ( ( times_3573771949741848930nteger @ Z2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ Z2 ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus1_right
% 0.33/0.59 thf(fact_4568_mult__minus1,axiom,
% 0.33/0.59 ! [Z2: real] :
% 0.33/0.59 ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z2 )
% 0.33/0.59 = ( uminus_uminus_real @ Z2 ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus1
% 0.33/0.59 thf(fact_4569_mult__minus1,axiom,
% 0.33/0.59 ! [Z2: int] :
% 0.33/0.59 ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z2 )
% 0.33/0.59 = ( uminus_uminus_int @ Z2 ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus1
% 0.33/0.59 thf(fact_4570_mult__minus1,axiom,
% 0.33/0.59 ! [Z2: complex] :
% 0.33/0.59 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z2 )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ Z2 ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus1
% 0.33/0.59 thf(fact_4571_mult__minus1,axiom,
% 0.33/0.59 ! [Z2: rat] :
% 0.33/0.59 ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z2 )
% 0.33/0.59 = ( uminus_uminus_rat @ Z2 ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus1
% 0.33/0.59 thf(fact_4572_mult__minus1,axiom,
% 0.33/0.59 ! [Z2: code_integer] :
% 0.33/0.59 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z2 )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ Z2 ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_minus1
% 0.33/0.59 thf(fact_4573_diff__minus__eq__add,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
% 0.33/0.59 = ( plus_plus_real @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_minus_eq_add
% 0.33/0.59 thf(fact_4574_diff__minus__eq__add,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 0.33/0.59 = ( plus_plus_int @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_minus_eq_add
% 0.33/0.59 thf(fact_4575_diff__minus__eq__add,axiom,
% 0.33/0.59 ! [A: complex,B: complex] :
% 0.33/0.59 ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.33/0.59 = ( plus_plus_complex @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_minus_eq_add
% 0.33/0.59 thf(fact_4576_diff__minus__eq__add,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.33/0.59 = ( plus_plus_rat @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_minus_eq_add
% 0.33/0.59 thf(fact_4577_diff__minus__eq__add,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.59 = ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_minus_eq_add
% 0.33/0.59 thf(fact_4578_uminus__add__conv__diff,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
% 0.33/0.59 = ( minus_minus_real @ B @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % uminus_add_conv_diff
% 0.33/0.59 thf(fact_4579_uminus__add__conv__diff,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.59 = ( minus_minus_int @ B @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % uminus_add_conv_diff
% 0.33/0.59 thf(fact_4580_uminus__add__conv__diff,axiom,
% 0.33/0.59 ! [A: complex,B: complex] :
% 0.33/0.59 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.33/0.59 = ( minus_minus_complex @ B @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % uminus_add_conv_diff
% 0.33/0.59 thf(fact_4581_uminus__add__conv__diff,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.33/0.59 = ( minus_minus_rat @ B @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % uminus_add_conv_diff
% 0.33/0.59 thf(fact_4582_uminus__add__conv__diff,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.33/0.59 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % uminus_add_conv_diff
% 0.33/0.59 thf(fact_4583_divide__minus1,axiom,
% 0.33/0.59 ! [X: real] :
% 0.33/0.59 ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.59 = ( uminus_uminus_real @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % divide_minus1
% 0.33/0.59 thf(fact_4584_divide__minus1,axiom,
% 0.33/0.59 ! [X: complex] :
% 0.33/0.59 ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % divide_minus1
% 0.33/0.59 thf(fact_4585_divide__minus1,axiom,
% 0.33/0.59 ! [X: rat] :
% 0.33/0.59 ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.59 = ( uminus_uminus_rat @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % divide_minus1
% 0.33/0.59 thf(fact_4586_div__minus1__right,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.59 = ( uminus_uminus_int @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % div_minus1_right
% 0.33/0.59 thf(fact_4587_div__minus1__right,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % div_minus1_right
% 0.33/0.59 thf(fact_4588_minus__mod__self1,axiom,
% 0.33/0.59 ! [B: int,A: int] :
% 0.33/0.59 ( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
% 0.33/0.59 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_mod_self1
% 0.33/0.59 thf(fact_4589_minus__mod__self1,axiom,
% 0.33/0.59 ! [B: code_integer,A: code_integer] :
% 0.33/0.59 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
% 0.33/0.59 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_mod_self1
% 0.33/0.59 thf(fact_4590_ceiling__zero,axiom,
% 0.33/0.59 ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
% 0.33/0.59 = zero_zero_int ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_zero
% 0.33/0.59 thf(fact_4591_ceiling__zero,axiom,
% 0.33/0.59 ( ( archim7802044766580827645g_real @ zero_zero_real )
% 0.33/0.59 = zero_zero_int ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_zero
% 0.33/0.59 thf(fact_4592_signed__take__bit__of__minus__1,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.59
% 0.33/0.59 % signed_take_bit_of_minus_1
% 0.33/0.59 thf(fact_4593_signed__take__bit__of__minus__1,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.59 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.59
% 0.33/0.59 % signed_take_bit_of_minus_1
% 0.33/0.59 thf(fact_4594_real__add__minus__iff,axiom,
% 0.33/0.59 ! [X: real,A: real] :
% 0.33/0.59 ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
% 0.33/0.59 = zero_zero_real )
% 0.33/0.59 = ( X = A ) ) ).
% 0.33/0.59
% 0.33/0.59 % real_add_minus_iff
% 0.33/0.59 thf(fact_4595_ceiling__numeral,axiom,
% 0.33/0.59 ! [V: num] :
% 0.33/0.59 ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
% 0.33/0.59 = ( numeral_numeral_int @ V ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_numeral
% 0.33/0.59 thf(fact_4596_ceiling__numeral,axiom,
% 0.33/0.59 ! [V: num] :
% 0.33/0.59 ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
% 0.33/0.59 = ( numeral_numeral_int @ V ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_numeral
% 0.33/0.59 thf(fact_4597_ceiling__one,axiom,
% 0.33/0.59 ( ( archim2889992004027027881ng_rat @ one_one_rat )
% 0.33/0.59 = one_one_int ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_one
% 0.33/0.59 thf(fact_4598_ceiling__one,axiom,
% 0.33/0.59 ( ( archim7802044766580827645g_real @ one_one_real )
% 0.33/0.59 = one_one_int ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_one
% 0.33/0.59 thf(fact_4599_ceiling__of__nat,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( archim7802044766580827645g_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.33/0.59 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_of_nat
% 0.33/0.59 thf(fact_4600_ceiling__of__nat,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( archim2889992004027027881ng_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.33/0.59 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_of_nat
% 0.33/0.59 thf(fact_4601_dbl__simps_I1_J,axiom,
% 0.33/0.59 ! [K: num] :
% 0.33/0.59 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.33/0.59 = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % dbl_simps(1)
% 0.33/0.59 thf(fact_4602_dbl__simps_I1_J,axiom,
% 0.33/0.59 ! [K: num] :
% 0.33/0.59 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.59 = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % dbl_simps(1)
% 0.33/0.59 thf(fact_4603_dbl__simps_I1_J,axiom,
% 0.33/0.59 ! [K: num] :
% 0.33/0.59 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % dbl_simps(1)
% 0.33/0.59 thf(fact_4604_dbl__simps_I1_J,axiom,
% 0.33/0.59 ! [K: num] :
% 0.33/0.59 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.33/0.59 = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % dbl_simps(1)
% 0.33/0.59 thf(fact_4605_dbl__simps_I1_J,axiom,
% 0.33/0.59 ! [K: num] :
% 0.33/0.59 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % dbl_simps(1)
% 0.33/0.59 thf(fact_4606_add__neg__numeral__special_I7_J,axiom,
% 0.33/0.59 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.59 = zero_zero_real ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_special(7)
% 0.33/0.59 thf(fact_4607_add__neg__numeral__special_I7_J,axiom,
% 0.33/0.59 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.59 = zero_zero_int ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_special(7)
% 0.33/0.59 thf(fact_4608_add__neg__numeral__special_I7_J,axiom,
% 0.33/0.59 ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.59 = zero_zero_complex ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_special(7)
% 0.33/0.59 thf(fact_4609_add__neg__numeral__special_I7_J,axiom,
% 0.33/0.59 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.59 = zero_zero_rat ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_special(7)
% 0.33/0.59 thf(fact_4610_add__neg__numeral__special_I7_J,axiom,
% 0.33/0.59 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.59 = zero_z3403309356797280102nteger ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_special(7)
% 0.33/0.59 thf(fact_4611_add__neg__numeral__special_I8_J,axiom,
% 0.33/0.59 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.33/0.59 = zero_zero_real ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_special(8)
% 0.33/0.59 thf(fact_4612_add__neg__numeral__special_I8_J,axiom,
% 0.33/0.59 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.33/0.59 = zero_zero_int ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_special(8)
% 0.33/0.59 thf(fact_4613_add__neg__numeral__special_I8_J,axiom,
% 0.33/0.59 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.33/0.59 = zero_zero_complex ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_special(8)
% 0.33/0.59 thf(fact_4614_add__neg__numeral__special_I8_J,axiom,
% 0.33/0.59 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.33/0.59 = zero_zero_rat ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_special(8)
% 0.33/0.59 thf(fact_4615_add__neg__numeral__special_I8_J,axiom,
% 0.33/0.59 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.33/0.59 = zero_z3403309356797280102nteger ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_special(8)
% 0.33/0.59 thf(fact_4616_diff__numeral__special_I12_J,axiom,
% 0.33/0.59 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.59 = zero_zero_real ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(12)
% 0.33/0.59 thf(fact_4617_diff__numeral__special_I12_J,axiom,
% 0.33/0.59 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.59 = zero_zero_int ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(12)
% 0.33/0.59 thf(fact_4618_diff__numeral__special_I12_J,axiom,
% 0.33/0.59 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.59 = zero_zero_complex ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(12)
% 0.33/0.59 thf(fact_4619_diff__numeral__special_I12_J,axiom,
% 0.33/0.59 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.59 = zero_zero_rat ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(12)
% 0.33/0.59 thf(fact_4620_diff__numeral__special_I12_J,axiom,
% 0.33/0.59 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.59 = zero_z3403309356797280102nteger ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(12)
% 0.33/0.59 thf(fact_4621_numeral__eq__neg__one__iff,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
% 0.33/0.59 = ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.59 = ( N = one ) ) ).
% 0.33/0.59
% 0.33/0.59 % numeral_eq_neg_one_iff
% 0.33/0.59 thf(fact_4622_numeral__eq__neg__one__iff,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
% 0.33/0.59 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.59 = ( N = one ) ) ).
% 0.33/0.59
% 0.33/0.59 % numeral_eq_neg_one_iff
% 0.33/0.59 thf(fact_4623_numeral__eq__neg__one__iff,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.59 = ( N = one ) ) ).
% 0.33/0.59
% 0.33/0.59 % numeral_eq_neg_one_iff
% 0.33/0.59 thf(fact_4624_numeral__eq__neg__one__iff,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
% 0.33/0.59 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.59 = ( N = one ) ) ).
% 0.33/0.59
% 0.33/0.59 % numeral_eq_neg_one_iff
% 0.33/0.59 thf(fact_4625_numeral__eq__neg__one__iff,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.59 = ( N = one ) ) ).
% 0.33/0.59
% 0.33/0.59 % numeral_eq_neg_one_iff
% 0.33/0.59 thf(fact_4626_neg__one__eq__numeral__iff,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( ( uminus_uminus_real @ one_one_real )
% 0.33/0.59 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.33/0.59 = ( N = one ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_one_eq_numeral_iff
% 0.33/0.59 thf(fact_4627_neg__one__eq__numeral__iff,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( ( uminus_uminus_int @ one_one_int )
% 0.33/0.59 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.59 = ( N = one ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_one_eq_numeral_iff
% 0.33/0.59 thf(fact_4628_neg__one__eq__numeral__iff,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( ( uminus1482373934393186551omplex @ one_one_complex )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.33/0.59 = ( N = one ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_one_eq_numeral_iff
% 0.33/0.59 thf(fact_4629_neg__one__eq__numeral__iff,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( ( uminus_uminus_rat @ one_one_rat )
% 0.33/0.59 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.33/0.59 = ( N = one ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_one_eq_numeral_iff
% 0.33/0.59 thf(fact_4630_neg__one__eq__numeral__iff,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.33/0.59 = ( N = one ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_one_eq_numeral_iff
% 0.33/0.59 thf(fact_4631_minus__one__mult__self,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
% 0.33/0.59 = one_one_real ) ).
% 0.33/0.59
% 0.33/0.59 % minus_one_mult_self
% 0.33/0.59 thf(fact_4632_minus__one__mult__self,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
% 0.33/0.59 = one_one_int ) ).
% 0.33/0.59
% 0.33/0.59 % minus_one_mult_self
% 0.33/0.59 thf(fact_4633_minus__one__mult__self,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
% 0.33/0.59 = one_one_complex ) ).
% 0.33/0.59
% 0.33/0.59 % minus_one_mult_self
% 0.33/0.59 thf(fact_4634_minus__one__mult__self,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
% 0.33/0.59 = one_one_rat ) ).
% 0.33/0.59
% 0.33/0.59 % minus_one_mult_self
% 0.33/0.59 thf(fact_4635_minus__one__mult__self,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
% 0.33/0.59 = one_one_Code_integer ) ).
% 0.33/0.59
% 0.33/0.59 % minus_one_mult_self
% 0.33/0.59 thf(fact_4636_left__minus__one__mult__self,axiom,
% 0.33/0.59 ! [N: nat,A: real] :
% 0.33/0.59 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
% 0.33/0.59 = A ) ).
% 0.33/0.59
% 0.33/0.59 % left_minus_one_mult_self
% 0.33/0.59 thf(fact_4637_left__minus__one__mult__self,axiom,
% 0.33/0.59 ! [N: nat,A: int] :
% 0.33/0.59 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
% 0.33/0.59 = A ) ).
% 0.33/0.59
% 0.33/0.59 % left_minus_one_mult_self
% 0.33/0.59 thf(fact_4638_left__minus__one__mult__self,axiom,
% 0.33/0.59 ! [N: nat,A: complex] :
% 0.33/0.59 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
% 0.33/0.59 = A ) ).
% 0.33/0.59
% 0.33/0.59 % left_minus_one_mult_self
% 0.33/0.59 thf(fact_4639_left__minus__one__mult__self,axiom,
% 0.33/0.59 ! [N: nat,A: rat] :
% 0.33/0.59 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
% 0.33/0.59 = A ) ).
% 0.33/0.59
% 0.33/0.59 % left_minus_one_mult_self
% 0.33/0.59 thf(fact_4640_left__minus__one__mult__self,axiom,
% 0.33/0.59 ! [N: nat,A: code_integer] :
% 0.33/0.59 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
% 0.33/0.59 = A ) ).
% 0.33/0.59
% 0.33/0.59 % left_minus_one_mult_self
% 0.33/0.59 thf(fact_4641_mod__minus1__right,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.59 = zero_zero_int ) ).
% 0.33/0.59
% 0.33/0.59 % mod_minus1_right
% 0.33/0.59 thf(fact_4642_mod__minus1__right,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.59 = zero_z3403309356797280102nteger ) ).
% 0.33/0.59
% 0.33/0.59 % mod_minus1_right
% 0.33/0.59 thf(fact_4643_norm__neg__numeral,axiom,
% 0.33/0.59 ! [W: num] :
% 0.33/0.59 ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.59 = ( numeral_numeral_real @ W ) ) ).
% 0.33/0.59
% 0.33/0.59 % norm_neg_numeral
% 0.33/0.59 thf(fact_4644_norm__neg__numeral,axiom,
% 0.33/0.59 ! [W: num] :
% 0.33/0.59 ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.33/0.59 = ( numeral_numeral_real @ W ) ) ).
% 0.33/0.59
% 0.33/0.59 % norm_neg_numeral
% 0.33/0.59 thf(fact_4645_semiring__norm_I168_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: real] :
% 0.33/0.59 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 0.33/0.59 = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(168)
% 0.33/0.59 thf(fact_4646_semiring__norm_I168_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: int] :
% 0.33/0.59 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 0.33/0.59 = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(168)
% 0.33/0.59 thf(fact_4647_semiring__norm_I168_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: complex] :
% 0.33/0.59 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 0.33/0.59 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(168)
% 0.33/0.59 thf(fact_4648_semiring__norm_I168_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: rat] :
% 0.33/0.59 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 0.33/0.59 = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(168)
% 0.33/0.59 thf(fact_4649_semiring__norm_I168_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: code_integer] :
% 0.33/0.59 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 0.33/0.59 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(168)
% 0.33/0.59 thf(fact_4650_diff__numeral__simps_I3_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 0.33/0.59 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_simps(3)
% 0.33/0.59 thf(fact_4651_diff__numeral__simps_I3_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.59 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_simps(3)
% 0.33/0.59 thf(fact_4652_diff__numeral__simps_I3_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_simps(3)
% 0.33/0.59 thf(fact_4653_diff__numeral__simps_I3_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 0.33/0.59 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_simps(3)
% 0.33/0.59 thf(fact_4654_diff__numeral__simps_I3_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_simps(3)
% 0.33/0.59 thf(fact_4655_diff__numeral__simps_I2_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.33/0.59 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_simps(2)
% 0.33/0.59 thf(fact_4656_diff__numeral__simps_I2_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.59 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_simps(2)
% 0.33/0.59 thf(fact_4657_diff__numeral__simps_I2_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.33/0.59 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_simps(2)
% 0.33/0.59 thf(fact_4658_diff__numeral__simps_I2_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.33/0.59 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_simps(2)
% 0.33/0.59 thf(fact_4659_diff__numeral__simps_I2_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.33/0.59 = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_simps(2)
% 0.33/0.59 thf(fact_4660_ceiling__neg__numeral,axiom,
% 0.33/0.59 ! [V: num] :
% 0.33/0.59 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.33/0.59 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_neg_numeral
% 0.33/0.59 thf(fact_4661_ceiling__neg__numeral,axiom,
% 0.33/0.59 ! [V: num] :
% 0.33/0.59 ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.33/0.59 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_neg_numeral
% 0.33/0.59 thf(fact_4662_semiring__norm_I172_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: real] :
% 0.33/0.59 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 0.33/0.59 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(172)
% 0.33/0.59 thf(fact_4663_semiring__norm_I172_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: int] :
% 0.33/0.59 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 0.33/0.59 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(172)
% 0.33/0.59 thf(fact_4664_semiring__norm_I172_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: complex] :
% 0.33/0.59 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 0.33/0.59 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(172)
% 0.33/0.59 thf(fact_4665_semiring__norm_I172_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: rat] :
% 0.33/0.59 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 0.33/0.59 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(172)
% 0.33/0.59 thf(fact_4666_semiring__norm_I172_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: code_integer] :
% 0.33/0.59 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 0.33/0.59 = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(172)
% 0.33/0.59 thf(fact_4667_semiring__norm_I171_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: real] :
% 0.33/0.59 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 0.33/0.59 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(171)
% 0.33/0.59 thf(fact_4668_semiring__norm_I171_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: int] :
% 0.33/0.59 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 0.33/0.59 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(171)
% 0.33/0.59 thf(fact_4669_semiring__norm_I171_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: complex] :
% 0.33/0.59 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 0.33/0.59 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(171)
% 0.33/0.59 thf(fact_4670_semiring__norm_I171_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: rat] :
% 0.33/0.59 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 0.33/0.59 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(171)
% 0.33/0.59 thf(fact_4671_semiring__norm_I171_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: code_integer] :
% 0.33/0.59 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 0.33/0.59 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(171)
% 0.33/0.59 thf(fact_4672_semiring__norm_I170_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: real] :
% 0.33/0.59 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y ) )
% 0.33/0.59 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(170)
% 0.33/0.59 thf(fact_4673_semiring__norm_I170_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: int] :
% 0.33/0.59 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
% 0.33/0.59 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(170)
% 0.33/0.59 thf(fact_4674_semiring__norm_I170_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: complex] :
% 0.33/0.59 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y ) )
% 0.33/0.59 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(170)
% 0.33/0.59 thf(fact_4675_semiring__norm_I170_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: rat] :
% 0.33/0.59 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
% 0.33/0.59 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(170)
% 0.33/0.59 thf(fact_4676_semiring__norm_I170_J,axiom,
% 0.33/0.59 ! [V: num,W: num,Y: code_integer] :
% 0.33/0.59 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
% 0.33/0.59 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.33/0.59
% 0.33/0.59 % semiring_norm(170)
% 0.33/0.59 thf(fact_4677_mult__neg__numeral__simps_I3_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.33/0.59 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_neg_numeral_simps(3)
% 0.33/0.59 thf(fact_4678_mult__neg__numeral__simps_I3_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.59 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_neg_numeral_simps(3)
% 0.33/0.59 thf(fact_4679_mult__neg__numeral__simps_I3_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_neg_numeral_simps(3)
% 0.33/0.59 thf(fact_4680_mult__neg__numeral__simps_I3_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.33/0.59 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_neg_numeral_simps(3)
% 0.33/0.59 thf(fact_4681_mult__neg__numeral__simps_I3_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_neg_numeral_simps(3)
% 0.33/0.59 thf(fact_4682_mult__neg__numeral__simps_I2_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 0.33/0.59 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_neg_numeral_simps(2)
% 0.33/0.59 thf(fact_4683_mult__neg__numeral__simps_I2_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.59 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_neg_numeral_simps(2)
% 0.33/0.59 thf(fact_4684_mult__neg__numeral__simps_I2_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_neg_numeral_simps(2)
% 0.33/0.59 thf(fact_4685_mult__neg__numeral__simps_I2_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 0.33/0.59 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_neg_numeral_simps(2)
% 0.33/0.59 thf(fact_4686_mult__neg__numeral__simps_I2_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_neg_numeral_simps(2)
% 0.33/0.59 thf(fact_4687_mult__neg__numeral__simps_I1_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.33/0.59 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_neg_numeral_simps(1)
% 0.33/0.59 thf(fact_4688_mult__neg__numeral__simps_I1_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.59 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_neg_numeral_simps(1)
% 0.33/0.59 thf(fact_4689_mult__neg__numeral__simps_I1_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.33/0.59 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_neg_numeral_simps(1)
% 0.33/0.59 thf(fact_4690_mult__neg__numeral__simps_I1_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.33/0.59 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_neg_numeral_simps(1)
% 0.33/0.59 thf(fact_4691_mult__neg__numeral__simps_I1_J,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.33/0.59 = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % mult_neg_numeral_simps(1)
% 0.33/0.59 thf(fact_4692_neg__numeral__le__iff,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.33/0.59 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_le_iff
% 0.33/0.59 thf(fact_4693_neg__numeral__le__iff,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.33/0.59 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_le_iff
% 0.33/0.59 thf(fact_4694_neg__numeral__le__iff,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.33/0.59 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_le_iff
% 0.33/0.59 thf(fact_4695_neg__numeral__le__iff,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.59 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_le_iff
% 0.33/0.59 thf(fact_4696_neg__numeral__less__iff,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.33/0.59 = ( ord_less_num @ N @ M ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_less_iff
% 0.33/0.59 thf(fact_4697_neg__numeral__less__iff,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.59 = ( ord_less_num @ N @ M ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_less_iff
% 0.33/0.59 thf(fact_4698_neg__numeral__less__iff,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.33/0.59 = ( ord_less_num @ N @ M ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_less_iff
% 0.33/0.59 thf(fact_4699_neg__numeral__less__iff,axiom,
% 0.33/0.59 ! [M: num,N: num] :
% 0.33/0.59 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.33/0.59 = ( ord_less_num @ N @ M ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_less_iff
% 0.33/0.59 thf(fact_4700_not__neg__one__le__neg__numeral__iff,axiom,
% 0.33/0.59 ! [M: num] :
% 0.33/0.59 ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 0.33/0.59 = ( M != one ) ) ).
% 0.33/0.59
% 0.33/0.59 % not_neg_one_le_neg_numeral_iff
% 0.33/0.59 thf(fact_4701_not__neg__one__le__neg__numeral__iff,axiom,
% 0.33/0.59 ! [M: num] :
% 0.33/0.59 ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 0.33/0.59 = ( M != one ) ) ).
% 0.33/0.59
% 0.33/0.59 % not_neg_one_le_neg_numeral_iff
% 0.33/0.59 thf(fact_4702_not__neg__one__le__neg__numeral__iff,axiom,
% 0.33/0.59 ! [M: num] :
% 0.33/0.59 ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 0.33/0.59 = ( M != one ) ) ).
% 0.33/0.59
% 0.33/0.59 % not_neg_one_le_neg_numeral_iff
% 0.33/0.59 thf(fact_4703_not__neg__one__le__neg__numeral__iff,axiom,
% 0.33/0.59 ! [M: num] :
% 0.33/0.59 ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 0.33/0.59 = ( M != one ) ) ).
% 0.33/0.59
% 0.33/0.59 % not_neg_one_le_neg_numeral_iff
% 0.33/0.59 thf(fact_4704_le__divide__eq__numeral1_I2_J,axiom,
% 0.33/0.59 ! [A: real,B: real,W: num] :
% 0.33/0.59 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.33/0.59 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % le_divide_eq_numeral1(2)
% 0.33/0.59 thf(fact_4705_le__divide__eq__numeral1_I2_J,axiom,
% 0.33/0.59 ! [A: rat,B: rat,W: num] :
% 0.33/0.59 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.33/0.59 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % le_divide_eq_numeral1(2)
% 0.33/0.59 thf(fact_4706_divide__le__eq__numeral1_I2_J,axiom,
% 0.33/0.59 ! [B: real,W: num,A: real] :
% 0.33/0.59 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 0.33/0.59 = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % divide_le_eq_numeral1(2)
% 0.33/0.59 thf(fact_4707_divide__le__eq__numeral1_I2_J,axiom,
% 0.33/0.59 ! [B: rat,W: num,A: rat] :
% 0.33/0.59 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 0.33/0.59 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % divide_le_eq_numeral1(2)
% 0.33/0.59 thf(fact_4708_eq__divide__eq__numeral1_I2_J,axiom,
% 0.33/0.59 ! [A: real,B: real,W: num] :
% 0.33/0.59 ( ( A
% 0.33/0.59 = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.33/0.59 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.33/0.59 != zero_zero_real )
% 0.33/0.59 => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.59 = B ) )
% 0.33/0.59 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.33/0.59 = zero_zero_real )
% 0.33/0.59 => ( A = zero_zero_real ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % eq_divide_eq_numeral1(2)
% 0.33/0.59 thf(fact_4709_eq__divide__eq__numeral1_I2_J,axiom,
% 0.33/0.59 ! [A: complex,B: complex,W: num] :
% 0.33/0.59 ( ( A
% 0.33/0.59 = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
% 0.33/0.59 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.33/0.59 != zero_zero_complex )
% 0.33/0.59 => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.33/0.59 = B ) )
% 0.33/0.59 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.33/0.59 = zero_zero_complex )
% 0.33/0.59 => ( A = zero_zero_complex ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % eq_divide_eq_numeral1(2)
% 0.33/0.59 thf(fact_4710_eq__divide__eq__numeral1_I2_J,axiom,
% 0.33/0.59 ! [A: rat,B: rat,W: num] :
% 0.33/0.59 ( ( A
% 0.33/0.59 = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.33/0.59 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.33/0.59 != zero_zero_rat )
% 0.33/0.59 => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.33/0.59 = B ) )
% 0.33/0.59 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.33/0.59 = zero_zero_rat )
% 0.33/0.59 => ( A = zero_zero_rat ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % eq_divide_eq_numeral1(2)
% 0.33/0.59 thf(fact_4711_divide__eq__eq__numeral1_I2_J,axiom,
% 0.33/0.59 ! [B: real,W: num,A: real] :
% 0.33/0.59 ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.59 = A )
% 0.33/0.59 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.33/0.59 != zero_zero_real )
% 0.33/0.59 => ( B
% 0.33/0.59 = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
% 0.33/0.59 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.33/0.59 = zero_zero_real )
% 0.33/0.59 => ( A = zero_zero_real ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % divide_eq_eq_numeral1(2)
% 0.33/0.59 thf(fact_4712_divide__eq__eq__numeral1_I2_J,axiom,
% 0.33/0.59 ! [B: complex,W: num,A: complex] :
% 0.33/0.59 ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.33/0.59 = A )
% 0.33/0.59 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.33/0.59 != zero_zero_complex )
% 0.33/0.59 => ( B
% 0.33/0.59 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
% 0.33/0.59 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.33/0.59 = zero_zero_complex )
% 0.33/0.59 => ( A = zero_zero_complex ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % divide_eq_eq_numeral1(2)
% 0.33/0.59 thf(fact_4713_divide__eq__eq__numeral1_I2_J,axiom,
% 0.33/0.59 ! [B: rat,W: num,A: rat] :
% 0.33/0.59 ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.33/0.59 = A )
% 0.33/0.59 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.33/0.59 != zero_zero_rat )
% 0.33/0.59 => ( B
% 0.33/0.59 = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
% 0.33/0.59 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.33/0.59 = zero_zero_rat )
% 0.33/0.59 => ( A = zero_zero_rat ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % divide_eq_eq_numeral1(2)
% 0.33/0.59 thf(fact_4714_neg__numeral__less__neg__one__iff,axiom,
% 0.33/0.59 ! [M: num] :
% 0.33/0.59 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.59 = ( M != one ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_less_neg_one_iff
% 0.33/0.59 thf(fact_4715_neg__numeral__less__neg__one__iff,axiom,
% 0.33/0.59 ! [M: num] :
% 0.33/0.59 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.59 = ( M != one ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_less_neg_one_iff
% 0.33/0.59 thf(fact_4716_neg__numeral__less__neg__one__iff,axiom,
% 0.33/0.59 ! [M: num] :
% 0.33/0.59 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.59 = ( M != one ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_less_neg_one_iff
% 0.33/0.59 thf(fact_4717_neg__numeral__less__neg__one__iff,axiom,
% 0.33/0.59 ! [M: num] :
% 0.33/0.59 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.59 = ( M != one ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_less_neg_one_iff
% 0.33/0.59 thf(fact_4718_less__divide__eq__numeral1_I2_J,axiom,
% 0.33/0.59 ! [A: real,B: real,W: num] :
% 0.33/0.59 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.33/0.59 = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % less_divide_eq_numeral1(2)
% 0.33/0.59 thf(fact_4719_less__divide__eq__numeral1_I2_J,axiom,
% 0.33/0.59 ! [A: rat,B: rat,W: num] :
% 0.33/0.59 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.33/0.59 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % less_divide_eq_numeral1(2)
% 0.33/0.59 thf(fact_4720_divide__less__eq__numeral1_I2_J,axiom,
% 0.33/0.59 ! [B: real,W: num,A: real] :
% 0.33/0.59 ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 0.33/0.59 = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % divide_less_eq_numeral1(2)
% 0.33/0.59 thf(fact_4721_divide__less__eq__numeral1_I2_J,axiom,
% 0.33/0.59 ! [B: rat,W: num,A: rat] :
% 0.33/0.59 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 0.33/0.59 = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 0.33/0.59
% 0.33/0.59 % divide_less_eq_numeral1(2)
% 0.33/0.59 thf(fact_4722_power2__minus,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.59 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % power2_minus
% 0.33/0.59 thf(fact_4723_power2__minus,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.59 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % power2_minus
% 0.33/0.59 thf(fact_4724_power2__minus,axiom,
% 0.33/0.59 ! [A: complex] :
% 0.33/0.59 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.59 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % power2_minus
% 0.33/0.59 thf(fact_4725_power2__minus,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.59 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % power2_minus
% 0.33/0.59 thf(fact_4726_power2__minus,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.59 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % power2_minus
% 0.33/0.59 thf(fact_4727_sgn__neg,axiom,
% 0.33/0.59 ! [A: real] :
% 0.33/0.59 ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.59 => ( ( sgn_sgn_real @ A )
% 0.33/0.59 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % sgn_neg
% 0.33/0.59 thf(fact_4728_sgn__neg,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( ord_less_int @ A @ zero_zero_int )
% 0.33/0.59 => ( ( sgn_sgn_int @ A )
% 0.33/0.59 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % sgn_neg
% 0.33/0.59 thf(fact_4729_sgn__neg,axiom,
% 0.33/0.59 ! [A: rat] :
% 0.33/0.59 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.59 => ( ( sgn_sgn_rat @ A )
% 0.33/0.59 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % sgn_neg
% 0.33/0.59 thf(fact_4730_sgn__neg,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 0.33/0.59 => ( ( sgn_sgn_Code_integer @ A )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % sgn_neg
% 0.33/0.59 thf(fact_4731_push__bit__Suc__minus__numeral,axiom,
% 0.33/0.59 ! [N: nat,K: num] :
% 0.33/0.59 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.59 = ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % push_bit_Suc_minus_numeral
% 0.33/0.59 thf(fact_4732_push__bit__Suc__minus__numeral,axiom,
% 0.33/0.59 ! [N: nat,K: num] :
% 0.33/0.59 ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.33/0.59 = ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % push_bit_Suc_minus_numeral
% 0.33/0.59 thf(fact_4733_ceiling__le__zero,axiom,
% 0.33/0.59 ! [X: real] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
% 0.33/0.59 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_le_zero
% 0.33/0.59 thf(fact_4734_ceiling__le__zero,axiom,
% 0.33/0.59 ! [X: rat] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
% 0.33/0.59 = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_le_zero
% 0.33/0.59 thf(fact_4735_zero__less__ceiling,axiom,
% 0.33/0.59 ! [X: rat] :
% 0.33/0.59 ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
% 0.33/0.59 = ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % zero_less_ceiling
% 0.33/0.59 thf(fact_4736_zero__less__ceiling,axiom,
% 0.33/0.59 ! [X: real] :
% 0.33/0.59 ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
% 0.33/0.59 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % zero_less_ceiling
% 0.33/0.59 thf(fact_4737_ceiling__le__numeral,axiom,
% 0.33/0.59 ! [X: real,V: num] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.33/0.59 = ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_le_numeral
% 0.33/0.59 thf(fact_4738_ceiling__le__numeral,axiom,
% 0.33/0.59 ! [X: rat,V: num] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 0.33/0.59 = ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_le_numeral
% 0.33/0.59 thf(fact_4739_ceiling__less__one,axiom,
% 0.33/0.59 ! [X: real] :
% 0.33/0.59 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
% 0.33/0.59 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_less_one
% 0.33/0.59 thf(fact_4740_ceiling__less__one,axiom,
% 0.33/0.59 ! [X: rat] :
% 0.33/0.59 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
% 0.33/0.59 = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_less_one
% 0.33/0.59 thf(fact_4741_one__le__ceiling,axiom,
% 0.33/0.59 ! [X: rat] :
% 0.33/0.59 ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
% 0.33/0.59 = ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % one_le_ceiling
% 0.33/0.59 thf(fact_4742_one__le__ceiling,axiom,
% 0.33/0.59 ! [X: real] :
% 0.33/0.59 ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
% 0.33/0.59 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % one_le_ceiling
% 0.33/0.59 thf(fact_4743_numeral__less__ceiling,axiom,
% 0.33/0.59 ! [V: num,X: real] :
% 0.33/0.59 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
% 0.33/0.59 = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % numeral_less_ceiling
% 0.33/0.59 thf(fact_4744_numeral__less__ceiling,axiom,
% 0.33/0.59 ! [V: num,X: rat] :
% 0.33/0.59 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
% 0.33/0.59 = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % numeral_less_ceiling
% 0.33/0.59 thf(fact_4745_ceiling__le__one,axiom,
% 0.33/0.59 ! [X: real] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
% 0.33/0.59 = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_le_one
% 0.33/0.59 thf(fact_4746_ceiling__le__one,axiom,
% 0.33/0.59 ! [X: rat] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
% 0.33/0.59 = ( ord_less_eq_rat @ X @ one_one_rat ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_le_one
% 0.33/0.59 thf(fact_4747_one__less__ceiling,axiom,
% 0.33/0.59 ! [X: rat] :
% 0.33/0.59 ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
% 0.33/0.59 = ( ord_less_rat @ one_one_rat @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % one_less_ceiling
% 0.33/0.59 thf(fact_4748_one__less__ceiling,axiom,
% 0.33/0.59 ! [X: real] :
% 0.33/0.59 ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
% 0.33/0.59 = ( ord_less_real @ one_one_real @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % one_less_ceiling
% 0.33/0.59 thf(fact_4749_ceiling__add__numeral,axiom,
% 0.33/0.59 ! [X: real,V: num] :
% 0.33/0.59 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 0.33/0.59 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_add_numeral
% 0.33/0.59 thf(fact_4750_ceiling__add__numeral,axiom,
% 0.33/0.59 ! [X: rat,V: num] :
% 0.33/0.59 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 0.33/0.59 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_add_numeral
% 0.33/0.59 thf(fact_4751_ceiling__add__one,axiom,
% 0.33/0.59 ! [X: rat] :
% 0.33/0.59 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
% 0.33/0.59 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_add_one
% 0.33/0.59 thf(fact_4752_ceiling__add__one,axiom,
% 0.33/0.59 ! [X: real] :
% 0.33/0.59 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ one_one_real ) )
% 0.33/0.59 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_add_one
% 0.33/0.59 thf(fact_4753_ceiling__diff__numeral,axiom,
% 0.33/0.59 ! [X: real,V: num] :
% 0.33/0.59 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 0.33/0.59 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_diff_numeral
% 0.33/0.59 thf(fact_4754_ceiling__diff__numeral,axiom,
% 0.33/0.59 ! [X: rat,V: num] :
% 0.33/0.59 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 0.33/0.59 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_diff_numeral
% 0.33/0.59 thf(fact_4755_ceiling__diff__one,axiom,
% 0.33/0.59 ! [X: rat] :
% 0.33/0.59 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
% 0.33/0.59 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_diff_one
% 0.33/0.59 thf(fact_4756_ceiling__diff__one,axiom,
% 0.33/0.59 ! [X: real] :
% 0.33/0.59 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ one_one_real ) )
% 0.33/0.59 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_diff_one
% 0.33/0.59 thf(fact_4757_ceiling__numeral__power,axiom,
% 0.33/0.59 ! [X: num,N: nat] :
% 0.33/0.59 ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.33/0.59 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_numeral_power
% 0.33/0.59 thf(fact_4758_ceiling__numeral__power,axiom,
% 0.33/0.59 ! [X: num,N: nat] :
% 0.33/0.59 ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.33/0.59 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_numeral_power
% 0.33/0.59 thf(fact_4759_add__neg__numeral__special_I9_J,axiom,
% 0.33/0.59 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.59 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_special(9)
% 0.33/0.59 thf(fact_4760_add__neg__numeral__special_I9_J,axiom,
% 0.33/0.59 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.59 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_special(9)
% 0.33/0.59 thf(fact_4761_add__neg__numeral__special_I9_J,axiom,
% 0.33/0.59 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_special(9)
% 0.33/0.59 thf(fact_4762_add__neg__numeral__special_I9_J,axiom,
% 0.33/0.59 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.59 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_special(9)
% 0.33/0.59 thf(fact_4763_add__neg__numeral__special_I9_J,axiom,
% 0.33/0.59 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % add_neg_numeral_special(9)
% 0.33/0.59 thf(fact_4764_diff__numeral__special_I10_J,axiom,
% 0.33/0.59 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.33/0.59 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(10)
% 0.33/0.59 thf(fact_4765_diff__numeral__special_I10_J,axiom,
% 0.33/0.59 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.33/0.59 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(10)
% 0.33/0.59 thf(fact_4766_diff__numeral__special_I10_J,axiom,
% 0.33/0.59 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(10)
% 0.33/0.59 thf(fact_4767_diff__numeral__special_I10_J,axiom,
% 0.33/0.59 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.33/0.59 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(10)
% 0.33/0.59 thf(fact_4768_diff__numeral__special_I10_J,axiom,
% 0.33/0.59 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(10)
% 0.33/0.59 thf(fact_4769_diff__numeral__special_I11_J,axiom,
% 0.33/0.59 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.59 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(11)
% 0.33/0.59 thf(fact_4770_diff__numeral__special_I11_J,axiom,
% 0.33/0.59 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.59 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(11)
% 0.33/0.59 thf(fact_4771_diff__numeral__special_I11_J,axiom,
% 0.33/0.59 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.59 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(11)
% 0.33/0.59 thf(fact_4772_diff__numeral__special_I11_J,axiom,
% 0.33/0.59 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.59 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(11)
% 0.33/0.59 thf(fact_4773_diff__numeral__special_I11_J,axiom,
% 0.33/0.59 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.59 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(11)
% 0.33/0.59 thf(fact_4774_minus__1__div__2__eq,axiom,
% 0.33/0.59 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.59 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_1_div_2_eq
% 0.33/0.59 thf(fact_4775_minus__1__div__2__eq,axiom,
% 0.33/0.59 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_1_div_2_eq
% 0.33/0.59 thf(fact_4776_minus__1__mod__2__eq,axiom,
% 0.33/0.59 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.59 = one_one_int ) ).
% 0.33/0.59
% 0.33/0.59 % minus_1_mod_2_eq
% 0.33/0.59 thf(fact_4777_minus__1__mod__2__eq,axiom,
% 0.33/0.59 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.59 = one_one_Code_integer ) ).
% 0.33/0.59
% 0.33/0.59 % minus_1_mod_2_eq
% 0.33/0.59 thf(fact_4778_bits__minus__1__mod__2__eq,axiom,
% 0.33/0.59 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.59 = one_one_int ) ).
% 0.33/0.59
% 0.33/0.59 % bits_minus_1_mod_2_eq
% 0.33/0.59 thf(fact_4779_bits__minus__1__mod__2__eq,axiom,
% 0.33/0.59 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.59 = one_one_Code_integer ) ).
% 0.33/0.59
% 0.33/0.59 % bits_minus_1_mod_2_eq
% 0.33/0.59 thf(fact_4780_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.33/0.59 ! [A: real,N: nat] :
% 0.33/0.59 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.59 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % Power.ring_1_class.power_minus_even
% 0.33/0.59 thf(fact_4781_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.33/0.59 ! [A: int,N: nat] :
% 0.33/0.59 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.59 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % Power.ring_1_class.power_minus_even
% 0.33/0.59 thf(fact_4782_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.33/0.59 ! [A: complex,N: nat] :
% 0.33/0.59 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.59 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % Power.ring_1_class.power_minus_even
% 0.33/0.59 thf(fact_4783_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.33/0.59 ! [A: rat,N: nat] :
% 0.33/0.59 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.59 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % Power.ring_1_class.power_minus_even
% 0.33/0.59 thf(fact_4784_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.33/0.59 ! [A: code_integer,N: nat] :
% 0.33/0.59 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.59 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % Power.ring_1_class.power_minus_even
% 0.33/0.59 thf(fact_4785_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.33/0.59 ! [N: nat,A: real] :
% 0.33/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.33/0.59 = ( power_power_real @ A @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % Parity.ring_1_class.power_minus_even
% 0.33/0.59 thf(fact_4786_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.33/0.59 ! [N: nat,A: int] :
% 0.33/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.33/0.59 = ( power_power_int @ A @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % Parity.ring_1_class.power_minus_even
% 0.33/0.59 thf(fact_4787_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.33/0.59 ! [N: nat,A: complex] :
% 0.33/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.33/0.59 = ( power_power_complex @ A @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % Parity.ring_1_class.power_minus_even
% 0.33/0.59 thf(fact_4788_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.33/0.59 ! [N: nat,A: rat] :
% 0.33/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.33/0.59 = ( power_power_rat @ A @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % Parity.ring_1_class.power_minus_even
% 0.33/0.59 thf(fact_4789_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.33/0.59 ! [N: nat,A: code_integer] :
% 0.33/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.33/0.59 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % Parity.ring_1_class.power_minus_even
% 0.33/0.59 thf(fact_4790_power__minus__odd,axiom,
% 0.33/0.59 ! [N: nat,A: real] :
% 0.33/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.33/0.59 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % power_minus_odd
% 0.33/0.59 thf(fact_4791_power__minus__odd,axiom,
% 0.33/0.59 ! [N: nat,A: int] :
% 0.33/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.33/0.59 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % power_minus_odd
% 0.33/0.59 thf(fact_4792_power__minus__odd,axiom,
% 0.33/0.59 ! [N: nat,A: complex] :
% 0.33/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % power_minus_odd
% 0.33/0.59 thf(fact_4793_power__minus__odd,axiom,
% 0.33/0.59 ! [N: nat,A: rat] :
% 0.33/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.33/0.59 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % power_minus_odd
% 0.33/0.59 thf(fact_4794_power__minus__odd,axiom,
% 0.33/0.59 ! [N: nat,A: code_integer] :
% 0.33/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % power_minus_odd
% 0.33/0.59 thf(fact_4795_diff__numeral__special_I4_J,axiom,
% 0.33/0.59 ! [M: num] :
% 0.33/0.59 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 0.33/0.59 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(4)
% 0.33/0.59 thf(fact_4796_diff__numeral__special_I4_J,axiom,
% 0.33/0.59 ! [M: num] :
% 0.33/0.59 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 0.33/0.59 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(4)
% 0.33/0.59 thf(fact_4797_diff__numeral__special_I4_J,axiom,
% 0.33/0.59 ! [M: num] :
% 0.33/0.59 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(4)
% 0.33/0.59 thf(fact_4798_diff__numeral__special_I4_J,axiom,
% 0.33/0.59 ! [M: num] :
% 0.33/0.59 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 0.33/0.59 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(4)
% 0.33/0.59 thf(fact_4799_diff__numeral__special_I4_J,axiom,
% 0.33/0.59 ! [M: num] :
% 0.33/0.59 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(4)
% 0.33/0.59 thf(fact_4800_diff__numeral__special_I3_J,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.33/0.59 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(3)
% 0.33/0.59 thf(fact_4801_diff__numeral__special_I3_J,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.59 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(3)
% 0.33/0.59 thf(fact_4802_diff__numeral__special_I3_J,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.33/0.59 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(3)
% 0.33/0.59 thf(fact_4803_diff__numeral__special_I3_J,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.33/0.59 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(3)
% 0.33/0.59 thf(fact_4804_diff__numeral__special_I3_J,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.33/0.59 = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % diff_numeral_special(3)
% 0.33/0.59 thf(fact_4805_ceiling__le__neg__numeral,axiom,
% 0.33/0.59 ! [X: real,V: num] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.33/0.59 = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_le_neg_numeral
% 0.33/0.59 thf(fact_4806_ceiling__le__neg__numeral,axiom,
% 0.33/0.59 ! [X: rat,V: num] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.33/0.59 = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_le_neg_numeral
% 0.33/0.59 thf(fact_4807_ceiling__less__zero,axiom,
% 0.33/0.59 ! [X: real] :
% 0.33/0.59 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
% 0.33/0.59 = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_less_zero
% 0.33/0.59 thf(fact_4808_ceiling__less__zero,axiom,
% 0.33/0.59 ! [X: rat] :
% 0.33/0.59 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
% 0.33/0.59 = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_less_zero
% 0.33/0.59 thf(fact_4809_zero__le__ceiling,axiom,
% 0.33/0.59 ! [X: real] :
% 0.33/0.59 ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
% 0.33/0.59 = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % zero_le_ceiling
% 0.33/0.59 thf(fact_4810_zero__le__ceiling,axiom,
% 0.33/0.59 ! [X: rat] :
% 0.33/0.59 ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
% 0.33/0.59 = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % zero_le_ceiling
% 0.33/0.59 thf(fact_4811_neg__numeral__less__ceiling,axiom,
% 0.33/0.59 ! [V: num,X: real] :
% 0.33/0.59 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
% 0.33/0.59 = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_less_ceiling
% 0.33/0.59 thf(fact_4812_neg__numeral__less__ceiling,axiom,
% 0.33/0.59 ! [V: num,X: rat] :
% 0.33/0.59 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
% 0.33/0.59 = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_less_ceiling
% 0.33/0.59 thf(fact_4813_dbl__simps_I4_J,axiom,
% 0.33/0.59 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.59 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % dbl_simps(4)
% 0.33/0.59 thf(fact_4814_dbl__simps_I4_J,axiom,
% 0.33/0.59 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.59 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % dbl_simps(4)
% 0.33/0.59 thf(fact_4815_dbl__simps_I4_J,axiom,
% 0.33/0.59 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % dbl_simps(4)
% 0.33/0.59 thf(fact_4816_dbl__simps_I4_J,axiom,
% 0.33/0.59 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.59 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % dbl_simps(4)
% 0.33/0.59 thf(fact_4817_dbl__simps_I4_J,axiom,
% 0.33/0.59 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % dbl_simps(4)
% 0.33/0.59 thf(fact_4818_power__minus1__even,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.59 = one_one_real ) ).
% 0.33/0.59
% 0.33/0.59 % power_minus1_even
% 0.33/0.59 thf(fact_4819_power__minus1__even,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.59 = one_one_int ) ).
% 0.33/0.59
% 0.33/0.59 % power_minus1_even
% 0.33/0.59 thf(fact_4820_power__minus1__even,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.59 = one_one_complex ) ).
% 0.33/0.59
% 0.33/0.59 % power_minus1_even
% 0.33/0.59 thf(fact_4821_power__minus1__even,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.59 = one_one_rat ) ).
% 0.33/0.59
% 0.33/0.59 % power_minus1_even
% 0.33/0.59 thf(fact_4822_power__minus1__even,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.59 = one_one_Code_integer ) ).
% 0.33/0.59
% 0.33/0.59 % power_minus1_even
% 0.33/0.59 thf(fact_4823_neg__one__even__power,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.33/0.59 = one_one_real ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_one_even_power
% 0.33/0.59 thf(fact_4824_neg__one__even__power,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.33/0.59 = one_one_int ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_one_even_power
% 0.33/0.59 thf(fact_4825_neg__one__even__power,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.33/0.59 = one_one_complex ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_one_even_power
% 0.33/0.59 thf(fact_4826_neg__one__even__power,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.33/0.59 = one_one_rat ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_one_even_power
% 0.33/0.59 thf(fact_4827_neg__one__even__power,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.33/0.59 = one_one_Code_integer ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_one_even_power
% 0.33/0.59 thf(fact_4828_neg__one__odd__power,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.33/0.59 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_one_odd_power
% 0.33/0.59 thf(fact_4829_neg__one__odd__power,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.33/0.59 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_one_odd_power
% 0.33/0.59 thf(fact_4830_neg__one__odd__power,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.33/0.59 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_one_odd_power
% 0.33/0.59 thf(fact_4831_neg__one__odd__power,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.33/0.59 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_one_odd_power
% 0.33/0.59 thf(fact_4832_neg__one__odd__power,axiom,
% 0.33/0.59 ! [N: nat] :
% 0.33/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.59 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_one_odd_power
% 0.33/0.59 thf(fact_4833_ceiling__less__numeral,axiom,
% 0.33/0.59 ! [X: real,V: num] :
% 0.33/0.59 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.33/0.59 = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_less_numeral
% 0.33/0.59 thf(fact_4834_ceiling__less__numeral,axiom,
% 0.33/0.59 ! [X: rat,V: num] :
% 0.33/0.59 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 0.33/0.59 = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_less_numeral
% 0.33/0.59 thf(fact_4835_numeral__le__ceiling,axiom,
% 0.33/0.59 ! [V: num,X: real] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
% 0.33/0.59 = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % numeral_le_ceiling
% 0.33/0.59 thf(fact_4836_numeral__le__ceiling,axiom,
% 0.33/0.59 ! [V: num,X: rat] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
% 0.33/0.59 = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % numeral_le_ceiling
% 0.33/0.59 thf(fact_4837_signed__take__bit__0,axiom,
% 0.33/0.59 ! [A: code_integer] :
% 0.33/0.59 ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % signed_take_bit_0
% 0.33/0.59 thf(fact_4838_signed__take__bit__0,axiom,
% 0.33/0.59 ! [A: int] :
% 0.33/0.59 ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 0.33/0.59 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % signed_take_bit_0
% 0.33/0.59 thf(fact_4839_push__bit__numeral__minus__1,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.59 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % push_bit_numeral_minus_1
% 0.33/0.59 thf(fact_4840_push__bit__numeral__minus__1,axiom,
% 0.33/0.59 ! [N: num] :
% 0.33/0.59 ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % push_bit_numeral_minus_1
% 0.33/0.59 thf(fact_4841_ceiling__less__neg__numeral,axiom,
% 0.33/0.59 ! [X: real,V: num] :
% 0.33/0.59 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.33/0.59 = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_less_neg_numeral
% 0.33/0.59 thf(fact_4842_ceiling__less__neg__numeral,axiom,
% 0.33/0.59 ! [X: rat,V: num] :
% 0.33/0.59 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.33/0.59 = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % ceiling_less_neg_numeral
% 0.33/0.59 thf(fact_4843_neg__numeral__le__ceiling,axiom,
% 0.33/0.59 ! [V: num,X: real] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
% 0.33/0.59 = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_le_ceiling
% 0.33/0.59 thf(fact_4844_neg__numeral__le__ceiling,axiom,
% 0.33/0.59 ! [V: num,X: rat] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
% 0.33/0.59 = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % neg_numeral_le_ceiling
% 0.33/0.59 thf(fact_4845_verit__negate__coefficient_I3_J,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( A = B )
% 0.33/0.59 => ( ( uminus_uminus_real @ A )
% 0.33/0.59 = ( uminus_uminus_real @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % verit_negate_coefficient(3)
% 0.33/0.59 thf(fact_4846_verit__negate__coefficient_I3_J,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( A = B )
% 0.33/0.59 => ( ( uminus_uminus_int @ A )
% 0.33/0.59 = ( uminus_uminus_int @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % verit_negate_coefficient(3)
% 0.33/0.59 thf(fact_4847_verit__negate__coefficient_I3_J,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( A = B )
% 0.33/0.59 => ( ( uminus_uminus_rat @ A )
% 0.33/0.59 = ( uminus_uminus_rat @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % verit_negate_coefficient(3)
% 0.33/0.59 thf(fact_4848_verit__negate__coefficient_I3_J,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( A = B )
% 0.33/0.59 => ( ( uminus1351360451143612070nteger @ A )
% 0.33/0.59 = ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % verit_negate_coefficient(3)
% 0.33/0.59 thf(fact_4849_minus__equation__iff,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( ( uminus_uminus_real @ A )
% 0.33/0.59 = B )
% 0.33/0.59 = ( ( uminus_uminus_real @ B )
% 0.33/0.59 = A ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_equation_iff
% 0.33/0.59 thf(fact_4850_minus__equation__iff,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( ( uminus_uminus_int @ A )
% 0.33/0.59 = B )
% 0.33/0.59 = ( ( uminus_uminus_int @ B )
% 0.33/0.59 = A ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_equation_iff
% 0.33/0.59 thf(fact_4851_minus__equation__iff,axiom,
% 0.33/0.59 ! [A: complex,B: complex] :
% 0.33/0.59 ( ( ( uminus1482373934393186551omplex @ A )
% 0.33/0.59 = B )
% 0.33/0.59 = ( ( uminus1482373934393186551omplex @ B )
% 0.33/0.59 = A ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_equation_iff
% 0.33/0.59 thf(fact_4852_minus__equation__iff,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( ( uminus_uminus_rat @ A )
% 0.33/0.59 = B )
% 0.33/0.59 = ( ( uminus_uminus_rat @ B )
% 0.33/0.59 = A ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_equation_iff
% 0.33/0.59 thf(fact_4853_minus__equation__iff,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( ( uminus1351360451143612070nteger @ A )
% 0.33/0.59 = B )
% 0.33/0.59 = ( ( uminus1351360451143612070nteger @ B )
% 0.33/0.59 = A ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_equation_iff
% 0.33/0.59 thf(fact_4854_equation__minus__iff,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( A
% 0.33/0.59 = ( uminus_uminus_real @ B ) )
% 0.33/0.59 = ( B
% 0.33/0.59 = ( uminus_uminus_real @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % equation_minus_iff
% 0.33/0.59 thf(fact_4855_equation__minus__iff,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( A
% 0.33/0.59 = ( uminus_uminus_int @ B ) )
% 0.33/0.59 = ( B
% 0.33/0.59 = ( uminus_uminus_int @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % equation_minus_iff
% 0.33/0.59 thf(fact_4856_equation__minus__iff,axiom,
% 0.33/0.59 ! [A: complex,B: complex] :
% 0.33/0.59 ( ( A
% 0.33/0.59 = ( uminus1482373934393186551omplex @ B ) )
% 0.33/0.59 = ( B
% 0.33/0.59 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % equation_minus_iff
% 0.33/0.59 thf(fact_4857_equation__minus__iff,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( A
% 0.33/0.59 = ( uminus_uminus_rat @ B ) )
% 0.33/0.59 = ( B
% 0.33/0.59 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % equation_minus_iff
% 0.33/0.59 thf(fact_4858_equation__minus__iff,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( A
% 0.33/0.59 = ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.59 = ( B
% 0.33/0.59 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % equation_minus_iff
% 0.33/0.59 thf(fact_4859_tanh__real__gt__neg1,axiom,
% 0.33/0.59 ! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).
% 0.33/0.59
% 0.33/0.59 % tanh_real_gt_neg1
% 0.33/0.59 thf(fact_4860_le__imp__neg__le,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( ord_less_eq_real @ A @ B )
% 0.33/0.59 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % le_imp_neg_le
% 0.33/0.59 thf(fact_4861_le__imp__neg__le,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.33/0.59 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % le_imp_neg_le
% 0.33/0.59 thf(fact_4862_le__imp__neg__le,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.59 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % le_imp_neg_le
% 0.33/0.59 thf(fact_4863_le__imp__neg__le,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( ord_less_eq_int @ A @ B )
% 0.33/0.59 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % le_imp_neg_le
% 0.33/0.59 thf(fact_4864_minus__le__iff,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.33/0.59 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_le_iff
% 0.33/0.59 thf(fact_4865_minus__le__iff,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.33/0.59 = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_le_iff
% 0.33/0.59 thf(fact_4866_minus__le__iff,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.33/0.59 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_le_iff
% 0.33/0.59 thf(fact_4867_minus__le__iff,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.59 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.33/0.59
% 0.33/0.59 % minus_le_iff
% 0.33/0.59 thf(fact_4868_le__minus__iff,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 0.33/0.59 = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % le_minus_iff
% 0.33/0.59 thf(fact_4869_le__minus__iff,axiom,
% 0.33/0.59 ! [A: code_integer,B: code_integer] :
% 0.33/0.59 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.59 = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % le_minus_iff
% 0.33/0.59 thf(fact_4870_le__minus__iff,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.33/0.59 = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % le_minus_iff
% 0.33/0.59 thf(fact_4871_le__minus__iff,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 0.33/0.59 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % le_minus_iff
% 0.33/0.59 thf(fact_4872_verit__negate__coefficient_I2_J,axiom,
% 0.33/0.59 ! [A: real,B: real] :
% 0.33/0.59 ( ( ord_less_real @ A @ B )
% 0.33/0.59 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % verit_negate_coefficient(2)
% 0.33/0.59 thf(fact_4873_verit__negate__coefficient_I2_J,axiom,
% 0.33/0.59 ! [A: int,B: int] :
% 0.33/0.59 ( ( ord_less_int @ A @ B )
% 0.33/0.59 => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % verit_negate_coefficient(2)
% 0.33/0.59 thf(fact_4874_verit__negate__coefficient_I2_J,axiom,
% 0.33/0.59 ! [A: rat,B: rat] :
% 0.33/0.59 ( ( ord_less_rat @ A @ B )
% 0.33/0.59 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.33/0.59
% 0.33/0.59 % verit_negate_coefficient(2)
% 0.33/0.59 thf(fact_4875_verit__negate__coefficient_I2_J,axiom,
% 0.33/0.60 ! [A: code_integer,B: code_integer] :
% 0.33/0.60 ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.33/0.60 => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % verit_negate_coefficient(2)
% 0.33/0.60 thf(fact_4876_minus__less__iff,axiom,
% 0.33/0.60 ! [A: real,B: real] :
% 0.33/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 0.33/0.60 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_less_iff
% 0.33/0.60 thf(fact_4877_minus__less__iff,axiom,
% 0.33/0.60 ! [A: int,B: int] :
% 0.33/0.60 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.60 = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_less_iff
% 0.33/0.60 thf(fact_4878_minus__less__iff,axiom,
% 0.33/0.60 ! [A: rat,B: rat] :
% 0.33/0.60 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.33/0.60 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_less_iff
% 0.33/0.60 thf(fact_4879_minus__less__iff,axiom,
% 0.33/0.60 ! [A: code_integer,B: code_integer] :
% 0.33/0.60 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.33/0.60 = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_less_iff
% 0.33/0.60 thf(fact_4880_less__minus__iff,axiom,
% 0.33/0.60 ! [A: real,B: real] :
% 0.33/0.60 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 0.33/0.60 = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % less_minus_iff
% 0.33/0.60 thf(fact_4881_less__minus__iff,axiom,
% 0.33/0.60 ! [A: int,B: int] :
% 0.33/0.60 ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
% 0.33/0.60 = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % less_minus_iff
% 0.33/0.60 thf(fact_4882_less__minus__iff,axiom,
% 0.33/0.60 ! [A: rat,B: rat] :
% 0.33/0.60 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.33/0.60 = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % less_minus_iff
% 0.33/0.60 thf(fact_4883_less__minus__iff,axiom,
% 0.33/0.60 ! [A: code_integer,B: code_integer] :
% 0.33/0.60 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.60 = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % less_minus_iff
% 0.33/0.60 thf(fact_4884_numeral__neq__neg__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ( ( numeral_numeral_real @ M )
% 0.33/0.60 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % numeral_neq_neg_numeral
% 0.33/0.60 thf(fact_4885_numeral__neq__neg__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ( ( numeral_numeral_int @ M )
% 0.33/0.60 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % numeral_neq_neg_numeral
% 0.33/0.60 thf(fact_4886_numeral__neq__neg__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ( ( numera6690914467698888265omplex @ M )
% 0.33/0.60 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % numeral_neq_neg_numeral
% 0.33/0.60 thf(fact_4887_numeral__neq__neg__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ( ( numeral_numeral_rat @ M )
% 0.33/0.60 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % numeral_neq_neg_numeral
% 0.33/0.60 thf(fact_4888_numeral__neq__neg__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ( ( numera6620942414471956472nteger @ M )
% 0.33/0.60 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % numeral_neq_neg_numeral
% 0.33/0.60 thf(fact_4889_neg__numeral__neq__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 0.33/0.60 != ( numeral_numeral_real @ N ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_neq_numeral
% 0.33/0.60 thf(fact_4890_neg__numeral__neq__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 0.33/0.60 != ( numeral_numeral_int @ N ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_neq_numeral
% 0.33/0.60 thf(fact_4891_neg__numeral__neq__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 0.33/0.60 != ( numera6690914467698888265omplex @ N ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_neq_numeral
% 0.33/0.60 thf(fact_4892_neg__numeral__neq__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 0.33/0.60 != ( numeral_numeral_rat @ N ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_neq_numeral
% 0.33/0.60 thf(fact_4893_neg__numeral__neq__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 0.33/0.60 != ( numera6620942414471956472nteger @ N ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_neq_numeral
% 0.33/0.60 thf(fact_4894_minus__mult__commute,axiom,
% 0.33/0.60 ! [A: real,B: real] :
% 0.33/0.60 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 0.33/0.60 = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_mult_commute
% 0.33/0.60 thf(fact_4895_minus__mult__commute,axiom,
% 0.33/0.60 ! [A: int,B: int] :
% 0.33/0.60 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.60 = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_mult_commute
% 0.33/0.60 thf(fact_4896_minus__mult__commute,axiom,
% 0.33/0.60 ! [A: complex,B: complex] :
% 0.33/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.33/0.60 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_mult_commute
% 0.33/0.60 thf(fact_4897_minus__mult__commute,axiom,
% 0.33/0.60 ! [A: rat,B: rat] :
% 0.33/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.33/0.60 = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_mult_commute
% 0.33/0.60 thf(fact_4898_minus__mult__commute,axiom,
% 0.33/0.60 ! [A: code_integer,B: code_integer] :
% 0.33/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.33/0.60 = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_mult_commute
% 0.33/0.60 thf(fact_4899_square__eq__iff,axiom,
% 0.33/0.60 ! [A: real,B: real] :
% 0.33/0.60 ( ( ( times_times_real @ A @ A )
% 0.33/0.60 = ( times_times_real @ B @ B ) )
% 0.33/0.60 = ( ( A = B )
% 0.33/0.60 | ( A
% 0.33/0.60 = ( uminus_uminus_real @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % square_eq_iff
% 0.33/0.60 thf(fact_4900_square__eq__iff,axiom,
% 0.33/0.60 ! [A: int,B: int] :
% 0.33/0.60 ( ( ( times_times_int @ A @ A )
% 0.33/0.60 = ( times_times_int @ B @ B ) )
% 0.33/0.60 = ( ( A = B )
% 0.33/0.60 | ( A
% 0.33/0.60 = ( uminus_uminus_int @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % square_eq_iff
% 0.33/0.60 thf(fact_4901_square__eq__iff,axiom,
% 0.33/0.60 ! [A: complex,B: complex] :
% 0.33/0.60 ( ( ( times_times_complex @ A @ A )
% 0.33/0.60 = ( times_times_complex @ B @ B ) )
% 0.33/0.60 = ( ( A = B )
% 0.33/0.60 | ( A
% 0.33/0.60 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % square_eq_iff
% 0.33/0.60 thf(fact_4902_square__eq__iff,axiom,
% 0.33/0.60 ! [A: rat,B: rat] :
% 0.33/0.60 ( ( ( times_times_rat @ A @ A )
% 0.33/0.60 = ( times_times_rat @ B @ B ) )
% 0.33/0.60 = ( ( A = B )
% 0.33/0.60 | ( A
% 0.33/0.60 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % square_eq_iff
% 0.33/0.60 thf(fact_4903_square__eq__iff,axiom,
% 0.33/0.60 ! [A: code_integer,B: code_integer] :
% 0.33/0.60 ( ( ( times_3573771949741848930nteger @ A @ A )
% 0.33/0.60 = ( times_3573771949741848930nteger @ B @ B ) )
% 0.33/0.60 = ( ( A = B )
% 0.33/0.60 | ( A
% 0.33/0.60 = ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % square_eq_iff
% 0.33/0.60 thf(fact_4904_one__neq__neg__one,axiom,
% 0.33/0.60 ( one_one_real
% 0.33/0.60 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % one_neq_neg_one
% 0.33/0.60 thf(fact_4905_one__neq__neg__one,axiom,
% 0.33/0.60 ( one_one_int
% 0.33/0.60 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % one_neq_neg_one
% 0.33/0.60 thf(fact_4906_one__neq__neg__one,axiom,
% 0.33/0.60 ( one_one_complex
% 0.33/0.60 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.60
% 0.33/0.60 % one_neq_neg_one
% 0.33/0.60 thf(fact_4907_one__neq__neg__one,axiom,
% 0.33/0.60 ( one_one_rat
% 0.33/0.60 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % one_neq_neg_one
% 0.33/0.60 thf(fact_4908_one__neq__neg__one,axiom,
% 0.33/0.60 ( one_one_Code_integer
% 0.33/0.60 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.60
% 0.33/0.60 % one_neq_neg_one
% 0.33/0.60 thf(fact_4909_is__num__normalize_I8_J,axiom,
% 0.33/0.60 ! [A: real,B: real] :
% 0.33/0.60 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.33/0.60 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % is_num_normalize(8)
% 0.33/0.60 thf(fact_4910_is__num__normalize_I8_J,axiom,
% 0.33/0.60 ! [A: int,B: int] :
% 0.33/0.60 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.33/0.60 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % is_num_normalize(8)
% 0.33/0.60 thf(fact_4911_is__num__normalize_I8_J,axiom,
% 0.33/0.60 ! [A: complex,B: complex] :
% 0.33/0.60 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.33/0.60 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % is_num_normalize(8)
% 0.33/0.60 thf(fact_4912_is__num__normalize_I8_J,axiom,
% 0.33/0.60 ! [A: rat,B: rat] :
% 0.33/0.60 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.33/0.60 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % is_num_normalize(8)
% 0.33/0.60 thf(fact_4913_is__num__normalize_I8_J,axiom,
% 0.33/0.60 ! [A: code_integer,B: code_integer] :
% 0.33/0.60 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.33/0.60 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % is_num_normalize(8)
% 0.33/0.60 thf(fact_4914_add_Oinverse__distrib__swap,axiom,
% 0.33/0.60 ! [A: real,B: real] :
% 0.33/0.60 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.33/0.60 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add.inverse_distrib_swap
% 0.33/0.60 thf(fact_4915_add_Oinverse__distrib__swap,axiom,
% 0.33/0.60 ! [A: int,B: int] :
% 0.33/0.60 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.33/0.60 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add.inverse_distrib_swap
% 0.33/0.60 thf(fact_4916_add_Oinverse__distrib__swap,axiom,
% 0.33/0.60 ! [A: complex,B: complex] :
% 0.33/0.60 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.33/0.60 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add.inverse_distrib_swap
% 0.33/0.60 thf(fact_4917_add_Oinverse__distrib__swap,axiom,
% 0.33/0.60 ! [A: rat,B: rat] :
% 0.33/0.60 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.33/0.60 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add.inverse_distrib_swap
% 0.33/0.60 thf(fact_4918_add_Oinverse__distrib__swap,axiom,
% 0.33/0.60 ! [A: code_integer,B: code_integer] :
% 0.33/0.60 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.33/0.60 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add.inverse_distrib_swap
% 0.33/0.60 thf(fact_4919_group__cancel_Oneg1,axiom,
% 0.33/0.60 ! [A3: real,K: real,A: real] :
% 0.33/0.60 ( ( A3
% 0.33/0.60 = ( plus_plus_real @ K @ A ) )
% 0.33/0.60 => ( ( uminus_uminus_real @ A3 )
% 0.33/0.60 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % group_cancel.neg1
% 0.33/0.60 thf(fact_4920_group__cancel_Oneg1,axiom,
% 0.33/0.60 ! [A3: int,K: int,A: int] :
% 0.33/0.60 ( ( A3
% 0.33/0.60 = ( plus_plus_int @ K @ A ) )
% 0.33/0.60 => ( ( uminus_uminus_int @ A3 )
% 0.33/0.60 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % group_cancel.neg1
% 0.33/0.60 thf(fact_4921_group__cancel_Oneg1,axiom,
% 0.33/0.60 ! [A3: complex,K: complex,A: complex] :
% 0.33/0.60 ( ( A3
% 0.33/0.60 = ( plus_plus_complex @ K @ A ) )
% 0.33/0.60 => ( ( uminus1482373934393186551omplex @ A3 )
% 0.33/0.60 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % group_cancel.neg1
% 0.33/0.60 thf(fact_4922_group__cancel_Oneg1,axiom,
% 0.33/0.60 ! [A3: rat,K: rat,A: rat] :
% 0.33/0.60 ( ( A3
% 0.33/0.60 = ( plus_plus_rat @ K @ A ) )
% 0.33/0.60 => ( ( uminus_uminus_rat @ A3 )
% 0.33/0.60 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % group_cancel.neg1
% 0.33/0.60 thf(fact_4923_group__cancel_Oneg1,axiom,
% 0.33/0.60 ! [A3: code_integer,K: code_integer,A: code_integer] :
% 0.33/0.60 ( ( A3
% 0.33/0.60 = ( plus_p5714425477246183910nteger @ K @ A ) )
% 0.33/0.60 => ( ( uminus1351360451143612070nteger @ A3 )
% 0.33/0.60 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % group_cancel.neg1
% 0.33/0.60 thf(fact_4924_minus__diff__commute,axiom,
% 0.33/0.60 ! [B: real,A: real] :
% 0.33/0.60 ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 0.33/0.60 = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_diff_commute
% 0.33/0.60 thf(fact_4925_minus__diff__commute,axiom,
% 0.33/0.60 ! [B: int,A: int] :
% 0.33/0.60 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 0.33/0.60 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_diff_commute
% 0.33/0.60 thf(fact_4926_minus__diff__commute,axiom,
% 0.33/0.60 ! [B: complex,A: complex] :
% 0.33/0.60 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
% 0.33/0.60 = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_diff_commute
% 0.33/0.60 thf(fact_4927_minus__diff__commute,axiom,
% 0.33/0.60 ! [B: rat,A: rat] :
% 0.33/0.60 ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
% 0.33/0.60 = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_diff_commute
% 0.33/0.60 thf(fact_4928_minus__diff__commute,axiom,
% 0.33/0.60 ! [B: code_integer,A: code_integer] :
% 0.33/0.60 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
% 0.33/0.60 = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_diff_commute
% 0.33/0.60 thf(fact_4929_minus__diff__minus,axiom,
% 0.33/0.60 ! [A: real,B: real] :
% 0.33/0.60 ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.33/0.60 = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_diff_minus
% 0.33/0.60 thf(fact_4930_minus__diff__minus,axiom,
% 0.33/0.60 ! [A: int,B: int] :
% 0.33/0.60 ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.33/0.60 = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_diff_minus
% 0.33/0.60 thf(fact_4931_minus__diff__minus,axiom,
% 0.33/0.60 ! [A: complex,B: complex] :
% 0.33/0.60 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.33/0.60 = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_diff_minus
% 0.33/0.60 thf(fact_4932_minus__diff__minus,axiom,
% 0.33/0.60 ! [A: rat,B: rat] :
% 0.33/0.60 ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.33/0.60 = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_diff_minus
% 0.33/0.60 thf(fact_4933_minus__diff__minus,axiom,
% 0.33/0.60 ! [A: code_integer,B: code_integer] :
% 0.33/0.60 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.60 = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_diff_minus
% 0.33/0.60 thf(fact_4934_minus__divide__left,axiom,
% 0.33/0.60 ! [A: real,B: real] :
% 0.33/0.60 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.60 = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_left
% 0.33/0.60 thf(fact_4935_minus__divide__left,axiom,
% 0.33/0.60 ! [A: complex,B: complex] :
% 0.33/0.60 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.60 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_left
% 0.33/0.60 thf(fact_4936_minus__divide__left,axiom,
% 0.33/0.60 ! [A: rat,B: rat] :
% 0.33/0.60 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.33/0.60 = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_left
% 0.33/0.60 thf(fact_4937_minus__divide__divide,axiom,
% 0.33/0.60 ! [A: real,B: real] :
% 0.33/0.60 ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.33/0.60 = ( divide_divide_real @ A @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_divide
% 0.33/0.60 thf(fact_4938_minus__divide__divide,axiom,
% 0.33/0.60 ! [A: complex,B: complex] :
% 0.33/0.60 ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.33/0.60 = ( divide1717551699836669952omplex @ A @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_divide
% 0.33/0.60 thf(fact_4939_minus__divide__divide,axiom,
% 0.33/0.60 ! [A: rat,B: rat] :
% 0.33/0.60 ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.33/0.60 = ( divide_divide_rat @ A @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_divide
% 0.33/0.60 thf(fact_4940_minus__divide__right,axiom,
% 0.33/0.60 ! [A: real,B: real] :
% 0.33/0.60 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.60 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_right
% 0.33/0.60 thf(fact_4941_minus__divide__right,axiom,
% 0.33/0.60 ! [A: complex,B: complex] :
% 0.33/0.60 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.60 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_right
% 0.33/0.60 thf(fact_4942_minus__divide__right,axiom,
% 0.33/0.60 ! [A: rat,B: rat] :
% 0.33/0.60 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.33/0.60 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_right
% 0.33/0.60 thf(fact_4943_div__minus__right,axiom,
% 0.33/0.60 ! [A: int,B: int] :
% 0.33/0.60 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.33/0.60 = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % div_minus_right
% 0.33/0.60 thf(fact_4944_div__minus__right,axiom,
% 0.33/0.60 ! [A: code_integer,B: code_integer] :
% 0.33/0.60 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.60 = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % div_minus_right
% 0.33/0.60 thf(fact_4945_mod__minus__right,axiom,
% 0.33/0.60 ! [A: int,B: int] :
% 0.33/0.60 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.33/0.60 = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % mod_minus_right
% 0.33/0.60 thf(fact_4946_mod__minus__right,axiom,
% 0.33/0.60 ! [A: code_integer,B: code_integer] :
% 0.33/0.60 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.60 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % mod_minus_right
% 0.33/0.60 thf(fact_4947_mod__minus__cong,axiom,
% 0.33/0.60 ! [A: int,B: int,A5: int] :
% 0.33/0.60 ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.60 = ( modulo_modulo_int @ A5 @ B ) )
% 0.33/0.60 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.60 = ( modulo_modulo_int @ ( uminus_uminus_int @ A5 ) @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % mod_minus_cong
% 0.33/0.60 thf(fact_4948_mod__minus__cong,axiom,
% 0.33/0.60 ! [A: code_integer,B: code_integer,A5: code_integer] :
% 0.33/0.60 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.33/0.60 = ( modulo364778990260209775nteger @ A5 @ B ) )
% 0.33/0.60 => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.33/0.60 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A5 ) @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % mod_minus_cong
% 0.33/0.60 thf(fact_4949_mod__minus__eq,axiom,
% 0.33/0.60 ! [A: int,B: int] :
% 0.33/0.60 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
% 0.33/0.60 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % mod_minus_eq
% 0.33/0.60 thf(fact_4950_mod__minus__eq,axiom,
% 0.33/0.60 ! [A: code_integer,B: code_integer] :
% 0.33/0.60 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
% 0.33/0.60 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % mod_minus_eq
% 0.33/0.60 thf(fact_4951_push__bit__minus,axiom,
% 0.33/0.60 ! [N: nat,A: int] :
% 0.33/0.60 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ A ) )
% 0.33/0.60 = ( uminus_uminus_int @ ( bit_se545348938243370406it_int @ N @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % push_bit_minus
% 0.33/0.60 thf(fact_4952_push__bit__minus,axiom,
% 0.33/0.60 ! [N: nat,A: code_integer] :
% 0.33/0.60 ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ A ) )
% 0.33/0.60 = ( uminus1351360451143612070nteger @ ( bit_se7788150548672797655nteger @ N @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % push_bit_minus
% 0.33/0.60 thf(fact_4953_sgn__real__def,axiom,
% 0.33/0.60 ( sgn_sgn_real
% 0.33/0.60 = ( ^ [A2: real] : ( if_real @ ( A2 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A2 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_real_def
% 0.33/0.60 thf(fact_4954_ceiling__mono,axiom,
% 0.33/0.60 ! [Y: real,X: real] :
% 0.33/0.60 ( ( ord_less_eq_real @ Y @ X )
% 0.33/0.60 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % ceiling_mono
% 0.33/0.60 thf(fact_4955_ceiling__mono,axiom,
% 0.33/0.60 ! [Y: rat,X: rat] :
% 0.33/0.60 ( ( ord_less_eq_rat @ Y @ X )
% 0.33/0.60 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % ceiling_mono
% 0.33/0.60 thf(fact_4956_ceiling__less__cancel,axiom,
% 0.33/0.60 ! [X: rat,Y: rat] :
% 0.33/0.60 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) )
% 0.33/0.60 => ( ord_less_rat @ X @ Y ) ) ).
% 0.33/0.60
% 0.33/0.60 % ceiling_less_cancel
% 0.33/0.60 thf(fact_4957_ceiling__less__cancel,axiom,
% 0.33/0.60 ! [X: real,Y: real] :
% 0.33/0.60 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
% 0.33/0.60 => ( ord_less_real @ X @ Y ) ) ).
% 0.33/0.60
% 0.33/0.60 % ceiling_less_cancel
% 0.33/0.60 thf(fact_4958_not__numeral__le__neg__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_numeral_le_neg_numeral
% 0.33/0.60 thf(fact_4959_not__numeral__le__neg__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_numeral_le_neg_numeral
% 0.33/0.60 thf(fact_4960_not__numeral__le__neg__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_numeral_le_neg_numeral
% 0.33/0.60 thf(fact_4961_not__numeral__le__neg__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_numeral_le_neg_numeral
% 0.33/0.60 thf(fact_4962_neg__numeral__le__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_le_numeral
% 0.33/0.60 thf(fact_4963_neg__numeral__le__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_le_numeral
% 0.33/0.60 thf(fact_4964_neg__numeral__le__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_le_numeral
% 0.33/0.60 thf(fact_4965_neg__numeral__le__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_le_numeral
% 0.33/0.60 thf(fact_4966_zero__neq__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ( zero_zero_real
% 0.33/0.60 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % zero_neq_neg_numeral
% 0.33/0.60 thf(fact_4967_zero__neq__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ( zero_zero_int
% 0.33/0.60 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % zero_neq_neg_numeral
% 0.33/0.60 thf(fact_4968_zero__neq__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ( zero_zero_complex
% 0.33/0.60 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % zero_neq_neg_numeral
% 0.33/0.60 thf(fact_4969_zero__neq__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ( zero_zero_rat
% 0.33/0.60 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % zero_neq_neg_numeral
% 0.33/0.60 thf(fact_4970_zero__neq__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ( zero_z3403309356797280102nteger
% 0.33/0.60 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % zero_neq_neg_numeral
% 0.33/0.60 thf(fact_4971_not__numeral__less__neg__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_numeral_less_neg_numeral
% 0.33/0.60 thf(fact_4972_not__numeral__less__neg__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_numeral_less_neg_numeral
% 0.33/0.60 thf(fact_4973_not__numeral__less__neg__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_numeral_less_neg_numeral
% 0.33/0.60 thf(fact_4974_not__numeral__less__neg__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] :
% 0.33/0.60 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_numeral_less_neg_numeral
% 0.33/0.60 thf(fact_4975_neg__numeral__less__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_less_numeral
% 0.33/0.60 thf(fact_4976_neg__numeral__less__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_less_numeral
% 0.33/0.60 thf(fact_4977_neg__numeral__less__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_less_numeral
% 0.33/0.60 thf(fact_4978_neg__numeral__less__numeral,axiom,
% 0.33/0.60 ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_less_numeral
% 0.33/0.60 thf(fact_4979_le__minus__one__simps_I2_J,axiom,
% 0.33/0.60 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.33/0.60
% 0.33/0.60 % le_minus_one_simps(2)
% 0.33/0.60 thf(fact_4980_le__minus__one__simps_I2_J,axiom,
% 0.33/0.60 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.33/0.60
% 0.33/0.60 % le_minus_one_simps(2)
% 0.33/0.60 thf(fact_4981_le__minus__one__simps_I2_J,axiom,
% 0.33/0.60 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.33/0.60
% 0.33/0.60 % le_minus_one_simps(2)
% 0.33/0.60 thf(fact_4982_le__minus__one__simps_I2_J,axiom,
% 0.33/0.60 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.33/0.60
% 0.33/0.60 % le_minus_one_simps(2)
% 0.33/0.60 thf(fact_4983_le__minus__one__simps_I4_J,axiom,
% 0.33/0.60 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % le_minus_one_simps(4)
% 0.33/0.60 thf(fact_4984_le__minus__one__simps_I4_J,axiom,
% 0.33/0.60 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.60
% 0.33/0.60 % le_minus_one_simps(4)
% 0.33/0.60 thf(fact_4985_le__minus__one__simps_I4_J,axiom,
% 0.33/0.60 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % le_minus_one_simps(4)
% 0.33/0.60 thf(fact_4986_le__minus__one__simps_I4_J,axiom,
% 0.33/0.60 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % le_minus_one_simps(4)
% 0.33/0.60 thf(fact_4987_zero__neq__neg__one,axiom,
% 0.33/0.60 ( zero_zero_real
% 0.33/0.60 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % zero_neq_neg_one
% 0.33/0.60 thf(fact_4988_zero__neq__neg__one,axiom,
% 0.33/0.60 ( zero_zero_int
% 0.33/0.60 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % zero_neq_neg_one
% 0.33/0.60 thf(fact_4989_zero__neq__neg__one,axiom,
% 0.33/0.60 ( zero_zero_complex
% 0.33/0.60 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.60
% 0.33/0.60 % zero_neq_neg_one
% 0.33/0.60 thf(fact_4990_zero__neq__neg__one,axiom,
% 0.33/0.60 ( zero_zero_rat
% 0.33/0.60 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % zero_neq_neg_one
% 0.33/0.60 thf(fact_4991_zero__neq__neg__one,axiom,
% 0.33/0.60 ( zero_z3403309356797280102nteger
% 0.33/0.60 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.60
% 0.33/0.60 % zero_neq_neg_one
% 0.33/0.60 thf(fact_4992_neg__eq__iff__add__eq__0,axiom,
% 0.33/0.60 ! [A: real,B: real] :
% 0.33/0.60 ( ( ( uminus_uminus_real @ A )
% 0.33/0.60 = B )
% 0.33/0.60 = ( ( plus_plus_real @ A @ B )
% 0.33/0.60 = zero_zero_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_eq_iff_add_eq_0
% 0.33/0.60 thf(fact_4993_neg__eq__iff__add__eq__0,axiom,
% 0.33/0.60 ! [A: int,B: int] :
% 0.33/0.60 ( ( ( uminus_uminus_int @ A )
% 0.33/0.60 = B )
% 0.33/0.60 = ( ( plus_plus_int @ A @ B )
% 0.33/0.60 = zero_zero_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_eq_iff_add_eq_0
% 0.33/0.60 thf(fact_4994_neg__eq__iff__add__eq__0,axiom,
% 0.33/0.60 ! [A: complex,B: complex] :
% 0.33/0.60 ( ( ( uminus1482373934393186551omplex @ A )
% 0.33/0.60 = B )
% 0.33/0.60 = ( ( plus_plus_complex @ A @ B )
% 0.33/0.60 = zero_zero_complex ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_eq_iff_add_eq_0
% 0.33/0.60 thf(fact_4995_neg__eq__iff__add__eq__0,axiom,
% 0.33/0.60 ! [A: rat,B: rat] :
% 0.33/0.60 ( ( ( uminus_uminus_rat @ A )
% 0.33/0.60 = B )
% 0.33/0.60 = ( ( plus_plus_rat @ A @ B )
% 0.33/0.60 = zero_zero_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_eq_iff_add_eq_0
% 0.33/0.60 thf(fact_4996_neg__eq__iff__add__eq__0,axiom,
% 0.33/0.60 ! [A: code_integer,B: code_integer] :
% 0.33/0.60 ( ( ( uminus1351360451143612070nteger @ A )
% 0.33/0.60 = B )
% 0.33/0.60 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.33/0.60 = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_eq_iff_add_eq_0
% 0.33/0.60 thf(fact_4997_eq__neg__iff__add__eq__0,axiom,
% 0.33/0.60 ! [A: real,B: real] :
% 0.33/0.60 ( ( A
% 0.33/0.60 = ( uminus_uminus_real @ B ) )
% 0.33/0.60 = ( ( plus_plus_real @ A @ B )
% 0.33/0.60 = zero_zero_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % eq_neg_iff_add_eq_0
% 0.33/0.60 thf(fact_4998_eq__neg__iff__add__eq__0,axiom,
% 0.33/0.60 ! [A: int,B: int] :
% 0.33/0.60 ( ( A
% 0.33/0.60 = ( uminus_uminus_int @ B ) )
% 0.33/0.60 = ( ( plus_plus_int @ A @ B )
% 0.33/0.60 = zero_zero_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % eq_neg_iff_add_eq_0
% 0.33/0.60 thf(fact_4999_eq__neg__iff__add__eq__0,axiom,
% 0.33/0.60 ! [A: complex,B: complex] :
% 0.33/0.60 ( ( A
% 0.33/0.60 = ( uminus1482373934393186551omplex @ B ) )
% 0.33/0.60 = ( ( plus_plus_complex @ A @ B )
% 0.33/0.60 = zero_zero_complex ) ) ).
% 0.33/0.60
% 0.33/0.60 % eq_neg_iff_add_eq_0
% 0.33/0.60 thf(fact_5000_eq__neg__iff__add__eq__0,axiom,
% 0.33/0.60 ! [A: rat,B: rat] :
% 0.33/0.60 ( ( A
% 0.33/0.60 = ( uminus_uminus_rat @ B ) )
% 0.33/0.60 = ( ( plus_plus_rat @ A @ B )
% 0.33/0.60 = zero_zero_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % eq_neg_iff_add_eq_0
% 0.33/0.60 thf(fact_5001_eq__neg__iff__add__eq__0,axiom,
% 0.33/0.60 ! [A: code_integer,B: code_integer] :
% 0.33/0.60 ( ( A
% 0.33/0.60 = ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.60 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.33/0.60 = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.60
% 0.33/0.60 % eq_neg_iff_add_eq_0
% 0.33/0.60 thf(fact_5002_add_Oinverse__unique,axiom,
% 0.33/0.60 ! [A: real,B: real] :
% 0.33/0.60 ( ( ( plus_plus_real @ A @ B )
% 0.33/0.60 = zero_zero_real )
% 0.33/0.60 => ( ( uminus_uminus_real @ A )
% 0.33/0.60 = B ) ) ).
% 0.33/0.60
% 0.33/0.60 % add.inverse_unique
% 0.33/0.60 thf(fact_5003_add_Oinverse__unique,axiom,
% 0.33/0.60 ! [A: int,B: int] :
% 0.33/0.60 ( ( ( plus_plus_int @ A @ B )
% 0.33/0.60 = zero_zero_int )
% 0.33/0.60 => ( ( uminus_uminus_int @ A )
% 0.33/0.60 = B ) ) ).
% 0.33/0.60
% 0.33/0.60 % add.inverse_unique
% 0.33/0.60 thf(fact_5004_add_Oinverse__unique,axiom,
% 0.33/0.60 ! [A: complex,B: complex] :
% 0.33/0.60 ( ( ( plus_plus_complex @ A @ B )
% 0.33/0.60 = zero_zero_complex )
% 0.33/0.60 => ( ( uminus1482373934393186551omplex @ A )
% 0.33/0.60 = B ) ) ).
% 0.33/0.60
% 0.33/0.60 % add.inverse_unique
% 0.33/0.60 thf(fact_5005_add_Oinverse__unique,axiom,
% 0.33/0.60 ! [A: rat,B: rat] :
% 0.33/0.60 ( ( ( plus_plus_rat @ A @ B )
% 0.33/0.60 = zero_zero_rat )
% 0.33/0.60 => ( ( uminus_uminus_rat @ A )
% 0.33/0.60 = B ) ) ).
% 0.33/0.60
% 0.33/0.60 % add.inverse_unique
% 0.33/0.60 thf(fact_5006_add_Oinverse__unique,axiom,
% 0.33/0.60 ! [A: code_integer,B: code_integer] :
% 0.33/0.60 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.33/0.60 = zero_z3403309356797280102nteger )
% 0.33/0.60 => ( ( uminus1351360451143612070nteger @ A )
% 0.33/0.60 = B ) ) ).
% 0.33/0.60
% 0.33/0.60 % add.inverse_unique
% 0.33/0.60 thf(fact_5007_ab__group__add__class_Oab__left__minus,axiom,
% 0.33/0.60 ! [A: real] :
% 0.33/0.60 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.33/0.60 = zero_zero_real ) ).
% 0.33/0.60
% 0.33/0.60 % ab_group_add_class.ab_left_minus
% 0.33/0.60 thf(fact_5008_ab__group__add__class_Oab__left__minus,axiom,
% 0.33/0.60 ! [A: int] :
% 0.33/0.60 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.33/0.60 = zero_zero_int ) ).
% 0.33/0.60
% 0.33/0.60 % ab_group_add_class.ab_left_minus
% 0.33/0.60 thf(fact_5009_ab__group__add__class_Oab__left__minus,axiom,
% 0.33/0.60 ! [A: complex] :
% 0.33/0.60 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.33/0.60 = zero_zero_complex ) ).
% 0.33/0.60
% 0.33/0.60 % ab_group_add_class.ab_left_minus
% 0.33/0.60 thf(fact_5010_ab__group__add__class_Oab__left__minus,axiom,
% 0.33/0.60 ! [A: rat] :
% 0.33/0.60 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.33/0.60 = zero_zero_rat ) ).
% 0.33/0.60
% 0.33/0.60 % ab_group_add_class.ab_left_minus
% 0.33/0.60 thf(fact_5011_ab__group__add__class_Oab__left__minus,axiom,
% 0.33/0.60 ! [A: code_integer] :
% 0.33/0.60 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.33/0.60 = zero_z3403309356797280102nteger ) ).
% 0.33/0.60
% 0.33/0.60 % ab_group_add_class.ab_left_minus
% 0.33/0.60 thf(fact_5012_add__eq__0__iff,axiom,
% 0.33/0.60 ! [A: real,B: real] :
% 0.33/0.60 ( ( ( plus_plus_real @ A @ B )
% 0.33/0.60 = zero_zero_real )
% 0.33/0.60 = ( B
% 0.33/0.60 = ( uminus_uminus_real @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add_eq_0_iff
% 0.33/0.60 thf(fact_5013_add__eq__0__iff,axiom,
% 0.33/0.60 ! [A: int,B: int] :
% 0.33/0.60 ( ( ( plus_plus_int @ A @ B )
% 0.33/0.60 = zero_zero_int )
% 0.33/0.60 = ( B
% 0.33/0.60 = ( uminus_uminus_int @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add_eq_0_iff
% 0.33/0.60 thf(fact_5014_add__eq__0__iff,axiom,
% 0.33/0.60 ! [A: complex,B: complex] :
% 0.33/0.60 ( ( ( plus_plus_complex @ A @ B )
% 0.33/0.60 = zero_zero_complex )
% 0.33/0.60 = ( B
% 0.33/0.60 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add_eq_0_iff
% 0.33/0.60 thf(fact_5015_add__eq__0__iff,axiom,
% 0.33/0.60 ! [A: rat,B: rat] :
% 0.33/0.60 ( ( ( plus_plus_rat @ A @ B )
% 0.33/0.60 = zero_zero_rat )
% 0.33/0.60 = ( B
% 0.33/0.60 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add_eq_0_iff
% 0.33/0.60 thf(fact_5016_add__eq__0__iff,axiom,
% 0.33/0.60 ! [A: code_integer,B: code_integer] :
% 0.33/0.60 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.33/0.60 = zero_z3403309356797280102nteger )
% 0.33/0.60 = ( B
% 0.33/0.60 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add_eq_0_iff
% 0.33/0.60 thf(fact_5017_less__minus__one__simps_I4_J,axiom,
% 0.33/0.60 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % less_minus_one_simps(4)
% 0.33/0.60 thf(fact_5018_less__minus__one__simps_I4_J,axiom,
% 0.33/0.60 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % less_minus_one_simps(4)
% 0.33/0.60 thf(fact_5019_less__minus__one__simps_I4_J,axiom,
% 0.33/0.60 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % less_minus_one_simps(4)
% 0.33/0.60 thf(fact_5020_less__minus__one__simps_I4_J,axiom,
% 0.33/0.60 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.60
% 0.33/0.60 % less_minus_one_simps(4)
% 0.33/0.60 thf(fact_5021_less__minus__one__simps_I2_J,axiom,
% 0.33/0.60 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.33/0.60
% 0.33/0.60 % less_minus_one_simps(2)
% 0.33/0.60 thf(fact_5022_less__minus__one__simps_I2_J,axiom,
% 0.33/0.60 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.33/0.60
% 0.33/0.60 % less_minus_one_simps(2)
% 0.33/0.60 thf(fact_5023_less__minus__one__simps_I2_J,axiom,
% 0.33/0.60 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.33/0.60
% 0.33/0.60 % less_minus_one_simps(2)
% 0.33/0.60 thf(fact_5024_less__minus__one__simps_I2_J,axiom,
% 0.33/0.60 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.33/0.60
% 0.33/0.60 % less_minus_one_simps(2)
% 0.33/0.60 thf(fact_5025_numeral__times__minus__swap,axiom,
% 0.33/0.60 ! [W: num,X: real] :
% 0.33/0.60 ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
% 0.33/0.60 = ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % numeral_times_minus_swap
% 0.33/0.60 thf(fact_5026_numeral__times__minus__swap,axiom,
% 0.33/0.60 ! [W: num,X: int] :
% 0.33/0.60 ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
% 0.33/0.60 = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % numeral_times_minus_swap
% 0.33/0.60 thf(fact_5027_numeral__times__minus__swap,axiom,
% 0.33/0.60 ! [W: num,X: complex] :
% 0.33/0.60 ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
% 0.33/0.60 = ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % numeral_times_minus_swap
% 0.33/0.60 thf(fact_5028_numeral__times__minus__swap,axiom,
% 0.33/0.60 ! [W: num,X: rat] :
% 0.33/0.60 ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
% 0.33/0.60 = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % numeral_times_minus_swap
% 0.33/0.60 thf(fact_5029_numeral__times__minus__swap,axiom,
% 0.33/0.60 ! [W: num,X: code_integer] :
% 0.33/0.60 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
% 0.33/0.60 = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % numeral_times_minus_swap
% 0.33/0.60 thf(fact_5030_nonzero__minus__divide__divide,axiom,
% 0.33/0.60 ! [B: real,A: real] :
% 0.33/0.60 ( ( B != zero_zero_real )
% 0.33/0.60 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.33/0.60 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % nonzero_minus_divide_divide
% 0.33/0.60 thf(fact_5031_nonzero__minus__divide__divide,axiom,
% 0.33/0.60 ! [B: complex,A: complex] :
% 0.33/0.60 ( ( B != zero_zero_complex )
% 0.33/0.60 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.33/0.60 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % nonzero_minus_divide_divide
% 0.33/0.60 thf(fact_5032_nonzero__minus__divide__divide,axiom,
% 0.33/0.60 ! [B: rat,A: rat] :
% 0.33/0.60 ( ( B != zero_zero_rat )
% 0.33/0.60 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.33/0.60 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % nonzero_minus_divide_divide
% 0.33/0.60 thf(fact_5033_nonzero__minus__divide__right,axiom,
% 0.33/0.60 ! [B: real,A: real] :
% 0.33/0.60 ( ( B != zero_zero_real )
% 0.33/0.60 => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.60 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % nonzero_minus_divide_right
% 0.33/0.60 thf(fact_5034_nonzero__minus__divide__right,axiom,
% 0.33/0.60 ! [B: complex,A: complex] :
% 0.33/0.60 ( ( B != zero_zero_complex )
% 0.33/0.60 => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.60 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % nonzero_minus_divide_right
% 0.33/0.60 thf(fact_5035_nonzero__minus__divide__right,axiom,
% 0.33/0.60 ! [B: rat,A: rat] :
% 0.33/0.60 ( ( B != zero_zero_rat )
% 0.33/0.60 => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.33/0.60 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % nonzero_minus_divide_right
% 0.33/0.60 thf(fact_5036_numeral__neq__neg__one,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ( ( numeral_numeral_real @ N )
% 0.33/0.60 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % numeral_neq_neg_one
% 0.33/0.60 thf(fact_5037_numeral__neq__neg__one,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ( ( numeral_numeral_int @ N )
% 0.33/0.60 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % numeral_neq_neg_one
% 0.33/0.60 thf(fact_5038_numeral__neq__neg__one,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ( ( numera6690914467698888265omplex @ N )
% 0.33/0.60 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.60
% 0.33/0.60 % numeral_neq_neg_one
% 0.33/0.60 thf(fact_5039_numeral__neq__neg__one,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ( ( numeral_numeral_rat @ N )
% 0.33/0.60 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % numeral_neq_neg_one
% 0.33/0.60 thf(fact_5040_numeral__neq__neg__one,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ( ( numera6620942414471956472nteger @ N )
% 0.33/0.60 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.60
% 0.33/0.60 % numeral_neq_neg_one
% 0.33/0.60 thf(fact_5041_one__neq__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ( one_one_real
% 0.33/0.60 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % one_neq_neg_numeral
% 0.33/0.60 thf(fact_5042_one__neq__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ( one_one_int
% 0.33/0.60 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % one_neq_neg_numeral
% 0.33/0.60 thf(fact_5043_one__neq__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ( one_one_complex
% 0.33/0.60 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % one_neq_neg_numeral
% 0.33/0.60 thf(fact_5044_one__neq__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ( one_one_rat
% 0.33/0.60 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % one_neq_neg_numeral
% 0.33/0.60 thf(fact_5045_one__neq__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ( one_one_Code_integer
% 0.33/0.60 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % one_neq_neg_numeral
% 0.33/0.60 thf(fact_5046_square__eq__1__iff,axiom,
% 0.33/0.60 ! [X: real] :
% 0.33/0.60 ( ( ( times_times_real @ X @ X )
% 0.33/0.60 = one_one_real )
% 0.33/0.60 = ( ( X = one_one_real )
% 0.33/0.60 | ( X
% 0.33/0.60 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % square_eq_1_iff
% 0.33/0.60 thf(fact_5047_square__eq__1__iff,axiom,
% 0.33/0.60 ! [X: int] :
% 0.33/0.60 ( ( ( times_times_int @ X @ X )
% 0.33/0.60 = one_one_int )
% 0.33/0.60 = ( ( X = one_one_int )
% 0.33/0.60 | ( X
% 0.33/0.60 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % square_eq_1_iff
% 0.33/0.60 thf(fact_5048_square__eq__1__iff,axiom,
% 0.33/0.60 ! [X: complex] :
% 0.33/0.60 ( ( ( times_times_complex @ X @ X )
% 0.33/0.60 = one_one_complex )
% 0.33/0.60 = ( ( X = one_one_complex )
% 0.33/0.60 | ( X
% 0.33/0.60 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % square_eq_1_iff
% 0.33/0.60 thf(fact_5049_square__eq__1__iff,axiom,
% 0.33/0.60 ! [X: rat] :
% 0.33/0.60 ( ( ( times_times_rat @ X @ X )
% 0.33/0.60 = one_one_rat )
% 0.33/0.60 = ( ( X = one_one_rat )
% 0.33/0.60 | ( X
% 0.33/0.60 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % square_eq_1_iff
% 0.33/0.60 thf(fact_5050_square__eq__1__iff,axiom,
% 0.33/0.60 ! [X: code_integer] :
% 0.33/0.60 ( ( ( times_3573771949741848930nteger @ X @ X )
% 0.33/0.60 = one_one_Code_integer )
% 0.33/0.60 = ( ( X = one_one_Code_integer )
% 0.33/0.60 | ( X
% 0.33/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % square_eq_1_iff
% 0.33/0.60 thf(fact_5051_group__cancel_Osub2,axiom,
% 0.33/0.60 ! [B5: real,K: real,B: real,A: real] :
% 0.33/0.60 ( ( B5
% 0.33/0.60 = ( plus_plus_real @ K @ B ) )
% 0.33/0.60 => ( ( minus_minus_real @ A @ B5 )
% 0.33/0.60 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % group_cancel.sub2
% 0.33/0.60 thf(fact_5052_group__cancel_Osub2,axiom,
% 0.33/0.60 ! [B5: int,K: int,B: int,A: int] :
% 0.33/0.60 ( ( B5
% 0.33/0.60 = ( plus_plus_int @ K @ B ) )
% 0.33/0.60 => ( ( minus_minus_int @ A @ B5 )
% 0.33/0.60 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % group_cancel.sub2
% 0.33/0.60 thf(fact_5053_group__cancel_Osub2,axiom,
% 0.33/0.60 ! [B5: complex,K: complex,B: complex,A: complex] :
% 0.33/0.60 ( ( B5
% 0.33/0.60 = ( plus_plus_complex @ K @ B ) )
% 0.33/0.60 => ( ( minus_minus_complex @ A @ B5 )
% 0.33/0.60 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % group_cancel.sub2
% 0.33/0.60 thf(fact_5054_group__cancel_Osub2,axiom,
% 0.33/0.60 ! [B5: rat,K: rat,B: rat,A: rat] :
% 0.33/0.60 ( ( B5
% 0.33/0.60 = ( plus_plus_rat @ K @ B ) )
% 0.33/0.60 => ( ( minus_minus_rat @ A @ B5 )
% 0.33/0.60 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % group_cancel.sub2
% 0.33/0.60 thf(fact_5055_group__cancel_Osub2,axiom,
% 0.33/0.60 ! [B5: code_integer,K: code_integer,B: code_integer,A: code_integer] :
% 0.33/0.60 ( ( B5
% 0.33/0.60 = ( plus_p5714425477246183910nteger @ K @ B ) )
% 0.33/0.60 => ( ( minus_8373710615458151222nteger @ A @ B5 )
% 0.33/0.60 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % group_cancel.sub2
% 0.33/0.60 thf(fact_5056_diff__conv__add__uminus,axiom,
% 0.33/0.60 ( minus_minus_real
% 0.33/0.60 = ( ^ [A2: real,B2: real] : ( plus_plus_real @ A2 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % diff_conv_add_uminus
% 0.33/0.60 thf(fact_5057_diff__conv__add__uminus,axiom,
% 0.33/0.60 ( minus_minus_int
% 0.33/0.60 = ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % diff_conv_add_uminus
% 0.33/0.60 thf(fact_5058_diff__conv__add__uminus,axiom,
% 0.33/0.60 ( minus_minus_complex
% 0.33/0.60 = ( ^ [A2: complex,B2: complex] : ( plus_plus_complex @ A2 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % diff_conv_add_uminus
% 0.33/0.60 thf(fact_5059_diff__conv__add__uminus,axiom,
% 0.33/0.60 ( minus_minus_rat
% 0.33/0.60 = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % diff_conv_add_uminus
% 0.33/0.60 thf(fact_5060_diff__conv__add__uminus,axiom,
% 0.33/0.60 ( minus_8373710615458151222nteger
% 0.33/0.60 = ( ^ [A2: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % diff_conv_add_uminus
% 0.33/0.60 thf(fact_5061_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.33/0.60 ( minus_minus_real
% 0.33/0.60 = ( ^ [A2: real,B2: real] : ( plus_plus_real @ A2 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.33/0.60 thf(fact_5062_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.33/0.60 ( minus_minus_int
% 0.33/0.60 = ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.33/0.60 thf(fact_5063_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.33/0.60 ( minus_minus_complex
% 0.33/0.60 = ( ^ [A2: complex,B2: complex] : ( plus_plus_complex @ A2 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.33/0.60 thf(fact_5064_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.33/0.60 ( minus_minus_rat
% 0.33/0.60 = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.33/0.60 thf(fact_5065_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.33/0.60 ( minus_8373710615458151222nteger
% 0.33/0.60 = ( ^ [A2: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.33/0.60 thf(fact_5066_dvd__neg__div,axiom,
% 0.33/0.60 ! [B: real,A: real] :
% 0.33/0.60 ( ( dvd_dvd_real @ B @ A )
% 0.33/0.60 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
% 0.33/0.60 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % dvd_neg_div
% 0.33/0.60 thf(fact_5067_dvd__neg__div,axiom,
% 0.33/0.60 ! [B: int,A: int] :
% 0.33/0.60 ( ( dvd_dvd_int @ B @ A )
% 0.33/0.60 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.60 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % dvd_neg_div
% 0.33/0.60 thf(fact_5068_dvd__neg__div,axiom,
% 0.33/0.60 ! [B: complex,A: complex] :
% 0.33/0.60 ( ( dvd_dvd_complex @ B @ A )
% 0.33/0.60 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.33/0.60 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % dvd_neg_div
% 0.33/0.60 thf(fact_5069_dvd__neg__div,axiom,
% 0.33/0.60 ! [B: rat,A: rat] :
% 0.33/0.60 ( ( dvd_dvd_rat @ B @ A )
% 0.33/0.60 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.33/0.60 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % dvd_neg_div
% 0.33/0.60 thf(fact_5070_dvd__neg__div,axiom,
% 0.33/0.60 ! [B: code_integer,A: code_integer] :
% 0.33/0.60 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.33/0.60 => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.33/0.60 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % dvd_neg_div
% 0.33/0.60 thf(fact_5071_dvd__div__neg,axiom,
% 0.33/0.60 ! [B: real,A: real] :
% 0.33/0.60 ( ( dvd_dvd_real @ B @ A )
% 0.33/0.60 => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
% 0.33/0.60 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % dvd_div_neg
% 0.33/0.60 thf(fact_5072_dvd__div__neg,axiom,
% 0.33/0.60 ! [B: int,A: int] :
% 0.33/0.60 ( ( dvd_dvd_int @ B @ A )
% 0.33/0.60 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.33/0.60 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % dvd_div_neg
% 0.33/0.60 thf(fact_5073_dvd__div__neg,axiom,
% 0.33/0.60 ! [B: complex,A: complex] :
% 0.33/0.60 ( ( dvd_dvd_complex @ B @ A )
% 0.33/0.60 => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.33/0.60 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % dvd_div_neg
% 0.33/0.60 thf(fact_5074_dvd__div__neg,axiom,
% 0.33/0.60 ! [B: rat,A: rat] :
% 0.33/0.60 ( ( dvd_dvd_rat @ B @ A )
% 0.33/0.60 => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.33/0.60 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % dvd_div_neg
% 0.33/0.60 thf(fact_5075_dvd__div__neg,axiom,
% 0.33/0.60 ! [B: code_integer,A: code_integer] :
% 0.33/0.60 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.33/0.60 => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.60 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % dvd_div_neg
% 0.33/0.60 thf(fact_5076_sgn__not__eq__imp,axiom,
% 0.33/0.60 ! [B: real,A: real] :
% 0.33/0.60 ( ( ( sgn_sgn_real @ B )
% 0.33/0.60 != ( sgn_sgn_real @ A ) )
% 0.33/0.60 => ( ( ( sgn_sgn_real @ A )
% 0.33/0.60 != zero_zero_real )
% 0.33/0.60 => ( ( ( sgn_sgn_real @ B )
% 0.33/0.60 != zero_zero_real )
% 0.33/0.60 => ( ( sgn_sgn_real @ A )
% 0.33/0.60 = ( uminus_uminus_real @ ( sgn_sgn_real @ B ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_not_eq_imp
% 0.33/0.60 thf(fact_5077_sgn__not__eq__imp,axiom,
% 0.33/0.60 ! [B: int,A: int] :
% 0.33/0.60 ( ( ( sgn_sgn_int @ B )
% 0.33/0.60 != ( sgn_sgn_int @ A ) )
% 0.33/0.60 => ( ( ( sgn_sgn_int @ A )
% 0.33/0.60 != zero_zero_int )
% 0.33/0.60 => ( ( ( sgn_sgn_int @ B )
% 0.33/0.60 != zero_zero_int )
% 0.33/0.60 => ( ( sgn_sgn_int @ A )
% 0.33/0.60 = ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_not_eq_imp
% 0.33/0.60 thf(fact_5078_sgn__not__eq__imp,axiom,
% 0.33/0.60 ! [B: rat,A: rat] :
% 0.33/0.60 ( ( ( sgn_sgn_rat @ B )
% 0.33/0.60 != ( sgn_sgn_rat @ A ) )
% 0.33/0.60 => ( ( ( sgn_sgn_rat @ A )
% 0.33/0.60 != zero_zero_rat )
% 0.33/0.60 => ( ( ( sgn_sgn_rat @ B )
% 0.33/0.60 != zero_zero_rat )
% 0.33/0.60 => ( ( sgn_sgn_rat @ A )
% 0.33/0.60 = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_not_eq_imp
% 0.33/0.60 thf(fact_5079_sgn__not__eq__imp,axiom,
% 0.33/0.60 ! [B: code_integer,A: code_integer] :
% 0.33/0.60 ( ( ( sgn_sgn_Code_integer @ B )
% 0.33/0.60 != ( sgn_sgn_Code_integer @ A ) )
% 0.33/0.60 => ( ( ( sgn_sgn_Code_integer @ A )
% 0.33/0.60 != zero_z3403309356797280102nteger )
% 0.33/0.60 => ( ( ( sgn_sgn_Code_integer @ B )
% 0.33/0.60 != zero_z3403309356797280102nteger )
% 0.33/0.60 => ( ( sgn_sgn_Code_integer @ A )
% 0.33/0.60 = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_not_eq_imp
% 0.33/0.60 thf(fact_5080_sgn__minus__1,axiom,
% 0.33/0.60 ( ( sgn_sgn_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.60 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_minus_1
% 0.33/0.60 thf(fact_5081_sgn__minus__1,axiom,
% 0.33/0.60 ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.60 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_minus_1
% 0.33/0.60 thf(fact_5082_sgn__minus__1,axiom,
% 0.33/0.60 ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.60 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_minus_1
% 0.33/0.60 thf(fact_5083_sgn__minus__1,axiom,
% 0.33/0.60 ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.60 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_minus_1
% 0.33/0.60 thf(fact_5084_sgn__minus__1,axiom,
% 0.33/0.60 ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_minus_1
% 0.33/0.60 thf(fact_5085_real__minus__mult__self__le,axiom,
% 0.33/0.60 ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% 0.33/0.60
% 0.33/0.60 % real_minus_mult_self_le
% 0.33/0.60 thf(fact_5086_minus__real__def,axiom,
% 0.33/0.60 ( minus_minus_real
% 0.33/0.60 = ( ^ [X3: real,Y6: real] : ( plus_plus_real @ X3 @ ( uminus_uminus_real @ Y6 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_real_def
% 0.33/0.60 thf(fact_5087_tanh__real__lt__1,axiom,
% 0.33/0.60 ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).
% 0.33/0.60
% 0.33/0.60 % tanh_real_lt_1
% 0.33/0.60 thf(fact_5088_ceiling__add__le,axiom,
% 0.33/0.60 ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % ceiling_add_le
% 0.33/0.60 thf(fact_5089_ceiling__add__le,axiom,
% 0.33/0.60 ! [X: real,Y: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % ceiling_add_le
% 0.33/0.60 thf(fact_5090_not__zero__le__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_zero_le_neg_numeral
% 0.33/0.60 thf(fact_5091_not__zero__le__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_zero_le_neg_numeral
% 0.33/0.60 thf(fact_5092_not__zero__le__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_zero_le_neg_numeral
% 0.33/0.60 thf(fact_5093_not__zero__le__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_zero_le_neg_numeral
% 0.33/0.60 thf(fact_5094_neg__numeral__le__zero,axiom,
% 0.33/0.60 ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_le_zero
% 0.33/0.60 thf(fact_5095_neg__numeral__le__zero,axiom,
% 0.33/0.60 ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_le_zero
% 0.33/0.60 thf(fact_5096_neg__numeral__le__zero,axiom,
% 0.33/0.60 ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_le_zero
% 0.33/0.60 thf(fact_5097_neg__numeral__le__zero,axiom,
% 0.33/0.60 ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_le_zero
% 0.33/0.60 thf(fact_5098_not__zero__less__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_zero_less_neg_numeral
% 0.33/0.60 thf(fact_5099_not__zero__less__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_zero_less_neg_numeral
% 0.33/0.60 thf(fact_5100_not__zero__less__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_zero_less_neg_numeral
% 0.33/0.60 thf(fact_5101_not__zero__less__neg__numeral,axiom,
% 0.33/0.60 ! [N: num] :
% 0.33/0.60 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_zero_less_neg_numeral
% 0.33/0.60 thf(fact_5102_neg__numeral__less__zero,axiom,
% 0.33/0.60 ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_less_zero
% 0.33/0.60 thf(fact_5103_neg__numeral__less__zero,axiom,
% 0.33/0.60 ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_less_zero
% 0.33/0.60 thf(fact_5104_neg__numeral__less__zero,axiom,
% 0.33/0.60 ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_less_zero
% 0.33/0.60 thf(fact_5105_neg__numeral__less__zero,axiom,
% 0.33/0.60 ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_less_zero
% 0.33/0.60 thf(fact_5106_le__minus__one__simps_I1_J,axiom,
% 0.33/0.60 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.33/0.60
% 0.33/0.60 % le_minus_one_simps(1)
% 0.33/0.60 thf(fact_5107_le__minus__one__simps_I1_J,axiom,
% 0.33/0.60 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.33/0.60
% 0.33/0.60 % le_minus_one_simps(1)
% 0.33/0.60 thf(fact_5108_le__minus__one__simps_I1_J,axiom,
% 0.33/0.60 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.33/0.60
% 0.33/0.60 % le_minus_one_simps(1)
% 0.33/0.60 thf(fact_5109_le__minus__one__simps_I1_J,axiom,
% 0.33/0.60 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.33/0.60
% 0.33/0.60 % le_minus_one_simps(1)
% 0.33/0.60 thf(fact_5110_le__minus__one__simps_I3_J,axiom,
% 0.33/0.60 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % le_minus_one_simps(3)
% 0.33/0.60 thf(fact_5111_le__minus__one__simps_I3_J,axiom,
% 0.33/0.60 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.60
% 0.33/0.60 % le_minus_one_simps(3)
% 0.33/0.60 thf(fact_5112_le__minus__one__simps_I3_J,axiom,
% 0.33/0.60 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % le_minus_one_simps(3)
% 0.33/0.60 thf(fact_5113_le__minus__one__simps_I3_J,axiom,
% 0.33/0.60 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % le_minus_one_simps(3)
% 0.33/0.60 thf(fact_5114_less__minus__one__simps_I1_J,axiom,
% 0.33/0.60 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.33/0.60
% 0.33/0.60 % less_minus_one_simps(1)
% 0.33/0.60 thf(fact_5115_less__minus__one__simps_I1_J,axiom,
% 0.33/0.60 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.33/0.60
% 0.33/0.60 % less_minus_one_simps(1)
% 0.33/0.60 thf(fact_5116_less__minus__one__simps_I1_J,axiom,
% 0.33/0.60 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.33/0.60
% 0.33/0.60 % less_minus_one_simps(1)
% 0.33/0.60 thf(fact_5117_less__minus__one__simps_I1_J,axiom,
% 0.33/0.60 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.33/0.60
% 0.33/0.60 % less_minus_one_simps(1)
% 0.33/0.60 thf(fact_5118_less__minus__one__simps_I3_J,axiom,
% 0.33/0.60 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % less_minus_one_simps(3)
% 0.33/0.60 thf(fact_5119_less__minus__one__simps_I3_J,axiom,
% 0.33/0.60 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % less_minus_one_simps(3)
% 0.33/0.60 thf(fact_5120_less__minus__one__simps_I3_J,axiom,
% 0.33/0.60 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % less_minus_one_simps(3)
% 0.33/0.60 thf(fact_5121_less__minus__one__simps_I3_J,axiom,
% 0.33/0.60 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.60
% 0.33/0.60 % less_minus_one_simps(3)
% 0.33/0.60 thf(fact_5122_neg__numeral__le__one,axiom,
% 0.33/0.60 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_le_one
% 0.33/0.60 thf(fact_5123_neg__numeral__le__one,axiom,
% 0.33/0.60 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_le_one
% 0.33/0.60 thf(fact_5124_neg__numeral__le__one,axiom,
% 0.33/0.60 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_le_one
% 0.33/0.60 thf(fact_5125_neg__numeral__le__one,axiom,
% 0.33/0.60 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_le_one
% 0.33/0.60 thf(fact_5126_neg__one__le__numeral,axiom,
% 0.33/0.60 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_one_le_numeral
% 0.33/0.60 thf(fact_5127_neg__one__le__numeral,axiom,
% 0.33/0.60 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_one_le_numeral
% 0.33/0.60 thf(fact_5128_neg__one__le__numeral,axiom,
% 0.33/0.60 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_one_le_numeral
% 0.33/0.60 thf(fact_5129_neg__one__le__numeral,axiom,
% 0.33/0.60 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_one_le_numeral
% 0.33/0.60 thf(fact_5130_neg__numeral__le__neg__one,axiom,
% 0.33/0.60 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_le_neg_one
% 0.33/0.60 thf(fact_5131_neg__numeral__le__neg__one,axiom,
% 0.33/0.60 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_le_neg_one
% 0.33/0.60 thf(fact_5132_neg__numeral__le__neg__one,axiom,
% 0.33/0.60 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_le_neg_one
% 0.33/0.60 thf(fact_5133_neg__numeral__le__neg__one,axiom,
% 0.33/0.60 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_le_neg_one
% 0.33/0.60 thf(fact_5134_not__numeral__le__neg__one,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_numeral_le_neg_one
% 0.33/0.60 thf(fact_5135_not__numeral__le__neg__one,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_numeral_le_neg_one
% 0.33/0.60 thf(fact_5136_not__numeral__le__neg__one,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_numeral_le_neg_one
% 0.33/0.60 thf(fact_5137_not__numeral__le__neg__one,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_numeral_le_neg_one
% 0.33/0.60 thf(fact_5138_not__one__le__neg__numeral,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_one_le_neg_numeral
% 0.33/0.60 thf(fact_5139_not__one__le__neg__numeral,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_one_le_neg_numeral
% 0.33/0.60 thf(fact_5140_not__one__le__neg__numeral,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_one_le_neg_numeral
% 0.33/0.60 thf(fact_5141_not__one__le__neg__numeral,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_one_le_neg_numeral
% 0.33/0.60 thf(fact_5142_neg__numeral__less__one,axiom,
% 0.33/0.60 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_less_one
% 0.33/0.60 thf(fact_5143_neg__numeral__less__one,axiom,
% 0.33/0.60 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_less_one
% 0.33/0.60 thf(fact_5144_neg__numeral__less__one,axiom,
% 0.33/0.60 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_less_one
% 0.33/0.60 thf(fact_5145_neg__numeral__less__one,axiom,
% 0.33/0.60 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.33/0.60
% 0.33/0.60 % neg_numeral_less_one
% 0.33/0.60 thf(fact_5146_neg__one__less__numeral,axiom,
% 0.33/0.60 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_one_less_numeral
% 0.33/0.60 thf(fact_5147_neg__one__less__numeral,axiom,
% 0.33/0.60 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_one_less_numeral
% 0.33/0.60 thf(fact_5148_neg__one__less__numeral,axiom,
% 0.33/0.60 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_one_less_numeral
% 0.33/0.60 thf(fact_5149_neg__one__less__numeral,axiom,
% 0.33/0.60 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_one_less_numeral
% 0.33/0.60 thf(fact_5150_not__numeral__less__neg__one,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_numeral_less_neg_one
% 0.33/0.60 thf(fact_5151_not__numeral__less__neg__one,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_numeral_less_neg_one
% 0.33/0.60 thf(fact_5152_not__numeral__less__neg__one,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_numeral_less_neg_one
% 0.33/0.60 thf(fact_5153_not__numeral__less__neg__one,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_numeral_less_neg_one
% 0.33/0.60 thf(fact_5154_not__one__less__neg__numeral,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_one_less_neg_numeral
% 0.33/0.60 thf(fact_5155_not__one__less__neg__numeral,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_one_less_neg_numeral
% 0.33/0.60 thf(fact_5156_not__one__less__neg__numeral,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_one_less_neg_numeral
% 0.33/0.60 thf(fact_5157_not__one__less__neg__numeral,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_one_less_neg_numeral
% 0.33/0.60 thf(fact_5158_not__neg__one__less__neg__numeral,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_neg_one_less_neg_numeral
% 0.33/0.60 thf(fact_5159_not__neg__one__less__neg__numeral,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_neg_one_less_neg_numeral
% 0.33/0.60 thf(fact_5160_not__neg__one__less__neg__numeral,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_neg_one_less_neg_numeral
% 0.33/0.60 thf(fact_5161_not__neg__one__less__neg__numeral,axiom,
% 0.33/0.60 ! [M: num] :
% 0.33/0.60 ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % not_neg_one_less_neg_numeral
% 0.33/0.60 thf(fact_5162_nonzero__neg__divide__eq__eq2,axiom,
% 0.33/0.60 ! [B: real,C: real,A: real] :
% 0.33/0.60 ( ( B != zero_zero_real )
% 0.33/0.60 => ( ( C
% 0.33/0.60 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
% 0.33/0.60 = ( ( times_times_real @ C @ B )
% 0.33/0.60 = ( uminus_uminus_real @ A ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % nonzero_neg_divide_eq_eq2
% 0.33/0.60 thf(fact_5163_nonzero__neg__divide__eq__eq2,axiom,
% 0.33/0.60 ! [B: complex,C: complex,A: complex] :
% 0.33/0.60 ( ( B != zero_zero_complex )
% 0.33/0.60 => ( ( C
% 0.33/0.60 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.33/0.60 = ( ( times_times_complex @ C @ B )
% 0.33/0.60 = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % nonzero_neg_divide_eq_eq2
% 0.33/0.60 thf(fact_5164_nonzero__neg__divide__eq__eq2,axiom,
% 0.33/0.60 ! [B: rat,C: rat,A: rat] :
% 0.33/0.60 ( ( B != zero_zero_rat )
% 0.33/0.60 => ( ( C
% 0.33/0.60 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
% 0.33/0.60 = ( ( times_times_rat @ C @ B )
% 0.33/0.60 = ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % nonzero_neg_divide_eq_eq2
% 0.33/0.60 thf(fact_5165_nonzero__neg__divide__eq__eq,axiom,
% 0.33/0.60 ! [B: real,A: real,C: real] :
% 0.33/0.60 ( ( B != zero_zero_real )
% 0.33/0.60 => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.60 = C )
% 0.33/0.60 = ( ( uminus_uminus_real @ A )
% 0.33/0.60 = ( times_times_real @ C @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % nonzero_neg_divide_eq_eq
% 0.33/0.60 thf(fact_5166_nonzero__neg__divide__eq__eq,axiom,
% 0.33/0.60 ! [B: complex,A: complex,C: complex] :
% 0.33/0.60 ( ( B != zero_zero_complex )
% 0.33/0.60 => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.60 = C )
% 0.33/0.60 = ( ( uminus1482373934393186551omplex @ A )
% 0.33/0.60 = ( times_times_complex @ C @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % nonzero_neg_divide_eq_eq
% 0.33/0.60 thf(fact_5167_nonzero__neg__divide__eq__eq,axiom,
% 0.33/0.60 ! [B: rat,A: rat,C: rat] :
% 0.33/0.60 ( ( B != zero_zero_rat )
% 0.33/0.60 => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.33/0.60 = C )
% 0.33/0.60 = ( ( uminus_uminus_rat @ A )
% 0.33/0.60 = ( times_times_rat @ C @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % nonzero_neg_divide_eq_eq
% 0.33/0.60 thf(fact_5168_minus__divide__eq__eq,axiom,
% 0.33/0.60 ! [B: real,C: real,A: real] :
% 0.33/0.60 ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
% 0.33/0.60 = A )
% 0.33/0.60 = ( ( ( C != zero_zero_real )
% 0.33/0.60 => ( ( uminus_uminus_real @ B )
% 0.33/0.60 = ( times_times_real @ A @ C ) ) )
% 0.33/0.60 & ( ( C = zero_zero_real )
% 0.33/0.60 => ( A = zero_zero_real ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_eq_eq
% 0.33/0.60 thf(fact_5169_minus__divide__eq__eq,axiom,
% 0.33/0.60 ! [B: complex,C: complex,A: complex] :
% 0.33/0.60 ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.33/0.60 = A )
% 0.33/0.60 = ( ( ( C != zero_zero_complex )
% 0.33/0.60 => ( ( uminus1482373934393186551omplex @ B )
% 0.33/0.60 = ( times_times_complex @ A @ C ) ) )
% 0.33/0.60 & ( ( C = zero_zero_complex )
% 0.33/0.60 => ( A = zero_zero_complex ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_eq_eq
% 0.33/0.60 thf(fact_5170_minus__divide__eq__eq,axiom,
% 0.33/0.60 ! [B: rat,C: rat,A: rat] :
% 0.33/0.60 ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
% 0.33/0.60 = A )
% 0.33/0.60 = ( ( ( C != zero_zero_rat )
% 0.33/0.60 => ( ( uminus_uminus_rat @ B )
% 0.33/0.60 = ( times_times_rat @ A @ C ) ) )
% 0.33/0.60 & ( ( C = zero_zero_rat )
% 0.33/0.60 => ( A = zero_zero_rat ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_eq_eq
% 0.33/0.60 thf(fact_5171_eq__minus__divide__eq,axiom,
% 0.33/0.60 ! [A: real,B: real,C: real] :
% 0.33/0.60 ( ( A
% 0.33/0.60 = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.33/0.60 = ( ( ( C != zero_zero_real )
% 0.33/0.60 => ( ( times_times_real @ A @ C )
% 0.33/0.60 = ( uminus_uminus_real @ B ) ) )
% 0.33/0.60 & ( ( C = zero_zero_real )
% 0.33/0.60 => ( A = zero_zero_real ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % eq_minus_divide_eq
% 0.33/0.60 thf(fact_5172_eq__minus__divide__eq,axiom,
% 0.33/0.60 ! [A: complex,B: complex,C: complex] :
% 0.33/0.60 ( ( A
% 0.33/0.60 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
% 0.33/0.60 = ( ( ( C != zero_zero_complex )
% 0.33/0.60 => ( ( times_times_complex @ A @ C )
% 0.33/0.60 = ( uminus1482373934393186551omplex @ B ) ) )
% 0.33/0.60 & ( ( C = zero_zero_complex )
% 0.33/0.60 => ( A = zero_zero_complex ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % eq_minus_divide_eq
% 0.33/0.60 thf(fact_5173_eq__minus__divide__eq,axiom,
% 0.33/0.60 ! [A: rat,B: rat,C: rat] :
% 0.33/0.60 ( ( A
% 0.33/0.60 = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.33/0.60 = ( ( ( C != zero_zero_rat )
% 0.33/0.60 => ( ( times_times_rat @ A @ C )
% 0.33/0.60 = ( uminus_uminus_rat @ B ) ) )
% 0.33/0.60 & ( ( C = zero_zero_rat )
% 0.33/0.60 => ( A = zero_zero_rat ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % eq_minus_divide_eq
% 0.33/0.60 thf(fact_5174_divide__eq__minus__1__iff,axiom,
% 0.33/0.60 ! [A: real,B: real] :
% 0.33/0.60 ( ( ( divide_divide_real @ A @ B )
% 0.33/0.60 = ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.60 = ( ( B != zero_zero_real )
% 0.33/0.60 & ( A
% 0.33/0.60 = ( uminus_uminus_real @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % divide_eq_minus_1_iff
% 0.33/0.60 thf(fact_5175_divide__eq__minus__1__iff,axiom,
% 0.33/0.60 ! [A: complex,B: complex] :
% 0.33/0.60 ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.33/0.60 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.60 = ( ( B != zero_zero_complex )
% 0.33/0.60 & ( A
% 0.33/0.60 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % divide_eq_minus_1_iff
% 0.33/0.60 thf(fact_5176_divide__eq__minus__1__iff,axiom,
% 0.33/0.60 ! [A: rat,B: rat] :
% 0.33/0.60 ( ( ( divide_divide_rat @ A @ B )
% 0.33/0.60 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.60 = ( ( B != zero_zero_rat )
% 0.33/0.60 & ( A
% 0.33/0.60 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % divide_eq_minus_1_iff
% 0.33/0.60 thf(fact_5177_mult__1s__ring__1_I1_J,axiom,
% 0.33/0.60 ! [B: real] :
% 0.33/0.60 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
% 0.33/0.60 = ( uminus_uminus_real @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % mult_1s_ring_1(1)
% 0.33/0.60 thf(fact_5178_mult__1s__ring__1_I1_J,axiom,
% 0.33/0.60 ! [B: int] :
% 0.33/0.60 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
% 0.33/0.60 = ( uminus_uminus_int @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % mult_1s_ring_1(1)
% 0.33/0.60 thf(fact_5179_mult__1s__ring__1_I1_J,axiom,
% 0.33/0.60 ! [B: complex] :
% 0.33/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
% 0.33/0.60 = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % mult_1s_ring_1(1)
% 0.33/0.60 thf(fact_5180_mult__1s__ring__1_I1_J,axiom,
% 0.33/0.60 ! [B: rat] :
% 0.33/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
% 0.33/0.60 = ( uminus_uminus_rat @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % mult_1s_ring_1(1)
% 0.33/0.60 thf(fact_5181_mult__1s__ring__1_I1_J,axiom,
% 0.33/0.60 ! [B: code_integer] :
% 0.33/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
% 0.33/0.60 = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % mult_1s_ring_1(1)
% 0.33/0.60 thf(fact_5182_mult__1s__ring__1_I2_J,axiom,
% 0.33/0.60 ! [B: real] :
% 0.33/0.60 ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
% 0.33/0.60 = ( uminus_uminus_real @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % mult_1s_ring_1(2)
% 0.33/0.60 thf(fact_5183_mult__1s__ring__1_I2_J,axiom,
% 0.33/0.60 ! [B: int] :
% 0.33/0.60 ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
% 0.33/0.60 = ( uminus_uminus_int @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % mult_1s_ring_1(2)
% 0.33/0.60 thf(fact_5184_mult__1s__ring__1_I2_J,axiom,
% 0.33/0.60 ! [B: complex] :
% 0.33/0.60 ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
% 0.33/0.60 = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % mult_1s_ring_1(2)
% 0.33/0.60 thf(fact_5185_mult__1s__ring__1_I2_J,axiom,
% 0.33/0.60 ! [B: rat] :
% 0.33/0.60 ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
% 0.33/0.60 = ( uminus_uminus_rat @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % mult_1s_ring_1(2)
% 0.33/0.60 thf(fact_5186_mult__1s__ring__1_I2_J,axiom,
% 0.33/0.60 ! [B: code_integer] :
% 0.33/0.60 ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
% 0.33/0.60 = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.33/0.60
% 0.33/0.60 % mult_1s_ring_1(2)
% 0.33/0.60 thf(fact_5187_uminus__numeral__One,axiom,
% 0.33/0.60 ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
% 0.33/0.60 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % uminus_numeral_One
% 0.33/0.60 thf(fact_5188_uminus__numeral__One,axiom,
% 0.33/0.60 ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
% 0.33/0.60 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % uminus_numeral_One
% 0.33/0.60 thf(fact_5189_uminus__numeral__One,axiom,
% 0.33/0.60 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
% 0.33/0.60 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.60
% 0.33/0.60 % uminus_numeral_One
% 0.33/0.60 thf(fact_5190_uminus__numeral__One,axiom,
% 0.33/0.60 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
% 0.33/0.60 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % uminus_numeral_One
% 0.33/0.60 thf(fact_5191_uminus__numeral__One,axiom,
% 0.33/0.60 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
% 0.33/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.60
% 0.33/0.60 % uminus_numeral_One
% 0.33/0.60 thf(fact_5192_power__minus,axiom,
% 0.33/0.60 ! [A: real,N: nat] :
% 0.33/0.60 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.33/0.60 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power_minus
% 0.33/0.60 thf(fact_5193_power__minus,axiom,
% 0.33/0.60 ! [A: int,N: nat] :
% 0.33/0.60 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.33/0.60 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power_minus
% 0.33/0.60 thf(fact_5194_power__minus,axiom,
% 0.33/0.60 ! [A: complex,N: nat] :
% 0.33/0.60 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.33/0.60 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power_minus
% 0.33/0.60 thf(fact_5195_power__minus,axiom,
% 0.33/0.60 ! [A: rat,N: nat] :
% 0.33/0.60 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.33/0.60 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power_minus
% 0.33/0.60 thf(fact_5196_power__minus,axiom,
% 0.33/0.60 ! [A: code_integer,N: nat] :
% 0.33/0.60 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.33/0.60 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power_minus
% 0.33/0.60 thf(fact_5197_power__minus__Bit0,axiom,
% 0.33/0.60 ! [X: real,K: num] :
% 0.33/0.60 ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.33/0.60 = ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power_minus_Bit0
% 0.33/0.60 thf(fact_5198_power__minus__Bit0,axiom,
% 0.33/0.60 ! [X: int,K: num] :
% 0.33/0.60 ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.33/0.60 = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power_minus_Bit0
% 0.33/0.60 thf(fact_5199_power__minus__Bit0,axiom,
% 0.33/0.60 ! [X: complex,K: num] :
% 0.33/0.60 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.33/0.60 = ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power_minus_Bit0
% 0.33/0.60 thf(fact_5200_power__minus__Bit0,axiom,
% 0.33/0.60 ! [X: rat,K: num] :
% 0.33/0.60 ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.33/0.60 = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power_minus_Bit0
% 0.33/0.60 thf(fact_5201_power__minus__Bit0,axiom,
% 0.33/0.60 ! [X: code_integer,K: num] :
% 0.33/0.60 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.33/0.60 = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power_minus_Bit0
% 0.33/0.60 thf(fact_5202_norm__uminus__minus,axiom,
% 0.33/0.60 ! [X: real,Y: real] :
% 0.33/0.60 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y ) )
% 0.33/0.60 = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % norm_uminus_minus
% 0.33/0.60 thf(fact_5203_norm__uminus__minus,axiom,
% 0.33/0.60 ! [X: complex,Y: complex] :
% 0.33/0.60 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y ) )
% 0.33/0.60 = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % norm_uminus_minus
% 0.33/0.60 thf(fact_5204_real__0__less__add__iff,axiom,
% 0.33/0.60 ! [X: real,Y: real] :
% 0.33/0.60 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.60 = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% 0.33/0.60
% 0.33/0.60 % real_0_less_add_iff
% 0.33/0.60 thf(fact_5205_real__add__less__0__iff,axiom,
% 0.33/0.60 ! [X: real,Y: real] :
% 0.33/0.60 ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 0.33/0.60 = ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % real_add_less_0_iff
% 0.33/0.60 thf(fact_5206_real__0__le__add__iff,axiom,
% 0.33/0.60 ! [X: real,Y: real] :
% 0.33/0.60 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.60 = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% 0.33/0.60
% 0.33/0.60 % real_0_le_add_iff
% 0.33/0.60 thf(fact_5207_real__add__le__0__iff,axiom,
% 0.33/0.60 ! [X: real,Y: real] :
% 0.33/0.60 ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 0.33/0.60 = ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % real_add_le_0_iff
% 0.33/0.60 thf(fact_5208_pos__minus__divide__less__eq,axiom,
% 0.33/0.60 ! [C: real,B: real,A: real] :
% 0.33/0.60 ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.33/0.60 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pos_minus_divide_less_eq
% 0.33/0.60 thf(fact_5209_pos__minus__divide__less__eq,axiom,
% 0.33/0.60 ! [C: rat,B: rat,A: rat] :
% 0.33/0.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.33/0.60 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pos_minus_divide_less_eq
% 0.33/0.60 thf(fact_5210_pos__less__minus__divide__eq,axiom,
% 0.33/0.60 ! [C: real,A: real,B: real] :
% 0.33/0.60 ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.33/0.60 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pos_less_minus_divide_eq
% 0.33/0.60 thf(fact_5211_pos__less__minus__divide__eq,axiom,
% 0.33/0.60 ! [C: rat,A: rat,B: rat] :
% 0.33/0.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.33/0.60 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pos_less_minus_divide_eq
% 0.33/0.60 thf(fact_5212_neg__minus__divide__less__eq,axiom,
% 0.33/0.60 ! [C: real,B: real,A: real] :
% 0.33/0.60 ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.33/0.60 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_minus_divide_less_eq
% 0.33/0.60 thf(fact_5213_neg__minus__divide__less__eq,axiom,
% 0.33/0.60 ! [C: rat,B: rat,A: rat] :
% 0.33/0.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.33/0.60 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_minus_divide_less_eq
% 0.33/0.60 thf(fact_5214_neg__less__minus__divide__eq,axiom,
% 0.33/0.60 ! [C: real,A: real,B: real] :
% 0.33/0.60 ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.33/0.60 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_less_minus_divide_eq
% 0.33/0.60 thf(fact_5215_neg__less__minus__divide__eq,axiom,
% 0.33/0.60 ! [C: rat,A: rat,B: rat] :
% 0.33/0.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.33/0.60 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_less_minus_divide_eq
% 0.33/0.60 thf(fact_5216_minus__divide__less__eq,axiom,
% 0.33/0.60 ! [B: real,C: real,A: real] :
% 0.33/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.33/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.33/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.33/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_less_eq
% 0.33/0.60 thf(fact_5217_minus__divide__less__eq,axiom,
% 0.33/0.60 ! [B: rat,C: rat,A: rat] :
% 0.33/0.60 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.33/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.33/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.33/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_less_eq
% 0.33/0.60 thf(fact_5218_less__minus__divide__eq,axiom,
% 0.33/0.60 ! [A: real,B: real,C: real] :
% 0.33/0.60 ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.33/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.33/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.33/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % less_minus_divide_eq
% 0.33/0.60 thf(fact_5219_less__minus__divide__eq,axiom,
% 0.33/0.60 ! [A: rat,B: rat,C: rat] :
% 0.33/0.60 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.33/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.33/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.33/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % less_minus_divide_eq
% 0.33/0.60 thf(fact_5220_eq__divide__eq__numeral_I2_J,axiom,
% 0.33/0.60 ! [W: num,B: real,C: real] :
% 0.33/0.60 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.33/0.60 = ( divide_divide_real @ B @ C ) )
% 0.33/0.60 = ( ( ( C != zero_zero_real )
% 0.33/0.60 => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
% 0.33/0.60 = B ) )
% 0.33/0.60 & ( ( C = zero_zero_real )
% 0.33/0.60 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.33/0.60 = zero_zero_real ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % eq_divide_eq_numeral(2)
% 0.33/0.60 thf(fact_5221_eq__divide__eq__numeral_I2_J,axiom,
% 0.33/0.60 ! [W: num,B: complex,C: complex] :
% 0.33/0.60 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.33/0.60 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.33/0.60 = ( ( ( C != zero_zero_complex )
% 0.33/0.60 => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
% 0.33/0.60 = B ) )
% 0.33/0.60 & ( ( C = zero_zero_complex )
% 0.33/0.60 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.33/0.60 = zero_zero_complex ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % eq_divide_eq_numeral(2)
% 0.33/0.60 thf(fact_5222_eq__divide__eq__numeral_I2_J,axiom,
% 0.33/0.60 ! [W: num,B: rat,C: rat] :
% 0.33/0.60 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.33/0.60 = ( divide_divide_rat @ B @ C ) )
% 0.33/0.60 = ( ( ( C != zero_zero_rat )
% 0.33/0.60 => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
% 0.33/0.60 = B ) )
% 0.33/0.60 & ( ( C = zero_zero_rat )
% 0.33/0.60 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.33/0.60 = zero_zero_rat ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % eq_divide_eq_numeral(2)
% 0.33/0.60 thf(fact_5223_divide__eq__eq__numeral_I2_J,axiom,
% 0.33/0.60 ! [B: real,C: real,W: num] :
% 0.33/0.60 ( ( ( divide_divide_real @ B @ C )
% 0.33/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.60 = ( ( ( C != zero_zero_real )
% 0.33/0.60 => ( B
% 0.33/0.60 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.33/0.60 & ( ( C = zero_zero_real )
% 0.33/0.60 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.33/0.60 = zero_zero_real ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % divide_eq_eq_numeral(2)
% 0.33/0.60 thf(fact_5224_divide__eq__eq__numeral_I2_J,axiom,
% 0.33/0.60 ! [B: complex,C: complex,W: num] :
% 0.33/0.60 ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.33/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.33/0.60 = ( ( ( C != zero_zero_complex )
% 0.33/0.60 => ( B
% 0.33/0.60 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
% 0.33/0.60 & ( ( C = zero_zero_complex )
% 0.33/0.60 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.33/0.60 = zero_zero_complex ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % divide_eq_eq_numeral(2)
% 0.33/0.60 thf(fact_5225_divide__eq__eq__numeral_I2_J,axiom,
% 0.33/0.60 ! [B: rat,C: rat,W: num] :
% 0.33/0.60 ( ( ( divide_divide_rat @ B @ C )
% 0.33/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.33/0.60 = ( ( ( C != zero_zero_rat )
% 0.33/0.60 => ( B
% 0.33/0.60 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.33/0.60 & ( ( C = zero_zero_rat )
% 0.33/0.60 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.33/0.60 = zero_zero_rat ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % divide_eq_eq_numeral(2)
% 0.33/0.60 thf(fact_5226_minus__divide__add__eq__iff,axiom,
% 0.33/0.60 ! [Z2: real,X: real,Y: real] :
% 0.33/0.60 ( ( Z2 != zero_zero_real )
% 0.33/0.60 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z2 ) ) @ Y )
% 0.33/0.60 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_add_eq_iff
% 0.33/0.60 thf(fact_5227_minus__divide__add__eq__iff,axiom,
% 0.33/0.60 ! [Z2: complex,X: complex,Y: complex] :
% 0.33/0.60 ( ( Z2 != zero_zero_complex )
% 0.33/0.60 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z2 ) ) @ Y )
% 0.33/0.60 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_add_eq_iff
% 0.33/0.60 thf(fact_5228_minus__divide__add__eq__iff,axiom,
% 0.33/0.60 ! [Z2: rat,X: rat,Y: rat] :
% 0.33/0.60 ( ( Z2 != zero_zero_rat )
% 0.33/0.60 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y )
% 0.33/0.60 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_add_eq_iff
% 0.33/0.60 thf(fact_5229_add__divide__eq__if__simps_I3_J,axiom,
% 0.33/0.60 ! [Z2: real,A: real,B: real] :
% 0.33/0.60 ( ( ( Z2 = zero_zero_real )
% 0.33/0.60 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 0.33/0.60 = B ) )
% 0.33/0.60 & ( ( Z2 != zero_zero_real )
% 0.33/0.60 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 0.33/0.60 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add_divide_eq_if_simps(3)
% 0.33/0.60 thf(fact_5230_add__divide__eq__if__simps_I3_J,axiom,
% 0.33/0.60 ! [Z2: complex,A: complex,B: complex] :
% 0.33/0.60 ( ( ( Z2 = zero_zero_complex )
% 0.33/0.60 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 0.33/0.60 = B ) )
% 0.33/0.60 & ( ( Z2 != zero_zero_complex )
% 0.33/0.60 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 0.33/0.60 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add_divide_eq_if_simps(3)
% 0.33/0.60 thf(fact_5231_add__divide__eq__if__simps_I3_J,axiom,
% 0.33/0.60 ! [Z2: rat,A: rat,B: rat] :
% 0.33/0.60 ( ( ( Z2 = zero_zero_rat )
% 0.33/0.60 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 0.33/0.60 = B ) )
% 0.33/0.60 & ( ( Z2 != zero_zero_rat )
% 0.33/0.60 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 0.33/0.60 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add_divide_eq_if_simps(3)
% 0.33/0.60 thf(fact_5232_minus__divide__diff__eq__iff,axiom,
% 0.33/0.60 ! [Z2: real,X: real,Y: real] :
% 0.33/0.60 ( ( Z2 != zero_zero_real )
% 0.33/0.60 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z2 ) ) @ Y )
% 0.33/0.60 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_diff_eq_iff
% 0.33/0.60 thf(fact_5233_minus__divide__diff__eq__iff,axiom,
% 0.33/0.60 ! [Z2: complex,X: complex,Y: complex] :
% 0.33/0.60 ( ( Z2 != zero_zero_complex )
% 0.33/0.60 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z2 ) ) @ Y )
% 0.33/0.60 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_diff_eq_iff
% 0.33/0.60 thf(fact_5234_minus__divide__diff__eq__iff,axiom,
% 0.33/0.60 ! [Z2: rat,X: rat,Y: rat] :
% 0.33/0.60 ( ( Z2 != zero_zero_rat )
% 0.33/0.60 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y )
% 0.33/0.60 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_diff_eq_iff
% 0.33/0.60 thf(fact_5235_add__divide__eq__if__simps_I5_J,axiom,
% 0.33/0.60 ! [Z2: real,A: real,B: real] :
% 0.33/0.60 ( ( ( Z2 = zero_zero_real )
% 0.33/0.60 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 0.33/0.60 = ( uminus_uminus_real @ B ) ) )
% 0.33/0.60 & ( ( Z2 != zero_zero_real )
% 0.33/0.60 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 0.33/0.60 = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add_divide_eq_if_simps(5)
% 0.33/0.60 thf(fact_5236_add__divide__eq__if__simps_I5_J,axiom,
% 0.33/0.60 ! [Z2: complex,A: complex,B: complex] :
% 0.33/0.60 ( ( ( Z2 = zero_zero_complex )
% 0.33/0.60 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 0.33/0.60 = ( uminus1482373934393186551omplex @ B ) ) )
% 0.33/0.60 & ( ( Z2 != zero_zero_complex )
% 0.33/0.60 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 0.33/0.60 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add_divide_eq_if_simps(5)
% 0.33/0.60 thf(fact_5237_add__divide__eq__if__simps_I5_J,axiom,
% 0.33/0.60 ! [Z2: rat,A: rat,B: rat] :
% 0.33/0.60 ( ( ( Z2 = zero_zero_rat )
% 0.33/0.60 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 0.33/0.60 = ( uminus_uminus_rat @ B ) ) )
% 0.33/0.60 & ( ( Z2 != zero_zero_rat )
% 0.33/0.60 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 0.33/0.60 = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add_divide_eq_if_simps(5)
% 0.33/0.60 thf(fact_5238_add__divide__eq__if__simps_I6_J,axiom,
% 0.33/0.60 ! [Z2: real,A: real,B: real] :
% 0.33/0.60 ( ( ( Z2 = zero_zero_real )
% 0.33/0.60 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 0.33/0.60 = ( uminus_uminus_real @ B ) ) )
% 0.33/0.60 & ( ( Z2 != zero_zero_real )
% 0.33/0.60 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 0.33/0.60 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add_divide_eq_if_simps(6)
% 0.33/0.60 thf(fact_5239_add__divide__eq__if__simps_I6_J,axiom,
% 0.33/0.60 ! [Z2: complex,A: complex,B: complex] :
% 0.33/0.60 ( ( ( Z2 = zero_zero_complex )
% 0.33/0.60 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 0.33/0.60 = ( uminus1482373934393186551omplex @ B ) ) )
% 0.33/0.60 & ( ( Z2 != zero_zero_complex )
% 0.33/0.60 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 0.33/0.60 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add_divide_eq_if_simps(6)
% 0.33/0.60 thf(fact_5240_add__divide__eq__if__simps_I6_J,axiom,
% 0.33/0.60 ! [Z2: rat,A: rat,B: rat] :
% 0.33/0.60 ( ( ( Z2 = zero_zero_rat )
% 0.33/0.60 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 0.33/0.60 = ( uminus_uminus_rat @ B ) ) )
% 0.33/0.60 & ( ( Z2 != zero_zero_rat )
% 0.33/0.60 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 0.33/0.60 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % add_divide_eq_if_simps(6)
% 0.33/0.60 thf(fact_5241_even__minus,axiom,
% 0.33/0.60 ! [A: int] :
% 0.33/0.60 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
% 0.33/0.60 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.33/0.60
% 0.33/0.60 % even_minus
% 0.33/0.60 thf(fact_5242_even__minus,axiom,
% 0.33/0.60 ! [A: code_integer] :
% 0.33/0.60 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.33/0.60 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.33/0.60
% 0.33/0.60 % even_minus
% 0.33/0.60 thf(fact_5243_power2__eq__iff,axiom,
% 0.33/0.60 ! [X: real,Y: real] :
% 0.33/0.60 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.60 = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.60 = ( ( X = Y )
% 0.33/0.60 | ( X
% 0.33/0.60 = ( uminus_uminus_real @ Y ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power2_eq_iff
% 0.33/0.60 thf(fact_5244_power2__eq__iff,axiom,
% 0.33/0.60 ! [X: int,Y: int] :
% 0.33/0.60 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.60 = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.60 = ( ( X = Y )
% 0.33/0.60 | ( X
% 0.33/0.60 = ( uminus_uminus_int @ Y ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power2_eq_iff
% 0.33/0.60 thf(fact_5245_power2__eq__iff,axiom,
% 0.33/0.60 ! [X: complex,Y: complex] :
% 0.33/0.60 ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.60 = ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.60 = ( ( X = Y )
% 0.33/0.60 | ( X
% 0.33/0.60 = ( uminus1482373934393186551omplex @ Y ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power2_eq_iff
% 0.33/0.60 thf(fact_5246_power2__eq__iff,axiom,
% 0.33/0.60 ! [X: rat,Y: rat] :
% 0.33/0.60 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.60 = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.60 = ( ( X = Y )
% 0.33/0.60 | ( X
% 0.33/0.60 = ( uminus_uminus_rat @ Y ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power2_eq_iff
% 0.33/0.60 thf(fact_5247_power2__eq__iff,axiom,
% 0.33/0.60 ! [X: code_integer,Y: code_integer] :
% 0.33/0.60 ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.60 = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.60 = ( ( X = Y )
% 0.33/0.60 | ( X
% 0.33/0.60 = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power2_eq_iff
% 0.33/0.60 thf(fact_5248_sgn__if,axiom,
% 0.33/0.60 ( sgn_sgn_real
% 0.33/0.60 = ( ^ [X3: real] : ( if_real @ ( X3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_if
% 0.33/0.60 thf(fact_5249_sgn__if,axiom,
% 0.33/0.60 ( sgn_sgn_int
% 0.33/0.60 = ( ^ [X3: int] : ( if_int @ ( X3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_if
% 0.33/0.60 thf(fact_5250_sgn__if,axiom,
% 0.33/0.60 ( sgn_sgn_rat
% 0.33/0.60 = ( ^ [X3: rat] : ( if_rat @ ( X3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_if
% 0.33/0.60 thf(fact_5251_sgn__if,axiom,
% 0.33/0.60 ( sgn_sgn_Code_integer
% 0.33/0.60 = ( ^ [X3: code_integer] : ( if_Code_integer @ ( X3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X3 ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_if
% 0.33/0.60 thf(fact_5252_sgn__1__neg,axiom,
% 0.33/0.60 ! [A: real] :
% 0.33/0.60 ( ( ( sgn_sgn_real @ A )
% 0.33/0.60 = ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.60 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_1_neg
% 0.33/0.60 thf(fact_5253_sgn__1__neg,axiom,
% 0.33/0.60 ! [A: int] :
% 0.33/0.60 ( ( ( sgn_sgn_int @ A )
% 0.33/0.60 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.60 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_1_neg
% 0.33/0.60 thf(fact_5254_sgn__1__neg,axiom,
% 0.33/0.60 ! [A: rat] :
% 0.33/0.60 ( ( ( sgn_sgn_rat @ A )
% 0.33/0.60 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.60 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_1_neg
% 0.33/0.60 thf(fact_5255_sgn__1__neg,axiom,
% 0.33/0.60 ! [A: code_integer] :
% 0.33/0.60 ( ( ( sgn_sgn_Code_integer @ A )
% 0.33/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.60 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.33/0.60
% 0.33/0.60 % sgn_1_neg
% 0.33/0.60 thf(fact_5256_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.33/0.60 ! [N: nat,K: nat] :
% 0.33/0.60 ( ( ord_less_nat @ N @ K )
% 0.33/0.60 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.33/0.60 = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_of_nat_eq_0_lemma
% 0.33/0.60 thf(fact_5257_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.33/0.60 ! [N: nat,K: nat] :
% 0.33/0.60 ( ( ord_less_nat @ N @ K )
% 0.33/0.60 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.33/0.60 = zero_zero_complex ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_of_nat_eq_0_lemma
% 0.33/0.60 thf(fact_5258_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.33/0.60 ! [N: nat,K: nat] :
% 0.33/0.60 ( ( ord_less_nat @ N @ K )
% 0.33/0.60 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.33/0.60 = zero_zero_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_of_nat_eq_0_lemma
% 0.33/0.60 thf(fact_5259_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.33/0.60 ! [N: nat,K: nat] :
% 0.33/0.60 ( ( ord_less_nat @ N @ K )
% 0.33/0.60 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.33/0.60 = zero_zero_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_of_nat_eq_0_lemma
% 0.33/0.60 thf(fact_5260_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.33/0.60 ! [N: nat,K: nat] :
% 0.33/0.60 ( ( ord_less_nat @ N @ K )
% 0.33/0.60 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.33/0.60 = zero_zero_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_of_nat_eq_0_lemma
% 0.33/0.60 thf(fact_5261_pochhammer__of__nat__eq__0__iff,axiom,
% 0.33/0.60 ! [N: nat,K: nat] :
% 0.33/0.60 ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.33/0.60 = zero_z3403309356797280102nteger )
% 0.33/0.60 = ( ord_less_nat @ N @ K ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_of_nat_eq_0_iff
% 0.33/0.60 thf(fact_5262_pochhammer__of__nat__eq__0__iff,axiom,
% 0.33/0.60 ! [N: nat,K: nat] :
% 0.33/0.60 ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.33/0.60 = zero_zero_complex )
% 0.33/0.60 = ( ord_less_nat @ N @ K ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_of_nat_eq_0_iff
% 0.33/0.60 thf(fact_5263_pochhammer__of__nat__eq__0__iff,axiom,
% 0.33/0.60 ! [N: nat,K: nat] :
% 0.33/0.60 ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.33/0.60 = zero_zero_real )
% 0.33/0.60 = ( ord_less_nat @ N @ K ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_of_nat_eq_0_iff
% 0.33/0.60 thf(fact_5264_pochhammer__of__nat__eq__0__iff,axiom,
% 0.33/0.60 ! [N: nat,K: nat] :
% 0.33/0.60 ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.33/0.60 = zero_zero_rat )
% 0.33/0.60 = ( ord_less_nat @ N @ K ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_of_nat_eq_0_iff
% 0.33/0.60 thf(fact_5265_pochhammer__of__nat__eq__0__iff,axiom,
% 0.33/0.60 ! [N: nat,K: nat] :
% 0.33/0.60 ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.33/0.60 = zero_zero_int )
% 0.33/0.60 = ( ord_less_nat @ N @ K ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_of_nat_eq_0_iff
% 0.33/0.60 thf(fact_5266_pochhammer__eq__0__iff,axiom,
% 0.33/0.60 ! [A: complex,N: nat] :
% 0.33/0.60 ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.33/0.60 = zero_zero_complex )
% 0.33/0.60 = ( ? [K2: nat] :
% 0.33/0.60 ( ( ord_less_nat @ K2 @ N )
% 0.33/0.60 & ( A
% 0.33/0.60 = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_eq_0_iff
% 0.33/0.60 thf(fact_5267_pochhammer__eq__0__iff,axiom,
% 0.33/0.60 ! [A: real,N: nat] :
% 0.33/0.60 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.33/0.60 = zero_zero_real )
% 0.33/0.60 = ( ? [K2: nat] :
% 0.33/0.60 ( ( ord_less_nat @ K2 @ N )
% 0.33/0.60 & ( A
% 0.33/0.60 = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_eq_0_iff
% 0.33/0.60 thf(fact_5268_pochhammer__eq__0__iff,axiom,
% 0.33/0.60 ! [A: rat,N: nat] :
% 0.33/0.60 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.33/0.60 = zero_zero_rat )
% 0.33/0.60 = ( ? [K2: nat] :
% 0.33/0.60 ( ( ord_less_nat @ K2 @ N )
% 0.33/0.60 & ( A
% 0.33/0.60 = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K2 ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_eq_0_iff
% 0.33/0.60 thf(fact_5269_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.33/0.60 ! [K: nat,N: nat] :
% 0.33/0.60 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.60 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.33/0.60 != zero_z3403309356797280102nteger ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_of_nat_eq_0_lemma'
% 0.33/0.60 thf(fact_5270_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.33/0.60 ! [K: nat,N: nat] :
% 0.33/0.60 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.60 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.33/0.60 != zero_zero_complex ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_of_nat_eq_0_lemma'
% 0.33/0.60 thf(fact_5271_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.33/0.60 ! [K: nat,N: nat] :
% 0.33/0.60 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.60 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.33/0.60 != zero_zero_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_of_nat_eq_0_lemma'
% 0.33/0.60 thf(fact_5272_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.33/0.60 ! [K: nat,N: nat] :
% 0.33/0.60 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.60 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.33/0.60 != zero_zero_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_of_nat_eq_0_lemma'
% 0.33/0.60 thf(fact_5273_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.33/0.60 ! [K: nat,N: nat] :
% 0.33/0.60 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.60 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.33/0.60 != zero_zero_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_of_nat_eq_0_lemma'
% 0.33/0.60 thf(fact_5274_ln__add__one__self__le__self2,axiom,
% 0.33/0.60 ! [X: real] :
% 0.33/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.60 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 0.33/0.60
% 0.33/0.60 % ln_add_one_self_le_self2
% 0.33/0.60 thf(fact_5275_mult__ceiling__le,axiom,
% 0.33/0.60 ! [A: real,B: real] :
% 0.33/0.60 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.60 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % mult_ceiling_le
% 0.33/0.60 thf(fact_5276_mult__ceiling__le,axiom,
% 0.33/0.60 ! [A: rat,B: rat] :
% 0.33/0.60 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.60 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % mult_ceiling_le
% 0.33/0.60 thf(fact_5277_fold__atLeastAtMost__nat_Oelims,axiom,
% 0.33/0.60 ! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
% 0.33/0.60 ( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
% 0.33/0.60 = Y )
% 0.33/0.60 => ( ( ( ord_less_nat @ Xb @ Xa )
% 0.33/0.60 => ( Y = Xc ) )
% 0.33/0.60 & ( ~ ( ord_less_nat @ Xb @ Xa )
% 0.33/0.60 => ( Y
% 0.33/0.60 = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % fold_atLeastAtMost_nat.elims
% 0.33/0.60 thf(fact_5278_fold__atLeastAtMost__nat_Osimps,axiom,
% 0.33/0.60 ( set_fo2584398358068434914at_nat
% 0.33/0.60 = ( ^ [F2: nat > nat > nat,A2: nat,B2: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B2 @ A2 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F2 @ ( plus_plus_nat @ A2 @ one_one_nat ) @ B2 @ ( F2 @ A2 @ Acc ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % fold_atLeastAtMost_nat.simps
% 0.33/0.60 thf(fact_5279_pos__minus__divide__le__eq,axiom,
% 0.33/0.60 ! [C: real,B: real,A: real] :
% 0.33/0.60 ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.33/0.60 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pos_minus_divide_le_eq
% 0.33/0.60 thf(fact_5280_pos__minus__divide__le__eq,axiom,
% 0.33/0.60 ! [C: rat,B: rat,A: rat] :
% 0.33/0.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.33/0.60 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pos_minus_divide_le_eq
% 0.33/0.60 thf(fact_5281_pos__le__minus__divide__eq,axiom,
% 0.33/0.60 ! [C: real,A: real,B: real] :
% 0.33/0.60 ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.33/0.60 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pos_le_minus_divide_eq
% 0.33/0.60 thf(fact_5282_pos__le__minus__divide__eq,axiom,
% 0.33/0.60 ! [C: rat,A: rat,B: rat] :
% 0.33/0.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.33/0.60 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pos_le_minus_divide_eq
% 0.33/0.60 thf(fact_5283_neg__minus__divide__le__eq,axiom,
% 0.33/0.60 ! [C: real,B: real,A: real] :
% 0.33/0.60 ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.33/0.60 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_minus_divide_le_eq
% 0.33/0.60 thf(fact_5284_neg__minus__divide__le__eq,axiom,
% 0.33/0.60 ! [C: rat,B: rat,A: rat] :
% 0.33/0.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.33/0.60 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_minus_divide_le_eq
% 0.33/0.60 thf(fact_5285_neg__le__minus__divide__eq,axiom,
% 0.33/0.60 ! [C: real,A: real,B: real] :
% 0.33/0.60 ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.33/0.60 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_le_minus_divide_eq
% 0.33/0.60 thf(fact_5286_neg__le__minus__divide__eq,axiom,
% 0.33/0.60 ! [C: rat,A: rat,B: rat] :
% 0.33/0.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.33/0.60 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_le_minus_divide_eq
% 0.33/0.60 thf(fact_5287_minus__divide__le__eq,axiom,
% 0.33/0.60 ! [B: real,C: real,A: real] :
% 0.33/0.60 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.33/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.33/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.33/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_le_eq
% 0.33/0.60 thf(fact_5288_minus__divide__le__eq,axiom,
% 0.33/0.60 ! [B: rat,C: rat,A: rat] :
% 0.33/0.60 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.33/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.33/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.33/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_divide_le_eq
% 0.33/0.60 thf(fact_5289_le__minus__divide__eq,axiom,
% 0.33/0.60 ! [A: real,B: real,C: real] :
% 0.33/0.60 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.33/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.33/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.33/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % le_minus_divide_eq
% 0.33/0.60 thf(fact_5290_le__minus__divide__eq,axiom,
% 0.33/0.60 ! [A: rat,B: rat,C: rat] :
% 0.33/0.60 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.33/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.33/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.33/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % le_minus_divide_eq
% 0.33/0.60 thf(fact_5291_divide__less__eq__numeral_I2_J,axiom,
% 0.33/0.60 ! [B: real,C: real,W: num] :
% 0.33/0.60 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.33/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.33/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % divide_less_eq_numeral(2)
% 0.33/0.60 thf(fact_5292_divide__less__eq__numeral_I2_J,axiom,
% 0.33/0.60 ! [B: rat,C: rat,W: num] :
% 0.33/0.60 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.33/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.33/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.33/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % divide_less_eq_numeral(2)
% 0.33/0.60 thf(fact_5293_less__divide__eq__numeral_I2_J,axiom,
% 0.33/0.60 ! [W: num,B: real,C: real] :
% 0.33/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 0.33/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.33/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.33/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % less_divide_eq_numeral(2)
% 0.33/0.60 thf(fact_5294_less__divide__eq__numeral_I2_J,axiom,
% 0.33/0.60 ! [W: num,B: rat,C: rat] :
% 0.33/0.60 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 0.33/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.33/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.33/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % less_divide_eq_numeral(2)
% 0.33/0.60 thf(fact_5295_power2__eq__1__iff,axiom,
% 0.33/0.60 ! [A: real] :
% 0.33/0.60 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.60 = one_one_real )
% 0.33/0.60 = ( ( A = one_one_real )
% 0.33/0.60 | ( A
% 0.33/0.60 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power2_eq_1_iff
% 0.33/0.60 thf(fact_5296_power2__eq__1__iff,axiom,
% 0.33/0.60 ! [A: int] :
% 0.33/0.60 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.60 = one_one_int )
% 0.33/0.60 = ( ( A = one_one_int )
% 0.33/0.60 | ( A
% 0.33/0.60 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power2_eq_1_iff
% 0.33/0.60 thf(fact_5297_power2__eq__1__iff,axiom,
% 0.33/0.60 ! [A: complex] :
% 0.33/0.60 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.60 = one_one_complex )
% 0.33/0.60 = ( ( A = one_one_complex )
% 0.33/0.60 | ( A
% 0.33/0.60 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power2_eq_1_iff
% 0.33/0.60 thf(fact_5298_power2__eq__1__iff,axiom,
% 0.33/0.60 ! [A: rat] :
% 0.33/0.60 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.60 = one_one_rat )
% 0.33/0.60 = ( ( A = one_one_rat )
% 0.33/0.60 | ( A
% 0.33/0.60 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power2_eq_1_iff
% 0.33/0.60 thf(fact_5299_power2__eq__1__iff,axiom,
% 0.33/0.60 ! [A: code_integer] :
% 0.33/0.60 ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.60 = one_one_Code_integer )
% 0.33/0.60 = ( ( A = one_one_Code_integer )
% 0.33/0.60 | ( A
% 0.33/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % power2_eq_1_iff
% 0.33/0.60 thf(fact_5300_uminus__power__if,axiom,
% 0.33/0.60 ! [N: nat,A: real] :
% 0.33/0.60 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.33/0.60 = ( power_power_real @ A @ N ) ) )
% 0.33/0.60 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.33/0.60 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % uminus_power_if
% 0.33/0.60 thf(fact_5301_uminus__power__if,axiom,
% 0.33/0.60 ! [N: nat,A: int] :
% 0.33/0.60 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.33/0.60 = ( power_power_int @ A @ N ) ) )
% 0.33/0.60 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.33/0.60 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % uminus_power_if
% 0.33/0.60 thf(fact_5302_uminus__power__if,axiom,
% 0.33/0.60 ! [N: nat,A: complex] :
% 0.33/0.60 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.33/0.60 = ( power_power_complex @ A @ N ) ) )
% 0.33/0.60 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.33/0.60 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % uminus_power_if
% 0.33/0.60 thf(fact_5303_uminus__power__if,axiom,
% 0.33/0.60 ! [N: nat,A: rat] :
% 0.33/0.60 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.33/0.60 = ( power_power_rat @ A @ N ) ) )
% 0.33/0.60 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.33/0.60 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % uminus_power_if
% 0.33/0.60 thf(fact_5304_uminus__power__if,axiom,
% 0.33/0.60 ! [N: nat,A: code_integer] :
% 0.33/0.60 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.33/0.60 = ( power_8256067586552552935nteger @ A @ N ) ) )
% 0.33/0.60 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.33/0.60 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % uminus_power_if
% 0.33/0.60 thf(fact_5305_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.33/0.60 ! [K: nat,N: nat] :
% 0.33/0.60 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.60 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 0.33/0.60 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_one_power_add_eq_neg_one_power_diff
% 0.33/0.60 thf(fact_5306_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.33/0.60 ! [K: nat,N: nat] :
% 0.33/0.60 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.60 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 0.33/0.60 = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_one_power_add_eq_neg_one_power_diff
% 0.33/0.60 thf(fact_5307_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.33/0.60 ! [K: nat,N: nat] :
% 0.33/0.60 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.60 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 0.33/0.60 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_one_power_add_eq_neg_one_power_diff
% 0.33/0.60 thf(fact_5308_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.33/0.60 ! [K: nat,N: nat] :
% 0.33/0.60 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.60 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 0.33/0.60 = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_one_power_add_eq_neg_one_power_diff
% 0.33/0.60 thf(fact_5309_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.33/0.60 ! [K: nat,N: nat] :
% 0.33/0.60 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.60 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 0.33/0.60 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % neg_one_power_add_eq_neg_one_power_diff
% 0.33/0.60 thf(fact_5310_realpow__square__minus__le,axiom,
% 0.33/0.60 ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % realpow_square_minus_le
% 0.33/0.60 thf(fact_5311_ln__one__minus__pos__upper__bound,axiom,
% 0.33/0.60 ! [X: real] :
% 0.33/0.60 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.60 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.60 => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % ln_one_minus_pos_upper_bound
% 0.33/0.60 thf(fact_5312_le__divide__eq__numeral_I2_J,axiom,
% 0.33/0.60 ! [W: num,B: real,C: real] :
% 0.33/0.60 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 0.33/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.33/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.33/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % le_divide_eq_numeral(2)
% 0.33/0.60 thf(fact_5313_le__divide__eq__numeral_I2_J,axiom,
% 0.33/0.60 ! [W: num,B: rat,C: rat] :
% 0.33/0.60 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 0.33/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.33/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.33/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % le_divide_eq_numeral(2)
% 0.33/0.60 thf(fact_5314_divide__le__eq__numeral_I2_J,axiom,
% 0.33/0.60 ! [B: real,C: real,W: num] :
% 0.33/0.60 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.33/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.33/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.60 => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % divide_le_eq_numeral(2)
% 0.33/0.60 thf(fact_5315_divide__le__eq__numeral_I2_J,axiom,
% 0.33/0.60 ! [B: rat,C: rat,W: num] :
% 0.33/0.60 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.33/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.33/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.33/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.60 => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % divide_le_eq_numeral(2)
% 0.33/0.60 thf(fact_5316_square__le__1,axiom,
% 0.33/0.60 ! [X: real] :
% 0.33/0.60 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.60 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.60 => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % square_le_1
% 0.33/0.60 thf(fact_5317_square__le__1,axiom,
% 0.33/0.60 ! [X: code_integer] :
% 0.33/0.60 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 0.33/0.60 => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
% 0.33/0.60 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % square_le_1
% 0.33/0.60 thf(fact_5318_square__le__1,axiom,
% 0.33/0.60 ! [X: rat] :
% 0.33/0.60 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
% 0.33/0.60 => ( ( ord_less_eq_rat @ X @ one_one_rat )
% 0.33/0.60 => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % square_le_1
% 0.33/0.60 thf(fact_5319_square__le__1,axiom,
% 0.33/0.60 ! [X: int] :
% 0.33/0.60 ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 0.33/0.60 => ( ( ord_less_eq_int @ X @ one_one_int )
% 0.33/0.60 => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % square_le_1
% 0.33/0.60 thf(fact_5320_minus__power__mult__self,axiom,
% 0.33/0.60 ! [A: real,N: nat] :
% 0.33/0.60 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 0.33/0.60 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_power_mult_self
% 0.33/0.60 thf(fact_5321_minus__power__mult__self,axiom,
% 0.33/0.60 ! [A: int,N: nat] :
% 0.33/0.60 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 0.33/0.60 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_power_mult_self
% 0.33/0.60 thf(fact_5322_minus__power__mult__self,axiom,
% 0.33/0.60 ! [A: complex,N: nat] :
% 0.33/0.60 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
% 0.33/0.60 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_power_mult_self
% 0.33/0.60 thf(fact_5323_minus__power__mult__self,axiom,
% 0.33/0.60 ! [A: rat,N: nat] :
% 0.33/0.60 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 0.33/0.60 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_power_mult_self
% 0.33/0.60 thf(fact_5324_minus__power__mult__self,axiom,
% 0.33/0.60 ! [A: code_integer,N: nat] :
% 0.33/0.60 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 0.33/0.60 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_power_mult_self
% 0.33/0.60 thf(fact_5325_minus__one__power__iff,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.33/0.60 = one_one_real ) )
% 0.33/0.60 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.33/0.60 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_one_power_iff
% 0.33/0.60 thf(fact_5326_minus__one__power__iff,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.33/0.60 = one_one_int ) )
% 0.33/0.60 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.33/0.60 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_one_power_iff
% 0.33/0.60 thf(fact_5327_minus__one__power__iff,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.33/0.60 = one_one_complex ) )
% 0.33/0.60 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.33/0.60 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_one_power_iff
% 0.33/0.60 thf(fact_5328_minus__one__power__iff,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.33/0.60 = one_one_rat ) )
% 0.33/0.60 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.33/0.60 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_one_power_iff
% 0.33/0.60 thf(fact_5329_minus__one__power__iff,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.33/0.60 = one_one_Code_integer ) )
% 0.33/0.60 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.60 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.33/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % minus_one_power_iff
% 0.33/0.60 thf(fact_5330_pochhammer__absorb__comp,axiom,
% 0.33/0.60 ! [R: code_integer,K: nat] :
% 0.33/0.60 ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R ) @ K ) )
% 0.33/0.60 = ( times_3573771949741848930nteger @ R @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R ) @ one_one_Code_integer ) @ K ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_absorb_comp
% 0.33/0.60 thf(fact_5331_pochhammer__absorb__comp,axiom,
% 0.33/0.60 ! [R: complex,K: nat] :
% 0.33/0.60 ( ( times_times_complex @ ( minus_minus_complex @ R @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R ) @ K ) )
% 0.33/0.60 = ( times_times_complex @ R @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R ) @ one_one_complex ) @ K ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_absorb_comp
% 0.33/0.60 thf(fact_5332_pochhammer__absorb__comp,axiom,
% 0.33/0.60 ! [R: real,K: nat] :
% 0.33/0.60 ( ( times_times_real @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R ) @ K ) )
% 0.33/0.60 = ( times_times_real @ R @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R ) @ one_one_real ) @ K ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_absorb_comp
% 0.33/0.60 thf(fact_5333_pochhammer__absorb__comp,axiom,
% 0.33/0.60 ! [R: rat,K: nat] :
% 0.33/0.60 ( ( times_times_rat @ ( minus_minus_rat @ R @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R ) @ K ) )
% 0.33/0.60 = ( times_times_rat @ R @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R ) @ one_one_rat ) @ K ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_absorb_comp
% 0.33/0.60 thf(fact_5334_pochhammer__absorb__comp,axiom,
% 0.33/0.60 ! [R: int,K: nat] :
% 0.33/0.60 ( ( times_times_int @ ( minus_minus_int @ R @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R ) @ K ) )
% 0.33/0.60 = ( times_times_int @ R @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R ) @ one_one_int ) @ K ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_absorb_comp
% 0.33/0.60 thf(fact_5335_pochhammer__same,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
% 0.33/0.60 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_same
% 0.33/0.60 thf(fact_5336_pochhammer__same,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
% 0.33/0.60 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_same
% 0.33/0.60 thf(fact_5337_pochhammer__same,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
% 0.33/0.60 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_same
% 0.33/0.60 thf(fact_5338_pochhammer__same,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
% 0.33/0.60 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_same
% 0.33/0.60 thf(fact_5339_pochhammer__same,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
% 0.33/0.60 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_same
% 0.33/0.60 thf(fact_5340_Bernoulli__inequality,axiom,
% 0.33/0.60 ! [X: real,N: nat] :
% 0.33/0.60 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.60 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % Bernoulli_inequality
% 0.33/0.60 thf(fact_5341_power__minus1__odd,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.33/0.60 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.60
% 0.33/0.60 % power_minus1_odd
% 0.33/0.60 thf(fact_5342_power__minus1__odd,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.33/0.60 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % power_minus1_odd
% 0.33/0.60 thf(fact_5343_power__minus1__odd,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.33/0.60 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.60
% 0.33/0.60 % power_minus1_odd
% 0.33/0.60 thf(fact_5344_power__minus1__odd,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.33/0.60 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.60
% 0.33/0.60 % power_minus1_odd
% 0.33/0.60 thf(fact_5345_power__minus1__odd,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.33/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.60
% 0.33/0.60 % power_minus1_odd
% 0.33/0.60 thf(fact_5346_pochhammer__minus,axiom,
% 0.33/0.60 ! [B: code_integer,K: nat] :
% 0.33/0.60 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
% 0.33/0.60 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_minus
% 0.33/0.60 thf(fact_5347_pochhammer__minus,axiom,
% 0.33/0.60 ! [B: complex,K: nat] :
% 0.33/0.60 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
% 0.33/0.60 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_minus
% 0.33/0.60 thf(fact_5348_pochhammer__minus,axiom,
% 0.33/0.60 ! [B: real,K: nat] :
% 0.33/0.60 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
% 0.33/0.60 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_minus
% 0.33/0.60 thf(fact_5349_pochhammer__minus,axiom,
% 0.33/0.60 ! [B: rat,K: nat] :
% 0.33/0.60 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
% 0.33/0.60 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_minus
% 0.33/0.60 thf(fact_5350_pochhammer__minus,axiom,
% 0.33/0.60 ! [B: int,K: nat] :
% 0.33/0.60 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
% 0.33/0.60 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_minus
% 0.33/0.60 thf(fact_5351_pochhammer__minus_H,axiom,
% 0.33/0.60 ! [B: code_integer,K: nat] :
% 0.33/0.60 ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
% 0.33/0.60 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_minus'
% 0.33/0.60 thf(fact_5352_pochhammer__minus_H,axiom,
% 0.33/0.60 ! [B: complex,K: nat] :
% 0.33/0.60 ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
% 0.33/0.60 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_minus'
% 0.33/0.60 thf(fact_5353_pochhammer__minus_H,axiom,
% 0.33/0.60 ! [B: real,K: nat] :
% 0.33/0.60 ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
% 0.33/0.60 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_minus'
% 0.33/0.60 thf(fact_5354_pochhammer__minus_H,axiom,
% 0.33/0.60 ! [B: rat,K: nat] :
% 0.33/0.60 ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
% 0.33/0.60 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_minus'
% 0.33/0.60 thf(fact_5355_pochhammer__minus_H,axiom,
% 0.33/0.60 ! [B: int,K: nat] :
% 0.33/0.60 ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
% 0.33/0.60 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % pochhammer_minus'
% 0.33/0.60 thf(fact_5356_eq__diff__eq_H,axiom,
% 0.33/0.60 ! [X: real,Y: real,Z2: real] :
% 0.33/0.60 ( ( X
% 0.33/0.60 = ( minus_minus_real @ Y @ Z2 ) )
% 0.33/0.60 = ( Y
% 0.33/0.60 = ( plus_plus_real @ X @ Z2 ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % eq_diff_eq'
% 0.33/0.60 thf(fact_5357_take__bit__Suc__minus__1__eq,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.60 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).
% 0.33/0.60
% 0.33/0.60 % take_bit_Suc_minus_1_eq
% 0.33/0.60 thf(fact_5358_take__bit__Suc__minus__1__eq,axiom,
% 0.33/0.60 ! [N: nat] :
% 0.33/0.60 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.60 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % take_bit_Suc_minus_1_eq
% 0.33/0.60 thf(fact_5359_take__bit__numeral__minus__1__eq,axiom,
% 0.33/0.60 ! [K: num] :
% 0.33/0.60 ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.60 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).
% 0.33/0.60
% 0.33/0.60 % take_bit_numeral_minus_1_eq
% 0.33/0.60 thf(fact_5360_take__bit__numeral__minus__1__eq,axiom,
% 0.33/0.60 ! [K: num] :
% 0.33/0.60 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.60 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).
% 0.33/0.60
% 0.33/0.60 % take_bit_numeral_minus_1_eq
% 0.33/0.60 thf(fact_5361_fact__code,axiom,
% 0.33/0.60 ( semiri5044797733671781792omplex
% 0.33/0.60 = ( ^ [N3: nat] : ( semiri8010041392384452111omplex @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % fact_code
% 0.33/0.60 thf(fact_5362_fact__code,axiom,
% 0.33/0.60 ( semiri773545260158071498ct_rat
% 0.33/0.60 = ( ^ [N3: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % fact_code
% 0.33/0.60 thf(fact_5363_fact__code,axiom,
% 0.33/0.60 ( semiri1406184849735516958ct_int
% 0.33/0.60 = ( ^ [N3: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % fact_code
% 0.33/0.60 thf(fact_5364_fact__code,axiom,
% 0.33/0.60 ( semiri1408675320244567234ct_nat
% 0.33/0.60 = ( ^ [N3: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % fact_code
% 0.33/0.60 thf(fact_5365_fact__code,axiom,
% 0.33/0.60 ( semiri2265585572941072030t_real
% 0.33/0.60 = ( ^ [N3: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % fact_code
% 0.33/0.60 thf(fact_5366_signed__take__bit__rec,axiom,
% 0.33/0.60 ( bit_ri6519982836138164636nteger
% 0.33/0.60 = ( ^ [N3: nat,A2: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % signed_take_bit_rec
% 0.33/0.60 thf(fact_5367_signed__take__bit__rec,axiom,
% 0.33/0.60 ( bit_ri631733984087533419it_int
% 0.33/0.60 = ( ^ [N3: nat,A2: int] : ( if_int @ ( N3 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % signed_take_bit_rec
% 0.33/0.60 thf(fact_5368_sin__coeff__def,axiom,
% 0.33/0.60 ( sin_coeff
% 0.33/0.60 = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % sin_coeff_def
% 0.33/0.60 thf(fact_5369_cos__coeff__def,axiom,
% 0.33/0.60 ( cos_coeff
% 0.33/0.60 = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ zero_zero_real ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % cos_coeff_def
% 0.33/0.60 thf(fact_5370_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 0.33/0.60 ! [X: real] :
% 0.33/0.60 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.60 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.60 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % abs_ln_one_plus_x_minus_x_bound_nonpos
% 0.33/0.60 thf(fact_5371_abs__ln__one__plus__x__minus__x__bound,axiom,
% 0.33/0.60 ! [X: real] :
% 0.33/0.60 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.60 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % abs_ln_one_plus_x_minus_x_bound
% 0.33/0.60 thf(fact_5372_ceiling__log__eq__powr__iff,axiom,
% 0.33/0.60 ! [X: real,B: real,K: nat] :
% 0.33/0.60 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.60 => ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.60 => ( ( ( archim7802044766580827645g_real @ ( log2 @ B @ X ) )
% 0.33/0.60 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 0.33/0.60 = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
% 0.33/0.60 & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % ceiling_log_eq_powr_iff
% 0.33/0.60 thf(fact_5373_floor__log__nat__eq__powr__iff,axiom,
% 0.33/0.60 ! [B: nat,K: nat,N: nat] :
% 0.33/0.60 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.33/0.60 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.60 => ( ( ( archim6058952711729229775r_real @ ( log2 @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.33/0.60 = ( semiri1314217659103216013at_int @ N ) )
% 0.33/0.60 = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.33/0.60 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % floor_log_nat_eq_powr_iff
% 0.33/0.60 thf(fact_5374_divide__int__unfold,axiom,
% 0.33/0.60 ! [L: int,K: int,N: nat,M: nat] :
% 0.33/0.60 ( ( ( ( ( sgn_sgn_int @ L )
% 0.33/0.60 = zero_zero_int )
% 0.33/0.60 | ( ( sgn_sgn_int @ K )
% 0.33/0.60 = zero_zero_int )
% 0.33/0.60 | ( N = zero_zero_nat ) )
% 0.33/0.60 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.33/0.60 = zero_zero_int ) )
% 0.33/0.60 & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.33/0.60 = zero_zero_int )
% 0.33/0.60 | ( ( sgn_sgn_int @ K )
% 0.33/0.60 = zero_zero_int )
% 0.33/0.60 | ( N = zero_zero_nat ) )
% 0.33/0.60 => ( ( ( ( sgn_sgn_int @ K )
% 0.33/0.60 = ( sgn_sgn_int @ L ) )
% 0.33/0.60 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.33/0.60 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 0.33/0.60 & ( ( ( sgn_sgn_int @ K )
% 0.33/0.60 != ( sgn_sgn_int @ L ) )
% 0.33/0.60 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.33/0.60 = ( uminus_uminus_int
% 0.33/0.60 @ ( semiri1314217659103216013at_int
% 0.33/0.60 @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 0.33/0.60 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.60 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.60
% 0.33/0.60 % divide_int_unfold
% 0.33/0.60 thf(fact_5375_abs__idempotent,axiom,
% 0.33/0.60 ! [A: real] :
% 0.33/0.60 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.33/0.60 = ( abs_abs_real @ A ) ) ).
% 0.33/0.60
% 0.33/0.60 % abs_idempotent
% 0.33/0.60 thf(fact_5376_abs__idempotent,axiom,
% 0.33/0.60 ! [A: int] :
% 0.33/0.60 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.33/0.60 = ( abs_abs_int @ A ) ) ).
% 0.33/0.60
% 0.33/0.60 % abs_idempotent
% 0.33/0.60 thf(fact_5377_abs__idempotent,axiom,
% 0.33/0.60 ! [A: code_integer] :
% 0.33/0.60 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.33/0.60 = ( abs_abs_Code_integer @ A ) ) ).
% 0.33/0.60
% 0.33/0.60 % abs_idempotent
% 0.33/0.60 thf(fact_5378_abs__idempotent,axiom,
% 0.33/0.60 ! [A: rat] :
% 0.33/0.60 ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 0.33/0.60 = ( abs_abs_rat @ A ) ) ).
% 0.33/0.60
% 0.33/0.60 % abs_idempotent
% 0.33/0.60 thf(fact_5379_abs__abs,axiom,
% 0.33/0.60 ! [A: real] :
% 0.33/0.60 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.33/0.60 = ( abs_abs_real @ A ) ) ).
% 0.33/0.60
% 0.33/0.60 % abs_abs
% 0.33/0.60 thf(fact_5380_abs__abs,axiom,
% 0.33/0.60 ! [A: int] :
% 0.33/0.60 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.33/0.60 = ( abs_abs_int @ A ) ) ).
% 0.33/0.60
% 0.33/0.60 % abs_abs
% 0.33/0.60 thf(fact_5381_abs__abs,axiom,
% 0.33/0.60 ! [A: code_integer] :
% 0.33/0.60 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.33/0.60 = ( abs_abs_Code_integer @ A ) ) ).
% 0.33/0.60
% 0.33/0.60 % abs_abs
% 0.33/0.60 thf(fact_5382_abs__abs,axiom,
% 0.33/0.60 ! [A: rat] :
% 0.33/0.60 ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 0.33/0.60 = ( abs_abs_rat @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_abs
% 0.33/0.61 thf(fact_5383_abs__0,axiom,
% 0.33/0.61 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.33/0.61 = zero_z3403309356797280102nteger ) ).
% 0.33/0.61
% 0.33/0.61 % abs_0
% 0.33/0.61 thf(fact_5384_abs__0,axiom,
% 0.33/0.61 ( ( abs_abs_real @ zero_zero_real )
% 0.33/0.61 = zero_zero_real ) ).
% 0.33/0.61
% 0.33/0.61 % abs_0
% 0.33/0.61 thf(fact_5385_abs__0,axiom,
% 0.33/0.61 ( ( abs_abs_rat @ zero_zero_rat )
% 0.33/0.61 = zero_zero_rat ) ).
% 0.33/0.61
% 0.33/0.61 % abs_0
% 0.33/0.61 thf(fact_5386_abs__0,axiom,
% 0.33/0.61 ( ( abs_abs_int @ zero_zero_int )
% 0.33/0.61 = zero_zero_int ) ).
% 0.33/0.61
% 0.33/0.61 % abs_0
% 0.33/0.61 thf(fact_5387_abs__zero,axiom,
% 0.33/0.61 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.33/0.61 = zero_z3403309356797280102nteger ) ).
% 0.33/0.61
% 0.33/0.61 % abs_zero
% 0.33/0.61 thf(fact_5388_abs__zero,axiom,
% 0.33/0.61 ( ( abs_abs_real @ zero_zero_real )
% 0.33/0.61 = zero_zero_real ) ).
% 0.33/0.61
% 0.33/0.61 % abs_zero
% 0.33/0.61 thf(fact_5389_abs__zero,axiom,
% 0.33/0.61 ( ( abs_abs_rat @ zero_zero_rat )
% 0.33/0.61 = zero_zero_rat ) ).
% 0.33/0.61
% 0.33/0.61 % abs_zero
% 0.33/0.61 thf(fact_5390_abs__zero,axiom,
% 0.33/0.61 ( ( abs_abs_int @ zero_zero_int )
% 0.33/0.61 = zero_zero_int ) ).
% 0.33/0.61
% 0.33/0.61 % abs_zero
% 0.33/0.61 thf(fact_5391_abs__eq__0,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( ( abs_abs_Code_integer @ A )
% 0.33/0.61 = zero_z3403309356797280102nteger )
% 0.33/0.61 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_0
% 0.33/0.61 thf(fact_5392_abs__eq__0,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( ( abs_abs_real @ A )
% 0.33/0.61 = zero_zero_real )
% 0.33/0.61 = ( A = zero_zero_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_0
% 0.33/0.61 thf(fact_5393_abs__eq__0,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( ( abs_abs_rat @ A )
% 0.33/0.61 = zero_zero_rat )
% 0.33/0.61 = ( A = zero_zero_rat ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_0
% 0.33/0.61 thf(fact_5394_abs__eq__0,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( ( abs_abs_int @ A )
% 0.33/0.61 = zero_zero_int )
% 0.33/0.61 = ( A = zero_zero_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_0
% 0.33/0.61 thf(fact_5395_abs__0__eq,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( zero_z3403309356797280102nteger
% 0.33/0.61 = ( abs_abs_Code_integer @ A ) )
% 0.33/0.61 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_0_eq
% 0.33/0.61 thf(fact_5396_abs__0__eq,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( zero_zero_real
% 0.33/0.61 = ( abs_abs_real @ A ) )
% 0.33/0.61 = ( A = zero_zero_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_0_eq
% 0.33/0.61 thf(fact_5397_abs__0__eq,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( zero_zero_rat
% 0.33/0.61 = ( abs_abs_rat @ A ) )
% 0.33/0.61 = ( A = zero_zero_rat ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_0_eq
% 0.33/0.61 thf(fact_5398_abs__0__eq,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( zero_zero_int
% 0.33/0.61 = ( abs_abs_int @ A ) )
% 0.33/0.61 = ( A = zero_zero_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_0_eq
% 0.33/0.61 thf(fact_5399_abs__numeral,axiom,
% 0.33/0.61 ! [N: num] :
% 0.33/0.61 ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
% 0.33/0.61 = ( numera6620942414471956472nteger @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_numeral
% 0.33/0.61 thf(fact_5400_abs__numeral,axiom,
% 0.33/0.61 ! [N: num] :
% 0.33/0.61 ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
% 0.33/0.61 = ( numeral_numeral_real @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_numeral
% 0.33/0.61 thf(fact_5401_abs__numeral,axiom,
% 0.33/0.61 ! [N: num] :
% 0.33/0.61 ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
% 0.33/0.61 = ( numeral_numeral_rat @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_numeral
% 0.33/0.61 thf(fact_5402_abs__numeral,axiom,
% 0.33/0.61 ! [N: num] :
% 0.33/0.61 ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
% 0.33/0.61 = ( numeral_numeral_int @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_numeral
% 0.33/0.61 thf(fact_5403_abs__mult__self__eq,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 0.33/0.61 = ( times_3573771949741848930nteger @ A @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_self_eq
% 0.33/0.61 thf(fact_5404_abs__mult__self__eq,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 0.33/0.61 = ( times_times_real @ A @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_self_eq
% 0.33/0.61 thf(fact_5405_abs__mult__self__eq,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
% 0.33/0.61 = ( times_times_rat @ A @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_self_eq
% 0.33/0.61 thf(fact_5406_abs__mult__self__eq,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
% 0.33/0.61 = ( times_times_int @ A @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_self_eq
% 0.33/0.61 thf(fact_5407_abs__1,axiom,
% 0.33/0.61 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.33/0.61 = one_one_Code_integer ) ).
% 0.33/0.61
% 0.33/0.61 % abs_1
% 0.33/0.61 thf(fact_5408_abs__1,axiom,
% 0.33/0.61 ( ( abs_abs_complex @ one_one_complex )
% 0.33/0.61 = one_one_complex ) ).
% 0.33/0.61
% 0.33/0.61 % abs_1
% 0.33/0.61 thf(fact_5409_abs__1,axiom,
% 0.33/0.61 ( ( abs_abs_real @ one_one_real )
% 0.33/0.61 = one_one_real ) ).
% 0.33/0.61
% 0.33/0.61 % abs_1
% 0.33/0.61 thf(fact_5410_abs__1,axiom,
% 0.33/0.61 ( ( abs_abs_rat @ one_one_rat )
% 0.33/0.61 = one_one_rat ) ).
% 0.33/0.61
% 0.33/0.61 % abs_1
% 0.33/0.61 thf(fact_5411_abs__1,axiom,
% 0.33/0.61 ( ( abs_abs_int @ one_one_int )
% 0.33/0.61 = one_one_int ) ).
% 0.33/0.61
% 0.33/0.61 % abs_1
% 0.33/0.61 thf(fact_5412_abs__add__abs,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer] :
% 0.33/0.61 ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
% 0.33/0.61 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_add_abs
% 0.33/0.61 thf(fact_5413_abs__add__abs,axiom,
% 0.33/0.61 ! [A: real,B: real] :
% 0.33/0.61 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 0.33/0.61 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_add_abs
% 0.33/0.61 thf(fact_5414_abs__add__abs,axiom,
% 0.33/0.61 ! [A: rat,B: rat] :
% 0.33/0.61 ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
% 0.33/0.61 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_add_abs
% 0.33/0.61 thf(fact_5415_abs__add__abs,axiom,
% 0.33/0.61 ! [A: int,B: int] :
% 0.33/0.61 ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
% 0.33/0.61 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_add_abs
% 0.33/0.61 thf(fact_5416_abs__minus__cancel,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.33/0.61 = ( abs_abs_real @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus_cancel
% 0.33/0.61 thf(fact_5417_abs__minus__cancel,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 0.33/0.61 = ( abs_abs_int @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus_cancel
% 0.33/0.61 thf(fact_5418_abs__minus__cancel,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 0.33/0.61 = ( abs_abs_rat @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus_cancel
% 0.33/0.61 thf(fact_5419_abs__minus__cancel,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.33/0.61 = ( abs_abs_Code_integer @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus_cancel
% 0.33/0.61 thf(fact_5420_abs__minus,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.33/0.61 = ( abs_abs_real @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus
% 0.33/0.61 thf(fact_5421_abs__minus,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 0.33/0.61 = ( abs_abs_int @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus
% 0.33/0.61 thf(fact_5422_abs__minus,axiom,
% 0.33/0.61 ! [A: complex] :
% 0.33/0.61 ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
% 0.33/0.61 = ( abs_abs_complex @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus
% 0.33/0.61 thf(fact_5423_abs__minus,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 0.33/0.61 = ( abs_abs_rat @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus
% 0.33/0.61 thf(fact_5424_abs__minus,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.33/0.61 = ( abs_abs_Code_integer @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus
% 0.33/0.61 thf(fact_5425_abs__divide,axiom,
% 0.33/0.61 ! [A: complex,B: complex] :
% 0.33/0.61 ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.61 = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_divide
% 0.33/0.61 thf(fact_5426_abs__divide,axiom,
% 0.33/0.61 ! [A: real,B: real] :
% 0.33/0.61 ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.61 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_divide
% 0.33/0.61 thf(fact_5427_abs__divide,axiom,
% 0.33/0.61 ! [A: rat,B: rat] :
% 0.33/0.61 ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 0.33/0.61 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_divide
% 0.33/0.61 thf(fact_5428_dvd__abs__iff,axiom,
% 0.33/0.61 ! [M: real,K: real] :
% 0.33/0.61 ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
% 0.33/0.61 = ( dvd_dvd_real @ M @ K ) ) ).
% 0.33/0.61
% 0.33/0.61 % dvd_abs_iff
% 0.33/0.61 thf(fact_5429_dvd__abs__iff,axiom,
% 0.33/0.61 ! [M: int,K: int] :
% 0.33/0.61 ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
% 0.33/0.61 = ( dvd_dvd_int @ M @ K ) ) ).
% 0.33/0.61
% 0.33/0.61 % dvd_abs_iff
% 0.33/0.61 thf(fact_5430_dvd__abs__iff,axiom,
% 0.33/0.61 ! [M: code_integer,K: code_integer] :
% 0.33/0.61 ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
% 0.33/0.61 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 0.33/0.61
% 0.33/0.61 % dvd_abs_iff
% 0.33/0.61 thf(fact_5431_dvd__abs__iff,axiom,
% 0.33/0.61 ! [M: rat,K: rat] :
% 0.33/0.61 ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
% 0.33/0.61 = ( dvd_dvd_rat @ M @ K ) ) ).
% 0.33/0.61
% 0.33/0.61 % dvd_abs_iff
% 0.33/0.61 thf(fact_5432_abs__dvd__iff,axiom,
% 0.33/0.61 ! [M: real,K: real] :
% 0.33/0.61 ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
% 0.33/0.61 = ( dvd_dvd_real @ M @ K ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_dvd_iff
% 0.33/0.61 thf(fact_5433_abs__dvd__iff,axiom,
% 0.33/0.61 ! [M: int,K: int] :
% 0.33/0.61 ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
% 0.33/0.61 = ( dvd_dvd_int @ M @ K ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_dvd_iff
% 0.33/0.61 thf(fact_5434_abs__dvd__iff,axiom,
% 0.33/0.61 ! [M: code_integer,K: code_integer] :
% 0.33/0.61 ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
% 0.33/0.61 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_dvd_iff
% 0.33/0.61 thf(fact_5435_abs__dvd__iff,axiom,
% 0.33/0.61 ! [M: rat,K: rat] :
% 0.33/0.61 ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
% 0.33/0.61 = ( dvd_dvd_rat @ M @ K ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_dvd_iff
% 0.33/0.61 thf(fact_5436_abs__of__nat,axiom,
% 0.33/0.61 ! [N: nat] :
% 0.33/0.61 ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.33/0.61 = ( semiri4939895301339042750nteger @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_nat
% 0.33/0.61 thf(fact_5437_abs__of__nat,axiom,
% 0.33/0.61 ! [N: nat] :
% 0.33/0.61 ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.33/0.61 = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_nat
% 0.33/0.61 thf(fact_5438_abs__of__nat,axiom,
% 0.33/0.61 ! [N: nat] :
% 0.33/0.61 ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.33/0.61 = ( semiri681578069525770553at_rat @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_nat
% 0.33/0.61 thf(fact_5439_abs__of__nat,axiom,
% 0.33/0.61 ! [N: nat] :
% 0.33/0.61 ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.33/0.61 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_nat
% 0.33/0.61 thf(fact_5440_powr__eq__0__iff,axiom,
% 0.33/0.61 ! [W: real,Z2: real] :
% 0.33/0.61 ( ( ( powr_real @ W @ Z2 )
% 0.33/0.61 = zero_zero_real )
% 0.33/0.61 = ( W = zero_zero_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_eq_0_iff
% 0.33/0.61 thf(fact_5441_powr__0,axiom,
% 0.33/0.61 ! [Z2: real] :
% 0.33/0.61 ( ( powr_real @ zero_zero_real @ Z2 )
% 0.33/0.61 = zero_zero_real ) ).
% 0.33/0.61
% 0.33/0.61 % powr_0
% 0.33/0.61 thf(fact_5442_powr__one__eq__one,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( powr_real @ one_one_real @ A )
% 0.33/0.61 = one_one_real ) ).
% 0.33/0.61
% 0.33/0.61 % powr_one_eq_one
% 0.33/0.61 thf(fact_5443_abs__bool__eq,axiom,
% 0.33/0.61 ! [P: $o] :
% 0.33/0.61 ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.33/0.61 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_bool_eq
% 0.33/0.61 thf(fact_5444_abs__bool__eq,axiom,
% 0.33/0.61 ! [P: $o] :
% 0.33/0.61 ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 0.33/0.61 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_bool_eq
% 0.33/0.61 thf(fact_5445_abs__bool__eq,axiom,
% 0.33/0.61 ! [P: $o] :
% 0.33/0.61 ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.33/0.61 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_bool_eq
% 0.33/0.61 thf(fact_5446_abs__bool__eq,axiom,
% 0.33/0.61 ! [P: $o] :
% 0.33/0.61 ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
% 0.33/0.61 = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_bool_eq
% 0.33/0.61 thf(fact_5447_abs__of__nonneg,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.33/0.61 => ( ( abs_abs_Code_integer @ A )
% 0.33/0.61 = A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_nonneg
% 0.33/0.61 thf(fact_5448_abs__of__nonneg,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.61 => ( ( abs_abs_real @ A )
% 0.33/0.61 = A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_nonneg
% 0.33/0.61 thf(fact_5449_abs__of__nonneg,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.61 => ( ( abs_abs_rat @ A )
% 0.33/0.61 = A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_nonneg
% 0.33/0.61 thf(fact_5450_abs__of__nonneg,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.61 => ( ( abs_abs_int @ A )
% 0.33/0.61 = A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_nonneg
% 0.33/0.61 thf(fact_5451_abs__le__self__iff,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 0.33/0.61 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_self_iff
% 0.33/0.61 thf(fact_5452_abs__le__self__iff,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 0.33/0.61 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_self_iff
% 0.33/0.61 thf(fact_5453_abs__le__self__iff,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 0.33/0.61 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_self_iff
% 0.33/0.61 thf(fact_5454_abs__le__self__iff,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 0.33/0.61 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_self_iff
% 0.33/0.61 thf(fact_5455_abs__le__zero__iff,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 0.33/0.61 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_zero_iff
% 0.33/0.61 thf(fact_5456_abs__le__zero__iff,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 0.33/0.61 = ( A = zero_zero_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_zero_iff
% 0.33/0.61 thf(fact_5457_abs__le__zero__iff,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 0.33/0.61 = ( A = zero_zero_rat ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_zero_iff
% 0.33/0.61 thf(fact_5458_abs__le__zero__iff,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 0.33/0.61 = ( A = zero_zero_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_zero_iff
% 0.33/0.61 thf(fact_5459_zero__less__abs__iff,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 0.33/0.61 = ( A != zero_z3403309356797280102nteger ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_less_abs_iff
% 0.33/0.61 thf(fact_5460_zero__less__abs__iff,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 0.33/0.61 = ( A != zero_zero_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_less_abs_iff
% 0.33/0.61 thf(fact_5461_zero__less__abs__iff,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 0.33/0.61 = ( A != zero_zero_rat ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_less_abs_iff
% 0.33/0.61 thf(fact_5462_zero__less__abs__iff,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 0.33/0.61 = ( A != zero_zero_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_less_abs_iff
% 0.33/0.61 thf(fact_5463_abs__neg__numeral,axiom,
% 0.33/0.61 ! [N: num] :
% 0.33/0.61 ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.33/0.61 = ( numeral_numeral_real @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_neg_numeral
% 0.33/0.61 thf(fact_5464_abs__neg__numeral,axiom,
% 0.33/0.61 ! [N: num] :
% 0.33/0.61 ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.61 = ( numeral_numeral_int @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_neg_numeral
% 0.33/0.61 thf(fact_5465_abs__neg__numeral,axiom,
% 0.33/0.61 ! [N: num] :
% 0.33/0.61 ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.33/0.61 = ( numeral_numeral_rat @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_neg_numeral
% 0.33/0.61 thf(fact_5466_abs__neg__numeral,axiom,
% 0.33/0.61 ! [N: num] :
% 0.33/0.61 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.33/0.61 = ( numera6620942414471956472nteger @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_neg_numeral
% 0.33/0.61 thf(fact_5467_abs__neg__one,axiom,
% 0.33/0.61 ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.61 = one_one_real ) ).
% 0.33/0.61
% 0.33/0.61 % abs_neg_one
% 0.33/0.61 thf(fact_5468_abs__neg__one,axiom,
% 0.33/0.61 ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.61 = one_one_int ) ).
% 0.33/0.61
% 0.33/0.61 % abs_neg_one
% 0.33/0.61 thf(fact_5469_abs__neg__one,axiom,
% 0.33/0.61 ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.61 = one_one_rat ) ).
% 0.33/0.61
% 0.33/0.61 % abs_neg_one
% 0.33/0.61 thf(fact_5470_abs__neg__one,axiom,
% 0.33/0.61 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.61 = one_one_Code_integer ) ).
% 0.33/0.61
% 0.33/0.61 % abs_neg_one
% 0.33/0.61 thf(fact_5471_abs__power__minus,axiom,
% 0.33/0.61 ! [A: real,N: nat] :
% 0.33/0.61 ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 0.33/0.61 = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_power_minus
% 0.33/0.61 thf(fact_5472_abs__power__minus,axiom,
% 0.33/0.61 ! [A: int,N: nat] :
% 0.33/0.61 ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 0.33/0.61 = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_power_minus
% 0.33/0.61 thf(fact_5473_abs__power__minus,axiom,
% 0.33/0.61 ! [A: rat,N: nat] :
% 0.33/0.61 ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 0.33/0.61 = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_power_minus
% 0.33/0.61 thf(fact_5474_abs__power__minus,axiom,
% 0.33/0.61 ! [A: code_integer,N: nat] :
% 0.33/0.61 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 0.33/0.61 = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_power_minus
% 0.33/0.61 thf(fact_5475_powr__zero__eq__one,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ( X = zero_zero_real )
% 0.33/0.61 => ( ( powr_real @ X @ zero_zero_real )
% 0.33/0.61 = zero_zero_real ) )
% 0.33/0.61 & ( ( X != zero_zero_real )
% 0.33/0.61 => ( ( powr_real @ X @ zero_zero_real )
% 0.33/0.61 = one_one_real ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_zero_eq_one
% 0.33/0.61 thf(fact_5476_floor__zero,axiom,
% 0.33/0.61 ( ( archim6058952711729229775r_real @ zero_zero_real )
% 0.33/0.61 = zero_zero_int ) ).
% 0.33/0.61
% 0.33/0.61 % floor_zero
% 0.33/0.61 thf(fact_5477_floor__zero,axiom,
% 0.33/0.61 ( ( archim3151403230148437115or_rat @ zero_zero_rat )
% 0.33/0.61 = zero_zero_int ) ).
% 0.33/0.61
% 0.33/0.61 % floor_zero
% 0.33/0.61 thf(fact_5478_floor__numeral,axiom,
% 0.33/0.61 ! [V: num] :
% 0.33/0.61 ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
% 0.33/0.61 = ( numeral_numeral_int @ V ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_numeral
% 0.33/0.61 thf(fact_5479_floor__numeral,axiom,
% 0.33/0.61 ! [V: num] :
% 0.33/0.61 ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
% 0.33/0.61 = ( numeral_numeral_int @ V ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_numeral
% 0.33/0.61 thf(fact_5480_negative__eq__positive,axiom,
% 0.33/0.61 ! [N: nat,M: nat] :
% 0.33/0.61 ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.33/0.61 = ( semiri1314217659103216013at_int @ M ) )
% 0.33/0.61 = ( ( N = zero_zero_nat )
% 0.33/0.61 & ( M = zero_zero_nat ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % negative_eq_positive
% 0.33/0.61 thf(fact_5481_floor__one,axiom,
% 0.33/0.61 ( ( archim6058952711729229775r_real @ one_one_real )
% 0.33/0.61 = one_one_int ) ).
% 0.33/0.61
% 0.33/0.61 % floor_one
% 0.33/0.61 thf(fact_5482_floor__one,axiom,
% 0.33/0.61 ( ( archim3151403230148437115or_rat @ one_one_rat )
% 0.33/0.61 = one_one_int ) ).
% 0.33/0.61
% 0.33/0.61 % floor_one
% 0.33/0.61 thf(fact_5483_powr__less__cancel__iff,axiom,
% 0.33/0.61 ! [X: real,A: real,B: real] :
% 0.33/0.61 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.61 => ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 0.33/0.61 = ( ord_less_real @ A @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_less_cancel_iff
% 0.33/0.61 thf(fact_5484_negative__zle,axiom,
% 0.33/0.61 ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.33/0.61
% 0.33/0.61 % negative_zle
% 0.33/0.61 thf(fact_5485_floor__of__nat,axiom,
% 0.33/0.61 ! [N: nat] :
% 0.33/0.61 ( ( archim6058952711729229775r_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.33/0.61 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_of_nat
% 0.33/0.61 thf(fact_5486_floor__of__nat,axiom,
% 0.33/0.61 ! [N: nat] :
% 0.33/0.61 ( ( archim3151403230148437115or_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.33/0.61 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_of_nat
% 0.33/0.61 thf(fact_5487_sin__coeff__0,axiom,
% 0.33/0.61 ( ( sin_coeff @ zero_zero_nat )
% 0.33/0.61 = zero_zero_real ) ).
% 0.33/0.61
% 0.33/0.61 % sin_coeff_0
% 0.33/0.61 thf(fact_5488_cos__coeff__0,axiom,
% 0.33/0.61 ( ( cos_coeff @ zero_zero_nat )
% 0.33/0.61 = one_one_real ) ).
% 0.33/0.61
% 0.33/0.61 % cos_coeff_0
% 0.33/0.61 thf(fact_5489_abs__of__nonpos,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.61 => ( ( abs_abs_real @ A )
% 0.33/0.61 = ( uminus_uminus_real @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_nonpos
% 0.33/0.61 thf(fact_5490_abs__of__nonpos,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 0.33/0.61 => ( ( abs_abs_Code_integer @ A )
% 0.33/0.61 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_nonpos
% 0.33/0.61 thf(fact_5491_abs__of__nonpos,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.33/0.61 => ( ( abs_abs_rat @ A )
% 0.33/0.61 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_nonpos
% 0.33/0.61 thf(fact_5492_abs__of__nonpos,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.33/0.61 => ( ( abs_abs_int @ A )
% 0.33/0.61 = ( uminus_uminus_int @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_nonpos
% 0.33/0.61 thf(fact_5493_divide__le__0__abs__iff,axiom,
% 0.33/0.61 ! [A: real,B: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 0.33/0.61 = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.61 | ( B = zero_zero_real ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % divide_le_0_abs_iff
% 0.33/0.61 thf(fact_5494_divide__le__0__abs__iff,axiom,
% 0.33/0.61 ! [A: rat,B: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
% 0.33/0.61 = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.33/0.61 | ( B = zero_zero_rat ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % divide_le_0_abs_iff
% 0.33/0.61 thf(fact_5495_zero__le__divide__abs__iff,axiom,
% 0.33/0.61 ! [A: real,B: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 0.33/0.61 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.61 | ( B = zero_zero_real ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_le_divide_abs_iff
% 0.33/0.61 thf(fact_5496_zero__le__divide__abs__iff,axiom,
% 0.33/0.61 ! [A: rat,B: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
% 0.33/0.61 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.61 | ( B = zero_zero_rat ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_le_divide_abs_iff
% 0.33/0.61 thf(fact_5497_abs__sgn__eq__1,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( A != zero_z3403309356797280102nteger )
% 0.33/0.61 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.33/0.61 = one_one_Code_integer ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_sgn_eq_1
% 0.33/0.61 thf(fact_5498_abs__sgn__eq__1,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( A != zero_zero_real )
% 0.33/0.61 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.33/0.61 = one_one_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_sgn_eq_1
% 0.33/0.61 thf(fact_5499_abs__sgn__eq__1,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( A != zero_zero_rat )
% 0.33/0.61 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.33/0.61 = one_one_rat ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_sgn_eq_1
% 0.33/0.61 thf(fact_5500_abs__sgn__eq__1,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( A != zero_zero_int )
% 0.33/0.61 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.33/0.61 = one_one_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_sgn_eq_1
% 0.33/0.61 thf(fact_5501_powr__eq__one__iff,axiom,
% 0.33/0.61 ! [A: real,X: real] :
% 0.33/0.61 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.61 => ( ( ( powr_real @ A @ X )
% 0.33/0.61 = one_one_real )
% 0.33/0.61 = ( X = zero_zero_real ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_eq_one_iff
% 0.33/0.61 thf(fact_5502_powr__one__gt__zero__iff,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ( powr_real @ X @ one_one_real )
% 0.33/0.61 = X )
% 0.33/0.61 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_one_gt_zero_iff
% 0.33/0.61 thf(fact_5503_powr__one,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( powr_real @ X @ one_one_real )
% 0.33/0.61 = X ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_one
% 0.33/0.61 thf(fact_5504_powr__le__cancel__iff,axiom,
% 0.33/0.61 ! [X: real,A: real,B: real] :
% 0.33/0.61 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.61 => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 0.33/0.61 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_le_cancel_iff
% 0.33/0.61 thf(fact_5505_negative__zless,axiom,
% 0.33/0.61 ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.33/0.61
% 0.33/0.61 % negative_zless
% 0.33/0.61 thf(fact_5506_artanh__minus__real,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.61 => ( ( artanh_real @ ( uminus_uminus_real @ X ) )
% 0.33/0.61 = ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % artanh_minus_real
% 0.33/0.61 thf(fact_5507_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.33/0.61 ! [A: complex] :
% 0.33/0.61 ( ( sgn_sgn_complex @ ( abs_abs_complex @ A ) )
% 0.33/0.61 = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % idom_abs_sgn_class.abs_sgn
% 0.33/0.61 thf(fact_5508_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( sgn_sgn_real @ ( abs_abs_real @ A ) )
% 0.33/0.61 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % idom_abs_sgn_class.abs_sgn
% 0.33/0.61 thf(fact_5509_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( sgn_sgn_rat @ ( abs_abs_rat @ A ) )
% 0.33/0.61 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % idom_abs_sgn_class.abs_sgn
% 0.33/0.61 thf(fact_5510_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( sgn_sgn_int @ ( abs_abs_int @ A ) )
% 0.33/0.61 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % idom_abs_sgn_class.abs_sgn
% 0.33/0.61 thf(fact_5511_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( sgn_sgn_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.33/0.61 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % idom_abs_sgn_class.abs_sgn
% 0.33/0.61 thf(fact_5512_sgn__abs,axiom,
% 0.33/0.61 ! [A: complex] :
% 0.33/0.61 ( ( abs_abs_complex @ ( sgn_sgn_complex @ A ) )
% 0.33/0.61 = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % sgn_abs
% 0.33/0.61 thf(fact_5513_sgn__abs,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.33/0.61 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % sgn_abs
% 0.33/0.61 thf(fact_5514_sgn__abs,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.33/0.61 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % sgn_abs
% 0.33/0.61 thf(fact_5515_sgn__abs,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.33/0.61 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % sgn_abs
% 0.33/0.61 thf(fact_5516_sgn__abs,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.33/0.61 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % sgn_abs
% 0.33/0.61 thf(fact_5517_numeral__powr__numeral__real,axiom,
% 0.33/0.61 ! [M: num,N: num] :
% 0.33/0.61 ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.33/0.61 = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % numeral_powr_numeral_real
% 0.33/0.61 thf(fact_5518_zero__less__power__abs__iff,axiom,
% 0.33/0.61 ! [A: code_integer,N: nat] :
% 0.33/0.61 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
% 0.33/0.61 = ( ( A != zero_z3403309356797280102nteger )
% 0.33/0.61 | ( N = zero_zero_nat ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_less_power_abs_iff
% 0.33/0.61 thf(fact_5519_zero__less__power__abs__iff,axiom,
% 0.33/0.61 ! [A: real,N: nat] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.33/0.61 = ( ( A != zero_zero_real )
% 0.33/0.61 | ( N = zero_zero_nat ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_less_power_abs_iff
% 0.33/0.61 thf(fact_5520_zero__less__power__abs__iff,axiom,
% 0.33/0.61 ! [A: rat,N: nat] :
% 0.33/0.61 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
% 0.33/0.61 = ( ( A != zero_zero_rat )
% 0.33/0.61 | ( N = zero_zero_nat ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_less_power_abs_iff
% 0.33/0.61 thf(fact_5521_zero__less__power__abs__iff,axiom,
% 0.33/0.61 ! [A: int,N: nat] :
% 0.33/0.61 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 0.33/0.61 = ( ( A != zero_zero_int )
% 0.33/0.61 | ( N = zero_zero_nat ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_less_power_abs_iff
% 0.33/0.61 thf(fact_5522_abs__power2,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.61 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_power2
% 0.33/0.61 thf(fact_5523_abs__power2,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.61 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_power2
% 0.33/0.61 thf(fact_5524_abs__power2,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.61 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_power2
% 0.33/0.61 thf(fact_5525_abs__power2,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.61 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_power2
% 0.33/0.61 thf(fact_5526_power2__abs,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.61 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power2_abs
% 0.33/0.61 thf(fact_5527_power2__abs,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.61 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power2_abs
% 0.33/0.61 thf(fact_5528_power2__abs,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.61 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power2_abs
% 0.33/0.61 thf(fact_5529_power2__abs,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.61 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power2_abs
% 0.33/0.61 thf(fact_5530_zero__le__floor,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.61 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_le_floor
% 0.33/0.61 thf(fact_5531_zero__le__floor,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
% 0.33/0.61 = ( ord_less_eq_rat @ zero_zero_rat @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_le_floor
% 0.33/0.61 thf(fact_5532_floor__less__zero,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
% 0.33/0.61 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_less_zero
% 0.33/0.61 thf(fact_5533_floor__less__zero,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
% 0.33/0.61 = ( ord_less_rat @ X @ zero_zero_rat ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_less_zero
% 0.33/0.61 thf(fact_5534_numeral__le__floor,axiom,
% 0.33/0.61 ! [V: num,X: real] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.61 = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % numeral_le_floor
% 0.33/0.61 thf(fact_5535_numeral__le__floor,axiom,
% 0.33/0.61 ! [V: num,X: rat] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
% 0.33/0.61 = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % numeral_le_floor
% 0.33/0.61 thf(fact_5536_zero__less__floor,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.61 = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_less_floor
% 0.33/0.61 thf(fact_5537_zero__less__floor,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
% 0.33/0.61 = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_less_floor
% 0.33/0.61 thf(fact_5538_floor__le__zero,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
% 0.33/0.61 = ( ord_less_real @ X @ one_one_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_le_zero
% 0.33/0.61 thf(fact_5539_floor__le__zero,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
% 0.33/0.61 = ( ord_less_rat @ X @ one_one_rat ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_le_zero
% 0.33/0.61 thf(fact_5540_floor__less__numeral,axiom,
% 0.33/0.61 ! [X: real,V: num] :
% 0.33/0.61 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.33/0.61 = ( ord_less_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_less_numeral
% 0.33/0.61 thf(fact_5541_floor__less__numeral,axiom,
% 0.33/0.61 ! [X: rat,V: num] :
% 0.33/0.61 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 0.33/0.61 = ( ord_less_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_less_numeral
% 0.33/0.61 thf(fact_5542_one__le__floor,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.61 = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % one_le_floor
% 0.33/0.61 thf(fact_5543_one__le__floor,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
% 0.33/0.61 = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % one_le_floor
% 0.33/0.61 thf(fact_5544_floor__less__one,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 0.33/0.61 = ( ord_less_real @ X @ one_one_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_less_one
% 0.33/0.61 thf(fact_5545_floor__less__one,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 0.33/0.61 = ( ord_less_rat @ X @ one_one_rat ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_less_one
% 0.33/0.61 thf(fact_5546_floor__neg__numeral,axiom,
% 0.33/0.61 ! [V: num] :
% 0.33/0.61 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.33/0.61 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_neg_numeral
% 0.33/0.61 thf(fact_5547_floor__neg__numeral,axiom,
% 0.33/0.61 ! [V: num] :
% 0.33/0.61 ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.33/0.61 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_neg_numeral
% 0.33/0.61 thf(fact_5548_floor__diff__numeral,axiom,
% 0.33/0.61 ! [X: real,V: num] :
% 0.33/0.61 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 0.33/0.61 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_diff_numeral
% 0.33/0.61 thf(fact_5549_floor__diff__numeral,axiom,
% 0.33/0.61 ! [X: rat,V: num] :
% 0.33/0.61 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 0.33/0.61 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_diff_numeral
% 0.33/0.61 thf(fact_5550_floor__diff__one,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ one_one_real ) )
% 0.33/0.61 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_diff_one
% 0.33/0.61 thf(fact_5551_floor__diff__one,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
% 0.33/0.61 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_diff_one
% 0.33/0.61 thf(fact_5552_floor__numeral__power,axiom,
% 0.33/0.61 ! [X: num,N: nat] :
% 0.33/0.61 ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.33/0.61 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_numeral_power
% 0.33/0.61 thf(fact_5553_floor__numeral__power,axiom,
% 0.33/0.61 ! [X: num,N: nat] :
% 0.33/0.61 ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.33/0.61 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_numeral_power
% 0.33/0.61 thf(fact_5554_log__powr__cancel,axiom,
% 0.33/0.61 ! [A: real,Y: real] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.61 => ( ( A != one_one_real )
% 0.33/0.61 => ( ( log2 @ A @ ( powr_real @ A @ Y ) )
% 0.33/0.61 = Y ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % log_powr_cancel
% 0.33/0.61 thf(fact_5555_powr__log__cancel,axiom,
% 0.33/0.61 ! [A: real,X: real] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.61 => ( ( A != one_one_real )
% 0.33/0.61 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( powr_real @ A @ ( log2 @ A @ X ) )
% 0.33/0.61 = X ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_log_cancel
% 0.33/0.61 thf(fact_5556_floor__divide__eq__div__numeral,axiom,
% 0.33/0.61 ! [A: num,B: num] :
% 0.33/0.61 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 0.33/0.61 = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_divide_eq_div_numeral
% 0.33/0.61 thf(fact_5557_power__even__abs__numeral,axiom,
% 0.33/0.61 ! [W: num,A: code_integer] :
% 0.33/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.33/0.61 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.33/0.61 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power_even_abs_numeral
% 0.33/0.61 thf(fact_5558_power__even__abs__numeral,axiom,
% 0.33/0.61 ! [W: num,A: rat] :
% 0.33/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.33/0.61 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.33/0.61 = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power_even_abs_numeral
% 0.33/0.61 thf(fact_5559_power__even__abs__numeral,axiom,
% 0.33/0.61 ! [W: num,A: real] :
% 0.33/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.33/0.61 => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.33/0.61 = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power_even_abs_numeral
% 0.33/0.61 thf(fact_5560_power__even__abs__numeral,axiom,
% 0.33/0.61 ! [W: num,A: int] :
% 0.33/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.33/0.61 => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.33/0.61 = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power_even_abs_numeral
% 0.33/0.61 thf(fact_5561_powr__numeral,axiom,
% 0.33/0.61 ! [X: real,N: num] :
% 0.33/0.61 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
% 0.33/0.61 = ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_numeral
% 0.33/0.61 thf(fact_5562_ceiling__divide__eq__div__numeral,axiom,
% 0.33/0.61 ! [A: num,B: num] :
% 0.33/0.61 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 0.33/0.61 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % ceiling_divide_eq_div_numeral
% 0.33/0.61 thf(fact_5563_numeral__less__floor,axiom,
% 0.33/0.61 ! [V: num,X: real] :
% 0.33/0.61 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.61 = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % numeral_less_floor
% 0.33/0.61 thf(fact_5564_numeral__less__floor,axiom,
% 0.33/0.61 ! [V: num,X: rat] :
% 0.33/0.61 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
% 0.33/0.61 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % numeral_less_floor
% 0.33/0.61 thf(fact_5565_floor__le__numeral,axiom,
% 0.33/0.61 ! [X: real,V: num] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.33/0.61 = ( ord_less_real @ X @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_le_numeral
% 0.33/0.61 thf(fact_5566_floor__le__numeral,axiom,
% 0.33/0.61 ! [X: rat,V: num] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 0.33/0.61 = ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_le_numeral
% 0.33/0.61 thf(fact_5567_one__less__floor,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.61 = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % one_less_floor
% 0.33/0.61 thf(fact_5568_one__less__floor,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
% 0.33/0.61 = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % one_less_floor
% 0.33/0.61 thf(fact_5569_floor__le__one,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 0.33/0.61 = ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_le_one
% 0.33/0.61 thf(fact_5570_floor__le__one,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 0.33/0.61 = ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_le_one
% 0.33/0.61 thf(fact_5571_neg__numeral__le__floor,axiom,
% 0.33/0.61 ! [V: num,X: real] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.61 = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % neg_numeral_le_floor
% 0.33/0.61 thf(fact_5572_neg__numeral__le__floor,axiom,
% 0.33/0.61 ! [V: num,X: rat] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
% 0.33/0.61 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % neg_numeral_le_floor
% 0.33/0.61 thf(fact_5573_floor__less__neg__numeral,axiom,
% 0.33/0.61 ! [X: real,V: num] :
% 0.33/0.61 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.33/0.61 = ( ord_less_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_less_neg_numeral
% 0.33/0.61 thf(fact_5574_floor__less__neg__numeral,axiom,
% 0.33/0.61 ! [X: rat,V: num] :
% 0.33/0.61 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.33/0.61 = ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_less_neg_numeral
% 0.33/0.61 thf(fact_5575_signed__take__bit__Suc__minus__bit0,axiom,
% 0.33/0.61 ! [N: nat,K: num] :
% 0.33/0.61 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.33/0.61 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % signed_take_bit_Suc_minus_bit0
% 0.33/0.61 thf(fact_5576_floor__one__divide__eq__div__numeral,axiom,
% 0.33/0.61 ! [B: num] :
% 0.33/0.61 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
% 0.33/0.61 = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_one_divide_eq_div_numeral
% 0.33/0.61 thf(fact_5577_floor__minus__divide__eq__div__numeral,axiom,
% 0.33/0.61 ! [A: num,B: num] :
% 0.33/0.61 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 0.33/0.61 = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_minus_divide_eq_div_numeral
% 0.33/0.61 thf(fact_5578_ceiling__minus__divide__eq__div__numeral,axiom,
% 0.33/0.61 ! [A: num,B: num] :
% 0.33/0.61 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 0.33/0.61 = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % ceiling_minus_divide_eq_div_numeral
% 0.33/0.61 thf(fact_5579_neg__numeral__less__floor,axiom,
% 0.33/0.61 ! [V: num,X: real] :
% 0.33/0.61 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.61 = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % neg_numeral_less_floor
% 0.33/0.61 thf(fact_5580_neg__numeral__less__floor,axiom,
% 0.33/0.61 ! [V: num,X: rat] :
% 0.33/0.61 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
% 0.33/0.61 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % neg_numeral_less_floor
% 0.33/0.61 thf(fact_5581_floor__le__neg__numeral,axiom,
% 0.33/0.61 ! [X: real,V: num] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.33/0.61 = ( ord_less_real @ X @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_le_neg_numeral
% 0.33/0.61 thf(fact_5582_floor__le__neg__numeral,axiom,
% 0.33/0.61 ! [X: rat,V: num] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.33/0.61 = ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_le_neg_numeral
% 0.33/0.61 thf(fact_5583_square__powr__half,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.61 = ( abs_abs_real @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % square_powr_half
% 0.33/0.61 thf(fact_5584_floor__minus__one__divide__eq__div__numeral,axiom,
% 0.33/0.61 ! [B: num] :
% 0.33/0.61 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
% 0.33/0.61 = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_minus_one_divide_eq_div_numeral
% 0.33/0.61 thf(fact_5585_abs__le__D1,axiom,
% 0.33/0.61 ! [A: real,B: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.33/0.61 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_D1
% 0.33/0.61 thf(fact_5586_abs__le__D1,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer] :
% 0.33/0.61 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.33/0.61 => ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_D1
% 0.33/0.61 thf(fact_5587_abs__le__D1,axiom,
% 0.33/0.61 ! [A: rat,B: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.33/0.61 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_D1
% 0.33/0.61 thf(fact_5588_abs__le__D1,axiom,
% 0.33/0.61 ! [A: int,B: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.33/0.61 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_D1
% 0.33/0.61 thf(fact_5589_abs__ge__self,axiom,
% 0.33/0.61 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_ge_self
% 0.33/0.61 thf(fact_5590_abs__ge__self,axiom,
% 0.33/0.61 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_ge_self
% 0.33/0.61 thf(fact_5591_abs__ge__self,axiom,
% 0.33/0.61 ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_ge_self
% 0.33/0.61 thf(fact_5592_abs__ge__self,axiom,
% 0.33/0.61 ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_ge_self
% 0.33/0.61 thf(fact_5593_abs__eq__0__iff,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( ( abs_abs_Code_integer @ A )
% 0.33/0.61 = zero_z3403309356797280102nteger )
% 0.33/0.61 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_0_iff
% 0.33/0.61 thf(fact_5594_abs__eq__0__iff,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( ( abs_abs_real @ A )
% 0.33/0.61 = zero_zero_real )
% 0.33/0.61 = ( A = zero_zero_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_0_iff
% 0.33/0.61 thf(fact_5595_abs__eq__0__iff,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( ( abs_abs_rat @ A )
% 0.33/0.61 = zero_zero_rat )
% 0.33/0.61 = ( A = zero_zero_rat ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_0_iff
% 0.33/0.61 thf(fact_5596_abs__eq__0__iff,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( ( abs_abs_int @ A )
% 0.33/0.61 = zero_zero_int )
% 0.33/0.61 = ( A = zero_zero_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_0_iff
% 0.33/0.61 thf(fact_5597_abs__mult,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer] :
% 0.33/0.61 ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.33/0.61 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult
% 0.33/0.61 thf(fact_5598_abs__mult,axiom,
% 0.33/0.61 ! [A: real,B: real] :
% 0.33/0.61 ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 0.33/0.61 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult
% 0.33/0.61 thf(fact_5599_abs__mult,axiom,
% 0.33/0.61 ! [A: rat,B: rat] :
% 0.33/0.61 ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 0.33/0.61 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult
% 0.33/0.61 thf(fact_5600_abs__mult,axiom,
% 0.33/0.61 ! [A: int,B: int] :
% 0.33/0.61 ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 0.33/0.61 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult
% 0.33/0.61 thf(fact_5601_abs__one,axiom,
% 0.33/0.61 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.33/0.61 = one_one_Code_integer ) ).
% 0.33/0.61
% 0.33/0.61 % abs_one
% 0.33/0.61 thf(fact_5602_abs__one,axiom,
% 0.33/0.61 ( ( abs_abs_real @ one_one_real )
% 0.33/0.61 = one_one_real ) ).
% 0.33/0.61
% 0.33/0.61 % abs_one
% 0.33/0.61 thf(fact_5603_abs__one,axiom,
% 0.33/0.61 ( ( abs_abs_rat @ one_one_rat )
% 0.33/0.61 = one_one_rat ) ).
% 0.33/0.61
% 0.33/0.61 % abs_one
% 0.33/0.61 thf(fact_5604_abs__one,axiom,
% 0.33/0.61 ( ( abs_abs_int @ one_one_int )
% 0.33/0.61 = one_one_int ) ).
% 0.33/0.61
% 0.33/0.61 % abs_one
% 0.33/0.61 thf(fact_5605_abs__minus__commute,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer] :
% 0.33/0.61 ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.33/0.61 = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus_commute
% 0.33/0.61 thf(fact_5606_abs__minus__commute,axiom,
% 0.33/0.61 ! [A: real,B: real] :
% 0.33/0.61 ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 0.33/0.61 = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus_commute
% 0.33/0.61 thf(fact_5607_abs__minus__commute,axiom,
% 0.33/0.61 ! [A: rat,B: rat] :
% 0.33/0.61 ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
% 0.33/0.61 = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus_commute
% 0.33/0.61 thf(fact_5608_abs__minus__commute,axiom,
% 0.33/0.61 ! [A: int,B: int] :
% 0.33/0.61 ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 0.33/0.61 = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus_commute
% 0.33/0.61 thf(fact_5609_abs__eq__iff,axiom,
% 0.33/0.61 ! [X: real,Y: real] :
% 0.33/0.61 ( ( ( abs_abs_real @ X )
% 0.33/0.61 = ( abs_abs_real @ Y ) )
% 0.33/0.61 = ( ( X = Y )
% 0.33/0.61 | ( X
% 0.33/0.61 = ( uminus_uminus_real @ Y ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_iff
% 0.33/0.61 thf(fact_5610_abs__eq__iff,axiom,
% 0.33/0.61 ! [X: int,Y: int] :
% 0.33/0.61 ( ( ( abs_abs_int @ X )
% 0.33/0.61 = ( abs_abs_int @ Y ) )
% 0.33/0.61 = ( ( X = Y )
% 0.33/0.61 | ( X
% 0.33/0.61 = ( uminus_uminus_int @ Y ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_iff
% 0.33/0.61 thf(fact_5611_abs__eq__iff,axiom,
% 0.33/0.61 ! [X: rat,Y: rat] :
% 0.33/0.61 ( ( ( abs_abs_rat @ X )
% 0.33/0.61 = ( abs_abs_rat @ Y ) )
% 0.33/0.61 = ( ( X = Y )
% 0.33/0.61 | ( X
% 0.33/0.61 = ( uminus_uminus_rat @ Y ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_iff
% 0.33/0.61 thf(fact_5612_abs__eq__iff,axiom,
% 0.33/0.61 ! [X: code_integer,Y: code_integer] :
% 0.33/0.61 ( ( ( abs_abs_Code_integer @ X )
% 0.33/0.61 = ( abs_abs_Code_integer @ Y ) )
% 0.33/0.61 = ( ( X = Y )
% 0.33/0.61 | ( X
% 0.33/0.61 = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_iff
% 0.33/0.61 thf(fact_5613_power__abs,axiom,
% 0.33/0.61 ! [A: code_integer,N: nat] :
% 0.33/0.61 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.33/0.61 = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % power_abs
% 0.33/0.61 thf(fact_5614_power__abs,axiom,
% 0.33/0.61 ! [A: rat,N: nat] :
% 0.33/0.61 ( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
% 0.33/0.61 = ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % power_abs
% 0.33/0.61 thf(fact_5615_power__abs,axiom,
% 0.33/0.61 ! [A: real,N: nat] :
% 0.33/0.61 ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
% 0.33/0.61 = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % power_abs
% 0.33/0.61 thf(fact_5616_power__abs,axiom,
% 0.33/0.61 ! [A: int,N: nat] :
% 0.33/0.61 ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
% 0.33/0.61 = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % power_abs
% 0.33/0.61 thf(fact_5617_dvd__if__abs__eq,axiom,
% 0.33/0.61 ! [L: real,K: real] :
% 0.33/0.61 ( ( ( abs_abs_real @ L )
% 0.33/0.61 = ( abs_abs_real @ K ) )
% 0.33/0.61 => ( dvd_dvd_real @ L @ K ) ) ).
% 0.33/0.61
% 0.33/0.61 % dvd_if_abs_eq
% 0.33/0.61 thf(fact_5618_dvd__if__abs__eq,axiom,
% 0.33/0.61 ! [L: int,K: int] :
% 0.33/0.61 ( ( ( abs_abs_int @ L )
% 0.33/0.61 = ( abs_abs_int @ K ) )
% 0.33/0.61 => ( dvd_dvd_int @ L @ K ) ) ).
% 0.33/0.61
% 0.33/0.61 % dvd_if_abs_eq
% 0.33/0.61 thf(fact_5619_dvd__if__abs__eq,axiom,
% 0.33/0.61 ! [L: code_integer,K: code_integer] :
% 0.33/0.61 ( ( ( abs_abs_Code_integer @ L )
% 0.33/0.61 = ( abs_abs_Code_integer @ K ) )
% 0.33/0.61 => ( dvd_dvd_Code_integer @ L @ K ) ) ).
% 0.33/0.61
% 0.33/0.61 % dvd_if_abs_eq
% 0.33/0.61 thf(fact_5620_dvd__if__abs__eq,axiom,
% 0.33/0.61 ! [L: rat,K: rat] :
% 0.33/0.61 ( ( ( abs_abs_rat @ L )
% 0.33/0.61 = ( abs_abs_rat @ K ) )
% 0.33/0.61 => ( dvd_dvd_rat @ L @ K ) ) ).
% 0.33/0.61
% 0.33/0.61 % dvd_if_abs_eq
% 0.33/0.61 thf(fact_5621_powr__powr,axiom,
% 0.33/0.61 ! [X: real,A: real,B: real] :
% 0.33/0.61 ( ( powr_real @ ( powr_real @ X @ A ) @ B )
% 0.33/0.61 = ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_powr
% 0.33/0.61 thf(fact_5622_real__sgn__eq,axiom,
% 0.33/0.61 ( sgn_sgn_real
% 0.33/0.61 = ( ^ [X3: real] : ( divide_divide_real @ X3 @ ( abs_abs_real @ X3 ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % real_sgn_eq
% 0.33/0.61 thf(fact_5623_uminus__int__code_I1_J,axiom,
% 0.33/0.61 ( ( uminus_uminus_int @ zero_zero_int )
% 0.33/0.61 = zero_zero_int ) ).
% 0.33/0.61
% 0.33/0.61 % uminus_int_code(1)
% 0.33/0.61 thf(fact_5624_int__cases2,axiom,
% 0.33/0.61 ! [Z2: int] :
% 0.33/0.61 ( ! [N2: nat] :
% 0.33/0.61 ( Z2
% 0.33/0.61 != ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.61 => ~ ! [N2: nat] :
% 0.33/0.61 ( Z2
% 0.33/0.61 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % int_cases2
% 0.33/0.61 thf(fact_5625_uminus__dvd__conv_I1_J,axiom,
% 0.33/0.61 ( dvd_dvd_int
% 0.33/0.61 = ( ^ [D4: int] : ( dvd_dvd_int @ ( uminus_uminus_int @ D4 ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % uminus_dvd_conv(1)
% 0.33/0.61 thf(fact_5626_uminus__dvd__conv_I2_J,axiom,
% 0.33/0.61 ( dvd_dvd_int
% 0.33/0.61 = ( ^ [D4: int,T2: int] : ( dvd_dvd_int @ D4 @ ( uminus_uminus_int @ T2 ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % uminus_dvd_conv(2)
% 0.33/0.61 thf(fact_5627_take__bit__minus,axiom,
% 0.33/0.61 ! [N: nat,K: int] :
% 0.33/0.61 ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.33/0.61 = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % take_bit_minus
% 0.33/0.61 thf(fact_5628_signed__take__bit__minus,axiom,
% 0.33/0.61 ! [N: nat,K: int] :
% 0.33/0.61 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
% 0.33/0.61 = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % signed_take_bit_minus
% 0.33/0.61 thf(fact_5629_abs__ge__zero,axiom,
% 0.33/0.61 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_ge_zero
% 0.33/0.61 thf(fact_5630_abs__ge__zero,axiom,
% 0.33/0.61 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_ge_zero
% 0.33/0.61 thf(fact_5631_abs__ge__zero,axiom,
% 0.33/0.61 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_ge_zero
% 0.33/0.61 thf(fact_5632_abs__ge__zero,axiom,
% 0.33/0.61 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_ge_zero
% 0.33/0.61 thf(fact_5633_floor__mono,axiom,
% 0.33/0.61 ! [X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.61 => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_mono
% 0.33/0.61 thf(fact_5634_floor__mono,axiom,
% 0.33/0.61 ! [X: rat,Y: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ X @ Y )
% 0.33/0.61 => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_mono
% 0.33/0.61 thf(fact_5635_abs__not__less__zero,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 0.33/0.61
% 0.33/0.61 % abs_not_less_zero
% 0.33/0.61 thf(fact_5636_abs__not__less__zero,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 0.33/0.61
% 0.33/0.61 % abs_not_less_zero
% 0.33/0.61 thf(fact_5637_abs__not__less__zero,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 0.33/0.61
% 0.33/0.61 % abs_not_less_zero
% 0.33/0.61 thf(fact_5638_abs__not__less__zero,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 0.33/0.61
% 0.33/0.61 % abs_not_less_zero
% 0.33/0.61 thf(fact_5639_abs__of__pos,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 0.33/0.61 => ( ( abs_abs_Code_integer @ A )
% 0.33/0.61 = A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_pos
% 0.33/0.61 thf(fact_5640_abs__of__pos,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.61 => ( ( abs_abs_real @ A )
% 0.33/0.61 = A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_pos
% 0.33/0.61 thf(fact_5641_abs__of__pos,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.61 => ( ( abs_abs_rat @ A )
% 0.33/0.61 = A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_pos
% 0.33/0.61 thf(fact_5642_abs__of__pos,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.61 => ( ( abs_abs_int @ A )
% 0.33/0.61 = A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_pos
% 0.33/0.61 thf(fact_5643_floor__less__cancel,axiom,
% 0.33/0.61 ! [X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) )
% 0.33/0.61 => ( ord_less_real @ X @ Y ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_less_cancel
% 0.33/0.61 thf(fact_5644_floor__less__cancel,axiom,
% 0.33/0.61 ! [X: rat,Y: rat] :
% 0.33/0.61 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) )
% 0.33/0.61 => ( ord_less_rat @ X @ Y ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_less_cancel
% 0.33/0.61 thf(fact_5645_abs__triangle__ineq,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq
% 0.33/0.61 thf(fact_5646_abs__triangle__ineq,axiom,
% 0.33/0.61 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq
% 0.33/0.61 thf(fact_5647_abs__triangle__ineq,axiom,
% 0.33/0.61 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq
% 0.33/0.61 thf(fact_5648_abs__triangle__ineq,axiom,
% 0.33/0.61 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq
% 0.33/0.61 thf(fact_5649_abs__mult__less,axiom,
% 0.33/0.61 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 0.33/0.61 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 0.33/0.61 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
% 0.33/0.61 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_less
% 0.33/0.61 thf(fact_5650_abs__mult__less,axiom,
% 0.33/0.61 ! [A: real,C: real,B: real,D: real] :
% 0.33/0.61 ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 0.33/0.61 => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
% 0.33/0.61 => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_less
% 0.33/0.61 thf(fact_5651_abs__mult__less,axiom,
% 0.33/0.61 ! [A: rat,C: rat,B: rat,D: rat] :
% 0.33/0.61 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 0.33/0.61 => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
% 0.33/0.61 => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_less
% 0.33/0.61 thf(fact_5652_abs__mult__less,axiom,
% 0.33/0.61 ! [A: int,C: int,B: int,D: int] :
% 0.33/0.61 ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 0.33/0.61 => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
% 0.33/0.61 => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_less
% 0.33/0.61 thf(fact_5653_abs__triangle__ineq2,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq2
% 0.33/0.61 thf(fact_5654_abs__triangle__ineq2,axiom,
% 0.33/0.61 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq2
% 0.33/0.61 thf(fact_5655_abs__triangle__ineq2,axiom,
% 0.33/0.61 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq2
% 0.33/0.61 thf(fact_5656_abs__triangle__ineq2,axiom,
% 0.33/0.61 ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq2
% 0.33/0.61 thf(fact_5657_abs__triangle__ineq3,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq3
% 0.33/0.61 thf(fact_5658_abs__triangle__ineq3,axiom,
% 0.33/0.61 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq3
% 0.33/0.61 thf(fact_5659_abs__triangle__ineq3,axiom,
% 0.33/0.61 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq3
% 0.33/0.61 thf(fact_5660_abs__triangle__ineq3,axiom,
% 0.33/0.61 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq3
% 0.33/0.61 thf(fact_5661_abs__triangle__ineq2__sym,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq2_sym
% 0.33/0.61 thf(fact_5662_abs__triangle__ineq2__sym,axiom,
% 0.33/0.61 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq2_sym
% 0.33/0.61 thf(fact_5663_abs__triangle__ineq2__sym,axiom,
% 0.33/0.61 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq2_sym
% 0.33/0.61 thf(fact_5664_abs__triangle__ineq2__sym,axiom,
% 0.33/0.61 ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq2_sym
% 0.33/0.61 thf(fact_5665_abs__leI,axiom,
% 0.33/0.61 ! [A: real,B: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ A @ B )
% 0.33/0.61 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.33/0.61 => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_leI
% 0.33/0.61 thf(fact_5666_abs__leI,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer] :
% 0.33/0.61 ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.33/0.61 => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.33/0.61 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_leI
% 0.33/0.61 thf(fact_5667_abs__leI,axiom,
% 0.33/0.61 ! [A: rat,B: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.61 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.33/0.61 => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_leI
% 0.33/0.61 thf(fact_5668_abs__leI,axiom,
% 0.33/0.61 ! [A: int,B: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ A @ B )
% 0.33/0.61 => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.61 => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_leI
% 0.33/0.61 thf(fact_5669_abs__le__D2,axiom,
% 0.33/0.61 ! [A: real,B: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.33/0.61 => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_D2
% 0.33/0.61 thf(fact_5670_abs__le__D2,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer] :
% 0.33/0.61 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.33/0.61 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_D2
% 0.33/0.61 thf(fact_5671_abs__le__D2,axiom,
% 0.33/0.61 ! [A: rat,B: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.33/0.61 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_D2
% 0.33/0.61 thf(fact_5672_abs__le__D2,axiom,
% 0.33/0.61 ! [A: int,B: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.33/0.61 => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_D2
% 0.33/0.61 thf(fact_5673_abs__le__iff,axiom,
% 0.33/0.61 ! [A: real,B: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.33/0.61 = ( ( ord_less_eq_real @ A @ B )
% 0.33/0.61 & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_iff
% 0.33/0.61 thf(fact_5674_abs__le__iff,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer] :
% 0.33/0.61 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.33/0.61 = ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.33/0.61 & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_iff
% 0.33/0.61 thf(fact_5675_abs__le__iff,axiom,
% 0.33/0.61 ! [A: rat,B: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.33/0.61 = ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.61 & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_iff
% 0.33/0.61 thf(fact_5676_abs__le__iff,axiom,
% 0.33/0.61 ! [A: int,B: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.33/0.61 = ( ( ord_less_eq_int @ A @ B )
% 0.33/0.61 & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_iff
% 0.33/0.61 thf(fact_5677_abs__ge__minus__self,axiom,
% 0.33/0.61 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_ge_minus_self
% 0.33/0.61 thf(fact_5678_abs__ge__minus__self,axiom,
% 0.33/0.61 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_ge_minus_self
% 0.33/0.61 thf(fact_5679_abs__ge__minus__self,axiom,
% 0.33/0.61 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_ge_minus_self
% 0.33/0.61 thf(fact_5680_abs__ge__minus__self,axiom,
% 0.33/0.61 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_ge_minus_self
% 0.33/0.61 thf(fact_5681_nonzero__abs__divide,axiom,
% 0.33/0.61 ! [B: real,A: real] :
% 0.33/0.61 ( ( B != zero_zero_real )
% 0.33/0.61 => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.61 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % nonzero_abs_divide
% 0.33/0.61 thf(fact_5682_nonzero__abs__divide,axiom,
% 0.33/0.61 ! [B: rat,A: rat] :
% 0.33/0.61 ( ( B != zero_zero_rat )
% 0.33/0.61 => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 0.33/0.61 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % nonzero_abs_divide
% 0.33/0.61 thf(fact_5683_abs__less__iff,axiom,
% 0.33/0.61 ! [A: real,B: real] :
% 0.33/0.61 ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 0.33/0.61 = ( ( ord_less_real @ A @ B )
% 0.33/0.61 & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_less_iff
% 0.33/0.61 thf(fact_5684_abs__less__iff,axiom,
% 0.33/0.61 ! [A: int,B: int] :
% 0.33/0.61 ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
% 0.33/0.61 = ( ( ord_less_int @ A @ B )
% 0.33/0.61 & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_less_iff
% 0.33/0.61 thf(fact_5685_abs__less__iff,axiom,
% 0.33/0.61 ! [A: rat,B: rat] :
% 0.33/0.61 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
% 0.33/0.61 = ( ( ord_less_rat @ A @ B )
% 0.33/0.61 & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_less_iff
% 0.33/0.61 thf(fact_5686_abs__less__iff,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer] :
% 0.33/0.61 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.33/0.61 = ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.33/0.61 & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_less_iff
% 0.33/0.61 thf(fact_5687_powr__less__cancel,axiom,
% 0.33/0.61 ! [X: real,A: real,B: real] :
% 0.33/0.61 ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 0.33/0.61 => ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.61 => ( ord_less_real @ A @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_less_cancel
% 0.33/0.61 thf(fact_5688_powr__less__mono,axiom,
% 0.33/0.61 ! [A: real,B: real,X: real] :
% 0.33/0.61 ( ( ord_less_real @ A @ B )
% 0.33/0.61 => ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.61 => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_less_mono
% 0.33/0.61 thf(fact_5689_powr__mono,axiom,
% 0.33/0.61 ! [A: real,B: real,X: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ A @ B )
% 0.33/0.61 => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.33/0.61 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_mono
% 0.33/0.61 thf(fact_5690_ceiling__def,axiom,
% 0.33/0.61 ( archim7802044766580827645g_real
% 0.33/0.61 = ( ^ [X3: real] : ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % ceiling_def
% 0.33/0.61 thf(fact_5691_ceiling__def,axiom,
% 0.33/0.61 ( archim2889992004027027881ng_rat
% 0.33/0.61 = ( ^ [X3: rat] : ( uminus_uminus_int @ ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ X3 ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % ceiling_def
% 0.33/0.61 thf(fact_5692_floor__minus,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X ) )
% 0.33/0.61 = ( uminus_uminus_int @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_minus
% 0.33/0.61 thf(fact_5693_floor__minus,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ X ) )
% 0.33/0.61 = ( uminus_uminus_int @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_minus
% 0.33/0.61 thf(fact_5694_ceiling__minus,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ X ) )
% 0.33/0.61 = ( uminus_uminus_int @ ( archim6058952711729229775r_real @ X ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % ceiling_minus
% 0.33/0.61 thf(fact_5695_ceiling__minus,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ X ) )
% 0.33/0.61 = ( uminus_uminus_int @ ( archim3151403230148437115or_rat @ X ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % ceiling_minus
% 0.33/0.61 thf(fact_5696_linordered__idom__class_Oabs__sgn,axiom,
% 0.33/0.61 ( abs_abs_Code_integer
% 0.33/0.61 = ( ^ [K2: code_integer] : ( times_3573771949741848930nteger @ K2 @ ( sgn_sgn_Code_integer @ K2 ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % linordered_idom_class.abs_sgn
% 0.33/0.61 thf(fact_5697_linordered__idom__class_Oabs__sgn,axiom,
% 0.33/0.61 ( abs_abs_real
% 0.33/0.61 = ( ^ [K2: real] : ( times_times_real @ K2 @ ( sgn_sgn_real @ K2 ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % linordered_idom_class.abs_sgn
% 0.33/0.61 thf(fact_5698_linordered__idom__class_Oabs__sgn,axiom,
% 0.33/0.61 ( abs_abs_rat
% 0.33/0.61 = ( ^ [K2: rat] : ( times_times_rat @ K2 @ ( sgn_sgn_rat @ K2 ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % linordered_idom_class.abs_sgn
% 0.33/0.61 thf(fact_5699_linordered__idom__class_Oabs__sgn,axiom,
% 0.33/0.61 ( abs_abs_int
% 0.33/0.61 = ( ^ [K2: int] : ( times_times_int @ K2 @ ( sgn_sgn_int @ K2 ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % linordered_idom_class.abs_sgn
% 0.33/0.61 thf(fact_5700_abs__mult__sgn,axiom,
% 0.33/0.61 ! [A: complex] :
% 0.33/0.61 ( ( times_times_complex @ ( abs_abs_complex @ A ) @ ( sgn_sgn_complex @ A ) )
% 0.33/0.61 = A ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_sgn
% 0.33/0.61 thf(fact_5701_abs__mult__sgn,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
% 0.33/0.61 = A ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_sgn
% 0.33/0.61 thf(fact_5702_abs__mult__sgn,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( sgn_sgn_real @ A ) )
% 0.33/0.61 = A ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_sgn
% 0.33/0.61 thf(fact_5703_abs__mult__sgn,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( sgn_sgn_rat @ A ) )
% 0.33/0.61 = A ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_sgn
% 0.33/0.61 thf(fact_5704_abs__mult__sgn,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
% 0.33/0.61 = A ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_sgn
% 0.33/0.61 thf(fact_5705_sgn__mult__abs,axiom,
% 0.33/0.61 ! [A: complex] :
% 0.33/0.61 ( ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( abs_abs_complex @ A ) )
% 0.33/0.61 = A ) ).
% 0.33/0.61
% 0.33/0.61 % sgn_mult_abs
% 0.33/0.61 thf(fact_5706_sgn__mult__abs,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 0.33/0.61 = A ) ).
% 0.33/0.61
% 0.33/0.61 % sgn_mult_abs
% 0.33/0.61 thf(fact_5707_sgn__mult__abs,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( abs_abs_real @ A ) )
% 0.33/0.61 = A ) ).
% 0.33/0.61
% 0.33/0.61 % sgn_mult_abs
% 0.33/0.61 thf(fact_5708_sgn__mult__abs,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( abs_abs_rat @ A ) )
% 0.33/0.61 = A ) ).
% 0.33/0.61
% 0.33/0.61 % sgn_mult_abs
% 0.33/0.61 thf(fact_5709_sgn__mult__abs,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
% 0.33/0.61 = A ) ).
% 0.33/0.61
% 0.33/0.61 % sgn_mult_abs
% 0.33/0.61 thf(fact_5710_mult__sgn__abs,axiom,
% 0.33/0.61 ! [X: code_integer] :
% 0.33/0.61 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ X ) @ ( abs_abs_Code_integer @ X ) )
% 0.33/0.61 = X ) ).
% 0.33/0.61
% 0.33/0.61 % mult_sgn_abs
% 0.33/0.61 thf(fact_5711_mult__sgn__abs,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( times_times_real @ ( sgn_sgn_real @ X ) @ ( abs_abs_real @ X ) )
% 0.33/0.61 = X ) ).
% 0.33/0.61
% 0.33/0.61 % mult_sgn_abs
% 0.33/0.61 thf(fact_5712_mult__sgn__abs,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ( times_times_rat @ ( sgn_sgn_rat @ X ) @ ( abs_abs_rat @ X ) )
% 0.33/0.61 = X ) ).
% 0.33/0.61
% 0.33/0.61 % mult_sgn_abs
% 0.33/0.61 thf(fact_5713_mult__sgn__abs,axiom,
% 0.33/0.61 ! [X: int] :
% 0.33/0.61 ( ( times_times_int @ ( sgn_sgn_int @ X ) @ ( abs_abs_int @ X ) )
% 0.33/0.61 = X ) ).
% 0.33/0.61
% 0.33/0.61 % mult_sgn_abs
% 0.33/0.61 thf(fact_5714_same__sgn__abs__add,axiom,
% 0.33/0.61 ! [B: code_integer,A: code_integer] :
% 0.33/0.61 ( ( ( sgn_sgn_Code_integer @ B )
% 0.33/0.61 = ( sgn_sgn_Code_integer @ A ) )
% 0.33/0.61 => ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.33/0.61 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % same_sgn_abs_add
% 0.33/0.61 thf(fact_5715_same__sgn__abs__add,axiom,
% 0.33/0.61 ! [B: real,A: real] :
% 0.33/0.61 ( ( ( sgn_sgn_real @ B )
% 0.33/0.61 = ( sgn_sgn_real @ A ) )
% 0.33/0.61 => ( ( abs_abs_real @ ( plus_plus_real @ A @ B ) )
% 0.33/0.61 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % same_sgn_abs_add
% 0.33/0.61 thf(fact_5716_same__sgn__abs__add,axiom,
% 0.33/0.61 ! [B: rat,A: rat] :
% 0.33/0.61 ( ( ( sgn_sgn_rat @ B )
% 0.33/0.61 = ( sgn_sgn_rat @ A ) )
% 0.33/0.61 => ( ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) )
% 0.33/0.61 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % same_sgn_abs_add
% 0.33/0.61 thf(fact_5717_same__sgn__abs__add,axiom,
% 0.33/0.61 ! [B: int,A: int] :
% 0.33/0.61 ( ( ( sgn_sgn_int @ B )
% 0.33/0.61 = ( sgn_sgn_int @ A ) )
% 0.33/0.61 => ( ( abs_abs_int @ ( plus_plus_int @ A @ B ) )
% 0.33/0.61 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % same_sgn_abs_add
% 0.33/0.61 thf(fact_5718_floor__le__ceiling,axiom,
% 0.33/0.61 ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_le_ceiling
% 0.33/0.61 thf(fact_5719_floor__le__ceiling,axiom,
% 0.33/0.61 ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim2889992004027027881ng_rat @ X ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_le_ceiling
% 0.33/0.61 thf(fact_5720_int__of__nat__induct,axiom,
% 0.33/0.61 ! [P: int > $o,Z2: int] :
% 0.33/0.61 ( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.61 => ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
% 0.33/0.61 => ( P @ Z2 ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % int_of_nat_induct
% 0.33/0.61 thf(fact_5721_int__cases,axiom,
% 0.33/0.61 ! [Z2: int] :
% 0.33/0.61 ( ! [N2: nat] :
% 0.33/0.61 ( Z2
% 0.33/0.61 != ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.61 => ~ ! [N2: nat] :
% 0.33/0.61 ( Z2
% 0.33/0.61 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % int_cases
% 0.33/0.61 thf(fact_5722_pos__zmult__eq__1__iff__lemma,axiom,
% 0.33/0.61 ! [M: int,N: int] :
% 0.33/0.61 ( ( ( times_times_int @ M @ N )
% 0.33/0.61 = one_one_int )
% 0.33/0.61 => ( ( M = one_one_int )
% 0.33/0.61 | ( M
% 0.33/0.61 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % pos_zmult_eq_1_iff_lemma
% 0.33/0.61 thf(fact_5723_zmult__eq__1__iff,axiom,
% 0.33/0.61 ! [M: int,N: int] :
% 0.33/0.61 ( ( ( times_times_int @ M @ N )
% 0.33/0.61 = one_one_int )
% 0.33/0.61 = ( ( ( M = one_one_int )
% 0.33/0.61 & ( N = one_one_int ) )
% 0.33/0.61 | ( ( M
% 0.33/0.61 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.61 & ( N
% 0.33/0.61 = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % zmult_eq_1_iff
% 0.33/0.61 thf(fact_5724_minus__int__code_I2_J,axiom,
% 0.33/0.61 ! [L: int] :
% 0.33/0.61 ( ( minus_minus_int @ zero_zero_int @ L )
% 0.33/0.61 = ( uminus_uminus_int @ L ) ) ).
% 0.33/0.61
% 0.33/0.61 % minus_int_code(2)
% 0.33/0.61 thf(fact_5725_not__int__zless__negative,axiom,
% 0.33/0.61 ! [N: nat,M: nat] :
% 0.33/0.61 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % not_int_zless_negative
% 0.33/0.61 thf(fact_5726_zmod__zminus1__not__zero,axiom,
% 0.33/0.61 ! [K: int,L: int] :
% 0.33/0.61 ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 0.33/0.61 != zero_zero_int )
% 0.33/0.61 => ( ( modulo_modulo_int @ K @ L )
% 0.33/0.61 != zero_zero_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % zmod_zminus1_not_zero
% 0.33/0.61 thf(fact_5727_zmod__zminus2__not__zero,axiom,
% 0.33/0.61 ! [K: int,L: int] :
% 0.33/0.61 ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
% 0.33/0.61 != zero_zero_int )
% 0.33/0.61 => ( ( modulo_modulo_int @ K @ L )
% 0.33/0.61 != zero_zero_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % zmod_zminus2_not_zero
% 0.33/0.61 thf(fact_5728_sin__coeff__Suc,axiom,
% 0.33/0.61 ! [N: nat] :
% 0.33/0.61 ( ( sin_coeff @ ( suc @ N ) )
% 0.33/0.61 = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % sin_coeff_Suc
% 0.33/0.61 thf(fact_5729_dense__eq0__I,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ! [E2: real] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.33/0.61 => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E2 ) )
% 0.33/0.61 => ( X = zero_zero_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % dense_eq0_I
% 0.33/0.61 thf(fact_5730_dense__eq0__I,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ! [E2: rat] :
% 0.33/0.61 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.33/0.61 => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E2 ) )
% 0.33/0.61 => ( X = zero_zero_rat ) ) ).
% 0.33/0.61
% 0.33/0.61 % dense_eq0_I
% 0.33/0.61 thf(fact_5731_abs__eq__mult,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer] :
% 0.33/0.61 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.33/0.61 | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 0.33/0.61 & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.33/0.61 | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 0.33/0.61 => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.33/0.61 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_mult
% 0.33/0.61 thf(fact_5732_abs__eq__mult,axiom,
% 0.33/0.61 ! [A: real,B: real] :
% 0.33/0.61 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.61 | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.33/0.61 & ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.61 | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.33/0.61 => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 0.33/0.61 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_mult
% 0.33/0.61 thf(fact_5733_abs__eq__mult,axiom,
% 0.33/0.61 ! [A: rat,B: rat] :
% 0.33/0.61 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.61 | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.33/0.61 & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.61 | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.33/0.61 => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 0.33/0.61 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_mult
% 0.33/0.61 thf(fact_5734_abs__eq__mult,axiom,
% 0.33/0.61 ! [A: int,B: int] :
% 0.33/0.61 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.61 | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.33/0.61 & ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.61 | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.33/0.61 => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 0.33/0.61 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_mult
% 0.33/0.61 thf(fact_5735_abs__mult__pos,axiom,
% 0.33/0.61 ! [X: code_integer,Y: code_integer] :
% 0.33/0.61 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 0.33/0.61 => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
% 0.33/0.61 = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_pos
% 0.33/0.61 thf(fact_5736_abs__mult__pos,axiom,
% 0.33/0.61 ! [X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
% 0.33/0.61 = ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_pos
% 0.33/0.61 thf(fact_5737_abs__mult__pos,axiom,
% 0.33/0.61 ! [X: rat,Y: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.33/0.61 => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
% 0.33/0.61 = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_pos
% 0.33/0.61 thf(fact_5738_abs__mult__pos,axiom,
% 0.33/0.61 ! [X: int,Y: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.61 => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
% 0.33/0.61 = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_mult_pos
% 0.33/0.61 thf(fact_5739_abs__eq__iff_H,axiom,
% 0.33/0.61 ! [A: real,B: real] :
% 0.33/0.61 ( ( ( abs_abs_real @ A )
% 0.33/0.61 = B )
% 0.33/0.61 = ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.61 & ( ( A = B )
% 0.33/0.61 | ( A
% 0.33/0.61 = ( uminus_uminus_real @ B ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_iff'
% 0.33/0.61 thf(fact_5740_abs__eq__iff_H,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer] :
% 0.33/0.61 ( ( ( abs_abs_Code_integer @ A )
% 0.33/0.61 = B )
% 0.33/0.61 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.33/0.61 & ( ( A = B )
% 0.33/0.61 | ( A
% 0.33/0.61 = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_iff'
% 0.33/0.61 thf(fact_5741_abs__eq__iff_H,axiom,
% 0.33/0.61 ! [A: rat,B: rat] :
% 0.33/0.61 ( ( ( abs_abs_rat @ A )
% 0.33/0.61 = B )
% 0.33/0.61 = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.61 & ( ( A = B )
% 0.33/0.61 | ( A
% 0.33/0.61 = ( uminus_uminus_rat @ B ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_iff'
% 0.33/0.61 thf(fact_5742_abs__eq__iff_H,axiom,
% 0.33/0.61 ! [A: int,B: int] :
% 0.33/0.61 ( ( ( abs_abs_int @ A )
% 0.33/0.61 = B )
% 0.33/0.61 = ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.61 & ( ( A = B )
% 0.33/0.61 | ( A
% 0.33/0.61 = ( uminus_uminus_int @ B ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_eq_iff'
% 0.33/0.61 thf(fact_5743_eq__abs__iff_H,axiom,
% 0.33/0.61 ! [A: real,B: real] :
% 0.33/0.61 ( ( A
% 0.33/0.61 = ( abs_abs_real @ B ) )
% 0.33/0.61 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.61 & ( ( B = A )
% 0.33/0.61 | ( B
% 0.33/0.61 = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % eq_abs_iff'
% 0.33/0.61 thf(fact_5744_eq__abs__iff_H,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer] :
% 0.33/0.61 ( ( A
% 0.33/0.61 = ( abs_abs_Code_integer @ B ) )
% 0.33/0.61 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.33/0.61 & ( ( B = A )
% 0.33/0.61 | ( B
% 0.33/0.61 = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % eq_abs_iff'
% 0.33/0.61 thf(fact_5745_eq__abs__iff_H,axiom,
% 0.33/0.61 ! [A: rat,B: rat] :
% 0.33/0.61 ( ( A
% 0.33/0.61 = ( abs_abs_rat @ B ) )
% 0.33/0.61 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.61 & ( ( B = A )
% 0.33/0.61 | ( B
% 0.33/0.61 = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % eq_abs_iff'
% 0.33/0.61 thf(fact_5746_eq__abs__iff_H,axiom,
% 0.33/0.61 ! [A: int,B: int] :
% 0.33/0.61 ( ( A
% 0.33/0.61 = ( abs_abs_int @ B ) )
% 0.33/0.61 = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.61 & ( ( B = A )
% 0.33/0.61 | ( B
% 0.33/0.61 = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % eq_abs_iff'
% 0.33/0.61 thf(fact_5747_abs__minus__le__zero,axiom,
% 0.33/0.61 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus_le_zero
% 0.33/0.61 thf(fact_5748_abs__minus__le__zero,axiom,
% 0.33/0.61 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus_le_zero
% 0.33/0.61 thf(fact_5749_abs__minus__le__zero,axiom,
% 0.33/0.61 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus_le_zero
% 0.33/0.61 thf(fact_5750_abs__minus__le__zero,axiom,
% 0.33/0.61 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 0.33/0.61
% 0.33/0.61 % abs_minus_le_zero
% 0.33/0.61 thf(fact_5751_zero__le__power__abs,axiom,
% 0.33/0.61 ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_le_power_abs
% 0.33/0.61 thf(fact_5752_zero__le__power__abs,axiom,
% 0.33/0.61 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_le_power_abs
% 0.33/0.61 thf(fact_5753_zero__le__power__abs,axiom,
% 0.33/0.61 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_le_power_abs
% 0.33/0.61 thf(fact_5754_zero__le__power__abs,axiom,
% 0.33/0.61 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % zero_le_power_abs
% 0.33/0.61 thf(fact_5755_abs__if,axiom,
% 0.33/0.61 ( abs_abs_real
% 0.33/0.61 = ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_if
% 0.33/0.61 thf(fact_5756_abs__if,axiom,
% 0.33/0.61 ( abs_abs_int
% 0.33/0.61 = ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_if
% 0.33/0.61 thf(fact_5757_abs__if,axiom,
% 0.33/0.61 ( abs_abs_rat
% 0.33/0.61 = ( ^ [A2: rat] : ( if_rat @ ( ord_less_rat @ A2 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A2 ) @ A2 ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_if
% 0.33/0.61 thf(fact_5758_abs__if,axiom,
% 0.33/0.61 ( abs_abs_Code_integer
% 0.33/0.61 = ( ^ [A2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A2 ) @ A2 ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_if
% 0.33/0.61 thf(fact_5759_abs__of__neg,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.61 => ( ( abs_abs_real @ A )
% 0.33/0.61 = ( uminus_uminus_real @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_neg
% 0.33/0.61 thf(fact_5760_abs__of__neg,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( ord_less_int @ A @ zero_zero_int )
% 0.33/0.61 => ( ( abs_abs_int @ A )
% 0.33/0.61 = ( uminus_uminus_int @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_neg
% 0.33/0.61 thf(fact_5761_abs__of__neg,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.61 => ( ( abs_abs_rat @ A )
% 0.33/0.61 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_neg
% 0.33/0.61 thf(fact_5762_abs__of__neg,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 0.33/0.61 => ( ( abs_abs_Code_integer @ A )
% 0.33/0.61 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_of_neg
% 0.33/0.61 thf(fact_5763_abs__if__raw,axiom,
% 0.33/0.61 ( abs_abs_real
% 0.33/0.61 = ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_if_raw
% 0.33/0.61 thf(fact_5764_abs__if__raw,axiom,
% 0.33/0.61 ( abs_abs_int
% 0.33/0.61 = ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_if_raw
% 0.33/0.61 thf(fact_5765_abs__if__raw,axiom,
% 0.33/0.61 ( abs_abs_rat
% 0.33/0.61 = ( ^ [A2: rat] : ( if_rat @ ( ord_less_rat @ A2 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A2 ) @ A2 ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_if_raw
% 0.33/0.61 thf(fact_5766_abs__if__raw,axiom,
% 0.33/0.61 ( abs_abs_Code_integer
% 0.33/0.61 = ( ^ [A2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A2 ) @ A2 ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_if_raw
% 0.33/0.61 thf(fact_5767_abs__div__pos,axiom,
% 0.33/0.61 ! [Y: real,X: real] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.61 => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
% 0.33/0.61 = ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_div_pos
% 0.33/0.61 thf(fact_5768_abs__div__pos,axiom,
% 0.33/0.61 ! [Y: rat,X: rat] :
% 0.33/0.61 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.33/0.61 => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
% 0.33/0.61 = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_div_pos
% 0.33/0.61 thf(fact_5769_abs__diff__triangle__ineq,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_diff_triangle_ineq
% 0.33/0.61 thf(fact_5770_abs__diff__triangle__ineq,axiom,
% 0.33/0.61 ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_diff_triangle_ineq
% 0.33/0.61 thf(fact_5771_abs__diff__triangle__ineq,axiom,
% 0.33/0.61 ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_diff_triangle_ineq
% 0.33/0.61 thf(fact_5772_abs__diff__triangle__ineq,axiom,
% 0.33/0.61 ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_diff_triangle_ineq
% 0.33/0.61 thf(fact_5773_abs__triangle__ineq4,axiom,
% 0.33/0.61 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq4
% 0.33/0.61 thf(fact_5774_abs__triangle__ineq4,axiom,
% 0.33/0.61 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq4
% 0.33/0.61 thf(fact_5775_abs__triangle__ineq4,axiom,
% 0.33/0.61 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq4
% 0.33/0.61 thf(fact_5776_abs__triangle__ineq4,axiom,
% 0.33/0.61 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_triangle_ineq4
% 0.33/0.61 thf(fact_5777_abs__diff__le__iff,axiom,
% 0.33/0.61 ! [X: code_integer,A: code_integer,R: code_integer] :
% 0.33/0.61 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R )
% 0.33/0.61 = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X )
% 0.33/0.61 & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_diff_le_iff
% 0.33/0.61 thf(fact_5778_abs__diff__le__iff,axiom,
% 0.33/0.61 ! [X: real,A: real,R: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
% 0.33/0.61 = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X )
% 0.33/0.61 & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_diff_le_iff
% 0.33/0.61 thf(fact_5779_abs__diff__le__iff,axiom,
% 0.33/0.61 ! [X: rat,A: rat,R: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R )
% 0.33/0.61 = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R ) @ X )
% 0.33/0.61 & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_diff_le_iff
% 0.33/0.61 thf(fact_5780_abs__diff__le__iff,axiom,
% 0.33/0.61 ! [X: int,A: int,R: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
% 0.33/0.61 = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R ) @ X )
% 0.33/0.61 & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_diff_le_iff
% 0.33/0.61 thf(fact_5781_abs__diff__less__iff,axiom,
% 0.33/0.61 ! [X: code_integer,A: code_integer,R: code_integer] :
% 0.33/0.61 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R )
% 0.33/0.61 = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X )
% 0.33/0.61 & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_diff_less_iff
% 0.33/0.61 thf(fact_5782_abs__diff__less__iff,axiom,
% 0.33/0.61 ! [X: real,A: real,R: real] :
% 0.33/0.61 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
% 0.33/0.61 = ( ( ord_less_real @ ( minus_minus_real @ A @ R ) @ X )
% 0.33/0.61 & ( ord_less_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_diff_less_iff
% 0.33/0.61 thf(fact_5783_abs__diff__less__iff,axiom,
% 0.33/0.61 ! [X: rat,A: rat,R: rat] :
% 0.33/0.61 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R )
% 0.33/0.61 = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R ) @ X )
% 0.33/0.61 & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_diff_less_iff
% 0.33/0.61 thf(fact_5784_abs__diff__less__iff,axiom,
% 0.33/0.61 ! [X: int,A: int,R: int] :
% 0.33/0.61 ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
% 0.33/0.61 = ( ( ord_less_int @ ( minus_minus_int @ A @ R ) @ X )
% 0.33/0.61 & ( ord_less_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_diff_less_iff
% 0.33/0.61 thf(fact_5785_le__floor__add,axiom,
% 0.33/0.61 ! [X: real,Y: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % le_floor_add
% 0.33/0.61 thf(fact_5786_le__floor__add,axiom,
% 0.33/0.61 ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % le_floor_add
% 0.33/0.61 thf(fact_5787_gr__one__powr,axiom,
% 0.33/0.61 ! [X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.61 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.61 => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % gr_one_powr
% 0.33/0.61 thf(fact_5788_powr__inj,axiom,
% 0.33/0.61 ! [A: real,X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.61 => ( ( A != one_one_real )
% 0.33/0.61 => ( ( ( powr_real @ A @ X )
% 0.33/0.61 = ( powr_real @ A @ Y ) )
% 0.33/0.61 = ( X = Y ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_inj
% 0.33/0.61 thf(fact_5789_ge__one__powr__ge__zero,axiom,
% 0.33/0.61 ! [X: real,A: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.33/0.61 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.61 => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % ge_one_powr_ge_zero
% 0.33/0.61 thf(fact_5790_powr__mono__both,axiom,
% 0.33/0.61 ! [A: real,B: real,X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.61 => ( ( ord_less_eq_real @ A @ B )
% 0.33/0.61 => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.33/0.61 => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.61 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_mono_both
% 0.33/0.61 thf(fact_5791_powr__le1,axiom,
% 0.33/0.61 ! [A: real,X: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.61 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.61 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_le1
% 0.33/0.61 thf(fact_5792_powr__divide,axiom,
% 0.33/0.61 ! [X: real,Y: real,A: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.61 => ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A )
% 0.33/0.61 = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_divide
% 0.33/0.61 thf(fact_5793_powr__mult,axiom,
% 0.33/0.61 ! [X: real,Y: real,A: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.61 => ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
% 0.33/0.61 = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_mult
% 0.33/0.61 thf(fact_5794_abs__sgn__eq,axiom,
% 0.33/0.61 ! [A: code_integer] :
% 0.33/0.61 ( ( ( A = zero_z3403309356797280102nteger )
% 0.33/0.61 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.33/0.61 = zero_z3403309356797280102nteger ) )
% 0.33/0.61 & ( ( A != zero_z3403309356797280102nteger )
% 0.33/0.61 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.33/0.61 = one_one_Code_integer ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_sgn_eq
% 0.33/0.61 thf(fact_5795_abs__sgn__eq,axiom,
% 0.33/0.61 ! [A: real] :
% 0.33/0.61 ( ( ( A = zero_zero_real )
% 0.33/0.61 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.33/0.61 = zero_zero_real ) )
% 0.33/0.61 & ( ( A != zero_zero_real )
% 0.33/0.61 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.33/0.61 = one_one_real ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_sgn_eq
% 0.33/0.61 thf(fact_5796_abs__sgn__eq,axiom,
% 0.33/0.61 ! [A: rat] :
% 0.33/0.61 ( ( ( A = zero_zero_rat )
% 0.33/0.61 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.33/0.61 = zero_zero_rat ) )
% 0.33/0.61 & ( ( A != zero_zero_rat )
% 0.33/0.61 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.33/0.61 = one_one_rat ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_sgn_eq
% 0.33/0.61 thf(fact_5797_abs__sgn__eq,axiom,
% 0.33/0.61 ! [A: int] :
% 0.33/0.61 ( ( ( A = zero_zero_int )
% 0.33/0.61 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.33/0.61 = zero_zero_int ) )
% 0.33/0.61 & ( ( A != zero_zero_int )
% 0.33/0.61 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.33/0.61 = one_one_int ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_sgn_eq
% 0.33/0.61 thf(fact_5798_divide__powr__uminus,axiom,
% 0.33/0.61 ! [A: real,B: real,C: real] :
% 0.33/0.61 ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
% 0.33/0.61 = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % divide_powr_uminus
% 0.33/0.61 thf(fact_5799_log__base__powr,axiom,
% 0.33/0.61 ! [A: real,B: real,X: real] :
% 0.33/0.61 ( ( A != zero_zero_real )
% 0.33/0.61 => ( ( log2 @ ( powr_real @ A @ B ) @ X )
% 0.33/0.61 = ( divide_divide_real @ ( log2 @ A @ X ) @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % log_base_powr
% 0.33/0.61 thf(fact_5800_ln__powr,axiom,
% 0.33/0.61 ! [X: real,Y: real] :
% 0.33/0.61 ( ( X != zero_zero_real )
% 0.33/0.61 => ( ( ln_ln_real @ ( powr_real @ X @ Y ) )
% 0.33/0.61 = ( times_times_real @ Y @ ( ln_ln_real @ X ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % ln_powr
% 0.33/0.61 thf(fact_5801_log__powr,axiom,
% 0.33/0.61 ! [X: real,B: real,Y: real] :
% 0.33/0.61 ( ( X != zero_zero_real )
% 0.33/0.61 => ( ( log2 @ B @ ( powr_real @ X @ Y ) )
% 0.33/0.61 = ( times_times_real @ Y @ ( log2 @ B @ X ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % log_powr
% 0.33/0.61 thf(fact_5802_abs__real__def,axiom,
% 0.33/0.61 ( abs_abs_real
% 0.33/0.61 = ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_real_def
% 0.33/0.61 thf(fact_5803_lemma__interval__lt,axiom,
% 0.33/0.61 ! [A: real,X: real,B: real] :
% 0.33/0.61 ( ( ord_less_real @ A @ X )
% 0.33/0.61 => ( ( ord_less_real @ X @ B )
% 0.33/0.61 => ? [D2: real] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.33/0.61 & ! [Y5: real] :
% 0.33/0.61 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D2 )
% 0.33/0.61 => ( ( ord_less_real @ A @ Y5 )
% 0.33/0.61 & ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % lemma_interval_lt
% 0.33/0.61 thf(fact_5804_powr__add,axiom,
% 0.33/0.61 ! [X: real,A: real,B: real] :
% 0.33/0.61 ( ( powr_real @ X @ ( plus_plus_real @ A @ B ) )
% 0.33/0.61 = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_add
% 0.33/0.61 thf(fact_5805_powr__diff,axiom,
% 0.33/0.61 ! [W: real,Z1: real,Z22: real] :
% 0.33/0.61 ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
% 0.33/0.61 = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_diff
% 0.33/0.61 thf(fact_5806_sin__bound__lemma,axiom,
% 0.33/0.61 ! [X: real,Y: real,U: real,V: real] :
% 0.33/0.61 ( ( X = Y )
% 0.33/0.61 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 0.33/0.61 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % sin_bound_lemma
% 0.33/0.61 thf(fact_5807_cos__coeff__Suc,axiom,
% 0.33/0.61 ! [N: nat] :
% 0.33/0.61 ( ( cos_coeff @ ( suc @ N ) )
% 0.33/0.61 = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % cos_coeff_Suc
% 0.33/0.61 thf(fact_5808_int__cases4,axiom,
% 0.33/0.61 ! [M: int] :
% 0.33/0.61 ( ! [N2: nat] :
% 0.33/0.61 ( M
% 0.33/0.61 != ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.61 => ~ ! [N2: nat] :
% 0.33/0.61 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.61 => ( M
% 0.33/0.61 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % int_cases4
% 0.33/0.61 thf(fact_5809_int__zle__neg,axiom,
% 0.33/0.61 ! [N: nat,M: nat] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.33/0.61 = ( ( N = zero_zero_nat )
% 0.33/0.61 & ( M = zero_zero_nat ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % int_zle_neg
% 0.33/0.61 thf(fact_5810_negative__zle__0,axiom,
% 0.33/0.61 ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% 0.33/0.61
% 0.33/0.61 % negative_zle_0
% 0.33/0.61 thf(fact_5811_nonpos__int__cases,axiom,
% 0.33/0.61 ! [K: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.33/0.61 => ~ ! [N2: nat] :
% 0.33/0.61 ( K
% 0.33/0.61 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % nonpos_int_cases
% 0.33/0.61 thf(fact_5812_zmod__zminus2__eq__if,axiom,
% 0.33/0.61 ! [A: int,B: int] :
% 0.33/0.61 ( ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.61 = zero_zero_int )
% 0.33/0.61 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.33/0.61 = zero_zero_int ) )
% 0.33/0.61 & ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.61 != zero_zero_int )
% 0.33/0.61 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.33/0.61 = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % zmod_zminus2_eq_if
% 0.33/0.61 thf(fact_5813_zmod__zminus1__eq__if,axiom,
% 0.33/0.61 ! [A: int,B: int] :
% 0.33/0.61 ( ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.61 = zero_zero_int )
% 0.33/0.61 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.61 = zero_zero_int ) )
% 0.33/0.61 & ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.61 != zero_zero_int )
% 0.33/0.61 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.61 = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % zmod_zminus1_eq_if
% 0.33/0.61 thf(fact_5814_abs__add__one__gt__zero,axiom,
% 0.33/0.61 ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_add_one_gt_zero
% 0.33/0.61 thf(fact_5815_abs__add__one__gt__zero,axiom,
% 0.33/0.61 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_add_one_gt_zero
% 0.33/0.61 thf(fact_5816_abs__add__one__gt__zero,axiom,
% 0.33/0.61 ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_add_one_gt_zero
% 0.33/0.61 thf(fact_5817_abs__add__one__gt__zero,axiom,
% 0.33/0.61 ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_add_one_gt_zero
% 0.33/0.61 thf(fact_5818_one__add__floor,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 0.33/0.61 = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % one_add_floor
% 0.33/0.61 thf(fact_5819_one__add__floor,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 0.33/0.61 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ one_one_rat ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % one_add_floor
% 0.33/0.61 thf(fact_5820_floor__divide__of__nat__eq,axiom,
% 0.33/0.61 ! [M: nat,N: nat] :
% 0.33/0.61 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.33/0.61 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_divide_of_nat_eq
% 0.33/0.61 thf(fact_5821_floor__divide__of__nat__eq,axiom,
% 0.33/0.61 ! [M: nat,N: nat] :
% 0.33/0.61 ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) )
% 0.33/0.61 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_divide_of_nat_eq
% 0.33/0.61 thf(fact_5822_powr__realpow,axiom,
% 0.33/0.61 ! [X: real,N: nat] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 0.33/0.61 = ( power_power_real @ X @ N ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_realpow
% 0.33/0.61 thf(fact_5823_less__log__iff,axiom,
% 0.33/0.61 ! [B: real,X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.61 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( ord_less_real @ Y @ ( log2 @ B @ X ) )
% 0.33/0.61 = ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % less_log_iff
% 0.33/0.61 thf(fact_5824_log__less__iff,axiom,
% 0.33/0.61 ! [B: real,X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.61 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( ord_less_real @ ( log2 @ B @ X ) @ Y )
% 0.33/0.61 = ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % log_less_iff
% 0.33/0.61 thf(fact_5825_less__powr__iff,axiom,
% 0.33/0.61 ! [B: real,X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.61 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( ord_less_real @ X @ ( powr_real @ B @ Y ) )
% 0.33/0.61 = ( ord_less_real @ ( log2 @ B @ X ) @ Y ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % less_powr_iff
% 0.33/0.61 thf(fact_5826_powr__less__iff,axiom,
% 0.33/0.61 ! [B: real,X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.61 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X )
% 0.33/0.61 = ( ord_less_real @ Y @ ( log2 @ B @ X ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_less_iff
% 0.33/0.61 thf(fact_5827_ceiling__diff__floor__le__1,axiom,
% 0.33/0.61 ! [X: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) @ one_one_int ) ).
% 0.33/0.61
% 0.33/0.61 % ceiling_diff_floor_le_1
% 0.33/0.61 thf(fact_5828_ceiling__diff__floor__le__1,axiom,
% 0.33/0.61 ! [X: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) @ one_one_int ) ).
% 0.33/0.61
% 0.33/0.61 % ceiling_diff_floor_le_1
% 0.33/0.61 thf(fact_5829_sgn__power__injE,axiom,
% 0.33/0.61 ! [A: real,N: nat,X: real,B: real] :
% 0.33/0.61 ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.33/0.61 = X )
% 0.33/0.61 => ( ( X
% 0.33/0.61 = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
% 0.33/0.61 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.61 => ( A = B ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % sgn_power_injE
% 0.33/0.61 thf(fact_5830_lemma__interval,axiom,
% 0.33/0.61 ! [A: real,X: real,B: real] :
% 0.33/0.61 ( ( ord_less_real @ A @ X )
% 0.33/0.61 => ( ( ord_less_real @ X @ B )
% 0.33/0.61 => ? [D2: real] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.33/0.61 & ! [Y5: real] :
% 0.33/0.61 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D2 )
% 0.33/0.61 => ( ( ord_less_eq_real @ A @ Y5 )
% 0.33/0.61 & ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % lemma_interval
% 0.33/0.61 thf(fact_5831_norm__triangle__ineq3,axiom,
% 0.33/0.61 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % norm_triangle_ineq3
% 0.33/0.61 thf(fact_5832_norm__triangle__ineq3,axiom,
% 0.33/0.61 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % norm_triangle_ineq3
% 0.33/0.61 thf(fact_5833_int__cases3,axiom,
% 0.33/0.61 ! [K: int] :
% 0.33/0.61 ( ( K != zero_zero_int )
% 0.33/0.61 => ( ! [N2: nat] :
% 0.33/0.61 ( ( K
% 0.33/0.61 = ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.61 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 0.33/0.61 => ~ ! [N2: nat] :
% 0.33/0.61 ( ( K
% 0.33/0.61 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.33/0.61 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % int_cases3
% 0.33/0.61 thf(fact_5834_not__zle__0__negative,axiom,
% 0.33/0.61 ! [N: nat] :
% 0.33/0.61 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % not_zle_0_negative
% 0.33/0.61 thf(fact_5835_negD,axiom,
% 0.33/0.61 ! [X: int] :
% 0.33/0.61 ( ( ord_less_int @ X @ zero_zero_int )
% 0.33/0.61 => ? [N2: nat] :
% 0.33/0.61 ( X
% 0.33/0.61 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % negD
% 0.33/0.61 thf(fact_5836_negative__zless__0,axiom,
% 0.33/0.61 ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 0.33/0.61
% 0.33/0.61 % negative_zless_0
% 0.33/0.61 thf(fact_5837_verit__less__mono__div__int2,axiom,
% 0.33/0.61 ! [A3: int,B5: int,N: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ A3 @ B5 )
% 0.33/0.61 => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 0.33/0.61 => ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A3 @ N ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % verit_less_mono_div_int2
% 0.33/0.61 thf(fact_5838_div__eq__minus1,axiom,
% 0.33/0.61 ! [B: int] :
% 0.33/0.61 ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.61 => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 0.33/0.61 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % div_eq_minus1
% 0.33/0.61 thf(fact_5839_zsgn__def,axiom,
% 0.33/0.61 ( sgn_sgn_int
% 0.33/0.61 = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % zsgn_def
% 0.33/0.61 thf(fact_5840_powr__minus__divide,axiom,
% 0.33/0.61 ! [X: real,A: real] :
% 0.33/0.61 ( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
% 0.33/0.61 = ( divide_divide_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_minus_divide
% 0.33/0.61 thf(fact_5841_le__mult__floor,axiom,
% 0.33/0.61 ! [A: real,B: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.61 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.61 => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % le_mult_floor
% 0.33/0.61 thf(fact_5842_le__mult__floor,axiom,
% 0.33/0.61 ! [A: rat,B: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.61 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.61 => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % le_mult_floor
% 0.33/0.61 thf(fact_5843_abs__le__square__iff,axiom,
% 0.33/0.61 ! [X: code_integer,Y: code_integer] :
% 0.33/0.61 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
% 0.33/0.61 = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_square_iff
% 0.33/0.61 thf(fact_5844_abs__le__square__iff,axiom,
% 0.33/0.61 ! [X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
% 0.33/0.61 = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_square_iff
% 0.33/0.61 thf(fact_5845_abs__le__square__iff,axiom,
% 0.33/0.61 ! [X: rat,Y: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
% 0.33/0.61 = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_square_iff
% 0.33/0.61 thf(fact_5846_abs__le__square__iff,axiom,
% 0.33/0.61 ! [X: int,Y: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
% 0.33/0.61 = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_le_square_iff
% 0.33/0.61 thf(fact_5847_abs__square__eq__1,axiom,
% 0.33/0.61 ! [X: code_integer] :
% 0.33/0.61 ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.61 = one_one_Code_integer )
% 0.33/0.61 = ( ( abs_abs_Code_integer @ X )
% 0.33/0.61 = one_one_Code_integer ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_square_eq_1
% 0.33/0.61 thf(fact_5848_abs__square__eq__1,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.61 = one_one_rat )
% 0.33/0.61 = ( ( abs_abs_rat @ X )
% 0.33/0.61 = one_one_rat ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_square_eq_1
% 0.33/0.61 thf(fact_5849_abs__square__eq__1,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.61 = one_one_real )
% 0.33/0.61 = ( ( abs_abs_real @ X )
% 0.33/0.61 = one_one_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_square_eq_1
% 0.33/0.61 thf(fact_5850_abs__square__eq__1,axiom,
% 0.33/0.61 ! [X: int] :
% 0.33/0.61 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.61 = one_one_int )
% 0.33/0.61 = ( ( abs_abs_int @ X )
% 0.33/0.61 = one_one_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_square_eq_1
% 0.33/0.61 thf(fact_5851_power__even__abs,axiom,
% 0.33/0.61 ! [N: nat,A: code_integer] :
% 0.33/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.61 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
% 0.33/0.61 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power_even_abs
% 0.33/0.61 thf(fact_5852_power__even__abs,axiom,
% 0.33/0.61 ! [N: nat,A: rat] :
% 0.33/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.61 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
% 0.33/0.61 = ( power_power_rat @ A @ N ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power_even_abs
% 0.33/0.61 thf(fact_5853_power__even__abs,axiom,
% 0.33/0.61 ! [N: nat,A: real] :
% 0.33/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.61 => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
% 0.33/0.61 = ( power_power_real @ A @ N ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power_even_abs
% 0.33/0.61 thf(fact_5854_power__even__abs,axiom,
% 0.33/0.61 ! [N: nat,A: int] :
% 0.33/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.61 => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
% 0.33/0.61 = ( power_power_int @ A @ N ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power_even_abs
% 0.33/0.61 thf(fact_5855_powr__neg__one,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.61 = ( divide_divide_real @ one_one_real @ X ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_neg_one
% 0.33/0.61 thf(fact_5856_powr__mult__base,axiom,
% 0.33/0.61 ! [X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
% 0.33/0.61 = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_mult_base
% 0.33/0.61 thf(fact_5857_powr__le__iff,axiom,
% 0.33/0.61 ! [B: real,X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.61 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
% 0.33/0.61 = ( ord_less_eq_real @ Y @ ( log2 @ B @ X ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_le_iff
% 0.33/0.61 thf(fact_5858_le__powr__iff,axiom,
% 0.33/0.61 ! [B: real,X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.61 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
% 0.33/0.61 = ( ord_less_eq_real @ ( log2 @ B @ X ) @ Y ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % le_powr_iff
% 0.33/0.61 thf(fact_5859_log__le__iff,axiom,
% 0.33/0.61 ! [B: real,X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.61 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( ord_less_eq_real @ ( log2 @ B @ X ) @ Y )
% 0.33/0.61 = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % log_le_iff
% 0.33/0.61 thf(fact_5860_le__log__iff,axiom,
% 0.33/0.61 ! [B: real,X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.61 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( ord_less_eq_real @ Y @ ( log2 @ B @ X ) )
% 0.33/0.61 = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % le_log_iff
% 0.33/0.61 thf(fact_5861_neg__int__cases,axiom,
% 0.33/0.61 ! [K: int] :
% 0.33/0.61 ( ( ord_less_int @ K @ zero_zero_int )
% 0.33/0.61 => ~ ! [N2: nat] :
% 0.33/0.61 ( ( K
% 0.33/0.61 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.33/0.61 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % neg_int_cases
% 0.33/0.61 thf(fact_5862_minus__mod__int__eq,axiom,
% 0.33/0.61 ! [L: int,K: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ zero_zero_int @ L )
% 0.33/0.61 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 0.33/0.61 = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % minus_mod_int_eq
% 0.33/0.61 thf(fact_5863_zmod__minus1,axiom,
% 0.33/0.61 ! [B: int] :
% 0.33/0.61 ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.61 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 0.33/0.61 = ( minus_minus_int @ B @ one_one_int ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % zmod_minus1
% 0.33/0.61 thf(fact_5864_zdiv__zminus1__eq__if,axiom,
% 0.33/0.61 ! [B: int,A: int] :
% 0.33/0.61 ( ( B != zero_zero_int )
% 0.33/0.61 => ( ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.61 = zero_zero_int )
% 0.33/0.61 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.61 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 0.33/0.61 & ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.61 != zero_zero_int )
% 0.33/0.61 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.61 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % zdiv_zminus1_eq_if
% 0.33/0.61 thf(fact_5865_zdiv__zminus2__eq__if,axiom,
% 0.33/0.61 ! [B: int,A: int] :
% 0.33/0.61 ( ( B != zero_zero_int )
% 0.33/0.61 => ( ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.61 = zero_zero_int )
% 0.33/0.61 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.33/0.61 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 0.33/0.61 & ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.61 != zero_zero_int )
% 0.33/0.61 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.33/0.61 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % zdiv_zminus2_eq_if
% 0.33/0.61 thf(fact_5866_power2__le__iff__abs__le,axiom,
% 0.33/0.61 ! [Y: code_integer,X: code_integer] :
% 0.33/0.61 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
% 0.33/0.61 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.61 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power2_le_iff_abs_le
% 0.33/0.61 thf(fact_5867_power2__le__iff__abs__le,axiom,
% 0.33/0.61 ! [Y: real,X: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.61 => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.61 = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power2_le_iff_abs_le
% 0.33/0.61 thf(fact_5868_power2__le__iff__abs__le,axiom,
% 0.33/0.61 ! [Y: rat,X: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.33/0.61 => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.61 = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power2_le_iff_abs_le
% 0.33/0.61 thf(fact_5869_power2__le__iff__abs__le,axiom,
% 0.33/0.61 ! [Y: int,X: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.61 => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.61 = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power2_le_iff_abs_le
% 0.33/0.61 thf(fact_5870_abs__square__le__1,axiom,
% 0.33/0.61 ! [X: code_integer] :
% 0.33/0.61 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.33/0.61 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_square_le_1
% 0.33/0.61 thf(fact_5871_abs__square__le__1,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.33/0.61 = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_square_le_1
% 0.33/0.61 thf(fact_5872_abs__square__le__1,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.33/0.61 = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_square_le_1
% 0.33/0.61 thf(fact_5873_abs__square__le__1,axiom,
% 0.33/0.61 ! [X: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.33/0.61 = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_square_le_1
% 0.33/0.61 thf(fact_5874_abs__square__less__1,axiom,
% 0.33/0.61 ! [X: code_integer] :
% 0.33/0.61 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.33/0.61 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_square_less_1
% 0.33/0.61 thf(fact_5875_abs__square__less__1,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.33/0.61 = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_square_less_1
% 0.33/0.61 thf(fact_5876_abs__square__less__1,axiom,
% 0.33/0.61 ! [X: rat] :
% 0.33/0.61 ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.33/0.61 = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_square_less_1
% 0.33/0.61 thf(fact_5877_abs__square__less__1,axiom,
% 0.33/0.61 ! [X: int] :
% 0.33/0.61 ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.33/0.61 = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_square_less_1
% 0.33/0.61 thf(fact_5878_power__mono__even,axiom,
% 0.33/0.61 ! [N: nat,A: code_integer,B: code_integer] :
% 0.33/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.61 => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
% 0.33/0.61 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power_mono_even
% 0.33/0.61 thf(fact_5879_power__mono__even,axiom,
% 0.33/0.61 ! [N: nat,A: real,B: real] :
% 0.33/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.61 => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
% 0.33/0.61 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power_mono_even
% 0.33/0.61 thf(fact_5880_power__mono__even,axiom,
% 0.33/0.61 ! [N: nat,A: rat,B: rat] :
% 0.33/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.61 => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
% 0.33/0.61 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power_mono_even
% 0.33/0.61 thf(fact_5881_power__mono__even,axiom,
% 0.33/0.61 ! [N: nat,A: int,B: int] :
% 0.33/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.61 => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
% 0.33/0.61 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % power_mono_even
% 0.33/0.61 thf(fact_5882_ln__powr__bound,axiom,
% 0.33/0.61 ! [X: real,A: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.33/0.61 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.61 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % ln_powr_bound
% 0.33/0.61 thf(fact_5883_ln__powr__bound2,axiom,
% 0.33/0.61 ! [X: real,A: real] :
% 0.33/0.61 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.61 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.61 => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % ln_powr_bound2
% 0.33/0.61 thf(fact_5884_add__log__eq__powr,axiom,
% 0.33/0.61 ! [B: real,X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.61 => ( ( B != one_one_real )
% 0.33/0.61 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( plus_plus_real @ Y @ ( log2 @ B @ X ) )
% 0.33/0.61 = ( log2 @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % add_log_eq_powr
% 0.33/0.61 thf(fact_5885_log__add__eq__powr,axiom,
% 0.33/0.61 ! [B: real,X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.61 => ( ( B != one_one_real )
% 0.33/0.61 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( plus_plus_real @ ( log2 @ B @ X ) @ Y )
% 0.33/0.61 = ( log2 @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % log_add_eq_powr
% 0.33/0.61 thf(fact_5886_minus__log__eq__powr,axiom,
% 0.33/0.61 ! [B: real,X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.61 => ( ( B != one_one_real )
% 0.33/0.61 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( minus_minus_real @ Y @ ( log2 @ B @ X ) )
% 0.33/0.61 = ( log2 @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % minus_log_eq_powr
% 0.33/0.61 thf(fact_5887_take__bit__Suc__minus__bit0,axiom,
% 0.33/0.61 ! [N: nat,K: num] :
% 0.33/0.61 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.33/0.61 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % take_bit_Suc_minus_bit0
% 0.33/0.61 thf(fact_5888_minus__1__div__exp__eq__int,axiom,
% 0.33/0.61 ! [N: nat] :
% 0.33/0.61 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.61 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % minus_1_div_exp_eq_int
% 0.33/0.61 thf(fact_5889_div__pos__neg__trivial,axiom,
% 0.33/0.61 ! [K: int,L: int] :
% 0.33/0.61 ( ( ord_less_int @ zero_zero_int @ K )
% 0.33/0.61 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 0.33/0.61 => ( ( divide_divide_int @ K @ L )
% 0.33/0.61 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % div_pos_neg_trivial
% 0.33/0.61 thf(fact_5890_signed__take__bit__int__less__eq__self__iff,axiom,
% 0.33/0.61 ! [N: nat,K: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.33/0.61 = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% 0.33/0.61
% 0.33/0.61 % signed_take_bit_int_less_eq_self_iff
% 0.33/0.61 thf(fact_5891_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 0.33/0.61 ! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).
% 0.33/0.61
% 0.33/0.61 % signed_take_bit_int_greater_eq_minus_exp
% 0.33/0.61 thf(fact_5892_signed__take__bit__int__greater__self__iff,axiom,
% 0.33/0.61 ! [K: int,N: nat] :
% 0.33/0.61 ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.33/0.61 = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % signed_take_bit_int_greater_self_iff
% 0.33/0.61 thf(fact_5893_push__bit__minus__one,axiom,
% 0.33/0.61 ! [N: nat] :
% 0.33/0.61 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.61 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % push_bit_minus_one
% 0.33/0.61 thf(fact_5894_log__minus__eq__powr,axiom,
% 0.33/0.61 ! [B: real,X: real,Y: real] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.61 => ( ( B != one_one_real )
% 0.33/0.61 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( minus_minus_real @ ( log2 @ B @ X ) @ Y )
% 0.33/0.61 = ( log2 @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % log_minus_eq_powr
% 0.33/0.61 thf(fact_5895_int__bit__induct,axiom,
% 0.33/0.61 ! [P: int > $o,K: int] :
% 0.33/0.61 ( ( P @ zero_zero_int )
% 0.33/0.61 => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.61 => ( ! [K3: int] :
% 0.33/0.61 ( ( P @ K3 )
% 0.33/0.61 => ( ( K3 != zero_zero_int )
% 0.33/0.61 => ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.61 => ( ! [K3: int] :
% 0.33/0.61 ( ( P @ K3 )
% 0.33/0.61 => ( ( K3
% 0.33/0.61 != ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.61 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 0.33/0.61 => ( P @ K ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % int_bit_induct
% 0.33/0.61 thf(fact_5896_signed__take__bit__int__eq__self,axiom,
% 0.33/0.61 ! [N: nat,K: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.33/0.61 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.61 => ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.33/0.61 = K ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % signed_take_bit_int_eq_self
% 0.33/0.61 thf(fact_5897_signed__take__bit__int__eq__self__iff,axiom,
% 0.33/0.61 ! [N: nat,K: int] :
% 0.33/0.61 ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.33/0.61 = K )
% 0.33/0.61 = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.33/0.61 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % signed_take_bit_int_eq_self_iff
% 0.33/0.61 thf(fact_5898_powr__neg__numeral,axiom,
% 0.33/0.61 ! [X: real,N: num] :
% 0.33/0.61 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.33/0.61 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % powr_neg_numeral
% 0.33/0.61 thf(fact_5899_floor__log2__div2,axiom,
% 0.33/0.61 ! [N: nat] :
% 0.33/0.61 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.61 => ( ( archim6058952711729229775r_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.33/0.61 = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_log2_div2
% 0.33/0.61 thf(fact_5900_signed__take__bit__int__greater__eq,axiom,
% 0.33/0.61 ! [K: int,N: nat] :
% 0.33/0.61 ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.33/0.61 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % signed_take_bit_int_greater_eq
% 0.33/0.61 thf(fact_5901_take__bit__minus__small__eq,axiom,
% 0.33/0.61 ! [K: int,N: nat] :
% 0.33/0.61 ( ( ord_less_int @ zero_zero_int @ K )
% 0.33/0.61 => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.61 => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 0.33/0.61 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % take_bit_minus_small_eq
% 0.33/0.61 thf(fact_5902_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.61 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.61 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_ln_one_plus_x_minus_x_bound_nonneg
% 0.33/0.61 thf(fact_5903_floor__log__nat__eq__if,axiom,
% 0.33/0.61 ! [B: nat,N: nat,K: nat] :
% 0.33/0.61 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.33/0.61 => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.33/0.61 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.33/0.61 => ( ( archim6058952711729229775r_real @ ( log2 @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.33/0.61 = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % floor_log_nat_eq_if
% 0.33/0.61 thf(fact_5904_abs__sqrt__wlog,axiom,
% 0.33/0.61 ! [P: code_integer > code_integer > $o,X: code_integer] :
% 0.33/0.61 ( ! [X4: code_integer] :
% 0.33/0.61 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
% 0.33/0.61 => ( P @ X4 @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.61 => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_sqrt_wlog
% 0.33/0.61 thf(fact_5905_abs__sqrt__wlog,axiom,
% 0.33/0.61 ! [P: real > real > $o,X: real] :
% 0.33/0.61 ( ! [X4: real] :
% 0.33/0.61 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.33/0.61 => ( P @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.61 => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_sqrt_wlog
% 0.33/0.61 thf(fact_5906_abs__sqrt__wlog,axiom,
% 0.33/0.61 ! [P: rat > rat > $o,X: rat] :
% 0.33/0.61 ( ! [X4: rat] :
% 0.33/0.61 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.33/0.61 => ( P @ X4 @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.61 => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_sqrt_wlog
% 0.33/0.61 thf(fact_5907_abs__sqrt__wlog,axiom,
% 0.33/0.61 ! [P: int > int > $o,X: int] :
% 0.33/0.61 ( ! [X4: int] :
% 0.33/0.61 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.33/0.61 => ( P @ X4 @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.61 => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % abs_sqrt_wlog
% 0.33/0.61 thf(fact_5908_signed__take__bit__Suc__minus__bit1,axiom,
% 0.33/0.61 ! [N: nat,K: num] :
% 0.33/0.61 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.33/0.61 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % signed_take_bit_Suc_minus_bit1
% 0.33/0.61 thf(fact_5909_arcosh__def,axiom,
% 0.33/0.61 ( arcosh_real
% 0.33/0.61 = ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % arcosh_def
% 0.33/0.61 thf(fact_5910_arctan__double,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.61 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
% 0.33/0.61 = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % arctan_double
% 0.33/0.61 thf(fact_5911_arsinh__def,axiom,
% 0.33/0.61 ( arsinh_real
% 0.33/0.61 = ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % arsinh_def
% 0.33/0.61 thf(fact_5912_signed__take__bit__Suc__bit1,axiom,
% 0.33/0.61 ! [N: nat,K: num] :
% 0.33/0.61 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.33/0.61 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % signed_take_bit_Suc_bit1
% 0.33/0.61 thf(fact_5913_verit__eq__simplify_I9_J,axiom,
% 0.33/0.61 ! [X32: num,Y32: num] :
% 0.33/0.61 ( ( ( bit1 @ X32 )
% 0.33/0.61 = ( bit1 @ Y32 ) )
% 0.33/0.61 = ( X32 = Y32 ) ) ).
% 0.33/0.61
% 0.33/0.61 % verit_eq_simplify(9)
% 0.33/0.61 thf(fact_5914_semiring__norm_I90_J,axiom,
% 0.33/0.61 ! [M: num,N: num] :
% 0.33/0.61 ( ( ( bit1 @ M )
% 0.33/0.61 = ( bit1 @ N ) )
% 0.33/0.61 = ( M = N ) ) ).
% 0.33/0.61
% 0.33/0.61 % semiring_norm(90)
% 0.33/0.61 thf(fact_5915_semiring__norm_I88_J,axiom,
% 0.33/0.61 ! [M: num,N: num] :
% 0.33/0.61 ( ( bit0 @ M )
% 0.33/0.61 != ( bit1 @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % semiring_norm(88)
% 0.33/0.61 thf(fact_5916_semiring__norm_I89_J,axiom,
% 0.33/0.61 ! [M: num,N: num] :
% 0.33/0.61 ( ( bit1 @ M )
% 0.33/0.61 != ( bit0 @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % semiring_norm(89)
% 0.33/0.61 thf(fact_5917_semiring__norm_I84_J,axiom,
% 0.33/0.61 ! [N: num] :
% 0.33/0.61 ( one
% 0.33/0.61 != ( bit1 @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % semiring_norm(84)
% 0.33/0.61 thf(fact_5918_semiring__norm_I86_J,axiom,
% 0.33/0.61 ! [M: num] :
% 0.33/0.61 ( ( bit1 @ M )
% 0.33/0.61 != one ) ).
% 0.33/0.61
% 0.33/0.61 % semiring_norm(86)
% 0.33/0.61 thf(fact_5919_zdvd1__eq,axiom,
% 0.33/0.61 ! [X: int] :
% 0.33/0.61 ( ( dvd_dvd_int @ X @ one_one_int )
% 0.33/0.61 = ( ( abs_abs_int @ X )
% 0.33/0.61 = one_one_int ) ) ).
% 0.33/0.61
% 0.33/0.61 % zdvd1_eq
% 0.33/0.61 thf(fact_5920_semiring__norm_I80_J,axiom,
% 0.33/0.61 ! [M: num,N: num] :
% 0.33/0.61 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.33/0.61 = ( ord_less_num @ M @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % semiring_norm(80)
% 0.33/0.61 thf(fact_5921_semiring__norm_I73_J,axiom,
% 0.33/0.61 ! [M: num,N: num] :
% 0.33/0.61 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.33/0.61 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.33/0.61
% 0.33/0.61 % semiring_norm(73)
% 0.33/0.61 thf(fact_5922_of__real__0,axiom,
% 0.33/0.61 ( ( real_V1803761363581548252l_real @ zero_zero_real )
% 0.33/0.61 = zero_zero_real ) ).
% 0.33/0.61
% 0.33/0.61 % of_real_0
% 0.33/0.61 thf(fact_5923_of__real__0,axiom,
% 0.33/0.61 ( ( real_V4546457046886955230omplex @ zero_zero_real )
% 0.33/0.61 = zero_zero_complex ) ).
% 0.33/0.61
% 0.33/0.61 % of_real_0
% 0.33/0.61 thf(fact_5924_of__real__eq__0__iff,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ( real_V1803761363581548252l_real @ X )
% 0.33/0.61 = zero_zero_real )
% 0.33/0.61 = ( X = zero_zero_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % of_real_eq_0_iff
% 0.33/0.61 thf(fact_5925_of__real__eq__0__iff,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ( real_V4546457046886955230omplex @ X )
% 0.33/0.61 = zero_zero_complex )
% 0.33/0.61 = ( X = zero_zero_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % of_real_eq_0_iff
% 0.33/0.61 thf(fact_5926_of__real__1,axiom,
% 0.33/0.61 ( ( real_V1803761363581548252l_real @ one_one_real )
% 0.33/0.61 = one_one_real ) ).
% 0.33/0.61
% 0.33/0.61 % of_real_1
% 0.33/0.61 thf(fact_5927_of__real__1,axiom,
% 0.33/0.61 ( ( real_V4546457046886955230omplex @ one_one_real )
% 0.33/0.61 = one_one_complex ) ).
% 0.33/0.61
% 0.33/0.61 % of_real_1
% 0.33/0.61 thf(fact_5928_of__real__eq__1__iff,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ( real_V1803761363581548252l_real @ X )
% 0.33/0.61 = one_one_real )
% 0.33/0.61 = ( X = one_one_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % of_real_eq_1_iff
% 0.33/0.61 thf(fact_5929_of__real__eq__1__iff,axiom,
% 0.33/0.61 ! [X: real] :
% 0.33/0.61 ( ( ( real_V4546457046886955230omplex @ X )
% 0.33/0.61 = one_one_complex )
% 0.33/0.61 = ( X = one_one_real ) ) ).
% 0.33/0.61
% 0.33/0.61 % of_real_eq_1_iff
% 0.33/0.61 thf(fact_5930_of__real__mult,axiom,
% 0.33/0.61 ! [X: real,Y: real] :
% 0.33/0.61 ( ( real_V1803761363581548252l_real @ ( times_times_real @ X @ Y ) )
% 0.33/0.61 = ( times_times_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % of_real_mult
% 0.33/0.61 thf(fact_5931_of__real__mult,axiom,
% 0.33/0.61 ! [X: real,Y: real] :
% 0.33/0.61 ( ( real_V4546457046886955230omplex @ ( times_times_real @ X @ Y ) )
% 0.33/0.61 = ( times_times_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% 0.33/0.61
% 0.33/0.61 % of_real_mult
% 0.33/0.61 thf(fact_5932_of__real__numeral,axiom,
% 0.33/0.61 ! [W: num] :
% 0.33/0.61 ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
% 0.33/0.61 = ( numeral_numeral_real @ W ) ) ).
% 0.33/0.61
% 0.33/0.61 % of_real_numeral
% 0.33/0.61 thf(fact_5933_of__real__numeral,axiom,
% 0.33/0.61 ! [W: num] :
% 0.33/0.61 ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
% 0.33/0.62 = ( numera6690914467698888265omplex @ W ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_real_numeral
% 0.33/0.62 thf(fact_5934_zabs__less__one__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
% 0.33/0.62 = ( Z2 = zero_zero_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % zabs_less_one_iff
% 0.33/0.62 thf(fact_5935_of__real__divide,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y ) )
% 0.33/0.62 = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_real_divide
% 0.33/0.62 thf(fact_5936_of__real__divide,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y ) )
% 0.33/0.62 = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_real_divide
% 0.33/0.62 thf(fact_5937_of__real__add,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.62 = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_real_add
% 0.33/0.62 thf(fact_5938_of__real__add,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.62 = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_real_add
% 0.33/0.62 thf(fact_5939_of__real__power,axiom,
% 0.33/0.62 ! [X: real,N: nat] :
% 0.33/0.62 ( ( real_V1803761363581548252l_real @ ( power_power_real @ X @ N ) )
% 0.33/0.62 = ( power_power_real @ ( real_V1803761363581548252l_real @ X ) @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_real_power
% 0.33/0.62 thf(fact_5940_of__real__power,axiom,
% 0.33/0.62 ! [X: real,N: nat] :
% 0.33/0.62 ( ( real_V4546457046886955230omplex @ ( power_power_real @ X @ N ) )
% 0.33/0.62 = ( power_power_complex @ ( real_V4546457046886955230omplex @ X ) @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_real_power
% 0.33/0.62 thf(fact_5941_of__real__diff,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( real_V1803761363581548252l_real @ ( minus_minus_real @ X @ Y ) )
% 0.33/0.62 = ( minus_minus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_real_diff
% 0.33/0.62 thf(fact_5942_of__real__diff,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( real_V4546457046886955230omplex @ ( minus_minus_real @ X @ Y ) )
% 0.33/0.62 = ( minus_minus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_real_diff
% 0.33/0.62 thf(fact_5943_semiring__norm_I9_J,axiom,
% 0.33/0.62 ! [M: num,N: num] :
% 0.33/0.62 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.33/0.62 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % semiring_norm(9)
% 0.33/0.62 thf(fact_5944_semiring__norm_I7_J,axiom,
% 0.33/0.62 ! [M: num,N: num] :
% 0.33/0.62 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.33/0.62 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % semiring_norm(7)
% 0.33/0.62 thf(fact_5945_of__real__of__nat__eq,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( real_V4546457046886955230omplex @ ( semiri5074537144036343181t_real @ N ) )
% 0.33/0.62 = ( semiri8010041392384452111omplex @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_real_of_nat_eq
% 0.33/0.62 thf(fact_5946_of__real__of__nat__eq,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( real_V1803761363581548252l_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.33/0.62 = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_real_of_nat_eq
% 0.33/0.62 thf(fact_5947_semiring__norm_I15_J,axiom,
% 0.33/0.62 ! [M: num,N: num] :
% 0.33/0.62 ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.33/0.62 = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % semiring_norm(15)
% 0.33/0.62 thf(fact_5948_semiring__norm_I14_J,axiom,
% 0.33/0.62 ! [M: num,N: num] :
% 0.33/0.62 ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.33/0.62 = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % semiring_norm(14)
% 0.33/0.62 thf(fact_5949_semiring__norm_I81_J,axiom,
% 0.33/0.62 ! [M: num,N: num] :
% 0.33/0.62 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.33/0.62 = ( ord_less_num @ M @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % semiring_norm(81)
% 0.33/0.62 thf(fact_5950_semiring__norm_I72_J,axiom,
% 0.33/0.62 ! [M: num,N: num] :
% 0.33/0.62 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.33/0.62 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % semiring_norm(72)
% 0.33/0.62 thf(fact_5951_semiring__norm_I77_J,axiom,
% 0.33/0.62 ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % semiring_norm(77)
% 0.33/0.62 thf(fact_5952_semiring__norm_I70_J,axiom,
% 0.33/0.62 ! [M: num] :
% 0.33/0.62 ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 0.33/0.62
% 0.33/0.62 % semiring_norm(70)
% 0.33/0.62 thf(fact_5953_zdiv__numeral__Bit1,axiom,
% 0.33/0.62 ! [V: num,W: num] :
% 0.33/0.62 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.33/0.62 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % zdiv_numeral_Bit1
% 0.33/0.62 thf(fact_5954_semiring__norm_I3_J,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( plus_plus_num @ one @ ( bit0 @ N ) )
% 0.33/0.62 = ( bit1 @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % semiring_norm(3)
% 0.33/0.62 thf(fact_5955_semiring__norm_I4_J,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( plus_plus_num @ one @ ( bit1 @ N ) )
% 0.33/0.62 = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % semiring_norm(4)
% 0.33/0.62 thf(fact_5956_semiring__norm_I5_J,axiom,
% 0.33/0.62 ! [M: num] :
% 0.33/0.62 ( ( plus_plus_num @ ( bit0 @ M ) @ one )
% 0.33/0.62 = ( bit1 @ M ) ) ).
% 0.33/0.62
% 0.33/0.62 % semiring_norm(5)
% 0.33/0.62 thf(fact_5957_semiring__norm_I8_J,axiom,
% 0.33/0.62 ! [M: num] :
% 0.33/0.62 ( ( plus_plus_num @ ( bit1 @ M ) @ one )
% 0.33/0.62 = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % semiring_norm(8)
% 0.33/0.62 thf(fact_5958_semiring__norm_I10_J,axiom,
% 0.33/0.62 ! [M: num,N: num] :
% 0.33/0.62 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.33/0.62 = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % semiring_norm(10)
% 0.33/0.62 thf(fact_5959_semiring__norm_I16_J,axiom,
% 0.33/0.62 ! [M: num,N: num] :
% 0.33/0.62 ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.33/0.62 = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % semiring_norm(16)
% 0.33/0.62 thf(fact_5960_semiring__norm_I74_J,axiom,
% 0.33/0.62 ! [M: num,N: num] :
% 0.33/0.62 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.33/0.62 = ( ord_less_num @ M @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % semiring_norm(74)
% 0.33/0.62 thf(fact_5961_semiring__norm_I79_J,axiom,
% 0.33/0.62 ! [M: num,N: num] :
% 0.33/0.62 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.33/0.62 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % semiring_norm(79)
% 0.33/0.62 thf(fact_5962_of__real__neg__numeral,axiom,
% 0.33/0.62 ! [W: num] :
% 0.33/0.62 ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_real_neg_numeral
% 0.33/0.62 thf(fact_5963_of__real__neg__numeral,axiom,
% 0.33/0.62 ! [W: num] :
% 0.33/0.62 ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.62 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_real_neg_numeral
% 0.33/0.62 thf(fact_5964_Suc__div__eq__add3__div__numeral,axiom,
% 0.33/0.62 ! [M: nat,V: num] :
% 0.33/0.62 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.33/0.62 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % Suc_div_eq_add3_div_numeral
% 0.33/0.62 thf(fact_5965_div__Suc__eq__div__add3,axiom,
% 0.33/0.62 ! [M: nat,N: nat] :
% 0.33/0.62 ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.33/0.62 = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % div_Suc_eq_div_add3
% 0.33/0.62 thf(fact_5966_Suc__mod__eq__add3__mod__numeral,axiom,
% 0.33/0.62 ! [M: nat,V: num] :
% 0.33/0.62 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.33/0.62 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % Suc_mod_eq_add3_mod_numeral
% 0.33/0.62 thf(fact_5967_mod__Suc__eq__mod__add3,axiom,
% 0.33/0.62 ! [M: nat,N: nat] :
% 0.33/0.62 ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.33/0.62 = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % mod_Suc_eq_mod_add3
% 0.33/0.62 thf(fact_5968_norm__of__real__add1,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ one_one_real ) )
% 0.33/0.62 = ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % norm_of_real_add1
% 0.33/0.62 thf(fact_5969_norm__of__real__add1,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ one_one_complex ) )
% 0.33/0.62 = ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % norm_of_real_add1
% 0.33/0.62 thf(fact_5970_norm__of__real__addn,axiom,
% 0.33/0.62 ! [X: real,B: num] :
% 0.33/0.62 ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( numeral_numeral_real @ B ) ) )
% 0.33/0.62 = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % norm_of_real_addn
% 0.33/0.62 thf(fact_5971_norm__of__real__addn,axiom,
% 0.33/0.62 ! [X: real,B: num] :
% 0.33/0.62 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( numera6690914467698888265omplex @ B ) ) )
% 0.33/0.62 = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % norm_of_real_addn
% 0.33/0.62 thf(fact_5972_zmod__numeral__Bit1,axiom,
% 0.33/0.62 ! [V: num,W: num] :
% 0.33/0.62 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.33/0.62 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % zmod_numeral_Bit1
% 0.33/0.62 thf(fact_5973_verit__eq__simplify_I14_J,axiom,
% 0.33/0.62 ! [X2: num,X32: num] :
% 0.33/0.62 ( ( bit0 @ X2 )
% 0.33/0.62 != ( bit1 @ X32 ) ) ).
% 0.33/0.62
% 0.33/0.62 % verit_eq_simplify(14)
% 0.33/0.62 thf(fact_5974_verit__eq__simplify_I12_J,axiom,
% 0.33/0.62 ! [X32: num] :
% 0.33/0.62 ( one
% 0.33/0.62 != ( bit1 @ X32 ) ) ).
% 0.33/0.62
% 0.33/0.62 % verit_eq_simplify(12)
% 0.33/0.62 thf(fact_5975_zdvd__antisym__abs,axiom,
% 0.33/0.62 ! [A: int,B: int] :
% 0.33/0.62 ( ( dvd_dvd_int @ A @ B )
% 0.33/0.62 => ( ( dvd_dvd_int @ B @ A )
% 0.33/0.62 => ( ( abs_abs_int @ A )
% 0.33/0.62 = ( abs_abs_int @ B ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % zdvd_antisym_abs
% 0.33/0.62 thf(fact_5976_num_Oexhaust,axiom,
% 0.33/0.62 ! [Y: num] :
% 0.33/0.62 ( ( Y != one )
% 0.33/0.62 => ( ! [X22: num] :
% 0.33/0.62 ( Y
% 0.33/0.62 != ( bit0 @ X22 ) )
% 0.33/0.62 => ~ ! [X33: num] :
% 0.33/0.62 ( Y
% 0.33/0.62 != ( bit1 @ X33 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % num.exhaust
% 0.33/0.62 thf(fact_5977_abs__zmult__eq__1,axiom,
% 0.33/0.62 ! [M: int,N: int] :
% 0.33/0.62 ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
% 0.33/0.62 = one_one_int )
% 0.33/0.62 => ( ( abs_abs_int @ M )
% 0.33/0.62 = one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % abs_zmult_eq_1
% 0.33/0.62 thf(fact_5978_abs__div,axiom,
% 0.33/0.62 ! [Y: int,X: int] :
% 0.33/0.62 ( ( dvd_dvd_int @ Y @ X )
% 0.33/0.62 => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y ) )
% 0.33/0.62 = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % abs_div
% 0.33/0.62 thf(fact_5979_div__eq__sgn__abs,axiom,
% 0.33/0.62 ! [K: int,L: int] :
% 0.33/0.62 ( ( ( sgn_sgn_int @ K )
% 0.33/0.62 = ( sgn_sgn_int @ L ) )
% 0.33/0.62 => ( ( divide_divide_int @ K @ L )
% 0.33/0.62 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % div_eq_sgn_abs
% 0.33/0.62 thf(fact_5980_nonzero__of__real__divide,axiom,
% 0.33/0.62 ! [Y: real,X: real] :
% 0.33/0.62 ( ( Y != zero_zero_real )
% 0.33/0.62 => ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y ) )
% 0.33/0.62 = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % nonzero_of_real_divide
% 0.33/0.62 thf(fact_5981_nonzero__of__real__divide,axiom,
% 0.33/0.62 ! [Y: real,X: real] :
% 0.33/0.62 ( ( Y != zero_zero_real )
% 0.33/0.62 => ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y ) )
% 0.33/0.62 = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % nonzero_of_real_divide
% 0.33/0.62 thf(fact_5982_numeral__Bit1,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 0.33/0.62 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_Bit1
% 0.33/0.62 thf(fact_5983_numeral__Bit1,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 0.33/0.62 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_Bit1
% 0.33/0.62 thf(fact_5984_numeral__Bit1,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 0.33/0.62 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_Bit1
% 0.33/0.62 thf(fact_5985_numeral__Bit1,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.33/0.62 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_Bit1
% 0.33/0.62 thf(fact_5986_numeral__Bit1,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 0.33/0.62 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_Bit1
% 0.33/0.62 thf(fact_5987_eval__nat__numeral_I3_J,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.33/0.62 = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % eval_nat_numeral(3)
% 0.33/0.62 thf(fact_5988_cong__exp__iff__simps_I10_J,axiom,
% 0.33/0.62 ! [M: num,Q2: num,N: num] :
% 0.33/0.62 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(10)
% 0.33/0.62 thf(fact_5989_cong__exp__iff__simps_I10_J,axiom,
% 0.33/0.62 ! [M: num,Q2: num,N: num] :
% 0.33/0.62 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(10)
% 0.33/0.62 thf(fact_5990_cong__exp__iff__simps_I10_J,axiom,
% 0.33/0.62 ! [M: num,Q2: num,N: num] :
% 0.33/0.62 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(10)
% 0.33/0.62 thf(fact_5991_cong__exp__iff__simps_I12_J,axiom,
% 0.33/0.62 ! [M: num,Q2: num,N: num] :
% 0.33/0.62 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(12)
% 0.33/0.62 thf(fact_5992_cong__exp__iff__simps_I12_J,axiom,
% 0.33/0.62 ! [M: num,Q2: num,N: num] :
% 0.33/0.62 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(12)
% 0.33/0.62 thf(fact_5993_cong__exp__iff__simps_I12_J,axiom,
% 0.33/0.62 ! [M: num,Q2: num,N: num] :
% 0.33/0.62 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(12)
% 0.33/0.62 thf(fact_5994_cong__exp__iff__simps_I13_J,axiom,
% 0.33/0.62 ! [M: num,Q2: num,N: num] :
% 0.33/0.62 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.62 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.33/0.62 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(13)
% 0.33/0.62 thf(fact_5995_cong__exp__iff__simps_I13_J,axiom,
% 0.33/0.62 ! [M: num,Q2: num,N: num] :
% 0.33/0.62 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.62 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.33/0.62 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(13)
% 0.33/0.62 thf(fact_5996_cong__exp__iff__simps_I13_J,axiom,
% 0.33/0.62 ! [M: num,Q2: num,N: num] :
% 0.33/0.62 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.62 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.33/0.62 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(13)
% 0.33/0.62 thf(fact_5997_power__minus__Bit1,axiom,
% 0.33/0.62 ! [X: real,K: num] :
% 0.33/0.62 ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % power_minus_Bit1
% 0.33/0.62 thf(fact_5998_power__minus__Bit1,axiom,
% 0.33/0.62 ! [X: int,K: num] :
% 0.33/0.62 ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.33/0.62 = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % power_minus_Bit1
% 0.33/0.62 thf(fact_5999_power__minus__Bit1,axiom,
% 0.33/0.62 ! [X: complex,K: num] :
% 0.33/0.62 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.33/0.62 = ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % power_minus_Bit1
% 0.33/0.62 thf(fact_6000_power__minus__Bit1,axiom,
% 0.33/0.62 ! [X: rat,K: num] :
% 0.33/0.62 ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.33/0.62 = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % power_minus_Bit1
% 0.33/0.62 thf(fact_6001_power__minus__Bit1,axiom,
% 0.33/0.62 ! [X: code_integer,K: num] :
% 0.33/0.62 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.33/0.62 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % power_minus_Bit1
% 0.33/0.62 thf(fact_6002_zabs__def,axiom,
% 0.33/0.62 ( abs_abs_int
% 0.33/0.62 = ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % zabs_def
% 0.33/0.62 thf(fact_6003_dvd__imp__le__int,axiom,
% 0.33/0.62 ! [I: int,D: int] :
% 0.33/0.62 ( ( I != zero_zero_int )
% 0.33/0.62 => ( ( dvd_dvd_int @ D @ I )
% 0.33/0.62 => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dvd_imp_le_int
% 0.33/0.62 thf(fact_6004_abs__mod__less,axiom,
% 0.33/0.62 ! [L: int,K: int] :
% 0.33/0.62 ( ( L != zero_zero_int )
% 0.33/0.62 => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % abs_mod_less
% 0.33/0.62 thf(fact_6005_norm__less__p1,axiom,
% 0.33/0.62 ! [X: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X ) ) @ one_one_real ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % norm_less_p1
% 0.33/0.62 thf(fact_6006_norm__less__p1,axiom,
% 0.33/0.62 ! [X: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X ) ) @ one_one_complex ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % norm_less_p1
% 0.33/0.62 thf(fact_6007_numeral__Bit1__div__2,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.62 = ( numeral_numeral_nat @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_Bit1_div_2
% 0.33/0.62 thf(fact_6008_numeral__Bit1__div__2,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.62 = ( numeral_numeral_int @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_Bit1_div_2
% 0.33/0.62 thf(fact_6009_odd__numeral,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % odd_numeral
% 0.33/0.62 thf(fact_6010_odd__numeral,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % odd_numeral
% 0.33/0.62 thf(fact_6011_odd__numeral,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % odd_numeral
% 0.33/0.62 thf(fact_6012_cong__exp__iff__simps_I3_J,axiom,
% 0.33/0.62 ! [N: num,Q2: num] :
% 0.33/0.62 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 != zero_zero_nat ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(3)
% 0.33/0.62 thf(fact_6013_cong__exp__iff__simps_I3_J,axiom,
% 0.33/0.62 ! [N: num,Q2: num] :
% 0.33/0.62 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 != zero_zero_int ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(3)
% 0.33/0.62 thf(fact_6014_cong__exp__iff__simps_I3_J,axiom,
% 0.33/0.62 ! [N: num,Q2: num] :
% 0.33/0.62 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 != zero_z3403309356797280102nteger ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(3)
% 0.33/0.62 thf(fact_6015_power3__eq__cube,axiom,
% 0.33/0.62 ! [A: complex] :
% 0.33/0.62 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.33/0.62 = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % power3_eq_cube
% 0.33/0.62 thf(fact_6016_power3__eq__cube,axiom,
% 0.33/0.62 ! [A: real] :
% 0.33/0.62 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.33/0.62 = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % power3_eq_cube
% 0.33/0.62 thf(fact_6017_power3__eq__cube,axiom,
% 0.33/0.62 ! [A: rat] :
% 0.33/0.62 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.33/0.62 = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % power3_eq_cube
% 0.33/0.62 thf(fact_6018_power3__eq__cube,axiom,
% 0.33/0.62 ! [A: nat] :
% 0.33/0.62 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.33/0.62 = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % power3_eq_cube
% 0.33/0.62 thf(fact_6019_power3__eq__cube,axiom,
% 0.33/0.62 ! [A: int] :
% 0.33/0.62 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.33/0.62 = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % power3_eq_cube
% 0.33/0.62 thf(fact_6020_numeral__3__eq__3,axiom,
% 0.33/0.62 ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 0.33/0.62 = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_3_eq_3
% 0.33/0.62 thf(fact_6021_Suc3__eq__add__3,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( suc @ ( suc @ ( suc @ N ) ) )
% 0.33/0.62 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % Suc3_eq_add_3
% 0.33/0.62 thf(fact_6022_zdvd__mult__cancel1,axiom,
% 0.33/0.62 ! [M: int,N: int] :
% 0.33/0.62 ( ( M != zero_zero_int )
% 0.33/0.62 => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
% 0.33/0.62 = ( ( abs_abs_int @ N )
% 0.33/0.62 = one_one_int ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % zdvd_mult_cancel1
% 0.33/0.62 thf(fact_6023_div__sgn__abs__cancel,axiom,
% 0.33/0.62 ! [V: int,K: int,L: int] :
% 0.33/0.62 ( ( V != zero_zero_int )
% 0.33/0.62 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
% 0.33/0.62 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % div_sgn_abs_cancel
% 0.33/0.62 thf(fact_6024_div__dvd__sgn__abs,axiom,
% 0.33/0.62 ! [L: int,K: int] :
% 0.33/0.62 ( ( dvd_dvd_int @ L @ K )
% 0.33/0.62 => ( ( divide_divide_int @ K @ L )
% 0.33/0.62 = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % div_dvd_sgn_abs
% 0.33/0.62 thf(fact_6025_num_Osize__gen_I3_J,axiom,
% 0.33/0.62 ! [X32: num] :
% 0.33/0.62 ( ( size_num @ ( bit1 @ X32 ) )
% 0.33/0.62 = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % num.size_gen(3)
% 0.33/0.62 thf(fact_6026_norm__of__real__diff,axiom,
% 0.33/0.62 ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( real_V1803761363581548252l_real @ B ) @ ( real_V1803761363581548252l_real @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % norm_of_real_diff
% 0.33/0.62 thf(fact_6027_norm__of__real__diff,axiom,
% 0.33/0.62 ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( real_V4546457046886955230omplex @ B ) @ ( real_V4546457046886955230omplex @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % norm_of_real_diff
% 0.33/0.62 thf(fact_6028_cong__exp__iff__simps_I7_J,axiom,
% 0.33/0.62 ! [Q2: num,N: num] :
% 0.33/0.62 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.62 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.33/0.62 = zero_zero_nat ) ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(7)
% 0.33/0.62 thf(fact_6029_cong__exp__iff__simps_I7_J,axiom,
% 0.33/0.62 ! [Q2: num,N: num] :
% 0.33/0.62 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.62 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 0.33/0.62 = zero_zero_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(7)
% 0.33/0.62 thf(fact_6030_cong__exp__iff__simps_I7_J,axiom,
% 0.33/0.62 ! [Q2: num,N: num] :
% 0.33/0.62 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.62 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.33/0.62 = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(7)
% 0.33/0.62 thf(fact_6031_cong__exp__iff__simps_I11_J,axiom,
% 0.33/0.62 ! [M: num,Q2: num] :
% 0.33/0.62 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.62 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.33/0.62 = zero_zero_nat ) ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(11)
% 0.33/0.62 thf(fact_6032_cong__exp__iff__simps_I11_J,axiom,
% 0.33/0.62 ! [M: num,Q2: num] :
% 0.33/0.62 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.62 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.33/0.62 = zero_zero_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(11)
% 0.33/0.62 thf(fact_6033_cong__exp__iff__simps_I11_J,axiom,
% 0.33/0.62 ! [M: num,Q2: num] :
% 0.33/0.62 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.33/0.62 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.62 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.33/0.62 = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.62
% 0.33/0.62 % cong_exp_iff_simps(11)
% 0.33/0.62 thf(fact_6034_even__abs__add__iff,axiom,
% 0.33/0.62 ! [K: int,L: int] :
% 0.33/0.62 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
% 0.33/0.62 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % even_abs_add_iff
% 0.33/0.62 thf(fact_6035_even__add__abs__iff,axiom,
% 0.33/0.62 ! [K: int,L: int] :
% 0.33/0.62 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
% 0.33/0.62 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % even_add_abs_iff
% 0.33/0.62 thf(fact_6036_Suc__div__eq__add3__div,axiom,
% 0.33/0.62 ! [M: nat,N: nat] :
% 0.33/0.62 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 0.33/0.62 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % Suc_div_eq_add3_div
% 0.33/0.62 thf(fact_6037_Suc__mod__eq__add3__mod,axiom,
% 0.33/0.62 ! [M: nat,N: nat] :
% 0.33/0.62 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 0.33/0.62 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % Suc_mod_eq_add3_mod
% 0.33/0.62 thf(fact_6038_nat__intermed__int__val,axiom,
% 0.33/0.62 ! [M: nat,N: nat,F: nat > int,K: int] :
% 0.33/0.62 ( ! [I2: nat] :
% 0.33/0.62 ( ( ( ord_less_eq_nat @ M @ I2 )
% 0.33/0.62 & ( ord_less_nat @ I2 @ N ) )
% 0.33/0.62 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 0.33/0.62 => ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.62 => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 0.33/0.62 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.33/0.62 => ? [I2: nat] :
% 0.33/0.62 ( ( ord_less_eq_nat @ M @ I2 )
% 0.33/0.62 & ( ord_less_eq_nat @ I2 @ N )
% 0.33/0.62 & ( ( F @ I2 )
% 0.33/0.62 = K ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % nat_intermed_int_val
% 0.33/0.62 thf(fact_6039_incr__lemma,axiom,
% 0.33/0.62 ! [D: int,Z2: int,X: int] :
% 0.33/0.62 ( ( ord_less_int @ zero_zero_int @ D )
% 0.33/0.62 => ( ord_less_int @ Z2 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % incr_lemma
% 0.33/0.62 thf(fact_6040_decr__lemma,axiom,
% 0.33/0.62 ! [D: int,X: int,Z2: int] :
% 0.33/0.62 ( ( ord_less_int @ zero_zero_int @ D )
% 0.33/0.62 => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % decr_lemma
% 0.33/0.62 thf(fact_6041_mod__exhaust__less__4,axiom,
% 0.33/0.62 ! [M: nat] :
% 0.33/0.62 ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = zero_zero_nat )
% 0.33/0.62 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = one_one_nat )
% 0.33/0.62 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.62 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % mod_exhaust_less_4
% 0.33/0.62 thf(fact_6042_div__noneq__sgn__abs,axiom,
% 0.33/0.62 ! [L: int,K: int] :
% 0.33/0.62 ( ( L != zero_zero_int )
% 0.33/0.62 => ( ( ( sgn_sgn_int @ K )
% 0.33/0.62 != ( sgn_sgn_int @ L ) )
% 0.33/0.62 => ( ( divide_divide_int @ K @ L )
% 0.33/0.62 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
% 0.33/0.62 @ ( zero_n2684676970156552555ol_int
% 0.33/0.62 @ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % div_noneq_sgn_abs
% 0.33/0.62 thf(fact_6043_nat__ivt__aux,axiom,
% 0.33/0.62 ! [N: nat,F: nat > int,K: int] :
% 0.33/0.62 ( ! [I2: nat] :
% 0.33/0.62 ( ( ord_less_nat @ I2 @ N )
% 0.33/0.62 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 0.33/0.62 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.33/0.62 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.33/0.62 => ? [I2: nat] :
% 0.33/0.62 ( ( ord_less_eq_nat @ I2 @ N )
% 0.33/0.62 & ( ( F @ I2 )
% 0.33/0.62 = K ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % nat_ivt_aux
% 0.33/0.62 thf(fact_6044_complex__mod__triangle__ineq2,axiom,
% 0.33/0.62 ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % complex_mod_triangle_ineq2
% 0.33/0.62 thf(fact_6045_take__bit__Suc__bit1,axiom,
% 0.33/0.62 ! [N: nat,K: num] :
% 0.33/0.62 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.33/0.62 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % take_bit_Suc_bit1
% 0.33/0.62 thf(fact_6046_take__bit__Suc__bit1,axiom,
% 0.33/0.62 ! [N: nat,K: num] :
% 0.33/0.62 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.33/0.62 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 0.33/0.62
% 0.33/0.62 % take_bit_Suc_bit1
% 0.33/0.62 thf(fact_6047_nat0__intermed__int__val,axiom,
% 0.33/0.62 ! [N: nat,F: nat > int,K: int] :
% 0.33/0.62 ( ! [I2: nat] :
% 0.33/0.62 ( ( ord_less_nat @ I2 @ N )
% 0.33/0.62 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 0.33/0.62 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.33/0.62 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.33/0.62 => ? [I2: nat] :
% 0.33/0.62 ( ( ord_less_eq_nat @ I2 @ N )
% 0.33/0.62 & ( ( F @ I2 )
% 0.33/0.62 = K ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % nat0_intermed_int_val
% 0.33/0.62 thf(fact_6048_arctan__add,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.62 => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.33/0.62 => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 0.33/0.62 = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % arctan_add
% 0.33/0.62 thf(fact_6049_odd__mod__4__div__2,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.33/0.62 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % odd_mod_4_div_2
% 0.33/0.62 thf(fact_6050_signed__take__bit__numeral__minus__bit1,axiom,
% 0.33/0.62 ! [L: num,K: num] :
% 0.33/0.62 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.33/0.62 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % signed_take_bit_numeral_minus_bit1
% 0.33/0.62 thf(fact_6051_dbl__dec__simps_I4_J,axiom,
% 0.33/0.62 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(4)
% 0.33/0.62 thf(fact_6052_dbl__dec__simps_I4_J,axiom,
% 0.33/0.62 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(4)
% 0.33/0.62 thf(fact_6053_dbl__dec__simps_I4_J,axiom,
% 0.33/0.62 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.62 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(4)
% 0.33/0.62 thf(fact_6054_dbl__dec__simps_I4_J,axiom,
% 0.33/0.62 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(4)
% 0.33/0.62 thf(fact_6055_dbl__dec__simps_I4_J,axiom,
% 0.33/0.62 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.62 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(4)
% 0.33/0.62 thf(fact_6056_take__bit__Suc__minus__bit1,axiom,
% 0.33/0.62 ! [N: nat,K: num] :
% 0.33/0.62 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.33/0.62 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % take_bit_Suc_minus_bit1
% 0.33/0.62 thf(fact_6057_signed__take__bit__numeral__bit1,axiom,
% 0.33/0.62 ! [L: num,K: num] :
% 0.33/0.62 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.33/0.62 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % signed_take_bit_numeral_bit1
% 0.33/0.62 thf(fact_6058_dbl__inc__simps_I3_J,axiom,
% 0.33/0.62 ( ( neg_nu8557863876264182079omplex @ one_one_complex )
% 0.33/0.62 = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(3)
% 0.33/0.62 thf(fact_6059_dbl__inc__simps_I3_J,axiom,
% 0.33/0.62 ( ( neg_nu8295874005876285629c_real @ one_one_real )
% 0.33/0.62 = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(3)
% 0.33/0.62 thf(fact_6060_dbl__inc__simps_I3_J,axiom,
% 0.33/0.62 ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
% 0.33/0.62 = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(3)
% 0.33/0.62 thf(fact_6061_dbl__inc__simps_I3_J,axiom,
% 0.33/0.62 ( ( neg_nu5851722552734809277nc_int @ one_one_int )
% 0.33/0.62 = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(3)
% 0.33/0.62 thf(fact_6062_arctan__inverse,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( X != zero_zero_real )
% 0.33/0.62 => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
% 0.33/0.62 = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % arctan_inverse
% 0.33/0.62 thf(fact_6063_dbl__dec__simps_I3_J,axiom,
% 0.33/0.62 ( ( neg_nu6511756317524482435omplex @ one_one_complex )
% 0.33/0.62 = one_one_complex ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(3)
% 0.33/0.62 thf(fact_6064_dbl__dec__simps_I3_J,axiom,
% 0.33/0.62 ( ( neg_nu6075765906172075777c_real @ one_one_real )
% 0.33/0.62 = one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(3)
% 0.33/0.62 thf(fact_6065_dbl__dec__simps_I3_J,axiom,
% 0.33/0.62 ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
% 0.33/0.62 = one_one_rat ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(3)
% 0.33/0.62 thf(fact_6066_dbl__dec__simps_I3_J,axiom,
% 0.33/0.62 ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 0.33/0.62 = one_one_int ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(3)
% 0.33/0.62 thf(fact_6067_pred__numeral__simps_I1_J,axiom,
% 0.33/0.62 ( ( pred_numeral @ one )
% 0.33/0.62 = zero_zero_nat ) ).
% 0.33/0.62
% 0.33/0.62 % pred_numeral_simps(1)
% 0.33/0.62 thf(fact_6068_Suc__eq__numeral,axiom,
% 0.33/0.62 ! [N: nat,K: num] :
% 0.33/0.62 ( ( ( suc @ N )
% 0.33/0.62 = ( numeral_numeral_nat @ K ) )
% 0.33/0.62 = ( N
% 0.33/0.62 = ( pred_numeral @ K ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % Suc_eq_numeral
% 0.33/0.62 thf(fact_6069_eq__numeral__Suc,axiom,
% 0.33/0.62 ! [K: num,N: nat] :
% 0.33/0.62 ( ( ( numeral_numeral_nat @ K )
% 0.33/0.62 = ( suc @ N ) )
% 0.33/0.62 = ( ( pred_numeral @ K )
% 0.33/0.62 = N ) ) ).
% 0.33/0.62
% 0.33/0.62 % eq_numeral_Suc
% 0.33/0.62 thf(fact_6070_dbl__inc__simps_I2_J,axiom,
% 0.33/0.62 ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 0.33/0.62 = one_one_complex ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(2)
% 0.33/0.62 thf(fact_6071_dbl__inc__simps_I2_J,axiom,
% 0.33/0.62 ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 0.33/0.62 = one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(2)
% 0.33/0.62 thf(fact_6072_dbl__inc__simps_I2_J,axiom,
% 0.33/0.62 ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 0.33/0.62 = one_one_rat ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(2)
% 0.33/0.62 thf(fact_6073_dbl__inc__simps_I2_J,axiom,
% 0.33/0.62 ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 0.33/0.62 = one_one_int ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(2)
% 0.33/0.62 thf(fact_6074_pred__numeral__inc,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( pred_numeral @ ( inc @ K ) )
% 0.33/0.62 = ( numeral_numeral_nat @ K ) ) ).
% 0.33/0.62
% 0.33/0.62 % pred_numeral_inc
% 0.33/0.62 thf(fact_6075_dbl__inc__simps_I4_J,axiom,
% 0.33/0.62 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.62 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(4)
% 0.33/0.62 thf(fact_6076_dbl__inc__simps_I4_J,axiom,
% 0.33/0.62 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.62 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(4)
% 0.33/0.62 thf(fact_6077_dbl__inc__simps_I4_J,axiom,
% 0.33/0.62 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.62 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(4)
% 0.33/0.62 thf(fact_6078_dbl__inc__simps_I4_J,axiom,
% 0.33/0.62 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.62 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(4)
% 0.33/0.62 thf(fact_6079_dbl__inc__simps_I4_J,axiom,
% 0.33/0.62 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.62 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(4)
% 0.33/0.62 thf(fact_6080_dbl__inc__simps_I5_J,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.33/0.62 = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(5)
% 0.33/0.62 thf(fact_6081_dbl__inc__simps_I5_J,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
% 0.33/0.62 = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(5)
% 0.33/0.62 thf(fact_6082_dbl__inc__simps_I5_J,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
% 0.33/0.62 = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(5)
% 0.33/0.62 thf(fact_6083_dbl__inc__simps_I5_J,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
% 0.33/0.62 = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(5)
% 0.33/0.62 thf(fact_6084_pred__numeral__simps_I3_J,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( pred_numeral @ ( bit1 @ K ) )
% 0.33/0.62 = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % pred_numeral_simps(3)
% 0.33/0.62 thf(fact_6085_less__Suc__numeral,axiom,
% 0.33/0.62 ! [N: nat,K: num] :
% 0.33/0.62 ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.33/0.62 = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % less_Suc_numeral
% 0.33/0.62 thf(fact_6086_less__numeral__Suc,axiom,
% 0.33/0.62 ! [K: num,N: nat] :
% 0.33/0.62 ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.33/0.62 = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % less_numeral_Suc
% 0.33/0.62 thf(fact_6087_le__Suc__numeral,axiom,
% 0.33/0.62 ! [N: nat,K: num] :
% 0.33/0.62 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.33/0.62 = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % le_Suc_numeral
% 0.33/0.62 thf(fact_6088_le__numeral__Suc,axiom,
% 0.33/0.62 ! [K: num,N: nat] :
% 0.33/0.62 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.33/0.62 = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % le_numeral_Suc
% 0.33/0.62 thf(fact_6089_diff__Suc__numeral,axiom,
% 0.33/0.62 ! [N: nat,K: num] :
% 0.33/0.62 ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.33/0.62 = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % diff_Suc_numeral
% 0.33/0.62 thf(fact_6090_diff__numeral__Suc,axiom,
% 0.33/0.62 ! [K: num,N: nat] :
% 0.33/0.62 ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.33/0.62 = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % diff_numeral_Suc
% 0.33/0.62 thf(fact_6091_dbl__dec__simps_I2_J,axiom,
% 0.33/0.62 ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 0.33/0.62 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(2)
% 0.33/0.62 thf(fact_6092_dbl__dec__simps_I2_J,axiom,
% 0.33/0.62 ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 0.33/0.62 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(2)
% 0.33/0.62 thf(fact_6093_dbl__dec__simps_I2_J,axiom,
% 0.33/0.62 ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 0.33/0.62 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(2)
% 0.33/0.62 thf(fact_6094_dbl__dec__simps_I2_J,axiom,
% 0.33/0.62 ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 0.33/0.62 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(2)
% 0.33/0.62 thf(fact_6095_dbl__dec__simps_I2_J,axiom,
% 0.33/0.62 ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 0.33/0.62 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(2)
% 0.33/0.62 thf(fact_6096_dbl__dec__simps_I1_J,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(1)
% 0.33/0.62 thf(fact_6097_dbl__dec__simps_I1_J,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.62 = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(1)
% 0.33/0.62 thf(fact_6098_dbl__dec__simps_I1_J,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.33/0.62 = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(1)
% 0.33/0.62 thf(fact_6099_dbl__dec__simps_I1_J,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.33/0.62 = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(1)
% 0.33/0.62 thf(fact_6100_dbl__dec__simps_I1_J,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.33/0.62 = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_simps(1)
% 0.33/0.62 thf(fact_6101_dbl__inc__simps_I1_J,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(1)
% 0.33/0.62 thf(fact_6102_dbl__inc__simps_I1_J,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.62 = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(1)
% 0.33/0.62 thf(fact_6103_dbl__inc__simps_I1_J,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.33/0.62 = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(1)
% 0.33/0.62 thf(fact_6104_dbl__inc__simps_I1_J,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.33/0.62 = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(1)
% 0.33/0.62 thf(fact_6105_dbl__inc__simps_I1_J,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.33/0.62 = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_simps(1)
% 0.33/0.62 thf(fact_6106_add__neg__numeral__special_I5_J,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % add_neg_numeral_special(5)
% 0.33/0.62 thf(fact_6107_add__neg__numeral__special_I5_J,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % add_neg_numeral_special(5)
% 0.33/0.62 thf(fact_6108_add__neg__numeral__special_I5_J,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.33/0.62 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % add_neg_numeral_special(5)
% 0.33/0.62 thf(fact_6109_add__neg__numeral__special_I5_J,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.33/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % add_neg_numeral_special(5)
% 0.33/0.62 thf(fact_6110_add__neg__numeral__special_I5_J,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.33/0.62 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % add_neg_numeral_special(5)
% 0.33/0.62 thf(fact_6111_add__neg__numeral__special_I6_J,axiom,
% 0.33/0.62 ! [M: num] :
% 0.33/0.62 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % add_neg_numeral_special(6)
% 0.33/0.62 thf(fact_6112_add__neg__numeral__special_I6_J,axiom,
% 0.33/0.62 ! [M: num] :
% 0.33/0.62 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % add_neg_numeral_special(6)
% 0.33/0.62 thf(fact_6113_add__neg__numeral__special_I6_J,axiom,
% 0.33/0.62 ! [M: num] :
% 0.33/0.62 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.62 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % add_neg_numeral_special(6)
% 0.33/0.62 thf(fact_6114_add__neg__numeral__special_I6_J,axiom,
% 0.33/0.62 ! [M: num] :
% 0.33/0.62 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % add_neg_numeral_special(6)
% 0.33/0.62 thf(fact_6115_add__neg__numeral__special_I6_J,axiom,
% 0.33/0.62 ! [M: num] :
% 0.33/0.62 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.62 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % add_neg_numeral_special(6)
% 0.33/0.62 thf(fact_6116_diff__numeral__special_I5_J,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % diff_numeral_special(5)
% 0.33/0.62 thf(fact_6117_diff__numeral__special_I5_J,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % diff_numeral_special(5)
% 0.33/0.62 thf(fact_6118_diff__numeral__special_I5_J,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
% 0.33/0.62 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % diff_numeral_special(5)
% 0.33/0.62 thf(fact_6119_diff__numeral__special_I5_J,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
% 0.33/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % diff_numeral_special(5)
% 0.33/0.62 thf(fact_6120_diff__numeral__special_I5_J,axiom,
% 0.33/0.62 ! [N: num] :
% 0.33/0.62 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
% 0.33/0.62 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % diff_numeral_special(5)
% 0.33/0.62 thf(fact_6121_diff__numeral__special_I6_J,axiom,
% 0.33/0.62 ! [M: num] :
% 0.33/0.62 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.62 = ( numeral_numeral_real @ ( inc @ M ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % diff_numeral_special(6)
% 0.33/0.62 thf(fact_6122_diff__numeral__special_I6_J,axiom,
% 0.33/0.62 ! [M: num] :
% 0.33/0.62 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.62 = ( numeral_numeral_int @ ( inc @ M ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % diff_numeral_special(6)
% 0.33/0.62 thf(fact_6123_diff__numeral__special_I6_J,axiom,
% 0.33/0.62 ! [M: num] :
% 0.33/0.62 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.62 = ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % diff_numeral_special(6)
% 0.33/0.62 thf(fact_6124_diff__numeral__special_I6_J,axiom,
% 0.33/0.62 ! [M: num] :
% 0.33/0.62 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.62 = ( numeral_numeral_rat @ ( inc @ M ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % diff_numeral_special(6)
% 0.33/0.62 thf(fact_6125_diff__numeral__special_I6_J,axiom,
% 0.33/0.62 ! [M: num] :
% 0.33/0.62 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.62 = ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % diff_numeral_special(6)
% 0.33/0.62 thf(fact_6126_push__bit__numeral,axiom,
% 0.33/0.62 ! [L: num,K: num] :
% 0.33/0.62 ( ( bit_se547839408752420682it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ K ) )
% 0.33/0.62 = ( bit_se547839408752420682it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % push_bit_numeral
% 0.33/0.62 thf(fact_6127_push__bit__numeral,axiom,
% 0.33/0.62 ! [L: num,K: num] :
% 0.33/0.62 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ K ) )
% 0.33/0.62 = ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % push_bit_numeral
% 0.33/0.62 thf(fact_6128_push__bit__numeral,axiom,
% 0.33/0.62 ! [L: num,K: num] :
% 0.33/0.62 ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ L ) @ ( numera6620942414471956472nteger @ K ) )
% 0.33/0.62 = ( bit_se7788150548672797655nteger @ ( pred_numeral @ L ) @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % push_bit_numeral
% 0.33/0.62 thf(fact_6129_push__bit__numeral,axiom,
% 0.33/0.62 ! [L: num,K: num] :
% 0.33/0.62 ( ( bit_se6611745700429515170atural @ ( numeral_numeral_nat @ L ) @ ( numera5444537566228673987atural @ K ) )
% 0.33/0.62 = ( bit_se6611745700429515170atural @ ( pred_numeral @ L ) @ ( numera5444537566228673987atural @ ( bit0 @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % push_bit_numeral
% 0.33/0.62 thf(fact_6130_push__bit__minus__numeral,axiom,
% 0.33/0.62 ! [L: num,K: num] :
% 0.33/0.62 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.62 = ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % push_bit_minus_numeral
% 0.33/0.62 thf(fact_6131_push__bit__minus__numeral,axiom,
% 0.33/0.62 ! [L: num,K: num] :
% 0.33/0.62 ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.33/0.62 = ( bit_se7788150548672797655nteger @ ( pred_numeral @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % push_bit_minus_numeral
% 0.33/0.62 thf(fact_6132_signed__take__bit__numeral__bit0,axiom,
% 0.33/0.62 ! [L: num,K: num] :
% 0.33/0.62 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.33/0.62 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % signed_take_bit_numeral_bit0
% 0.33/0.62 thf(fact_6133_signed__take__bit__numeral__minus__bit0,axiom,
% 0.33/0.62 ! [L: num,K: num] :
% 0.33/0.62 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.33/0.62 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % signed_take_bit_numeral_minus_bit0
% 0.33/0.62 thf(fact_6134_num__induct,axiom,
% 0.33/0.62 ! [P: num > $o,X: num] :
% 0.33/0.62 ( ( P @ one )
% 0.33/0.62 => ( ! [X4: num] :
% 0.33/0.62 ( ( P @ X4 )
% 0.33/0.62 => ( P @ ( inc @ X4 ) ) )
% 0.33/0.62 => ( P @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % num_induct
% 0.33/0.62 thf(fact_6135_add__inc,axiom,
% 0.33/0.62 ! [X: num,Y: num] :
% 0.33/0.62 ( ( plus_plus_num @ X @ ( inc @ Y ) )
% 0.33/0.62 = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % add_inc
% 0.33/0.62 thf(fact_6136_numeral__eq__Suc,axiom,
% 0.33/0.62 ( numeral_numeral_nat
% 0.33/0.62 = ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_eq_Suc
% 0.33/0.62 thf(fact_6137_inc_Osimps_I1_J,axiom,
% 0.33/0.62 ( ( inc @ one )
% 0.33/0.62 = ( bit0 @ one ) ) ).
% 0.33/0.62
% 0.33/0.62 % inc.simps(1)
% 0.33/0.62 thf(fact_6138_inc_Osimps_I3_J,axiom,
% 0.33/0.62 ! [X: num] :
% 0.33/0.62 ( ( inc @ ( bit1 @ X ) )
% 0.33/0.62 = ( bit0 @ ( inc @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % inc.simps(3)
% 0.33/0.62 thf(fact_6139_inc_Osimps_I2_J,axiom,
% 0.33/0.62 ! [X: num] :
% 0.33/0.62 ( ( inc @ ( bit0 @ X ) )
% 0.33/0.62 = ( bit1 @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % inc.simps(2)
% 0.33/0.62 thf(fact_6140_add__One,axiom,
% 0.33/0.62 ! [X: num] :
% 0.33/0.62 ( ( plus_plus_num @ X @ one )
% 0.33/0.62 = ( inc @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % add_One
% 0.33/0.62 thf(fact_6141_mult__inc,axiom,
% 0.33/0.62 ! [X: num,Y: num] :
% 0.33/0.62 ( ( times_times_num @ X @ ( inc @ Y ) )
% 0.33/0.62 = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % mult_inc
% 0.33/0.62 thf(fact_6142_pred__numeral__def,axiom,
% 0.33/0.62 ( pred_numeral
% 0.33/0.62 = ( ^ [K2: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ one_one_nat ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % pred_numeral_def
% 0.33/0.62 thf(fact_6143_numeral__inc,axiom,
% 0.33/0.62 ! [X: num] :
% 0.33/0.62 ( ( numera6690914467698888265omplex @ ( inc @ X ) )
% 0.33/0.62 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_inc
% 0.33/0.62 thf(fact_6144_numeral__inc,axiom,
% 0.33/0.62 ! [X: num] :
% 0.33/0.62 ( ( numeral_numeral_real @ ( inc @ X ) )
% 0.33/0.62 = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_inc
% 0.33/0.62 thf(fact_6145_numeral__inc,axiom,
% 0.33/0.62 ! [X: num] :
% 0.33/0.62 ( ( numeral_numeral_rat @ ( inc @ X ) )
% 0.33/0.62 = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_inc
% 0.33/0.62 thf(fact_6146_numeral__inc,axiom,
% 0.33/0.62 ! [X: num] :
% 0.33/0.62 ( ( numeral_numeral_nat @ ( inc @ X ) )
% 0.33/0.62 = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_inc
% 0.33/0.62 thf(fact_6147_numeral__inc,axiom,
% 0.33/0.62 ! [X: num] :
% 0.33/0.62 ( ( numeral_numeral_int @ ( inc @ X ) )
% 0.33/0.62 = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_inc
% 0.33/0.62 thf(fact_6148_pi__less__4,axiom,
% 0.33/0.62 ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % pi_less_4
% 0.33/0.62 thf(fact_6149_pi__ge__two,axiom,
% 0.33/0.62 ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 0.33/0.62
% 0.33/0.62 % pi_ge_two
% 0.33/0.62 thf(fact_6150_pi__half__neq__two,axiom,
% 0.33/0.62 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.62 != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % pi_half_neq_two
% 0.33/0.62 thf(fact_6151_dbl__inc__def,axiom,
% 0.33/0.62 ( neg_nu8557863876264182079omplex
% 0.33/0.62 = ( ^ [X3: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_def
% 0.33/0.62 thf(fact_6152_dbl__inc__def,axiom,
% 0.33/0.62 ( neg_nu8295874005876285629c_real
% 0.33/0.62 = ( ^ [X3: real] : ( plus_plus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_def
% 0.33/0.62 thf(fact_6153_dbl__inc__def,axiom,
% 0.33/0.62 ( neg_nu5219082963157363817nc_rat
% 0.33/0.62 = ( ^ [X3: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_def
% 0.33/0.62 thf(fact_6154_dbl__inc__def,axiom,
% 0.33/0.62 ( neg_nu5851722552734809277nc_int
% 0.33/0.62 = ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_inc_def
% 0.33/0.62 thf(fact_6155_fact__numeral,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
% 0.33/0.62 = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % fact_numeral
% 0.33/0.62 thf(fact_6156_fact__numeral,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
% 0.33/0.62 = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % fact_numeral
% 0.33/0.62 thf(fact_6157_fact__numeral,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
% 0.33/0.62 = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % fact_numeral
% 0.33/0.62 thf(fact_6158_fact__numeral,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
% 0.33/0.62 = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % fact_numeral
% 0.33/0.62 thf(fact_6159_fact__numeral,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
% 0.33/0.62 = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % fact_numeral
% 0.33/0.62 thf(fact_6160_pi__half__neq__zero,axiom,
% 0.33/0.62 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.62 != zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % pi_half_neq_zero
% 0.33/0.62 thf(fact_6161_pi__half__less__two,axiom,
% 0.33/0.62 ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.33/0.62
% 0.33/0.62 % pi_half_less_two
% 0.33/0.62 thf(fact_6162_pi__half__le__two,axiom,
% 0.33/0.62 ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.33/0.62
% 0.33/0.62 % pi_half_le_two
% 0.33/0.62 thf(fact_6163_take__bit__numeral__minus__bit1,axiom,
% 0.33/0.62 ! [L: num,K: num] :
% 0.33/0.62 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.33/0.62 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % take_bit_numeral_minus_bit1
% 0.33/0.62 thf(fact_6164_dbl__dec__def,axiom,
% 0.33/0.62 ( neg_nu6511756317524482435omplex
% 0.33/0.62 = ( ^ [X3: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_def
% 0.33/0.62 thf(fact_6165_dbl__dec__def,axiom,
% 0.33/0.62 ( neg_nu6075765906172075777c_real
% 0.33/0.62 = ( ^ [X3: real] : ( minus_minus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_def
% 0.33/0.62 thf(fact_6166_dbl__dec__def,axiom,
% 0.33/0.62 ( neg_nu3179335615603231917ec_rat
% 0.33/0.62 = ( ^ [X3: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_def
% 0.33/0.62 thf(fact_6167_dbl__dec__def,axiom,
% 0.33/0.62 ( neg_nu3811975205180677377ec_int
% 0.33/0.62 = ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % dbl_dec_def
% 0.33/0.62 thf(fact_6168_pi__half__gt__zero,axiom,
% 0.33/0.62 ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % pi_half_gt_zero
% 0.33/0.62 thf(fact_6169_pi__half__ge__zero,axiom,
% 0.33/0.62 ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % pi_half_ge_zero
% 0.33/0.62 thf(fact_6170_m2pi__less__pi,axiom,
% 0.33/0.62 ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 0.33/0.62
% 0.33/0.62 % m2pi_less_pi
% 0.33/0.62 thf(fact_6171_arctan__ubound,axiom,
% 0.33/0.62 ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % arctan_ubound
% 0.33/0.62 thf(fact_6172_arctan__one,axiom,
% 0.33/0.62 ( ( arctan @ one_one_real )
% 0.33/0.62 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % arctan_one
% 0.33/0.62 thf(fact_6173_take__bit__numeral__bit0,axiom,
% 0.33/0.62 ! [L: num,K: num] :
% 0.33/0.62 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.33/0.62 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % take_bit_numeral_bit0
% 0.33/0.62 thf(fact_6174_take__bit__numeral__bit0,axiom,
% 0.33/0.62 ! [L: num,K: num] :
% 0.33/0.62 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.33/0.62 = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % take_bit_numeral_bit0
% 0.33/0.62 thf(fact_6175_minus__pi__half__less__zero,axiom,
% 0.33/0.62 ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 0.33/0.62
% 0.33/0.62 % minus_pi_half_less_zero
% 0.33/0.62 thf(fact_6176_arctan__bounded,axiom,
% 0.33/0.62 ! [Y: real] :
% 0.33/0.62 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
% 0.33/0.62 & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % arctan_bounded
% 0.33/0.62 thf(fact_6177_arctan__lbound,axiom,
% 0.33/0.62 ! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).
% 0.33/0.62
% 0.33/0.62 % arctan_lbound
% 0.33/0.62 thf(fact_6178_take__bit__numeral__minus__bit0,axiom,
% 0.33/0.62 ! [L: num,K: num] :
% 0.33/0.62 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.33/0.62 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % take_bit_numeral_minus_bit0
% 0.33/0.62 thf(fact_6179_machin__Euler,axiom,
% 0.33/0.62 ( ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.33/0.62 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % machin_Euler
% 0.33/0.62 thf(fact_6180_machin,axiom,
% 0.33/0.62 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % machin
% 0.33/0.62 thf(fact_6181_take__bit__numeral__bit1,axiom,
% 0.33/0.62 ! [L: num,K: num] :
% 0.33/0.62 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.33/0.62 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % take_bit_numeral_bit1
% 0.33/0.62 thf(fact_6182_take__bit__numeral__bit1,axiom,
% 0.33/0.62 ! [L: num,K: num] :
% 0.33/0.62 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.33/0.62 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 0.33/0.62
% 0.33/0.62 % take_bit_numeral_bit1
% 0.33/0.62 thf(fact_6183_sin__cos__npi,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_cos_npi
% 0.33/0.62 thf(fact_6184_cos__pi__eq__zero,axiom,
% 0.33/0.62 ! [M: nat] :
% 0.33/0.62 ( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % cos_pi_eq_zero
% 0.33/0.62 thf(fact_6185_cos__zero__iff,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ( cos_real @ X )
% 0.33/0.62 = zero_zero_real )
% 0.33/0.62 = ( ? [N3: nat] :
% 0.33/0.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 0.33/0.62 & ( X
% 0.33/0.62 = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.62 | ? [N3: nat] :
% 0.33/0.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 0.33/0.62 & ( X
% 0.33/0.62 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_zero_iff
% 0.33/0.62 thf(fact_6186_cot__less__zero,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.33/0.62 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.62 => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cot_less_zero
% 0.33/0.62 thf(fact_6187_cos__zero__lemma,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.62 => ( ( ( cos_real @ X )
% 0.33/0.62 = zero_zero_real )
% 0.33/0.62 => ? [N2: nat] :
% 0.33/0.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.62 & ( X
% 0.33/0.62 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_zero_lemma
% 0.33/0.62 thf(fact_6188_sin__zero__iff,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ( sin_real @ X )
% 0.33/0.62 = zero_zero_real )
% 0.33/0.62 = ( ? [N3: nat] :
% 0.33/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 0.33/0.62 & ( X
% 0.33/0.62 = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.62 | ? [N3: nat] :
% 0.33/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 0.33/0.62 & ( X
% 0.33/0.62 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_zero_iff
% 0.33/0.62 thf(fact_6189_sin__zero,axiom,
% 0.33/0.62 ( ( sin_real @ zero_zero_real )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % sin_zero
% 0.33/0.62 thf(fact_6190_cot__zero,axiom,
% 0.33/0.62 ( ( cot_real @ zero_zero_real )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % cot_zero
% 0.33/0.62 thf(fact_6191_cos__zero,axiom,
% 0.33/0.62 ( ( cos_complex @ zero_zero_complex )
% 0.33/0.62 = one_one_complex ) ).
% 0.33/0.62
% 0.33/0.62 % cos_zero
% 0.33/0.62 thf(fact_6192_cos__zero,axiom,
% 0.33/0.62 ( ( cos_real @ zero_zero_real )
% 0.33/0.62 = one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % cos_zero
% 0.33/0.62 thf(fact_6193_sin__pi__minus,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( sin_real @ ( minus_minus_real @ pi @ X ) )
% 0.33/0.62 = ( sin_real @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_pi_minus
% 0.33/0.62 thf(fact_6194_sin__of__real__pi,axiom,
% 0.33/0.62 ( ( sin_real @ ( real_V1803761363581548252l_real @ pi ) )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % sin_of_real_pi
% 0.33/0.62 thf(fact_6195_sin__of__real__pi,axiom,
% 0.33/0.62 ( ( sin_complex @ ( real_V4546457046886955230omplex @ pi ) )
% 0.33/0.62 = zero_zero_complex ) ).
% 0.33/0.62
% 0.33/0.62 % sin_of_real_pi
% 0.33/0.62 thf(fact_6196_cos__pi,axiom,
% 0.33/0.62 ( ( cos_real @ pi )
% 0.33/0.62 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_pi
% 0.33/0.62 thf(fact_6197_cos__periodic__pi,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( cos_real @ ( plus_plus_real @ X @ pi ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_periodic_pi
% 0.33/0.62 thf(fact_6198_cos__periodic__pi2,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( cos_real @ ( plus_plus_real @ pi @ X ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_periodic_pi2
% 0.33/0.62 thf(fact_6199_sin__periodic__pi,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( sin_real @ ( plus_plus_real @ X @ pi ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_periodic_pi
% 0.33/0.62 thf(fact_6200_sin__periodic__pi2,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( sin_real @ ( plus_plus_real @ pi @ X ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_periodic_pi2
% 0.33/0.62 thf(fact_6201_cos__minus__pi,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( cos_real @ ( minus_minus_real @ X @ pi ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_minus_pi
% 0.33/0.62 thf(fact_6202_cos__pi__minus,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( cos_real @ ( minus_minus_real @ pi @ X ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_pi_minus
% 0.33/0.62 thf(fact_6203_sin__minus__pi,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( sin_real @ ( minus_minus_real @ X @ pi ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_minus_pi
% 0.33/0.62 thf(fact_6204_sin__cos__squared__add3,axiom,
% 0.33/0.62 ! [X: complex] :
% 0.33/0.62 ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
% 0.33/0.62 = one_one_complex ) ).
% 0.33/0.62
% 0.33/0.62 % sin_cos_squared_add3
% 0.33/0.62 thf(fact_6205_sin__cos__squared__add3,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
% 0.33/0.62 = one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % sin_cos_squared_add3
% 0.33/0.62 thf(fact_6206_cos__of__real__pi,axiom,
% 0.33/0.62 ( ( cos_real @ ( real_V1803761363581548252l_real @ pi ) )
% 0.33/0.62 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_of_real_pi
% 0.33/0.62 thf(fact_6207_cos__of__real__pi,axiom,
% 0.33/0.62 ( ( cos_complex @ ( real_V4546457046886955230omplex @ pi ) )
% 0.33/0.62 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_of_real_pi
% 0.33/0.62 thf(fact_6208_sin__npi2,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % sin_npi2
% 0.33/0.62 thf(fact_6209_sin__npi,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % sin_npi
% 0.33/0.62 thf(fact_6210_cot__npi,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % cot_npi
% 0.33/0.62 thf(fact_6211_cos__pi__half,axiom,
% 0.33/0.62 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % cos_pi_half
% 0.33/0.62 thf(fact_6212_sin__two__pi,axiom,
% 0.33/0.62 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % sin_two_pi
% 0.33/0.62 thf(fact_6213_sin__pi__half,axiom,
% 0.33/0.62 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % sin_pi_half
% 0.33/0.62 thf(fact_6214_cos__two__pi,axiom,
% 0.33/0.62 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.33/0.62 = one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % cos_two_pi
% 0.33/0.62 thf(fact_6215_cos__periodic,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.33/0.62 = ( cos_real @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_periodic
% 0.33/0.62 thf(fact_6216_sin__periodic,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.33/0.62 = ( sin_real @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_periodic
% 0.33/0.62 thf(fact_6217_cos__2pi__minus,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 0.33/0.62 = ( cos_real @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_2pi_minus
% 0.33/0.62 thf(fact_6218_cos__npi2,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.33/0.62 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_npi2
% 0.33/0.62 thf(fact_6219_cos__npi,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.33/0.62 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_npi
% 0.33/0.62 thf(fact_6220_cot__periodic,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.33/0.62 = ( cot_real @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % cot_periodic
% 0.33/0.62 thf(fact_6221_sin__cos__squared__add,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % sin_cos_squared_add
% 0.33/0.62 thf(fact_6222_sin__cos__squared__add,axiom,
% 0.33/0.62 ! [X: complex] :
% 0.33/0.62 ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = one_one_complex ) ).
% 0.33/0.62
% 0.33/0.62 % sin_cos_squared_add
% 0.33/0.62 thf(fact_6223_sin__cos__squared__add2,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % sin_cos_squared_add2
% 0.33/0.62 thf(fact_6224_sin__cos__squared__add2,axiom,
% 0.33/0.62 ! [X: complex] :
% 0.33/0.62 ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = one_one_complex ) ).
% 0.33/0.62
% 0.33/0.62 % sin_cos_squared_add2
% 0.33/0.62 thf(fact_6225_cos__of__real__pi__half,axiom,
% 0.33/0.62 ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % cos_of_real_pi_half
% 0.33/0.62 thf(fact_6226_cos__of__real__pi__half,axiom,
% 0.33/0.62 ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = zero_zero_complex ) ).
% 0.33/0.62
% 0.33/0.62 % cos_of_real_pi_half
% 0.33/0.62 thf(fact_6227_sin__of__real__pi__half,axiom,
% 0.33/0.62 ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % sin_of_real_pi_half
% 0.33/0.62 thf(fact_6228_sin__of__real__pi__half,axiom,
% 0.33/0.62 ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = one_one_complex ) ).
% 0.33/0.62
% 0.33/0.62 % sin_of_real_pi_half
% 0.33/0.62 thf(fact_6229_sin__2npi,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % sin_2npi
% 0.33/0.62 thf(fact_6230_cos__2npi,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 0.33/0.62 = one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % cos_2npi
% 0.33/0.62 thf(fact_6231_sin__2pi__minus,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_2pi_minus
% 0.33/0.62 thf(fact_6232_cos__3over2__pi,axiom,
% 0.33/0.62 ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % cos_3over2_pi
% 0.33/0.62 thf(fact_6233_sin__3over2__pi,axiom,
% 0.33/0.62 ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.33/0.62 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_3over2_pi
% 0.33/0.62 thf(fact_6234_sin__add,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( sin_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.62 = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_add
% 0.33/0.62 thf(fact_6235_cos__one__sin__zero,axiom,
% 0.33/0.62 ! [X: complex] :
% 0.33/0.62 ( ( ( cos_complex @ X )
% 0.33/0.62 = one_one_complex )
% 0.33/0.62 => ( ( sin_complex @ X )
% 0.33/0.62 = zero_zero_complex ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_one_sin_zero
% 0.33/0.62 thf(fact_6236_cos__one__sin__zero,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ( cos_real @ X )
% 0.33/0.62 = one_one_real )
% 0.33/0.62 => ( ( sin_real @ X )
% 0.33/0.62 = zero_zero_real ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_one_sin_zero
% 0.33/0.62 thf(fact_6237_cot__def,axiom,
% 0.33/0.62 ( cot_complex
% 0.33/0.62 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( cos_complex @ X3 ) @ ( sin_complex @ X3 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cot_def
% 0.33/0.62 thf(fact_6238_cot__def,axiom,
% 0.33/0.62 ( cot_real
% 0.33/0.62 = ( ^ [X3: real] : ( divide_divide_real @ ( cos_real @ X3 ) @ ( sin_real @ X3 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cot_def
% 0.33/0.62 thf(fact_6239_polar__Ex,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ? [R2: real,A4: real] :
% 0.33/0.62 ( ( X
% 0.33/0.62 = ( times_times_real @ R2 @ ( cos_real @ A4 ) ) )
% 0.33/0.62 & ( Y
% 0.33/0.62 = ( times_times_real @ R2 @ ( sin_real @ A4 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % polar_Ex
% 0.33/0.62 thf(fact_6240_sin__diff,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( sin_real @ ( minus_minus_real @ X @ Y ) )
% 0.33/0.62 = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_diff
% 0.33/0.62 thf(fact_6241_cos__add,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.62 = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_add
% 0.33/0.62 thf(fact_6242_cos__diff,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
% 0.33/0.62 = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_diff
% 0.33/0.62 thf(fact_6243_sin__zero__norm__cos__one,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ( sin_real @ X )
% 0.33/0.62 = zero_zero_real )
% 0.33/0.62 => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
% 0.33/0.62 = one_one_real ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_zero_norm_cos_one
% 0.33/0.62 thf(fact_6244_sin__zero__norm__cos__one,axiom,
% 0.33/0.62 ! [X: complex] :
% 0.33/0.62 ( ( ( sin_complex @ X )
% 0.33/0.62 = zero_zero_complex )
% 0.33/0.62 => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
% 0.33/0.62 = one_one_real ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_zero_norm_cos_one
% 0.33/0.62 thf(fact_6245_sin__zero__abs__cos__one,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ( sin_real @ X )
% 0.33/0.62 = zero_zero_real )
% 0.33/0.62 => ( ( abs_abs_real @ ( cos_real @ X ) )
% 0.33/0.62 = one_one_real ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_zero_abs_cos_one
% 0.33/0.62 thf(fact_6246_sin__double,axiom,
% 0.33/0.62 ! [X: complex] :
% 0.33/0.62 ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.62 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X ) ) @ ( cos_complex @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_double
% 0.33/0.62 thf(fact_6247_sin__double,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.62 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X ) ) @ ( cos_real @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_double
% 0.33/0.62 thf(fact_6248_sin__le__one,axiom,
% 0.33/0.62 ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % sin_le_one
% 0.33/0.62 thf(fact_6249_cos__le__one,axiom,
% 0.33/0.62 ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % cos_le_one
% 0.33/0.62 thf(fact_6250_sin__cos__le1,axiom,
% 0.33/0.62 ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % sin_cos_le1
% 0.33/0.62 thf(fact_6251_cos__squared__eq,axiom,
% 0.33/0.62 ! [X: complex] :
% 0.33/0.62 ( ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.62 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_squared_eq
% 0.33/0.62 thf(fact_6252_cos__squared__eq,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.62 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_squared_eq
% 0.33/0.62 thf(fact_6253_sin__squared__eq,axiom,
% 0.33/0.62 ! [X: complex] :
% 0.33/0.62 ( ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.62 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_squared_eq
% 0.33/0.62 thf(fact_6254_sin__squared__eq,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.62 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_squared_eq
% 0.33/0.62 thf(fact_6255_sin__ge__minus__one,axiom,
% 0.33/0.62 ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_ge_minus_one
% 0.33/0.62 thf(fact_6256_cos__ge__minus__one,axiom,
% 0.33/0.62 ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_ge_minus_one
% 0.33/0.62 thf(fact_6257_abs__sin__le__one,axiom,
% 0.33/0.62 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % abs_sin_le_one
% 0.33/0.62 thf(fact_6258_abs__cos__le__one,axiom,
% 0.33/0.62 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % abs_cos_le_one
% 0.33/0.62 thf(fact_6259_sin__times__sin,axiom,
% 0.33/0.62 ! [W: complex,Z2: complex] :
% 0.33/0.62 ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z2 ) )
% 0.33/0.62 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_times_sin
% 0.33/0.62 thf(fact_6260_sin__times__sin,axiom,
% 0.33/0.62 ! [W: real,Z2: real] :
% 0.33/0.62 ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z2 ) )
% 0.33/0.62 = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_times_sin
% 0.33/0.62 thf(fact_6261_sin__times__cos,axiom,
% 0.33/0.62 ! [W: complex,Z2: complex] :
% 0.33/0.62 ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z2 ) )
% 0.33/0.62 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_times_cos
% 0.33/0.62 thf(fact_6262_sin__times__cos,axiom,
% 0.33/0.62 ! [W: real,Z2: real] :
% 0.33/0.62 ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z2 ) )
% 0.33/0.62 = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_times_cos
% 0.33/0.62 thf(fact_6263_cos__times__sin,axiom,
% 0.33/0.62 ! [W: complex,Z2: complex] :
% 0.33/0.62 ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z2 ) )
% 0.33/0.62 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_times_sin
% 0.33/0.62 thf(fact_6264_cos__times__sin,axiom,
% 0.33/0.62 ! [W: real,Z2: real] :
% 0.33/0.62 ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z2 ) )
% 0.33/0.62 = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_times_sin
% 0.33/0.62 thf(fact_6265_sin__plus__sin,axiom,
% 0.33/0.62 ! [W: complex,Z2: complex] :
% 0.33/0.62 ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z2 ) )
% 0.33/0.62 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_plus_sin
% 0.33/0.62 thf(fact_6266_sin__plus__sin,axiom,
% 0.33/0.62 ! [W: real,Z2: real] :
% 0.33/0.62 ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z2 ) )
% 0.33/0.62 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_plus_sin
% 0.33/0.62 thf(fact_6267_sin__diff__sin,axiom,
% 0.33/0.62 ! [W: complex,Z2: complex] :
% 0.33/0.62 ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z2 ) )
% 0.33/0.62 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_diff_sin
% 0.33/0.62 thf(fact_6268_sin__diff__sin,axiom,
% 0.33/0.62 ! [W: real,Z2: real] :
% 0.33/0.62 ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z2 ) )
% 0.33/0.62 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_diff_sin
% 0.33/0.62 thf(fact_6269_cos__diff__cos,axiom,
% 0.33/0.62 ! [W: complex,Z2: complex] :
% 0.33/0.62 ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z2 ) )
% 0.33/0.62 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z2 @ W ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_diff_cos
% 0.33/0.62 thf(fact_6270_cos__diff__cos,axiom,
% 0.33/0.62 ! [W: real,Z2: real] :
% 0.33/0.62 ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z2 ) )
% 0.33/0.62 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z2 @ W ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_diff_cos
% 0.33/0.62 thf(fact_6271_cos__double,axiom,
% 0.33/0.62 ! [X: complex] :
% 0.33/0.62 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.62 = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_double
% 0.33/0.62 thf(fact_6272_cos__double,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.62 = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_double
% 0.33/0.62 thf(fact_6273_sin__cos__eq,axiom,
% 0.33/0.62 ( sin_real
% 0.33/0.62 = ( ^ [X3: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_cos_eq
% 0.33/0.62 thf(fact_6274_sin__cos__eq,axiom,
% 0.33/0.62 ( sin_complex
% 0.33/0.62 = ( ^ [X3: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_cos_eq
% 0.33/0.62 thf(fact_6275_cos__sin__eq,axiom,
% 0.33/0.62 ( cos_real
% 0.33/0.62 = ( ^ [X3: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_sin_eq
% 0.33/0.62 thf(fact_6276_cos__sin__eq,axiom,
% 0.33/0.62 ( cos_complex
% 0.33/0.62 = ( ^ [X3: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_sin_eq
% 0.33/0.62 thf(fact_6277_cos__double__sin,axiom,
% 0.33/0.62 ! [W: complex] :
% 0.33/0.62 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 0.33/0.62 = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_double_sin
% 0.33/0.62 thf(fact_6278_cos__double__sin,axiom,
% 0.33/0.62 ! [W: real] :
% 0.33/0.62 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 0.33/0.62 = ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_double_sin
% 0.33/0.62 thf(fact_6279_minus__sin__cos__eq,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( uminus_uminus_real @ ( sin_real @ X ) )
% 0.33/0.62 = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % minus_sin_cos_eq
% 0.33/0.62 thf(fact_6280_minus__sin__cos__eq,axiom,
% 0.33/0.62 ! [X: complex] :
% 0.33/0.62 ( ( uminus1482373934393186551omplex @ ( sin_complex @ X ) )
% 0.33/0.62 = ( cos_complex @ ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % minus_sin_cos_eq
% 0.33/0.62 thf(fact_6281_cos__two__neq__zero,axiom,
% 0.33/0.62 ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.62 != zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % cos_two_neq_zero
% 0.33/0.62 thf(fact_6282_sincos__total__pi,axiom,
% 0.33/0.62 ! [Y: real,X: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.62 => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = one_one_real )
% 0.33/0.62 => ? [T3: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.33/0.62 & ( ord_less_eq_real @ T3 @ pi )
% 0.33/0.62 & ( X
% 0.33/0.62 = ( cos_real @ T3 ) )
% 0.33/0.62 & ( Y
% 0.33/0.62 = ( sin_real @ T3 ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sincos_total_pi
% 0.33/0.62 thf(fact_6283_sin__expansion__lemma,axiom,
% 0.33/0.62 ! [X: real,M: nat] :
% 0.33/0.62 ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.33/0.62 = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_expansion_lemma
% 0.33/0.62 thf(fact_6284_cos__expansion__lemma,axiom,
% 0.33/0.62 ! [X: real,M: nat] :
% 0.33/0.62 ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_expansion_lemma
% 0.33/0.62 thf(fact_6285_sin__gt__zero__02,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.62 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.62 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_gt_zero_02
% 0.33/0.62 thf(fact_6286_cos__two__less__zero,axiom,
% 0.33/0.62 ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.33/0.62
% 0.33/0.62 % cos_two_less_zero
% 0.33/0.62 thf(fact_6287_cos__is__zero,axiom,
% 0.33/0.62 ? [X4: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.33/0.62 & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.62 & ( ( cos_real @ X4 )
% 0.33/0.62 = zero_zero_real )
% 0.33/0.62 & ! [Y5: real] :
% 0.33/0.62 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 0.33/0.62 & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.62 & ( ( cos_real @ Y5 )
% 0.33/0.62 = zero_zero_real ) )
% 0.33/0.62 => ( Y5 = X4 ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_is_zero
% 0.33/0.62 thf(fact_6288_cos__two__le__zero,axiom,
% 0.33/0.62 ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.33/0.62
% 0.33/0.62 % cos_two_le_zero
% 0.33/0.62 thf(fact_6289_cos__total,axiom,
% 0.33/0.62 ! [Y: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.62 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.62 => ? [X4: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.33/0.62 & ( ord_less_eq_real @ X4 @ pi )
% 0.33/0.62 & ( ( cos_real @ X4 )
% 0.33/0.62 = Y )
% 0.33/0.62 & ! [Y5: real] :
% 0.33/0.62 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 0.33/0.62 & ( ord_less_eq_real @ Y5 @ pi )
% 0.33/0.62 & ( ( cos_real @ Y5 )
% 0.33/0.62 = Y ) )
% 0.33/0.62 => ( Y5 = X4 ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_total
% 0.33/0.62 thf(fact_6290_sincos__total__pi__half,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.62 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.62 => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = one_one_real )
% 0.33/0.62 => ? [T3: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.33/0.62 & ( ord_less_eq_real @ T3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 & ( X
% 0.33/0.62 = ( cos_real @ T3 ) )
% 0.33/0.62 & ( Y
% 0.33/0.62 = ( sin_real @ T3 ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sincos_total_pi_half
% 0.33/0.62 thf(fact_6291_sincos__total__2pi__le,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = one_one_real )
% 0.33/0.62 => ? [T3: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.33/0.62 & ( ord_less_eq_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.33/0.62 & ( X
% 0.33/0.62 = ( cos_real @ T3 ) )
% 0.33/0.62 & ( Y
% 0.33/0.62 = ( sin_real @ T3 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sincos_total_2pi_le
% 0.33/0.62 thf(fact_6292_sincos__total__2pi,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.62 = one_one_real )
% 0.33/0.62 => ~ ! [T3: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.33/0.62 => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.33/0.62 => ( ( X
% 0.33/0.62 = ( cos_real @ T3 ) )
% 0.33/0.62 => ( Y
% 0.33/0.62 != ( sin_real @ T3 ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sincos_total_2pi
% 0.33/0.62 thf(fact_6293_sin__pi__divide__n__ge__0,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( N != zero_zero_nat )
% 0.33/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_pi_divide_n_ge_0
% 0.33/0.62 thf(fact_6294_cos__times__cos,axiom,
% 0.33/0.62 ! [W: complex,Z2: complex] :
% 0.33/0.62 ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z2 ) )
% 0.33/0.62 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_times_cos
% 0.33/0.62 thf(fact_6295_cos__times__cos,axiom,
% 0.33/0.62 ! [W: real,Z2: real] :
% 0.33/0.62 ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z2 ) )
% 0.33/0.62 = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_times_cos
% 0.33/0.62 thf(fact_6296_cos__plus__cos,axiom,
% 0.33/0.62 ! [W: complex,Z2: complex] :
% 0.33/0.62 ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z2 ) )
% 0.33/0.62 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_plus_cos
% 0.33/0.62 thf(fact_6297_cos__plus__cos,axiom,
% 0.33/0.62 ! [W: real,Z2: real] :
% 0.33/0.62 ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z2 ) )
% 0.33/0.62 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_plus_cos
% 0.33/0.62 thf(fact_6298_sin__gt__zero2,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.62 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_gt_zero2
% 0.33/0.62 thf(fact_6299_sin__lt__zero,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ord_less_real @ pi @ X )
% 0.33/0.62 => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.33/0.62 => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_lt_zero
% 0.33/0.62 thf(fact_6300_cos__double__less__one,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.62 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.62 => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_double_less_one
% 0.33/0.62 thf(fact_6301_sin__30,axiom,
% 0.33/0.62 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.33/0.62 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_30
% 0.33/0.62 thf(fact_6302_cos__gt__zero,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.62 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_gt_zero
% 0.33/0.62 thf(fact_6303_sin__inj__pi,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.62 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.62 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 => ( ( ( sin_real @ X )
% 0.33/0.62 = ( sin_real @ Y ) )
% 0.33/0.62 => ( X = Y ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_inj_pi
% 0.33/0.62 thf(fact_6304_sin__mono__le__eq,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.62 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.62 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
% 0.33/0.62 = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_mono_le_eq
% 0.33/0.62 thf(fact_6305_sin__monotone__2pi__le,axiom,
% 0.33/0.62 ! [Y: real,X: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.62 => ( ( ord_less_eq_real @ Y @ X )
% 0.33/0.62 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 => ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_monotone_2pi_le
% 0.33/0.62 thf(fact_6306_cos__60,axiom,
% 0.33/0.62 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.33/0.62 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_60
% 0.33/0.62 thf(fact_6307_cos__double__cos,axiom,
% 0.33/0.62 ! [W: complex] :
% 0.33/0.62 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 0.33/0.62 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_double_cos
% 0.33/0.62 thf(fact_6308_cos__double__cos,axiom,
% 0.33/0.62 ! [W: real] :
% 0.33/0.62 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 0.33/0.62 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_double_cos
% 0.33/0.62 thf(fact_6309_cos__treble__cos,axiom,
% 0.33/0.62 ! [X: complex] :
% 0.33/0.62 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X ) )
% 0.33/0.62 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_treble_cos
% 0.33/0.62 thf(fact_6310_cos__treble__cos,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X ) )
% 0.33/0.62 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_treble_cos
% 0.33/0.62 thf(fact_6311_sin__le__zero,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ pi @ X )
% 0.33/0.62 => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.33/0.62 => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_le_zero
% 0.33/0.62 thf(fact_6312_sin__less__zero,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.33/0.62 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.62 => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_less_zero
% 0.33/0.62 thf(fact_6313_sin__monotone__2pi,axiom,
% 0.33/0.62 ! [Y: real,X: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.62 => ( ( ord_less_real @ Y @ X )
% 0.33/0.62 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_monotone_2pi
% 0.33/0.62 thf(fact_6314_sin__mono__less__eq,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.62 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.62 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
% 0.33/0.62 = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_mono_less_eq
% 0.33/0.62 thf(fact_6315_sin__total,axiom,
% 0.33/0.62 ! [Y: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.62 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.62 => ? [X4: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.33/0.62 & ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 & ( ( sin_real @ X4 )
% 0.33/0.62 = Y )
% 0.33/0.62 & ! [Y5: real] :
% 0.33/0.62 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 0.33/0.62 & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 & ( ( sin_real @ Y5 )
% 0.33/0.62 = Y ) )
% 0.33/0.62 => ( Y5 = X4 ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_total
% 0.33/0.62 thf(fact_6316_cos__gt__zero__pi,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.62 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_gt_zero_pi
% 0.33/0.62 thf(fact_6317_cos__ge__zero,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.62 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_ge_zero
% 0.33/0.62 thf(fact_6318_cos__one__2pi,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ( cos_real @ X )
% 0.33/0.62 = one_one_real )
% 0.33/0.62 = ( ? [X3: nat] :
% 0.33/0.62 ( X
% 0.33/0.62 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.33/0.62 | ? [X3: nat] :
% 0.33/0.62 ( X
% 0.33/0.62 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_one_2pi
% 0.33/0.62 thf(fact_6319_sin__pi__divide__n__gt__0,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.62 => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_pi_divide_n_gt_0
% 0.33/0.62 thf(fact_6320_cot__gt__zero,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.62 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cot_gt_zero
% 0.33/0.62 thf(fact_6321_sin__zero__lemma,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.62 => ( ( ( sin_real @ X )
% 0.33/0.62 = zero_zero_real )
% 0.33/0.62 => ? [N2: nat] :
% 0.33/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.62 & ( X
% 0.33/0.62 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_zero_lemma
% 0.33/0.62 thf(fact_6322_tan__double,axiom,
% 0.33/0.62 ! [X: complex] :
% 0.33/0.62 ( ( ( cos_complex @ X )
% 0.33/0.62 != zero_zero_complex )
% 0.33/0.62 => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.62 != zero_zero_complex )
% 0.33/0.62 => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.62 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % tan_double
% 0.33/0.62 thf(fact_6323_tan__double,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ( cos_real @ X )
% 0.33/0.62 != zero_zero_real )
% 0.33/0.62 => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.62 != zero_zero_real )
% 0.33/0.62 => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.62 = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % tan_double
% 0.33/0.62 thf(fact_6324_complex__unimodular__polar,axiom,
% 0.33/0.62 ! [Z2: complex] :
% 0.33/0.62 ( ( ( real_V1022390504157884413omplex @ Z2 )
% 0.33/0.62 = one_one_real )
% 0.33/0.62 => ~ ! [T3: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.33/0.62 => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.33/0.62 => ( Z2
% 0.33/0.62 != ( complex2 @ ( cos_real @ T3 ) @ ( sin_real @ T3 ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % complex_unimodular_polar
% 0.33/0.62 thf(fact_6325_cos__npi__int,axiom,
% 0.33/0.62 ! [N: int] :
% 0.33/0.62 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.33/0.62 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.33/0.62 = one_one_real ) )
% 0.33/0.62 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.33/0.62 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.33/0.62 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % cos_npi_int
% 0.33/0.62 thf(fact_6326_le__arcsin__iff,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.62 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.62 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
% 0.33/0.62 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 => ( ( ord_less_eq_real @ Y @ ( arcsin @ X ) )
% 0.33/0.62 = ( ord_less_eq_real @ ( sin_real @ Y ) @ X ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % le_arcsin_iff
% 0.33/0.62 thf(fact_6327_arcsin__le__iff,axiom,
% 0.33/0.62 ! [X: real,Y: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.62 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.62 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
% 0.33/0.62 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.62 => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y )
% 0.33/0.62 = ( ord_less_eq_real @ X @ ( sin_real @ Y ) ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % arcsin_le_iff
% 0.33/0.62 thf(fact_6328_arcsin__pi,axiom,
% 0.33/0.62 ! [Y: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.62 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.62 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 0.33/0.62 & ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
% 0.33/0.62 & ( ( sin_real @ ( arcsin @ Y ) )
% 0.33/0.62 = Y ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % arcsin_pi
% 0.33/0.62 thf(fact_6329_of__int__eq__iff,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ( ring_1_of_int_real @ W )
% 0.33/0.62 = ( ring_1_of_int_real @ Z2 ) )
% 0.33/0.62 = ( W = Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_iff
% 0.33/0.62 thf(fact_6330_of__int__eq__iff,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ( ring_1_of_int_rat @ W )
% 0.33/0.62 = ( ring_1_of_int_rat @ Z2 ) )
% 0.33/0.62 = ( W = Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_iff
% 0.33/0.62 thf(fact_6331_tan__zero,axiom,
% 0.33/0.62 ( ( tan_real @ zero_zero_real )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % tan_zero
% 0.33/0.62 thf(fact_6332_tan__periodic__pi,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( tan_real @ ( plus_plus_real @ X @ pi ) )
% 0.33/0.62 = ( tan_real @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % tan_periodic_pi
% 0.33/0.62 thf(fact_6333_of__int__floor__cancel,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.62 = X )
% 0.33/0.62 = ( ? [N3: int] :
% 0.33/0.62 ( X
% 0.33/0.62 = ( ring_1_of_int_real @ N3 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_floor_cancel
% 0.33/0.62 thf(fact_6334_of__int__floor__cancel,axiom,
% 0.33/0.62 ! [X: rat] :
% 0.33/0.62 ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) )
% 0.33/0.62 = X )
% 0.33/0.62 = ( ? [N3: int] :
% 0.33/0.62 ( X
% 0.33/0.62 = ( ring_1_of_int_rat @ N3 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_floor_cancel
% 0.33/0.62 thf(fact_6335_floor__of__int,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( archim6058952711729229775r_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.33/0.62 = Z2 ) ).
% 0.33/0.62
% 0.33/0.62 % floor_of_int
% 0.33/0.62 thf(fact_6336_floor__of__int,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( archim3151403230148437115or_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.33/0.62 = Z2 ) ).
% 0.33/0.62
% 0.33/0.62 % floor_of_int
% 0.33/0.62 thf(fact_6337_of__int__ceiling__cancel,axiom,
% 0.33/0.62 ! [X: rat] :
% 0.33/0.62 ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) )
% 0.33/0.62 = X )
% 0.33/0.62 = ( ? [N3: int] :
% 0.33/0.62 ( X
% 0.33/0.62 = ( ring_1_of_int_rat @ N3 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_ceiling_cancel
% 0.33/0.62 thf(fact_6338_of__int__ceiling__cancel,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
% 0.33/0.62 = X )
% 0.33/0.62 = ( ? [N3: int] :
% 0.33/0.62 ( X
% 0.33/0.62 = ( ring_1_of_int_real @ N3 ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_ceiling_cancel
% 0.33/0.62 thf(fact_6339_ceiling__of__int,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( archim2889992004027027881ng_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.33/0.62 = Z2 ) ).
% 0.33/0.62
% 0.33/0.62 % ceiling_of_int
% 0.33/0.62 thf(fact_6340_ceiling__of__int,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( archim7802044766580827645g_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.33/0.62 = Z2 ) ).
% 0.33/0.62
% 0.33/0.62 % ceiling_of_int
% 0.33/0.62 thf(fact_6341_of__int__eq__0__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ( ring_1_of_int_int @ Z2 )
% 0.33/0.62 = zero_zero_int )
% 0.33/0.62 = ( Z2 = zero_zero_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_0_iff
% 0.33/0.62 thf(fact_6342_of__int__eq__0__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ( ring_1_of_int_real @ Z2 )
% 0.33/0.62 = zero_zero_real )
% 0.33/0.62 = ( Z2 = zero_zero_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_0_iff
% 0.33/0.62 thf(fact_6343_of__int__eq__0__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ( ring_1_of_int_rat @ Z2 )
% 0.33/0.62 = zero_zero_rat )
% 0.33/0.62 = ( Z2 = zero_zero_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_0_iff
% 0.33/0.62 thf(fact_6344_of__int__0__eq__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( zero_zero_int
% 0.33/0.62 = ( ring_1_of_int_int @ Z2 ) )
% 0.33/0.62 = ( Z2 = zero_zero_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_0_eq_iff
% 0.33/0.62 thf(fact_6345_of__int__0__eq__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( zero_zero_real
% 0.33/0.62 = ( ring_1_of_int_real @ Z2 ) )
% 0.33/0.62 = ( Z2 = zero_zero_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_0_eq_iff
% 0.33/0.62 thf(fact_6346_of__int__0__eq__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( zero_zero_rat
% 0.33/0.62 = ( ring_1_of_int_rat @ Z2 ) )
% 0.33/0.62 = ( Z2 = zero_zero_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_0_eq_iff
% 0.33/0.62 thf(fact_6347_of__int__0,axiom,
% 0.33/0.62 ( ( ring_1_of_int_int @ zero_zero_int )
% 0.33/0.62 = zero_zero_int ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_0
% 0.33/0.62 thf(fact_6348_of__int__0,axiom,
% 0.33/0.62 ( ( ring_1_of_int_real @ zero_zero_int )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_0
% 0.33/0.62 thf(fact_6349_of__int__0,axiom,
% 0.33/0.62 ( ( ring_1_of_int_rat @ zero_zero_int )
% 0.33/0.62 = zero_zero_rat ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_0
% 0.33/0.62 thf(fact_6350_of__int__le__iff,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) )
% 0.33/0.62 = ( ord_less_eq_int @ W @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_iff
% 0.33/0.62 thf(fact_6351_of__int__le__iff,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z2 ) )
% 0.33/0.62 = ( ord_less_eq_int @ W @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_iff
% 0.33/0.62 thf(fact_6352_of__int__le__iff,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) )
% 0.33/0.62 = ( ord_less_eq_int @ W @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_iff
% 0.33/0.62 thf(fact_6353_of__int__numeral,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
% 0.33/0.62 = ( numera6690914467698888265omplex @ K ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_numeral
% 0.33/0.62 thf(fact_6354_of__int__numeral,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 0.33/0.62 = ( numeral_numeral_real @ K ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_numeral
% 0.33/0.62 thf(fact_6355_of__int__numeral,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 0.33/0.62 = ( numeral_numeral_rat @ K ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_numeral
% 0.33/0.62 thf(fact_6356_of__int__numeral,axiom,
% 0.33/0.62 ! [K: num] :
% 0.33/0.62 ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 0.33/0.62 = ( numeral_numeral_int @ K ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_numeral
% 0.33/0.62 thf(fact_6357_of__int__eq__numeral__iff,axiom,
% 0.33/0.62 ! [Z2: int,N: num] :
% 0.33/0.62 ( ( ( ring_17405671764205052669omplex @ Z2 )
% 0.33/0.62 = ( numera6690914467698888265omplex @ N ) )
% 0.33/0.62 = ( Z2
% 0.33/0.62 = ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_numeral_iff
% 0.33/0.62 thf(fact_6358_of__int__eq__numeral__iff,axiom,
% 0.33/0.62 ! [Z2: int,N: num] :
% 0.33/0.62 ( ( ( ring_1_of_int_real @ Z2 )
% 0.33/0.62 = ( numeral_numeral_real @ N ) )
% 0.33/0.62 = ( Z2
% 0.33/0.62 = ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_numeral_iff
% 0.33/0.62 thf(fact_6359_of__int__eq__numeral__iff,axiom,
% 0.33/0.62 ! [Z2: int,N: num] :
% 0.33/0.62 ( ( ( ring_1_of_int_rat @ Z2 )
% 0.33/0.62 = ( numeral_numeral_rat @ N ) )
% 0.33/0.62 = ( Z2
% 0.33/0.62 = ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_numeral_iff
% 0.33/0.62 thf(fact_6360_of__int__eq__numeral__iff,axiom,
% 0.33/0.62 ! [Z2: int,N: num] :
% 0.33/0.62 ( ( ( ring_1_of_int_int @ Z2 )
% 0.33/0.62 = ( numeral_numeral_int @ N ) )
% 0.33/0.62 = ( Z2
% 0.33/0.62 = ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_numeral_iff
% 0.33/0.62 thf(fact_6361_of__int__less__iff,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) )
% 0.33/0.62 = ( ord_less_int @ W @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_iff
% 0.33/0.62 thf(fact_6362_of__int__less__iff,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z2 ) )
% 0.33/0.62 = ( ord_less_int @ W @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_iff
% 0.33/0.62 thf(fact_6363_of__int__less__iff,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) )
% 0.33/0.62 = ( ord_less_int @ W @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_iff
% 0.33/0.62 thf(fact_6364_of__int__1,axiom,
% 0.33/0.62 ( ( ring_17405671764205052669omplex @ one_one_int )
% 0.33/0.62 = one_one_complex ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_1
% 0.33/0.62 thf(fact_6365_of__int__1,axiom,
% 0.33/0.62 ( ( ring_1_of_int_int @ one_one_int )
% 0.33/0.62 = one_one_int ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_1
% 0.33/0.62 thf(fact_6366_of__int__1,axiom,
% 0.33/0.62 ( ( ring_1_of_int_real @ one_one_int )
% 0.33/0.62 = one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_1
% 0.33/0.62 thf(fact_6367_of__int__1,axiom,
% 0.33/0.62 ( ( ring_1_of_int_rat @ one_one_int )
% 0.33/0.62 = one_one_rat ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_1
% 0.33/0.62 thf(fact_6368_of__int__eq__1__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ( ring_17405671764205052669omplex @ Z2 )
% 0.33/0.62 = one_one_complex )
% 0.33/0.62 = ( Z2 = one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_1_iff
% 0.33/0.62 thf(fact_6369_of__int__eq__1__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ( ring_1_of_int_int @ Z2 )
% 0.33/0.62 = one_one_int )
% 0.33/0.62 = ( Z2 = one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_1_iff
% 0.33/0.62 thf(fact_6370_of__int__eq__1__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ( ring_1_of_int_real @ Z2 )
% 0.33/0.62 = one_one_real )
% 0.33/0.62 = ( Z2 = one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_1_iff
% 0.33/0.62 thf(fact_6371_of__int__eq__1__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ( ring_1_of_int_rat @ Z2 )
% 0.33/0.62 = one_one_rat )
% 0.33/0.62 = ( Z2 = one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_1_iff
% 0.33/0.62 thf(fact_6372_of__int__mult,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z2 ) )
% 0.33/0.62 = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_mult
% 0.33/0.62 thf(fact_6373_of__int__mult,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z2 ) )
% 0.33/0.62 = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_mult
% 0.33/0.62 thf(fact_6374_of__int__mult,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z2 ) )
% 0.33/0.62 = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_mult
% 0.33/0.62 thf(fact_6375_of__int__add,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z2 ) )
% 0.33/0.62 = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_add
% 0.33/0.62 thf(fact_6376_of__int__add,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z2 ) )
% 0.33/0.62 = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_add
% 0.33/0.62 thf(fact_6377_of__int__add,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z2 ) )
% 0.33/0.62 = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_add
% 0.33/0.62 thf(fact_6378_of__int__minus,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z2 ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_minus
% 0.33/0.62 thf(fact_6379_of__int__minus,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z2 ) )
% 0.33/0.62 = ( uminus_uminus_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_minus
% 0.33/0.62 thf(fact_6380_of__int__minus,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ Z2 ) )
% 0.33/0.62 = ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_minus
% 0.33/0.62 thf(fact_6381_of__int__minus,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z2 ) )
% 0.33/0.62 = ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_minus
% 0.33/0.62 thf(fact_6382_of__int__minus,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ Z2 ) )
% 0.33/0.62 = ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_minus
% 0.33/0.62 thf(fact_6383_of__int__diff,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z2 ) )
% 0.33/0.62 = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_diff
% 0.33/0.62 thf(fact_6384_of__int__diff,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z2 ) )
% 0.33/0.62 = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_diff
% 0.33/0.62 thf(fact_6385_of__int__diff,axiom,
% 0.33/0.62 ! [W: int,Z2: int] :
% 0.33/0.62 ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z2 ) )
% 0.33/0.62 = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_diff
% 0.33/0.62 thf(fact_6386_of__int__of__nat__eq,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( ring_17405671764205052669omplex @ ( semiri1314217659103216013at_int @ N ) )
% 0.33/0.62 = ( semiri8010041392384452111omplex @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_of_nat_eq
% 0.33/0.62 thf(fact_6387_of__int__of__nat__eq,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
% 0.33/0.62 = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_of_nat_eq
% 0.33/0.62 thf(fact_6388_of__int__of__nat__eq,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N ) )
% 0.33/0.62 = ( semiri681578069525770553at_rat @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_of_nat_eq
% 0.33/0.62 thf(fact_6389_of__int__of__nat__eq,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.33/0.62 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_of_nat_eq
% 0.33/0.62 thf(fact_6390_of__int__abs,axiom,
% 0.33/0.62 ! [X: int] :
% 0.33/0.62 ( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
% 0.33/0.62 = ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_abs
% 0.33/0.62 thf(fact_6391_of__int__abs,axiom,
% 0.33/0.62 ! [X: int] :
% 0.33/0.62 ( ( ring_18347121197199848620nteger @ ( abs_abs_int @ X ) )
% 0.33/0.62 = ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_abs
% 0.33/0.62 thf(fact_6392_of__int__abs,axiom,
% 0.33/0.62 ! [X: int] :
% 0.33/0.62 ( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
% 0.33/0.62 = ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_abs
% 0.33/0.62 thf(fact_6393_of__int__abs,axiom,
% 0.33/0.62 ! [X: int] :
% 0.33/0.62 ( ( ring_1_of_int_rat @ ( abs_abs_int @ X ) )
% 0.33/0.62 = ( abs_abs_rat @ ( ring_1_of_int_rat @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_abs
% 0.33/0.62 thf(fact_6394_of__int__power,axiom,
% 0.33/0.62 ! [Z2: int,N: nat] :
% 0.33/0.62 ( ( ring_1_of_int_rat @ ( power_power_int @ Z2 @ N ) )
% 0.33/0.62 = ( power_power_rat @ ( ring_1_of_int_rat @ Z2 ) @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_power
% 0.33/0.62 thf(fact_6395_of__int__power,axiom,
% 0.33/0.62 ! [Z2: int,N: nat] :
% 0.33/0.62 ( ( ring_1_of_int_real @ ( power_power_int @ Z2 @ N ) )
% 0.33/0.62 = ( power_power_real @ ( ring_1_of_int_real @ Z2 ) @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_power
% 0.33/0.62 thf(fact_6396_of__int__power,axiom,
% 0.33/0.62 ! [Z2: int,N: nat] :
% 0.33/0.62 ( ( ring_1_of_int_int @ ( power_power_int @ Z2 @ N ) )
% 0.33/0.62 = ( power_power_int @ ( ring_1_of_int_int @ Z2 ) @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_power
% 0.33/0.62 thf(fact_6397_of__int__power,axiom,
% 0.33/0.62 ! [Z2: int,N: nat] :
% 0.33/0.62 ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z2 @ N ) )
% 0.33/0.62 = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z2 ) @ N ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_power
% 0.33/0.62 thf(fact_6398_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.33/0.62 ! [B: int,W: nat,X: int] :
% 0.33/0.62 ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
% 0.33/0.62 = ( ring_1_of_int_rat @ X ) )
% 0.33/0.62 = ( ( power_power_int @ B @ W )
% 0.33/0.62 = X ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_of_int_power_cancel_iff
% 0.33/0.62 thf(fact_6399_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.33/0.62 ! [B: int,W: nat,X: int] :
% 0.33/0.62 ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 0.33/0.62 = ( ring_1_of_int_real @ X ) )
% 0.33/0.62 = ( ( power_power_int @ B @ W )
% 0.33/0.62 = X ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_of_int_power_cancel_iff
% 0.33/0.62 thf(fact_6400_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.33/0.62 ! [B: int,W: nat,X: int] :
% 0.33/0.62 ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 0.33/0.62 = ( ring_1_of_int_int @ X ) )
% 0.33/0.62 = ( ( power_power_int @ B @ W )
% 0.33/0.62 = X ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_of_int_power_cancel_iff
% 0.33/0.62 thf(fact_6401_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.33/0.62 ! [B: int,W: nat,X: int] :
% 0.33/0.62 ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
% 0.33/0.62 = ( ring_17405671764205052669omplex @ X ) )
% 0.33/0.62 = ( ( power_power_int @ B @ W )
% 0.33/0.62 = X ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_of_int_power_cancel_iff
% 0.33/0.62 thf(fact_6402_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: int,B: int,W: nat] :
% 0.33/0.62 ( ( ( ring_1_of_int_rat @ X )
% 0.33/0.62 = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.33/0.62 = ( X
% 0.33/0.62 = ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_power_eq_of_int_cancel_iff
% 0.33/0.62 thf(fact_6403_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: int,B: int,W: nat] :
% 0.33/0.62 ( ( ( ring_1_of_int_real @ X )
% 0.33/0.62 = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.33/0.62 = ( X
% 0.33/0.62 = ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_power_eq_of_int_cancel_iff
% 0.33/0.62 thf(fact_6404_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: int,B: int,W: nat] :
% 0.33/0.62 ( ( ( ring_1_of_int_int @ X )
% 0.33/0.62 = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.33/0.62 = ( X
% 0.33/0.62 = ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_power_eq_of_int_cancel_iff
% 0.33/0.62 thf(fact_6405_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: int,B: int,W: nat] :
% 0.33/0.62 ( ( ( ring_17405671764205052669omplex @ X )
% 0.33/0.62 = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
% 0.33/0.62 = ( X
% 0.33/0.62 = ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_power_eq_of_int_cancel_iff
% 0.33/0.62 thf(fact_6406_of__int__of__bool,axiom,
% 0.33/0.62 ! [P: $o] :
% 0.33/0.62 ( ( ring_1_of_int_real @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.33/0.62 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_of_bool
% 0.33/0.62 thf(fact_6407_of__int__of__bool,axiom,
% 0.33/0.62 ! [P: $o] :
% 0.33/0.62 ( ( ring_1_of_int_rat @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.33/0.62 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_of_bool
% 0.33/0.62 thf(fact_6408_of__int__of__bool,axiom,
% 0.33/0.62 ! [P: $o] :
% 0.33/0.62 ( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.33/0.62 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_of_bool
% 0.33/0.62 thf(fact_6409_of__int__of__bool,axiom,
% 0.33/0.62 ! [P: $o] :
% 0.33/0.62 ( ( ring_18347121197199848620nteger @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.33/0.62 = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_of_bool
% 0.33/0.62 thf(fact_6410_tan__npi,axiom,
% 0.33/0.62 ! [N: nat] :
% 0.33/0.62 ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % tan_npi
% 0.33/0.62 thf(fact_6411_floor__uminus__of__int,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( ring_1_of_int_real @ Z2 ) ) )
% 0.33/0.62 = ( uminus_uminus_int @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % floor_uminus_of_int
% 0.33/0.62 thf(fact_6412_floor__uminus__of__int,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z2 ) ) )
% 0.33/0.62 = ( uminus_uminus_int @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % floor_uminus_of_int
% 0.33/0.62 thf(fact_6413_tan__periodic__n,axiom,
% 0.33/0.62 ! [X: real,N: num] :
% 0.33/0.62 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
% 0.33/0.62 = ( tan_real @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % tan_periodic_n
% 0.33/0.62 thf(fact_6414_ceiling__add__of__int,axiom,
% 0.33/0.62 ! [X: rat,Z2: int] :
% 0.33/0.62 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) )
% 0.33/0.62 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % ceiling_add_of_int
% 0.33/0.62 thf(fact_6415_ceiling__add__of__int,axiom,
% 0.33/0.62 ! [X: real,Z2: int] :
% 0.33/0.62 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z2 ) ) )
% 0.33/0.62 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % ceiling_add_of_int
% 0.33/0.62 thf(fact_6416_floor__diff__of__int,axiom,
% 0.33/0.62 ! [X: real,Z2: int] :
% 0.33/0.62 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z2 ) ) )
% 0.33/0.62 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % floor_diff_of_int
% 0.33/0.62 thf(fact_6417_floor__diff__of__int,axiom,
% 0.33/0.62 ! [X: rat,Z2: int] :
% 0.33/0.62 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) )
% 0.33/0.62 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % floor_diff_of_int
% 0.33/0.62 thf(fact_6418_ceiling__diff__of__int,axiom,
% 0.33/0.62 ! [X: rat,Z2: int] :
% 0.33/0.62 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) )
% 0.33/0.62 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % ceiling_diff_of_int
% 0.33/0.62 thf(fact_6419_ceiling__diff__of__int,axiom,
% 0.33/0.62 ! [X: real,Z2: int] :
% 0.33/0.62 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z2 ) ) )
% 0.33/0.62 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % ceiling_diff_of_int
% 0.33/0.62 thf(fact_6420_tan__periodic__nat,axiom,
% 0.33/0.62 ! [X: real,N: nat] :
% 0.33/0.62 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
% 0.33/0.62 = ( tan_real @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % tan_periodic_nat
% 0.33/0.62 thf(fact_6421_tan__periodic__int,axiom,
% 0.33/0.62 ! [X: real,I: int] :
% 0.33/0.62 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
% 0.33/0.62 = ( tan_real @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % tan_periodic_int
% 0.33/0.62 thf(fact_6422_of__int__le__0__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
% 0.33/0.62 = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_0_iff
% 0.33/0.62 thf(fact_6423_of__int__le__0__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
% 0.33/0.62 = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_0_iff
% 0.33/0.62 thf(fact_6424_of__int__le__0__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
% 0.33/0.62 = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_0_iff
% 0.33/0.62 thf(fact_6425_of__int__0__le__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.33/0.62 = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_0_le_iff
% 0.33/0.62 thf(fact_6426_of__int__0__le__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.33/0.62 = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_0_le_iff
% 0.33/0.62 thf(fact_6427_of__int__0__le__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
% 0.33/0.62 = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_0_le_iff
% 0.33/0.62 thf(fact_6428_of__int__less__0__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
% 0.33/0.62 = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_0_iff
% 0.33/0.62 thf(fact_6429_of__int__less__0__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
% 0.33/0.62 = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_0_iff
% 0.33/0.62 thf(fact_6430_of__int__less__0__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
% 0.33/0.62 = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_0_iff
% 0.33/0.62 thf(fact_6431_of__int__0__less__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.33/0.62 = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_0_less_iff
% 0.33/0.62 thf(fact_6432_of__int__0__less__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.33/0.62 = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_0_less_iff
% 0.33/0.62 thf(fact_6433_of__int__0__less__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
% 0.33/0.62 = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_0_less_iff
% 0.33/0.62 thf(fact_6434_of__int__le__numeral__iff,axiom,
% 0.33/0.62 ! [Z2: int,N: num] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
% 0.33/0.62 = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_numeral_iff
% 0.33/0.62 thf(fact_6435_of__int__le__numeral__iff,axiom,
% 0.33/0.62 ! [Z2: int,N: num] :
% 0.33/0.62 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
% 0.33/0.62 = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_numeral_iff
% 0.33/0.62 thf(fact_6436_of__int__le__numeral__iff,axiom,
% 0.33/0.62 ! [Z2: int,N: num] :
% 0.33/0.62 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.62 = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_numeral_iff
% 0.33/0.62 thf(fact_6437_of__int__numeral__le__iff,axiom,
% 0.33/0.62 ! [N: num,Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
% 0.33/0.62 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_numeral_le_iff
% 0.33/0.62 thf(fact_6438_of__int__numeral__le__iff,axiom,
% 0.33/0.62 ! [N: num,Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
% 0.33/0.62 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_numeral_le_iff
% 0.33/0.62 thf(fact_6439_of__int__numeral__le__iff,axiom,
% 0.33/0.62 ! [N: num,Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
% 0.33/0.62 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_numeral_le_iff
% 0.33/0.62 thf(fact_6440_of__int__less__numeral__iff,axiom,
% 0.33/0.62 ! [Z2: int,N: num] :
% 0.33/0.62 ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
% 0.33/0.62 = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_numeral_iff
% 0.33/0.62 thf(fact_6441_of__int__less__numeral__iff,axiom,
% 0.33/0.62 ! [Z2: int,N: num] :
% 0.33/0.62 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
% 0.33/0.62 = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_numeral_iff
% 0.33/0.62 thf(fact_6442_of__int__less__numeral__iff,axiom,
% 0.33/0.62 ! [Z2: int,N: num] :
% 0.33/0.62 ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.62 = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_numeral_iff
% 0.33/0.62 thf(fact_6443_of__int__numeral__less__iff,axiom,
% 0.33/0.62 ! [N: num,Z2: int] :
% 0.33/0.62 ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
% 0.33/0.62 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_numeral_less_iff
% 0.33/0.62 thf(fact_6444_of__int__numeral__less__iff,axiom,
% 0.33/0.62 ! [N: num,Z2: int] :
% 0.33/0.62 ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
% 0.33/0.62 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_numeral_less_iff
% 0.33/0.62 thf(fact_6445_of__int__numeral__less__iff,axiom,
% 0.33/0.62 ! [N: num,Z2: int] :
% 0.33/0.62 ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
% 0.33/0.62 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_numeral_less_iff
% 0.33/0.62 thf(fact_6446_of__int__le__1__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
% 0.33/0.62 = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_1_iff
% 0.33/0.62 thf(fact_6447_of__int__le__1__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
% 0.33/0.62 = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_1_iff
% 0.33/0.62 thf(fact_6448_of__int__le__1__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
% 0.33/0.62 = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_1_iff
% 0.33/0.62 thf(fact_6449_of__int__1__le__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.33/0.62 = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_1_le_iff
% 0.33/0.62 thf(fact_6450_of__int__1__le__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.33/0.62 = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_1_le_iff
% 0.33/0.62 thf(fact_6451_of__int__1__le__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
% 0.33/0.62 = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_1_le_iff
% 0.33/0.62 thf(fact_6452_of__int__less__1__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
% 0.33/0.62 = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_1_iff
% 0.33/0.62 thf(fact_6453_of__int__less__1__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
% 0.33/0.62 = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_1_iff
% 0.33/0.62 thf(fact_6454_of__int__less__1__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
% 0.33/0.62 = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_1_iff
% 0.33/0.62 thf(fact_6455_of__int__1__less__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.33/0.62 = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_1_less_iff
% 0.33/0.62 thf(fact_6456_of__int__1__less__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.33/0.62 = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_1_less_iff
% 0.33/0.62 thf(fact_6457_of__int__1__less__iff,axiom,
% 0.33/0.62 ! [Z2: int] :
% 0.33/0.62 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
% 0.33/0.62 = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_1_less_iff
% 0.33/0.62 thf(fact_6458_of__int__power__le__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: int,B: int,W: nat] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.33/0.62 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_power_le_of_int_cancel_iff
% 0.33/0.62 thf(fact_6459_of__int__power__le__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: int,B: int,W: nat] :
% 0.33/0.62 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.33/0.62 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_power_le_of_int_cancel_iff
% 0.33/0.62 thf(fact_6460_of__int__power__le__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: int,B: int,W: nat] :
% 0.33/0.62 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.33/0.62 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_power_le_of_int_cancel_iff
% 0.33/0.62 thf(fact_6461_of__int__le__of__int__power__cancel__iff,axiom,
% 0.33/0.62 ! [B: int,W: nat,X: int] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 0.33/0.62 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_of_int_power_cancel_iff
% 0.33/0.62 thf(fact_6462_of__int__le__of__int__power__cancel__iff,axiom,
% 0.33/0.62 ! [B: int,W: nat,X: int] :
% 0.33/0.62 ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 0.33/0.62 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_of_int_power_cancel_iff
% 0.33/0.62 thf(fact_6463_of__int__le__of__int__power__cancel__iff,axiom,
% 0.33/0.62 ! [B: int,W: nat,X: int] :
% 0.33/0.62 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 0.33/0.62 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_of_int_power_cancel_iff
% 0.33/0.62 thf(fact_6464_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,Y: int] :
% 0.33/0.62 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
% 0.33/0.62 = ( ring_17405671764205052669omplex @ Y ) )
% 0.33/0.62 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.33/0.62 = Y ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_power_eq_of_int_cancel_iff
% 0.33/0.62 thf(fact_6465_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,Y: int] :
% 0.33/0.62 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 0.33/0.62 = ( ring_1_of_int_real @ Y ) )
% 0.33/0.62 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.33/0.62 = Y ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_power_eq_of_int_cancel_iff
% 0.33/0.62 thf(fact_6466_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,Y: int] :
% 0.33/0.62 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
% 0.33/0.62 = ( ring_1_of_int_rat @ Y ) )
% 0.33/0.62 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.33/0.62 = Y ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_power_eq_of_int_cancel_iff
% 0.33/0.62 thf(fact_6467_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,Y: int] :
% 0.33/0.62 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.33/0.62 = ( ring_1_of_int_int @ Y ) )
% 0.33/0.62 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.33/0.62 = Y ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_power_eq_of_int_cancel_iff
% 0.33/0.62 thf(fact_6468_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [Y: int,X: num,N: nat] :
% 0.33/0.62 ( ( ( ring_17405671764205052669omplex @ Y )
% 0.33/0.62 = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
% 0.33/0.62 = ( Y
% 0.33/0.62 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6469_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [Y: int,X: num,N: nat] :
% 0.33/0.62 ( ( ( ring_1_of_int_real @ Y )
% 0.33/0.62 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.33/0.62 = ( Y
% 0.33/0.62 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6470_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [Y: int,X: num,N: nat] :
% 0.33/0.62 ( ( ( ring_1_of_int_rat @ Y )
% 0.33/0.62 = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.33/0.62 = ( Y
% 0.33/0.62 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6471_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [Y: int,X: num,N: nat] :
% 0.33/0.62 ( ( ( ring_1_of_int_int @ Y )
% 0.33/0.62 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.33/0.62 = ( Y
% 0.33/0.62 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6472_of__int__power__less__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: int,B: int,W: nat] :
% 0.33/0.62 ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.33/0.62 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_power_less_of_int_cancel_iff
% 0.33/0.62 thf(fact_6473_of__int__power__less__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: int,B: int,W: nat] :
% 0.33/0.62 ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.33/0.62 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_power_less_of_int_cancel_iff
% 0.33/0.62 thf(fact_6474_of__int__power__less__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: int,B: int,W: nat] :
% 0.33/0.62 ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.33/0.62 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_power_less_of_int_cancel_iff
% 0.33/0.62 thf(fact_6475_of__int__less__of__int__power__cancel__iff,axiom,
% 0.33/0.62 ! [B: int,W: nat,X: int] :
% 0.33/0.62 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 0.33/0.62 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_of_int_power_cancel_iff
% 0.33/0.62 thf(fact_6476_of__int__less__of__int__power__cancel__iff,axiom,
% 0.33/0.62 ! [B: int,W: nat,X: int] :
% 0.33/0.62 ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 0.33/0.62 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_of_int_power_cancel_iff
% 0.33/0.62 thf(fact_6477_of__int__less__of__int__power__cancel__iff,axiom,
% 0.33/0.62 ! [B: int,W: nat,X: int] :
% 0.33/0.62 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 0.33/0.62 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_of_int_power_cancel_iff
% 0.33/0.62 thf(fact_6478_sin__npi__int,axiom,
% 0.33/0.62 ! [N: int] :
% 0.33/0.62 ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % sin_npi_int
% 0.33/0.62 thf(fact_6479_sin__arcsin,axiom,
% 0.33/0.62 ! [Y: real] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.62 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.62 => ( ( sin_real @ ( arcsin @ Y ) )
% 0.33/0.62 = Y ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sin_arcsin
% 0.33/0.62 thf(fact_6480_norm__cos__sin,axiom,
% 0.33/0.62 ! [T: real] :
% 0.33/0.62 ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
% 0.33/0.62 = one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % norm_cos_sin
% 0.33/0.62 thf(fact_6481_tan__periodic,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.33/0.62 = ( tan_real @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % tan_periodic
% 0.33/0.62 thf(fact_6482_numeral__power__le__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,A: int] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.33/0.62 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_power_le_of_int_cancel_iff
% 0.33/0.62 thf(fact_6483_numeral__power__le__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,A: int] :
% 0.33/0.62 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.33/0.62 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_power_le_of_int_cancel_iff
% 0.33/0.62 thf(fact_6484_numeral__power__le__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,A: int] :
% 0.33/0.62 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.33/0.62 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_power_le_of_int_cancel_iff
% 0.33/0.62 thf(fact_6485_of__int__le__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [A: int,X: num,N: nat] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.33/0.62 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6486_of__int__le__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [A: int,X: num,N: nat] :
% 0.33/0.62 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.33/0.62 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6487_of__int__le__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [A: int,X: num,N: nat] :
% 0.33/0.62 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.33/0.62 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6488_numeral__power__less__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,A: int] :
% 0.33/0.62 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.33/0.62 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_power_less_of_int_cancel_iff
% 0.33/0.62 thf(fact_6489_numeral__power__less__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,A: int] :
% 0.33/0.62 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.33/0.62 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_power_less_of_int_cancel_iff
% 0.33/0.62 thf(fact_6490_numeral__power__less__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,A: int] :
% 0.33/0.62 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.33/0.62 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % numeral_power_less_of_int_cancel_iff
% 0.33/0.62 thf(fact_6491_of__int__less__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [A: int,X: num,N: nat] :
% 0.33/0.62 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.33/0.62 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6492_of__int__less__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [A: int,X: num,N: nat] :
% 0.33/0.62 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.33/0.62 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6493_of__int__less__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [A: int,X: num,N: nat] :
% 0.33/0.62 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.33/0.62 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6494_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,Y: int] :
% 0.33/0.62 ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N )
% 0.33/0.62 = ( ring_1_of_int_real @ Y ) )
% 0.33/0.62 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.33/0.62 = Y ) ) ).
% 0.33/0.62
% 0.33/0.62 % neg_numeral_power_eq_of_int_cancel_iff
% 0.33/0.62 thf(fact_6495_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,Y: int] :
% 0.33/0.62 ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.33/0.62 = ( ring_1_of_int_int @ Y ) )
% 0.33/0.62 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.33/0.62 = Y ) ) ).
% 0.33/0.62
% 0.33/0.62 % neg_numeral_power_eq_of_int_cancel_iff
% 0.33/0.62 thf(fact_6496_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,Y: int] :
% 0.33/0.62 ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N )
% 0.33/0.62 = ( ring_17405671764205052669omplex @ Y ) )
% 0.33/0.62 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.33/0.62 = Y ) ) ).
% 0.33/0.62
% 0.33/0.62 % neg_numeral_power_eq_of_int_cancel_iff
% 0.33/0.62 thf(fact_6497_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,Y: int] :
% 0.33/0.62 ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
% 0.33/0.62 = ( ring_1_of_int_rat @ Y ) )
% 0.33/0.62 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.33/0.62 = Y ) ) ).
% 0.33/0.62
% 0.33/0.62 % neg_numeral_power_eq_of_int_cancel_iff
% 0.33/0.62 thf(fact_6498_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,Y: int] :
% 0.33/0.62 ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
% 0.33/0.62 = ( ring_18347121197199848620nteger @ Y ) )
% 0.33/0.62 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.33/0.62 = Y ) ) ).
% 0.33/0.62
% 0.33/0.62 % neg_numeral_power_eq_of_int_cancel_iff
% 0.33/0.62 thf(fact_6499_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [Y: int,X: num,N: nat] :
% 0.33/0.62 ( ( ( ring_1_of_int_real @ Y )
% 0.33/0.62 = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 0.33/0.62 = ( Y
% 0.33/0.62 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_neg_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6500_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [Y: int,X: num,N: nat] :
% 0.33/0.62 ( ( ( ring_1_of_int_int @ Y )
% 0.33/0.62 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 0.33/0.62 = ( Y
% 0.33/0.62 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_neg_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6501_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [Y: int,X: num,N: nat] :
% 0.33/0.62 ( ( ( ring_17405671764205052669omplex @ Y )
% 0.33/0.62 = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N ) )
% 0.33/0.62 = ( Y
% 0.33/0.62 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_neg_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6502_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [Y: int,X: num,N: nat] :
% 0.33/0.62 ( ( ( ring_1_of_int_rat @ Y )
% 0.33/0.62 = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 0.33/0.62 = ( Y
% 0.33/0.62 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_neg_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6503_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [Y: int,X: num,N: nat] :
% 0.33/0.62 ( ( ( ring_18347121197199848620nteger @ Y )
% 0.33/0.62 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 0.33/0.62 = ( Y
% 0.33/0.62 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_eq_neg_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6504_arcsin__1,axiom,
% 0.33/0.62 ( ( arcsin @ one_one_real )
% 0.33/0.62 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % arcsin_1
% 0.33/0.62 thf(fact_6505_sin__int__2pin,axiom,
% 0.33/0.62 ! [N: int] :
% 0.33/0.62 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 0.33/0.62 = zero_zero_real ) ).
% 0.33/0.62
% 0.33/0.62 % sin_int_2pin
% 0.33/0.62 thf(fact_6506_cos__int__2pin,axiom,
% 0.33/0.62 ! [N: int] :
% 0.33/0.62 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 0.33/0.62 = one_one_real ) ).
% 0.33/0.62
% 0.33/0.62 % cos_int_2pin
% 0.33/0.62 thf(fact_6507_arcsin__minus__1,axiom,
% 0.33/0.62 ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.62 = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % arcsin_minus_1
% 0.33/0.62 thf(fact_6508_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,A: int] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.33/0.62 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % neg_numeral_power_le_of_int_cancel_iff
% 0.33/0.62 thf(fact_6509_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,A: int] :
% 0.33/0.62 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.33/0.62 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % neg_numeral_power_le_of_int_cancel_iff
% 0.33/0.62 thf(fact_6510_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,A: int] :
% 0.33/0.62 ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.33/0.62 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % neg_numeral_power_le_of_int_cancel_iff
% 0.33/0.62 thf(fact_6511_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,A: int] :
% 0.33/0.62 ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.33/0.62 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % neg_numeral_power_le_of_int_cancel_iff
% 0.33/0.62 thf(fact_6512_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [A: int,X: num,N: nat] :
% 0.33/0.62 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 0.33/0.62 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_neg_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6513_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [A: int,X: num,N: nat] :
% 0.33/0.62 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 0.33/0.62 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_neg_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6514_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [A: int,X: num,N: nat] :
% 0.33/0.62 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 0.33/0.62 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_neg_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6515_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [A: int,X: num,N: nat] :
% 0.33/0.62 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 0.33/0.62 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_le_neg_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6516_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,A: int] :
% 0.33/0.62 ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.33/0.62 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % neg_numeral_power_less_of_int_cancel_iff
% 0.33/0.62 thf(fact_6517_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,A: int] :
% 0.33/0.62 ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.33/0.62 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % neg_numeral_power_less_of_int_cancel_iff
% 0.33/0.62 thf(fact_6518_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,A: int] :
% 0.33/0.62 ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.33/0.62 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % neg_numeral_power_less_of_int_cancel_iff
% 0.33/0.62 thf(fact_6519_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.33/0.62 ! [X: num,N: nat,A: int] :
% 0.33/0.62 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.33/0.62 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.33/0.62
% 0.33/0.62 % neg_numeral_power_less_of_int_cancel_iff
% 0.33/0.62 thf(fact_6520_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [A: int,X: num,N: nat] :
% 0.33/0.62 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 0.33/0.62 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_neg_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6521_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [A: int,X: num,N: nat] :
% 0.33/0.62 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 0.33/0.62 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_neg_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6522_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [A: int,X: num,N: nat] :
% 0.33/0.62 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 0.33/0.62 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_neg_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6523_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.62 ! [A: int,X: num,N: nat] :
% 0.33/0.62 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 0.33/0.62 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_less_neg_numeral_power_cancel_iff
% 0.33/0.62 thf(fact_6524_ex__le__of__int,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ? [Z4: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z4 ) ) ).
% 0.33/0.62
% 0.33/0.62 % ex_le_of_int
% 0.33/0.62 thf(fact_6525_ex__le__of__int,axiom,
% 0.33/0.62 ! [X: rat] :
% 0.33/0.62 ? [Z4: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z4 ) ) ).
% 0.33/0.62
% 0.33/0.62 % ex_le_of_int
% 0.33/0.62 thf(fact_6526_ex__less__of__int,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ? [Z4: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z4 ) ) ).
% 0.33/0.62
% 0.33/0.62 % ex_less_of_int
% 0.33/0.62 thf(fact_6527_ex__less__of__int,axiom,
% 0.33/0.62 ! [X: rat] :
% 0.33/0.62 ? [Z4: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z4 ) ) ).
% 0.33/0.62
% 0.33/0.62 % ex_less_of_int
% 0.33/0.62 thf(fact_6528_ex__of__int__less,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ? [Z4: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z4 ) @ X ) ).
% 0.33/0.62
% 0.33/0.62 % ex_of_int_less
% 0.33/0.62 thf(fact_6529_ex__of__int__less,axiom,
% 0.33/0.62 ! [X: rat] :
% 0.33/0.62 ? [Z4: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z4 ) @ X ) ).
% 0.33/0.62
% 0.33/0.62 % ex_of_int_less
% 0.33/0.62 thf(fact_6530_mult__of__int__commute,axiom,
% 0.33/0.62 ! [X: int,Y: real] :
% 0.33/0.62 ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
% 0.33/0.62 = ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % mult_of_int_commute
% 0.33/0.62 thf(fact_6531_mult__of__int__commute,axiom,
% 0.33/0.62 ! [X: int,Y: rat] :
% 0.33/0.62 ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y )
% 0.33/0.62 = ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % mult_of_int_commute
% 0.33/0.62 thf(fact_6532_mult__of__int__commute,axiom,
% 0.33/0.62 ! [X: int,Y: int] :
% 0.33/0.62 ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
% 0.33/0.62 = ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % mult_of_int_commute
% 0.33/0.62 thf(fact_6533_Complex__mult__complex__of__real,axiom,
% 0.33/0.62 ! [X: real,Y: real,R: real] :
% 0.33/0.62 ( ( times_times_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R ) )
% 0.33/0.62 = ( complex2 @ ( times_times_real @ X @ R ) @ ( times_times_real @ Y @ R ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % Complex_mult_complex_of_real
% 0.33/0.62 thf(fact_6534_complex__of__real__mult__Complex,axiom,
% 0.33/0.62 ! [R: real,X: real,Y: real] :
% 0.33/0.62 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X @ Y ) )
% 0.33/0.62 = ( complex2 @ ( times_times_real @ R @ X ) @ ( times_times_real @ R @ Y ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % complex_of_real_mult_Complex
% 0.33/0.62 thf(fact_6535_complex__diff,axiom,
% 0.33/0.62 ! [A: real,B: real,C: real,D: real] :
% 0.33/0.62 ( ( minus_minus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 0.33/0.62 = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % complex_diff
% 0.33/0.62 thf(fact_6536_of__int__floor__le,axiom,
% 0.33/0.62 ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_floor_le
% 0.33/0.62 thf(fact_6537_of__int__floor__le,axiom,
% 0.33/0.62 ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X ) ).
% 0.33/0.62
% 0.33/0.62 % of_int_floor_le
% 0.33/0.62 thf(fact_6538_Complex__eq__1,axiom,
% 0.33/0.62 ! [A: real,B: real] :
% 0.33/0.62 ( ( ( complex2 @ A @ B )
% 0.33/0.62 = one_one_complex )
% 0.33/0.62 = ( ( A = one_one_real )
% 0.33/0.62 & ( B = zero_zero_real ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % Complex_eq_1
% 0.33/0.62 thf(fact_6539_one__complex_Ocode,axiom,
% 0.33/0.62 ( one_one_complex
% 0.33/0.62 = ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% 0.33/0.62
% 0.33/0.62 % one_complex.code
% 0.33/0.62 thf(fact_6540_le__of__int__ceiling,axiom,
% 0.33/0.62 ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % le_of_int_ceiling
% 0.33/0.62 thf(fact_6541_le__of__int__ceiling,axiom,
% 0.33/0.62 ! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % le_of_int_ceiling
% 0.33/0.62 thf(fact_6542_take__bit__of__int,axiom,
% 0.33/0.62 ! [N: nat,K: int] :
% 0.33/0.62 ( ( bit_se2923211474154528505it_int @ N @ ( ring_1_of_int_int @ K ) )
% 0.33/0.62 = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % take_bit_of_int
% 0.33/0.62 thf(fact_6543_push__bit__of__int,axiom,
% 0.33/0.62 ! [N: nat,K: int] :
% 0.33/0.62 ( ( bit_se545348938243370406it_int @ N @ ( ring_1_of_int_int @ K ) )
% 0.33/0.62 = ( ring_1_of_int_int @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % push_bit_of_int
% 0.33/0.62 thf(fact_6544_push__bit__of__int,axiom,
% 0.33/0.62 ! [N: nat,K: int] :
% 0.33/0.62 ( ( bit_se7788150548672797655nteger @ N @ ( ring_18347121197199848620nteger @ K ) )
% 0.33/0.62 = ( ring_18347121197199848620nteger @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % push_bit_of_int
% 0.33/0.62 thf(fact_6545_complex__add,axiom,
% 0.33/0.62 ! [A: real,B: real,C: real,D: real] :
% 0.33/0.62 ( ( plus_plus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 0.33/0.63 = ( complex2 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % complex_add
% 0.33/0.63 thf(fact_6546_Complex__add__complex__of__real,axiom,
% 0.33/0.63 ! [X: real,Y: real,R: real] :
% 0.33/0.63 ( ( plus_plus_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R ) )
% 0.33/0.63 = ( complex2 @ ( plus_plus_real @ X @ R ) @ Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % Complex_add_complex_of_real
% 0.33/0.63 thf(fact_6547_complex__of__real__add__Complex,axiom,
% 0.33/0.63 ! [R: real,X: real,Y: real] :
% 0.33/0.63 ( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X @ Y ) )
% 0.33/0.63 = ( complex2 @ ( plus_plus_real @ R @ X ) @ Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % complex_of_real_add_Complex
% 0.33/0.63 thf(fact_6548_cos__int__times__real,axiom,
% 0.33/0.63 ! [M: int,X: real] :
% 0.33/0.63 ( ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
% 0.33/0.63 = ( real_V1803761363581548252l_real @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % cos_int_times_real
% 0.33/0.63 thf(fact_6549_cos__int__times__real,axiom,
% 0.33/0.63 ! [M: int,X: real] :
% 0.33/0.63 ( ( cos_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
% 0.33/0.63 = ( real_V4546457046886955230omplex @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % cos_int_times_real
% 0.33/0.63 thf(fact_6550_sin__int__times__real,axiom,
% 0.33/0.63 ! [M: int,X: real] :
% 0.33/0.63 ( ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
% 0.33/0.63 = ( real_V1803761363581548252l_real @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sin_int_times_real
% 0.33/0.63 thf(fact_6551_sin__int__times__real,axiom,
% 0.33/0.63 ! [M: int,X: real] :
% 0.33/0.63 ( ( sin_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
% 0.33/0.63 = ( real_V4546457046886955230omplex @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sin_int_times_real
% 0.33/0.63 thf(fact_6552_le__floor__iff,axiom,
% 0.33/0.63 ! [Z2: int,X: real] :
% 0.33/0.63 ( ( ord_less_eq_int @ Z2 @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.63 = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_floor_iff
% 0.33/0.63 thf(fact_6553_le__floor__iff,axiom,
% 0.33/0.63 ! [Z2: int,X: rat] :
% 0.33/0.63 ( ( ord_less_eq_int @ Z2 @ ( archim3151403230148437115or_rat @ X ) )
% 0.33/0.63 = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_floor_iff
% 0.33/0.63 thf(fact_6554_floor__less__iff,axiom,
% 0.33/0.63 ! [X: real,Z2: int] :
% 0.33/0.63 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z2 )
% 0.33/0.63 = ( ord_less_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_less_iff
% 0.33/0.63 thf(fact_6555_floor__less__iff,axiom,
% 0.33/0.63 ! [X: rat,Z2: int] :
% 0.33/0.63 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 )
% 0.33/0.63 = ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_less_iff
% 0.33/0.63 thf(fact_6556_ceiling__le__iff,axiom,
% 0.33/0.63 ! [X: real,Z2: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z2 )
% 0.33/0.63 = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_le_iff
% 0.33/0.63 thf(fact_6557_ceiling__le__iff,axiom,
% 0.33/0.63 ! [X: rat,Z2: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 )
% 0.33/0.63 = ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_le_iff
% 0.33/0.63 thf(fact_6558_ceiling__le,axiom,
% 0.33/0.63 ! [X: real,A: int] :
% 0.33/0.63 ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
% 0.33/0.63 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_le
% 0.33/0.63 thf(fact_6559_ceiling__le,axiom,
% 0.33/0.63 ! [X: rat,A: int] :
% 0.33/0.63 ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) )
% 0.33/0.63 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ A ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_le
% 0.33/0.63 thf(fact_6560_less__ceiling__iff,axiom,
% 0.33/0.63 ! [Z2: int,X: rat] :
% 0.33/0.63 ( ( ord_less_int @ Z2 @ ( archim2889992004027027881ng_rat @ X ) )
% 0.33/0.63 = ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_ceiling_iff
% 0.33/0.63 thf(fact_6561_less__ceiling__iff,axiom,
% 0.33/0.63 ! [Z2: int,X: real] :
% 0.33/0.63 ( ( ord_less_int @ Z2 @ ( archim7802044766580827645g_real @ X ) )
% 0.33/0.63 = ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_ceiling_iff
% 0.33/0.63 thf(fact_6562_int__add__floor,axiom,
% 0.33/0.63 ! [Z2: int,X: real] :
% 0.33/0.63 ( ( plus_plus_int @ Z2 @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.63 = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % int_add_floor
% 0.33/0.63 thf(fact_6563_int__add__floor,axiom,
% 0.33/0.63 ! [Z2: int,X: rat] :
% 0.33/0.63 ( ( plus_plus_int @ Z2 @ ( archim3151403230148437115or_rat @ X ) )
% 0.33/0.63 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % int_add_floor
% 0.33/0.63 thf(fact_6564_floor__add__int,axiom,
% 0.33/0.63 ! [X: real,Z2: int] :
% 0.33/0.63 ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ Z2 )
% 0.33/0.63 = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_add_int
% 0.33/0.63 thf(fact_6565_floor__add__int,axiom,
% 0.33/0.63 ! [X: rat,Z2: int] :
% 0.33/0.63 ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 )
% 0.33/0.63 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_add_int
% 0.33/0.63 thf(fact_6566_Complex__eq__neg__1,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ( complex2 @ A @ B )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.63 = ( ( A
% 0.33/0.63 = ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.63 & ( B = zero_zero_real ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % Complex_eq_neg_1
% 0.33/0.63 thf(fact_6567_real__of__int__div4,axiom,
% 0.33/0.63 ! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_of_int_div4
% 0.33/0.63 thf(fact_6568_floor__divide__of__int__eq,axiom,
% 0.33/0.63 ! [K: int,L: int] :
% 0.33/0.63 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L ) ) )
% 0.33/0.63 = ( divide_divide_int @ K @ L ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_divide_of_int_eq
% 0.33/0.63 thf(fact_6569_floor__divide__of__int__eq,axiom,
% 0.33/0.63 ! [K: int,L: int] :
% 0.33/0.63 ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L ) ) )
% 0.33/0.63 = ( divide_divide_int @ K @ L ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_divide_of_int_eq
% 0.33/0.63 thf(fact_6570_floor__power,axiom,
% 0.33/0.63 ! [X: real,N: nat] :
% 0.33/0.63 ( ( X
% 0.33/0.63 = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
% 0.33/0.63 => ( ( archim6058952711729229775r_real @ ( power_power_real @ X @ N ) )
% 0.33/0.63 = ( power_power_int @ ( archim6058952711729229775r_real @ X ) @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_power
% 0.33/0.63 thf(fact_6571_floor__power,axiom,
% 0.33/0.63 ! [X: rat,N: nat] :
% 0.33/0.63 ( ( X
% 0.33/0.63 = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
% 0.33/0.63 => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X @ N ) )
% 0.33/0.63 = ( power_power_int @ ( archim3151403230148437115or_rat @ X ) @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_power
% 0.33/0.63 thf(fact_6572_complex__mult,axiom,
% 0.33/0.63 ! [A: real,B: real,C: real,D: real] :
% 0.33/0.63 ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 0.33/0.63 = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % complex_mult
% 0.33/0.63 thf(fact_6573_real__of__int__div,axiom,
% 0.33/0.63 ! [D: int,N: int] :
% 0.33/0.63 ( ( dvd_dvd_int @ D @ N )
% 0.33/0.63 => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
% 0.33/0.63 = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_of_int_div
% 0.33/0.63 thf(fact_6574_arcsin__le__arcsin,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.63 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.63 => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arcsin_le_arcsin
% 0.33/0.63 thf(fact_6575_arcsin__minus,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.63 => ( ( arcsin @ ( uminus_uminus_real @ X ) )
% 0.33/0.63 = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arcsin_minus
% 0.33/0.63 thf(fact_6576_arcsin__le__mono,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.63 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.33/0.63 => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
% 0.33/0.63 = ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arcsin_le_mono
% 0.33/0.63 thf(fact_6577_arcsin__eq__iff,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.63 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.33/0.63 => ( ( ( arcsin @ X )
% 0.33/0.63 = ( arcsin @ Y ) )
% 0.33/0.63 = ( X = Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arcsin_eq_iff
% 0.33/0.63 thf(fact_6578_tan__def,axiom,
% 0.33/0.63 ( tan_complex
% 0.33/0.63 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X3 ) @ ( cos_complex @ X3 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_def
% 0.33/0.63 thf(fact_6579_tan__def,axiom,
% 0.33/0.63 ( tan_real
% 0.33/0.63 = ( ^ [X3: real] : ( divide_divide_real @ ( sin_real @ X3 ) @ ( cos_real @ X3 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_def
% 0.33/0.63 thf(fact_6580_of__int__nonneg,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_nonneg
% 0.33/0.63 thf(fact_6581_of__int__nonneg,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_nonneg
% 0.33/0.63 thf(fact_6582_of__int__nonneg,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_nonneg
% 0.33/0.63 thf(fact_6583_of__int__leD,axiom,
% 0.33/0.63 ! [N: int,X: code_integer] :
% 0.33/0.63 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 0.33/0.63 => ( ( N = zero_zero_int )
% 0.33/0.63 | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_leD
% 0.33/0.63 thf(fact_6584_of__int__leD,axiom,
% 0.33/0.63 ! [N: int,X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 0.33/0.63 => ( ( N = zero_zero_int )
% 0.33/0.63 | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_leD
% 0.33/0.63 thf(fact_6585_of__int__leD,axiom,
% 0.33/0.63 ! [N: int,X: rat] :
% 0.33/0.63 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 0.33/0.63 => ( ( N = zero_zero_int )
% 0.33/0.63 | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_leD
% 0.33/0.63 thf(fact_6586_of__int__leD,axiom,
% 0.33/0.63 ! [N: int,X: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 0.33/0.63 => ( ( N = zero_zero_int )
% 0.33/0.63 | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_leD
% 0.33/0.63 thf(fact_6587_of__int__pos,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_pos
% 0.33/0.63 thf(fact_6588_of__int__pos,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_pos
% 0.33/0.63 thf(fact_6589_of__int__pos,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_pos
% 0.33/0.63 thf(fact_6590_of__int__lessD,axiom,
% 0.33/0.63 ! [N: int,X: code_integer] :
% 0.33/0.63 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 0.33/0.63 => ( ( N = zero_zero_int )
% 0.33/0.63 | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_lessD
% 0.33/0.63 thf(fact_6591_of__int__lessD,axiom,
% 0.33/0.63 ! [N: int,X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 0.33/0.63 => ( ( N = zero_zero_int )
% 0.33/0.63 | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_lessD
% 0.33/0.63 thf(fact_6592_of__int__lessD,axiom,
% 0.33/0.63 ! [N: int,X: rat] :
% 0.33/0.63 ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 0.33/0.63 => ( ( N = zero_zero_int )
% 0.33/0.63 | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_lessD
% 0.33/0.63 thf(fact_6593_of__int__lessD,axiom,
% 0.33/0.63 ! [N: int,X: int] :
% 0.33/0.63 ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 0.33/0.63 => ( ( N = zero_zero_int )
% 0.33/0.63 | ( ord_less_int @ one_one_int @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_lessD
% 0.33/0.63 thf(fact_6594_floor__exists1,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ? [X4: int] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X )
% 0.33/0.63 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 0.33/0.63 & ! [Y5: int] :
% 0.33/0.63 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X )
% 0.33/0.63 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 0.33/0.63 => ( Y5 = X4 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_exists1
% 0.33/0.63 thf(fact_6595_floor__exists1,axiom,
% 0.33/0.63 ! [X: rat] :
% 0.33/0.63 ? [X4: int] :
% 0.33/0.63 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X )
% 0.33/0.63 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 0.33/0.63 & ! [Y5: int] :
% 0.33/0.63 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X )
% 0.33/0.63 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 0.33/0.63 => ( Y5 = X4 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_exists1
% 0.33/0.63 thf(fact_6596_floor__exists,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ? [Z4: int] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z4 ) @ X )
% 0.33/0.63 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_exists
% 0.33/0.63 thf(fact_6597_floor__exists,axiom,
% 0.33/0.63 ! [X: rat] :
% 0.33/0.63 ? [Z4: int] :
% 0.33/0.63 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z4 ) @ X )
% 0.33/0.63 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_exists
% 0.33/0.63 thf(fact_6598_of__int__ceiling__le__add__one,axiom,
% 0.33/0.63 ! [R: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R ) ) @ ( plus_plus_real @ R @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_ceiling_le_add_one
% 0.33/0.63 thf(fact_6599_of__int__ceiling__le__add__one,axiom,
% 0.33/0.63 ! [R: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R ) ) @ ( plus_plus_rat @ R @ one_one_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_ceiling_le_add_one
% 0.33/0.63 thf(fact_6600_of__int__neg__numeral,axiom,
% 0.33/0.63 ! [K: num] :
% 0.33/0.63 ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.63 = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_neg_numeral
% 0.33/0.63 thf(fact_6601_of__int__neg__numeral,axiom,
% 0.33/0.63 ! [K: num] :
% 0.33/0.63 ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.63 = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_neg_numeral
% 0.33/0.63 thf(fact_6602_of__int__neg__numeral,axiom,
% 0.33/0.63 ! [K: num] :
% 0.33/0.63 ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_neg_numeral
% 0.33/0.63 thf(fact_6603_of__int__neg__numeral,axiom,
% 0.33/0.63 ! [K: num] :
% 0.33/0.63 ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.63 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_neg_numeral
% 0.33/0.63 thf(fact_6604_of__int__neg__numeral,axiom,
% 0.33/0.63 ! [K: num] :
% 0.33/0.63 ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.63 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_neg_numeral
% 0.33/0.63 thf(fact_6605_of__int__ceiling__diff__one__le,axiom,
% 0.33/0.63 ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R ) ) @ one_one_real ) @ R ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_ceiling_diff_one_le
% 0.33/0.63 thf(fact_6606_of__int__ceiling__diff__one__le,axiom,
% 0.33/0.63 ! [R: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R ) ) @ one_one_rat ) @ R ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_ceiling_diff_one_le
% 0.33/0.63 thf(fact_6607_of__nat__less__of__int__iff,axiom,
% 0.33/0.63 ! [N: nat,X: int] :
% 0.33/0.63 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
% 0.33/0.63 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_less_of_int_iff
% 0.33/0.63 thf(fact_6608_of__nat__less__of__int__iff,axiom,
% 0.33/0.63 ! [N: nat,X: int] :
% 0.33/0.63 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
% 0.33/0.63 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_less_of_int_iff
% 0.33/0.63 thf(fact_6609_of__nat__less__of__int__iff,axiom,
% 0.33/0.63 ! [N: nat,X: int] :
% 0.33/0.63 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
% 0.33/0.63 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_less_of_int_iff
% 0.33/0.63 thf(fact_6610_int__le__real__less,axiom,
% 0.33/0.63 ( ord_less_eq_int
% 0.33/0.63 = ( ^ [N3: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % int_le_real_less
% 0.33/0.63 thf(fact_6611_int__less__real__le,axiom,
% 0.33/0.63 ( ord_less_int
% 0.33/0.63 = ( ^ [N3: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % int_less_real_le
% 0.33/0.63 thf(fact_6612_ceiling__divide__eq__div,axiom,
% 0.33/0.63 ! [A: int,B: int] :
% 0.33/0.63 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
% 0.33/0.63 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_divide_eq_div
% 0.33/0.63 thf(fact_6613_ceiling__divide__eq__div,axiom,
% 0.33/0.63 ! [A: int,B: int] :
% 0.33/0.63 ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) )
% 0.33/0.63 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_divide_eq_div
% 0.33/0.63 thf(fact_6614_sin__zero__iff__int2,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ( sin_real @ X )
% 0.33/0.63 = zero_zero_real )
% 0.33/0.63 = ( ? [I4: int] :
% 0.33/0.63 ( X
% 0.33/0.63 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ pi ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sin_zero_iff_int2
% 0.33/0.63 thf(fact_6615_ceiling__altdef,axiom,
% 0.33/0.63 ( archim7802044766580827645g_real
% 0.33/0.63 = ( ^ [X3: real] :
% 0.33/0.63 ( if_int
% 0.33/0.63 @ ( X3
% 0.33/0.63 = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X3 ) ) )
% 0.33/0.63 @ ( archim6058952711729229775r_real @ X3 )
% 0.33/0.63 @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_altdef
% 0.33/0.63 thf(fact_6616_ceiling__altdef,axiom,
% 0.33/0.63 ( archim2889992004027027881ng_rat
% 0.33/0.63 = ( ^ [X3: rat] :
% 0.33/0.63 ( if_int
% 0.33/0.63 @ ( X3
% 0.33/0.63 = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) )
% 0.33/0.63 @ ( archim3151403230148437115or_rat @ X3 )
% 0.33/0.63 @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_altdef
% 0.33/0.63 thf(fact_6617_real__of__int__div__aux,axiom,
% 0.33/0.63 ! [X: int,D: int] :
% 0.33/0.63 ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
% 0.33/0.63 = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_of_int_div_aux
% 0.33/0.63 thf(fact_6618_real__of__int__floor__add__one__gt,axiom,
% 0.33/0.63 ! [R: real] : ( ord_less_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_of_int_floor_add_one_gt
% 0.33/0.63 thf(fact_6619_floor__eq,axiom,
% 0.33/0.63 ! [N: int,X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.33/0.63 => ( ( archim6058952711729229775r_real @ X )
% 0.33/0.63 = N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_eq
% 0.33/0.63 thf(fact_6620_real__of__int__floor__add__one__ge,axiom,
% 0.33/0.63 ! [R: real] : ( ord_less_eq_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_of_int_floor_add_one_ge
% 0.33/0.63 thf(fact_6621_real__of__int__floor__gt__diff__one,axiom,
% 0.33/0.63 ! [R: real] : ( ord_less_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_of_int_floor_gt_diff_one
% 0.33/0.63 thf(fact_6622_real__of__int__floor__ge__diff__one,axiom,
% 0.33/0.63 ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_of_int_floor_ge_diff_one
% 0.33/0.63 thf(fact_6623_arcsin__less__arcsin,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ Y )
% 0.33/0.63 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.63 => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arcsin_less_arcsin
% 0.33/0.63 thf(fact_6624_arcsin__less__mono,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.63 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.33/0.63 => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
% 0.33/0.63 = ( ord_less_real @ X @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arcsin_less_mono
% 0.33/0.63 thf(fact_6625_floor__split,axiom,
% 0.33/0.63 ! [P: int > $o,T: real] :
% 0.33/0.63 ( ( P @ ( archim6058952711729229775r_real @ T ) )
% 0.33/0.63 = ( ! [I4: int] :
% 0.33/0.63 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I4 ) @ T )
% 0.33/0.63 & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) ) )
% 0.33/0.63 => ( P @ I4 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_split
% 0.33/0.63 thf(fact_6626_floor__split,axiom,
% 0.33/0.63 ! [P: int > $o,T: rat] :
% 0.33/0.63 ( ( P @ ( archim3151403230148437115or_rat @ T ) )
% 0.33/0.63 = ( ! [I4: int] :
% 0.33/0.63 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I4 ) @ T )
% 0.33/0.63 & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) ) )
% 0.33/0.63 => ( P @ I4 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_split
% 0.33/0.63 thf(fact_6627_floor__eq__iff,axiom,
% 0.33/0.63 ! [X: real,A: int] :
% 0.33/0.63 ( ( ( archim6058952711729229775r_real @ X )
% 0.33/0.63 = A )
% 0.33/0.63 = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X )
% 0.33/0.63 & ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_eq_iff
% 0.33/0.63 thf(fact_6628_floor__eq__iff,axiom,
% 0.33/0.63 ! [X: rat,A: int] :
% 0.33/0.63 ( ( ( archim3151403230148437115or_rat @ X )
% 0.33/0.63 = A )
% 0.33/0.63 = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
% 0.33/0.63 & ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_eq_iff
% 0.33/0.63 thf(fact_6629_floor__unique,axiom,
% 0.33/0.63 ! [Z2: int,X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) )
% 0.33/0.63 => ( ( archim6058952711729229775r_real @ X )
% 0.33/0.63 = Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_unique
% 0.33/0.63 thf(fact_6630_floor__unique,axiom,
% 0.33/0.63 ! [Z2: int,X: rat] :
% 0.33/0.63 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
% 0.33/0.63 => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) )
% 0.33/0.63 => ( ( archim3151403230148437115or_rat @ X )
% 0.33/0.63 = Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_unique
% 0.33/0.63 thf(fact_6631_ceiling__correct,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) @ one_one_real ) @ X )
% 0.33/0.63 & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_correct
% 0.33/0.63 thf(fact_6632_ceiling__correct,axiom,
% 0.33/0.63 ! [X: rat] :
% 0.33/0.63 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
% 0.33/0.63 & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_correct
% 0.33/0.63 thf(fact_6633_ceiling__unique,axiom,
% 0.33/0.63 ! [Z2: int,X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) )
% 0.33/0.63 => ( ( archim7802044766580827645g_real @ X )
% 0.33/0.63 = Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_unique
% 0.33/0.63 thf(fact_6634_ceiling__unique,axiom,
% 0.33/0.63 ! [Z2: int,X: rat] :
% 0.33/0.63 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X )
% 0.33/0.63 => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) )
% 0.33/0.63 => ( ( archim2889992004027027881ng_rat @ X )
% 0.33/0.63 = Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_unique
% 0.33/0.63 thf(fact_6635_ceiling__eq__iff,axiom,
% 0.33/0.63 ! [X: real,A: int] :
% 0.33/0.63 ( ( ( archim7802044766580827645g_real @ X )
% 0.33/0.63 = A )
% 0.33/0.63 = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
% 0.33/0.63 & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_eq_iff
% 0.33/0.63 thf(fact_6636_ceiling__eq__iff,axiom,
% 0.33/0.63 ! [X: rat,A: int] :
% 0.33/0.63 ( ( ( archim2889992004027027881ng_rat @ X )
% 0.33/0.63 = A )
% 0.33/0.63 = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
% 0.33/0.63 & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_eq_iff
% 0.33/0.63 thf(fact_6637_ceiling__split,axiom,
% 0.33/0.63 ! [P: int > $o,T: real] :
% 0.33/0.63 ( ( P @ ( archim7802044766580827645g_real @ T ) )
% 0.33/0.63 = ( ! [I4: int] :
% 0.33/0.63 ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) @ T )
% 0.33/0.63 & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I4 ) ) )
% 0.33/0.63 => ( P @ I4 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_split
% 0.33/0.63 thf(fact_6638_ceiling__split,axiom,
% 0.33/0.63 ! [P: int > $o,T: rat] :
% 0.33/0.63 ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
% 0.33/0.63 = ( ! [I4: int] :
% 0.33/0.63 ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) @ T )
% 0.33/0.63 & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I4 ) ) )
% 0.33/0.63 => ( P @ I4 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_split
% 0.33/0.63 thf(fact_6639_less__floor__iff,axiom,
% 0.33/0.63 ! [Z2: int,X: real] :
% 0.33/0.63 ( ( ord_less_int @ Z2 @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.63 = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_floor_iff
% 0.33/0.63 thf(fact_6640_less__floor__iff,axiom,
% 0.33/0.63 ! [Z2: int,X: rat] :
% 0.33/0.63 ( ( ord_less_int @ Z2 @ ( archim3151403230148437115or_rat @ X ) )
% 0.33/0.63 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_floor_iff
% 0.33/0.63 thf(fact_6641_floor__le__iff,axiom,
% 0.33/0.63 ! [X: real,Z2: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z2 )
% 0.33/0.63 = ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_le_iff
% 0.33/0.63 thf(fact_6642_floor__le__iff,axiom,
% 0.33/0.63 ! [X: rat,Z2: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 )
% 0.33/0.63 = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_le_iff
% 0.33/0.63 thf(fact_6643_ceiling__less__iff,axiom,
% 0.33/0.63 ! [X: real,Z2: int] :
% 0.33/0.63 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z2 )
% 0.33/0.63 = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_less_iff
% 0.33/0.63 thf(fact_6644_ceiling__less__iff,axiom,
% 0.33/0.63 ! [X: rat,Z2: int] :
% 0.33/0.63 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 )
% 0.33/0.63 = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_less_iff
% 0.33/0.63 thf(fact_6645_le__ceiling__iff,axiom,
% 0.33/0.63 ! [Z2: int,X: rat] :
% 0.33/0.63 ( ( ord_less_eq_int @ Z2 @ ( archim2889992004027027881ng_rat @ X ) )
% 0.33/0.63 = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_ceiling_iff
% 0.33/0.63 thf(fact_6646_le__ceiling__iff,axiom,
% 0.33/0.63 ! [Z2: int,X: real] :
% 0.33/0.63 ( ( ord_less_eq_int @ Z2 @ ( archim7802044766580827645g_real @ X ) )
% 0.33/0.63 = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_ceiling_iff
% 0.33/0.63 thf(fact_6647_floor__correct,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X )
% 0.33/0.63 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_correct
% 0.33/0.63 thf(fact_6648_floor__correct,axiom,
% 0.33/0.63 ! [X: rat] :
% 0.33/0.63 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
% 0.33/0.63 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_correct
% 0.33/0.63 thf(fact_6649_real__of__int__div2,axiom,
% 0.33/0.63 ! [N: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_of_int_div2
% 0.33/0.63 thf(fact_6650_real__of__int__div3,axiom,
% 0.33/0.63 ! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) @ one_one_real ) ).
% 0.33/0.63
% 0.33/0.63 % real_of_int_div3
% 0.33/0.63 thf(fact_6651_floor__eq2,axiom,
% 0.33/0.63 ! [N: int,X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.33/0.63 => ( ( archim6058952711729229775r_real @ X )
% 0.33/0.63 = N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_eq2
% 0.33/0.63 thf(fact_6652_floor__divide__real__eq__div,axiom,
% 0.33/0.63 ! [B: int,A: real] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.63 => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
% 0.33/0.63 = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_divide_real_eq_div
% 0.33/0.63 thf(fact_6653_floor__divide__lower,axiom,
% 0.33/0.63 ! [Q2: real,P2: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 0.33/0.63 => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ Q2 ) @ P2 ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_divide_lower
% 0.33/0.63 thf(fact_6654_floor__divide__lower,axiom,
% 0.33/0.63 ! [Q2: rat,P2: rat] :
% 0.33/0.63 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 0.33/0.63 => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ Q2 ) @ P2 ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_divide_lower
% 0.33/0.63 thf(fact_6655_ceiling__divide__upper,axiom,
% 0.33/0.63 ! [Q2: real,P2: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 0.33/0.63 => ( ord_less_eq_real @ P2 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ Q2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_divide_upper
% 0.33/0.63 thf(fact_6656_ceiling__divide__upper,axiom,
% 0.33/0.63 ! [Q2: rat,P2: rat] :
% 0.33/0.63 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 0.33/0.63 => ( ord_less_eq_rat @ P2 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ Q2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_divide_upper
% 0.33/0.63 thf(fact_6657_tan__45,axiom,
% 0.33/0.63 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.33/0.63 = one_one_real ) ).
% 0.33/0.63
% 0.33/0.63 % tan_45
% 0.33/0.63 thf(fact_6658_even__of__int__iff,axiom,
% 0.33/0.63 ! [K: int] :
% 0.33/0.63 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
% 0.33/0.63 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 0.33/0.63
% 0.33/0.63 % even_of_int_iff
% 0.33/0.63 thf(fact_6659_even__of__int__iff,axiom,
% 0.33/0.63 ! [K: int] :
% 0.33/0.63 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
% 0.33/0.63 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 0.33/0.63
% 0.33/0.63 % even_of_int_iff
% 0.33/0.63 thf(fact_6660_cos__arcsin__nonzero,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.63 => ( ( cos_real @ ( arcsin @ X ) )
% 0.33/0.63 != zero_zero_real ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % cos_arcsin_nonzero
% 0.33/0.63 thf(fact_6661_lemma__tan__total,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.63 => ? [X4: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.33/0.63 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 & ( ord_less_real @ Y @ ( tan_real @ X4 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % lemma_tan_total
% 0.33/0.63 thf(fact_6662_tan__gt__zero,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_gt_zero
% 0.33/0.63 thf(fact_6663_floor__divide__upper,axiom,
% 0.33/0.63 ! [Q2: real,P2: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 0.33/0.63 => ( ord_less_real @ P2 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_divide_upper
% 0.33/0.63 thf(fact_6664_floor__divide__upper,axiom,
% 0.33/0.63 ! [Q2: rat,P2: rat] :
% 0.33/0.63 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 0.33/0.63 => ( ord_less_rat @ P2 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_divide_upper
% 0.33/0.63 thf(fact_6665_ceiling__divide__lower,axiom,
% 0.33/0.63 ! [Q2: real,P2: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 0.33/0.63 => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) @ P2 ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_divide_lower
% 0.33/0.63 thf(fact_6666_ceiling__divide__lower,axiom,
% 0.33/0.63 ! [Q2: rat,P2: rat] :
% 0.33/0.63 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 0.33/0.63 => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) @ P2 ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_divide_lower
% 0.33/0.63 thf(fact_6667_lemma__tan__total1,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ? [X4: real] :
% 0.33/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.33/0.63 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 & ( ( tan_real @ X4 )
% 0.33/0.63 = Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % lemma_tan_total1
% 0.33/0.63 thf(fact_6668_tan__mono__lt__eq,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.63 => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 0.33/0.63 = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_mono_lt_eq
% 0.33/0.63 thf(fact_6669_tan__monotone_H,axiom,
% 0.33/0.63 ! [Y: real,X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.63 => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ( ord_less_real @ Y @ X )
% 0.33/0.63 = ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_monotone'
% 0.33/0.63 thf(fact_6670_tan__monotone,axiom,
% 0.33/0.63 ! [Y: real,X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.63 => ( ( ord_less_real @ Y @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_monotone
% 0.33/0.63 thf(fact_6671_tan__total,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ? [X4: real] :
% 0.33/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.33/0.63 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 & ( ( tan_real @ X4 )
% 0.33/0.63 = Y )
% 0.33/0.63 & ! [Y5: real] :
% 0.33/0.63 ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 0.33/0.63 & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 & ( ( tan_real @ Y5 )
% 0.33/0.63 = Y ) )
% 0.33/0.63 => ( Y5 = X4 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_total
% 0.33/0.63 thf(fact_6672_tan__minus__45,axiom,
% 0.33/0.63 ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.63 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_minus_45
% 0.33/0.63 thf(fact_6673_tan__inverse,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y ) )
% 0.33/0.63 = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_inverse
% 0.33/0.63 thf(fact_6674_ceiling__eq,axiom,
% 0.33/0.63 ! [N: int,X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.33/0.63 => ( ( archim7802044766580827645g_real @ X )
% 0.33/0.63 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_eq
% 0.33/0.63 thf(fact_6675_ceiling__eq,axiom,
% 0.33/0.63 ! [N: int,X: rat] :
% 0.33/0.63 ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X )
% 0.33/0.63 => ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
% 0.33/0.63 => ( ( archim2889992004027027881ng_rat @ X )
% 0.33/0.63 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_eq
% 0.33/0.63 thf(fact_6676_cos__one__2pi__int,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ( cos_real @ X )
% 0.33/0.63 = one_one_real )
% 0.33/0.63 = ( ? [X3: int] :
% 0.33/0.63 ( X
% 0.33/0.63 = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % cos_one_2pi_int
% 0.33/0.63 thf(fact_6677_add__tan__eq,axiom,
% 0.33/0.63 ! [X: complex,Y: complex] :
% 0.33/0.63 ( ( ( cos_complex @ X )
% 0.33/0.63 != zero_zero_complex )
% 0.33/0.63 => ( ( ( cos_complex @ Y )
% 0.33/0.63 != zero_zero_complex )
% 0.33/0.63 => ( ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) )
% 0.33/0.63 = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % add_tan_eq
% 0.33/0.63 thf(fact_6678_add__tan__eq,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ( cos_real @ X )
% 0.33/0.63 != zero_zero_real )
% 0.33/0.63 => ( ( ( cos_real @ Y )
% 0.33/0.63 != zero_zero_real )
% 0.33/0.63 => ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 0.33/0.63 = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % add_tan_eq
% 0.33/0.63 thf(fact_6679_tan__cot_H,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 0.33/0.63 = ( cot_real @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_cot'
% 0.33/0.63 thf(fact_6680_tan__pos__pi2__le,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_pos_pi2_le
% 0.33/0.63 thf(fact_6681_tan__total__pos,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.63 => ? [X4: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.33/0.63 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 & ( ( tan_real @ X4 )
% 0.33/0.63 = Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_total_pos
% 0.33/0.63 thf(fact_6682_tan__less__zero,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.63 => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_less_zero
% 0.33/0.63 thf(fact_6683_tan__mono__le,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.63 => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_mono_le
% 0.33/0.63 thf(fact_6684_tan__mono__le__eq,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.63 => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 0.33/0.63 = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_mono_le_eq
% 0.33/0.63 thf(fact_6685_tan__bound__pi2,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.33/0.63 => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_bound_pi2
% 0.33/0.63 thf(fact_6686_arctan,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
% 0.33/0.63 & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 & ( ( tan_real @ ( arctan @ Y ) )
% 0.33/0.63 = Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % arctan
% 0.33/0.63 thf(fact_6687_arctan__tan,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ( arctan @ ( tan_real @ X ) )
% 0.33/0.63 = X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arctan_tan
% 0.33/0.63 thf(fact_6688_arctan__unique,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ( ( tan_real @ X )
% 0.33/0.63 = Y )
% 0.33/0.63 => ( ( arctan @ Y )
% 0.33/0.63 = X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arctan_unique
% 0.33/0.63 thf(fact_6689_tan__add,axiom,
% 0.33/0.63 ! [X: complex,Y: complex] :
% 0.33/0.63 ( ( ( cos_complex @ X )
% 0.33/0.63 != zero_zero_complex )
% 0.33/0.63 => ( ( ( cos_complex @ Y )
% 0.33/0.63 != zero_zero_complex )
% 0.33/0.63 => ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.33/0.63 != zero_zero_complex )
% 0.33/0.63 => ( ( tan_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.33/0.63 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_add
% 0.33/0.63 thf(fact_6690_tan__add,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ( cos_real @ X )
% 0.33/0.63 != zero_zero_real )
% 0.33/0.63 => ( ( ( cos_real @ Y )
% 0.33/0.63 != zero_zero_real )
% 0.33/0.63 => ( ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.63 != zero_zero_real )
% 0.33/0.63 => ( ( tan_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.63 = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_add
% 0.33/0.63 thf(fact_6691_tan__diff,axiom,
% 0.33/0.63 ! [X: complex,Y: complex] :
% 0.33/0.63 ( ( ( cos_complex @ X )
% 0.33/0.63 != zero_zero_complex )
% 0.33/0.63 => ( ( ( cos_complex @ Y )
% 0.33/0.63 != zero_zero_complex )
% 0.33/0.63 => ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
% 0.33/0.63 != zero_zero_complex )
% 0.33/0.63 => ( ( tan_complex @ ( minus_minus_complex @ X @ Y ) )
% 0.33/0.63 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_diff
% 0.33/0.63 thf(fact_6692_tan__diff,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ( cos_real @ X )
% 0.33/0.63 != zero_zero_real )
% 0.33/0.63 => ( ( ( cos_real @ Y )
% 0.33/0.63 != zero_zero_real )
% 0.33/0.63 => ( ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
% 0.33/0.63 != zero_zero_real )
% 0.33/0.63 => ( ( tan_real @ ( minus_minus_real @ X @ Y ) )
% 0.33/0.63 = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_diff
% 0.33/0.63 thf(fact_6693_lemma__tan__add1,axiom,
% 0.33/0.63 ! [X: complex,Y: complex] :
% 0.33/0.63 ( ( ( cos_complex @ X )
% 0.33/0.63 != zero_zero_complex )
% 0.33/0.63 => ( ( ( cos_complex @ Y )
% 0.33/0.63 != zero_zero_complex )
% 0.33/0.63 => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) )
% 0.33/0.63 = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % lemma_tan_add1
% 0.33/0.63 thf(fact_6694_lemma__tan__add1,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ( cos_real @ X )
% 0.33/0.63 != zero_zero_real )
% 0.33/0.63 => ( ( ( cos_real @ Y )
% 0.33/0.63 != zero_zero_real )
% 0.33/0.63 => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) )
% 0.33/0.63 = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % lemma_tan_add1
% 0.33/0.63 thf(fact_6695_tan__total__pi4,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.63 => ? [Z4: real] :
% 0.33/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z4 )
% 0.33/0.63 & ( ord_less_real @ Z4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.33/0.63 & ( ( tan_real @ Z4 )
% 0.33/0.63 = X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_total_pi4
% 0.33/0.63 thf(fact_6696_floor__log__eq__powr__iff,axiom,
% 0.33/0.63 ! [X: real,B: real,K: int] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.63 => ( ( ( archim6058952711729229775r_real @ ( log2 @ B @ X ) )
% 0.33/0.63 = K )
% 0.33/0.63 = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
% 0.33/0.63 & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_log_eq_powr_iff
% 0.33/0.63 thf(fact_6697_arcsin__lt__bounded,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.63 => ( ( ord_less_real @ Y @ one_one_real )
% 0.33/0.63 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 0.33/0.63 & ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arcsin_lt_bounded
% 0.33/0.63 thf(fact_6698_arcsin__lbound,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.63 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.63 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arcsin_lbound
% 0.33/0.63 thf(fact_6699_arcsin__ubound,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.63 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.63 => ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arcsin_ubound
% 0.33/0.63 thf(fact_6700_arcsin__bounded,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.63 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.63 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 0.33/0.63 & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arcsin_bounded
% 0.33/0.63 thf(fact_6701_arcsin__sin,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ( arcsin @ ( sin_real @ X ) )
% 0.33/0.63 = X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arcsin_sin
% 0.33/0.63 thf(fact_6702_cos__zero__iff__int,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ( cos_real @ X )
% 0.33/0.63 = zero_zero_real )
% 0.33/0.63 = ( ? [I4: int] :
% 0.33/0.63 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
% 0.33/0.63 & ( X
% 0.33/0.63 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % cos_zero_iff_int
% 0.33/0.63 thf(fact_6703_sin__zero__iff__int,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ( sin_real @ X )
% 0.33/0.63 = zero_zero_real )
% 0.33/0.63 = ( ? [I4: int] :
% 0.33/0.63 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
% 0.33/0.63 & ( X
% 0.33/0.63 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sin_zero_iff_int
% 0.33/0.63 thf(fact_6704_tan__half,axiom,
% 0.33/0.63 ( tan_complex
% 0.33/0.63 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_complex ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_half
% 0.33/0.63 thf(fact_6705_tan__half,axiom,
% 0.33/0.63 ( tan_real
% 0.33/0.63 = ( ^ [X3: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_real ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_half
% 0.33/0.63 thf(fact_6706_arcsin,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.63 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.63 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 0.33/0.63 & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 & ( ( sin_real @ ( arcsin @ Y ) )
% 0.33/0.63 = Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arcsin
% 0.33/0.63 thf(fact_6707_round__unique,axiom,
% 0.33/0.63 ! [X: real,Y: int] :
% 0.33/0.63 ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
% 0.33/0.63 => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.33/0.63 => ( ( archim8280529875227126926d_real @ X )
% 0.33/0.63 = Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_unique
% 0.33/0.63 thf(fact_6708_round__unique,axiom,
% 0.33/0.63 ! [X: rat,Y: int] :
% 0.33/0.63 ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
% 0.33/0.63 => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.63 => ( ( archim7778729529865785530nd_rat @ X )
% 0.33/0.63 = Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_unique
% 0.33/0.63 thf(fact_6709_round__unique_H,axiom,
% 0.33/0.63 ! [X: real,N: int] :
% 0.33/0.63 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ( archim8280529875227126926d_real @ X )
% 0.33/0.63 = N ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_unique'
% 0.33/0.63 thf(fact_6710_round__unique_H,axiom,
% 0.33/0.63 ! [X: rat,N: int] :
% 0.33/0.63 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ( archim7778729529865785530nd_rat @ X )
% 0.33/0.63 = N ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_unique'
% 0.33/0.63 thf(fact_6711_of__int__round__abs__le,axiom,
% 0.33/0.63 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_round_abs_le
% 0.33/0.63 thf(fact_6712_of__int__round__abs__le,axiom,
% 0.33/0.63 ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_round_abs_le
% 0.33/0.63 thf(fact_6713_sin__tan,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ( sin_real @ X )
% 0.33/0.63 = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sin_tan
% 0.33/0.63 thf(fact_6714_cos__tan,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ( cos_real @ X )
% 0.33/0.63 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % cos_tan
% 0.33/0.63 thf(fact_6715_of__int__round__gt,axiom,
% 0.33/0.63 ! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_round_gt
% 0.33/0.63 thf(fact_6716_of__int__round__gt,axiom,
% 0.33/0.63 ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_round_gt
% 0.33/0.63 thf(fact_6717_real__sqrt__eq__iff,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ( sqrt @ X )
% 0.33/0.63 = ( sqrt @ Y ) )
% 0.33/0.63 = ( X = Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_eq_iff
% 0.33/0.63 thf(fact_6718_real__sqrt__zero,axiom,
% 0.33/0.63 ( ( sqrt @ zero_zero_real )
% 0.33/0.63 = zero_zero_real ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_zero
% 0.33/0.63 thf(fact_6719_real__sqrt__eq__zero__cancel__iff,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ( sqrt @ X )
% 0.33/0.63 = zero_zero_real )
% 0.33/0.63 = ( X = zero_zero_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_eq_zero_cancel_iff
% 0.33/0.63 thf(fact_6720_real__sqrt__less__iff,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
% 0.33/0.63 = ( ord_less_real @ X @ Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_less_iff
% 0.33/0.63 thf(fact_6721_real__sqrt__le__iff,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
% 0.33/0.63 = ( ord_less_eq_real @ X @ Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_le_iff
% 0.33/0.63 thf(fact_6722_real__sqrt__one,axiom,
% 0.33/0.63 ( ( sqrt @ one_one_real )
% 0.33/0.63 = one_one_real ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_one
% 0.33/0.63 thf(fact_6723_real__sqrt__eq__1__iff,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ( sqrt @ X )
% 0.33/0.63 = one_one_real )
% 0.33/0.63 = ( X = one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_eq_1_iff
% 0.33/0.63 thf(fact_6724_round__of__int,axiom,
% 0.33/0.63 ! [N: int] :
% 0.33/0.63 ( ( archim8280529875227126926d_real @ ( ring_1_of_int_real @ N ) )
% 0.33/0.63 = N ) ).
% 0.33/0.63
% 0.33/0.63 % round_of_int
% 0.33/0.63 thf(fact_6725_round__of__int,axiom,
% 0.33/0.63 ! [N: int] :
% 0.33/0.63 ( ( archim7778729529865785530nd_rat @ ( ring_1_of_int_rat @ N ) )
% 0.33/0.63 = N ) ).
% 0.33/0.63
% 0.33/0.63 % round_of_int
% 0.33/0.63 thf(fact_6726_real__sqrt__lt__0__iff,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
% 0.33/0.63 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_lt_0_iff
% 0.33/0.63 thf(fact_6727_real__sqrt__gt__0__iff,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
% 0.33/0.63 = ( ord_less_real @ zero_zero_real @ Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_gt_0_iff
% 0.33/0.63 thf(fact_6728_real__sqrt__ge__0__iff,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
% 0.33/0.63 = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_ge_0_iff
% 0.33/0.63 thf(fact_6729_real__sqrt__le__0__iff,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
% 0.33/0.63 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_le_0_iff
% 0.33/0.63 thf(fact_6730_real__sqrt__gt__1__iff,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
% 0.33/0.63 = ( ord_less_real @ one_one_real @ Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_gt_1_iff
% 0.33/0.63 thf(fact_6731_real__sqrt__lt__1__iff,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
% 0.33/0.63 = ( ord_less_real @ X @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_lt_1_iff
% 0.33/0.63 thf(fact_6732_real__sqrt__ge__1__iff,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
% 0.33/0.63 = ( ord_less_eq_real @ one_one_real @ Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_ge_1_iff
% 0.33/0.63 thf(fact_6733_real__sqrt__le__1__iff,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
% 0.33/0.63 = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_le_1_iff
% 0.33/0.63 thf(fact_6734_real__sqrt__mult__self,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
% 0.33/0.63 = ( abs_abs_real @ A ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_mult_self
% 0.33/0.63 thf(fact_6735_real__sqrt__abs2,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( sqrt @ ( times_times_real @ X @ X ) )
% 0.33/0.63 = ( abs_abs_real @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_abs2
% 0.33/0.63 thf(fact_6736_round__0,axiom,
% 0.33/0.63 ( ( archim8280529875227126926d_real @ zero_zero_real )
% 0.33/0.63 = zero_zero_int ) ).
% 0.33/0.63
% 0.33/0.63 % round_0
% 0.33/0.63 thf(fact_6737_round__0,axiom,
% 0.33/0.63 ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
% 0.33/0.63 = zero_zero_int ) ).
% 0.33/0.63
% 0.33/0.63 % round_0
% 0.33/0.63 thf(fact_6738_round__numeral,axiom,
% 0.33/0.63 ! [N: num] :
% 0.33/0.63 ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
% 0.33/0.63 = ( numeral_numeral_int @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_numeral
% 0.33/0.63 thf(fact_6739_round__numeral,axiom,
% 0.33/0.63 ! [N: num] :
% 0.33/0.63 ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
% 0.33/0.63 = ( numeral_numeral_int @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_numeral
% 0.33/0.63 thf(fact_6740_round__1,axiom,
% 0.33/0.63 ( ( archim8280529875227126926d_real @ one_one_real )
% 0.33/0.63 = one_one_int ) ).
% 0.33/0.63
% 0.33/0.63 % round_1
% 0.33/0.63 thf(fact_6741_round__1,axiom,
% 0.33/0.63 ( ( archim7778729529865785530nd_rat @ one_one_rat )
% 0.33/0.63 = one_one_int ) ).
% 0.33/0.63
% 0.33/0.63 % round_1
% 0.33/0.63 thf(fact_6742_round__of__nat,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( archim8280529875227126926d_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.33/0.63 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_of_nat
% 0.33/0.63 thf(fact_6743_round__of__nat,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( archim7778729529865785530nd_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.33/0.63 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_of_nat
% 0.33/0.63 thf(fact_6744_real__sqrt__four,axiom,
% 0.33/0.63 ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.63 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_four
% 0.33/0.63 thf(fact_6745_round__neg__numeral,axiom,
% 0.33/0.63 ! [N: num] :
% 0.33/0.63 ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.33/0.63 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_neg_numeral
% 0.33/0.63 thf(fact_6746_round__neg__numeral,axiom,
% 0.33/0.63 ! [N: num] :
% 0.33/0.63 ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.33/0.63 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_neg_numeral
% 0.33/0.63 thf(fact_6747_real__sqrt__abs,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.63 = ( abs_abs_real @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_abs
% 0.33/0.63 thf(fact_6748_real__sqrt__pow2__iff,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.63 = X )
% 0.33/0.63 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_pow2_iff
% 0.33/0.63 thf(fact_6749_real__sqrt__pow2,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.63 = X ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_pow2
% 0.33/0.63 thf(fact_6750_real__sqrt__sum__squares__mult__squared__eq,axiom,
% 0.33/0.63 ! [X: real,Y: real,Xa: real,Ya: real] :
% 0.33/0.63 ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.63 = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_sum_squares_mult_squared_eq
% 0.33/0.63 thf(fact_6751_real__sqrt__less__mono,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_real @ X @ Y )
% 0.33/0.63 => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_less_mono
% 0.33/0.63 thf(fact_6752_real__sqrt__le__mono,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.63 => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_le_mono
% 0.33/0.63 thf(fact_6753_real__sqrt__divide,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( sqrt @ ( divide_divide_real @ X @ Y ) )
% 0.33/0.63 = ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_divide
% 0.33/0.63 thf(fact_6754_real__sqrt__mult,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( sqrt @ ( times_times_real @ X @ Y ) )
% 0.33/0.63 = ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_mult
% 0.33/0.63 thf(fact_6755_real__sqrt__power,axiom,
% 0.33/0.63 ! [X: real,K: nat] :
% 0.33/0.63 ( ( sqrt @ ( power_power_real @ X @ K ) )
% 0.33/0.63 = ( power_power_real @ ( sqrt @ X ) @ K ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_power
% 0.33/0.63 thf(fact_6756_real__sqrt__minus,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( sqrt @ ( uminus_uminus_real @ X ) )
% 0.33/0.63 = ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_minus
% 0.33/0.63 thf(fact_6757_real__sqrt__gt__zero,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_gt_zero
% 0.33/0.63 thf(fact_6758_real__sqrt__ge__zero,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_ge_zero
% 0.33/0.63 thf(fact_6759_real__sqrt__eq__zero__cancel,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ( sqrt @ X )
% 0.33/0.63 = zero_zero_real )
% 0.33/0.63 => ( X = zero_zero_real ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_eq_zero_cancel
% 0.33/0.63 thf(fact_6760_real__sqrt__ge__one,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.33/0.63 => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_ge_one
% 0.33/0.63 thf(fact_6761_real__div__sqrt,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
% 0.33/0.63 = ( sqrt @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_div_sqrt
% 0.33/0.63 thf(fact_6762_sqrt__add__le__add__sqrt,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.63 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sqrt_add_le_add_sqrt
% 0.33/0.63 thf(fact_6763_le__real__sqrt__sumsq,axiom,
% 0.33/0.63 ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_real_sqrt_sumsq
% 0.33/0.63 thf(fact_6764_round__mono,axiom,
% 0.33/0.63 ! [X: rat,Y: rat] :
% 0.33/0.63 ( ( ord_less_eq_rat @ X @ Y )
% 0.33/0.63 => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_mono
% 0.33/0.63 thf(fact_6765_floor__le__round,axiom,
% 0.33/0.63 ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim8280529875227126926d_real @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_le_round
% 0.33/0.63 thf(fact_6766_floor__le__round,axiom,
% 0.33/0.63 ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim7778729529865785530nd_rat @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_le_round
% 0.33/0.63 thf(fact_6767_ceiling__ge__round,axiom,
% 0.33/0.63 ! [X: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % ceiling_ge_round
% 0.33/0.63 thf(fact_6768_sqrt2__less__2,axiom,
% 0.33/0.63 ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.33/0.63
% 0.33/0.63 % sqrt2_less_2
% 0.33/0.63 thf(fact_6769_real__less__rsqrt,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
% 0.33/0.63 => ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_less_rsqrt
% 0.33/0.63 thf(fact_6770_real__le__rsqrt,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
% 0.33/0.63 => ( ord_less_eq_real @ X @ ( sqrt @ Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_le_rsqrt
% 0.33/0.63 thf(fact_6771_sqrt__le__D,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y )
% 0.33/0.63 => ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sqrt_le_D
% 0.33/0.63 thf(fact_6772_tan__60,axiom,
% 0.33/0.63 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.33/0.63 = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_60
% 0.33/0.63 thf(fact_6773_real__sqrt__unique,axiom,
% 0.33/0.63 ! [Y: real,X: real] :
% 0.33/0.63 ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.63 = X )
% 0.33/0.63 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.63 => ( ( sqrt @ X )
% 0.33/0.63 = Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_unique
% 0.33/0.63 thf(fact_6774_real__le__lsqrt,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.63 => ( ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ord_less_eq_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_le_lsqrt
% 0.33/0.63 thf(fact_6775_lemma__real__divide__sqrt__less,axiom,
% 0.33/0.63 ! [U: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ U )
% 0.33/0.63 => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 0.33/0.63
% 0.33/0.63 % lemma_real_divide_sqrt_less
% 0.33/0.63 thf(fact_6776_real__sqrt__sum__squares__eq__cancel2,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.63 = Y )
% 0.33/0.63 => ( X = zero_zero_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_sum_squares_eq_cancel2
% 0.33/0.63 thf(fact_6777_real__sqrt__sum__squares__eq__cancel,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.63 = X )
% 0.33/0.63 => ( Y = zero_zero_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_sum_squares_eq_cancel
% 0.33/0.63 thf(fact_6778_real__sqrt__sum__squares__triangle__ineq,axiom,
% 0.33/0.63 ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_sum_squares_triangle_ineq
% 0.33/0.63 thf(fact_6779_real__sqrt__sum__squares__ge2,axiom,
% 0.33/0.63 ! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_sum_squares_ge2
% 0.33/0.63 thf(fact_6780_real__sqrt__sum__squares__ge1,axiom,
% 0.33/0.63 ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_sum_squares_ge1
% 0.33/0.63 thf(fact_6781_sqrt__ge__absD,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y ) )
% 0.33/0.63 => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % sqrt_ge_absD
% 0.33/0.63 thf(fact_6782_cos__45,axiom,
% 0.33/0.63 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.33/0.63 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % cos_45
% 0.33/0.63 thf(fact_6783_sin__45,axiom,
% 0.33/0.63 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.33/0.63 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sin_45
% 0.33/0.63 thf(fact_6784_round__diff__minimal,axiom,
% 0.33/0.63 ! [Z2: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z2 ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ M ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_diff_minimal
% 0.33/0.63 thf(fact_6785_round__diff__minimal,axiom,
% 0.33/0.63 ! [Z2: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z2 ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ M ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_diff_minimal
% 0.33/0.63 thf(fact_6786_real__less__lsqrt,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.63 => ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_less_lsqrt
% 0.33/0.63 thf(fact_6787_sqrt__sum__squares__le__sum,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.63 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sqrt_sum_squares_le_sum
% 0.33/0.63 thf(fact_6788_tan__30,axiom,
% 0.33/0.63 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.33/0.63 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tan_30
% 0.33/0.63 thf(fact_6789_ln__sqrt,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ln_ln_real @ ( sqrt @ X ) )
% 0.33/0.63 = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ln_sqrt
% 0.33/0.63 thf(fact_6790_real__sqrt__ge__abs1,axiom,
% 0.33/0.63 ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_ge_abs1
% 0.33/0.63 thf(fact_6791_real__sqrt__ge__abs2,axiom,
% 0.33/0.63 ! [Y: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_ge_abs2
% 0.33/0.63 thf(fact_6792_sqrt__sum__squares__le__sum__abs,axiom,
% 0.33/0.63 ! [X: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sqrt_sum_squares_le_sum_abs
% 0.33/0.63 thf(fact_6793_sqrt__even__pow2,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.63 => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.63 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sqrt_even_pow2
% 0.33/0.63 thf(fact_6794_cos__30,axiom,
% 0.33/0.63 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.33/0.63 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % cos_30
% 0.33/0.63 thf(fact_6795_sin__60,axiom,
% 0.33/0.63 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.33/0.63 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sin_60
% 0.33/0.63 thf(fact_6796_arsinh__real__def,axiom,
% 0.33/0.63 ( arsinh_real
% 0.33/0.63 = ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arsinh_real_def
% 0.33/0.63 thf(fact_6797_complex__norm,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y ) )
% 0.33/0.63 = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % complex_norm
% 0.33/0.63 thf(fact_6798_arsinh__real__aux,axiom,
% 0.33/0.63 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arsinh_real_aux
% 0.33/0.63 thf(fact_6799_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 0.33/0.63 ! [X: real,Y: real,Xa: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_sum_squares_mult_ge_zero
% 0.33/0.63 thf(fact_6800_real__sqrt__power__even,axiom,
% 0.33/0.63 ! [N: nat,X: real] :
% 0.33/0.63 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.63 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( power_power_real @ ( sqrt @ X ) @ N )
% 0.33/0.63 = ( power_power_real @ X @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_power_even
% 0.33/0.63 thf(fact_6801_arith__geo__mean__sqrt,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.63 => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arith_geo_mean_sqrt
% 0.33/0.63 thf(fact_6802_powr__half__sqrt,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 = ( sqrt @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % powr_half_sqrt
% 0.33/0.63 thf(fact_6803_cos__x__y__le__one,axiom,
% 0.33/0.63 ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 0.33/0.63
% 0.33/0.63 % cos_x_y_le_one
% 0.33/0.63 thf(fact_6804_real__sqrt__sum__squares__less,axiom,
% 0.33/0.63 ! [X: real,U: real,Y: real] :
% 0.33/0.63 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.33/0.63 => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.33/0.63 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_sum_squares_less
% 0.33/0.63 thf(fact_6805_arcosh__real__def,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.33/0.63 => ( ( arcosh_real @ X )
% 0.33/0.63 = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arcosh_real_def
% 0.33/0.63 thf(fact_6806_cos__arctan,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( cos_real @ ( arctan @ X ) )
% 0.33/0.63 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % cos_arctan
% 0.33/0.63 thf(fact_6807_sin__arctan,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( sin_real @ ( arctan @ X ) )
% 0.33/0.63 = ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sin_arctan
% 0.33/0.63 thf(fact_6808_sqrt__sum__squares__half__less,axiom,
% 0.33/0.63 ! [X: real,U: real,Y: real] :
% 0.33/0.63 ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.63 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sqrt_sum_squares_half_less
% 0.33/0.63 thf(fact_6809_sin__cos__sqrt,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
% 0.33/0.63 => ( ( sin_real @ X )
% 0.33/0.63 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sin_cos_sqrt
% 0.33/0.63 thf(fact_6810_arctan__half,axiom,
% 0.33/0.63 ( arctan
% 0.33/0.63 = ( ^ [X3: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X3 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arctan_half
% 0.33/0.63 thf(fact_6811_cos__arcsin,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.63 => ( ( cos_real @ ( arcsin @ X ) )
% 0.33/0.63 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % cos_arcsin
% 0.33/0.63 thf(fact_6812_round__def,axiom,
% 0.33/0.63 ( archim8280529875227126926d_real
% 0.33/0.63 = ( ^ [X3: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_def
% 0.33/0.63 thf(fact_6813_round__def,axiom,
% 0.33/0.63 ( archim7778729529865785530nd_rat
% 0.33/0.63 = ( ^ [X3: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_def
% 0.33/0.63 thf(fact_6814_of__int__round__le,axiom,
% 0.33/0.63 ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_round_le
% 0.33/0.63 thf(fact_6815_of__int__round__le,axiom,
% 0.33/0.63 ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_round_le
% 0.33/0.63 thf(fact_6816_of__int__round__ge,axiom,
% 0.33/0.63 ! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_round_ge
% 0.33/0.63 thf(fact_6817_of__int__round__ge,axiom,
% 0.33/0.63 ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_round_ge
% 0.33/0.63 thf(fact_6818_round__altdef,axiom,
% 0.33/0.63 ( archim8280529875227126926d_real
% 0.33/0.63 = ( ^ [X3: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X3 ) ) @ ( archim7802044766580827645g_real @ X3 ) @ ( archim6058952711729229775r_real @ X3 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_altdef
% 0.33/0.63 thf(fact_6819_round__altdef,axiom,
% 0.33/0.63 ( archim7778729529865785530nd_rat
% 0.33/0.63 = ( ^ [X3: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X3 ) ) @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim3151403230148437115or_rat @ X3 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % round_altdef
% 0.33/0.63 thf(fact_6820_sin__arccos__abs,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.33/0.63 => ( ( sin_real @ ( arccos @ Y ) )
% 0.33/0.63 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sin_arccos_abs
% 0.33/0.63 thf(fact_6821_sin__arccos,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.63 => ( ( sin_real @ ( arccos @ X ) )
% 0.33/0.63 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sin_arccos
% 0.33/0.63 thf(fact_6822_arccos__cos__eq__abs__2pi,axiom,
% 0.33/0.63 ! [Theta: real] :
% 0.33/0.63 ~ ! [K3: int] :
% 0.33/0.63 ( ( arccos @ ( cos_real @ Theta ) )
% 0.33/0.63 != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arccos_cos_eq_abs_2pi
% 0.33/0.63 thf(fact_6823_log__base__10__eq1,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( log2 @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.63 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % log_base_10_eq1
% 0.33/0.63 thf(fact_6824_exp__zero,axiom,
% 0.33/0.63 ( ( exp_complex @ zero_zero_complex )
% 0.33/0.63 = one_one_complex ) ).
% 0.33/0.63
% 0.33/0.63 % exp_zero
% 0.33/0.63 thf(fact_6825_exp__zero,axiom,
% 0.33/0.63 ( ( exp_real @ zero_zero_real )
% 0.33/0.63 = one_one_real ) ).
% 0.33/0.63
% 0.33/0.63 % exp_zero
% 0.33/0.63 thf(fact_6826_exp__eq__one__iff,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ( exp_real @ X )
% 0.33/0.63 = one_one_real )
% 0.33/0.63 = ( X = zero_zero_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_eq_one_iff
% 0.33/0.63 thf(fact_6827_arccos__1,axiom,
% 0.33/0.63 ( ( arccos @ one_one_real )
% 0.33/0.63 = zero_zero_real ) ).
% 0.33/0.63
% 0.33/0.63 % arccos_1
% 0.33/0.63 thf(fact_6828_frac__of__int,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.33/0.63 = zero_zero_real ) ).
% 0.33/0.63
% 0.33/0.63 % frac_of_int
% 0.33/0.63 thf(fact_6829_frac__of__int,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.33/0.63 = zero_zero_rat ) ).
% 0.33/0.63
% 0.33/0.63 % frac_of_int
% 0.33/0.63 thf(fact_6830_one__less__exp__iff,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
% 0.33/0.63 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % one_less_exp_iff
% 0.33/0.63 thf(fact_6831_exp__less__one__iff,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
% 0.33/0.63 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_less_one_iff
% 0.33/0.63 thf(fact_6832_one__le__exp__iff,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
% 0.33/0.63 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % one_le_exp_iff
% 0.33/0.63 thf(fact_6833_exp__le__one__iff,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
% 0.33/0.63 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_le_one_iff
% 0.33/0.63 thf(fact_6834_arccos__minus__1,axiom,
% 0.33/0.63 ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.63 = pi ) ).
% 0.33/0.63
% 0.33/0.63 % arccos_minus_1
% 0.33/0.63 thf(fact_6835_cos__arccos,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.63 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.63 => ( ( cos_real @ ( arccos @ Y ) )
% 0.33/0.63 = Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % cos_arccos
% 0.33/0.63 thf(fact_6836_arccos__0,axiom,
% 0.33/0.63 ( ( arccos @ zero_zero_real )
% 0.33/0.63 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arccos_0
% 0.33/0.63 thf(fact_6837_exp__not__eq__zero,axiom,
% 0.33/0.63 ! [X: complex] :
% 0.33/0.63 ( ( exp_complex @ X )
% 0.33/0.63 != zero_zero_complex ) ).
% 0.33/0.63
% 0.33/0.63 % exp_not_eq_zero
% 0.33/0.63 thf(fact_6838_exp__not__eq__zero,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( exp_real @ X )
% 0.33/0.63 != zero_zero_real ) ).
% 0.33/0.63
% 0.33/0.63 % exp_not_eq_zero
% 0.33/0.63 thf(fact_6839_exp__times__arg__commute,axiom,
% 0.33/0.63 ! [A3: complex] :
% 0.33/0.63 ( ( times_times_complex @ ( exp_complex @ A3 ) @ A3 )
% 0.33/0.63 = ( times_times_complex @ A3 @ ( exp_complex @ A3 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_times_arg_commute
% 0.33/0.63 thf(fact_6840_exp__times__arg__commute,axiom,
% 0.33/0.63 ! [A3: real] :
% 0.33/0.63 ( ( times_times_real @ ( exp_real @ A3 ) @ A3 )
% 0.33/0.63 = ( times_times_real @ A3 @ ( exp_real @ A3 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_times_arg_commute
% 0.33/0.63 thf(fact_6841_mult__exp__exp,axiom,
% 0.33/0.63 ! [X: complex,Y: complex] :
% 0.33/0.63 ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) )
% 0.33/0.63 = ( exp_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_exp_exp
% 0.33/0.63 thf(fact_6842_mult__exp__exp,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 0.33/0.63 = ( exp_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_exp_exp
% 0.33/0.63 thf(fact_6843_exp__add__commuting,axiom,
% 0.33/0.63 ! [X: complex,Y: complex] :
% 0.33/0.63 ( ( ( times_times_complex @ X @ Y )
% 0.33/0.63 = ( times_times_complex @ Y @ X ) )
% 0.33/0.63 => ( ( exp_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.33/0.63 = ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_add_commuting
% 0.33/0.63 thf(fact_6844_exp__add__commuting,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ( times_times_real @ X @ Y )
% 0.33/0.63 = ( times_times_real @ Y @ X ) )
% 0.33/0.63 => ( ( exp_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.63 = ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_add_commuting
% 0.33/0.63 thf(fact_6845_exp__diff,axiom,
% 0.33/0.63 ! [X: complex,Y: complex] :
% 0.33/0.63 ( ( exp_complex @ ( minus_minus_complex @ X @ Y ) )
% 0.33/0.63 = ( divide1717551699836669952omplex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_diff
% 0.33/0.63 thf(fact_6846_exp__diff,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( exp_real @ ( minus_minus_real @ X @ Y ) )
% 0.33/0.63 = ( divide_divide_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_diff
% 0.33/0.63 thf(fact_6847_frac__ge__0,axiom,
% 0.33/0.63 ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % frac_ge_0
% 0.33/0.63 thf(fact_6848_frac__ge__0,axiom,
% 0.33/0.63 ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % frac_ge_0
% 0.33/0.63 thf(fact_6849_frac__lt__1,axiom,
% 0.33/0.63 ! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).
% 0.33/0.63
% 0.33/0.63 % frac_lt_1
% 0.33/0.63 thf(fact_6850_frac__lt__1,axiom,
% 0.33/0.63 ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).
% 0.33/0.63
% 0.33/0.63 % frac_lt_1
% 0.33/0.63 thf(fact_6851_frac__1__eq,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ one_one_real ) )
% 0.33/0.63 = ( archim2898591450579166408c_real @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % frac_1_eq
% 0.33/0.63 thf(fact_6852_frac__1__eq,axiom,
% 0.33/0.63 ! [X: rat] :
% 0.33/0.63 ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
% 0.33/0.63 = ( archimedean_frac_rat @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % frac_1_eq
% 0.33/0.63 thf(fact_6853_exp__gt__one,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_gt_one
% 0.33/0.63 thf(fact_6854_exp__ge__add__one__self,axiom,
% 0.33/0.63 ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_ge_add_one_self
% 0.33/0.63 thf(fact_6855_exp__minus__inverse,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
% 0.33/0.63 = one_one_real ) ).
% 0.33/0.63
% 0.33/0.63 % exp_minus_inverse
% 0.33/0.63 thf(fact_6856_exp__minus__inverse,axiom,
% 0.33/0.63 ! [X: complex] :
% 0.33/0.63 ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
% 0.33/0.63 = one_one_complex ) ).
% 0.33/0.63
% 0.33/0.63 % exp_minus_inverse
% 0.33/0.63 thf(fact_6857_exp__of__nat2__mult,axiom,
% 0.33/0.63 ! [X: complex,N: nat] :
% 0.33/0.63 ( ( exp_complex @ ( times_times_complex @ X @ ( semiri8010041392384452111omplex @ N ) ) )
% 0.33/0.63 = ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_of_nat2_mult
% 0.33/0.63 thf(fact_6858_exp__of__nat2__mult,axiom,
% 0.33/0.63 ! [X: real,N: nat] :
% 0.33/0.63 ( ( exp_real @ ( times_times_real @ X @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.33/0.63 = ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_of_nat2_mult
% 0.33/0.63 thf(fact_6859_exp__of__nat__mult,axiom,
% 0.33/0.63 ! [N: nat,X: complex] :
% 0.33/0.63 ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X ) )
% 0.33/0.63 = ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_of_nat_mult
% 0.33/0.63 thf(fact_6860_exp__of__nat__mult,axiom,
% 0.33/0.63 ! [N: nat,X: real] :
% 0.33/0.63 ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) )
% 0.33/0.63 = ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_of_nat_mult
% 0.33/0.63 thf(fact_6861_log__ln,axiom,
% 0.33/0.63 ( ln_ln_real
% 0.33/0.63 = ( log2 @ ( exp_real @ one_one_real ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % log_ln
% 0.33/0.63 thf(fact_6862_arccos__le__arccos,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.63 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.63 => ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arccos_le_arccos
% 0.33/0.63 thf(fact_6863_arccos__le__mono,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.63 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.33/0.63 => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y ) )
% 0.33/0.63 = ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arccos_le_mono
% 0.33/0.63 thf(fact_6864_arccos__eq__iff,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.63 & ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
% 0.33/0.63 => ( ( ( arccos @ X )
% 0.33/0.63 = ( arccos @ Y ) )
% 0.33/0.63 = ( X = Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arccos_eq_iff
% 0.33/0.63 thf(fact_6865_exp__ge__add__one__self__aux,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_ge_add_one_self_aux
% 0.33/0.63 thf(fact_6866_lemma__exp__total,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ one_one_real @ Y )
% 0.33/0.63 => ? [X4: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.33/0.63 & ( ord_less_eq_real @ X4 @ ( minus_minus_real @ Y @ one_one_real ) )
% 0.33/0.63 & ( ( exp_real @ X4 )
% 0.33/0.63 = Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % lemma_exp_total
% 0.33/0.63 thf(fact_6867_ln__x__over__x__mono,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.63 => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ln_x_over_x_mono
% 0.33/0.63 thf(fact_6868_frac__def,axiom,
% 0.33/0.63 ( archim2898591450579166408c_real
% 0.33/0.63 = ( ^ [X3: real] : ( minus_minus_real @ X3 @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % frac_def
% 0.33/0.63 thf(fact_6869_frac__def,axiom,
% 0.33/0.63 ( archimedean_frac_rat
% 0.33/0.63 = ( ^ [X3: rat] : ( minus_minus_rat @ X3 @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % frac_def
% 0.33/0.63 thf(fact_6870_powr__def,axiom,
% 0.33/0.63 ( powr_real
% 0.33/0.63 = ( ^ [X3: real,A2: real] : ( if_real @ ( X3 = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A2 @ ( ln_ln_real @ X3 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % powr_def
% 0.33/0.63 thf(fact_6871_arccos__lbound,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.63 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arccos_lbound
% 0.33/0.63 thf(fact_6872_arccos__less__arccos,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ Y )
% 0.33/0.63 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.63 => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arccos_less_arccos
% 0.33/0.63 thf(fact_6873_arccos__less__mono,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.63 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.33/0.63 => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
% 0.33/0.63 = ( ord_less_real @ Y @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arccos_less_mono
% 0.33/0.63 thf(fact_6874_arccos__ubound,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.63 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.63 => ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arccos_ubound
% 0.33/0.63 thf(fact_6875_exp__le,axiom,
% 0.33/0.63 ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_le
% 0.33/0.63 thf(fact_6876_cos__arccos__abs,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.33/0.63 => ( ( cos_real @ ( arccos @ Y ) )
% 0.33/0.63 = Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % cos_arccos_abs
% 0.33/0.63 thf(fact_6877_exp__divide__power__eq,axiom,
% 0.33/0.63 ! [N: nat,X: complex] :
% 0.33/0.63 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.63 => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
% 0.33/0.63 = ( exp_complex @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_divide_power_eq
% 0.33/0.63 thf(fact_6878_exp__divide__power__eq,axiom,
% 0.33/0.63 ! [N: nat,X: real] :
% 0.33/0.63 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.63 => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 0.33/0.63 = ( exp_real @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_divide_power_eq
% 0.33/0.63 thf(fact_6879_frac__eq,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ( archim2898591450579166408c_real @ X )
% 0.33/0.63 = X )
% 0.33/0.63 = ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 & ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % frac_eq
% 0.33/0.63 thf(fact_6880_frac__eq,axiom,
% 0.33/0.63 ! [X: rat] :
% 0.33/0.63 ( ( ( archimedean_frac_rat @ X )
% 0.33/0.63 = X )
% 0.33/0.63 = ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.33/0.63 & ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % frac_eq
% 0.33/0.63 thf(fact_6881_frac__add,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 0.33/0.63 => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.63 = ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) ) )
% 0.33/0.63 & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 0.33/0.63 => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.63 = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % frac_add
% 0.33/0.63 thf(fact_6882_frac__add,axiom,
% 0.33/0.63 ! [X: rat,Y: rat] :
% 0.33/0.63 ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 0.33/0.63 => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
% 0.33/0.63 = ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) ) )
% 0.33/0.63 & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 0.33/0.63 => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
% 0.33/0.63 = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % frac_add
% 0.33/0.63 thf(fact_6883_tanh__altdef,axiom,
% 0.33/0.63 ( tanh_real
% 0.33/0.63 = ( ^ [X3: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) @ ( plus_plus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tanh_altdef
% 0.33/0.63 thf(fact_6884_tanh__altdef,axiom,
% 0.33/0.63 ( tanh_complex
% 0.33/0.63 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tanh_altdef
% 0.33/0.63 thf(fact_6885_exp__half__le2,axiom,
% 0.33/0.63 ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_half_le2
% 0.33/0.63 thf(fact_6886_arccos__lt__bounded,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.63 => ( ( ord_less_real @ Y @ one_one_real )
% 0.33/0.63 => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
% 0.33/0.63 & ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arccos_lt_bounded
% 0.33/0.63 thf(fact_6887_arccos__bounded,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.63 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.63 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
% 0.33/0.63 & ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arccos_bounded
% 0.33/0.63 thf(fact_6888_sin__arccos__nonzero,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.63 => ( ( sin_real @ ( arccos @ X ) )
% 0.33/0.63 != zero_zero_real ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sin_arccos_nonzero
% 0.33/0.63 thf(fact_6889_exp__double,axiom,
% 0.33/0.63 ! [Z2: complex] :
% 0.33/0.63 ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 ) )
% 0.33/0.63 = ( power_power_complex @ ( exp_complex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_double
% 0.33/0.63 thf(fact_6890_exp__double,axiom,
% 0.33/0.63 ! [Z2: real] :
% 0.33/0.63 ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 ) )
% 0.33/0.63 = ( power_power_real @ ( exp_real @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_double
% 0.33/0.63 thf(fact_6891_arccos__minus,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.63 => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 0.33/0.63 = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arccos_minus
% 0.33/0.63 thf(fact_6892_arccos,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.63 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.63 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
% 0.33/0.63 & ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
% 0.33/0.63 & ( ( cos_real @ ( arccos @ Y ) )
% 0.33/0.63 = Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arccos
% 0.33/0.63 thf(fact_6893_exp__bound__half,axiom,
% 0.33/0.63 ! [Z2: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_bound_half
% 0.33/0.63 thf(fact_6894_exp__bound__half,axiom,
% 0.33/0.63 ! [Z2: complex] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_bound_half
% 0.33/0.63 thf(fact_6895_arccos__minus__abs,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.63 => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 0.33/0.63 = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arccos_minus_abs
% 0.33/0.63 thf(fact_6896_exp__bound,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.63 => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_bound
% 0.33/0.63 thf(fact_6897_floor__add,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 0.33/0.63 => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.63 = ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) )
% 0.33/0.63 & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 0.33/0.63 => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.63 = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ one_one_int ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_add
% 0.33/0.63 thf(fact_6898_floor__add,axiom,
% 0.33/0.63 ! [X: rat,Y: rat] :
% 0.33/0.63 ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 0.33/0.63 => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
% 0.33/0.63 = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) )
% 0.33/0.63 & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 0.33/0.63 => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
% 0.33/0.63 = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ one_one_int ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_add
% 0.33/0.63 thf(fact_6899_real__exp__bound__lemma,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_exp_bound_lemma
% 0.33/0.63 thf(fact_6900_exp__ge__one__plus__x__over__n__power__n,axiom,
% 0.33/0.63 ! [N: nat,X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
% 0.33/0.63 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.63 => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_ge_one_plus_x_over_n_power_n
% 0.33/0.63 thf(fact_6901_exp__ge__one__minus__x__over__n__power__n,axiom,
% 0.33/0.63 ! [X: real,N: nat] :
% 0.33/0.63 ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 0.33/0.63 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.63 => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_ge_one_minus_x_over_n_power_n
% 0.33/0.63 thf(fact_6902_arccos__le__pi2,axiom,
% 0.33/0.63 ! [Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.63 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.63 => ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % arccos_le_pi2
% 0.33/0.63 thf(fact_6903_exp__bound__lemma,axiom,
% 0.33/0.63 ! [Z2: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z2 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_bound_lemma
% 0.33/0.63 thf(fact_6904_exp__bound__lemma,axiom,
% 0.33/0.63 ! [Z2: complex] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z2 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_bound_lemma
% 0.33/0.63 thf(fact_6905_exp__lower__Taylor__quadratic,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_lower_Taylor_quadratic
% 0.33/0.63 thf(fact_6906_log__base__10__eq2,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( log2 @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.63 = ( times_times_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % log_base_10_eq2
% 0.33/0.63 thf(fact_6907_tanh__real__altdef,axiom,
% 0.33/0.63 ( tanh_real
% 0.33/0.63 = ( ^ [X3: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % tanh_real_altdef
% 0.33/0.63 thf(fact_6908_modulo__int__def,axiom,
% 0.33/0.63 ( modulo_modulo_int
% 0.33/0.63 = ( ^ [K2: int,L2: int] :
% 0.33/0.63 ( if_int @ ( L2 = zero_zero_int ) @ K2
% 0.33/0.63 @ ( if_int
% 0.33/0.63 @ ( ( sgn_sgn_int @ K2 )
% 0.33/0.63 = ( sgn_sgn_int @ L2 ) )
% 0.33/0.63 @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
% 0.33/0.63 @ ( times_times_int @ ( sgn_sgn_int @ L2 )
% 0.33/0.63 @ ( minus_minus_int
% 0.33/0.63 @ ( times_times_int @ ( abs_abs_int @ L2 )
% 0.33/0.63 @ ( zero_n2684676970156552555ol_int
% 0.33/0.63 @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) )
% 0.33/0.63 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % modulo_int_def
% 0.33/0.63 thf(fact_6909_divide__int__def,axiom,
% 0.33/0.63 ( divide_divide_int
% 0.33/0.63 = ( ^ [K2: int,L2: int] :
% 0.33/0.63 ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
% 0.33/0.63 @ ( if_int
% 0.33/0.63 @ ( ( sgn_sgn_int @ K2 )
% 0.33/0.63 = ( sgn_sgn_int @ L2 ) )
% 0.33/0.63 @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
% 0.33/0.63 @ ( uminus_uminus_int
% 0.33/0.63 @ ( semiri1314217659103216013at_int
% 0.33/0.63 @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
% 0.33/0.63 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.63 @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % divide_int_def
% 0.33/0.63 thf(fact_6910_signed__take__bit__eq__take__bit__minus,axiom,
% 0.33/0.63 ( bit_ri631733984087533419it_int
% 0.33/0.63 = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % signed_take_bit_eq_take_bit_minus
% 0.33/0.63 thf(fact_6911_mask__numeral,axiom,
% 0.33/0.63 ! [N: num] :
% 0.33/0.63 ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
% 0.33/0.63 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mask_numeral
% 0.33/0.63 thf(fact_6912_mask__numeral,axiom,
% 0.33/0.63 ! [N: num] :
% 0.33/0.63 ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
% 0.33/0.63 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mask_numeral
% 0.33/0.63 thf(fact_6913_powr__int,axiom,
% 0.33/0.63 ! [X: real,I: int] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.33/0.63 => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 0.33/0.63 = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
% 0.33/0.63 & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 0.33/0.63 => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 0.33/0.63 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % powr_int
% 0.33/0.63 thf(fact_6914_mask__nat__positive__iff,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.33/0.63 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % mask_nat_positive_iff
% 0.33/0.63 thf(fact_6915_nat__int,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
% 0.33/0.63 = N ) ).
% 0.33/0.63
% 0.33/0.63 % nat_int
% 0.33/0.63 thf(fact_6916_nat__numeral,axiom,
% 0.33/0.63 ! [K: num] :
% 0.33/0.63 ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 0.33/0.63 = ( numeral_numeral_nat @ K ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_numeral
% 0.33/0.63 thf(fact_6917_mask__0,axiom,
% 0.33/0.63 ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
% 0.33/0.63 = zero_zero_nat ) ).
% 0.33/0.63
% 0.33/0.63 % mask_0
% 0.33/0.63 thf(fact_6918_mask__0,axiom,
% 0.33/0.63 ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
% 0.33/0.63 = zero_zero_int ) ).
% 0.33/0.63
% 0.33/0.63 % mask_0
% 0.33/0.63 thf(fact_6919_mask__eq__0__iff,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( ( bit_se2002935070580805687sk_nat @ N )
% 0.33/0.63 = zero_zero_nat )
% 0.33/0.63 = ( N = zero_zero_nat ) ) ).
% 0.33/0.63
% 0.33/0.63 % mask_eq_0_iff
% 0.33/0.63 thf(fact_6920_mask__eq__0__iff,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( ( bit_se2000444600071755411sk_int @ N )
% 0.33/0.63 = zero_zero_int )
% 0.33/0.63 = ( N = zero_zero_nat ) ) ).
% 0.33/0.63
% 0.33/0.63 % mask_eq_0_iff
% 0.33/0.63 thf(fact_6921_nat__of__bool,axiom,
% 0.33/0.63 ! [P: $o] :
% 0.33/0.63 ( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.33/0.63 = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_of_bool
% 0.33/0.63 thf(fact_6922_bit__numeral__Bit0__Suc__iff,axiom,
% 0.33/0.63 ! [M: num,N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 0.33/0.63 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_numeral_Bit0_Suc_iff
% 0.33/0.63 thf(fact_6923_bit__numeral__Bit0__Suc__iff,axiom,
% 0.33/0.63 ! [M: num,N: nat] :
% 0.33/0.63 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 0.33/0.63 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_numeral_Bit0_Suc_iff
% 0.33/0.63 thf(fact_6924_bit__numeral__Bit1__Suc__iff,axiom,
% 0.33/0.63 ! [M: num,N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 0.33/0.63 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_numeral_Bit1_Suc_iff
% 0.33/0.63 thf(fact_6925_bit__numeral__Bit1__Suc__iff,axiom,
% 0.33/0.63 ! [M: num,N: nat] :
% 0.33/0.63 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 0.33/0.63 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_numeral_Bit1_Suc_iff
% 0.33/0.63 thf(fact_6926_nat__1,axiom,
% 0.33/0.63 ( ( nat2 @ one_one_int )
% 0.33/0.63 = ( suc @ zero_zero_nat ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_1
% 0.33/0.63 thf(fact_6927_mask__Suc__0,axiom,
% 0.33/0.63 ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
% 0.33/0.63 = one_one_nat ) ).
% 0.33/0.63
% 0.33/0.63 % mask_Suc_0
% 0.33/0.63 thf(fact_6928_mask__Suc__0,axiom,
% 0.33/0.63 ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
% 0.33/0.63 = one_one_int ) ).
% 0.33/0.63
% 0.33/0.63 % mask_Suc_0
% 0.33/0.63 thf(fact_6929_nat__0__iff,axiom,
% 0.33/0.63 ! [I: int] :
% 0.33/0.63 ( ( ( nat2 @ I )
% 0.33/0.63 = zero_zero_nat )
% 0.33/0.63 = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_0_iff
% 0.33/0.63 thf(fact_6930_nat__le__0,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
% 0.33/0.63 => ( ( nat2 @ Z2 )
% 0.33/0.63 = zero_zero_nat ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_le_0
% 0.33/0.63 thf(fact_6931_zless__nat__conj,axiom,
% 0.33/0.63 ! [W: int,Z2: int] :
% 0.33/0.63 ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
% 0.33/0.63 = ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.33/0.63 & ( ord_less_int @ W @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % zless_nat_conj
% 0.33/0.63 thf(fact_6932_nat__neg__numeral,axiom,
% 0.33/0.63 ! [K: num] :
% 0.33/0.63 ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.63 = zero_zero_nat ) ).
% 0.33/0.63
% 0.33/0.63 % nat_neg_numeral
% 0.33/0.63 thf(fact_6933_nat__zminus__int,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.33/0.63 = zero_zero_nat ) ).
% 0.33/0.63
% 0.33/0.63 % nat_zminus_int
% 0.33/0.63 thf(fact_6934_int__nat__eq,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
% 0.33/0.63 = Z2 ) )
% 0.33/0.63 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
% 0.33/0.63 = zero_zero_int ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % int_nat_eq
% 0.33/0.63 thf(fact_6935_take__bit__minus__one__eq__mask,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( bit_se1745604003318907178nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.63 = ( bit_se2119862282449309892nteger @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % take_bit_minus_one_eq_mask
% 0.33/0.63 thf(fact_6936_take__bit__minus__one__eq__mask,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.63 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % take_bit_minus_one_eq_mask
% 0.33/0.63 thf(fact_6937_of__nat__nat__take__bit__eq,axiom,
% 0.33/0.63 ! [N: nat,K: int] :
% 0.33/0.63 ( ( semiri8010041392384452111omplex @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.33/0.63 = ( ring_17405671764205052669omplex @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_nat_take_bit_eq
% 0.33/0.63 thf(fact_6938_of__nat__nat__take__bit__eq,axiom,
% 0.33/0.63 ! [N: nat,K: int] :
% 0.33/0.63 ( ( semiri5074537144036343181t_real @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.33/0.63 = ( ring_1_of_int_real @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_nat_take_bit_eq
% 0.33/0.63 thf(fact_6939_of__nat__nat__take__bit__eq,axiom,
% 0.33/0.63 ! [N: nat,K: int] :
% 0.33/0.63 ( ( semiri681578069525770553at_rat @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.33/0.63 = ( ring_1_of_int_rat @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_nat_take_bit_eq
% 0.33/0.63 thf(fact_6940_of__nat__nat__take__bit__eq,axiom,
% 0.33/0.63 ! [N: nat,K: int] :
% 0.33/0.63 ( ( semiri1314217659103216013at_int @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.33/0.63 = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_nat_take_bit_eq
% 0.33/0.63 thf(fact_6941_signed__take__bit__nonnegative__iff,axiom,
% 0.33/0.63 ! [N: nat,K: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.33/0.63 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % signed_take_bit_nonnegative_iff
% 0.33/0.63 thf(fact_6942_signed__take__bit__negative__iff,axiom,
% 0.33/0.63 ! [N: nat,K: int] :
% 0.33/0.63 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 0.33/0.63 = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % signed_take_bit_negative_iff
% 0.33/0.63 thf(fact_6943_zero__less__nat__eq,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
% 0.33/0.63 = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 0.33/0.63
% 0.33/0.63 % zero_less_nat_eq
% 0.33/0.63 thf(fact_6944_of__nat__nat,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ( semiri8010041392384452111omplex @ ( nat2 @ Z2 ) )
% 0.33/0.63 = ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_nat
% 0.33/0.63 thf(fact_6945_of__nat__nat,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ( semiri5074537144036343181t_real @ ( nat2 @ Z2 ) )
% 0.33/0.63 = ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_nat
% 0.33/0.63 thf(fact_6946_of__nat__nat,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ( semiri681578069525770553at_rat @ ( nat2 @ Z2 ) )
% 0.33/0.63 = ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_nat
% 0.33/0.63 thf(fact_6947_of__nat__nat,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
% 0.33/0.63 = ( ring_1_of_int_int @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_nat
% 0.33/0.63 thf(fact_6948_bit__numeral__simps_I2_J,axiom,
% 0.33/0.63 ! [W: num,N: num] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.33/0.63 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_numeral_simps(2)
% 0.33/0.63 thf(fact_6949_bit__numeral__simps_I2_J,axiom,
% 0.33/0.63 ! [W: num,N: num] :
% 0.33/0.63 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.33/0.63 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_numeral_simps(2)
% 0.33/0.63 thf(fact_6950_diff__nat__numeral,axiom,
% 0.33/0.63 ! [V: num,V2: num] :
% 0.33/0.63 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V2 ) )
% 0.33/0.63 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V2 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % diff_nat_numeral
% 0.33/0.63 thf(fact_6951_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 0.33/0.63 ! [W: num,N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
% 0.33/0.63 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_minus_numeral_Bit0_Suc_iff
% 0.33/0.63 thf(fact_6952_bit__numeral__simps_I3_J,axiom,
% 0.33/0.63 ! [W: num,N: num] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.33/0.63 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_numeral_simps(3)
% 0.33/0.63 thf(fact_6953_bit__numeral__simps_I3_J,axiom,
% 0.33/0.63 ! [W: num,N: num] :
% 0.33/0.63 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.33/0.63 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_numeral_simps(3)
% 0.33/0.63 thf(fact_6954_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 0.33/0.63 ! [W: num,N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
% 0.33/0.63 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_minus_numeral_Bit1_Suc_iff
% 0.33/0.63 thf(fact_6955_numeral__power__eq__nat__cancel__iff,axiom,
% 0.33/0.63 ! [X: num,N: nat,Y: int] :
% 0.33/0.63 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.33/0.63 = ( nat2 @ Y ) )
% 0.33/0.63 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.33/0.63 = Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % numeral_power_eq_nat_cancel_iff
% 0.33/0.63 thf(fact_6956_nat__eq__numeral__power__cancel__iff,axiom,
% 0.33/0.63 ! [Y: int,X: num,N: nat] :
% 0.33/0.63 ( ( ( nat2 @ Y )
% 0.33/0.63 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.33/0.63 = ( Y
% 0.33/0.63 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_eq_numeral_power_cancel_iff
% 0.33/0.63 thf(fact_6957_dvd__nat__abs__iff,axiom,
% 0.33/0.63 ! [N: nat,K: int] :
% 0.33/0.63 ( ( dvd_dvd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 0.33/0.63 = ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% 0.33/0.63
% 0.33/0.63 % dvd_nat_abs_iff
% 0.33/0.63 thf(fact_6958_nat__abs__dvd__iff,axiom,
% 0.33/0.63 ! [K: int,N: nat] :
% 0.33/0.63 ( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 0.33/0.63 = ( dvd_dvd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_abs_dvd_iff
% 0.33/0.63 thf(fact_6959_nat__ceiling__le__eq,axiom,
% 0.33/0.63 ! [X: real,A: nat] :
% 0.33/0.63 ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
% 0.33/0.63 = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_ceiling_le_eq
% 0.33/0.63 thf(fact_6960_bit__0,axiom,
% 0.33/0.63 ! [A: code_integer] :
% 0.33/0.63 ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
% 0.33/0.63 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_0
% 0.33/0.63 thf(fact_6961_bit__0,axiom,
% 0.33/0.63 ! [A: int] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
% 0.33/0.63 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_0
% 0.33/0.63 thf(fact_6962_bit__0,axiom,
% 0.33/0.63 ! [A: nat] :
% 0.33/0.63 ( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
% 0.33/0.63 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_0
% 0.33/0.63 thf(fact_6963_one__less__nat__eq,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
% 0.33/0.63 = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 0.33/0.63
% 0.33/0.63 % one_less_nat_eq
% 0.33/0.63 thf(fact_6964_bit__minus__numeral__int_I1_J,axiom,
% 0.33/0.63 ! [W: num,N: num] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 0.33/0.63 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_minus_numeral_int(1)
% 0.33/0.63 thf(fact_6965_bit__minus__numeral__int_I2_J,axiom,
% 0.33/0.63 ! [W: num,N: num] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 0.33/0.63 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_minus_numeral_int(2)
% 0.33/0.63 thf(fact_6966_bit__mod__2__iff,axiom,
% 0.33/0.63 ! [A: code_integer,N: nat] :
% 0.33/0.63 ( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N )
% 0.33/0.63 = ( ( N = zero_zero_nat )
% 0.33/0.63 & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_mod_2_iff
% 0.33/0.63 thf(fact_6967_bit__mod__2__iff,axiom,
% 0.33/0.63 ! [A: code_natural,N: nat] :
% 0.33/0.63 ( ( bit_se8040316288895769887atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ N )
% 0.33/0.63 = ( ( N = zero_zero_nat )
% 0.33/0.63 & ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_mod_2_iff
% 0.33/0.63 thf(fact_6968_bit__mod__2__iff,axiom,
% 0.33/0.63 ! [A: int,N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N )
% 0.33/0.63 = ( ( N = zero_zero_nat )
% 0.33/0.63 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_mod_2_iff
% 0.33/0.63 thf(fact_6969_bit__mod__2__iff,axiom,
% 0.33/0.63 ! [A: nat,N: nat] :
% 0.33/0.63 ( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 0.33/0.63 = ( ( N = zero_zero_nat )
% 0.33/0.63 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_mod_2_iff
% 0.33/0.63 thf(fact_6970_nat__numeral__diff__1,axiom,
% 0.33/0.63 ! [V: num] :
% 0.33/0.63 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 0.33/0.63 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_numeral_diff_1
% 0.33/0.63 thf(fact_6971_numeral__power__less__nat__cancel__iff,axiom,
% 0.33/0.63 ! [X: num,N: nat,A: int] :
% 0.33/0.63 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 0.33/0.63 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.33/0.63
% 0.33/0.63 % numeral_power_less_nat_cancel_iff
% 0.33/0.63 thf(fact_6972_nat__less__numeral__power__cancel__iff,axiom,
% 0.33/0.63 ! [A: int,X: num,N: nat] :
% 0.33/0.63 ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.33/0.63 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_less_numeral_power_cancel_iff
% 0.33/0.63 thf(fact_6973_numeral__power__le__nat__cancel__iff,axiom,
% 0.33/0.63 ! [X: num,N: nat,A: int] :
% 0.33/0.63 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 0.33/0.63 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.33/0.63
% 0.33/0.63 % numeral_power_le_nat_cancel_iff
% 0.33/0.63 thf(fact_6974_nat__le__numeral__power__cancel__iff,axiom,
% 0.33/0.63 ! [A: int,X: num,N: nat] :
% 0.33/0.63 ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.33/0.63 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_le_numeral_power_cancel_iff
% 0.33/0.63 thf(fact_6975_bit__numeral__iff,axiom,
% 0.33/0.63 ! [M: num,N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N )
% 0.33/0.63 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_numeral_iff
% 0.33/0.63 thf(fact_6976_bit__numeral__iff,axiom,
% 0.33/0.63 ! [M: num,N: nat] :
% 0.33/0.63 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N )
% 0.33/0.63 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_numeral_iff
% 0.33/0.63 thf(fact_6977_of__nat__mask__eq,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.33/0.63 = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_mask_eq
% 0.33/0.63 thf(fact_6978_of__nat__mask__eq,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.33/0.63 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_mask_eq
% 0.33/0.63 thf(fact_6979_nat__mask__eq,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.33/0.63 = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_mask_eq
% 0.33/0.63 thf(fact_6980_bit__of__nat__iff__bit,axiom,
% 0.33/0.63 ! [M: nat,N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( semiri1314217659103216013at_int @ M ) @ N )
% 0.33/0.63 = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_of_nat_iff_bit
% 0.33/0.63 thf(fact_6981_bit__of__nat__iff__bit,axiom,
% 0.33/0.63 ! [M: nat,N: nat] :
% 0.33/0.63 ( ( bit_se1148574629649215175it_nat @ ( semiri1316708129612266289at_nat @ M ) @ N )
% 0.33/0.63 = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_of_nat_iff_bit
% 0.33/0.63 thf(fact_6982_bit__disjunctive__add__iff,axiom,
% 0.33/0.63 ! [A: int,B: int,N: nat] :
% 0.33/0.63 ( ! [N2: nat] :
% 0.33/0.63 ( ~ ( bit_se1146084159140164899it_int @ A @ N2 )
% 0.33/0.63 | ~ ( bit_se1146084159140164899it_int @ B @ N2 ) )
% 0.33/0.63 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N )
% 0.33/0.63 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.33/0.63 | ( bit_se1146084159140164899it_int @ B @ N ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_disjunctive_add_iff
% 0.33/0.63 thf(fact_6983_bit__disjunctive__add__iff,axiom,
% 0.33/0.63 ! [A: nat,B: nat,N: nat] :
% 0.33/0.63 ( ! [N2: nat] :
% 0.33/0.63 ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 )
% 0.33/0.63 | ~ ( bit_se1148574629649215175it_nat @ B @ N2 ) )
% 0.33/0.63 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 0.33/0.63 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.33/0.63 | ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_disjunctive_add_iff
% 0.33/0.63 thf(fact_6984_of__int__mask__eq,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( ring_1_of_int_int @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.33/0.63 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_mask_eq
% 0.33/0.63 thf(fact_6985_bit__unset__bit__iff,axiom,
% 0.33/0.63 ! [M: nat,A: int,N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( bit_se4203085406695923979it_int @ M @ A ) @ N )
% 0.33/0.63 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.33/0.63 & ( M != N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_unset_bit_iff
% 0.33/0.63 thf(fact_6986_bit__unset__bit__iff,axiom,
% 0.33/0.63 ! [M: nat,A: nat,N: nat] :
% 0.33/0.63 ( ( bit_se1148574629649215175it_nat @ ( bit_se4205575877204974255it_nat @ M @ A ) @ N )
% 0.33/0.63 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.33/0.63 & ( M != N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_unset_bit_iff
% 0.33/0.63 thf(fact_6987_less__eq__mask,axiom,
% 0.33/0.63 ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_eq_mask
% 0.33/0.63 thf(fact_6988_not__bit__1__Suc,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_bit_1_Suc
% 0.33/0.63 thf(fact_6989_not__bit__1__Suc,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_bit_1_Suc
% 0.33/0.63 thf(fact_6990_bit__1__iff,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ one_one_int @ N )
% 0.33/0.63 = ( N = zero_zero_nat ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_1_iff
% 0.33/0.63 thf(fact_6991_bit__1__iff,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N )
% 0.33/0.63 = ( N = zero_zero_nat ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_1_iff
% 0.33/0.63 thf(fact_6992_bit__numeral__simps_I1_J,axiom,
% 0.33/0.63 ! [N: num] :
% 0.33/0.63 ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_numeral_simps(1)
% 0.33/0.63 thf(fact_6993_bit__numeral__simps_I1_J,axiom,
% 0.33/0.63 ! [N: num] :
% 0.33/0.63 ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_numeral_simps(1)
% 0.33/0.63 thf(fact_6994_bit__take__bit__iff,axiom,
% 0.33/0.63 ! [M: nat,A: int,N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N )
% 0.33/0.63 = ( ( ord_less_nat @ N @ M )
% 0.33/0.63 & ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_take_bit_iff
% 0.33/0.63 thf(fact_6995_bit__take__bit__iff,axiom,
% 0.33/0.63 ! [M: nat,A: nat,N: nat] :
% 0.33/0.63 ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N )
% 0.33/0.63 = ( ( ord_less_nat @ N @ M )
% 0.33/0.63 & ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_take_bit_iff
% 0.33/0.63 thf(fact_6996_bit__of__bool__iff,axiom,
% 0.33/0.63 ! [B: $o,N: nat] :
% 0.33/0.63 ( ( bit_se9216721137139052372nteger @ ( zero_n356916108424825756nteger @ B ) @ N )
% 0.33/0.63 = ( B
% 0.33/0.63 & ( N = zero_zero_nat ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_of_bool_iff
% 0.33/0.63 thf(fact_6997_bit__of__bool__iff,axiom,
% 0.33/0.63 ! [B: $o,N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B ) @ N )
% 0.33/0.63 = ( B
% 0.33/0.63 & ( N = zero_zero_nat ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_of_bool_iff
% 0.33/0.63 thf(fact_6998_bit__of__bool__iff,axiom,
% 0.33/0.63 ! [B: $o,N: nat] :
% 0.33/0.63 ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ N )
% 0.33/0.63 = ( B
% 0.33/0.63 & ( N = zero_zero_nat ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_of_bool_iff
% 0.33/0.63 thf(fact_6999_nat__zero__as__int,axiom,
% 0.33/0.63 ( zero_zero_nat
% 0.33/0.63 = ( nat2 @ zero_zero_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_zero_as_int
% 0.33/0.63 thf(fact_7000_nat__numeral__as__int,axiom,
% 0.33/0.63 ( numeral_numeral_nat
% 0.33/0.63 = ( ^ [I4: num] : ( nat2 @ ( numeral_numeral_int @ I4 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_numeral_as_int
% 0.33/0.63 thf(fact_7001_nat__mono,axiom,
% 0.33/0.63 ! [X: int,Y: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ X @ Y )
% 0.33/0.63 => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_mono
% 0.33/0.63 thf(fact_7002_ex__nat,axiom,
% 0.33/0.63 ( ( ^ [P5: nat > $o] :
% 0.33/0.63 ? [X7: nat] : ( P5 @ X7 ) )
% 0.33/0.63 = ( ^ [P6: nat > $o] :
% 0.33/0.63 ? [X3: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.33/0.63 & ( P6 @ ( nat2 @ X3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ex_nat
% 0.33/0.63 thf(fact_7003_all__nat,axiom,
% 0.33/0.63 ( ( ^ [P5: nat > $o] :
% 0.33/0.63 ! [X7: nat] : ( P5 @ X7 ) )
% 0.33/0.63 = ( ^ [P6: nat > $o] :
% 0.33/0.63 ! [X3: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.33/0.63 => ( P6 @ ( nat2 @ X3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % all_nat
% 0.33/0.63 thf(fact_7004_eq__nat__nat__iff,axiom,
% 0.33/0.63 ! [Z2: int,Z6: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 0.33/0.63 => ( ( ( nat2 @ Z2 )
% 0.33/0.63 = ( nat2 @ Z6 ) )
% 0.33/0.63 = ( Z2 = Z6 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % eq_nat_nat_iff
% 0.33/0.63 thf(fact_7005_nat__one__as__int,axiom,
% 0.33/0.63 ( one_one_nat
% 0.33/0.63 = ( nat2 @ one_one_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_one_as_int
% 0.33/0.63 thf(fact_7006_signed__take__bit__eq__if__positive,axiom,
% 0.33/0.63 ! [A: int,N: nat] :
% 0.33/0.63 ( ~ ( bit_se1146084159140164899it_int @ A @ N )
% 0.33/0.63 => ( ( bit_ri631733984087533419it_int @ N @ A )
% 0.33/0.63 = ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % signed_take_bit_eq_if_positive
% 0.33/0.63 thf(fact_7007_complex__exp__exists,axiom,
% 0.33/0.63 ! [Z2: complex] :
% 0.33/0.63 ? [A4: complex,R2: real] :
% 0.33/0.63 ( Z2
% 0.33/0.63 = ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( exp_complex @ A4 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % complex_exp_exists
% 0.33/0.63 thf(fact_7008_unset__bit__nat__def,axiom,
% 0.33/0.63 ( bit_se4205575877204974255it_nat
% 0.33/0.63 = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M3 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % unset_bit_nat_def
% 0.33/0.63 thf(fact_7009_mask__nonnegative__int,axiom,
% 0.33/0.63 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % mask_nonnegative_int
% 0.33/0.63 thf(fact_7010_not__mask__negative__int,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 0.33/0.63
% 0.33/0.63 % not_mask_negative_int
% 0.33/0.63 thf(fact_7011_push__bit__nat__eq,axiom,
% 0.33/0.63 ! [N: nat,K: int] :
% 0.33/0.63 ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
% 0.33/0.63 = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % push_bit_nat_eq
% 0.33/0.63 thf(fact_7012_nat__mono__iff,axiom,
% 0.33/0.63 ! [Z2: int,W: int] :
% 0.33/0.63 ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
% 0.33/0.63 = ( ord_less_int @ W @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_mono_iff
% 0.33/0.63 thf(fact_7013_of__nat__ceiling,axiom,
% 0.33/0.63 ! [R: real] : ( ord_less_eq_real @ R @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_ceiling
% 0.33/0.63 thf(fact_7014_of__nat__ceiling,axiom,
% 0.33/0.63 ! [R: rat] : ( ord_less_eq_rat @ R @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_ceiling
% 0.33/0.63 thf(fact_7015_zless__nat__eq__int__zless,axiom,
% 0.33/0.63 ! [M: nat,Z2: int] :
% 0.33/0.63 ( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
% 0.33/0.63 = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).
% 0.33/0.63
% 0.33/0.63 % zless_nat_eq_int_zless
% 0.33/0.63 thf(fact_7016_nat__le__iff,axiom,
% 0.33/0.63 ! [X: int,N: nat] :
% 0.33/0.63 ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
% 0.33/0.63 = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_le_iff
% 0.33/0.63 thf(fact_7017_int__eq__iff,axiom,
% 0.33/0.63 ! [M: nat,Z2: int] :
% 0.33/0.63 ( ( ( semiri1314217659103216013at_int @ M )
% 0.33/0.63 = Z2 )
% 0.33/0.63 = ( ( M
% 0.33/0.63 = ( nat2 @ Z2 ) )
% 0.33/0.63 & ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % int_eq_iff
% 0.33/0.63 thf(fact_7018_nat__0__le,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
% 0.33/0.63 = Z2 ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_0_le
% 0.33/0.63 thf(fact_7019_bit__not__int__iff_H,axiom,
% 0.33/0.63 ! [K: int,N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
% 0.33/0.63 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_not_int_iff'
% 0.33/0.63 thf(fact_7020_nat__int__add,axiom,
% 0.33/0.63 ! [A: nat,B: nat] :
% 0.33/0.63 ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
% 0.33/0.63 = ( plus_plus_nat @ A @ B ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_int_add
% 0.33/0.63 thf(fact_7021_int__minus,axiom,
% 0.33/0.63 ! [N: nat,M: nat] :
% 0.33/0.63 ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
% 0.33/0.63 = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % int_minus
% 0.33/0.63 thf(fact_7022_nat__abs__mult__distrib,axiom,
% 0.33/0.63 ! [W: int,Z2: int] :
% 0.33/0.63 ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z2 ) ) )
% 0.33/0.63 = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z2 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_abs_mult_distrib
% 0.33/0.63 thf(fact_7023_bit__push__bit__iff__int,axiom,
% 0.33/0.63 ! [M: nat,K: int,N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 0.33/0.63 = ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.63 & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_push_bit_iff_int
% 0.33/0.63 thf(fact_7024_flip__bit__eq__if,axiom,
% 0.33/0.63 ( bit_se2159334234014336723it_int
% 0.33/0.63 = ( ^ [N3: nat,A2: int] : ( if_nat_int_int @ ( bit_se1146084159140164899it_int @ A2 @ N3 ) @ bit_se4203085406695923979it_int @ bit_se7879613467334960850it_int @ N3 @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % flip_bit_eq_if
% 0.33/0.63 thf(fact_7025_flip__bit__eq__if,axiom,
% 0.33/0.63 ( bit_se2161824704523386999it_nat
% 0.33/0.63 = ( ^ [N3: nat,A2: nat] : ( if_nat_nat_nat @ ( bit_se1148574629649215175it_nat @ A2 @ N3 ) @ bit_se4205575877204974255it_nat @ bit_se7882103937844011126it_nat @ N3 @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % flip_bit_eq_if
% 0.33/0.63 thf(fact_7026_real__nat__ceiling__ge,axiom,
% 0.33/0.63 ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_nat_ceiling_ge
% 0.33/0.63 thf(fact_7027_less__mask,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.33/0.63 => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_mask
% 0.33/0.63 thf(fact_7028_of__nat__floor,axiom,
% 0.33/0.63 ! [R: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ R )
% 0.33/0.63 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R ) ) ) @ R ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_floor
% 0.33/0.63 thf(fact_7029_of__nat__floor,axiom,
% 0.33/0.63 ! [R: rat] :
% 0.33/0.63 ( ( ord_less_eq_rat @ zero_zero_rat @ R )
% 0.33/0.63 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R ) ) ) @ R ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_floor
% 0.33/0.63 thf(fact_7030_nat__less__eq__zless,axiom,
% 0.33/0.63 ! [W: int,Z2: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.33/0.63 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
% 0.33/0.63 = ( ord_less_int @ W @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_less_eq_zless
% 0.33/0.63 thf(fact_7031_nat__le__eq__zle,axiom,
% 0.33/0.63 ! [W: int,Z2: int] :
% 0.33/0.63 ( ( ( ord_less_int @ zero_zero_int @ W )
% 0.33/0.63 | ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
% 0.33/0.63 => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
% 0.33/0.63 = ( ord_less_eq_int @ W @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_le_eq_zle
% 0.33/0.63 thf(fact_7032_nat__eq__iff2,axiom,
% 0.33/0.63 ! [M: nat,W: int] :
% 0.33/0.63 ( ( M
% 0.33/0.63 = ( nat2 @ W ) )
% 0.33/0.63 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.33/0.63 => ( W
% 0.33/0.63 = ( semiri1314217659103216013at_int @ M ) ) )
% 0.33/0.63 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 0.33/0.63 => ( M = zero_zero_nat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_eq_iff2
% 0.33/0.63 thf(fact_7033_nat__eq__iff,axiom,
% 0.33/0.63 ! [W: int,M: nat] :
% 0.33/0.63 ( ( ( nat2 @ W )
% 0.33/0.63 = M )
% 0.33/0.63 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.33/0.63 => ( W
% 0.33/0.63 = ( semiri1314217659103216013at_int @ M ) ) )
% 0.33/0.63 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 0.33/0.63 => ( M = zero_zero_nat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_eq_iff
% 0.33/0.63 thf(fact_7034_split__nat,axiom,
% 0.33/0.63 ! [P: nat > $o,I: int] :
% 0.33/0.63 ( ( P @ ( nat2 @ I ) )
% 0.33/0.63 = ( ! [N3: nat] :
% 0.33/0.63 ( ( I
% 0.33/0.63 = ( semiri1314217659103216013at_int @ N3 ) )
% 0.33/0.63 => ( P @ N3 ) )
% 0.33/0.63 & ( ( ord_less_int @ I @ zero_zero_int )
% 0.33/0.63 => ( P @ zero_zero_nat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % split_nat
% 0.33/0.63 thf(fact_7035_le__mult__nat__floor,axiom,
% 0.33/0.63 ! [A: real,B: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_mult_nat_floor
% 0.33/0.63 thf(fact_7036_le__mult__nat__floor,axiom,
% 0.33/0.63 ! [A: rat,B: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_mult_nat_floor
% 0.33/0.63 thf(fact_7037_le__nat__iff,axiom,
% 0.33/0.63 ! [K: int,N: nat] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.63 => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 0.33/0.63 = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_nat_iff
% 0.33/0.63 thf(fact_7038_nat__add__distrib,axiom,
% 0.33/0.63 ! [Z2: int,Z6: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 0.33/0.63 => ( ( nat2 @ ( plus_plus_int @ Z2 @ Z6 ) )
% 0.33/0.63 = ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_add_distrib
% 0.33/0.63 thf(fact_7039_bit__imp__take__bit__positive,axiom,
% 0.33/0.63 ! [N: nat,M: nat,K: int] :
% 0.33/0.63 ( ( ord_less_nat @ N @ M )
% 0.33/0.63 => ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.33/0.63 => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_imp_take_bit_positive
% 0.33/0.63 thf(fact_7040_nat__mult__distrib,axiom,
% 0.33/0.63 ! [Z2: int,Z6: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ( nat2 @ ( times_times_int @ Z2 @ Z6 ) )
% 0.33/0.63 = ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_mult_distrib
% 0.33/0.63 thf(fact_7041_Suc__as__int,axiom,
% 0.33/0.63 ( suc
% 0.33/0.63 = ( ^ [A2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ one_one_int ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % Suc_as_int
% 0.33/0.63 thf(fact_7042_nat__diff__distrib,axiom,
% 0.33/0.63 ! [Z6: int,Z2: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 0.33/0.63 => ( ( ord_less_eq_int @ Z6 @ Z2 )
% 0.33/0.63 => ( ( nat2 @ ( minus_minus_int @ Z2 @ Z6 ) )
% 0.33/0.63 = ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_diff_distrib
% 0.33/0.63 thf(fact_7043_nat__diff__distrib_H,axiom,
% 0.33/0.63 ! [X: int,Y: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.63 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.63 => ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
% 0.33/0.63 = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_diff_distrib'
% 0.33/0.63 thf(fact_7044_nat__abs__triangle__ineq,axiom,
% 0.33/0.63 ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_abs_triangle_ineq
% 0.33/0.63 thf(fact_7045_nat__div__distrib_H,axiom,
% 0.33/0.63 ! [Y: int,X: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.63 => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
% 0.33/0.63 = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_div_distrib'
% 0.33/0.63 thf(fact_7046_nat__div__distrib,axiom,
% 0.33/0.63 ! [X: int,Y: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.63 => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
% 0.33/0.63 = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_div_distrib
% 0.33/0.63 thf(fact_7047_bit__concat__bit__iff,axiom,
% 0.33/0.63 ! [M: nat,K: int,L: int,N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
% 0.33/0.63 = ( ( ( ord_less_nat @ N @ M )
% 0.33/0.63 & ( bit_se1146084159140164899it_int @ K @ N ) )
% 0.33/0.63 | ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.63 & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_concat_bit_iff
% 0.33/0.63 thf(fact_7048_nat__floor__neg,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.63 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.63 = zero_zero_nat ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_floor_neg
% 0.33/0.63 thf(fact_7049_nat__power__eq,axiom,
% 0.33/0.63 ! [Z2: int,N: nat] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ( nat2 @ ( power_power_int @ Z2 @ N ) )
% 0.33/0.63 = ( power_power_nat @ ( nat2 @ Z2 ) @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_power_eq
% 0.33/0.63 thf(fact_7050_nat__mod__distrib,axiom,
% 0.33/0.63 ! [X: int,Y: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.63 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.63 => ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
% 0.33/0.63 = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_mod_distrib
% 0.33/0.63 thf(fact_7051_div__abs__eq__div__nat,axiom,
% 0.33/0.63 ! [K: int,L: int] :
% 0.33/0.63 ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
% 0.33/0.63 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % div_abs_eq_div_nat
% 0.33/0.63 thf(fact_7052_floor__eq3,axiom,
% 0.33/0.63 ! [N: nat,X: real] :
% 0.33/0.63 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.33/0.63 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.63 = N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_eq3
% 0.33/0.63 thf(fact_7053_le__nat__floor,axiom,
% 0.33/0.63 ! [X: nat,A: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
% 0.33/0.63 => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_nat_floor
% 0.33/0.63 thf(fact_7054_mod__abs__eq__div__nat,axiom,
% 0.33/0.63 ! [K: int,L: int] :
% 0.33/0.63 ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
% 0.33/0.63 = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mod_abs_eq_div_nat
% 0.33/0.63 thf(fact_7055_nat__take__bit__eq,axiom,
% 0.33/0.63 ! [K: int,N: nat] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.63 => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.33/0.63 = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_take_bit_eq
% 0.33/0.63 thf(fact_7056_take__bit__nat__eq,axiom,
% 0.33/0.63 ! [K: int,N: nat] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.63 => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
% 0.33/0.63 = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % take_bit_nat_eq
% 0.33/0.63 thf(fact_7057_signed__take__bit__eq__concat__bit,axiom,
% 0.33/0.63 ( bit_ri631733984087533419it_int
% 0.33/0.63 = ( ^ [N3: nat,K2: int] : ( bit_concat_bit @ N3 @ K2 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % signed_take_bit_eq_concat_bit
% 0.33/0.63 thf(fact_7058_exp__eq__0__imp__not__bit,axiom,
% 0.33/0.63 ! [N: nat,A: int] :
% 0.33/0.63 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.33/0.63 = zero_zero_int )
% 0.33/0.63 => ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_eq_0_imp_not_bit
% 0.33/0.63 thf(fact_7059_exp__eq__0__imp__not__bit,axiom,
% 0.33/0.63 ! [N: nat,A: nat] :
% 0.33/0.63 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.63 = zero_zero_nat )
% 0.33/0.63 => ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_eq_0_imp_not_bit
% 0.33/0.63 thf(fact_7060_bit__Suc,axiom,
% 0.33/0.63 ! [A: int,N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ A @ ( suc @ N ) )
% 0.33/0.63 = ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_Suc
% 0.33/0.63 thf(fact_7061_bit__Suc,axiom,
% 0.33/0.63 ! [A: nat,N: nat] :
% 0.33/0.63 ( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N ) )
% 0.33/0.63 = ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_Suc
% 0.33/0.63 thf(fact_7062_stable__imp__bit__iff__odd,axiom,
% 0.33/0.63 ! [A: code_integer,N: nat] :
% 0.33/0.63 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.63 = A )
% 0.33/0.63 => ( ( bit_se9216721137139052372nteger @ A @ N )
% 0.33/0.63 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % stable_imp_bit_iff_odd
% 0.33/0.63 thf(fact_7063_stable__imp__bit__iff__odd,axiom,
% 0.33/0.63 ! [A: int,N: nat] :
% 0.33/0.63 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.63 = A )
% 0.33/0.63 => ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.33/0.63 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % stable_imp_bit_iff_odd
% 0.33/0.63 thf(fact_7064_stable__imp__bit__iff__odd,axiom,
% 0.33/0.63 ! [A: nat,N: nat] :
% 0.33/0.63 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.63 = A )
% 0.33/0.63 => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.33/0.63 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % stable_imp_bit_iff_odd
% 0.33/0.63 thf(fact_7065_bit__iff__idd__imp__stable,axiom,
% 0.33/0.63 ! [A: code_integer] :
% 0.33/0.63 ( ! [N2: nat] :
% 0.33/0.63 ( ( bit_se9216721137139052372nteger @ A @ N2 )
% 0.33/0.63 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 0.33/0.63 => ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.63 = A ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_iff_idd_imp_stable
% 0.33/0.63 thf(fact_7066_bit__iff__idd__imp__stable,axiom,
% 0.33/0.63 ! [A: int] :
% 0.33/0.63 ( ! [N2: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ A @ N2 )
% 0.33/0.63 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 0.33/0.63 => ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.63 = A ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_iff_idd_imp_stable
% 0.33/0.63 thf(fact_7067_bit__iff__idd__imp__stable,axiom,
% 0.33/0.63 ! [A: nat] :
% 0.33/0.63 ( ! [N2: nat] :
% 0.33/0.63 ( ( bit_se1148574629649215175it_nat @ A @ N2 )
% 0.33/0.63 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 0.33/0.63 => ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.63 = A ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_iff_idd_imp_stable
% 0.33/0.63 thf(fact_7068_nat__2,axiom,
% 0.33/0.63 ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.63 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_2
% 0.33/0.63 thf(fact_7069_take__bit__eq__mask__iff,axiom,
% 0.33/0.63 ! [N: nat,K: int] :
% 0.33/0.63 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.33/0.63 = ( bit_se2000444600071755411sk_int @ N ) )
% 0.33/0.63 = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 0.33/0.63 = zero_zero_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % take_bit_eq_mask_iff
% 0.33/0.63 thf(fact_7070_int__bit__bound,axiom,
% 0.33/0.63 ! [K: int] :
% 0.33/0.63 ~ ! [N2: nat] :
% 0.33/0.63 ( ! [M4: nat] :
% 0.33/0.63 ( ( ord_less_eq_nat @ N2 @ M4 )
% 0.33/0.63 => ( ( bit_se1146084159140164899it_int @ K @ M4 )
% 0.33/0.63 = ( bit_se1146084159140164899it_int @ K @ N2 ) ) )
% 0.33/0.63 => ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.63 => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 0.33/0.63 = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % int_bit_bound
% 0.33/0.63 thf(fact_7071_Suc__nat__eq__nat__zadd1,axiom,
% 0.33/0.63 ! [Z2: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( ( suc @ ( nat2 @ Z2 ) )
% 0.33/0.63 = ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % Suc_nat_eq_nat_zadd1
% 0.33/0.63 thf(fact_7072_nat__less__iff,axiom,
% 0.33/0.63 ! [W: int,M: nat] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.33/0.63 => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
% 0.33/0.63 = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_less_iff
% 0.33/0.63 thf(fact_7073_nat__mult__distrib__neg,axiom,
% 0.33/0.63 ! [Z2: int,Z6: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
% 0.33/0.63 => ( ( nat2 @ ( times_times_int @ Z2 @ Z6 ) )
% 0.33/0.63 = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_mult_distrib_neg
% 0.33/0.63 thf(fact_7074_nat__abs__int__diff,axiom,
% 0.33/0.63 ! [A: nat,B: nat] :
% 0.33/0.63 ( ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.63 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 0.33/0.63 = ( minus_minus_nat @ B @ A ) ) )
% 0.33/0.63 & ( ~ ( ord_less_eq_nat @ A @ B )
% 0.33/0.63 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 0.33/0.63 = ( minus_minus_nat @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_abs_int_diff
% 0.33/0.63 thf(fact_7075_floor__eq4,axiom,
% 0.33/0.63 ! [N: nat,X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.33/0.63 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.63 = N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % floor_eq4
% 0.33/0.63 thf(fact_7076_bit__iff__odd,axiom,
% 0.33/0.63 ( bit_se9216721137139052372nteger
% 0.33/0.63 = ( ^ [A2: code_integer,N3: nat] :
% 0.33/0.63 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_iff_odd
% 0.33/0.63 thf(fact_7077_bit__iff__odd,axiom,
% 0.33/0.63 ( bit_se1146084159140164899it_int
% 0.33/0.63 = ( ^ [A2: int,N3: nat] :
% 0.33/0.63 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_iff_odd
% 0.33/0.63 thf(fact_7078_bit__iff__odd,axiom,
% 0.33/0.63 ( bit_se1148574629649215175it_nat
% 0.33/0.63 = ( ^ [A2: nat,N3: nat] :
% 0.33/0.63 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_iff_odd
% 0.33/0.63 thf(fact_7079_Suc__mask__eq__exp,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.33/0.63 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % Suc_mask_eq_exp
% 0.33/0.63 thf(fact_7080_mask__nat__less__exp,axiom,
% 0.33/0.63 ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % mask_nat_less_exp
% 0.33/0.63 thf(fact_7081_of__int__of__nat,axiom,
% 0.33/0.63 ( ring_18347121197199848620nteger
% 0.33/0.63 = ( ^ [K2: int] : ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri4939895301339042750nteger @ ( nat2 @ K2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_of_nat
% 0.33/0.63 thf(fact_7082_of__int__of__nat,axiom,
% 0.33/0.63 ( ring_17405671764205052669omplex
% 0.33/0.63 = ( ^ [K2: int] : ( if_complex @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri8010041392384452111omplex @ ( nat2 @ K2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_of_nat
% 0.33/0.63 thf(fact_7083_of__int__of__nat,axiom,
% 0.33/0.63 ( ring_1_of_int_real
% 0.33/0.63 = ( ^ [K2: int] : ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_of_nat
% 0.33/0.63 thf(fact_7084_of__int__of__nat,axiom,
% 0.33/0.63 ( ring_1_of_int_rat
% 0.33/0.63 = ( ^ [K2: int] : ( if_rat @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri681578069525770553at_rat @ ( nat2 @ K2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_of_nat
% 0.33/0.63 thf(fact_7085_of__int__of__nat,axiom,
% 0.33/0.63 ( ring_1_of_int_int
% 0.33/0.63 = ( ^ [K2: int] : ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_of_nat
% 0.33/0.63 thf(fact_7086_bit__int__def,axiom,
% 0.33/0.63 ( bit_se1146084159140164899it_int
% 0.33/0.63 = ( ^ [K2: int,N3: nat] :
% 0.33/0.63 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_int_def
% 0.33/0.63 thf(fact_7087_nat__dvd__iff,axiom,
% 0.33/0.63 ! [Z2: int,M: nat] :
% 0.33/0.63 ( ( dvd_dvd_nat @ ( nat2 @ Z2 ) @ M )
% 0.33/0.63 = ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( dvd_dvd_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.33/0.63 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.33/0.63 => ( M = zero_zero_nat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nat_dvd_iff
% 0.33/0.63 thf(fact_7088_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N ) )
% 0.33/0.63 = ( N = zero_zero_nat ) ) ).
% 0.33/0.63
% 0.33/0.63 % semiring_bit_operations_class.even_mask_iff
% 0.33/0.63 thf(fact_7089_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.33/0.63 = ( N = zero_zero_nat ) ) ).
% 0.33/0.63
% 0.33/0.63 % semiring_bit_operations_class.even_mask_iff
% 0.33/0.63 thf(fact_7090_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.33/0.63 = ( N = zero_zero_nat ) ) ).
% 0.33/0.63
% 0.33/0.63 % semiring_bit_operations_class.even_mask_iff
% 0.33/0.63 thf(fact_7091_even__bit__succ__iff,axiom,
% 0.33/0.63 ! [A: code_integer,N: nat] :
% 0.33/0.63 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.33/0.63 => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N )
% 0.33/0.63 = ( ( bit_se9216721137139052372nteger @ A @ N )
% 0.33/0.63 | ( N = zero_zero_nat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % even_bit_succ_iff
% 0.33/0.63 thf(fact_7092_even__bit__succ__iff,axiom,
% 0.33/0.63 ! [A: int,N: nat] :
% 0.33/0.63 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.33/0.63 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N )
% 0.33/0.63 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.33/0.63 | ( N = zero_zero_nat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % even_bit_succ_iff
% 0.33/0.63 thf(fact_7093_even__bit__succ__iff,axiom,
% 0.33/0.63 ! [A: nat,N: nat] :
% 0.33/0.63 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.33/0.63 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N )
% 0.33/0.63 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.33/0.63 | ( N = zero_zero_nat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % even_bit_succ_iff
% 0.33/0.63 thf(fact_7094_odd__bit__iff__bit__pred,axiom,
% 0.33/0.63 ! [A: code_integer,N: nat] :
% 0.33/0.63 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.33/0.63 => ( ( bit_se9216721137139052372nteger @ A @ N )
% 0.33/0.63 = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N )
% 0.33/0.63 | ( N = zero_zero_nat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % odd_bit_iff_bit_pred
% 0.33/0.63 thf(fact_7095_odd__bit__iff__bit__pred,axiom,
% 0.33/0.63 ! [A: int,N: nat] :
% 0.33/0.63 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.33/0.63 => ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.33/0.63 = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N )
% 0.33/0.63 | ( N = zero_zero_nat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % odd_bit_iff_bit_pred
% 0.33/0.63 thf(fact_7096_odd__bit__iff__bit__pred,axiom,
% 0.33/0.63 ! [A: nat,N: nat] :
% 0.33/0.63 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.33/0.63 => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.33/0.63 = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N )
% 0.33/0.63 | ( N = zero_zero_nat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % odd_bit_iff_bit_pred
% 0.33/0.63 thf(fact_7097_mask__half__int,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.63 = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mask_half_int
% 0.33/0.63 thf(fact_7098_mask__nat__def,axiom,
% 0.33/0.63 ( bit_se2002935070580805687sk_nat
% 0.33/0.63 = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mask_nat_def
% 0.33/0.63 thf(fact_7099_mask__int__def,axiom,
% 0.33/0.63 ( bit_se2000444600071755411sk_int
% 0.33/0.63 = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mask_int_def
% 0.33/0.63 thf(fact_7100_bit__sum__mult__2__cases,axiom,
% 0.33/0.63 ! [A: code_integer,B: code_integer,N: nat] :
% 0.33/0.63 ( ! [J2: nat] :
% 0.33/0.63 ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J2 ) )
% 0.33/0.63 => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N )
% 0.33/0.63 = ( ( ( N = zero_zero_nat )
% 0.33/0.63 => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.33/0.63 & ( ( N != zero_zero_nat )
% 0.33/0.63 => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_sum_mult_2_cases
% 0.33/0.63 thf(fact_7101_bit__sum__mult__2__cases,axiom,
% 0.33/0.63 ! [A: int,B: int,N: nat] :
% 0.33/0.63 ( ! [J2: nat] :
% 0.33/0.63 ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J2 ) )
% 0.33/0.63 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ N )
% 0.33/0.63 = ( ( ( N = zero_zero_nat )
% 0.33/0.63 => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.33/0.63 & ( ( N != zero_zero_nat )
% 0.33/0.63 => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_sum_mult_2_cases
% 0.33/0.63 thf(fact_7102_bit__sum__mult__2__cases,axiom,
% 0.33/0.63 ! [A: nat,B: nat,N: nat] :
% 0.33/0.63 ( ! [J2: nat] :
% 0.33/0.63 ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J2 ) )
% 0.33/0.63 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ N )
% 0.33/0.63 = ( ( ( N = zero_zero_nat )
% 0.33/0.63 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.33/0.63 & ( ( N != zero_zero_nat )
% 0.33/0.63 => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_sum_mult_2_cases
% 0.33/0.63 thf(fact_7103_bit__rec,axiom,
% 0.33/0.63 ( bit_se9216721137139052372nteger
% 0.33/0.63 = ( ^ [A2: code_integer,N3: nat] :
% 0.33/0.63 ( ( ( N3 = zero_zero_nat )
% 0.33/0.63 => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) )
% 0.33/0.63 & ( ( N3 != zero_zero_nat )
% 0.33/0.63 => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_rec
% 0.33/0.63 thf(fact_7104_bit__rec,axiom,
% 0.33/0.63 ( bit_se1146084159140164899it_int
% 0.33/0.63 = ( ^ [A2: int,N3: nat] :
% 0.33/0.63 ( ( ( N3 = zero_zero_nat )
% 0.33/0.63 => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 0.33/0.63 & ( ( N3 != zero_zero_nat )
% 0.33/0.63 => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_rec
% 0.33/0.63 thf(fact_7105_bit__rec,axiom,
% 0.33/0.63 ( bit_se1148574629649215175it_nat
% 0.33/0.63 = ( ^ [A2: nat,N3: nat] :
% 0.33/0.63 ( ( ( N3 = zero_zero_nat )
% 0.33/0.63 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
% 0.33/0.63 & ( ( N3 != zero_zero_nat )
% 0.33/0.63 => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_rec
% 0.33/0.63 thf(fact_7106_mask__eq__exp__minus__1,axiom,
% 0.33/0.63 ( bit_se2002935070580805687sk_nat
% 0.33/0.63 = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mask_eq_exp_minus_1
% 0.33/0.63 thf(fact_7107_mask__eq__exp__minus__1,axiom,
% 0.33/0.63 ( bit_se2000444600071755411sk_int
% 0.33/0.63 = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mask_eq_exp_minus_1
% 0.33/0.63 thf(fact_7108_even__nat__iff,axiom,
% 0.33/0.63 ! [K: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.63 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 0.33/0.63 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % even_nat_iff
% 0.33/0.63 thf(fact_7109_set__bit__eq,axiom,
% 0.33/0.63 ( bit_se7879613467334960850it_int
% 0.33/0.63 = ( ^ [N3: nat,K2: int] :
% 0.33/0.63 ( plus_plus_int @ K2
% 0.33/0.63 @ ( times_times_int
% 0.33/0.63 @ ( zero_n2684676970156552555ol_int
% 0.33/0.63 @ ~ ( bit_se1146084159140164899it_int @ K2 @ N3 ) )
% 0.33/0.63 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % set_bit_eq
% 0.33/0.63 thf(fact_7110_unset__bit__eq,axiom,
% 0.33/0.63 ( bit_se4203085406695923979it_int
% 0.33/0.63 = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ K2 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % unset_bit_eq
% 0.33/0.63 thf(fact_7111_take__bit__Suc__from__most,axiom,
% 0.33/0.63 ! [N: nat,K: int] :
% 0.33/0.63 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
% 0.33/0.63 = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % take_bit_Suc_from_most
% 0.33/0.63 thf(fact_7112_take__bit__eq__mask__iff__exp__dvd,axiom,
% 0.33/0.63 ! [N: nat,K: int] :
% 0.33/0.63 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.33/0.63 = ( bit_se2000444600071755411sk_int @ N ) )
% 0.33/0.63 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % take_bit_eq_mask_iff_exp_dvd
% 0.33/0.63 thf(fact_7113_and__int__unfold,axiom,
% 0.33/0.63 ( bit_se725231765392027082nd_int
% 0.33/0.63 = ( ^ [K2: int,L2: int] :
% 0.33/0.63 ( if_int
% 0.33/0.63 @ ( ( K2 = zero_zero_int )
% 0.33/0.63 | ( L2 = zero_zero_int ) )
% 0.33/0.63 @ zero_zero_int
% 0.33/0.63 @ ( if_int
% 0.33/0.63 @ ( K2
% 0.33/0.63 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.63 @ L2
% 0.33/0.63 @ ( if_int
% 0.33/0.63 @ ( L2
% 0.33/0.63 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.63 @ K2
% 0.33/0.63 @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and_int_unfold
% 0.33/0.63 thf(fact_7114_powr__real__of__int,axiom,
% 0.33/0.63 ! [X: real,N: int] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 0.33/0.63 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 0.33/0.63 = ( power_power_real @ X @ ( nat2 @ N ) ) ) )
% 0.33/0.63 & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 0.33/0.63 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 0.33/0.63 = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % powr_real_of_int
% 0.33/0.63 thf(fact_7115_cis__2pi,axiom,
% 0.33/0.63 ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.33/0.63 = one_one_complex ) ).
% 0.33/0.63
% 0.33/0.63 % cis_2pi
% 0.33/0.63 thf(fact_7116_exp__two__pi__i,axiom,
% 0.33/0.63 ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
% 0.33/0.63 = one_one_complex ) ).
% 0.33/0.63
% 0.33/0.63 % exp_two_pi_i
% 0.33/0.63 thf(fact_7117_exp__two__pi__i_H,axiom,
% 0.33/0.63 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
% 0.33/0.63 = one_one_complex ) ).
% 0.33/0.63
% 0.33/0.63 % exp_two_pi_i'
% 0.33/0.63 thf(fact_7118_inverse__eq__iff__eq,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ( inverse_inverse_real @ A )
% 0.33/0.63 = ( inverse_inverse_real @ B ) )
% 0.33/0.63 = ( A = B ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_iff_eq
% 0.33/0.63 thf(fact_7119_inverse__eq__iff__eq,axiom,
% 0.33/0.63 ! [A: complex,B: complex] :
% 0.33/0.63 ( ( ( invers8013647133539491842omplex @ A )
% 0.33/0.63 = ( invers8013647133539491842omplex @ B ) )
% 0.33/0.63 = ( A = B ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_iff_eq
% 0.33/0.63 thf(fact_7120_inverse__eq__iff__eq,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ( inverse_inverse_rat @ A )
% 0.33/0.63 = ( inverse_inverse_rat @ B ) )
% 0.33/0.63 = ( A = B ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_iff_eq
% 0.33/0.63 thf(fact_7121_inverse__inverse__eq,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 0.33/0.63 = A ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_inverse_eq
% 0.33/0.63 thf(fact_7122_inverse__inverse__eq,axiom,
% 0.33/0.63 ! [A: complex] :
% 0.33/0.63 ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
% 0.33/0.63 = A ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_inverse_eq
% 0.33/0.63 thf(fact_7123_inverse__inverse__eq,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
% 0.33/0.63 = A ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_inverse_eq
% 0.33/0.63 thf(fact_7124_and_Oidem,axiom,
% 0.33/0.63 ! [A: int] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ A @ A )
% 0.33/0.63 = A ) ).
% 0.33/0.63
% 0.33/0.63 % and.idem
% 0.33/0.63 thf(fact_7125_and_Oidem,axiom,
% 0.33/0.63 ! [A: nat] :
% 0.33/0.63 ( ( bit_se727722235901077358nd_nat @ A @ A )
% 0.33/0.63 = A ) ).
% 0.33/0.63
% 0.33/0.63 % and.idem
% 0.33/0.63 thf(fact_7126_and_Oleft__idem,axiom,
% 0.33/0.63 ! [A: int,B: int] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 0.33/0.63 = ( bit_se725231765392027082nd_int @ A @ B ) ) ).
% 0.33/0.63
% 0.33/0.63 % and.left_idem
% 0.33/0.63 thf(fact_7127_and_Oleft__idem,axiom,
% 0.33/0.63 ! [A: nat,B: nat] :
% 0.33/0.63 ( ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 0.33/0.63 = ( bit_se727722235901077358nd_nat @ A @ B ) ) ).
% 0.33/0.63
% 0.33/0.63 % and.left_idem
% 0.33/0.63 thf(fact_7128_and_Oright__idem,axiom,
% 0.33/0.63 ! [A: int,B: int] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ B )
% 0.33/0.63 = ( bit_se725231765392027082nd_int @ A @ B ) ) ).
% 0.33/0.63
% 0.33/0.63 % and.right_idem
% 0.33/0.63 thf(fact_7129_and_Oright__idem,axiom,
% 0.33/0.63 ! [A: nat,B: nat] :
% 0.33/0.63 ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ B )
% 0.33/0.63 = ( bit_se727722235901077358nd_nat @ A @ B ) ) ).
% 0.33/0.63
% 0.33/0.63 % and.right_idem
% 0.33/0.63 thf(fact_7130_inverse__nonzero__iff__nonzero,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( ( inverse_inverse_real @ A )
% 0.33/0.63 = zero_zero_real )
% 0.33/0.63 = ( A = zero_zero_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_nonzero_iff_nonzero
% 0.33/0.63 thf(fact_7131_inverse__nonzero__iff__nonzero,axiom,
% 0.33/0.63 ! [A: complex] :
% 0.33/0.63 ( ( ( invers8013647133539491842omplex @ A )
% 0.33/0.63 = zero_zero_complex )
% 0.33/0.63 = ( A = zero_zero_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_nonzero_iff_nonzero
% 0.33/0.63 thf(fact_7132_inverse__nonzero__iff__nonzero,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( ( inverse_inverse_rat @ A )
% 0.33/0.63 = zero_zero_rat )
% 0.33/0.63 = ( A = zero_zero_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_nonzero_iff_nonzero
% 0.33/0.63 thf(fact_7133_inverse__zero,axiom,
% 0.33/0.63 ( ( inverse_inverse_real @ zero_zero_real )
% 0.33/0.63 = zero_zero_real ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_zero
% 0.33/0.63 thf(fact_7134_inverse__zero,axiom,
% 0.33/0.63 ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 0.33/0.63 = zero_zero_complex ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_zero
% 0.33/0.63 thf(fact_7135_inverse__zero,axiom,
% 0.33/0.63 ( ( inverse_inverse_rat @ zero_zero_rat )
% 0.33/0.63 = zero_zero_rat ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_zero
% 0.33/0.63 thf(fact_7136_inverse__mult__distrib,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
% 0.33/0.63 = ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_mult_distrib
% 0.33/0.63 thf(fact_7137_inverse__mult__distrib,axiom,
% 0.33/0.63 ! [A: complex,B: complex] :
% 0.33/0.63 ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
% 0.33/0.63 = ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_mult_distrib
% 0.33/0.63 thf(fact_7138_inverse__mult__distrib,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
% 0.33/0.63 = ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_mult_distrib
% 0.33/0.63 thf(fact_7139_inverse__eq__1__iff,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ( inverse_inverse_real @ X )
% 0.33/0.63 = one_one_real )
% 0.33/0.63 = ( X = one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_1_iff
% 0.33/0.63 thf(fact_7140_inverse__eq__1__iff,axiom,
% 0.33/0.63 ! [X: complex] :
% 0.33/0.63 ( ( ( invers8013647133539491842omplex @ X )
% 0.33/0.63 = one_one_complex )
% 0.33/0.63 = ( X = one_one_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_1_iff
% 0.33/0.63 thf(fact_7141_inverse__eq__1__iff,axiom,
% 0.33/0.63 ! [X: rat] :
% 0.33/0.63 ( ( ( inverse_inverse_rat @ X )
% 0.33/0.63 = one_one_rat )
% 0.33/0.63 = ( X = one_one_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_1_iff
% 0.33/0.63 thf(fact_7142_inverse__1,axiom,
% 0.33/0.63 ( ( inverse_inverse_real @ one_one_real )
% 0.33/0.63 = one_one_real ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_1
% 0.33/0.63 thf(fact_7143_inverse__1,axiom,
% 0.33/0.63 ( ( invers8013647133539491842omplex @ one_one_complex )
% 0.33/0.63 = one_one_complex ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_1
% 0.33/0.63 thf(fact_7144_inverse__1,axiom,
% 0.33/0.63 ( ( inverse_inverse_rat @ one_one_rat )
% 0.33/0.63 = one_one_rat ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_1
% 0.33/0.63 thf(fact_7145_bit_Oconj__zero__right,axiom,
% 0.33/0.63 ! [X: int] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
% 0.33/0.63 = zero_zero_int ) ).
% 0.33/0.63
% 0.33/0.63 % bit.conj_zero_right
% 0.33/0.63 thf(fact_7146_bit_Oconj__zero__left,axiom,
% 0.33/0.63 ! [X: int] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
% 0.33/0.63 = zero_zero_int ) ).
% 0.33/0.63
% 0.33/0.63 % bit.conj_zero_left
% 0.33/0.63 thf(fact_7147_zero__and__eq,axiom,
% 0.33/0.63 ! [A: int] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
% 0.33/0.63 = zero_zero_int ) ).
% 0.33/0.63
% 0.33/0.63 % zero_and_eq
% 0.33/0.63 thf(fact_7148_zero__and__eq,axiom,
% 0.33/0.63 ! [A: nat] :
% 0.33/0.63 ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
% 0.33/0.63 = zero_zero_nat ) ).
% 0.33/0.63
% 0.33/0.63 % zero_and_eq
% 0.33/0.63 thf(fact_7149_and__zero__eq,axiom,
% 0.33/0.63 ! [A: int] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
% 0.33/0.63 = zero_zero_int ) ).
% 0.33/0.63
% 0.33/0.63 % and_zero_eq
% 0.33/0.63 thf(fact_7150_and__zero__eq,axiom,
% 0.33/0.63 ! [A: nat] :
% 0.33/0.63 ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
% 0.33/0.63 = zero_zero_nat ) ).
% 0.33/0.63
% 0.33/0.63 % and_zero_eq
% 0.33/0.63 thf(fact_7151_inverse__minus__eq,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 0.33/0.63 = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_minus_eq
% 0.33/0.63 thf(fact_7152_inverse__minus__eq,axiom,
% 0.33/0.63 ! [A: complex] :
% 0.33/0.63 ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_minus_eq
% 0.33/0.63 thf(fact_7153_inverse__minus__eq,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
% 0.33/0.63 = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_minus_eq
% 0.33/0.63 thf(fact_7154_inverse__divide,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.63 = ( divide_divide_real @ B @ A ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_divide
% 0.33/0.63 thf(fact_7155_inverse__divide,axiom,
% 0.33/0.63 ! [A: complex,B: complex] :
% 0.33/0.63 ( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.63 = ( divide1717551699836669952omplex @ B @ A ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_divide
% 0.33/0.63 thf(fact_7156_inverse__divide,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B ) )
% 0.33/0.63 = ( divide_divide_rat @ B @ A ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_divide
% 0.33/0.63 thf(fact_7157_abs__inverse,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
% 0.33/0.63 = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % abs_inverse
% 0.33/0.63 thf(fact_7158_abs__inverse,axiom,
% 0.33/0.63 ! [A: complex] :
% 0.33/0.63 ( ( abs_abs_complex @ ( invers8013647133539491842omplex @ A ) )
% 0.33/0.63 = ( invers8013647133539491842omplex @ ( abs_abs_complex @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % abs_inverse
% 0.33/0.63 thf(fact_7159_abs__inverse,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
% 0.33/0.63 = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % abs_inverse
% 0.33/0.63 thf(fact_7160_sgn__inverse,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( sgn_sgn_real @ ( inverse_inverse_real @ A ) )
% 0.33/0.63 = ( inverse_inverse_real @ ( sgn_sgn_real @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sgn_inverse
% 0.33/0.63 thf(fact_7161_sgn__inverse,axiom,
% 0.33/0.63 ! [A: complex] :
% 0.33/0.63 ( ( sgn_sgn_complex @ ( invers8013647133539491842omplex @ A ) )
% 0.33/0.63 = ( invers8013647133539491842omplex @ ( sgn_sgn_complex @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sgn_inverse
% 0.33/0.63 thf(fact_7162_sgn__inverse,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( sgn_sgn_rat @ ( inverse_inverse_rat @ A ) )
% 0.33/0.63 = ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sgn_inverse
% 0.33/0.63 thf(fact_7163_inverse__sgn,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( inverse_inverse_real @ ( sgn_sgn_real @ A ) )
% 0.33/0.63 = ( sgn_sgn_real @ A ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_sgn
% 0.33/0.63 thf(fact_7164_inverse__sgn,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) )
% 0.33/0.63 = ( sgn_sgn_rat @ A ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_sgn
% 0.33/0.63 thf(fact_7165_take__bit__and,axiom,
% 0.33/0.63 ! [N: nat,A: int,B: int] :
% 0.33/0.63 ( ( bit_se2923211474154528505it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 0.33/0.63 = ( bit_se725231765392027082nd_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % take_bit_and
% 0.33/0.63 thf(fact_7166_take__bit__and,axiom,
% 0.33/0.63 ! [N: nat,A: nat,B: nat] :
% 0.33/0.63 ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 0.33/0.63 = ( bit_se727722235901077358nd_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % take_bit_and
% 0.33/0.63 thf(fact_7167_push__bit__and,axiom,
% 0.33/0.63 ! [N: nat,A: nat,B: nat] :
% 0.33/0.63 ( ( bit_se547839408752420682it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 0.33/0.63 = ( bit_se727722235901077358nd_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % push_bit_and
% 0.33/0.63 thf(fact_7168_push__bit__and,axiom,
% 0.33/0.63 ! [N: nat,A: int,B: int] :
% 0.33/0.63 ( ( bit_se545348938243370406it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 0.33/0.63 = ( bit_se725231765392027082nd_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % push_bit_and
% 0.33/0.63 thf(fact_7169_push__bit__and,axiom,
% 0.33/0.63 ! [N: nat,A: code_integer,B: code_integer] :
% 0.33/0.63 ( ( bit_se7788150548672797655nteger @ N @ ( bit_se3949692690581998587nteger @ A @ B ) )
% 0.33/0.63 = ( bit_se3949692690581998587nteger @ ( bit_se7788150548672797655nteger @ N @ A ) @ ( bit_se7788150548672797655nteger @ N @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % push_bit_and
% 0.33/0.63 thf(fact_7170_push__bit__and,axiom,
% 0.33/0.63 ! [N: nat,A: code_natural,B: code_natural] :
% 0.33/0.63 ( ( bit_se6611745700429515170atural @ N @ ( bit_se2773287842338716102atural @ A @ B ) )
% 0.33/0.63 = ( bit_se2773287842338716102atural @ ( bit_se6611745700429515170atural @ N @ A ) @ ( bit_se6611745700429515170atural @ N @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % push_bit_and
% 0.33/0.63 thf(fact_7171_inverse__nonnegative__iff__nonnegative,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.33/0.63 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_nonnegative_iff_nonnegative
% 0.33/0.63 thf(fact_7172_inverse__nonnegative__iff__nonnegative,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.33/0.63 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_nonnegative_iff_nonnegative
% 0.33/0.63 thf(fact_7173_inverse__nonpositive__iff__nonpositive,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.33/0.63 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_nonpositive_iff_nonpositive
% 0.33/0.63 thf(fact_7174_inverse__nonpositive__iff__nonpositive,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.33/0.63 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_nonpositive_iff_nonpositive
% 0.33/0.63 thf(fact_7175_inverse__less__iff__less,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.63 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.63 => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.33/0.63 = ( ord_less_real @ B @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_less_iff_less
% 0.33/0.63 thf(fact_7176_inverse__less__iff__less,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.63 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.33/0.63 => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.33/0.63 = ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_less_iff_less
% 0.33/0.63 thf(fact_7177_inverse__less__iff__less__neg,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.63 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.33/0.63 => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.33/0.63 = ( ord_less_real @ B @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_less_iff_less_neg
% 0.33/0.63 thf(fact_7178_inverse__less__iff__less__neg,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.63 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.33/0.63 => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.33/0.63 = ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_less_iff_less_neg
% 0.33/0.63 thf(fact_7179_inverse__negative__iff__negative,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.33/0.63 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_negative_iff_negative
% 0.33/0.63 thf(fact_7180_inverse__negative__iff__negative,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.33/0.63 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_negative_iff_negative
% 0.33/0.63 thf(fact_7181_inverse__positive__iff__positive,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.33/0.63 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_positive_iff_positive
% 0.33/0.63 thf(fact_7182_inverse__positive__iff__positive,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.33/0.63 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_positive_iff_positive
% 0.33/0.63 thf(fact_7183_and_Oleft__neutral,axiom,
% 0.33/0.63 ! [A: code_integer] :
% 0.33/0.63 ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
% 0.33/0.63 = A ) ).
% 0.33/0.63
% 0.33/0.63 % and.left_neutral
% 0.33/0.63 thf(fact_7184_and_Oleft__neutral,axiom,
% 0.33/0.63 ! [A: int] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
% 0.33/0.63 = A ) ).
% 0.33/0.63
% 0.33/0.63 % and.left_neutral
% 0.33/0.63 thf(fact_7185_and_Oright__neutral,axiom,
% 0.33/0.63 ! [A: code_integer] :
% 0.33/0.63 ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.63 = A ) ).
% 0.33/0.63
% 0.33/0.63 % and.right_neutral
% 0.33/0.63 thf(fact_7186_and_Oright__neutral,axiom,
% 0.33/0.63 ! [A: int] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.63 = A ) ).
% 0.33/0.63
% 0.33/0.63 % and.right_neutral
% 0.33/0.63 thf(fact_7187_bit_Oconj__one__right,axiom,
% 0.33/0.63 ! [X: code_integer] :
% 0.33/0.63 ( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.63 = X ) ).
% 0.33/0.63
% 0.33/0.63 % bit.conj_one_right
% 0.33/0.63 thf(fact_7188_bit_Oconj__one__right,axiom,
% 0.33/0.63 ! [X: int] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.63 = X ) ).
% 0.33/0.63
% 0.33/0.63 % bit.conj_one_right
% 0.33/0.63 thf(fact_7189_and__nonnegative__int__iff,axiom,
% 0.33/0.63 ! [K: int,L: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 0.33/0.63 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.63 | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and_nonnegative_int_iff
% 0.33/0.63 thf(fact_7190_and__negative__int__iff,axiom,
% 0.33/0.63 ! [K: int,L: int] :
% 0.33/0.63 ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
% 0.33/0.63 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.33/0.63 & ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and_negative_int_iff
% 0.33/0.63 thf(fact_7191_norm__ii,axiom,
% 0.33/0.63 ( ( real_V1022390504157884413omplex @ imaginary_unit )
% 0.33/0.63 = one_one_real ) ).
% 0.33/0.63
% 0.33/0.63 % norm_ii
% 0.33/0.63 thf(fact_7192_norm__cis,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
% 0.33/0.63 = one_one_real ) ).
% 0.33/0.63
% 0.33/0.63 % norm_cis
% 0.33/0.63 thf(fact_7193_cis__zero,axiom,
% 0.33/0.63 ( ( cis @ zero_zero_real )
% 0.33/0.63 = one_one_complex ) ).
% 0.33/0.63
% 0.33/0.63 % cis_zero
% 0.33/0.63 thf(fact_7194_divide__i,axiom,
% 0.33/0.63 ! [X: complex] :
% 0.33/0.63 ( ( divide1717551699836669952omplex @ X @ imaginary_unit )
% 0.33/0.63 = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % divide_i
% 0.33/0.63 thf(fact_7195_complex__i__mult__minus,axiom,
% 0.33/0.63 ! [X: complex] :
% 0.33/0.63 ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X ) )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % complex_i_mult_minus
% 0.33/0.63 thf(fact_7196_inverse__le__iff__le,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.63 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.63 => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.33/0.63 = ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_le_iff_le
% 0.33/0.63 thf(fact_7197_inverse__le__iff__le,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.63 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.33/0.63 => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.33/0.63 = ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_le_iff_le
% 0.33/0.63 thf(fact_7198_inverse__le__iff__le__neg,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.63 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.33/0.63 => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.33/0.63 = ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_le_iff_le_neg
% 0.33/0.63 thf(fact_7199_inverse__le__iff__le__neg,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.63 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.33/0.63 => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.33/0.63 = ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_le_iff_le_neg
% 0.33/0.63 thf(fact_7200_right__inverse,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( A != zero_zero_real )
% 0.33/0.63 => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
% 0.33/0.63 = one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % right_inverse
% 0.33/0.63 thf(fact_7201_right__inverse,axiom,
% 0.33/0.63 ! [A: complex] :
% 0.33/0.63 ( ( A != zero_zero_complex )
% 0.33/0.63 => ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
% 0.33/0.63 = one_one_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % right_inverse
% 0.33/0.63 thf(fact_7202_right__inverse,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( A != zero_zero_rat )
% 0.33/0.63 => ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
% 0.33/0.63 = one_one_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % right_inverse
% 0.33/0.63 thf(fact_7203_left__inverse,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( A != zero_zero_real )
% 0.33/0.63 => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 0.33/0.63 = one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % left_inverse
% 0.33/0.63 thf(fact_7204_left__inverse,axiom,
% 0.33/0.63 ! [A: complex] :
% 0.33/0.63 ( ( A != zero_zero_complex )
% 0.33/0.63 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 0.33/0.63 = one_one_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % left_inverse
% 0.33/0.63 thf(fact_7205_left__inverse,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( A != zero_zero_rat )
% 0.33/0.63 => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 0.33/0.63 = one_one_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % left_inverse
% 0.33/0.63 thf(fact_7206_inverse__eq__divide__numeral,axiom,
% 0.33/0.63 ! [W: num] :
% 0.33/0.63 ( ( inverse_inverse_real @ ( numeral_numeral_real @ W ) )
% 0.33/0.63 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_divide_numeral
% 0.33/0.63 thf(fact_7207_inverse__eq__divide__numeral,axiom,
% 0.33/0.63 ! [W: num] :
% 0.33/0.63 ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.33/0.63 = ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_divide_numeral
% 0.33/0.63 thf(fact_7208_inverse__eq__divide__numeral,axiom,
% 0.33/0.63 ! [W: num] :
% 0.33/0.63 ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W ) )
% 0.33/0.63 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_divide_numeral
% 0.33/0.63 thf(fact_7209_and__numerals_I2_J,axiom,
% 0.33/0.63 ! [Y: num] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 0.33/0.63 = one_one_int ) ).
% 0.33/0.63
% 0.33/0.63 % and_numerals(2)
% 0.33/0.63 thf(fact_7210_and__numerals_I2_J,axiom,
% 0.33/0.63 ! [Y: num] :
% 0.33/0.63 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.33/0.63 = one_one_nat ) ).
% 0.33/0.63
% 0.33/0.63 % and_numerals(2)
% 0.33/0.63 thf(fact_7211_and__numerals_I8_J,axiom,
% 0.33/0.63 ! [X: num] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
% 0.33/0.63 = one_one_int ) ).
% 0.33/0.63
% 0.33/0.63 % and_numerals(8)
% 0.33/0.63 thf(fact_7212_and__numerals_I8_J,axiom,
% 0.33/0.63 ! [X: num] :
% 0.33/0.63 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
% 0.33/0.63 = one_one_nat ) ).
% 0.33/0.63
% 0.33/0.63 % and_numerals(8)
% 0.33/0.63 thf(fact_7213_cis__pi,axiom,
% 0.33/0.63 ( ( cis @ pi )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % cis_pi
% 0.33/0.63 thf(fact_7214_i__squared,axiom,
% 0.33/0.63 ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % i_squared
% 0.33/0.63 thf(fact_7215_divide__numeral__i,axiom,
% 0.33/0.63 ! [Z2: complex,N: num] :
% 0.33/0.63 ( ( divide1717551699836669952omplex @ Z2 @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
% 0.33/0.63 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z2 ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % divide_numeral_i
% 0.33/0.63 thf(fact_7216_and__numerals_I5_J,axiom,
% 0.33/0.63 ! [X: num] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 0.33/0.63 = zero_zero_int ) ).
% 0.33/0.63
% 0.33/0.63 % and_numerals(5)
% 0.33/0.63 thf(fact_7217_and__numerals_I5_J,axiom,
% 0.33/0.63 ! [X: num] :
% 0.33/0.63 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 0.33/0.63 = zero_zero_nat ) ).
% 0.33/0.63
% 0.33/0.63 % and_numerals(5)
% 0.33/0.63 thf(fact_7218_and__numerals_I1_J,axiom,
% 0.33/0.63 ! [Y: num] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 0.33/0.63 = zero_zero_int ) ).
% 0.33/0.63
% 0.33/0.63 % and_numerals(1)
% 0.33/0.63 thf(fact_7219_and__numerals_I1_J,axiom,
% 0.33/0.63 ! [Y: num] :
% 0.33/0.63 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.33/0.63 = zero_zero_nat ) ).
% 0.33/0.63
% 0.33/0.63 % and_numerals(1)
% 0.33/0.63 thf(fact_7220_inverse__eq__divide__neg__numeral,axiom,
% 0.33/0.63 ! [W: num] :
% 0.33/0.63 ( ( inverse_inverse_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.63 = ( divide_divide_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_divide_neg_numeral
% 0.33/0.63 thf(fact_7221_inverse__eq__divide__neg__numeral,axiom,
% 0.33/0.63 ! [W: num] :
% 0.33/0.63 ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.33/0.63 = ( divide1717551699836669952omplex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_divide_neg_numeral
% 0.33/0.63 thf(fact_7222_inverse__eq__divide__neg__numeral,axiom,
% 0.33/0.63 ! [W: num] :
% 0.33/0.63 ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.33/0.63 = ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_divide_neg_numeral
% 0.33/0.63 thf(fact_7223_and__numerals_I3_J,axiom,
% 0.33/0.63 ! [X: num,Y: num] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 0.33/0.63 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and_numerals(3)
% 0.33/0.63 thf(fact_7224_and__numerals_I3_J,axiom,
% 0.33/0.63 ! [X: num,Y: num] :
% 0.33/0.63 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.33/0.63 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and_numerals(3)
% 0.33/0.63 thf(fact_7225_and__minus__numerals_I2_J,axiom,
% 0.33/0.63 ! [N: num] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.33/0.63 = one_one_int ) ).
% 0.33/0.63
% 0.33/0.63 % and_minus_numerals(2)
% 0.33/0.63 thf(fact_7226_and__minus__numerals_I6_J,axiom,
% 0.33/0.63 ! [N: num] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 0.33/0.63 = one_one_int ) ).
% 0.33/0.63
% 0.33/0.63 % and_minus_numerals(6)
% 0.33/0.63 thf(fact_7227_and__numerals_I6_J,axiom,
% 0.33/0.63 ! [X: num,Y: num] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 0.33/0.63 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and_numerals(6)
% 0.33/0.63 thf(fact_7228_and__numerals_I6_J,axiom,
% 0.33/0.63 ! [X: num,Y: num] :
% 0.33/0.63 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.33/0.63 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and_numerals(6)
% 0.33/0.63 thf(fact_7229_and__numerals_I4_J,axiom,
% 0.33/0.63 ! [X: num,Y: num] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 0.33/0.63 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and_numerals(4)
% 0.33/0.63 thf(fact_7230_and__numerals_I4_J,axiom,
% 0.33/0.63 ! [X: num,Y: num] :
% 0.33/0.63 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.33/0.63 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and_numerals(4)
% 0.33/0.63 thf(fact_7231_and__minus__numerals_I5_J,axiom,
% 0.33/0.63 ! [N: num] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 0.33/0.63 = zero_zero_int ) ).
% 0.33/0.63
% 0.33/0.63 % and_minus_numerals(5)
% 0.33/0.63 thf(fact_7232_and__minus__numerals_I1_J,axiom,
% 0.33/0.63 ! [N: num] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.33/0.63 = zero_zero_int ) ).
% 0.33/0.63
% 0.33/0.63 % and_minus_numerals(1)
% 0.33/0.63 thf(fact_7233_power2__i,axiom,
% 0.33/0.63 ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % power2_i
% 0.33/0.63 thf(fact_7234_cis__pi__half,axiom,
% 0.33/0.63 ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.63 = imaginary_unit ) ).
% 0.33/0.63
% 0.33/0.63 % cis_pi_half
% 0.33/0.63 thf(fact_7235_i__even__power,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.63 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % i_even_power
% 0.33/0.63 thf(fact_7236_and__numerals_I7_J,axiom,
% 0.33/0.63 ! [X: num,Y: num] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 0.33/0.63 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and_numerals(7)
% 0.33/0.63 thf(fact_7237_and__numerals_I7_J,axiom,
% 0.33/0.63 ! [X: num,Y: num] :
% 0.33/0.63 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.33/0.63 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and_numerals(7)
% 0.33/0.63 thf(fact_7238_exp__pi__i,axiom,
% 0.33/0.63 ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_pi_i
% 0.33/0.63 thf(fact_7239_exp__pi__i_H,axiom,
% 0.33/0.63 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % exp_pi_i'
% 0.33/0.63 thf(fact_7240_cis__minus__pi__half,axiom,
% 0.33/0.63 ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% 0.33/0.63
% 0.33/0.63 % cis_minus_pi_half
% 0.33/0.63 thf(fact_7241_bit__and__iff,axiom,
% 0.33/0.63 ! [A: int,B: int,N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ N )
% 0.33/0.63 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.33/0.63 & ( bit_se1146084159140164899it_int @ B @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_and_iff
% 0.33/0.63 thf(fact_7242_bit__and__iff,axiom,
% 0.33/0.63 ! [A: nat,B: nat,N: nat] :
% 0.33/0.63 ( ( bit_se1148574629649215175it_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ N )
% 0.33/0.63 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.33/0.63 & ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_and_iff
% 0.33/0.63 thf(fact_7243_of__int__and__eq,axiom,
% 0.33/0.63 ! [K: int,L: int] :
% 0.33/0.63 ( ( ring_1_of_int_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 0.33/0.63 = ( bit_se725231765392027082nd_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_int_and_eq
% 0.33/0.63 thf(fact_7244_field__class_Ofield__inverse__zero,axiom,
% 0.33/0.63 ( ( inverse_inverse_real @ zero_zero_real )
% 0.33/0.63 = zero_zero_real ) ).
% 0.33/0.63
% 0.33/0.63 % field_class.field_inverse_zero
% 0.33/0.63 thf(fact_7245_field__class_Ofield__inverse__zero,axiom,
% 0.33/0.63 ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 0.33/0.63 = zero_zero_complex ) ).
% 0.33/0.63
% 0.33/0.63 % field_class.field_inverse_zero
% 0.33/0.63 thf(fact_7246_field__class_Ofield__inverse__zero,axiom,
% 0.33/0.63 ( ( inverse_inverse_rat @ zero_zero_rat )
% 0.33/0.63 = zero_zero_rat ) ).
% 0.33/0.63
% 0.33/0.63 % field_class.field_inverse_zero
% 0.33/0.63 thf(fact_7247_inverse__zero__imp__zero,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( ( inverse_inverse_real @ A )
% 0.33/0.63 = zero_zero_real )
% 0.33/0.63 => ( A = zero_zero_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_zero_imp_zero
% 0.33/0.63 thf(fact_7248_inverse__zero__imp__zero,axiom,
% 0.33/0.63 ! [A: complex] :
% 0.33/0.63 ( ( ( invers8013647133539491842omplex @ A )
% 0.33/0.63 = zero_zero_complex )
% 0.33/0.63 => ( A = zero_zero_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_zero_imp_zero
% 0.33/0.63 thf(fact_7249_inverse__zero__imp__zero,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( ( inverse_inverse_rat @ A )
% 0.33/0.63 = zero_zero_rat )
% 0.33/0.63 => ( A = zero_zero_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_zero_imp_zero
% 0.33/0.63 thf(fact_7250_nonzero__inverse__eq__imp__eq,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ( inverse_inverse_real @ A )
% 0.33/0.63 = ( inverse_inverse_real @ B ) )
% 0.33/0.63 => ( ( A != zero_zero_real )
% 0.33/0.63 => ( ( B != zero_zero_real )
% 0.33/0.63 => ( A = B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_inverse_eq_imp_eq
% 0.33/0.63 thf(fact_7251_nonzero__inverse__eq__imp__eq,axiom,
% 0.33/0.63 ! [A: complex,B: complex] :
% 0.33/0.63 ( ( ( invers8013647133539491842omplex @ A )
% 0.33/0.63 = ( invers8013647133539491842omplex @ B ) )
% 0.33/0.63 => ( ( A != zero_zero_complex )
% 0.33/0.63 => ( ( B != zero_zero_complex )
% 0.33/0.63 => ( A = B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_inverse_eq_imp_eq
% 0.33/0.63 thf(fact_7252_nonzero__inverse__eq__imp__eq,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ( inverse_inverse_rat @ A )
% 0.33/0.63 = ( inverse_inverse_rat @ B ) )
% 0.33/0.63 => ( ( A != zero_zero_rat )
% 0.33/0.63 => ( ( B != zero_zero_rat )
% 0.33/0.63 => ( A = B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_inverse_eq_imp_eq
% 0.33/0.63 thf(fact_7253_nonzero__inverse__inverse__eq,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( A != zero_zero_real )
% 0.33/0.63 => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 0.33/0.63 = A ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_inverse_inverse_eq
% 0.33/0.63 thf(fact_7254_nonzero__inverse__inverse__eq,axiom,
% 0.33/0.63 ! [A: complex] :
% 0.33/0.63 ( ( A != zero_zero_complex )
% 0.33/0.63 => ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
% 0.33/0.63 = A ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_inverse_inverse_eq
% 0.33/0.63 thf(fact_7255_nonzero__inverse__inverse__eq,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( A != zero_zero_rat )
% 0.33/0.63 => ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
% 0.33/0.63 = A ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_inverse_inverse_eq
% 0.33/0.63 thf(fact_7256_nonzero__imp__inverse__nonzero,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( A != zero_zero_real )
% 0.33/0.63 => ( ( inverse_inverse_real @ A )
% 0.33/0.63 != zero_zero_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_imp_inverse_nonzero
% 0.33/0.63 thf(fact_7257_nonzero__imp__inverse__nonzero,axiom,
% 0.33/0.63 ! [A: complex] :
% 0.33/0.63 ( ( A != zero_zero_complex )
% 0.33/0.63 => ( ( invers8013647133539491842omplex @ A )
% 0.33/0.63 != zero_zero_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_imp_inverse_nonzero
% 0.33/0.63 thf(fact_7258_nonzero__imp__inverse__nonzero,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( A != zero_zero_rat )
% 0.33/0.63 => ( ( inverse_inverse_rat @ A )
% 0.33/0.63 != zero_zero_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_imp_inverse_nonzero
% 0.33/0.63 thf(fact_7259_mult__commute__imp__mult__inverse__commute,axiom,
% 0.33/0.63 ! [Y: real,X: real] :
% 0.33/0.63 ( ( ( times_times_real @ Y @ X )
% 0.33/0.63 = ( times_times_real @ X @ Y ) )
% 0.33/0.63 => ( ( times_times_real @ ( inverse_inverse_real @ Y ) @ X )
% 0.33/0.63 = ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_commute_imp_mult_inverse_commute
% 0.33/0.63 thf(fact_7260_mult__commute__imp__mult__inverse__commute,axiom,
% 0.33/0.63 ! [Y: complex,X: complex] :
% 0.33/0.63 ( ( ( times_times_complex @ Y @ X )
% 0.33/0.63 = ( times_times_complex @ X @ Y ) )
% 0.33/0.63 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ Y ) @ X )
% 0.33/0.63 = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_commute_imp_mult_inverse_commute
% 0.33/0.63 thf(fact_7261_mult__commute__imp__mult__inverse__commute,axiom,
% 0.33/0.63 ! [Y: rat,X: rat] :
% 0.33/0.63 ( ( ( times_times_rat @ Y @ X )
% 0.33/0.63 = ( times_times_rat @ X @ Y ) )
% 0.33/0.63 => ( ( times_times_rat @ ( inverse_inverse_rat @ Y ) @ X )
% 0.33/0.63 = ( times_times_rat @ X @ ( inverse_inverse_rat @ Y ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_commute_imp_mult_inverse_commute
% 0.33/0.63 thf(fact_7262_of__nat__and__eq,axiom,
% 0.33/0.63 ! [M: nat,N: nat] :
% 0.33/0.63 ( ( semiri1314217659103216013at_int @ ( bit_se727722235901077358nd_nat @ M @ N ) )
% 0.33/0.63 = ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_and_eq
% 0.33/0.63 thf(fact_7263_of__nat__and__eq,axiom,
% 0.33/0.63 ! [M: nat,N: nat] :
% 0.33/0.63 ( ( semiri1316708129612266289at_nat @ ( bit_se727722235901077358nd_nat @ M @ N ) )
% 0.33/0.63 = ( bit_se727722235901077358nd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % of_nat_and_eq
% 0.33/0.63 thf(fact_7264_inverse__eq__imp__eq,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ( inverse_inverse_real @ A )
% 0.33/0.63 = ( inverse_inverse_real @ B ) )
% 0.33/0.63 => ( A = B ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_imp_eq
% 0.33/0.63 thf(fact_7265_inverse__eq__imp__eq,axiom,
% 0.33/0.63 ! [A: complex,B: complex] :
% 0.33/0.63 ( ( ( invers8013647133539491842omplex @ A )
% 0.33/0.63 = ( invers8013647133539491842omplex @ B ) )
% 0.33/0.63 => ( A = B ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_imp_eq
% 0.33/0.63 thf(fact_7266_inverse__eq__imp__eq,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ( inverse_inverse_rat @ A )
% 0.33/0.63 = ( inverse_inverse_rat @ B ) )
% 0.33/0.63 => ( A = B ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_imp_eq
% 0.33/0.63 thf(fact_7267_and_Oassoc,axiom,
% 0.33/0.63 ! [A: int,B: int,C: int] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ C )
% 0.33/0.63 = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and.assoc
% 0.33/0.63 thf(fact_7268_and_Oassoc,axiom,
% 0.33/0.63 ! [A: nat,B: nat,C: nat] :
% 0.33/0.63 ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ C )
% 0.33/0.63 = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and.assoc
% 0.33/0.63 thf(fact_7269_and_Ocommute,axiom,
% 0.33/0.63 ( bit_se725231765392027082nd_int
% 0.33/0.63 = ( ^ [A2: int,B2: int] : ( bit_se725231765392027082nd_int @ B2 @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and.commute
% 0.33/0.63 thf(fact_7270_and_Ocommute,axiom,
% 0.33/0.63 ( bit_se727722235901077358nd_nat
% 0.33/0.63 = ( ^ [A2: nat,B2: nat] : ( bit_se727722235901077358nd_nat @ B2 @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and.commute
% 0.33/0.63 thf(fact_7271_and_Oleft__commute,axiom,
% 0.33/0.63 ! [B: int,A: int,C: int] :
% 0.33/0.63 ( ( bit_se725231765392027082nd_int @ B @ ( bit_se725231765392027082nd_int @ A @ C ) )
% 0.33/0.63 = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and.left_commute
% 0.33/0.63 thf(fact_7272_and_Oleft__commute,axiom,
% 0.33/0.63 ! [B: nat,A: nat,C: nat] :
% 0.33/0.63 ( ( bit_se727722235901077358nd_nat @ B @ ( bit_se727722235901077358nd_nat @ A @ C ) )
% 0.33/0.63 = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and.left_commute
% 0.33/0.63 thf(fact_7273_nonzero__norm__inverse,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( A != zero_zero_real )
% 0.33/0.63 => ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
% 0.33/0.63 = ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_norm_inverse
% 0.33/0.63 thf(fact_7274_nonzero__norm__inverse,axiom,
% 0.33/0.63 ! [A: complex] :
% 0.33/0.63 ( ( A != zero_zero_complex )
% 0.33/0.63 => ( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
% 0.33/0.63 = ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_norm_inverse
% 0.33/0.63 thf(fact_7275_power__inverse,axiom,
% 0.33/0.63 ! [A: real,N: nat] :
% 0.33/0.63 ( ( power_power_real @ ( inverse_inverse_real @ A ) @ N )
% 0.33/0.63 = ( inverse_inverse_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_inverse
% 0.33/0.63 thf(fact_7276_power__inverse,axiom,
% 0.33/0.63 ! [A: complex,N: nat] :
% 0.33/0.63 ( ( power_power_complex @ ( invers8013647133539491842omplex @ A ) @ N )
% 0.33/0.63 = ( invers8013647133539491842omplex @ ( power_power_complex @ A @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_inverse
% 0.33/0.63 thf(fact_7277_power__inverse,axiom,
% 0.33/0.63 ! [A: rat,N: nat] :
% 0.33/0.63 ( ( power_power_rat @ ( inverse_inverse_rat @ A ) @ N )
% 0.33/0.63 = ( inverse_inverse_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_inverse
% 0.33/0.63 thf(fact_7278_real__sqrt__inverse,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( sqrt @ ( inverse_inverse_real @ X ) )
% 0.33/0.63 = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_sqrt_inverse
% 0.33/0.63 thf(fact_7279_bit__and__int__iff,axiom,
% 0.33/0.63 ! [K: int,L: int,N: nat] :
% 0.33/0.63 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ N )
% 0.33/0.63 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.33/0.63 & ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_and_int_iff
% 0.33/0.63 thf(fact_7280_cis__conv__exp,axiom,
% 0.33/0.63 ( cis
% 0.33/0.63 = ( ^ [B2: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % cis_conv_exp
% 0.33/0.63 thf(fact_7281_cis__divide,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
% 0.33/0.63 = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % cis_divide
% 0.33/0.63 thf(fact_7282_complex__i__not__one,axiom,
% 0.33/0.63 imaginary_unit != one_one_complex ).
% 0.33/0.63
% 0.33/0.63 % complex_i_not_one
% 0.33/0.63 thf(fact_7283_and__eq__minus__1__iff,axiom,
% 0.33/0.63 ! [A: code_integer,B: code_integer] :
% 0.33/0.63 ( ( ( bit_se3949692690581998587nteger @ A @ B )
% 0.33/0.63 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.63 = ( ( A
% 0.33/0.63 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.63 & ( B
% 0.33/0.63 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and_eq_minus_1_iff
% 0.33/0.63 thf(fact_7284_and__eq__minus__1__iff,axiom,
% 0.33/0.63 ! [A: int,B: int] :
% 0.33/0.63 ( ( ( bit_se725231765392027082nd_int @ A @ B )
% 0.33/0.63 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.63 = ( ( A
% 0.33/0.63 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.63 & ( B
% 0.33/0.63 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % and_eq_minus_1_iff
% 0.33/0.63 thf(fact_7285_positive__imp__inverse__positive,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.63 => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % positive_imp_inverse_positive
% 0.33/0.63 thf(fact_7286_positive__imp__inverse__positive,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.63 => ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % positive_imp_inverse_positive
% 0.33/0.63 thf(fact_7287_negative__imp__inverse__negative,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.63 => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % negative_imp_inverse_negative
% 0.33/0.63 thf(fact_7288_negative__imp__inverse__negative,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.63 => ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % negative_imp_inverse_negative
% 0.33/0.63 thf(fact_7289_inverse__positive__imp__positive,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.33/0.63 => ( ( A != zero_zero_real )
% 0.33/0.63 => ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_positive_imp_positive
% 0.33/0.63 thf(fact_7290_inverse__positive__imp__positive,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.33/0.63 => ( ( A != zero_zero_rat )
% 0.33/0.63 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_positive_imp_positive
% 0.33/0.63 thf(fact_7291_inverse__negative__imp__negative,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.33/0.63 => ( ( A != zero_zero_real )
% 0.33/0.63 => ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_negative_imp_negative
% 0.33/0.63 thf(fact_7292_inverse__negative__imp__negative,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.33/0.63 => ( ( A != zero_zero_rat )
% 0.33/0.63 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_negative_imp_negative
% 0.33/0.63 thf(fact_7293_less__imp__inverse__less__neg,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ord_less_real @ A @ B )
% 0.33/0.63 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.33/0.63 => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_imp_inverse_less_neg
% 0.33/0.63 thf(fact_7294_less__imp__inverse__less__neg,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ord_less_rat @ A @ B )
% 0.33/0.63 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.33/0.63 => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_imp_inverse_less_neg
% 0.33/0.63 thf(fact_7295_inverse__less__imp__less__neg,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.33/0.63 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.33/0.63 => ( ord_less_real @ B @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_less_imp_less_neg
% 0.33/0.63 thf(fact_7296_inverse__less__imp__less__neg,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.33/0.63 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.33/0.63 => ( ord_less_rat @ B @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_less_imp_less_neg
% 0.33/0.63 thf(fact_7297_less__imp__inverse__less,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ord_less_real @ A @ B )
% 0.33/0.63 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.63 => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_imp_inverse_less
% 0.33/0.63 thf(fact_7298_less__imp__inverse__less,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ord_less_rat @ A @ B )
% 0.33/0.63 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.63 => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_imp_inverse_less
% 0.33/0.63 thf(fact_7299_inverse__less__imp__less,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.33/0.63 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.63 => ( ord_less_real @ B @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_less_imp_less
% 0.33/0.63 thf(fact_7300_inverse__less__imp__less,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.33/0.63 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.63 => ( ord_less_rat @ B @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_less_imp_less
% 0.33/0.63 thf(fact_7301_nonzero__inverse__mult__distrib,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( A != zero_zero_real )
% 0.33/0.63 => ( ( B != zero_zero_real )
% 0.33/0.63 => ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
% 0.33/0.63 = ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_inverse_mult_distrib
% 0.33/0.63 thf(fact_7302_nonzero__inverse__mult__distrib,axiom,
% 0.33/0.63 ! [A: complex,B: complex] :
% 0.33/0.63 ( ( A != zero_zero_complex )
% 0.33/0.63 => ( ( B != zero_zero_complex )
% 0.33/0.63 => ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
% 0.33/0.63 = ( times_times_complex @ ( invers8013647133539491842omplex @ B ) @ ( invers8013647133539491842omplex @ A ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_inverse_mult_distrib
% 0.33/0.63 thf(fact_7303_nonzero__inverse__mult__distrib,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( A != zero_zero_rat )
% 0.33/0.63 => ( ( B != zero_zero_rat )
% 0.33/0.63 => ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
% 0.33/0.63 = ( times_times_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_inverse_mult_distrib
% 0.33/0.63 thf(fact_7304_not__bit__Suc__0__Suc,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_bit_Suc_0_Suc
% 0.33/0.63 thf(fact_7305_bit__Suc__0__iff,axiom,
% 0.33/0.63 ! [N: nat] :
% 0.33/0.63 ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.33/0.63 = ( N = zero_zero_nat ) ) ).
% 0.33/0.63
% 0.33/0.63 % bit_Suc_0_iff
% 0.33/0.63 thf(fact_7306_nonzero__inverse__minus__eq,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( A != zero_zero_real )
% 0.33/0.63 => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 0.33/0.63 = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_inverse_minus_eq
% 0.33/0.63 thf(fact_7307_nonzero__inverse__minus__eq,axiom,
% 0.33/0.63 ! [A: complex] :
% 0.33/0.63 ( ( A != zero_zero_complex )
% 0.33/0.63 => ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_inverse_minus_eq
% 0.33/0.63 thf(fact_7308_nonzero__inverse__minus__eq,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( A != zero_zero_rat )
% 0.33/0.63 => ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
% 0.33/0.63 = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_inverse_minus_eq
% 0.33/0.63 thf(fact_7309_inverse__unique,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ( times_times_real @ A @ B )
% 0.33/0.63 = one_one_real )
% 0.33/0.63 => ( ( inverse_inverse_real @ A )
% 0.33/0.63 = B ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_unique
% 0.33/0.63 thf(fact_7310_inverse__unique,axiom,
% 0.33/0.63 ! [A: complex,B: complex] :
% 0.33/0.63 ( ( ( times_times_complex @ A @ B )
% 0.33/0.63 = one_one_complex )
% 0.33/0.63 => ( ( invers8013647133539491842omplex @ A )
% 0.33/0.63 = B ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_unique
% 0.33/0.63 thf(fact_7311_inverse__unique,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ( times_times_rat @ A @ B )
% 0.33/0.63 = one_one_rat )
% 0.33/0.63 => ( ( inverse_inverse_rat @ A )
% 0.33/0.63 = B ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_unique
% 0.33/0.63 thf(fact_7312_inverse__numeral__1,axiom,
% 0.33/0.63 ( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
% 0.33/0.63 = ( numeral_numeral_real @ one ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_numeral_1
% 0.33/0.63 thf(fact_7313_inverse__numeral__1,axiom,
% 0.33/0.63 ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ one ) )
% 0.33/0.63 = ( numera6690914467698888265omplex @ one ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_numeral_1
% 0.33/0.63 thf(fact_7314_inverse__numeral__1,axiom,
% 0.33/0.63 ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ one ) )
% 0.33/0.63 = ( numeral_numeral_rat @ one ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_numeral_1
% 0.33/0.63 thf(fact_7315_divide__inverse__commute,axiom,
% 0.33/0.63 ( divide_divide_real
% 0.33/0.63 = ( ^ [A2: real,B2: real] : ( times_times_real @ ( inverse_inverse_real @ B2 ) @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % divide_inverse_commute
% 0.33/0.63 thf(fact_7316_divide__inverse__commute,axiom,
% 0.33/0.63 ( divide1717551699836669952omplex
% 0.33/0.63 = ( ^ [A2: complex,B2: complex] : ( times_times_complex @ ( invers8013647133539491842omplex @ B2 ) @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % divide_inverse_commute
% 0.33/0.63 thf(fact_7317_divide__inverse__commute,axiom,
% 0.33/0.63 ( divide_divide_rat
% 0.33/0.63 = ( ^ [A2: rat,B2: rat] : ( times_times_rat @ ( inverse_inverse_rat @ B2 ) @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % divide_inverse_commute
% 0.33/0.63 thf(fact_7318_divide__inverse,axiom,
% 0.33/0.63 ( divide_divide_real
% 0.33/0.63 = ( ^ [A2: real,B2: real] : ( times_times_real @ A2 @ ( inverse_inverse_real @ B2 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % divide_inverse
% 0.33/0.63 thf(fact_7319_divide__inverse,axiom,
% 0.33/0.63 ( divide1717551699836669952omplex
% 0.33/0.63 = ( ^ [A2: complex,B2: complex] : ( times_times_complex @ A2 @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % divide_inverse
% 0.33/0.63 thf(fact_7320_divide__inverse,axiom,
% 0.33/0.63 ( divide_divide_rat
% 0.33/0.63 = ( ^ [A2: rat,B2: rat] : ( times_times_rat @ A2 @ ( inverse_inverse_rat @ B2 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % divide_inverse
% 0.33/0.63 thf(fact_7321_field__class_Ofield__divide__inverse,axiom,
% 0.33/0.63 ( divide_divide_real
% 0.33/0.63 = ( ^ [A2: real,B2: real] : ( times_times_real @ A2 @ ( inverse_inverse_real @ B2 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % field_class.field_divide_inverse
% 0.33/0.63 thf(fact_7322_field__class_Ofield__divide__inverse,axiom,
% 0.33/0.63 ( divide1717551699836669952omplex
% 0.33/0.63 = ( ^ [A2: complex,B2: complex] : ( times_times_complex @ A2 @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % field_class.field_divide_inverse
% 0.33/0.63 thf(fact_7323_field__class_Ofield__divide__inverse,axiom,
% 0.33/0.63 ( divide_divide_rat
% 0.33/0.63 = ( ^ [A2: rat,B2: rat] : ( times_times_rat @ A2 @ ( inverse_inverse_rat @ B2 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % field_class.field_divide_inverse
% 0.33/0.63 thf(fact_7324_inverse__eq__divide,axiom,
% 0.33/0.63 ( inverse_inverse_real
% 0.33/0.63 = ( divide_divide_real @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_divide
% 0.33/0.63 thf(fact_7325_inverse__eq__divide,axiom,
% 0.33/0.63 ( invers8013647133539491842omplex
% 0.33/0.63 = ( divide1717551699836669952omplex @ one_one_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_divide
% 0.33/0.63 thf(fact_7326_inverse__eq__divide,axiom,
% 0.33/0.63 ( inverse_inverse_rat
% 0.33/0.63 = ( divide_divide_rat @ one_one_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_eq_divide
% 0.33/0.63 thf(fact_7327_power__mult__inverse__distrib,axiom,
% 0.33/0.63 ! [X: real,M: nat] :
% 0.33/0.63 ( ( times_times_real @ ( power_power_real @ X @ M ) @ ( inverse_inverse_real @ X ) )
% 0.33/0.63 = ( times_times_real @ ( inverse_inverse_real @ X ) @ ( power_power_real @ X @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_mult_inverse_distrib
% 0.33/0.63 thf(fact_7328_power__mult__inverse__distrib,axiom,
% 0.33/0.63 ! [X: complex,M: nat] :
% 0.33/0.63 ( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( invers8013647133539491842omplex @ X ) )
% 0.33/0.63 = ( times_times_complex @ ( invers8013647133539491842omplex @ X ) @ ( power_power_complex @ X @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_mult_inverse_distrib
% 0.33/0.63 thf(fact_7329_power__mult__inverse__distrib,axiom,
% 0.33/0.63 ! [X: rat,M: nat] :
% 0.33/0.63 ( ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( inverse_inverse_rat @ X ) )
% 0.33/0.63 = ( times_times_rat @ ( inverse_inverse_rat @ X ) @ ( power_power_rat @ X @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_mult_inverse_distrib
% 0.33/0.63 thf(fact_7330_power__mult__power__inverse__commute,axiom,
% 0.33/0.63 ! [X: real,M: nat,N: nat] :
% 0.33/0.63 ( ( times_times_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) )
% 0.33/0.63 = ( times_times_real @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) @ ( power_power_real @ X @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_mult_power_inverse_commute
% 0.33/0.63 thf(fact_7331_power__mult__power__inverse__commute,axiom,
% 0.33/0.63 ! [X: complex,M: nat,N: nat] :
% 0.33/0.63 ( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) )
% 0.33/0.63 = ( times_times_complex @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) @ ( power_power_complex @ X @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_mult_power_inverse_commute
% 0.33/0.63 thf(fact_7332_power__mult__power__inverse__commute,axiom,
% 0.33/0.63 ! [X: rat,M: nat,N: nat] :
% 0.33/0.63 ( ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) )
% 0.33/0.63 = ( times_times_rat @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) @ ( power_power_rat @ X @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_mult_power_inverse_commute
% 0.33/0.63 thf(fact_7333_mult__inverse__of__nat__commute,axiom,
% 0.33/0.63 ! [Xa: nat,X: real] :
% 0.33/0.63 ( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) @ X )
% 0.33/0.63 = ( times_times_real @ X @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_inverse_of_nat_commute
% 0.33/0.63 thf(fact_7334_mult__inverse__of__nat__commute,axiom,
% 0.33/0.63 ! [Xa: nat,X: complex] :
% 0.33/0.63 ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) @ X )
% 0.33/0.63 = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_inverse_of_nat_commute
% 0.33/0.63 thf(fact_7335_mult__inverse__of__nat__commute,axiom,
% 0.33/0.63 ! [Xa: nat,X: rat] :
% 0.33/0.63 ( ( times_times_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) @ X )
% 0.33/0.63 = ( times_times_rat @ X @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_inverse_of_nat_commute
% 0.33/0.63 thf(fact_7336_nonzero__abs__inverse,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( A != zero_zero_real )
% 0.33/0.63 => ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
% 0.33/0.63 = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_abs_inverse
% 0.33/0.63 thf(fact_7337_nonzero__abs__inverse,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( A != zero_zero_rat )
% 0.33/0.63 => ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
% 0.33/0.63 = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_abs_inverse
% 0.33/0.63 thf(fact_7338_mult__inverse__of__int__commute,axiom,
% 0.33/0.63 ! [Xa: int,X: real] :
% 0.33/0.63 ( ( times_times_real @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) @ X )
% 0.33/0.63 = ( times_times_real @ X @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_inverse_of_int_commute
% 0.33/0.63 thf(fact_7339_mult__inverse__of__int__commute,axiom,
% 0.33/0.63 ! [Xa: int,X: complex] :
% 0.33/0.63 ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) @ X )
% 0.33/0.63 = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_inverse_of_int_commute
% 0.33/0.63 thf(fact_7340_mult__inverse__of__int__commute,axiom,
% 0.33/0.63 ! [Xa: int,X: rat] :
% 0.33/0.63 ( ( times_times_rat @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) @ X )
% 0.33/0.63 = ( times_times_rat @ X @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_inverse_of_int_commute
% 0.33/0.63 thf(fact_7341_AND__lower,axiom,
% 0.33/0.63 ! [X: int,Y: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.63 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % AND_lower
% 0.33/0.63 thf(fact_7342_AND__upper1,axiom,
% 0.33/0.63 ! [X: int,Y: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.63 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % AND_upper1
% 0.33/0.63 thf(fact_7343_AND__upper2,axiom,
% 0.33/0.63 ! [Y: int,X: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.63 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % AND_upper2
% 0.33/0.63 thf(fact_7344_AND__upper1_H,axiom,
% 0.33/0.63 ! [Y: int,Z2: int,Ya: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.63 => ( ( ord_less_eq_int @ Y @ Z2 )
% 0.33/0.63 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % AND_upper1'
% 0.33/0.63 thf(fact_7345_AND__upper2_H,axiom,
% 0.33/0.63 ! [Y: int,Z2: int,X: int] :
% 0.33/0.63 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.63 => ( ( ord_less_eq_int @ Y @ Z2 )
% 0.33/0.63 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % AND_upper2'
% 0.33/0.63 thf(fact_7346_divide__real__def,axiom,
% 0.33/0.63 ( divide_divide_real
% 0.33/0.63 = ( ^ [X3: real,Y6: real] : ( times_times_real @ X3 @ ( inverse_inverse_real @ Y6 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % divide_real_def
% 0.33/0.63 thf(fact_7347_take__bit__eq__mask,axiom,
% 0.33/0.63 ( bit_se2923211474154528505it_int
% 0.33/0.63 = ( ^ [N3: nat,A2: int] : ( bit_se725231765392027082nd_int @ A2 @ ( bit_se2000444600071755411sk_int @ N3 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % take_bit_eq_mask
% 0.33/0.63 thf(fact_7348_take__bit__eq__mask,axiom,
% 0.33/0.63 ( bit_se2925701944663578781it_nat
% 0.33/0.63 = ( ^ [N3: nat,A2: nat] : ( bit_se727722235901077358nd_nat @ A2 @ ( bit_se2002935070580805687sk_nat @ N3 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % take_bit_eq_mask
% 0.33/0.63 thf(fact_7349_DeMoivre,axiom,
% 0.33/0.63 ! [A: real,N: nat] :
% 0.33/0.63 ( ( power_power_complex @ ( cis @ A ) @ N )
% 0.33/0.63 = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % DeMoivre
% 0.33/0.63 thf(fact_7350_cis__mult,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
% 0.33/0.63 = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % cis_mult
% 0.33/0.63 thf(fact_7351_i__times__eq__iff,axiom,
% 0.33/0.63 ! [W: complex,Z2: complex] :
% 0.33/0.63 ( ( ( times_times_complex @ imaginary_unit @ W )
% 0.33/0.63 = Z2 )
% 0.33/0.63 = ( W
% 0.33/0.63 = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z2 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % i_times_eq_iff
% 0.33/0.63 thf(fact_7352_inverse__le__imp__le,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.33/0.63 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.63 => ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_le_imp_le
% 0.33/0.63 thf(fact_7353_inverse__le__imp__le,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.33/0.63 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.63 => ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_le_imp_le
% 0.33/0.63 thf(fact_7354_le__imp__inverse__le,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ A @ B )
% 0.33/0.63 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.63 => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_imp_inverse_le
% 0.33/0.63 thf(fact_7355_le__imp__inverse__le,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.63 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.63 => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_imp_inverse_le
% 0.33/0.63 thf(fact_7356_inverse__le__imp__le__neg,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.33/0.63 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.33/0.63 => ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_le_imp_le_neg
% 0.33/0.63 thf(fact_7357_inverse__le__imp__le__neg,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.33/0.63 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.33/0.63 => ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_le_imp_le_neg
% 0.33/0.63 thf(fact_7358_le__imp__inverse__le__neg,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ A @ B )
% 0.33/0.63 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.33/0.63 => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_imp_inverse_le_neg
% 0.33/0.63 thf(fact_7359_le__imp__inverse__le__neg,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.63 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.33/0.63 => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_imp_inverse_le_neg
% 0.33/0.63 thf(fact_7360_inverse__le__1__iff,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( inverse_inverse_real @ X ) @ one_one_real )
% 0.33/0.63 = ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.63 | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_le_1_iff
% 0.33/0.63 thf(fact_7361_inverse__le__1__iff,axiom,
% 0.33/0.63 ! [X: rat] :
% 0.33/0.63 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
% 0.33/0.63 = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.33/0.63 | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % inverse_le_1_iff
% 0.33/0.63 thf(fact_7362_one__less__inverse__iff,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X ) )
% 0.33/0.63 = ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 & ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % one_less_inverse_iff
% 0.33/0.63 thf(fact_7363_one__less__inverse__iff,axiom,
% 0.33/0.63 ! [X: rat] :
% 0.33/0.63 ( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
% 0.33/0.63 = ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.33/0.63 & ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % one_less_inverse_iff
% 0.33/0.63 thf(fact_7364_one__less__inverse,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.63 => ( ( ord_less_real @ A @ one_one_real )
% 0.33/0.63 => ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % one_less_inverse
% 0.33/0.63 thf(fact_7365_one__less__inverse,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.63 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.33/0.63 => ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % one_less_inverse
% 0.33/0.63 thf(fact_7366_field__class_Ofield__inverse,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( A != zero_zero_real )
% 0.33/0.63 => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 0.33/0.63 = one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % field_class.field_inverse
% 0.33/0.63 thf(fact_7367_field__class_Ofield__inverse,axiom,
% 0.33/0.63 ! [A: complex] :
% 0.33/0.63 ( ( A != zero_zero_complex )
% 0.33/0.63 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 0.33/0.63 = one_one_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % field_class.field_inverse
% 0.33/0.63 thf(fact_7368_field__class_Ofield__inverse,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( A != zero_zero_rat )
% 0.33/0.63 => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 0.33/0.63 = one_one_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % field_class.field_inverse
% 0.33/0.63 thf(fact_7369_division__ring__inverse__add,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( A != zero_zero_real )
% 0.33/0.63 => ( ( B != zero_zero_real )
% 0.33/0.63 => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.33/0.63 = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % division_ring_inverse_add
% 0.33/0.63 thf(fact_7370_division__ring__inverse__add,axiom,
% 0.33/0.63 ! [A: complex,B: complex] :
% 0.33/0.63 ( ( A != zero_zero_complex )
% 0.33/0.64 => ( ( B != zero_zero_complex )
% 0.33/0.64 => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 0.33/0.64 = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( plus_plus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % division_ring_inverse_add
% 0.33/0.64 thf(fact_7371_division__ring__inverse__add,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( A != zero_zero_rat )
% 0.33/0.64 => ( ( B != zero_zero_rat )
% 0.33/0.64 => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.33/0.64 = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % division_ring_inverse_add
% 0.33/0.64 thf(fact_7372_inverse__add,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( A != zero_zero_real )
% 0.33/0.64 => ( ( B != zero_zero_real )
% 0.33/0.64 => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.33/0.64 = ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_add
% 0.33/0.64 thf(fact_7373_inverse__add,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( A != zero_zero_complex )
% 0.33/0.64 => ( ( B != zero_zero_complex )
% 0.33/0.64 => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 0.33/0.64 = ( times_times_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( invers8013647133539491842omplex @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_add
% 0.33/0.64 thf(fact_7374_inverse__add,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( A != zero_zero_rat )
% 0.33/0.64 => ( ( B != zero_zero_rat )
% 0.33/0.64 => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.33/0.64 = ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_add
% 0.33/0.64 thf(fact_7375_division__ring__inverse__diff,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( A != zero_zero_real )
% 0.33/0.64 => ( ( B != zero_zero_real )
% 0.33/0.64 => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.33/0.64 = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % division_ring_inverse_diff
% 0.33/0.64 thf(fact_7376_division__ring__inverse__diff,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( A != zero_zero_complex )
% 0.33/0.64 => ( ( B != zero_zero_complex )
% 0.33/0.64 => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 0.33/0.64 = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ B @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % division_ring_inverse_diff
% 0.33/0.64 thf(fact_7377_division__ring__inverse__diff,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( A != zero_zero_rat )
% 0.33/0.64 => ( ( B != zero_zero_rat )
% 0.33/0.64 => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.33/0.64 = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ B @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % division_ring_inverse_diff
% 0.33/0.64 thf(fact_7378_nonzero__inverse__eq__divide,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( A != zero_zero_real )
% 0.33/0.64 => ( ( inverse_inverse_real @ A )
% 0.33/0.64 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % nonzero_inverse_eq_divide
% 0.33/0.64 thf(fact_7379_nonzero__inverse__eq__divide,axiom,
% 0.33/0.64 ! [A: complex] :
% 0.33/0.64 ( ( A != zero_zero_complex )
% 0.33/0.64 => ( ( invers8013647133539491842omplex @ A )
% 0.33/0.64 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % nonzero_inverse_eq_divide
% 0.33/0.64 thf(fact_7380_nonzero__inverse__eq__divide,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( A != zero_zero_rat )
% 0.33/0.64 => ( ( inverse_inverse_rat @ A )
% 0.33/0.64 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % nonzero_inverse_eq_divide
% 0.33/0.64 thf(fact_7381_not__bit__Suc__0__numeral,axiom,
% 0.33/0.64 ! [N: num] :
% 0.33/0.64 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 0.33/0.64
% 0.33/0.64 % not_bit_Suc_0_numeral
% 0.33/0.64 thf(fact_7382_and__less__eq,axiom,
% 0.33/0.64 ! [L: int,K: int] :
% 0.33/0.64 ( ( ord_less_int @ L @ zero_zero_int )
% 0.33/0.64 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_less_eq
% 0.33/0.64 thf(fact_7383_AND__upper1_H_H,axiom,
% 0.33/0.64 ! [Y: int,Z2: int,Ya: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.64 => ( ( ord_less_int @ Y @ Z2 )
% 0.33/0.64 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % AND_upper1''
% 0.33/0.64 thf(fact_7384_AND__upper2_H_H,axiom,
% 0.33/0.64 ! [Y: int,Z2: int,X: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.64 => ( ( ord_less_int @ Y @ Z2 )
% 0.33/0.64 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % AND_upper2''
% 0.33/0.64 thf(fact_7385_bit__push__bit__iff__nat,axiom,
% 0.33/0.64 ! [M: nat,Q2: nat,N: nat] :
% 0.33/0.64 ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
% 0.33/0.64 = ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.64 & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit_push_bit_iff_nat
% 0.33/0.64 thf(fact_7386_imaginary__unit_Ocode,axiom,
% 0.33/0.64 ( imaginary_unit
% 0.33/0.64 = ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % imaginary_unit.code
% 0.33/0.64 thf(fact_7387_Complex__eq__i,axiom,
% 0.33/0.64 ! [X: real,Y: real] :
% 0.33/0.64 ( ( ( complex2 @ X @ Y )
% 0.33/0.64 = imaginary_unit )
% 0.33/0.64 = ( ( X = zero_zero_real )
% 0.33/0.64 & ( Y = one_one_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Complex_eq_i
% 0.33/0.64 thf(fact_7388_even__and__iff,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ A @ B ) )
% 0.33/0.64 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.33/0.64 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % even_and_iff
% 0.33/0.64 thf(fact_7389_even__and__iff,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 0.33/0.64 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.33/0.64 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % even_and_iff
% 0.33/0.64 thf(fact_7390_even__and__iff,axiom,
% 0.33/0.64 ! [A: nat,B: nat] :
% 0.33/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 0.33/0.64 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.33/0.64 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % even_and_iff
% 0.33/0.64 thf(fact_7391_inverse__less__iff,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.33/0.64 = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.33/0.64 => ( ord_less_real @ B @ A ) )
% 0.33/0.64 & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.33/0.64 => ( ord_less_real @ A @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_less_iff
% 0.33/0.64 thf(fact_7392_inverse__less__iff,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.33/0.64 = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.33/0.64 => ( ord_less_rat @ B @ A ) )
% 0.33/0.64 & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.33/0.64 => ( ord_less_rat @ A @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_less_iff
% 0.33/0.64 thf(fact_7393_inverse__le__iff,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.33/0.64 = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.33/0.64 => ( ord_less_eq_real @ B @ A ) )
% 0.33/0.64 & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.33/0.64 => ( ord_less_eq_real @ A @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_le_iff
% 0.33/0.64 thf(fact_7394_inverse__le__iff,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.33/0.64 = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.33/0.64 => ( ord_less_eq_rat @ B @ A ) )
% 0.33/0.64 & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.33/0.64 => ( ord_less_eq_rat @ A @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_le_iff
% 0.33/0.64 thf(fact_7395_one__le__inverse__iff,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X ) )
% 0.33/0.64 = ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.64 & ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % one_le_inverse_iff
% 0.33/0.64 thf(fact_7396_one__le__inverse__iff,axiom,
% 0.33/0.64 ! [X: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
% 0.33/0.64 = ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.33/0.64 & ( ord_less_eq_rat @ X @ one_one_rat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % one_le_inverse_iff
% 0.33/0.64 thf(fact_7397_inverse__less__1__iff,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_real @ ( inverse_inverse_real @ X ) @ one_one_real )
% 0.33/0.64 = ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.64 | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_less_1_iff
% 0.33/0.64 thf(fact_7398_inverse__less__1__iff,axiom,
% 0.33/0.64 ! [X: rat] :
% 0.33/0.64 ( ( ord_less_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
% 0.33/0.64 = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.33/0.64 | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_less_1_iff
% 0.33/0.64 thf(fact_7399_one__le__inverse,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.64 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.33/0.64 => ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % one_le_inverse
% 0.33/0.64 thf(fact_7400_one__le__inverse,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.64 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.33/0.64 => ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % one_le_inverse
% 0.33/0.64 thf(fact_7401_inverse__diff__inverse,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( A != zero_zero_real )
% 0.33/0.64 => ( ( B != zero_zero_real )
% 0.33/0.64 => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.33/0.64 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_diff_inverse
% 0.33/0.64 thf(fact_7402_inverse__diff__inverse,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( A != zero_zero_complex )
% 0.33/0.64 => ( ( B != zero_zero_complex )
% 0.33/0.64 => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 0.33/0.64 = ( uminus1482373934393186551omplex @ ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_diff_inverse
% 0.33/0.64 thf(fact_7403_inverse__diff__inverse,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( A != zero_zero_rat )
% 0.33/0.64 => ( ( B != zero_zero_rat )
% 0.33/0.64 => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.33/0.64 = ( uminus_uminus_rat @ ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_diff_inverse
% 0.33/0.64 thf(fact_7404_reals__Archimedean,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.64 => ? [N2: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % reals_Archimedean
% 0.33/0.64 thf(fact_7405_reals__Archimedean,axiom,
% 0.33/0.64 ! [X: rat] :
% 0.33/0.64 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.33/0.64 => ? [N2: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % reals_Archimedean
% 0.33/0.64 thf(fact_7406_i__mult__Complex,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B ) )
% 0.33/0.64 = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % i_mult_Complex
% 0.33/0.64 thf(fact_7407_Complex__mult__i,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( times_times_complex @ ( complex2 @ A @ B ) @ imaginary_unit )
% 0.33/0.64 = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % Complex_mult_i
% 0.33/0.64 thf(fact_7408_bit__iff__and__push__bit__not__eq__0,axiom,
% 0.33/0.64 ( bit_se1148574629649215175it_nat
% 0.33/0.64 = ( ^ [A2: nat,N3: nat] :
% 0.33/0.64 ( ( bit_se727722235901077358nd_nat @ A2 @ ( bit_se547839408752420682it_nat @ N3 @ one_one_nat ) )
% 0.33/0.64 != zero_zero_nat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit_iff_and_push_bit_not_eq_0
% 0.33/0.64 thf(fact_7409_bit__iff__and__push__bit__not__eq__0,axiom,
% 0.33/0.64 ( bit_se1146084159140164899it_int
% 0.33/0.64 = ( ^ [A2: int,N3: nat] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ A2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) )
% 0.33/0.64 != zero_zero_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit_iff_and_push_bit_not_eq_0
% 0.33/0.64 thf(fact_7410_bit__iff__and__push__bit__not__eq__0,axiom,
% 0.33/0.64 ( bit_se9216721137139052372nteger
% 0.33/0.64 = ( ^ [A2: code_integer,N3: nat] :
% 0.33/0.64 ( ( bit_se3949692690581998587nteger @ A2 @ ( bit_se7788150548672797655nteger @ N3 @ one_one_Code_integer ) )
% 0.33/0.64 != zero_z3403309356797280102nteger ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit_iff_and_push_bit_not_eq_0
% 0.33/0.64 thf(fact_7411_bit__iff__and__push__bit__not__eq__0,axiom,
% 0.33/0.64 ( bit_se8040316288895769887atural
% 0.33/0.64 = ( ^ [A2: code_natural,N3: nat] :
% 0.33/0.64 ( ( bit_se2773287842338716102atural @ A2 @ ( bit_se6611745700429515170atural @ N3 @ one_one_Code_natural ) )
% 0.33/0.64 != zero_z2226904508553997617atural ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit_iff_and_push_bit_not_eq_0
% 0.33/0.64 thf(fact_7412_even__and__iff__int,axiom,
% 0.33/0.64 ! [K: int,L: int] :
% 0.33/0.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 0.33/0.64 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.33/0.64 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % even_and_iff_int
% 0.33/0.64 thf(fact_7413_forall__pos__mono__1,axiom,
% 0.33/0.64 ! [P: real > $o,E: real] :
% 0.33/0.64 ( ! [D2: real,E2: real] :
% 0.33/0.64 ( ( ord_less_real @ D2 @ E2 )
% 0.33/0.64 => ( ( P @ D2 )
% 0.33/0.64 => ( P @ E2 ) ) )
% 0.33/0.64 => ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
% 0.33/0.64 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.33/0.64 => ( P @ E ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % forall_pos_mono_1
% 0.33/0.64 thf(fact_7414_forall__pos__mono,axiom,
% 0.33/0.64 ! [P: real > $o,E: real] :
% 0.33/0.64 ( ! [D2: real,E2: real] :
% 0.33/0.64 ( ( ord_less_real @ D2 @ E2 )
% 0.33/0.64 => ( ( P @ D2 )
% 0.33/0.64 => ( P @ E2 ) ) )
% 0.33/0.64 => ( ! [N2: nat] :
% 0.33/0.64 ( ( N2 != zero_zero_nat )
% 0.33/0.64 => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
% 0.33/0.64 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.33/0.64 => ( P @ E ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % forall_pos_mono
% 0.33/0.64 thf(fact_7415_real__arch__inverse,axiom,
% 0.33/0.64 ! [E: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ E )
% 0.33/0.64 = ( ? [N3: nat] :
% 0.33/0.64 ( ( N3 != zero_zero_nat )
% 0.33/0.64 & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 0.33/0.64 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % real_arch_inverse
% 0.33/0.64 thf(fact_7416_bit__nat__iff,axiom,
% 0.33/0.64 ! [K: int,N: nat] :
% 0.33/0.64 ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
% 0.33/0.64 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.64 & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit_nat_iff
% 0.33/0.64 thf(fact_7417_sqrt__divide__self__eq,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
% 0.33/0.64 = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sqrt_divide_self_eq
% 0.33/0.64 thf(fact_7418_and__one__eq,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
% 0.33/0.64 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_one_eq
% 0.33/0.64 thf(fact_7419_and__one__eq,axiom,
% 0.33/0.64 ! [A: code_natural] :
% 0.33/0.64 ( ( bit_se2773287842338716102atural @ A @ one_one_Code_natural )
% 0.33/0.64 = ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_one_eq
% 0.33/0.64 thf(fact_7420_and__one__eq,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ A @ one_one_int )
% 0.33/0.64 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_one_eq
% 0.33/0.64 thf(fact_7421_and__one__eq,axiom,
% 0.33/0.64 ! [A: nat] :
% 0.33/0.64 ( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
% 0.33/0.64 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_one_eq
% 0.33/0.64 thf(fact_7422_one__and__eq,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
% 0.33/0.64 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % one_and_eq
% 0.33/0.64 thf(fact_7423_one__and__eq,axiom,
% 0.33/0.64 ! [A: code_natural] :
% 0.33/0.64 ( ( bit_se2773287842338716102atural @ one_one_Code_natural @ A )
% 0.33/0.64 = ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % one_and_eq
% 0.33/0.64 thf(fact_7424_one__and__eq,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ one_one_int @ A )
% 0.33/0.64 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % one_and_eq
% 0.33/0.64 thf(fact_7425_one__and__eq,axiom,
% 0.33/0.64 ! [A: nat] :
% 0.33/0.64 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
% 0.33/0.64 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % one_and_eq
% 0.33/0.64 thf(fact_7426_ex__inverse__of__nat__less,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.64 => ? [N2: nat] :
% 0.33/0.64 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.64 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % ex_inverse_of_nat_less
% 0.33/0.64 thf(fact_7427_ex__inverse__of__nat__less,axiom,
% 0.33/0.64 ! [X: rat] :
% 0.33/0.64 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.33/0.64 => ? [N2: nat] :
% 0.33/0.64 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.64 & ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % ex_inverse_of_nat_less
% 0.33/0.64 thf(fact_7428_power__diff__conv__inverse,axiom,
% 0.33/0.64 ! [X: real,M: nat,N: nat] :
% 0.33/0.64 ( ( X != zero_zero_real )
% 0.33/0.64 => ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.64 => ( ( power_power_real @ X @ ( minus_minus_nat @ N @ M ) )
% 0.33/0.64 = ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ M ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_diff_conv_inverse
% 0.33/0.64 thf(fact_7429_power__diff__conv__inverse,axiom,
% 0.33/0.64 ! [X: complex,M: nat,N: nat] :
% 0.33/0.64 ( ( X != zero_zero_complex )
% 0.33/0.64 => ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.64 => ( ( power_power_complex @ X @ ( minus_minus_nat @ N @ M ) )
% 0.33/0.64 = ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ M ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_diff_conv_inverse
% 0.33/0.64 thf(fact_7430_power__diff__conv__inverse,axiom,
% 0.33/0.64 ! [X: rat,M: nat,N: nat] :
% 0.33/0.64 ( ( X != zero_zero_rat )
% 0.33/0.64 => ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.64 => ( ( power_power_rat @ X @ ( minus_minus_nat @ N @ M ) )
% 0.33/0.64 = ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ M ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_diff_conv_inverse
% 0.33/0.64 thf(fact_7431_complex__of__real__i,axiom,
% 0.33/0.64 ! [R: real] :
% 0.33/0.64 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ imaginary_unit )
% 0.33/0.64 = ( complex2 @ zero_zero_real @ R ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_of_real_i
% 0.33/0.64 thf(fact_7432_i__complex__of__real,axiom,
% 0.33/0.64 ! [R: real] :
% 0.33/0.64 ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R ) )
% 0.33/0.64 = ( complex2 @ zero_zero_real @ R ) ) ).
% 0.33/0.64
% 0.33/0.64 % i_complex_of_real
% 0.33/0.64 thf(fact_7433_log__inverse,axiom,
% 0.33/0.64 ! [A: real,X: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.64 => ( ( A != one_one_real )
% 0.33/0.64 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ( log2 @ A @ ( inverse_inverse_real @ X ) )
% 0.33/0.64 = ( uminus_uminus_real @ ( log2 @ A @ X ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % log_inverse
% 0.33/0.64 thf(fact_7434_Complex__eq,axiom,
% 0.33/0.64 ( complex2
% 0.33/0.64 = ( ^ [A2: real,B2: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A2 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Complex_eq
% 0.33/0.64 thf(fact_7435_and__exp__eq__0__iff__not__bit,axiom,
% 0.33/0.64 ! [A: int,N: nat] :
% 0.33/0.64 ( ( ( bit_se725231765392027082nd_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.64 = zero_zero_int )
% 0.33/0.64 = ( ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_exp_eq_0_iff_not_bit
% 0.33/0.64 thf(fact_7436_and__exp__eq__0__iff__not__bit,axiom,
% 0.33/0.64 ! [A: nat,N: nat] :
% 0.33/0.64 ( ( ( bit_se727722235901077358nd_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.64 = zero_zero_nat )
% 0.33/0.64 = ( ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_exp_eq_0_iff_not_bit
% 0.33/0.64 thf(fact_7437_bit__nat__def,axiom,
% 0.33/0.64 ( bit_se1148574629649215175it_nat
% 0.33/0.64 = ( ^ [M3: nat,N3: nat] :
% 0.33/0.64 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit_nat_def
% 0.33/0.64 thf(fact_7438_exp__plus__inverse__exp,axiom,
% 0.33/0.64 ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % exp_plus_inverse_exp
% 0.33/0.64 thf(fact_7439_complex__split__polar,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ? [R2: real,A4: real] :
% 0.33/0.64 ( Z2
% 0.33/0.64 = ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A4 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A4 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_split_polar
% 0.33/0.64 thf(fact_7440_plus__inverse__ge__2,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % plus_inverse_ge_2
% 0.33/0.64 thf(fact_7441_real__inv__sqrt__pow2,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( inverse_inverse_real @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % real_inv_sqrt_pow2
% 0.33/0.64 thf(fact_7442_tan__cot,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 0.33/0.64 = ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % tan_cot
% 0.33/0.64 thf(fact_7443_and__int__rec,axiom,
% 0.33/0.64 ( bit_se725231765392027082nd_int
% 0.33/0.64 = ( ^ [K2: int,L2: int] :
% 0.33/0.64 ( plus_plus_int
% 0.33/0.64 @ ( zero_n2684676970156552555ol_int
% 0.33/0.64 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.33/0.64 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.33/0.64 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_int_rec
% 0.33/0.64 thf(fact_7444_real__le__x__sinh,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % real_le_x_sinh
% 0.33/0.64 thf(fact_7445_real__le__abs__sinh,axiom,
% 0.33/0.64 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % real_le_abs_sinh
% 0.33/0.64 thf(fact_7446_tan__sec,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ( cos_real @ X )
% 0.33/0.64 != zero_zero_real )
% 0.33/0.64 => ( ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( power_power_real @ ( inverse_inverse_real @ ( cos_real @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % tan_sec
% 0.33/0.64 thf(fact_7447_tan__sec,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( ( cos_complex @ X )
% 0.33/0.64 != zero_zero_complex )
% 0.33/0.64 => ( ( plus_plus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( power_power_complex @ ( invers8013647133539491842omplex @ ( cos_complex @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % tan_sec
% 0.33/0.64 thf(fact_7448_cmod__unit__one,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
% 0.33/0.64 = one_one_real ) ).
% 0.33/0.64
% 0.33/0.64 % cmod_unit_one
% 0.33/0.64 thf(fact_7449_cmod__complex__polar,axiom,
% 0.33/0.64 ! [R: real,A: real] :
% 0.33/0.64 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
% 0.33/0.64 = ( abs_abs_real @ R ) ) ).
% 0.33/0.64
% 0.33/0.64 % cmod_complex_polar
% 0.33/0.64 thf(fact_7450_Arg__minus__ii,axiom,
% 0.33/0.64 ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
% 0.33/0.64 = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Arg_minus_ii
% 0.33/0.64 thf(fact_7451_csqrt__ii,axiom,
% 0.33/0.64 ( ( csqrt @ imaginary_unit )
% 0.33/0.64 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % csqrt_ii
% 0.33/0.64 thf(fact_7452_Arg__ii,axiom,
% 0.33/0.64 ( ( arg @ imaginary_unit )
% 0.33/0.64 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Arg_ii
% 0.33/0.64 thf(fact_7453_sinh__ln__real,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ( sinh_real @ ( ln_ln_real @ X ) )
% 0.33/0.64 = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sinh_ln_real
% 0.33/0.64 thf(fact_7454_cis__multiple__2pi,axiom,
% 0.33/0.64 ! [N: real] :
% 0.33/0.64 ( ( member_real @ N @ ring_1_Ints_real )
% 0.33/0.64 => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.33/0.64 = one_one_complex ) ) ).
% 0.33/0.64
% 0.33/0.64 % cis_multiple_2pi
% 0.33/0.64 thf(fact_7455_sinh__0,axiom,
% 0.33/0.64 ( ( sinh_real @ zero_zero_real )
% 0.33/0.64 = zero_zero_real ) ).
% 0.33/0.64
% 0.33/0.64 % sinh_0
% 0.33/0.64 thf(fact_7456_frac__in__Ints__iff,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( member_real @ ( archim2898591450579166408c_real @ X ) @ ring_1_Ints_real )
% 0.33/0.64 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % frac_in_Ints_iff
% 0.33/0.64 thf(fact_7457_csqrt__eq__1,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( ( csqrt @ Z2 )
% 0.33/0.64 = one_one_complex )
% 0.33/0.64 = ( Z2 = one_one_complex ) ) ).
% 0.33/0.64
% 0.33/0.64 % csqrt_eq_1
% 0.33/0.64 thf(fact_7458_csqrt__1,axiom,
% 0.33/0.64 ( ( csqrt @ one_one_complex )
% 0.33/0.64 = one_one_complex ) ).
% 0.33/0.64
% 0.33/0.64 % csqrt_1
% 0.33/0.64 thf(fact_7459_frac__eq__0__iff,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ( archim2898591450579166408c_real @ X )
% 0.33/0.64 = zero_zero_real )
% 0.33/0.64 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % frac_eq_0_iff
% 0.33/0.64 thf(fact_7460_frac__eq__0__iff,axiom,
% 0.33/0.64 ! [X: rat] :
% 0.33/0.64 ( ( ( archimedean_frac_rat @ X )
% 0.33/0.64 = zero_zero_rat )
% 0.33/0.64 = ( member_rat @ X @ ring_1_Ints_rat ) ) ).
% 0.33/0.64
% 0.33/0.64 % frac_eq_0_iff
% 0.33/0.64 thf(fact_7461_floor__add2,axiom,
% 0.33/0.64 ! [X: real,Y: real] :
% 0.33/0.64 ( ( ( member_real @ X @ ring_1_Ints_real )
% 0.33/0.64 | ( member_real @ Y @ ring_1_Ints_real ) )
% 0.33/0.64 => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.64 = ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % floor_add2
% 0.33/0.64 thf(fact_7462_floor__add2,axiom,
% 0.33/0.64 ! [X: rat,Y: rat] :
% 0.33/0.64 ( ( ( member_rat @ X @ ring_1_Ints_rat )
% 0.33/0.64 | ( member_rat @ Y @ ring_1_Ints_rat ) )
% 0.33/0.64 => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
% 0.33/0.64 = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % floor_add2
% 0.33/0.64 thf(fact_7463_frac__gt__0__iff,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) )
% 0.33/0.64 = ( ~ ( member_real @ X @ ring_1_Ints_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % frac_gt_0_iff
% 0.33/0.64 thf(fact_7464_frac__gt__0__iff,axiom,
% 0.33/0.64 ! [X: rat] :
% 0.33/0.64 ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) )
% 0.33/0.64 = ( ~ ( member_rat @ X @ ring_1_Ints_rat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % frac_gt_0_iff
% 0.33/0.64 thf(fact_7465_and__nat__numerals_I1_J,axiom,
% 0.33/0.64 ! [Y: num] :
% 0.33/0.64 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.33/0.64 = zero_zero_nat ) ).
% 0.33/0.64
% 0.33/0.64 % and_nat_numerals(1)
% 0.33/0.64 thf(fact_7466_and__nat__numerals_I3_J,axiom,
% 0.33/0.64 ! [X: num] :
% 0.33/0.64 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.33/0.64 = zero_zero_nat ) ).
% 0.33/0.64
% 0.33/0.64 % and_nat_numerals(3)
% 0.33/0.64 thf(fact_7467_power2__csqrt,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( power_power_complex @ ( csqrt @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = Z2 ) ).
% 0.33/0.64
% 0.33/0.64 % power2_csqrt
% 0.33/0.64 thf(fact_7468_and__nat__numerals_I2_J,axiom,
% 0.33/0.64 ! [Y: num] :
% 0.33/0.64 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.33/0.64 = one_one_nat ) ).
% 0.33/0.64
% 0.33/0.64 % and_nat_numerals(2)
% 0.33/0.64 thf(fact_7469_and__nat__numerals_I4_J,axiom,
% 0.33/0.64 ! [X: num] :
% 0.33/0.64 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.33/0.64 = one_one_nat ) ).
% 0.33/0.64
% 0.33/0.64 % and_nat_numerals(4)
% 0.33/0.64 thf(fact_7470_and__Suc__0__eq,axiom,
% 0.33/0.64 ! [N: nat] :
% 0.33/0.64 ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.33/0.64 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_Suc_0_eq
% 0.33/0.64 thf(fact_7471_Suc__0__and__eq,axiom,
% 0.33/0.64 ! [N: nat] :
% 0.33/0.64 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.33/0.64 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Suc_0_and_eq
% 0.33/0.64 thf(fact_7472_Ints__0,axiom,
% 0.33/0.64 member_complex @ zero_zero_complex @ ring_1_Ints_complex ).
% 0.33/0.64
% 0.33/0.64 % Ints_0
% 0.33/0.64 thf(fact_7473_Ints__0,axiom,
% 0.33/0.64 member_real @ zero_zero_real @ ring_1_Ints_real ).
% 0.33/0.64
% 0.33/0.64 % Ints_0
% 0.33/0.64 thf(fact_7474_Ints__0,axiom,
% 0.33/0.64 member_rat @ zero_zero_rat @ ring_1_Ints_rat ).
% 0.33/0.64
% 0.33/0.64 % Ints_0
% 0.33/0.64 thf(fact_7475_Ints__0,axiom,
% 0.33/0.64 member_int @ zero_zero_int @ ring_1_Ints_int ).
% 0.33/0.64
% 0.33/0.64 % Ints_0
% 0.33/0.64 thf(fact_7476_Ints__numeral,axiom,
% 0.33/0.64 ! [N: num] : ( member_complex @ ( numera6690914467698888265omplex @ N ) @ ring_1_Ints_complex ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_numeral
% 0.33/0.64 thf(fact_7477_Ints__numeral,axiom,
% 0.33/0.64 ! [N: num] : ( member_real @ ( numeral_numeral_real @ N ) @ ring_1_Ints_real ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_numeral
% 0.33/0.64 thf(fact_7478_Ints__numeral,axiom,
% 0.33/0.64 ! [N: num] : ( member_rat @ ( numeral_numeral_rat @ N ) @ ring_1_Ints_rat ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_numeral
% 0.33/0.64 thf(fact_7479_Ints__numeral,axiom,
% 0.33/0.64 ! [N: num] : ( member_int @ ( numeral_numeral_int @ N ) @ ring_1_Ints_int ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_numeral
% 0.33/0.64 thf(fact_7480_Ints__mult,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.33/0.64 => ( ( member_complex @ B @ ring_1_Ints_complex )
% 0.33/0.64 => ( member_complex @ ( times_times_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_mult
% 0.33/0.64 thf(fact_7481_Ints__mult,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( member_real @ A @ ring_1_Ints_real )
% 0.33/0.64 => ( ( member_real @ B @ ring_1_Ints_real )
% 0.33/0.64 => ( member_real @ ( times_times_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_mult
% 0.33/0.64 thf(fact_7482_Ints__mult,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.33/0.64 => ( ( member_rat @ B @ ring_1_Ints_rat )
% 0.33/0.64 => ( member_rat @ ( times_times_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_mult
% 0.33/0.64 thf(fact_7483_Ints__mult,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( member_int @ A @ ring_1_Ints_int )
% 0.33/0.64 => ( ( member_int @ B @ ring_1_Ints_int )
% 0.33/0.64 => ( member_int @ ( times_times_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_mult
% 0.33/0.64 thf(fact_7484_Ints__1,axiom,
% 0.33/0.64 member_complex @ one_one_complex @ ring_1_Ints_complex ).
% 0.33/0.64
% 0.33/0.64 % Ints_1
% 0.33/0.64 thf(fact_7485_Ints__1,axiom,
% 0.33/0.64 member_rat @ one_one_rat @ ring_1_Ints_rat ).
% 0.33/0.64
% 0.33/0.64 % Ints_1
% 0.33/0.64 thf(fact_7486_Ints__1,axiom,
% 0.33/0.64 member_int @ one_one_int @ ring_1_Ints_int ).
% 0.33/0.64
% 0.33/0.64 % Ints_1
% 0.33/0.64 thf(fact_7487_Ints__1,axiom,
% 0.33/0.64 member_real @ one_one_real @ ring_1_Ints_real ).
% 0.33/0.64
% 0.33/0.64 % Ints_1
% 0.33/0.64 thf(fact_7488_Ints__add,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.33/0.64 => ( ( member_complex @ B @ ring_1_Ints_complex )
% 0.33/0.64 => ( member_complex @ ( plus_plus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_add
% 0.33/0.64 thf(fact_7489_Ints__add,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( member_real @ A @ ring_1_Ints_real )
% 0.33/0.64 => ( ( member_real @ B @ ring_1_Ints_real )
% 0.33/0.64 => ( member_real @ ( plus_plus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_add
% 0.33/0.64 thf(fact_7490_Ints__add,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.33/0.64 => ( ( member_rat @ B @ ring_1_Ints_rat )
% 0.33/0.64 => ( member_rat @ ( plus_plus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_add
% 0.33/0.64 thf(fact_7491_Ints__add,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( member_int @ A @ ring_1_Ints_int )
% 0.33/0.64 => ( ( member_int @ B @ ring_1_Ints_int )
% 0.33/0.64 => ( member_int @ ( plus_plus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_add
% 0.33/0.64 thf(fact_7492_Ints__diff,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.33/0.64 => ( ( member_complex @ B @ ring_1_Ints_complex )
% 0.33/0.64 => ( member_complex @ ( minus_minus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_diff
% 0.33/0.64 thf(fact_7493_Ints__diff,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( member_real @ A @ ring_1_Ints_real )
% 0.33/0.64 => ( ( member_real @ B @ ring_1_Ints_real )
% 0.33/0.64 => ( member_real @ ( minus_minus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_diff
% 0.33/0.64 thf(fact_7494_Ints__diff,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.33/0.64 => ( ( member_rat @ B @ ring_1_Ints_rat )
% 0.33/0.64 => ( member_rat @ ( minus_minus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_diff
% 0.33/0.64 thf(fact_7495_Ints__diff,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( member_int @ A @ ring_1_Ints_int )
% 0.33/0.64 => ( ( member_int @ B @ ring_1_Ints_int )
% 0.33/0.64 => ( member_int @ ( minus_minus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_diff
% 0.33/0.64 thf(fact_7496_minus__in__Ints__iff,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( member_real @ ( uminus_uminus_real @ X ) @ ring_1_Ints_real )
% 0.33/0.64 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_in_Ints_iff
% 0.33/0.64 thf(fact_7497_minus__in__Ints__iff,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( member_int @ ( uminus_uminus_int @ X ) @ ring_1_Ints_int )
% 0.33/0.64 = ( member_int @ X @ ring_1_Ints_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_in_Ints_iff
% 0.33/0.64 thf(fact_7498_minus__in__Ints__iff,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( member_complex @ ( uminus1482373934393186551omplex @ X ) @ ring_1_Ints_complex )
% 0.33/0.64 = ( member_complex @ X @ ring_1_Ints_complex ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_in_Ints_iff
% 0.33/0.64 thf(fact_7499_minus__in__Ints__iff,axiom,
% 0.33/0.64 ! [X: rat] :
% 0.33/0.64 ( ( member_rat @ ( uminus_uminus_rat @ X ) @ ring_1_Ints_rat )
% 0.33/0.64 = ( member_rat @ X @ ring_1_Ints_rat ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_in_Ints_iff
% 0.33/0.64 thf(fact_7500_minus__in__Ints__iff,axiom,
% 0.33/0.64 ! [X: code_integer] :
% 0.33/0.64 ( ( member_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ ring_11222124179247155820nteger )
% 0.33/0.64 = ( member_Code_integer @ X @ ring_11222124179247155820nteger ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_in_Ints_iff
% 0.33/0.64 thf(fact_7501_Ints__minus,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( member_real @ A @ ring_1_Ints_real )
% 0.33/0.64 => ( member_real @ ( uminus_uminus_real @ A ) @ ring_1_Ints_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_minus
% 0.33/0.64 thf(fact_7502_Ints__minus,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( member_int @ A @ ring_1_Ints_int )
% 0.33/0.64 => ( member_int @ ( uminus_uminus_int @ A ) @ ring_1_Ints_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_minus
% 0.33/0.64 thf(fact_7503_Ints__minus,axiom,
% 0.33/0.64 ! [A: complex] :
% 0.33/0.64 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.33/0.64 => ( member_complex @ ( uminus1482373934393186551omplex @ A ) @ ring_1_Ints_complex ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_minus
% 0.33/0.64 thf(fact_7504_Ints__minus,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.33/0.64 => ( member_rat @ ( uminus_uminus_rat @ A ) @ ring_1_Ints_rat ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_minus
% 0.33/0.64 thf(fact_7505_Ints__minus,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
% 0.33/0.64 => ( member_Code_integer @ ( uminus1351360451143612070nteger @ A ) @ ring_11222124179247155820nteger ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_minus
% 0.33/0.64 thf(fact_7506_divide__complex__def,axiom,
% 0.33/0.64 ( divide1717551699836669952omplex
% 0.33/0.64 = ( ^ [X3: complex,Y6: complex] : ( times_times_complex @ X3 @ ( invers8013647133539491842omplex @ Y6 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divide_complex_def
% 0.33/0.64 thf(fact_7507_Ints__power,axiom,
% 0.33/0.64 ! [A: real,N: nat] :
% 0.33/0.64 ( ( member_real @ A @ ring_1_Ints_real )
% 0.33/0.64 => ( member_real @ ( power_power_real @ A @ N ) @ ring_1_Ints_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_power
% 0.33/0.64 thf(fact_7508_Ints__power,axiom,
% 0.33/0.64 ! [A: int,N: nat] :
% 0.33/0.64 ( ( member_int @ A @ ring_1_Ints_int )
% 0.33/0.64 => ( member_int @ ( power_power_int @ A @ N ) @ ring_1_Ints_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_power
% 0.33/0.64 thf(fact_7509_Ints__power,axiom,
% 0.33/0.64 ! [A: complex,N: nat] :
% 0.33/0.64 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.33/0.64 => ( member_complex @ ( power_power_complex @ A @ N ) @ ring_1_Ints_complex ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_power
% 0.33/0.64 thf(fact_7510_Ints__of__nat,axiom,
% 0.33/0.64 ! [N: nat] : ( member_complex @ ( semiri8010041392384452111omplex @ N ) @ ring_1_Ints_complex ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_of_nat
% 0.33/0.64 thf(fact_7511_Ints__of__nat,axiom,
% 0.33/0.64 ! [N: nat] : ( member_real @ ( semiri5074537144036343181t_real @ N ) @ ring_1_Ints_real ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_of_nat
% 0.33/0.64 thf(fact_7512_Ints__of__nat,axiom,
% 0.33/0.64 ! [N: nat] : ( member_rat @ ( semiri681578069525770553at_rat @ N ) @ ring_1_Ints_rat ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_of_nat
% 0.33/0.64 thf(fact_7513_Ints__of__nat,axiom,
% 0.33/0.64 ! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ring_1_Ints_int ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_of_nat
% 0.33/0.64 thf(fact_7514_Ints__abs,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( member_int @ A @ ring_1_Ints_int )
% 0.33/0.64 => ( member_int @ ( abs_abs_int @ A ) @ ring_1_Ints_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_abs
% 0.33/0.64 thf(fact_7515_Ints__abs,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
% 0.33/0.64 => ( member_Code_integer @ ( abs_abs_Code_integer @ A ) @ ring_11222124179247155820nteger ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_abs
% 0.33/0.64 thf(fact_7516_Ints__abs,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.33/0.64 => ( member_rat @ ( abs_abs_rat @ A ) @ ring_1_Ints_rat ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_abs
% 0.33/0.64 thf(fact_7517_Ints__abs,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( member_real @ A @ ring_1_Ints_real )
% 0.33/0.64 => ( member_real @ ( abs_abs_real @ A ) @ ring_1_Ints_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_abs
% 0.33/0.64 thf(fact_7518_Ints__of__int,axiom,
% 0.33/0.64 ! [Z2: int] : ( member_complex @ ( ring_17405671764205052669omplex @ Z2 ) @ ring_1_Ints_complex ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_of_int
% 0.33/0.64 thf(fact_7519_Ints__of__int,axiom,
% 0.33/0.64 ! [Z2: int] : ( member_int @ ( ring_1_of_int_int @ Z2 ) @ ring_1_Ints_int ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_of_int
% 0.33/0.64 thf(fact_7520_Ints__of__int,axiom,
% 0.33/0.64 ! [Z2: int] : ( member_real @ ( ring_1_of_int_real @ Z2 ) @ ring_1_Ints_real ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_of_int
% 0.33/0.64 thf(fact_7521_Ints__of__int,axiom,
% 0.33/0.64 ! [Z2: int] : ( member_rat @ ( ring_1_of_int_rat @ Z2 ) @ ring_1_Ints_rat ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_of_int
% 0.33/0.64 thf(fact_7522_Ints__induct,axiom,
% 0.33/0.64 ! [Q2: complex,P: complex > $o] :
% 0.33/0.64 ( ( member_complex @ Q2 @ ring_1_Ints_complex )
% 0.33/0.64 => ( ! [Z4: int] : ( P @ ( ring_17405671764205052669omplex @ Z4 ) )
% 0.33/0.64 => ( P @ Q2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_induct
% 0.33/0.64 thf(fact_7523_Ints__induct,axiom,
% 0.33/0.64 ! [Q2: int,P: int > $o] :
% 0.33/0.64 ( ( member_int @ Q2 @ ring_1_Ints_int )
% 0.33/0.64 => ( ! [Z4: int] : ( P @ ( ring_1_of_int_int @ Z4 ) )
% 0.33/0.64 => ( P @ Q2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_induct
% 0.33/0.64 thf(fact_7524_Ints__induct,axiom,
% 0.33/0.64 ! [Q2: real,P: real > $o] :
% 0.33/0.64 ( ( member_real @ Q2 @ ring_1_Ints_real )
% 0.33/0.64 => ( ! [Z4: int] : ( P @ ( ring_1_of_int_real @ Z4 ) )
% 0.33/0.64 => ( P @ Q2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_induct
% 0.33/0.64 thf(fact_7525_Ints__induct,axiom,
% 0.33/0.64 ! [Q2: rat,P: rat > $o] :
% 0.33/0.64 ( ( member_rat @ Q2 @ ring_1_Ints_rat )
% 0.33/0.64 => ( ! [Z4: int] : ( P @ ( ring_1_of_int_rat @ Z4 ) )
% 0.33/0.64 => ( P @ Q2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_induct
% 0.33/0.64 thf(fact_7526_Ints__cases,axiom,
% 0.33/0.64 ! [Q2: complex] :
% 0.33/0.64 ( ( member_complex @ Q2 @ ring_1_Ints_complex )
% 0.33/0.64 => ~ ! [Z4: int] :
% 0.33/0.64 ( Q2
% 0.33/0.64 != ( ring_17405671764205052669omplex @ Z4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_cases
% 0.33/0.64 thf(fact_7527_Ints__cases,axiom,
% 0.33/0.64 ! [Q2: int] :
% 0.33/0.64 ( ( member_int @ Q2 @ ring_1_Ints_int )
% 0.33/0.64 => ~ ! [Z4: int] :
% 0.33/0.64 ( Q2
% 0.33/0.64 != ( ring_1_of_int_int @ Z4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_cases
% 0.33/0.64 thf(fact_7528_Ints__cases,axiom,
% 0.33/0.64 ! [Q2: real] :
% 0.33/0.64 ( ( member_real @ Q2 @ ring_1_Ints_real )
% 0.33/0.64 => ~ ! [Z4: int] :
% 0.33/0.64 ( Q2
% 0.33/0.64 != ( ring_1_of_int_real @ Z4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_cases
% 0.33/0.64 thf(fact_7529_Ints__cases,axiom,
% 0.33/0.64 ! [Q2: rat] :
% 0.33/0.64 ( ( member_rat @ Q2 @ ring_1_Ints_rat )
% 0.33/0.64 => ~ ! [Z4: int] :
% 0.33/0.64 ( Q2
% 0.33/0.64 != ( ring_1_of_int_rat @ Z4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_cases
% 0.33/0.64 thf(fact_7530_Ints__double__eq__0__iff,axiom,
% 0.33/0.64 ! [A: complex] :
% 0.33/0.64 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.33/0.64 => ( ( ( plus_plus_complex @ A @ A )
% 0.33/0.64 = zero_zero_complex )
% 0.33/0.64 = ( A = zero_zero_complex ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_double_eq_0_iff
% 0.33/0.64 thf(fact_7531_Ints__double__eq__0__iff,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( member_real @ A @ ring_1_Ints_real )
% 0.33/0.64 => ( ( ( plus_plus_real @ A @ A )
% 0.33/0.64 = zero_zero_real )
% 0.33/0.64 = ( A = zero_zero_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_double_eq_0_iff
% 0.33/0.64 thf(fact_7532_Ints__double__eq__0__iff,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.33/0.64 => ( ( ( plus_plus_rat @ A @ A )
% 0.33/0.64 = zero_zero_rat )
% 0.33/0.64 = ( A = zero_zero_rat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_double_eq_0_iff
% 0.33/0.64 thf(fact_7533_Ints__double__eq__0__iff,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( member_int @ A @ ring_1_Ints_int )
% 0.33/0.64 => ( ( ( plus_plus_int @ A @ A )
% 0.33/0.64 = zero_zero_int )
% 0.33/0.64 = ( A = zero_zero_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_double_eq_0_iff
% 0.33/0.64 thf(fact_7534_and__nat__def,axiom,
% 0.33/0.64 ( bit_se727722235901077358nd_nat
% 0.33/0.64 = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_nat_def
% 0.33/0.64 thf(fact_7535_Ints__odd__nonzero,axiom,
% 0.33/0.64 ! [A: complex] :
% 0.33/0.64 ( ( member_complex @ A @ ring_1_Ints_complex )
% 0.33/0.64 => ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A ) @ A )
% 0.33/0.64 != zero_zero_complex ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_odd_nonzero
% 0.33/0.64 thf(fact_7536_Ints__odd__nonzero,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( member_real @ A @ ring_1_Ints_real )
% 0.33/0.64 => ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
% 0.33/0.64 != zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_odd_nonzero
% 0.33/0.64 thf(fact_7537_Ints__odd__nonzero,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.33/0.64 => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A )
% 0.33/0.64 != zero_zero_rat ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_odd_nonzero
% 0.33/0.64 thf(fact_7538_Ints__odd__nonzero,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( member_int @ A @ ring_1_Ints_int )
% 0.33/0.64 => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
% 0.33/0.64 != zero_zero_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_odd_nonzero
% 0.33/0.64 thf(fact_7539_of__int__divide__in__Ints,axiom,
% 0.33/0.64 ! [B: int,A: int] :
% 0.33/0.64 ( ( dvd_dvd_int @ B @ A )
% 0.33/0.64 => ( member_complex @ ( divide1717551699836669952omplex @ ( ring_17405671764205052669omplex @ A ) @ ( ring_17405671764205052669omplex @ B ) ) @ ring_1_Ints_complex ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_int_divide_in_Ints
% 0.33/0.64 thf(fact_7540_of__int__divide__in__Ints,axiom,
% 0.33/0.64 ! [B: int,A: int] :
% 0.33/0.64 ( ( dvd_dvd_int @ B @ A )
% 0.33/0.64 => ( member_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) @ ring_1_Ints_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_int_divide_in_Ints
% 0.33/0.64 thf(fact_7541_of__int__divide__in__Ints,axiom,
% 0.33/0.64 ! [B: int,A: int] :
% 0.33/0.64 ( ( dvd_dvd_int @ B @ A )
% 0.33/0.64 => ( member_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) @ ring_1_Ints_rat ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_int_divide_in_Ints
% 0.33/0.64 thf(fact_7542_of__int__divide__in__Ints,axiom,
% 0.33/0.64 ! [B: int,A: int] :
% 0.33/0.64 ( ( dvd_dvd_int @ B @ A )
% 0.33/0.64 => ( member_int @ ( divide_divide_int @ ( ring_1_of_int_int @ A ) @ ( ring_1_of_int_int @ B ) ) @ ring_1_Ints_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_int_divide_in_Ints
% 0.33/0.64 thf(fact_7543_Ints__odd__less__0,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( member_real @ A @ ring_1_Ints_real )
% 0.33/0.64 => ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A ) @ zero_zero_real )
% 0.33/0.64 = ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_odd_less_0
% 0.33/0.64 thf(fact_7544_Ints__odd__less__0,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.33/0.64 => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A ) @ zero_zero_rat )
% 0.33/0.64 = ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_odd_less_0
% 0.33/0.64 thf(fact_7545_Ints__odd__less__0,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( member_int @ A @ ring_1_Ints_int )
% 0.33/0.64 => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A ) @ zero_zero_int )
% 0.33/0.64 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_odd_less_0
% 0.33/0.64 thf(fact_7546_Ints__nonzero__abs__ge1,axiom,
% 0.33/0.64 ! [X: code_integer] :
% 0.33/0.64 ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 0.33/0.64 => ( ( X != zero_z3403309356797280102nteger )
% 0.33/0.64 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_nonzero_abs_ge1
% 0.33/0.64 thf(fact_7547_Ints__nonzero__abs__ge1,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( member_real @ X @ ring_1_Ints_real )
% 0.33/0.64 => ( ( X != zero_zero_real )
% 0.33/0.64 => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_nonzero_abs_ge1
% 0.33/0.64 thf(fact_7548_Ints__nonzero__abs__ge1,axiom,
% 0.33/0.64 ! [X: rat] :
% 0.33/0.64 ( ( member_rat @ X @ ring_1_Ints_rat )
% 0.33/0.64 => ( ( X != zero_zero_rat )
% 0.33/0.64 => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_nonzero_abs_ge1
% 0.33/0.64 thf(fact_7549_Ints__nonzero__abs__ge1,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( member_int @ X @ ring_1_Ints_int )
% 0.33/0.64 => ( ( X != zero_zero_int )
% 0.33/0.64 => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_nonzero_abs_ge1
% 0.33/0.64 thf(fact_7550_Ints__nonzero__abs__less1,axiom,
% 0.33/0.64 ! [X: code_integer] :
% 0.33/0.64 ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 0.33/0.64 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer )
% 0.33/0.64 => ( X = zero_z3403309356797280102nteger ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_nonzero_abs_less1
% 0.33/0.64 thf(fact_7551_Ints__nonzero__abs__less1,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( member_real @ X @ ring_1_Ints_real )
% 0.33/0.64 => ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.64 => ( X = zero_zero_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_nonzero_abs_less1
% 0.33/0.64 thf(fact_7552_Ints__nonzero__abs__less1,axiom,
% 0.33/0.64 ! [X: rat] :
% 0.33/0.64 ( ( member_rat @ X @ ring_1_Ints_rat )
% 0.33/0.64 => ( ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat )
% 0.33/0.64 => ( X = zero_zero_rat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_nonzero_abs_less1
% 0.33/0.64 thf(fact_7553_Ints__nonzero__abs__less1,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( member_int @ X @ ring_1_Ints_int )
% 0.33/0.64 => ( ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int )
% 0.33/0.64 => ( X = zero_zero_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_nonzero_abs_less1
% 0.33/0.64 thf(fact_7554_Ints__eq__abs__less1,axiom,
% 0.33/0.64 ! [X: code_integer,Y: code_integer] :
% 0.33/0.64 ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 0.33/0.64 => ( ( member_Code_integer @ Y @ ring_11222124179247155820nteger )
% 0.33/0.64 => ( ( X = Y )
% 0.33/0.64 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ Y ) ) @ one_one_Code_integer ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_eq_abs_less1
% 0.33/0.64 thf(fact_7555_Ints__eq__abs__less1,axiom,
% 0.33/0.64 ! [X: real,Y: real] :
% 0.33/0.64 ( ( member_real @ X @ ring_1_Ints_real )
% 0.33/0.64 => ( ( member_real @ Y @ ring_1_Ints_real )
% 0.33/0.64 => ( ( X = Y )
% 0.33/0.64 = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ one_one_real ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_eq_abs_less1
% 0.33/0.64 thf(fact_7556_Ints__eq__abs__less1,axiom,
% 0.33/0.64 ! [X: rat,Y: rat] :
% 0.33/0.64 ( ( member_rat @ X @ ring_1_Ints_rat )
% 0.33/0.64 => ( ( member_rat @ Y @ ring_1_Ints_rat )
% 0.33/0.64 => ( ( X = Y )
% 0.33/0.64 = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ Y ) ) @ one_one_rat ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_eq_abs_less1
% 0.33/0.64 thf(fact_7557_Ints__eq__abs__less1,axiom,
% 0.33/0.64 ! [X: int,Y: int] :
% 0.33/0.64 ( ( member_int @ X @ ring_1_Ints_int )
% 0.33/0.64 => ( ( member_int @ Y @ ring_1_Ints_int )
% 0.33/0.64 => ( ( X = Y )
% 0.33/0.64 = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y ) ) @ one_one_int ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Ints_eq_abs_less1
% 0.33/0.64 thf(fact_7558_sin__times__pi__eq__0,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
% 0.33/0.64 = zero_zero_real )
% 0.33/0.64 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % sin_times_pi_eq_0
% 0.33/0.64 thf(fact_7559_frac__neg,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ( member_real @ X @ ring_1_Ints_real )
% 0.33/0.64 => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
% 0.33/0.64 = zero_zero_real ) )
% 0.33/0.64 & ( ~ ( member_real @ X @ ring_1_Ints_real )
% 0.33/0.64 => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
% 0.33/0.64 = ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % frac_neg
% 0.33/0.64 thf(fact_7560_frac__neg,axiom,
% 0.33/0.64 ! [X: rat] :
% 0.33/0.64 ( ( ( member_rat @ X @ ring_1_Ints_rat )
% 0.33/0.64 => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
% 0.33/0.64 = zero_zero_rat ) )
% 0.33/0.64 & ( ~ ( member_rat @ X @ ring_1_Ints_rat )
% 0.33/0.64 => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
% 0.33/0.64 = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % frac_neg
% 0.33/0.64 thf(fact_7561_le__mult__floor__Ints,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.64 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.33/0.64 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % le_mult_floor_Ints
% 0.33/0.64 thf(fact_7562_le__mult__floor__Ints,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.64 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.33/0.64 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % le_mult_floor_Ints
% 0.33/0.64 thf(fact_7563_le__mult__floor__Ints,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.64 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.33/0.64 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_int @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % le_mult_floor_Ints
% 0.33/0.64 thf(fact_7564_le__mult__floor__Ints,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.64 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.33/0.64 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_real @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % le_mult_floor_Ints
% 0.33/0.64 thf(fact_7565_le__mult__floor__Ints,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.64 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.33/0.64 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % le_mult_floor_Ints
% 0.33/0.64 thf(fact_7566_le__mult__floor__Ints,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.64 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.33/0.64 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_int @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % le_mult_floor_Ints
% 0.33/0.64 thf(fact_7567_frac__unique__iff,axiom,
% 0.33/0.64 ! [X: real,A: real] :
% 0.33/0.64 ( ( ( archim2898591450579166408c_real @ X )
% 0.33/0.64 = A )
% 0.33/0.64 = ( ( member_real @ ( minus_minus_real @ X @ A ) @ ring_1_Ints_real )
% 0.33/0.64 & ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.64 & ( ord_less_real @ A @ one_one_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % frac_unique_iff
% 0.33/0.64 thf(fact_7568_frac__unique__iff,axiom,
% 0.33/0.64 ! [X: rat,A: rat] :
% 0.33/0.64 ( ( ( archimedean_frac_rat @ X )
% 0.33/0.64 = A )
% 0.33/0.64 = ( ( member_rat @ ( minus_minus_rat @ X @ A ) @ ring_1_Ints_rat )
% 0.33/0.64 & ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.64 & ( ord_less_rat @ A @ one_one_rat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % frac_unique_iff
% 0.33/0.64 thf(fact_7569_mult__ceiling__le__Ints,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.64 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.33/0.64 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mult_ceiling_le_Ints
% 0.33/0.64 thf(fact_7570_mult__ceiling__le__Ints,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.64 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.33/0.64 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mult_ceiling_le_Ints
% 0.33/0.64 thf(fact_7571_mult__ceiling__le__Ints,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.64 => ( ( member_real @ A @ ring_1_Ints_real )
% 0.33/0.64 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mult_ceiling_le_Ints
% 0.33/0.64 thf(fact_7572_mult__ceiling__le__Ints,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.64 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.33/0.64 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mult_ceiling_le_Ints
% 0.33/0.64 thf(fact_7573_mult__ceiling__le__Ints,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.64 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.33/0.64 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mult_ceiling_le_Ints
% 0.33/0.64 thf(fact_7574_mult__ceiling__le__Ints,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.64 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 0.33/0.64 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mult_ceiling_le_Ints
% 0.33/0.64 thf(fact_7575_sin__integer__2pi,axiom,
% 0.33/0.64 ! [N: real] :
% 0.33/0.64 ( ( member_real @ N @ ring_1_Ints_real )
% 0.33/0.64 => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.33/0.64 = zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % sin_integer_2pi
% 0.33/0.64 thf(fact_7576_cos__integer__2pi,axiom,
% 0.33/0.64 ! [N: real] :
% 0.33/0.64 ( ( member_real @ N @ ring_1_Ints_real )
% 0.33/0.64 => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.33/0.64 = one_one_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % cos_integer_2pi
% 0.33/0.64 thf(fact_7577_and__nat__unfold,axiom,
% 0.33/0.64 ( bit_se727722235901077358nd_nat
% 0.33/0.64 = ( ^ [M3: nat,N3: nat] :
% 0.33/0.64 ( if_nat
% 0.33/0.64 @ ( ( M3 = zero_zero_nat )
% 0.33/0.64 | ( N3 = zero_zero_nat ) )
% 0.33/0.64 @ zero_zero_nat
% 0.33/0.64 @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_nat_unfold
% 0.33/0.64 thf(fact_7578_and__nat__rec,axiom,
% 0.33/0.64 ( bit_se727722235901077358nd_nat
% 0.33/0.64 = ( ^ [M3: nat,N3: nat] :
% 0.33/0.64 ( plus_plus_nat
% 0.33/0.64 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.64 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 0.33/0.64 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.33/0.64 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_nat_rec
% 0.33/0.64 thf(fact_7579_complex__inverse,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
% 0.33/0.64 = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_inverse
% 0.33/0.64 thf(fact_7580_sinh__field__def,axiom,
% 0.33/0.64 ( sinh_real
% 0.33/0.64 = ( ^ [Z5: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sinh_field_def
% 0.33/0.64 thf(fact_7581_sinh__field__def,axiom,
% 0.33/0.64 ( sinh_complex
% 0.33/0.64 = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sinh_field_def
% 0.33/0.64 thf(fact_7582_cosh__ln__real,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ( cosh_real @ ( ln_ln_real @ X ) )
% 0.33/0.64 = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_ln_real
% 0.33/0.64 thf(fact_7583_xor__Suc__0__eq,axiom,
% 0.33/0.64 ! [N: nat] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.33/0.64 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.33/0.64 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.64 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_Suc_0_eq
% 0.33/0.64 thf(fact_7584_Suc__0__xor__eq,axiom,
% 0.33/0.64 ! [N: nat] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.33/0.64 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.33/0.64 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.64 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Suc_0_xor_eq
% 0.33/0.64 thf(fact_7585_gbinomial__absorption_H,axiom,
% 0.33/0.64 ! [K: nat,A: complex] :
% 0.33/0.64 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.64 => ( ( gbinomial_complex @ A @ K )
% 0.33/0.64 = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_absorption'
% 0.33/0.64 thf(fact_7586_gbinomial__absorption_H,axiom,
% 0.33/0.64 ! [K: nat,A: real] :
% 0.33/0.64 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.64 => ( ( gbinomial_real @ A @ K )
% 0.33/0.64 = ( times_times_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_absorption'
% 0.33/0.64 thf(fact_7587_gbinomial__absorption_H,axiom,
% 0.33/0.64 ! [K: nat,A: rat] :
% 0.33/0.64 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.64 => ( ( gbinomial_rat @ A @ K )
% 0.33/0.64 = ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_absorption'
% 0.33/0.64 thf(fact_7588_cosh__double,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( cosh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.64 = ( plus_plus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_double
% 0.33/0.64 thf(fact_7589_cosh__double,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( cosh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.64 = ( plus_plus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_double
% 0.33/0.64 thf(fact_7590_bit_Oxor__left__self,axiom,
% 0.33/0.64 ! [X: int,Y: int] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ X @ ( bit_se6526347334894502574or_int @ X @ Y ) )
% 0.33/0.64 = Y ) ).
% 0.33/0.64
% 0.33/0.64 % bit.xor_left_self
% 0.33/0.64 thf(fact_7591_xor_Oright__neutral,axiom,
% 0.33/0.64 ! [A: nat] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
% 0.33/0.64 = A ) ).
% 0.33/0.64
% 0.33/0.64 % xor.right_neutral
% 0.33/0.64 thf(fact_7592_xor_Oright__neutral,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
% 0.33/0.64 = A ) ).
% 0.33/0.64
% 0.33/0.64 % xor.right_neutral
% 0.33/0.64 thf(fact_7593_xor_Oleft__neutral,axiom,
% 0.33/0.64 ! [A: nat] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
% 0.33/0.64 = A ) ).
% 0.33/0.64
% 0.33/0.64 % xor.left_neutral
% 0.33/0.64 thf(fact_7594_xor_Oleft__neutral,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
% 0.33/0.64 = A ) ).
% 0.33/0.64
% 0.33/0.64 % xor.left_neutral
% 0.33/0.64 thf(fact_7595_xor__self__eq,axiom,
% 0.33/0.64 ! [A: nat] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ A @ A )
% 0.33/0.64 = zero_zero_nat ) ).
% 0.33/0.64
% 0.33/0.64 % xor_self_eq
% 0.33/0.64 thf(fact_7596_xor__self__eq,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ A @ A )
% 0.33/0.64 = zero_zero_int ) ).
% 0.33/0.64
% 0.33/0.64 % xor_self_eq
% 0.33/0.64 thf(fact_7597_bit_Oxor__self,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ X @ X )
% 0.33/0.64 = zero_zero_int ) ).
% 0.33/0.64
% 0.33/0.64 % bit.xor_self
% 0.33/0.64 thf(fact_7598_take__bit__xor,axiom,
% 0.33/0.64 ! [N: nat,A: int,B: int] :
% 0.33/0.64 ( ( bit_se2923211474154528505it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 0.33/0.64 = ( bit_se6526347334894502574or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % take_bit_xor
% 0.33/0.64 thf(fact_7599_take__bit__xor,axiom,
% 0.33/0.64 ! [N: nat,A: nat,B: nat] :
% 0.33/0.64 ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 0.33/0.64 = ( bit_se6528837805403552850or_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % take_bit_xor
% 0.33/0.64 thf(fact_7600_push__bit__xor,axiom,
% 0.33/0.64 ! [N: nat,A: nat,B: nat] :
% 0.33/0.64 ( ( bit_se547839408752420682it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 0.33/0.64 = ( bit_se6528837805403552850or_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % push_bit_xor
% 0.33/0.64 thf(fact_7601_push__bit__xor,axiom,
% 0.33/0.64 ! [N: nat,A: int,B: int] :
% 0.33/0.64 ( ( bit_se545348938243370406it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 0.33/0.64 = ( bit_se6526347334894502574or_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % push_bit_xor
% 0.33/0.64 thf(fact_7602_push__bit__xor,axiom,
% 0.33/0.64 ! [N: nat,A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( bit_se7788150548672797655nteger @ N @ ( bit_se3222712562003087583nteger @ A @ B ) )
% 0.33/0.64 = ( bit_se3222712562003087583nteger @ ( bit_se7788150548672797655nteger @ N @ A ) @ ( bit_se7788150548672797655nteger @ N @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % push_bit_xor
% 0.33/0.64 thf(fact_7603_push__bit__xor,axiom,
% 0.33/0.64 ! [N: nat,A: code_natural,B: code_natural] :
% 0.33/0.64 ( ( bit_se6611745700429515170atural @ N @ ( bit_se2046307713759805098atural @ A @ B ) )
% 0.33/0.64 = ( bit_se2046307713759805098atural @ ( bit_se6611745700429515170atural @ N @ A ) @ ( bit_se6611745700429515170atural @ N @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % push_bit_xor
% 0.33/0.64 thf(fact_7604_cosh__0,axiom,
% 0.33/0.64 ( ( cosh_complex @ zero_zero_complex )
% 0.33/0.64 = one_one_complex ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_0
% 0.33/0.64 thf(fact_7605_cosh__0,axiom,
% 0.33/0.64 ( ( cosh_real @ zero_zero_real )
% 0.33/0.64 = one_one_real ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_0
% 0.33/0.64 thf(fact_7606_gbinomial__0_I2_J,axiom,
% 0.33/0.64 ! [K: nat] :
% 0.33/0.64 ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
% 0.33/0.64 = zero_zero_real ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_0(2)
% 0.33/0.64 thf(fact_7607_gbinomial__0_I2_J,axiom,
% 0.33/0.64 ! [K: nat] :
% 0.33/0.64 ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
% 0.33/0.64 = zero_zero_rat ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_0(2)
% 0.33/0.64 thf(fact_7608_gbinomial__0_I2_J,axiom,
% 0.33/0.64 ! [K: nat] :
% 0.33/0.64 ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
% 0.33/0.64 = zero_zero_nat ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_0(2)
% 0.33/0.64 thf(fact_7609_gbinomial__0_I2_J,axiom,
% 0.33/0.64 ! [K: nat] :
% 0.33/0.64 ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
% 0.33/0.64 = zero_zero_int ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_0(2)
% 0.33/0.64 thf(fact_7610_gbinomial__0_I1_J,axiom,
% 0.33/0.64 ! [A: complex] :
% 0.33/0.64 ( ( gbinomial_complex @ A @ zero_zero_nat )
% 0.33/0.64 = one_one_complex ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_0(1)
% 0.33/0.64 thf(fact_7611_gbinomial__0_I1_J,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( gbinomial_real @ A @ zero_zero_nat )
% 0.33/0.64 = one_one_real ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_0(1)
% 0.33/0.64 thf(fact_7612_gbinomial__0_I1_J,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( gbinomial_rat @ A @ zero_zero_nat )
% 0.33/0.64 = one_one_rat ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_0(1)
% 0.33/0.64 thf(fact_7613_gbinomial__0_I1_J,axiom,
% 0.33/0.64 ! [A: nat] :
% 0.33/0.64 ( ( gbinomial_nat @ A @ zero_zero_nat )
% 0.33/0.64 = one_one_nat ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_0(1)
% 0.33/0.64 thf(fact_7614_gbinomial__0_I1_J,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( gbinomial_int @ A @ zero_zero_nat )
% 0.33/0.64 = one_one_int ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_0(1)
% 0.33/0.64 thf(fact_7615_xor__numerals_I3_J,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.33/0.64 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_numerals(3)
% 0.33/0.64 thf(fact_7616_xor__numerals_I3_J,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 0.33/0.64 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_numerals(3)
% 0.33/0.64 thf(fact_7617_xor__numerals_I8_J,axiom,
% 0.33/0.64 ! [X: num] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
% 0.33/0.64 = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_numerals(8)
% 0.33/0.64 thf(fact_7618_xor__numerals_I8_J,axiom,
% 0.33/0.64 ! [X: num] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
% 0.33/0.64 = ( numeral_numeral_int @ ( bit0 @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_numerals(8)
% 0.33/0.64 thf(fact_7619_xor__numerals_I5_J,axiom,
% 0.33/0.64 ! [X: num] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 0.33/0.64 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_numerals(5)
% 0.33/0.64 thf(fact_7620_xor__numerals_I5_J,axiom,
% 0.33/0.64 ! [X: num] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 0.33/0.64 = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_numerals(5)
% 0.33/0.64 thf(fact_7621_xor__numerals_I2_J,axiom,
% 0.33/0.64 ! [Y: num] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.33/0.64 = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_numerals(2)
% 0.33/0.64 thf(fact_7622_xor__numerals_I2_J,axiom,
% 0.33/0.64 ! [Y: num] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 0.33/0.64 = ( numeral_numeral_int @ ( bit0 @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_numerals(2)
% 0.33/0.64 thf(fact_7623_xor__numerals_I1_J,axiom,
% 0.33/0.64 ! [Y: num] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.33/0.64 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_numerals(1)
% 0.33/0.64 thf(fact_7624_xor__numerals_I1_J,axiom,
% 0.33/0.64 ! [Y: num] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 0.33/0.64 = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_numerals(1)
% 0.33/0.64 thf(fact_7625_xor__numerals_I7_J,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.33/0.64 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_numerals(7)
% 0.33/0.64 thf(fact_7626_xor__numerals_I7_J,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 0.33/0.64 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_numerals(7)
% 0.33/0.64 thf(fact_7627_xor__nat__numerals_I4_J,axiom,
% 0.33/0.64 ! [X: num] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.33/0.64 = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_nat_numerals(4)
% 0.33/0.64 thf(fact_7628_xor__nat__numerals_I3_J,axiom,
% 0.33/0.64 ! [X: num] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.33/0.64 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_nat_numerals(3)
% 0.33/0.64 thf(fact_7629_xor__nat__numerals_I2_J,axiom,
% 0.33/0.64 ! [Y: num] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.33/0.64 = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_nat_numerals(2)
% 0.33/0.64 thf(fact_7630_xor__nat__numerals_I1_J,axiom,
% 0.33/0.64 ! [Y: num] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.33/0.64 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_nat_numerals(1)
% 0.33/0.64 thf(fact_7631_xor__numerals_I6_J,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.33/0.64 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_numerals(6)
% 0.33/0.64 thf(fact_7632_xor__numerals_I6_J,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 0.33/0.64 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_numerals(6)
% 0.33/0.64 thf(fact_7633_xor__numerals_I4_J,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.33/0.64 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_numerals(4)
% 0.33/0.64 thf(fact_7634_xor__numerals_I4_J,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 0.33/0.64 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_numerals(4)
% 0.33/0.64 thf(fact_7635_of__int__xor__eq,axiom,
% 0.33/0.64 ! [K: int,L: int] :
% 0.33/0.64 ( ( ring_1_of_int_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 0.33/0.64 = ( bit_se6526347334894502574or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_int_xor_eq
% 0.33/0.64 thf(fact_7636_of__nat__gbinomial,axiom,
% 0.33/0.64 ! [N: nat,K: nat] :
% 0.33/0.64 ( ( semiri8010041392384452111omplex @ ( gbinomial_nat @ N @ K ) )
% 0.33/0.64 = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_nat_gbinomial
% 0.33/0.64 thf(fact_7637_of__nat__gbinomial,axiom,
% 0.33/0.64 ! [N: nat,K: nat] :
% 0.33/0.64 ( ( semiri5074537144036343181t_real @ ( gbinomial_nat @ N @ K ) )
% 0.33/0.64 = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_nat_gbinomial
% 0.33/0.64 thf(fact_7638_of__nat__gbinomial,axiom,
% 0.33/0.64 ! [N: nat,K: nat] :
% 0.33/0.64 ( ( semiri681578069525770553at_rat @ ( gbinomial_nat @ N @ K ) )
% 0.33/0.64 = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_nat_gbinomial
% 0.33/0.64 thf(fact_7639_xor_Oassoc,axiom,
% 0.33/0.64 ! [A: nat,B: nat,C: nat] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ ( bit_se6528837805403552850or_nat @ A @ B ) @ C )
% 0.33/0.64 = ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B @ C ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor.assoc
% 0.33/0.64 thf(fact_7640_xor_Oassoc,axiom,
% 0.33/0.64 ! [A: int,B: int,C: int] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ ( bit_se6526347334894502574or_int @ A @ B ) @ C )
% 0.33/0.64 = ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B @ C ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor.assoc
% 0.33/0.64 thf(fact_7641_xor_Ocommute,axiom,
% 0.33/0.64 ( bit_se6528837805403552850or_nat
% 0.33/0.64 = ( ^ [A2: nat,B2: nat] : ( bit_se6528837805403552850or_nat @ B2 @ A2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor.commute
% 0.33/0.64 thf(fact_7642_xor_Ocommute,axiom,
% 0.33/0.64 ( bit_se6526347334894502574or_int
% 0.33/0.64 = ( ^ [A2: int,B2: int] : ( bit_se6526347334894502574or_int @ B2 @ A2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor.commute
% 0.33/0.64 thf(fact_7643_xor_Oleft__commute,axiom,
% 0.33/0.64 ! [B: nat,A: nat,C: nat] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ B @ ( bit_se6528837805403552850or_nat @ A @ C ) )
% 0.33/0.64 = ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B @ C ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor.left_commute
% 0.33/0.64 thf(fact_7644_xor_Oleft__commute,axiom,
% 0.33/0.64 ! [B: int,A: int,C: int] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ B @ ( bit_se6526347334894502574or_int @ A @ C ) )
% 0.33/0.64 = ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B @ C ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor.left_commute
% 0.33/0.64 thf(fact_7645_of__nat__xor__eq,axiom,
% 0.33/0.64 ! [M: nat,N: nat] :
% 0.33/0.64 ( ( semiri1316708129612266289at_nat @ ( bit_se6528837805403552850or_nat @ M @ N ) )
% 0.33/0.64 = ( bit_se6528837805403552850or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_nat_xor_eq
% 0.33/0.64 thf(fact_7646_of__nat__xor__eq,axiom,
% 0.33/0.64 ! [M: nat,N: nat] :
% 0.33/0.64 ( ( semiri1314217659103216013at_int @ ( bit_se6528837805403552850or_nat @ M @ N ) )
% 0.33/0.64 = ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_nat_xor_eq
% 0.33/0.64 thf(fact_7647_bit_Oconj__xor__distrib,axiom,
% 0.33/0.64 ! [X: int,Y: int,Z2: int] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ X @ ( bit_se6526347334894502574or_int @ Y @ Z2 ) )
% 0.33/0.64 = ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se725231765392027082nd_int @ X @ Z2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.conj_xor_distrib
% 0.33/0.64 thf(fact_7648_bit_Oconj__xor__distrib2,axiom,
% 0.33/0.64 ! [Y: int,Z2: int,X: int] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ ( bit_se6526347334894502574or_int @ Y @ Z2 ) @ X )
% 0.33/0.64 = ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ Y @ X ) @ ( bit_se725231765392027082nd_int @ Z2 @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.conj_xor_distrib2
% 0.33/0.64 thf(fact_7649_bit__xor__iff,axiom,
% 0.33/0.64 ! [A: nat,B: nat,N: nat] :
% 0.33/0.64 ( ( bit_se1148574629649215175it_nat @ ( bit_se6528837805403552850or_nat @ A @ B ) @ N )
% 0.33/0.64 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.33/0.64 != ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit_xor_iff
% 0.33/0.64 thf(fact_7650_bit__xor__iff,axiom,
% 0.33/0.64 ! [A: int,B: int,N: nat] :
% 0.33/0.64 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ A @ B ) @ N )
% 0.33/0.64 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.33/0.64 != ( bit_se1146084159140164899it_int @ B @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit_xor_iff
% 0.33/0.64 thf(fact_7651_cosh__real__ge__1,axiom,
% 0.33/0.64 ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_real_ge_1
% 0.33/0.64 thf(fact_7652_binomial__gbinomial,axiom,
% 0.33/0.64 ! [N: nat,K: nat] :
% 0.33/0.64 ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 0.33/0.64 = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K ) ) ).
% 0.33/0.64
% 0.33/0.64 % binomial_gbinomial
% 0.33/0.64 thf(fact_7653_binomial__gbinomial,axiom,
% 0.33/0.64 ! [N: nat,K: nat] :
% 0.33/0.64 ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 0.33/0.64 = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K ) ) ).
% 0.33/0.64
% 0.33/0.64 % binomial_gbinomial
% 0.33/0.64 thf(fact_7654_binomial__gbinomial,axiom,
% 0.33/0.64 ! [N: nat,K: nat] :
% 0.33/0.64 ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 0.33/0.64 = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K ) ) ).
% 0.33/0.64
% 0.33/0.64 % binomial_gbinomial
% 0.33/0.64 thf(fact_7655_flip__bit__nat__def,axiom,
% 0.33/0.64 ( bit_se2161824704523386999it_nat
% 0.33/0.64 = ( ^ [M3: nat,N3: nat] : ( bit_se6528837805403552850or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % flip_bit_nat_def
% 0.33/0.64 thf(fact_7656_gbinomial__Suc__Suc,axiom,
% 0.33/0.64 ! [A: complex,K: nat] :
% 0.33/0.64 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.33/0.64 = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_Suc_Suc
% 0.33/0.64 thf(fact_7657_gbinomial__Suc__Suc,axiom,
% 0.33/0.64 ! [A: real,K: nat] :
% 0.33/0.64 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.33/0.64 = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_Suc_Suc
% 0.33/0.64 thf(fact_7658_gbinomial__Suc__Suc,axiom,
% 0.33/0.64 ! [A: rat,K: nat] :
% 0.33/0.64 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.33/0.64 = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_Suc_Suc
% 0.33/0.64 thf(fact_7659_gbinomial__of__nat__symmetric,axiom,
% 0.33/0.64 ! [K: nat,N: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.64 => ( ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K )
% 0.33/0.64 = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_of_nat_symmetric
% 0.33/0.64 thf(fact_7660_gbinomial__of__nat__symmetric,axiom,
% 0.33/0.64 ! [K: nat,N: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.64 => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
% 0.33/0.64 = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_of_nat_symmetric
% 0.33/0.64 thf(fact_7661_gbinomial__of__nat__symmetric,axiom,
% 0.33/0.64 ! [K: nat,N: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.33/0.64 => ( ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K )
% 0.33/0.64 = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_of_nat_symmetric
% 0.33/0.64 thf(fact_7662_flip__bit__eq__xor,axiom,
% 0.33/0.64 ( bit_se2161824704523386999it_nat
% 0.33/0.64 = ( ^ [N3: nat,A2: nat] : ( bit_se6528837805403552850or_nat @ A2 @ ( bit_se547839408752420682it_nat @ N3 @ one_one_nat ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % flip_bit_eq_xor
% 0.33/0.64 thf(fact_7663_flip__bit__eq__xor,axiom,
% 0.33/0.64 ( bit_se2159334234014336723it_int
% 0.33/0.64 = ( ^ [N3: nat,A2: int] : ( bit_se6526347334894502574or_int @ A2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % flip_bit_eq_xor
% 0.33/0.64 thf(fact_7664_flip__bit__eq__xor,axiom,
% 0.33/0.64 ( bit_se1345352211410354436nteger
% 0.33/0.64 = ( ^ [N3: nat,A2: code_integer] : ( bit_se3222712562003087583nteger @ A2 @ ( bit_se7788150548672797655nteger @ N3 @ one_one_Code_integer ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % flip_bit_eq_xor
% 0.33/0.64 thf(fact_7665_flip__bit__eq__xor,axiom,
% 0.33/0.64 ( bit_se168947363167071951atural
% 0.33/0.64 = ( ^ [N3: nat,A2: code_natural] : ( bit_se2046307713759805098atural @ A2 @ ( bit_se6611745700429515170atural @ N3 @ one_one_Code_natural ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % flip_bit_eq_xor
% 0.33/0.64 thf(fact_7666_sinh__add,axiom,
% 0.33/0.64 ! [X: real,Y: real] :
% 0.33/0.64 ( ( sinh_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.64 = ( plus_plus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sinh_add
% 0.33/0.64 thf(fact_7667_cosh__add,axiom,
% 0.33/0.64 ! [X: real,Y: real] :
% 0.33/0.64 ( ( cosh_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.64 = ( plus_plus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_add
% 0.33/0.64 thf(fact_7668_sinh__diff,axiom,
% 0.33/0.64 ! [X: real,Y: real] :
% 0.33/0.64 ( ( sinh_real @ ( minus_minus_real @ X @ Y ) )
% 0.33/0.64 = ( minus_minus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sinh_diff
% 0.33/0.64 thf(fact_7669_cosh__diff,axiom,
% 0.33/0.64 ! [X: real,Y: real] :
% 0.33/0.64 ( ( cosh_real @ ( minus_minus_real @ X @ Y ) )
% 0.33/0.64 = ( minus_minus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_diff
% 0.33/0.64 thf(fact_7670_sinh__plus__cosh,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( plus_plus_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) )
% 0.33/0.64 = ( exp_complex @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % sinh_plus_cosh
% 0.33/0.64 thf(fact_7671_sinh__plus__cosh,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( plus_plus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
% 0.33/0.64 = ( exp_real @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % sinh_plus_cosh
% 0.33/0.64 thf(fact_7672_cosh__plus__sinh,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( plus_plus_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ X ) )
% 0.33/0.64 = ( exp_complex @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_plus_sinh
% 0.33/0.64 thf(fact_7673_cosh__plus__sinh,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( plus_plus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
% 0.33/0.64 = ( exp_real @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_plus_sinh
% 0.33/0.64 thf(fact_7674_tanh__def,axiom,
% 0.33/0.64 ( tanh_complex
% 0.33/0.64 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sinh_complex @ X3 ) @ ( cosh_complex @ X3 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % tanh_def
% 0.33/0.64 thf(fact_7675_tanh__def,axiom,
% 0.33/0.64 ( tanh_real
% 0.33/0.64 = ( ^ [X3: real] : ( divide_divide_real @ ( sinh_real @ X3 ) @ ( cosh_real @ X3 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % tanh_def
% 0.33/0.64 thf(fact_7676_gbinomial__addition__formula,axiom,
% 0.33/0.64 ! [A: complex,K: nat] :
% 0.33/0.64 ( ( gbinomial_complex @ A @ ( suc @ K ) )
% 0.33/0.64 = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_addition_formula
% 0.33/0.64 thf(fact_7677_gbinomial__addition__formula,axiom,
% 0.33/0.64 ! [A: real,K: nat] :
% 0.33/0.64 ( ( gbinomial_real @ A @ ( suc @ K ) )
% 0.33/0.64 = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_addition_formula
% 0.33/0.64 thf(fact_7678_gbinomial__addition__formula,axiom,
% 0.33/0.64 ! [A: rat,K: nat] :
% 0.33/0.64 ( ( gbinomial_rat @ A @ ( suc @ K ) )
% 0.33/0.64 = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_addition_formula
% 0.33/0.64 thf(fact_7679_gbinomial__absorb__comp,axiom,
% 0.33/0.64 ! [A: complex,K: nat] :
% 0.33/0.64 ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
% 0.33/0.64 = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_absorb_comp
% 0.33/0.64 thf(fact_7680_gbinomial__absorb__comp,axiom,
% 0.33/0.64 ! [A: real,K: nat] :
% 0.33/0.64 ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
% 0.33/0.64 = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_absorb_comp
% 0.33/0.64 thf(fact_7681_gbinomial__absorb__comp,axiom,
% 0.33/0.64 ! [A: rat,K: nat] :
% 0.33/0.64 ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
% 0.33/0.64 = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_absorb_comp
% 0.33/0.64 thf(fact_7682_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.33/0.64 ! [K: nat,A: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
% 0.33/0.64 => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_ge_n_over_k_pow_k
% 0.33/0.64 thf(fact_7683_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.33/0.64 ! [K: nat,A: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
% 0.33/0.64 => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_ge_n_over_k_pow_k
% 0.33/0.64 thf(fact_7684_gbinomial__mult__1_H,axiom,
% 0.33/0.64 ! [A: complex,K: nat] :
% 0.33/0.64 ( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ A )
% 0.33/0.64 = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_mult_1'
% 0.33/0.64 thf(fact_7685_gbinomial__mult__1_H,axiom,
% 0.33/0.64 ! [A: real,K: nat] :
% 0.33/0.64 ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
% 0.33/0.64 = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_mult_1'
% 0.33/0.64 thf(fact_7686_gbinomial__mult__1_H,axiom,
% 0.33/0.64 ! [A: rat,K: nat] :
% 0.33/0.64 ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
% 0.33/0.64 = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_mult_1'
% 0.33/0.64 thf(fact_7687_gbinomial__mult__1,axiom,
% 0.33/0.64 ! [A: complex,K: nat] :
% 0.33/0.64 ( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K ) )
% 0.33/0.64 = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_mult_1
% 0.33/0.64 thf(fact_7688_gbinomial__mult__1,axiom,
% 0.33/0.64 ! [A: real,K: nat] :
% 0.33/0.64 ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
% 0.33/0.64 = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_mult_1
% 0.33/0.64 thf(fact_7689_gbinomial__mult__1,axiom,
% 0.33/0.64 ! [A: rat,K: nat] :
% 0.33/0.64 ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
% 0.33/0.64 = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_mult_1
% 0.33/0.64 thf(fact_7690_even__xor__iff,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ A @ B ) )
% 0.33/0.64 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.33/0.64 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % even_xor_iff
% 0.33/0.64 thf(fact_7691_even__xor__iff,axiom,
% 0.33/0.64 ! [A: nat,B: nat] :
% 0.33/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 0.33/0.64 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.33/0.64 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % even_xor_iff
% 0.33/0.64 thf(fact_7692_even__xor__iff,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 0.33/0.64 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.33/0.64 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % even_xor_iff
% 0.33/0.64 thf(fact_7693_cosh__minus__sinh,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( minus_minus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
% 0.33/0.64 = ( exp_real @ ( uminus_uminus_real @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_minus_sinh
% 0.33/0.64 thf(fact_7694_cosh__minus__sinh,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( minus_minus_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ X ) )
% 0.33/0.64 = ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_minus_sinh
% 0.33/0.64 thf(fact_7695_sinh__minus__cosh,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( minus_minus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
% 0.33/0.64 = ( uminus_uminus_real @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sinh_minus_cosh
% 0.33/0.64 thf(fact_7696_sinh__minus__cosh,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( minus_minus_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) )
% 0.33/0.64 = ( uminus1482373934393186551omplex @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sinh_minus_cosh
% 0.33/0.64 thf(fact_7697_Suc__times__gbinomial,axiom,
% 0.33/0.64 ! [K: nat,A: complex] :
% 0.33/0.64 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
% 0.33/0.64 = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Suc_times_gbinomial
% 0.33/0.64 thf(fact_7698_Suc__times__gbinomial,axiom,
% 0.33/0.64 ! [K: nat,A: real] :
% 0.33/0.64 ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
% 0.33/0.64 = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Suc_times_gbinomial
% 0.33/0.64 thf(fact_7699_Suc__times__gbinomial,axiom,
% 0.33/0.64 ! [K: nat,A: rat] :
% 0.33/0.64 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
% 0.33/0.64 = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Suc_times_gbinomial
% 0.33/0.64 thf(fact_7700_gbinomial__absorption,axiom,
% 0.33/0.64 ! [K: nat,A: complex] :
% 0.33/0.64 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
% 0.33/0.64 = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_absorption
% 0.33/0.64 thf(fact_7701_gbinomial__absorption,axiom,
% 0.33/0.64 ! [K: nat,A: real] :
% 0.33/0.64 ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
% 0.33/0.64 = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_absorption
% 0.33/0.64 thf(fact_7702_gbinomial__absorption,axiom,
% 0.33/0.64 ! [K: nat,A: rat] :
% 0.33/0.64 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
% 0.33/0.64 = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_absorption
% 0.33/0.64 thf(fact_7703_gbinomial__trinomial__revision,axiom,
% 0.33/0.64 ! [K: nat,M: nat,A: complex] :
% 0.33/0.64 ( ( ord_less_eq_nat @ K @ M )
% 0.33/0.64 => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
% 0.33/0.64 = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_trinomial_revision
% 0.33/0.64 thf(fact_7704_gbinomial__trinomial__revision,axiom,
% 0.33/0.64 ! [K: nat,M: nat,A: real] :
% 0.33/0.64 ( ( ord_less_eq_nat @ K @ M )
% 0.33/0.64 => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
% 0.33/0.64 = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_trinomial_revision
% 0.33/0.64 thf(fact_7705_gbinomial__trinomial__revision,axiom,
% 0.33/0.64 ! [K: nat,M: nat,A: rat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ K @ M )
% 0.33/0.64 => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
% 0.33/0.64 = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_trinomial_revision
% 0.33/0.64 thf(fact_7706_sinh__double,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( sinh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.64 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sinh_complex @ X ) ) @ ( cosh_complex @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sinh_double
% 0.33/0.64 thf(fact_7707_sinh__double,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( sinh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.64 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sinh_real @ X ) ) @ ( cosh_real @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sinh_double
% 0.33/0.64 thf(fact_7708_gbinomial__rec,axiom,
% 0.33/0.64 ! [A: complex,K: nat] :
% 0.33/0.64 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.33/0.64 = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_rec
% 0.33/0.64 thf(fact_7709_gbinomial__rec,axiom,
% 0.33/0.64 ! [A: real,K: nat] :
% 0.33/0.64 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.33/0.64 = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_rec
% 0.33/0.64 thf(fact_7710_gbinomial__rec,axiom,
% 0.33/0.64 ! [A: rat,K: nat] :
% 0.33/0.64 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.33/0.64 = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_rec
% 0.33/0.64 thf(fact_7711_gbinomial__factors,axiom,
% 0.33/0.64 ! [A: complex,K: nat] :
% 0.33/0.64 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.33/0.64 = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_factors
% 0.33/0.64 thf(fact_7712_gbinomial__factors,axiom,
% 0.33/0.64 ! [A: real,K: nat] :
% 0.33/0.64 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.33/0.64 = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_factors
% 0.33/0.64 thf(fact_7713_gbinomial__factors,axiom,
% 0.33/0.64 ! [A: rat,K: nat] :
% 0.33/0.64 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.33/0.64 = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_factors
% 0.33/0.64 thf(fact_7714_gbinomial__negated__upper,axiom,
% 0.33/0.64 ( gbinomial_complex
% 0.33/0.64 = ( ^ [A2: complex,K2: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K2 ) @ A2 ) @ one_one_complex ) @ K2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_negated_upper
% 0.33/0.64 thf(fact_7715_gbinomial__negated__upper,axiom,
% 0.33/0.64 ( gbinomial_real
% 0.33/0.64 = ( ^ [A2: real,K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K2 ) @ A2 ) @ one_one_real ) @ K2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_negated_upper
% 0.33/0.64 thf(fact_7716_gbinomial__negated__upper,axiom,
% 0.33/0.64 ( gbinomial_rat
% 0.33/0.64 = ( ^ [A2: rat,K2: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K2 ) @ A2 ) @ one_one_rat ) @ K2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_negated_upper
% 0.33/0.64 thf(fact_7717_gbinomial__index__swap,axiom,
% 0.33/0.64 ! [K: nat,N: nat] :
% 0.33/0.64 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
% 0.33/0.64 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_index_swap
% 0.33/0.64 thf(fact_7718_gbinomial__index__swap,axiom,
% 0.33/0.64 ! [K: nat,N: nat] :
% 0.33/0.64 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ K ) )
% 0.33/0.64 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_index_swap
% 0.33/0.64 thf(fact_7719_gbinomial__index__swap,axiom,
% 0.33/0.64 ! [K: nat,N: nat] :
% 0.33/0.64 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ K ) )
% 0.33/0.64 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_index_swap
% 0.33/0.64 thf(fact_7720_tanh__add,axiom,
% 0.33/0.64 ! [X: complex,Y: complex] :
% 0.33/0.64 ( ( ( cosh_complex @ X )
% 0.33/0.64 != zero_zero_complex )
% 0.33/0.64 => ( ( ( cosh_complex @ Y )
% 0.33/0.64 != zero_zero_complex )
% 0.33/0.64 => ( ( tanh_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.33/0.64 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % tanh_add
% 0.33/0.64 thf(fact_7721_tanh__add,axiom,
% 0.33/0.64 ! [X: real,Y: real] :
% 0.33/0.64 ( ( ( cosh_real @ X )
% 0.33/0.64 != zero_zero_real )
% 0.33/0.64 => ( ( ( cosh_real @ Y )
% 0.33/0.64 != zero_zero_real )
% 0.33/0.64 => ( ( tanh_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.64 = ( divide_divide_real @ ( plus_plus_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % tanh_add
% 0.33/0.64 thf(fact_7722_gbinomial__minus,axiom,
% 0.33/0.64 ! [A: complex,K: nat] :
% 0.33/0.64 ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
% 0.33/0.64 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_minus
% 0.33/0.64 thf(fact_7723_gbinomial__minus,axiom,
% 0.33/0.64 ! [A: real,K: nat] :
% 0.33/0.64 ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
% 0.33/0.64 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_minus
% 0.33/0.64 thf(fact_7724_gbinomial__minus,axiom,
% 0.33/0.64 ! [A: rat,K: nat] :
% 0.33/0.64 ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
% 0.33/0.64 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_minus
% 0.33/0.64 thf(fact_7725_gbinomial__reduce__nat,axiom,
% 0.33/0.64 ! [K: nat,A: complex] :
% 0.33/0.64 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.64 => ( ( gbinomial_complex @ A @ K )
% 0.33/0.64 = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_reduce_nat
% 0.33/0.64 thf(fact_7726_gbinomial__reduce__nat,axiom,
% 0.33/0.64 ! [K: nat,A: real] :
% 0.33/0.64 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.64 => ( ( gbinomial_real @ A @ K )
% 0.33/0.64 = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_reduce_nat
% 0.33/0.64 thf(fact_7727_gbinomial__reduce__nat,axiom,
% 0.33/0.64 ! [K: nat,A: rat] :
% 0.33/0.64 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.64 => ( ( gbinomial_rat @ A @ K )
% 0.33/0.64 = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_reduce_nat
% 0.33/0.64 thf(fact_7728_gbinomial__pochhammer,axiom,
% 0.33/0.64 ( gbinomial_complex
% 0.33/0.64 = ( ^ [A2: complex,K2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A2 ) @ K2 ) ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_pochhammer
% 0.33/0.64 thf(fact_7729_gbinomial__pochhammer,axiom,
% 0.33/0.64 ( gbinomial_rat
% 0.33/0.64 = ( ^ [A2: rat,K2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A2 ) @ K2 ) ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_pochhammer
% 0.33/0.64 thf(fact_7730_gbinomial__pochhammer,axiom,
% 0.33/0.64 ( gbinomial_real
% 0.33/0.64 = ( ^ [A2: real,K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A2 ) @ K2 ) ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_pochhammer
% 0.33/0.64 thf(fact_7731_gbinomial__pochhammer_H,axiom,
% 0.33/0.64 ( gbinomial_complex
% 0.33/0.64 = ( ^ [A2: complex,K2: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A2 @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_pochhammer'
% 0.33/0.64 thf(fact_7732_gbinomial__pochhammer_H,axiom,
% 0.33/0.64 ( gbinomial_rat
% 0.33/0.64 = ( ^ [A2: rat,K2: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_pochhammer'
% 0.33/0.64 thf(fact_7733_gbinomial__pochhammer_H,axiom,
% 0.33/0.64 ( gbinomial_real
% 0.33/0.64 = ( ^ [A2: real,K2: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A2 @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gbinomial_pochhammer'
% 0.33/0.64 thf(fact_7734_cosh__field__def,axiom,
% 0.33/0.64 ( cosh_real
% 0.33/0.64 = ( ^ [Z5: real] : ( divide_divide_real @ ( plus_plus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_field_def
% 0.33/0.64 thf(fact_7735_cosh__field__def,axiom,
% 0.33/0.64 ( cosh_complex
% 0.33/0.64 = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_field_def
% 0.33/0.64 thf(fact_7736_xor__nat__unfold,axiom,
% 0.33/0.64 ( bit_se6528837805403552850or_nat
% 0.33/0.64 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_nat_unfold
% 0.33/0.64 thf(fact_7737_cosh__square__eq,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( plus_plus_real @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_square_eq
% 0.33/0.64 thf(fact_7738_cosh__square__eq,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( plus_plus_complex @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_square_eq
% 0.33/0.64 thf(fact_7739_sinh__square__eq,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% 0.33/0.64
% 0.33/0.64 % sinh_square_eq
% 0.33/0.64 thf(fact_7740_sinh__square__eq,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % sinh_square_eq
% 0.33/0.64 thf(fact_7741_hyperbolic__pythagoras,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = one_one_complex ) ).
% 0.33/0.64
% 0.33/0.64 % hyperbolic_pythagoras
% 0.33/0.64 thf(fact_7742_hyperbolic__pythagoras,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = one_one_real ) ).
% 0.33/0.64
% 0.33/0.64 % hyperbolic_pythagoras
% 0.33/0.64 thf(fact_7743_xor__nat__rec,axiom,
% 0.33/0.64 ( bit_se6528837805403552850or_nat
% 0.33/0.64 = ( ^ [M3: nat,N3: nat] :
% 0.33/0.64 ( plus_plus_nat
% 0.33/0.64 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.64 @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 0.33/0.64 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 0.33/0.64 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_nat_rec
% 0.33/0.64 thf(fact_7744_xor__one__eq,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( bit_se3222712562003087583nteger @ A @ one_one_Code_integer )
% 0.33/0.64 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 0.33/0.64 @ ( zero_n356916108424825756nteger
% 0.33/0.64 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_one_eq
% 0.33/0.64 thf(fact_7745_xor__one__eq,axiom,
% 0.33/0.64 ! [A: nat] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
% 0.33/0.64 = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 0.33/0.64 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.64 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_one_eq
% 0.33/0.64 thf(fact_7746_xor__one__eq,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ A @ one_one_int )
% 0.33/0.64 = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 0.33/0.64 @ ( zero_n2684676970156552555ol_int
% 0.33/0.64 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_one_eq
% 0.33/0.64 thf(fact_7747_one__xor__eq,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A )
% 0.33/0.64 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 0.33/0.64 @ ( zero_n356916108424825756nteger
% 0.33/0.64 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % one_xor_eq
% 0.33/0.64 thf(fact_7748_one__xor__eq,axiom,
% 0.33/0.64 ! [A: nat] :
% 0.33/0.64 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
% 0.33/0.64 = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 0.33/0.64 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.64 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % one_xor_eq
% 0.33/0.64 thf(fact_7749_one__xor__eq,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ one_one_int @ A )
% 0.33/0.64 = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 0.33/0.64 @ ( zero_n2684676970156552555ol_int
% 0.33/0.64 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % one_xor_eq
% 0.33/0.64 thf(fact_7750_cosh__zero__iff,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ( cosh_real @ X )
% 0.33/0.64 = zero_zero_real )
% 0.33/0.64 = ( ( power_power_real @ ( exp_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_zero_iff
% 0.33/0.64 thf(fact_7751_cosh__zero__iff,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( ( cosh_complex @ X )
% 0.33/0.64 = zero_zero_complex )
% 0.33/0.64 = ( ( power_power_complex @ ( exp_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_zero_iff
% 0.33/0.64 thf(fact_7752_Cauchy__iff2,axiom,
% 0.33/0.64 ( topolo4055970368930404560y_real
% 0.33/0.64 = ( ^ [X8: nat > real] :
% 0.33/0.64 ! [J3: nat] :
% 0.33/0.64 ? [M7: nat] :
% 0.33/0.64 ! [M3: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M7 @ M3 )
% 0.33/0.64 => ! [N3: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M7 @ N3 )
% 0.33/0.64 => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Cauchy_iff2
% 0.33/0.64 thf(fact_7753_csqrt_Osimps_I1_J,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( re @ ( csqrt @ Z2 ) )
% 0.33/0.64 = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % csqrt.simps(1)
% 0.33/0.64 thf(fact_7754_sinh__def,axiom,
% 0.33/0.64 ( sinh_real
% 0.33/0.64 = ( ^ [X3: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sinh_def
% 0.33/0.64 thf(fact_7755_sinh__def,axiom,
% 0.33/0.64 ( sinh_complex
% 0.33/0.64 = ( ^ [X3: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( minus_minus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sinh_def
% 0.33/0.64 thf(fact_7756_cosh__def,axiom,
% 0.33/0.64 ( cosh_real
% 0.33/0.64 = ( ^ [X3: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_def
% 0.33/0.64 thf(fact_7757_cosh__def,axiom,
% 0.33/0.64 ( cosh_complex
% 0.33/0.64 = ( ^ [X3: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cosh_def
% 0.33/0.64 thf(fact_7758_complex__div__cnj,axiom,
% 0.33/0.64 ( divide1717551699836669952omplex
% 0.33/0.64 = ( ^ [A2: complex,B2: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ ( cnj @ B2 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_div_cnj
% 0.33/0.64 thf(fact_7759_scaleR__cancel__right,axiom,
% 0.33/0.64 ! [A: real,X: real,B: real] :
% 0.33/0.64 ( ( ( real_V1485227260804924795R_real @ A @ X )
% 0.33/0.64 = ( real_V1485227260804924795R_real @ B @ X ) )
% 0.33/0.64 = ( ( A = B )
% 0.33/0.64 | ( X = zero_zero_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_cancel_right
% 0.33/0.64 thf(fact_7760_scaleR__cancel__right,axiom,
% 0.33/0.64 ! [A: real,X: complex,B: real] :
% 0.33/0.64 ( ( ( real_V2046097035970521341omplex @ A @ X )
% 0.33/0.64 = ( real_V2046097035970521341omplex @ B @ X ) )
% 0.33/0.64 = ( ( A = B )
% 0.33/0.64 | ( X = zero_zero_complex ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_cancel_right
% 0.33/0.64 thf(fact_7761_scaleR__zero__right,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( real_V1485227260804924795R_real @ A @ zero_zero_real )
% 0.33/0.64 = zero_zero_real ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_zero_right
% 0.33/0.64 thf(fact_7762_scaleR__zero__right,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ A @ zero_zero_complex )
% 0.33/0.64 = zero_zero_complex ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_zero_right
% 0.33/0.64 thf(fact_7763_mult__scaleR__right,axiom,
% 0.33/0.64 ! [X: real,A: real,Y: real] :
% 0.33/0.64 ( ( times_times_real @ X @ ( real_V1485227260804924795R_real @ A @ Y ) )
% 0.33/0.64 = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mult_scaleR_right
% 0.33/0.64 thf(fact_7764_mult__scaleR__right,axiom,
% 0.33/0.64 ! [X: complex,A: real,Y: complex] :
% 0.33/0.64 ( ( times_times_complex @ X @ ( real_V2046097035970521341omplex @ A @ Y ) )
% 0.33/0.64 = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mult_scaleR_right
% 0.33/0.64 thf(fact_7765_mult__scaleR__left,axiom,
% 0.33/0.64 ! [A: real,X: real,Y: real] :
% 0.33/0.64 ( ( times_times_real @ ( real_V1485227260804924795R_real @ A @ X ) @ Y )
% 0.33/0.64 = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mult_scaleR_left
% 0.33/0.64 thf(fact_7766_mult__scaleR__left,axiom,
% 0.33/0.64 ! [A: real,X: complex,Y: complex] :
% 0.33/0.64 ( ( times_times_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ Y )
% 0.33/0.64 = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mult_scaleR_left
% 0.33/0.64 thf(fact_7767_scaleR__one,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( real_V1485227260804924795R_real @ one_one_real @ X )
% 0.33/0.64 = X ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_one
% 0.33/0.64 thf(fact_7768_scaleR__one,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ one_one_real @ X )
% 0.33/0.64 = X ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_one
% 0.33/0.64 thf(fact_7769_scaleR__scaleR,axiom,
% 0.33/0.64 ! [A: real,B: real,X: real] :
% 0.33/0.64 ( ( real_V1485227260804924795R_real @ A @ ( real_V1485227260804924795R_real @ B @ X ) )
% 0.33/0.64 = ( real_V1485227260804924795R_real @ ( times_times_real @ A @ B ) @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_scaleR
% 0.33/0.64 thf(fact_7770_scaleR__scaleR,axiom,
% 0.33/0.64 ! [A: real,B: real,X: complex] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ A @ ( real_V2046097035970521341omplex @ B @ X ) )
% 0.33/0.64 = ( real_V2046097035970521341omplex @ ( times_times_real @ A @ B ) @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_scaleR
% 0.33/0.64 thf(fact_7771_complex__Re__of__nat,axiom,
% 0.33/0.64 ! [N: nat] :
% 0.33/0.64 ( ( re @ ( semiri8010041392384452111omplex @ N ) )
% 0.33/0.64 = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_Re_of_nat
% 0.33/0.64 thf(fact_7772_complex__cnj__mult,axiom,
% 0.33/0.64 ! [X: complex,Y: complex] :
% 0.33/0.64 ( ( cnj @ ( times_times_complex @ X @ Y ) )
% 0.33/0.64 = ( times_times_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_cnj_mult
% 0.33/0.64 thf(fact_7773_complex__cnj__one__iff,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( ( cnj @ Z2 )
% 0.33/0.64 = one_one_complex )
% 0.33/0.64 = ( Z2 = one_one_complex ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_cnj_one_iff
% 0.33/0.64 thf(fact_7774_complex__cnj__one,axiom,
% 0.33/0.64 ( ( cnj @ one_one_complex )
% 0.33/0.64 = one_one_complex ) ).
% 0.33/0.64
% 0.33/0.64 % complex_cnj_one
% 0.33/0.64 thf(fact_7775_complex__cnj__power,axiom,
% 0.33/0.64 ! [X: complex,N: nat] :
% 0.33/0.64 ( ( cnj @ ( power_power_complex @ X @ N ) )
% 0.33/0.64 = ( power_power_complex @ ( cnj @ X ) @ N ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_cnj_power
% 0.33/0.64 thf(fact_7776_complex__cnj__add,axiom,
% 0.33/0.64 ! [X: complex,Y: complex] :
% 0.33/0.64 ( ( cnj @ ( plus_plus_complex @ X @ Y ) )
% 0.33/0.64 = ( plus_plus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_cnj_add
% 0.33/0.64 thf(fact_7777_complex__cnj__diff,axiom,
% 0.33/0.64 ! [X: complex,Y: complex] :
% 0.33/0.64 ( ( cnj @ ( minus_minus_complex @ X @ Y ) )
% 0.33/0.64 = ( minus_minus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_cnj_diff
% 0.33/0.64 thf(fact_7778_scaleR__eq__0__iff,axiom,
% 0.33/0.64 ! [A: real,X: real] :
% 0.33/0.64 ( ( ( real_V1485227260804924795R_real @ A @ X )
% 0.33/0.64 = zero_zero_real )
% 0.33/0.64 = ( ( A = zero_zero_real )
% 0.33/0.64 | ( X = zero_zero_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_eq_0_iff
% 0.33/0.64 thf(fact_7779_scaleR__eq__0__iff,axiom,
% 0.33/0.64 ! [A: real,X: complex] :
% 0.33/0.64 ( ( ( real_V2046097035970521341omplex @ A @ X )
% 0.33/0.64 = zero_zero_complex )
% 0.33/0.64 = ( ( A = zero_zero_real )
% 0.33/0.64 | ( X = zero_zero_complex ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_eq_0_iff
% 0.33/0.64 thf(fact_7780_scaleR__zero__left,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
% 0.33/0.64 = zero_zero_real ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_zero_left
% 0.33/0.64 thf(fact_7781_scaleR__zero__left,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ zero_zero_real @ X )
% 0.33/0.64 = zero_zero_complex ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_zero_left
% 0.33/0.64 thf(fact_7782_scaleR__eq__iff,axiom,
% 0.33/0.64 ! [B: real,U: real,A: real] :
% 0.33/0.64 ( ( ( plus_plus_real @ B @ ( real_V1485227260804924795R_real @ U @ A ) )
% 0.33/0.64 = ( plus_plus_real @ A @ ( real_V1485227260804924795R_real @ U @ B ) ) )
% 0.33/0.64 = ( ( A = B )
% 0.33/0.64 | ( U = one_one_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_eq_iff
% 0.33/0.64 thf(fact_7783_scaleR__eq__iff,axiom,
% 0.33/0.64 ! [B: complex,U: real,A: complex] :
% 0.33/0.64 ( ( ( plus_plus_complex @ B @ ( real_V2046097035970521341omplex @ U @ A ) )
% 0.33/0.64 = ( plus_plus_complex @ A @ ( real_V2046097035970521341omplex @ U @ B ) ) )
% 0.33/0.64 = ( ( A = B )
% 0.33/0.64 | ( U = one_one_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_eq_iff
% 0.33/0.64 thf(fact_7784_scaleR__power,axiom,
% 0.33/0.64 ! [X: real,Y: real,N: nat] :
% 0.33/0.64 ( ( power_power_real @ ( real_V1485227260804924795R_real @ X @ Y ) @ N )
% 0.33/0.64 = ( real_V1485227260804924795R_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_power
% 0.33/0.64 thf(fact_7785_scaleR__power,axiom,
% 0.33/0.64 ! [X: real,Y: complex,N: nat] :
% 0.33/0.64 ( ( power_power_complex @ ( real_V2046097035970521341omplex @ X @ Y ) @ N )
% 0.33/0.64 = ( real_V2046097035970521341omplex @ ( power_power_real @ X @ N ) @ ( power_power_complex @ Y @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_power
% 0.33/0.64 thf(fact_7786_Re__sgn,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( re @ ( sgn_sgn_complex @ Z2 ) )
% 0.33/0.64 = ( divide_divide_real @ ( re @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Re_sgn
% 0.33/0.64 thf(fact_7787_xor__nonnegative__int__iff,axiom,
% 0.33/0.64 ! [K: int,L: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 0.33/0.64 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.64 = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_nonnegative_int_iff
% 0.33/0.64 thf(fact_7788_xor__negative__int__iff,axiom,
% 0.33/0.64 ! [K: int,L: int] :
% 0.33/0.64 ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
% 0.33/0.64 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.33/0.64 != ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_negative_int_iff
% 0.33/0.64 thf(fact_7789_scaleR__minus1__left,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.64 = ( uminus_uminus_real @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_minus1_left
% 0.33/0.64 thf(fact_7790_scaleR__minus1__left,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.64 = ( uminus1482373934393186551omplex @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_minus1_left
% 0.33/0.64 thf(fact_7791_scaleR__collapse,axiom,
% 0.33/0.64 ! [U: real,A: real] :
% 0.33/0.64 ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V1485227260804924795R_real @ U @ A ) )
% 0.33/0.64 = A ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_collapse
% 0.33/0.64 thf(fact_7792_scaleR__collapse,axiom,
% 0.33/0.64 ! [U: real,A: complex] :
% 0.33/0.64 ( ( plus_plus_complex @ ( real_V2046097035970521341omplex @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V2046097035970521341omplex @ U @ A ) )
% 0.33/0.64 = A ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_collapse
% 0.33/0.64 thf(fact_7793_norm__scaleR,axiom,
% 0.33/0.64 ! [A: real,X: real] :
% 0.33/0.64 ( ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ A @ X ) )
% 0.33/0.64 = ( times_times_real @ ( abs_abs_real @ A ) @ ( real_V7735802525324610683m_real @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % norm_scaleR
% 0.33/0.64 thf(fact_7794_norm__scaleR,axiom,
% 0.33/0.64 ! [A: real,X: complex] :
% 0.33/0.64 ( ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
% 0.33/0.64 = ( times_times_real @ ( abs_abs_real @ A ) @ ( real_V1022390504157884413omplex @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % norm_scaleR
% 0.33/0.64 thf(fact_7795_Re__divide__of__nat,axiom,
% 0.33/0.64 ! [Z2: complex,N: nat] :
% 0.33/0.64 ( ( re @ ( divide1717551699836669952omplex @ Z2 @ ( semiri8010041392384452111omplex @ N ) ) )
% 0.33/0.64 = ( divide_divide_real @ ( re @ Z2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Re_divide_of_nat
% 0.33/0.64 thf(fact_7796_Re__divide__of__real,axiom,
% 0.33/0.64 ! [Z2: complex,R: real] :
% 0.33/0.64 ( ( re @ ( divide1717551699836669952omplex @ Z2 @ ( real_V4546457046886955230omplex @ R ) ) )
% 0.33/0.64 = ( divide_divide_real @ ( re @ Z2 ) @ R ) ) ).
% 0.33/0.64
% 0.33/0.64 % Re_divide_of_real
% 0.33/0.64 thf(fact_7797_scaleR__times,axiom,
% 0.33/0.64 ! [U: num,W: num,A: real] :
% 0.33/0.64 ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ U ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 0.33/0.64 = ( real_V1485227260804924795R_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_times
% 0.33/0.64 thf(fact_7798_scaleR__times,axiom,
% 0.33/0.64 ! [U: num,W: num,A: complex] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ U ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 0.33/0.64 = ( real_V2046097035970521341omplex @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_times
% 0.33/0.64 thf(fact_7799_Re__divide__numeral,axiom,
% 0.33/0.64 ! [Z2: complex,W: num] :
% 0.33/0.64 ( ( re @ ( divide1717551699836669952omplex @ Z2 @ ( numera6690914467698888265omplex @ W ) ) )
% 0.33/0.64 = ( divide_divide_real @ ( re @ Z2 ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Re_divide_numeral
% 0.33/0.64 thf(fact_7800_inverse__scaleR__times,axiom,
% 0.33/0.64 ! [V: num,W: num,A: real] :
% 0.33/0.64 ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 0.33/0.64 = ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_scaleR_times
% 0.33/0.64 thf(fact_7801_inverse__scaleR__times,axiom,
% 0.33/0.64 ! [V: num,W: num,A: complex] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 0.33/0.64 = ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_scaleR_times
% 0.33/0.64 thf(fact_7802_fraction__scaleR__times,axiom,
% 0.33/0.64 ! [U: num,V: num,W: num,A: real] :
% 0.33/0.64 ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 0.33/0.64 = ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % fraction_scaleR_times
% 0.33/0.64 thf(fact_7803_fraction__scaleR__times,axiom,
% 0.33/0.64 ! [U: num,V: num,W: num,A: complex] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 0.33/0.64 = ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % fraction_scaleR_times
% 0.33/0.64 thf(fact_7804_scaleR__half__double,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ A @ A ) )
% 0.33/0.64 = A ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_half_double
% 0.33/0.64 thf(fact_7805_scaleR__half__double,axiom,
% 0.33/0.64 ! [A: complex] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_complex @ A @ A ) )
% 0.33/0.64 = A ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_half_double
% 0.33/0.64 thf(fact_7806_bit__xor__int__iff,axiom,
% 0.33/0.64 ! [K: int,L: int,N: nat] :
% 0.33/0.64 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ N )
% 0.33/0.64 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.33/0.64 != ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit_xor_int_iff
% 0.33/0.64 thf(fact_7807_real__scaleR__def,axiom,
% 0.33/0.64 real_V1485227260804924795R_real = times_times_real ).
% 0.33/0.64
% 0.33/0.64 % real_scaleR_def
% 0.33/0.64 thf(fact_7808_scaleR__right__imp__eq,axiom,
% 0.33/0.64 ! [X: real,A: real,B: real] :
% 0.33/0.64 ( ( X != zero_zero_real )
% 0.33/0.64 => ( ( ( real_V1485227260804924795R_real @ A @ X )
% 0.33/0.64 = ( real_V1485227260804924795R_real @ B @ X ) )
% 0.33/0.64 => ( A = B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_right_imp_eq
% 0.33/0.64 thf(fact_7809_scaleR__right__imp__eq,axiom,
% 0.33/0.64 ! [X: complex,A: real,B: real] :
% 0.33/0.64 ( ( X != zero_zero_complex )
% 0.33/0.64 => ( ( ( real_V2046097035970521341omplex @ A @ X )
% 0.33/0.64 = ( real_V2046097035970521341omplex @ B @ X ) )
% 0.33/0.64 => ( A = B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_right_imp_eq
% 0.33/0.64 thf(fact_7810_scaleR__right__distrib,axiom,
% 0.33/0.64 ! [A: real,X: real,Y: real] :
% 0.33/0.64 ( ( real_V1485227260804924795R_real @ A @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.64 = ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_right_distrib
% 0.33/0.64 thf(fact_7811_scaleR__right__distrib,axiom,
% 0.33/0.64 ! [A: real,X: complex,Y: complex] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ A @ ( plus_plus_complex @ X @ Y ) )
% 0.33/0.64 = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ A @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_right_distrib
% 0.33/0.64 thf(fact_7812_scaleR__right__diff__distrib,axiom,
% 0.33/0.64 ! [A: real,X: real,Y: real] :
% 0.33/0.64 ( ( real_V1485227260804924795R_real @ A @ ( minus_minus_real @ X @ Y ) )
% 0.33/0.64 = ( minus_minus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_right_diff_distrib
% 0.33/0.64 thf(fact_7813_scaleR__right__diff__distrib,axiom,
% 0.33/0.64 ! [A: real,X: complex,Y: complex] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ A @ ( minus_minus_complex @ X @ Y ) )
% 0.33/0.64 = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ A @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_right_diff_distrib
% 0.33/0.64 thf(fact_7814_scaleR__complex_Osimps_I1_J,axiom,
% 0.33/0.64 ! [R: real,X: complex] :
% 0.33/0.64 ( ( re @ ( real_V2046097035970521341omplex @ R @ X ) )
% 0.33/0.64 = ( times_times_real @ R @ ( re @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_complex.simps(1)
% 0.33/0.64 thf(fact_7815_Re__complex__div__eq__0,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.64 = zero_zero_real )
% 0.33/0.64 = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 0.33/0.64 = zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % Re_complex_div_eq_0
% 0.33/0.64 thf(fact_7816_complex__mod__sqrt__Re__mult__cnj,axiom,
% 0.33/0.64 ( real_V1022390504157884413omplex
% 0.33/0.64 = ( ^ [Z5: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z5 @ ( cnj @ Z5 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_mod_sqrt_Re_mult_cnj
% 0.33/0.64 thf(fact_7817_Re__complex__div__lt__0,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.33/0.64 = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % Re_complex_div_lt_0
% 0.33/0.64 thf(fact_7818_Re__complex__div__gt__0,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.33/0.64 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Re_complex_div_gt_0
% 0.33/0.64 thf(fact_7819_Re__complex__div__le__0,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.33/0.64 = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % Re_complex_div_le_0
% 0.33/0.64 thf(fact_7820_Re__complex__div__ge__0,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.33/0.64 = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Re_complex_div_ge_0
% 0.33/0.64 thf(fact_7821_XOR__lower,axiom,
% 0.33/0.64 ! [X: int,Y: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.64 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.64 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % XOR_lower
% 0.33/0.64 thf(fact_7822_scaleR__left_Oadd,axiom,
% 0.33/0.64 ! [X: real,Y: real,Xa: real] :
% 0.33/0.64 ( ( real_V1485227260804924795R_real @ ( plus_plus_real @ X @ Y ) @ Xa )
% 0.33/0.64 = ( plus_plus_real @ ( real_V1485227260804924795R_real @ X @ Xa ) @ ( real_V1485227260804924795R_real @ Y @ Xa ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_left.add
% 0.33/0.64 thf(fact_7823_scaleR__left_Oadd,axiom,
% 0.33/0.64 ! [X: real,Y: real,Xa: complex] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ ( plus_plus_real @ X @ Y ) @ Xa )
% 0.33/0.64 = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ X @ Xa ) @ ( real_V2046097035970521341omplex @ Y @ Xa ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_left.add
% 0.33/0.64 thf(fact_7824_scaleR__left__distrib,axiom,
% 0.33/0.64 ! [A: real,B: real,X: real] :
% 0.33/0.64 ( ( real_V1485227260804924795R_real @ ( plus_plus_real @ A @ B ) @ X )
% 0.33/0.64 = ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_left_distrib
% 0.33/0.64 thf(fact_7825_scaleR__left__distrib,axiom,
% 0.33/0.64 ! [A: real,B: real,X: complex] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ ( plus_plus_real @ A @ B ) @ X )
% 0.33/0.64 = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ B @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_left_distrib
% 0.33/0.64 thf(fact_7826_scaleR__conv__of__real,axiom,
% 0.33/0.64 ( real_V1485227260804924795R_real
% 0.33/0.64 = ( ^ [R5: real] : ( times_times_real @ ( real_V1803761363581548252l_real @ R5 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_conv_of_real
% 0.33/0.64 thf(fact_7827_scaleR__conv__of__real,axiom,
% 0.33/0.64 ( real_V2046097035970521341omplex
% 0.33/0.64 = ( ^ [R5: real] : ( times_times_complex @ ( real_V4546457046886955230omplex @ R5 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_conv_of_real
% 0.33/0.64 thf(fact_7828_of__real__def,axiom,
% 0.33/0.64 ( real_V1803761363581548252l_real
% 0.33/0.64 = ( ^ [R5: real] : ( real_V1485227260804924795R_real @ R5 @ one_one_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_real_def
% 0.33/0.64 thf(fact_7829_of__real__def,axiom,
% 0.33/0.64 ( real_V4546457046886955230omplex
% 0.33/0.64 = ( ^ [R5: real] : ( real_V2046097035970521341omplex @ R5 @ one_one_complex ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_real_def
% 0.33/0.64 thf(fact_7830_scaleR__left_Odiff,axiom,
% 0.33/0.64 ! [X: real,Y: real,Xa: real] :
% 0.33/0.64 ( ( real_V1485227260804924795R_real @ ( minus_minus_real @ X @ Y ) @ Xa )
% 0.33/0.64 = ( minus_minus_real @ ( real_V1485227260804924795R_real @ X @ Xa ) @ ( real_V1485227260804924795R_real @ Y @ Xa ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_left.diff
% 0.33/0.64 thf(fact_7831_scaleR__left_Odiff,axiom,
% 0.33/0.64 ! [X: real,Y: real,Xa: complex] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ ( minus_minus_real @ X @ Y ) @ Xa )
% 0.33/0.64 = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ X @ Xa ) @ ( real_V2046097035970521341omplex @ Y @ Xa ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_left.diff
% 0.33/0.64 thf(fact_7832_scaleR__left__diff__distrib,axiom,
% 0.33/0.64 ! [A: real,B: real,X: real] :
% 0.33/0.64 ( ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ X )
% 0.33/0.64 = ( minus_minus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_left_diff_distrib
% 0.33/0.64 thf(fact_7833_scaleR__left__diff__distrib,axiom,
% 0.33/0.64 ! [A: real,B: real,X: complex] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ ( minus_minus_real @ A @ B ) @ X )
% 0.33/0.64 = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ B @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_left_diff_distrib
% 0.33/0.64 thf(fact_7834_complex__scaleR,axiom,
% 0.33/0.64 ! [R: real,A: real,B: real] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ R @ ( complex2 @ A @ B ) )
% 0.33/0.64 = ( complex2 @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_scaleR
% 0.33/0.64 thf(fact_7835_one__complex_Osimps_I1_J,axiom,
% 0.33/0.64 ( ( re @ one_one_complex )
% 0.33/0.64 = one_one_real ) ).
% 0.33/0.64
% 0.33/0.64 % one_complex.simps(1)
% 0.33/0.64 thf(fact_7836_plus__complex_Osimps_I1_J,axiom,
% 0.33/0.64 ! [X: complex,Y: complex] :
% 0.33/0.64 ( ( re @ ( plus_plus_complex @ X @ Y ) )
% 0.33/0.64 = ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % plus_complex.simps(1)
% 0.33/0.64 thf(fact_7837_minus__complex_Osimps_I1_J,axiom,
% 0.33/0.64 ! [X: complex,Y: complex] :
% 0.33/0.64 ( ( re @ ( minus_minus_complex @ X @ Y ) )
% 0.33/0.64 = ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_complex.simps(1)
% 0.33/0.64 thf(fact_7838_scaleR__right__mono__neg,axiom,
% 0.33/0.64 ! [B: real,A: real,C: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ B @ A )
% 0.33/0.64 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.33/0.64 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ C ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_right_mono_neg
% 0.33/0.64 thf(fact_7839_scaleR__right__mono,axiom,
% 0.33/0.64 ! [A: real,B: real,X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ A @ B )
% 0.33/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_right_mono
% 0.33/0.64 thf(fact_7840_complex__add__cnj,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( plus_plus_complex @ Z2 @ ( cnj @ Z2 ) )
% 0.33/0.64 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_add_cnj
% 0.33/0.64 thf(fact_7841_vector__fraction__eq__iff,axiom,
% 0.33/0.64 ! [U: real,V: real,A: real,X: real] :
% 0.33/0.64 ( ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A )
% 0.33/0.64 = X )
% 0.33/0.64 = ( ( ( V = zero_zero_real )
% 0.33/0.64 => ( X = zero_zero_real ) )
% 0.33/0.64 & ( ( V != zero_zero_real )
% 0.33/0.64 => ( ( real_V1485227260804924795R_real @ U @ A )
% 0.33/0.64 = ( real_V1485227260804924795R_real @ V @ X ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % vector_fraction_eq_iff
% 0.33/0.64 thf(fact_7842_vector__fraction__eq__iff,axiom,
% 0.33/0.64 ! [U: real,V: real,A: complex,X: complex] :
% 0.33/0.64 ( ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A )
% 0.33/0.64 = X )
% 0.33/0.64 = ( ( ( V = zero_zero_real )
% 0.33/0.64 => ( X = zero_zero_complex ) )
% 0.33/0.64 & ( ( V != zero_zero_real )
% 0.33/0.64 => ( ( real_V2046097035970521341omplex @ U @ A )
% 0.33/0.64 = ( real_V2046097035970521341omplex @ V @ X ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % vector_fraction_eq_iff
% 0.33/0.64 thf(fact_7843_eq__vector__fraction__iff,axiom,
% 0.33/0.64 ! [X: real,U: real,V: real,A: real] :
% 0.33/0.64 ( ( X
% 0.33/0.64 = ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A ) )
% 0.33/0.64 = ( ( ( V = zero_zero_real )
% 0.33/0.64 => ( X = zero_zero_real ) )
% 0.33/0.64 & ( ( V != zero_zero_real )
% 0.33/0.64 => ( ( real_V1485227260804924795R_real @ V @ X )
% 0.33/0.64 = ( real_V1485227260804924795R_real @ U @ A ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % eq_vector_fraction_iff
% 0.33/0.64 thf(fact_7844_eq__vector__fraction__iff,axiom,
% 0.33/0.64 ! [X: complex,U: real,V: real,A: complex] :
% 0.33/0.64 ( ( X
% 0.33/0.64 = ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A ) )
% 0.33/0.64 = ( ( ( V = zero_zero_real )
% 0.33/0.64 => ( X = zero_zero_complex ) )
% 0.33/0.64 & ( ( V != zero_zero_real )
% 0.33/0.64 => ( ( real_V2046097035970521341omplex @ V @ X )
% 0.33/0.64 = ( real_V2046097035970521341omplex @ U @ A ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % eq_vector_fraction_iff
% 0.33/0.64 thf(fact_7845_Real__Vector__Spaces_Ole__add__iff2,axiom,
% 0.33/0.64 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E ) @ D ) )
% 0.33/0.64 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Real_Vector_Spaces.le_add_iff2
% 0.33/0.64 thf(fact_7846_Real__Vector__Spaces_Ole__add__iff1,axiom,
% 0.33/0.64 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E ) @ D ) )
% 0.33/0.64 = ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.33/0.64
% 0.33/0.64 % Real_Vector_Spaces.le_add_iff1
% 0.33/0.64 thf(fact_7847_xor__nat__def,axiom,
% 0.33/0.64 ( bit_se6528837805403552850or_nat
% 0.33/0.64 = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_nat_def
% 0.33/0.64 thf(fact_7848_cnj__add__mult__eq__Re,axiom,
% 0.33/0.64 ! [Z2: complex,W: complex] :
% 0.33/0.64 ( ( plus_plus_complex @ ( times_times_complex @ Z2 @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z2 ) @ W ) )
% 0.33/0.64 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z2 @ ( cnj @ W ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cnj_add_mult_eq_Re
% 0.33/0.64 thf(fact_7849_flip__bit__int__def,axiom,
% 0.33/0.64 ( bit_se2159334234014336723it_int
% 0.33/0.64 = ( ^ [N3: nat,K2: int] : ( bit_se6526347334894502574or_int @ K2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % flip_bit_int_def
% 0.33/0.64 thf(fact_7850_zero__le__scaleR__iff,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) )
% 0.33/0.64 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.64 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.33/0.64 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.64 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.33/0.64 | ( A = zero_zero_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % zero_le_scaleR_iff
% 0.33/0.64 thf(fact_7851_scaleR__le__0__iff,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ B ) @ zero_zero_real )
% 0.33/0.64 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.64 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.33/0.64 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.64 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.33/0.64 | ( A = zero_zero_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_le_0_iff
% 0.33/0.64 thf(fact_7852_scaleR__nonpos__nonpos,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.64 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.33/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_nonpos_nonpos
% 0.33/0.64 thf(fact_7853_scaleR__nonpos__nonneg,axiom,
% 0.33/0.64 ! [A: real,X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_nonpos_nonneg
% 0.33/0.64 thf(fact_7854_scaleR__nonneg__nonpos,axiom,
% 0.33/0.64 ! [A: real,X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.64 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.64 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_nonneg_nonpos
% 0.33/0.64 thf(fact_7855_scaleR__nonneg__nonneg,axiom,
% 0.33/0.64 ! [A: real,X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_nonneg_nonneg
% 0.33/0.64 thf(fact_7856_split__scaleR__pos__le,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.64 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.33/0.64 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.64 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.33/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % split_scaleR_pos_le
% 0.33/0.64 thf(fact_7857_split__scaleR__neg__le,axiom,
% 0.33/0.64 ! [A: real,X: real] :
% 0.33/0.64 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.64 & ( ord_less_eq_real @ X @ zero_zero_real ) )
% 0.33/0.64 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.64 & ( ord_less_eq_real @ zero_zero_real @ X ) ) )
% 0.33/0.64 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % split_scaleR_neg_le
% 0.33/0.64 thf(fact_7858_scaleR__mono_H,axiom,
% 0.33/0.64 ! [A: real,B: real,C: real,D: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ A @ B )
% 0.33/0.64 => ( ( ord_less_eq_real @ C @ D )
% 0.33/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.33/0.64 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ D ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_mono'
% 0.33/0.64 thf(fact_7859_scaleR__mono,axiom,
% 0.33/0.64 ! [A: real,B: real,X: real,Y: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ A @ B )
% 0.33/0.64 => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ Y ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_mono
% 0.33/0.64 thf(fact_7860_scaleR__left__le__one__le,axiom,
% 0.33/0.64 ! [X: real,A: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.33/0.64 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_left_le_one_le
% 0.33/0.64 thf(fact_7861_scaleR__2,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
% 0.33/0.64 = ( plus_plus_real @ X @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_2
% 0.33/0.64 thf(fact_7862_scaleR__2,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
% 0.33/0.64 = ( plus_plus_complex @ X @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_2
% 0.33/0.64 thf(fact_7863_real__vector__eq__affinity,axiom,
% 0.33/0.64 ! [M: real,Y: real,X: real,C: real] :
% 0.33/0.64 ( ( M != zero_zero_real )
% 0.33/0.64 => ( ( Y
% 0.33/0.64 = ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X ) @ C ) )
% 0.33/0.64 = ( ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) )
% 0.33/0.64 = X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % real_vector_eq_affinity
% 0.33/0.64 thf(fact_7864_real__vector__eq__affinity,axiom,
% 0.33/0.64 ! [M: real,Y: complex,X: complex,C: complex] :
% 0.33/0.64 ( ( M != zero_zero_real )
% 0.33/0.64 => ( ( Y
% 0.33/0.64 = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ M @ X ) @ C ) )
% 0.33/0.64 = ( ( minus_minus_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ Y ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ C ) )
% 0.33/0.64 = X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % real_vector_eq_affinity
% 0.33/0.64 thf(fact_7865_real__vector__affinity__eq,axiom,
% 0.33/0.64 ! [M: real,X: real,C: real,Y: real] :
% 0.33/0.64 ( ( M != zero_zero_real )
% 0.33/0.64 => ( ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X ) @ C )
% 0.33/0.64 = Y )
% 0.33/0.64 = ( X
% 0.33/0.64 = ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % real_vector_affinity_eq
% 0.33/0.64 thf(fact_7866_real__vector__affinity__eq,axiom,
% 0.33/0.64 ! [M: real,X: complex,C: complex,Y: complex] :
% 0.33/0.64 ( ( M != zero_zero_real )
% 0.33/0.64 => ( ( ( plus_plus_complex @ ( real_V2046097035970521341omplex @ M @ X ) @ C )
% 0.33/0.64 = Y )
% 0.33/0.64 = ( X
% 0.33/0.64 = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ Y ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ C ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % real_vector_affinity_eq
% 0.33/0.64 thf(fact_7867_nonzero__inverse__scaleR__distrib,axiom,
% 0.33/0.64 ! [A: real,X: real] :
% 0.33/0.64 ( ( A != zero_zero_real )
% 0.33/0.64 => ( ( X != zero_zero_real )
% 0.33/0.64 => ( ( inverse_inverse_real @ ( real_V1485227260804924795R_real @ A @ X ) )
% 0.33/0.64 = ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ X ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % nonzero_inverse_scaleR_distrib
% 0.33/0.64 thf(fact_7868_nonzero__inverse__scaleR__distrib,axiom,
% 0.33/0.64 ! [A: real,X: complex] :
% 0.33/0.64 ( ( A != zero_zero_real )
% 0.33/0.64 => ( ( X != zero_zero_complex )
% 0.33/0.64 => ( ( invers8013647133539491842omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
% 0.33/0.64 = ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ A ) @ ( invers8013647133539491842omplex @ X ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % nonzero_inverse_scaleR_distrib
% 0.33/0.64 thf(fact_7869_cmod__plus__Re__le__0__iff,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ zero_zero_real )
% 0.33/0.64 = ( ( re @ Z2 )
% 0.33/0.64 = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cmod_plus_Re_le_0_iff
% 0.33/0.64 thf(fact_7870_cos__n__Re__cis__pow__n,axiom,
% 0.33/0.64 ! [N: nat,A: real] :
% 0.33/0.64 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 0.33/0.64 = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cos_n_Re_cis_pow_n
% 0.33/0.64 thf(fact_7871_complex__mod__mult__cnj,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) )
% 0.33/0.64 = ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_mod_mult_cnj
% 0.33/0.64 thf(fact_7872_Cauchy__iff,axiom,
% 0.33/0.64 ( topolo6517432010174082258omplex
% 0.33/0.64 = ( ^ [X8: nat > complex] :
% 0.33/0.64 ! [E3: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ E3 )
% 0.33/0.64 => ? [M7: nat] :
% 0.33/0.64 ! [M3: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M7 @ M3 )
% 0.33/0.64 => ! [N3: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M7 @ N3 )
% 0.33/0.64 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) ) @ E3 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Cauchy_iff
% 0.33/0.64 thf(fact_7873_Cauchy__iff,axiom,
% 0.33/0.64 ( topolo4055970368930404560y_real
% 0.33/0.64 = ( ^ [X8: nat > real] :
% 0.33/0.64 ! [E3: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ E3 )
% 0.33/0.64 => ? [M7: nat] :
% 0.33/0.64 ! [M3: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M7 @ M3 )
% 0.33/0.64 => ! [N3: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M7 @ N3 )
% 0.33/0.64 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) ) @ E3 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Cauchy_iff
% 0.33/0.64 thf(fact_7874_CauchyI,axiom,
% 0.33/0.64 ! [X9: nat > complex] :
% 0.33/0.64 ( ! [E2: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.33/0.64 => ? [M8: nat] :
% 0.33/0.64 ! [M2: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M8 @ M2 )
% 0.33/0.64 => ! [N2: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M8 @ N2 )
% 0.33/0.64 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X9 @ M2 ) @ ( X9 @ N2 ) ) ) @ E2 ) ) ) )
% 0.33/0.64 => ( topolo6517432010174082258omplex @ X9 ) ) ).
% 0.33/0.64
% 0.33/0.64 % CauchyI
% 0.33/0.64 thf(fact_7875_CauchyI,axiom,
% 0.33/0.64 ! [X9: nat > real] :
% 0.33/0.64 ( ! [E2: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.33/0.64 => ? [M8: nat] :
% 0.33/0.64 ! [M2: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M8 @ M2 )
% 0.33/0.64 => ! [N2: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M8 @ N2 )
% 0.33/0.64 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X9 @ M2 ) @ ( X9 @ N2 ) ) ) @ E2 ) ) ) )
% 0.33/0.64 => ( topolo4055970368930404560y_real @ X9 ) ) ).
% 0.33/0.64
% 0.33/0.64 % CauchyI
% 0.33/0.64 thf(fact_7876_CauchyD,axiom,
% 0.33/0.64 ! [X9: nat > complex,E: real] :
% 0.33/0.64 ( ( topolo6517432010174082258omplex @ X9 )
% 0.33/0.64 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.33/0.64 => ? [M9: nat] :
% 0.33/0.64 ! [M4: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M9 @ M4 )
% 0.33/0.64 => ! [N6: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M9 @ N6 )
% 0.33/0.64 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X9 @ M4 ) @ ( X9 @ N6 ) ) ) @ E ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % CauchyD
% 0.33/0.64 thf(fact_7877_CauchyD,axiom,
% 0.33/0.64 ! [X9: nat > real,E: real] :
% 0.33/0.64 ( ( topolo4055970368930404560y_real @ X9 )
% 0.33/0.64 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.33/0.64 => ? [M9: nat] :
% 0.33/0.64 ! [M4: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M9 @ M4 )
% 0.33/0.64 => ! [N6: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M9 @ N6 )
% 0.33/0.64 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X9 @ M4 ) @ ( X9 @ N6 ) ) ) @ E ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % CauchyD
% 0.33/0.64 thf(fact_7878_XOR__upper,axiom,
% 0.33/0.64 ! [X: int,N: nat,Y: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.64 => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.64 => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.64 => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % XOR_upper
% 0.33/0.64 thf(fact_7879_complex__norm__square,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_norm_square
% 0.33/0.64 thf(fact_7880_xor__int__rec,axiom,
% 0.33/0.64 ( bit_se6526347334894502574or_int
% 0.33/0.64 = ( ^ [K2: int,L2: int] :
% 0.33/0.64 ( plus_plus_int
% 0.33/0.64 @ ( zero_n2684676970156552555ol_int
% 0.33/0.64 @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) )
% 0.33/0.64 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 0.33/0.64 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_int_rec
% 0.33/0.64 thf(fact_7881_csqrt_Ocode,axiom,
% 0.33/0.64 ( csqrt
% 0.33/0.64 = ( ^ [Z5: complex] :
% 0.33/0.64 ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.64 @ ( times_times_real
% 0.33/0.64 @ ( if_real
% 0.33/0.64 @ ( ( im @ Z5 )
% 0.33/0.64 = zero_zero_real )
% 0.33/0.64 @ one_one_real
% 0.33/0.64 @ ( sgn_sgn_real @ ( im @ Z5 ) ) )
% 0.33/0.64 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % csqrt.code
% 0.33/0.64 thf(fact_7882_csqrt_Osimps_I2_J,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( im @ ( csqrt @ Z2 ) )
% 0.33/0.64 = ( times_times_real
% 0.33/0.64 @ ( if_real
% 0.33/0.64 @ ( ( im @ Z2 )
% 0.33/0.64 = zero_zero_real )
% 0.33/0.64 @ one_one_real
% 0.33/0.64 @ ( sgn_sgn_real @ ( im @ Z2 ) ) )
% 0.33/0.64 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % csqrt.simps(2)
% 0.33/0.64 thf(fact_7883_xor__int__unfold,axiom,
% 0.33/0.64 ( bit_se6526347334894502574or_int
% 0.33/0.64 = ( ^ [K2: int,L2: int] :
% 0.33/0.64 ( if_int
% 0.33/0.64 @ ( K2
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.64 @ ( bit_ri7919022796975470100ot_int @ L2 )
% 0.33/0.64 @ ( if_int
% 0.33/0.64 @ ( L2
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.64 @ ( bit_ri7919022796975470100ot_int @ K2 )
% 0.33/0.64 @ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_int_unfold
% 0.33/0.64 thf(fact_7884_csqrt__of__real__nonpos,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( ( im @ X )
% 0.33/0.64 = zero_zero_real )
% 0.33/0.64 => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
% 0.33/0.64 => ( ( csqrt @ X )
% 0.33/0.64 = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % csqrt_of_real_nonpos
% 0.33/0.64 thf(fact_7885_Complex__divide,axiom,
% 0.33/0.64 ( divide1717551699836669952omplex
% 0.33/0.64 = ( ^ [X3: complex,Y6: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Complex_divide
% 0.33/0.64 thf(fact_7886_bit_Odouble__compl,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( bit_ri7919022796975470100ot_int @ ( bit_ri7919022796975470100ot_int @ X ) )
% 0.33/0.64 = X ) ).
% 0.33/0.64
% 0.33/0.64 % bit.double_compl
% 0.33/0.64 thf(fact_7887_bit_Ocompl__eq__compl__iff,axiom,
% 0.33/0.64 ! [X: int,Y: int] :
% 0.33/0.64 ( ( ( bit_ri7919022796975470100ot_int @ X )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ Y ) )
% 0.33/0.64 = ( X = Y ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.compl_eq_compl_iff
% 0.33/0.64 thf(fact_7888_bit_Oxor__compl__left,axiom,
% 0.33/0.64 ! [X: int,Y: int] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ Y )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.xor_compl_left
% 0.33/0.64 thf(fact_7889_bit_Oxor__compl__right,axiom,
% 0.33/0.64 ! [X: int,Y: int] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ X @ ( bit_ri7919022796975470100ot_int @ Y ) )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.xor_compl_right
% 0.33/0.64 thf(fact_7890_bit_Oconj__cancel__left,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
% 0.33/0.64 = zero_zero_int ) ).
% 0.33/0.64
% 0.33/0.64 % bit.conj_cancel_left
% 0.33/0.64 thf(fact_7891_bit_Oconj__cancel__right,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
% 0.33/0.64 = zero_zero_int ) ).
% 0.33/0.64
% 0.33/0.64 % bit.conj_cancel_right
% 0.33/0.64 thf(fact_7892_Im__power__real,axiom,
% 0.33/0.64 ! [X: complex,N: nat] :
% 0.33/0.64 ( ( ( im @ X )
% 0.33/0.64 = zero_zero_real )
% 0.33/0.64 => ( ( im @ ( power_power_complex @ X @ N ) )
% 0.33/0.64 = zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % Im_power_real
% 0.33/0.64 thf(fact_7893_bit_Ocompl__zero,axiom,
% 0.33/0.64 ( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.compl_zero
% 0.33/0.64 thf(fact_7894_bit_Ocompl__zero,axiom,
% 0.33/0.64 ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.compl_zero
% 0.33/0.64 thf(fact_7895_bit_Ocompl__one,axiom,
% 0.33/0.64 ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.64 = zero_z3403309356797280102nteger ) ).
% 0.33/0.64
% 0.33/0.64 % bit.compl_one
% 0.33/0.64 thf(fact_7896_bit_Ocompl__one,axiom,
% 0.33/0.64 ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.64 = zero_zero_int ) ).
% 0.33/0.64
% 0.33/0.64 % bit.compl_one
% 0.33/0.64 thf(fact_7897_bit_Oxor__cancel__right,axiom,
% 0.33/0.64 ! [X: code_integer] :
% 0.33/0.64 ( ( bit_se3222712562003087583nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.xor_cancel_right
% 0.33/0.64 thf(fact_7898_bit_Oxor__cancel__right,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.xor_cancel_right
% 0.33/0.64 thf(fact_7899_bit_Oxor__cancel__left,axiom,
% 0.33/0.64 ! [X: code_integer] :
% 0.33/0.64 ( ( bit_se3222712562003087583nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.xor_cancel_left
% 0.33/0.64 thf(fact_7900_bit_Oxor__cancel__left,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.xor_cancel_left
% 0.33/0.64 thf(fact_7901_bit_Oxor__one__right,axiom,
% 0.33/0.64 ! [X: code_integer] :
% 0.33/0.64 ( ( bit_se3222712562003087583nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.64 = ( bit_ri7632146776885996613nteger @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.xor_one_right
% 0.33/0.64 thf(fact_7902_bit_Oxor__one__right,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.xor_one_right
% 0.33/0.64 thf(fact_7903_bit_Oxor__one__left,axiom,
% 0.33/0.64 ! [X: code_integer] :
% 0.33/0.64 ( ( bit_se3222712562003087583nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 0.33/0.64 = ( bit_ri7632146776885996613nteger @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.xor_one_left
% 0.33/0.64 thf(fact_7904_bit_Oxor__one__left,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.xor_one_left
% 0.33/0.64 thf(fact_7905_not__negative__int__iff,axiom,
% 0.33/0.64 ! [K: int] :
% 0.33/0.64 ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 0.33/0.64 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.33/0.64
% 0.33/0.64 % not_negative_int_iff
% 0.33/0.64 thf(fact_7906_not__nonnegative__int__iff,axiom,
% 0.33/0.64 ! [K: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.33/0.64 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % not_nonnegative_int_iff
% 0.33/0.64 thf(fact_7907_minus__not__numeral__eq,axiom,
% 0.33/0.64 ! [N: num] :
% 0.33/0.64 ( ( uminus1351360451143612070nteger @ ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.33/0.64 = ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_not_numeral_eq
% 0.33/0.64 thf(fact_7908_minus__not__numeral__eq,axiom,
% 0.33/0.64 ! [N: num] :
% 0.33/0.64 ( ( uminus_uminus_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.64 = ( numeral_numeral_int @ ( inc @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_not_numeral_eq
% 0.33/0.64 thf(fact_7909_complex__In__mult__cnj__zero,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( im @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) )
% 0.33/0.64 = zero_zero_real ) ).
% 0.33/0.64
% 0.33/0.64 % complex_In_mult_cnj_zero
% 0.33/0.64 thf(fact_7910_Im__i__times,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( im @ ( times_times_complex @ imaginary_unit @ Z2 ) )
% 0.33/0.64 = ( re @ Z2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % Im_i_times
% 0.33/0.64 thf(fact_7911_Im__divide__of__real,axiom,
% 0.33/0.64 ! [Z2: complex,R: real] :
% 0.33/0.64 ( ( im @ ( divide1717551699836669952omplex @ Z2 @ ( real_V4546457046886955230omplex @ R ) ) )
% 0.33/0.64 = ( divide_divide_real @ ( im @ Z2 ) @ R ) ) ).
% 0.33/0.64
% 0.33/0.64 % Im_divide_of_real
% 0.33/0.64 thf(fact_7912_Im__sgn,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( im @ ( sgn_sgn_complex @ Z2 ) )
% 0.33/0.64 = ( divide_divide_real @ ( im @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Im_sgn
% 0.33/0.64 thf(fact_7913_even__not__iff,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri7632146776885996613nteger @ A ) )
% 0.33/0.64 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % even_not_iff
% 0.33/0.64 thf(fact_7914_even__not__iff,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ A ) )
% 0.33/0.64 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % even_not_iff
% 0.33/0.64 thf(fact_7915_push__bit__minus__one__eq__not__mask,axiom,
% 0.33/0.64 ! [N: nat] :
% 0.33/0.64 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % push_bit_minus_one_eq_not_mask
% 0.33/0.64 thf(fact_7916_push__bit__minus__one__eq__not__mask,axiom,
% 0.33/0.64 ! [N: nat] :
% 0.33/0.64 ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.64 = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % push_bit_minus_one_eq_not_mask
% 0.33/0.64 thf(fact_7917_Re__power__real,axiom,
% 0.33/0.64 ! [X: complex,N: nat] :
% 0.33/0.64 ( ( ( im @ X )
% 0.33/0.64 = zero_zero_real )
% 0.33/0.64 => ( ( re @ ( power_power_complex @ X @ N ) )
% 0.33/0.64 = ( power_power_real @ ( re @ X ) @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Re_power_real
% 0.33/0.64 thf(fact_7918_Re__i__times,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( re @ ( times_times_complex @ imaginary_unit @ Z2 ) )
% 0.33/0.64 = ( uminus_uminus_real @ ( im @ Z2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Re_i_times
% 0.33/0.64 thf(fact_7919_Im__divide__numeral,axiom,
% 0.33/0.64 ! [Z2: complex,W: num] :
% 0.33/0.64 ( ( im @ ( divide1717551699836669952omplex @ Z2 @ ( numera6690914467698888265omplex @ W ) ) )
% 0.33/0.64 = ( divide_divide_real @ ( im @ Z2 ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Im_divide_numeral
% 0.33/0.64 thf(fact_7920_Im__divide__of__nat,axiom,
% 0.33/0.64 ! [Z2: complex,N: nat] :
% 0.33/0.64 ( ( im @ ( divide1717551699836669952omplex @ Z2 @ ( semiri8010041392384452111omplex @ N ) ) )
% 0.33/0.64 = ( divide_divide_real @ ( im @ Z2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Im_divide_of_nat
% 0.33/0.64 thf(fact_7921_not__one__eq,axiom,
% 0.33/0.64 ( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % not_one_eq
% 0.33/0.64 thf(fact_7922_not__one__eq,axiom,
% 0.33/0.64 ( ( bit_ri7919022796975470100ot_int @ one_one_int )
% 0.33/0.64 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % not_one_eq
% 0.33/0.64 thf(fact_7923_csqrt__minus,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
% 0.33/0.64 | ( ( ( im @ X )
% 0.33/0.64 = zero_zero_real )
% 0.33/0.64 & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
% 0.33/0.64 => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
% 0.33/0.64 = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % csqrt_minus
% 0.33/0.64 thf(fact_7924_scaleR__complex_Osimps_I2_J,axiom,
% 0.33/0.64 ! [R: real,X: complex] :
% 0.33/0.64 ( ( im @ ( real_V2046097035970521341omplex @ R @ X ) )
% 0.33/0.64 = ( times_times_real @ R @ ( im @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_complex.simps(2)
% 0.33/0.64 thf(fact_7925_take__bit__not__take__bit,axiom,
% 0.33/0.64 ! [N: nat,A: int] :
% 0.33/0.64 ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.33/0.64 = ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % take_bit_not_take_bit
% 0.33/0.64 thf(fact_7926_take__bit__not__iff,axiom,
% 0.33/0.64 ! [N: nat,A: int,B: int] :
% 0.33/0.64 ( ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) )
% 0.33/0.64 = ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ B ) ) )
% 0.33/0.64 = ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.33/0.64 = ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % take_bit_not_iff
% 0.33/0.64 thf(fact_7927_bit__not__int__iff,axiom,
% 0.33/0.64 ! [K: int,N: nat] :
% 0.33/0.64 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
% 0.33/0.64 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit_not_int_iff
% 0.33/0.64 thf(fact_7928_of__int__not__eq,axiom,
% 0.33/0.64 ! [K: int] :
% 0.33/0.64 ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ ( ring_1_of_int_int @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_int_not_eq
% 0.33/0.64 thf(fact_7929_of__int__not__numeral,axiom,
% 0.33/0.64 ! [K: num] :
% 0.33/0.64 ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_int_not_numeral
% 0.33/0.64 thf(fact_7930_not__add__distrib,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( bit_ri7919022796975470100ot_int @ ( plus_plus_int @ A @ B ) )
% 0.33/0.64 = ( minus_minus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % not_add_distrib
% 0.33/0.64 thf(fact_7931_not__diff__distrib,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A @ B ) )
% 0.33/0.64 = ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % not_diff_distrib
% 0.33/0.64 thf(fact_7932_imaginary__unit_Osimps_I2_J,axiom,
% 0.33/0.64 ( ( im @ imaginary_unit )
% 0.33/0.64 = one_one_real ) ).
% 0.33/0.64
% 0.33/0.64 % imaginary_unit.simps(2)
% 0.33/0.64 thf(fact_7933_one__complex_Osimps_I2_J,axiom,
% 0.33/0.64 ( ( im @ one_one_complex )
% 0.33/0.64 = zero_zero_real ) ).
% 0.33/0.64
% 0.33/0.64 % one_complex.simps(2)
% 0.33/0.64 thf(fact_7934_plus__complex_Osimps_I2_J,axiom,
% 0.33/0.64 ! [X: complex,Y: complex] :
% 0.33/0.64 ( ( im @ ( plus_plus_complex @ X @ Y ) )
% 0.33/0.64 = ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % plus_complex.simps(2)
% 0.33/0.64 thf(fact_7935_minus__complex_Osimps_I2_J,axiom,
% 0.33/0.64 ! [X: complex,Y: complex] :
% 0.33/0.64 ( ( im @ ( minus_minus_complex @ X @ Y ) )
% 0.33/0.64 = ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_complex.simps(2)
% 0.33/0.64 thf(fact_7936_minus__eq__not__plus__1,axiom,
% 0.33/0.64 ( uminus1351360451143612070nteger
% 0.33/0.64 = ( ^ [A2: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A2 ) @ one_one_Code_integer ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_eq_not_plus_1
% 0.33/0.64 thf(fact_7937_minus__eq__not__plus__1,axiom,
% 0.33/0.64 ( uminus_uminus_int
% 0.33/0.64 = ( ^ [A2: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A2 ) @ one_one_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_eq_not_plus_1
% 0.33/0.64 thf(fact_7938_minus__eq__not__minus__1,axiom,
% 0.33/0.64 ( uminus1351360451143612070nteger
% 0.33/0.64 = ( ^ [A2: code_integer] : ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A2 @ one_one_Code_integer ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_eq_not_minus_1
% 0.33/0.64 thf(fact_7939_minus__eq__not__minus__1,axiom,
% 0.33/0.64 ( uminus_uminus_int
% 0.33/0.64 = ( ^ [A2: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A2 @ one_one_int ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_eq_not_minus_1
% 0.33/0.64 thf(fact_7940_not__eq__complement,axiom,
% 0.33/0.64 ( bit_ri7632146776885996613nteger
% 0.33/0.64 = ( ^ [A2: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A2 ) @ one_one_Code_integer ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % not_eq_complement
% 0.33/0.64 thf(fact_7941_not__eq__complement,axiom,
% 0.33/0.64 ( bit_ri7919022796975470100ot_int
% 0.33/0.64 = ( ^ [A2: int] : ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ one_one_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % not_eq_complement
% 0.33/0.64 thf(fact_7942_not__int__def,axiom,
% 0.33/0.64 ( bit_ri7919022796975470100ot_int
% 0.33/0.64 = ( ^ [K2: int] : ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % not_int_def
% 0.33/0.64 thf(fact_7943_and__not__numerals_I1_J,axiom,
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.33/0.64 = zero_zero_int ) ).
% 0.33/0.64
% 0.33/0.64 % and_not_numerals(1)
% 0.33/0.64 thf(fact_7944_scaleR__complex_Ocode,axiom,
% 0.33/0.64 ( real_V2046097035970521341omplex
% 0.33/0.64 = ( ^ [R5: real,X3: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X3 ) ) @ ( times_times_real @ R5 @ ( im @ X3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % scaleR_complex.code
% 0.33/0.64 thf(fact_7945_disjunctive__diff,axiom,
% 0.33/0.64 ! [B: int,A: int] :
% 0.33/0.64 ( ! [N2: nat] :
% 0.33/0.64 ( ( bit_se1146084159140164899it_int @ B @ N2 )
% 0.33/0.64 => ( bit_se1146084159140164899it_int @ A @ N2 ) )
% 0.33/0.64 => ( ( minus_minus_int @ A @ B )
% 0.33/0.64 = ( bit_se725231765392027082nd_int @ A @ ( bit_ri7919022796975470100ot_int @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % disjunctive_diff
% 0.33/0.64 thf(fact_7946_take__bit__not__eq__mask__diff,axiom,
% 0.33/0.64 ! [N: nat,A: int] :
% 0.33/0.64 ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) )
% 0.33/0.64 = ( minus_minus_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % take_bit_not_eq_mask_diff
% 0.33/0.64 thf(fact_7947_minus__numeral__inc__eq,axiom,
% 0.33/0.64 ! [N: num] :
% 0.33/0.64 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) )
% 0.33/0.64 = ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_numeral_inc_eq
% 0.33/0.64 thf(fact_7948_minus__numeral__inc__eq,axiom,
% 0.33/0.64 ! [N: num] :
% 0.33/0.64 ( ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_numeral_inc_eq
% 0.33/0.64 thf(fact_7949_not__int__div__2,axiom,
% 0.33/0.64 ! [K: int] :
% 0.33/0.64 ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % not_int_div_2
% 0.33/0.64 thf(fact_7950_even__not__iff__int,axiom,
% 0.33/0.64 ! [K: int] :
% 0.33/0.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.33/0.64 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % even_not_iff_int
% 0.33/0.64 thf(fact_7951_and__not__numerals_I4_J,axiom,
% 0.33/0.64 ! [M: num] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.33/0.64 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_not_numerals(4)
% 0.33/0.64 thf(fact_7952_and__not__numerals_I2_J,axiom,
% 0.33/0.64 ! [N: num] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.33/0.64 = one_one_int ) ).
% 0.33/0.64
% 0.33/0.64 % and_not_numerals(2)
% 0.33/0.64 thf(fact_7953_not__numeral__Bit0__eq,axiom,
% 0.33/0.64 ! [N: num] :
% 0.33/0.64 ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % not_numeral_Bit0_eq
% 0.33/0.64 thf(fact_7954_not__numeral__Bit0__eq,axiom,
% 0.33/0.64 ! [N: num] :
% 0.33/0.64 ( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) )
% 0.33/0.64 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % not_numeral_Bit0_eq
% 0.33/0.64 thf(fact_7955_bit__minus__int__iff,axiom,
% 0.33/0.64 ! [K: int,N: nat] :
% 0.33/0.64 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
% 0.33/0.64 = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit_minus_int_iff
% 0.33/0.64 thf(fact_7956_times__complex_Osimps_I2_J,axiom,
% 0.33/0.64 ! [X: complex,Y: complex] :
% 0.33/0.64 ( ( im @ ( times_times_complex @ X @ Y ) )
% 0.33/0.64 = ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % times_complex.simps(2)
% 0.33/0.64 thf(fact_7957_times__complex_Osimps_I1_J,axiom,
% 0.33/0.64 ! [X: complex,Y: complex] :
% 0.33/0.64 ( ( re @ ( times_times_complex @ X @ Y ) )
% 0.33/0.64 = ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % times_complex.simps(1)
% 0.33/0.64 thf(fact_7958_take__bit__not__mask__eq__0,axiom,
% 0.33/0.64 ! [M: nat,N: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.64 => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) )
% 0.33/0.64 = zero_zero_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % take_bit_not_mask_eq_0
% 0.33/0.64 thf(fact_7959_Im__complex__div__eq__0,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.64 = zero_zero_real )
% 0.33/0.64 = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 0.33/0.64 = zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % Im_complex_div_eq_0
% 0.33/0.64 thf(fact_7960_push__bit__mask__eq,axiom,
% 0.33/0.64 ! [M: nat,N: nat] :
% 0.33/0.64 ( ( bit_se545348938243370406it_int @ M @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.33/0.64 = ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ N @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ M ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % push_bit_mask_eq
% 0.33/0.64 thf(fact_7961_push__bit__mask__eq,axiom,
% 0.33/0.64 ! [M: nat,N: nat] :
% 0.33/0.64 ( ( bit_se7788150548672797655nteger @ M @ ( bit_se2119862282449309892nteger @ N ) )
% 0.33/0.64 = ( bit_se3949692690581998587nteger @ ( bit_se2119862282449309892nteger @ ( plus_plus_nat @ N @ M ) ) @ ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ M ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % push_bit_mask_eq
% 0.33/0.64 thf(fact_7962_plus__complex_Ocode,axiom,
% 0.33/0.64 ( plus_plus_complex
% 0.33/0.64 = ( ^ [X3: complex,Y6: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X3 ) @ ( re @ Y6 ) ) @ ( plus_plus_real @ ( im @ X3 ) @ ( im @ Y6 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % plus_complex.code
% 0.33/0.64 thf(fact_7963_unset__bit__eq__and__not,axiom,
% 0.33/0.64 ( bit_se8260200283734997820nteger
% 0.33/0.64 = ( ^ [N3: nat,A2: code_integer] : ( bit_se3949692690581998587nteger @ A2 @ ( bit_ri7632146776885996613nteger @ ( bit_se7788150548672797655nteger @ N3 @ one_one_Code_integer ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % unset_bit_eq_and_not
% 0.33/0.64 thf(fact_7964_unset__bit__eq__and__not,axiom,
% 0.33/0.64 ( bit_se4203085406695923979it_int
% 0.33/0.64 = ( ^ [N3: nat,A2: int] : ( bit_se725231765392027082nd_int @ A2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % unset_bit_eq_and_not
% 0.33/0.64 thf(fact_7965_minus__complex_Ocode,axiom,
% 0.33/0.64 ( minus_minus_complex
% 0.33/0.64 = ( ^ [X3: complex,Y6: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X3 ) @ ( re @ Y6 ) ) @ ( minus_minus_real @ ( im @ X3 ) @ ( im @ Y6 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_complex.code
% 0.33/0.64 thf(fact_7966_unset__bit__int__def,axiom,
% 0.33/0.64 ( bit_se4203085406695923979it_int
% 0.33/0.64 = ( ^ [N3: nat,K2: int] : ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % unset_bit_int_def
% 0.33/0.64 thf(fact_7967_and__not__numerals_I5_J,axiom,
% 0.33/0.64 ! [M: num,N: num] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.33/0.64 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_not_numerals(5)
% 0.33/0.64 thf(fact_7968_and__not__numerals_I7_J,axiom,
% 0.33/0.64 ! [M: num] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.33/0.64 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_not_numerals(7)
% 0.33/0.64 thf(fact_7969_and__not__numerals_I3_J,axiom,
% 0.33/0.64 ! [N: num] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.33/0.64 = zero_zero_int ) ).
% 0.33/0.64
% 0.33/0.64 % and_not_numerals(3)
% 0.33/0.64 thf(fact_7970_cmod__le,axiom,
% 0.33/0.64 ! [Z2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z2 ) ) @ ( abs_abs_real @ ( im @ Z2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cmod_le
% 0.33/0.64 thf(fact_7971_Im__complex__div__gt__0,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.33/0.64 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Im_complex_div_gt_0
% 0.33/0.64 thf(fact_7972_Im__complex__div__lt__0,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.33/0.64 = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % Im_complex_div_lt_0
% 0.33/0.64 thf(fact_7973_Im__complex__div__ge__0,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.33/0.64 = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Im_complex_div_ge_0
% 0.33/0.64 thf(fact_7974_Im__complex__div__le__0,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.33/0.64 = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % Im_complex_div_le_0
% 0.33/0.64 thf(fact_7975_sin__n__Im__cis__pow__n,axiom,
% 0.33/0.64 ! [N: nat,A: real] :
% 0.33/0.64 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 0.33/0.64 = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sin_n_Im_cis_pow_n
% 0.33/0.64 thf(fact_7976_Re__exp,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( re @ ( exp_complex @ Z2 ) )
% 0.33/0.64 = ( times_times_real @ ( exp_real @ ( re @ Z2 ) ) @ ( cos_real @ ( im @ Z2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Re_exp
% 0.33/0.64 thf(fact_7977_Im__exp,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( im @ ( exp_complex @ Z2 ) )
% 0.33/0.64 = ( times_times_real @ ( exp_real @ ( re @ Z2 ) ) @ ( sin_real @ ( im @ Z2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Im_exp
% 0.33/0.64 thf(fact_7978_complex__eq,axiom,
% 0.33/0.64 ! [A: complex] :
% 0.33/0.64 ( A
% 0.33/0.64 = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_eq
% 0.33/0.64 thf(fact_7979_and__not__numerals_I9_J,axiom,
% 0.33/0.64 ! [M: num,N: num] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.33/0.64 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_not_numerals(9)
% 0.33/0.64 thf(fact_7980_and__not__numerals_I6_J,axiom,
% 0.33/0.64 ! [M: num,N: num] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.33/0.64 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_not_numerals(6)
% 0.33/0.64 thf(fact_7981_bit__not__iff__eq,axiom,
% 0.33/0.64 ! [A: int,N: nat] :
% 0.33/0.64 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ A ) @ N )
% 0.33/0.64 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.33/0.64 != zero_zero_int )
% 0.33/0.64 & ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit_not_iff_eq
% 0.33/0.64 thf(fact_7982_minus__exp__eq__not__mask,axiom,
% 0.33/0.64 ! [N: nat] :
% 0.33/0.64 ( ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.64 = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_exp_eq_not_mask
% 0.33/0.64 thf(fact_7983_minus__exp__eq__not__mask,axiom,
% 0.33/0.64 ! [N: nat] :
% 0.33/0.64 ( ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_exp_eq_not_mask
% 0.33/0.64 thf(fact_7984_times__complex_Ocode,axiom,
% 0.33/0.64 ( times_times_complex
% 0.33/0.64 = ( ^ [X3: complex,Y6: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y6 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y6 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % times_complex.code
% 0.33/0.64 thf(fact_7985_complex__div__gt__0,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.33/0.64 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
% 0.33/0.64 & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.33/0.64 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_div_gt_0
% 0.33/0.64 thf(fact_7986_exp__eq__polar,axiom,
% 0.33/0.64 ( exp_complex
% 0.33/0.64 = ( ^ [Z5: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z5 ) ) ) @ ( cis @ ( im @ Z5 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % exp_eq_polar
% 0.33/0.64 thf(fact_7987_complex__eq__0,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( Z2 = zero_zero_complex )
% 0.33/0.64 = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_eq_0
% 0.33/0.64 thf(fact_7988_cmod__power2,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cmod_power2
% 0.33/0.64 thf(fact_7989_Im__power2,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Im_power2
% 0.33/0.64 thf(fact_7990_Re__power2,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( re @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( minus_minus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Re_power2
% 0.33/0.64 thf(fact_7991_complex__neq__0,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( Z2 != zero_zero_complex )
% 0.33/0.64 = ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_neq_0
% 0.33/0.64 thf(fact_7992_and__not__numerals_I8_J,axiom,
% 0.33/0.64 ! [M: num,N: num] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.33/0.64 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_not_numerals(8)
% 0.33/0.64 thf(fact_7993_norm__complex__def,axiom,
% 0.33/0.64 ( real_V1022390504157884413omplex
% 0.33/0.64 = ( ^ [Z5: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % norm_complex_def
% 0.33/0.64 thf(fact_7994_not__int__rec,axiom,
% 0.33/0.64 ( bit_ri7919022796975470100ot_int
% 0.33/0.64 = ( ^ [K2: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % not_int_rec
% 0.33/0.64 thf(fact_7995_inverse__complex_Osimps_I1_J,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( re @ ( invers8013647133539491842omplex @ X ) )
% 0.33/0.64 = ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_complex.simps(1)
% 0.33/0.64 thf(fact_7996_complex__unit__circle,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( Z2 != zero_zero_complex )
% 0.33/0.64 => ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = one_one_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_unit_circle
% 0.33/0.64 thf(fact_7997_Re__divide,axiom,
% 0.33/0.64 ! [X: complex,Y: complex] :
% 0.33/0.64 ( ( re @ ( divide1717551699836669952omplex @ X @ Y ) )
% 0.33/0.64 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Re_divide
% 0.33/0.64 thf(fact_7998_complex__mult__cnj,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( times_times_complex @ Z2 @ ( cnj @ Z2 ) )
% 0.33/0.64 = ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_mult_cnj
% 0.33/0.64 thf(fact_7999_csqrt__unique,axiom,
% 0.33/0.64 ! [W: complex,Z2: complex] :
% 0.33/0.64 ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = Z2 )
% 0.33/0.64 => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
% 0.33/0.64 | ( ( ( re @ W )
% 0.33/0.64 = zero_zero_real )
% 0.33/0.64 & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
% 0.33/0.64 => ( ( csqrt @ Z2 )
% 0.33/0.64 = W ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % csqrt_unique
% 0.33/0.64 thf(fact_8000_csqrt__square,axiom,
% 0.33/0.64 ! [B: complex] :
% 0.33/0.64 ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
% 0.33/0.64 | ( ( ( re @ B )
% 0.33/0.64 = zero_zero_real )
% 0.33/0.64 & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
% 0.33/0.64 => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = B ) ) ).
% 0.33/0.64
% 0.33/0.64 % csqrt_square
% 0.33/0.64 thf(fact_8001_inverse__complex_Osimps_I2_J,axiom,
% 0.33/0.64 ! [X: complex] :
% 0.33/0.64 ( ( im @ ( invers8013647133539491842omplex @ X ) )
% 0.33/0.64 = ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_complex.simps(2)
% 0.33/0.64 thf(fact_8002_complex__diff__cnj,axiom,
% 0.33/0.64 ! [Z2: complex] :
% 0.33/0.64 ( ( minus_minus_complex @ Z2 @ ( cnj @ Z2 ) )
% 0.33/0.64 = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z2 ) ) ) @ imaginary_unit ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_diff_cnj
% 0.33/0.64 thf(fact_8003_Im__divide,axiom,
% 0.33/0.64 ! [X: complex,Y: complex] :
% 0.33/0.64 ( ( im @ ( divide1717551699836669952omplex @ X @ Y ) )
% 0.33/0.64 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Im_divide
% 0.33/0.64 thf(fact_8004_complex__abs__le__norm,axiom,
% 0.33/0.64 ! [Z2: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z2 ) ) @ ( abs_abs_real @ ( im @ Z2 ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % complex_abs_le_norm
% 0.33/0.64 thf(fact_8005_inverse__complex_Ocode,axiom,
% 0.33/0.64 ( invers8013647133539491842omplex
% 0.33/0.64 = ( ^ [X3: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X3 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X3 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % inverse_complex.code
% 0.33/0.64 thf(fact_8006_Im__Reals__divide,axiom,
% 0.33/0.64 ! [R: complex,Z2: complex] :
% 0.33/0.64 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( ( im @ ( divide1717551699836669952omplex @ R @ Z2 ) )
% 0.33/0.64 = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R ) ) @ ( im @ Z2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Im_Reals_divide
% 0.33/0.64 thf(fact_8007_Re__Reals__divide,axiom,
% 0.33/0.64 ! [R: complex,Z2: complex] :
% 0.33/0.64 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( ( re @ ( divide1717551699836669952omplex @ R @ Z2 ) )
% 0.33/0.64 = ( divide_divide_real @ ( times_times_real @ ( re @ R ) @ ( re @ Z2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Re_Reals_divide
% 0.33/0.64 thf(fact_8008_num_Osize_I6_J,axiom,
% 0.33/0.64 ! [X32: num] :
% 0.33/0.64 ( ( size_size_num @ ( bit1 @ X32 ) )
% 0.33/0.64 = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % num.size(6)
% 0.33/0.64 thf(fact_8009_num_Osize_I5_J,axiom,
% 0.33/0.64 ! [X2: num] :
% 0.33/0.64 ( ( size_size_num @ ( bit0 @ X2 ) )
% 0.33/0.64 = ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % num.size(5)
% 0.33/0.64 thf(fact_8010_or__int__rec,axiom,
% 0.33/0.64 ( bit_se1409905431419307370or_int
% 0.33/0.64 = ( ^ [K2: int,L2: int] :
% 0.33/0.64 ( plus_plus_int
% 0.33/0.64 @ ( zero_n2684676970156552555ol_int
% 0.33/0.64 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.33/0.64 | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.33/0.64 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_int_rec
% 0.33/0.64 thf(fact_8011_or_Oright__idem,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ B )
% 0.33/0.64 = ( bit_se1409905431419307370or_int @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % or.right_idem
% 0.33/0.64 thf(fact_8012_or_Oright__idem,axiom,
% 0.33/0.64 ! [A: nat,B: nat] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ B )
% 0.33/0.64 = ( bit_se1412395901928357646or_nat @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % or.right_idem
% 0.33/0.64 thf(fact_8013_or_Oleft__idem,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 0.33/0.64 = ( bit_se1409905431419307370or_int @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % or.left_idem
% 0.33/0.64 thf(fact_8014_or_Oleft__idem,axiom,
% 0.33/0.64 ! [A: nat,B: nat] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 0.33/0.64 = ( bit_se1412395901928357646or_nat @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % or.left_idem
% 0.33/0.64 thf(fact_8015_or_Oidem,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ A @ A )
% 0.33/0.64 = A ) ).
% 0.33/0.64
% 0.33/0.64 % or.idem
% 0.33/0.64 thf(fact_8016_or_Oidem,axiom,
% 0.33/0.64 ! [A: nat] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ A @ A )
% 0.33/0.64 = A ) ).
% 0.33/0.64
% 0.33/0.64 % or.idem
% 0.33/0.64 thf(fact_8017_or_Oright__neutral,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ A @ zero_zero_int )
% 0.33/0.64 = A ) ).
% 0.33/0.64
% 0.33/0.64 % or.right_neutral
% 0.33/0.64 thf(fact_8018_or_Oright__neutral,axiom,
% 0.33/0.64 ! [A: nat] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ A @ zero_zero_nat )
% 0.33/0.64 = A ) ).
% 0.33/0.64
% 0.33/0.64 % or.right_neutral
% 0.33/0.64 thf(fact_8019_or_Oleft__neutral,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ zero_zero_int @ A )
% 0.33/0.64 = A ) ).
% 0.33/0.64
% 0.33/0.64 % or.left_neutral
% 0.33/0.64 thf(fact_8020_or_Oleft__neutral,axiom,
% 0.33/0.64 ! [A: nat] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ zero_zero_nat @ A )
% 0.33/0.64 = A ) ).
% 0.33/0.64
% 0.33/0.64 % or.left_neutral
% 0.33/0.64 thf(fact_8021_take__bit__or,axiom,
% 0.33/0.64 ! [N: nat,A: int,B: int] :
% 0.33/0.64 ( ( bit_se2923211474154528505it_int @ N @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 0.33/0.64 = ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % take_bit_or
% 0.33/0.64 thf(fact_8022_take__bit__or,axiom,
% 0.33/0.64 ! [N: nat,A: nat,B: nat] :
% 0.33/0.64 ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 0.33/0.64 = ( bit_se1412395901928357646or_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % take_bit_or
% 0.33/0.64 thf(fact_8023_push__bit__or,axiom,
% 0.33/0.64 ! [N: nat,A: nat,B: nat] :
% 0.33/0.64 ( ( bit_se547839408752420682it_nat @ N @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 0.33/0.64 = ( bit_se1412395901928357646or_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % push_bit_or
% 0.33/0.64 thf(fact_8024_push__bit__or,axiom,
% 0.33/0.64 ! [N: nat,A: int,B: int] :
% 0.33/0.64 ( ( bit_se545348938243370406it_int @ N @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 0.33/0.64 = ( bit_se1409905431419307370or_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % push_bit_or
% 0.33/0.64 thf(fact_8025_push__bit__or,axiom,
% 0.33/0.64 ! [N: nat,A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( bit_se7788150548672797655nteger @ N @ ( bit_se1080825931792720795nteger @ A @ B ) )
% 0.33/0.64 = ( bit_se1080825931792720795nteger @ ( bit_se7788150548672797655nteger @ N @ A ) @ ( bit_se7788150548672797655nteger @ N @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % push_bit_or
% 0.33/0.64 thf(fact_8026_push__bit__or,axiom,
% 0.33/0.64 ! [N: nat,A: code_natural,B: code_natural] :
% 0.33/0.64 ( ( bit_se6611745700429515170atural @ N @ ( bit_se9127793120404214118atural @ A @ B ) )
% 0.33/0.64 = ( bit_se9127793120404214118atural @ ( bit_se6611745700429515170atural @ N @ A ) @ ( bit_se6611745700429515170atural @ N @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % push_bit_or
% 0.33/0.64 thf(fact_8027_bit_Odisj__one__right,axiom,
% 0.33/0.64 ! [X: code_integer] :
% 0.33/0.64 ( ( bit_se1080825931792720795nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.disj_one_right
% 0.33/0.64 thf(fact_8028_bit_Odisj__one__right,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.disj_one_right
% 0.33/0.64 thf(fact_8029_bit_Odisj__one__left,axiom,
% 0.33/0.64 ! [X: code_integer] :
% 0.33/0.64 ( ( bit_se1080825931792720795nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.disj_one_left
% 0.33/0.64 thf(fact_8030_bit_Odisj__one__left,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.disj_one_left
% 0.33/0.64 thf(fact_8031_or__nonnegative__int__iff,axiom,
% 0.33/0.64 ! [K: int,L: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
% 0.33/0.64 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.64 & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_nonnegative_int_iff
% 0.33/0.64 thf(fact_8032_or__negative__int__iff,axiom,
% 0.33/0.64 ! [K: int,L: int] :
% 0.33/0.64 ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
% 0.33/0.64 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.33/0.64 | ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_negative_int_iff
% 0.33/0.64 thf(fact_8033_bit_Ode__Morgan__disj,axiom,
% 0.33/0.64 ! [X: int,Y: int] :
% 0.33/0.64 ( ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ X @ Y ) )
% 0.33/0.64 = ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X ) @ ( bit_ri7919022796975470100ot_int @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.de_Morgan_disj
% 0.33/0.64 thf(fact_8034_bit_Ode__Morgan__conj,axiom,
% 0.33/0.64 ! [X: int,Y: int] :
% 0.33/0.64 ( ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ X @ Y ) )
% 0.33/0.64 = ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ ( bit_ri7919022796975470100ot_int @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.de_Morgan_conj
% 0.33/0.64 thf(fact_8035_or__numerals_I8_J,axiom,
% 0.33/0.64 ! [X: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
% 0.33/0.64 = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_numerals(8)
% 0.33/0.64 thf(fact_8036_or__numerals_I8_J,axiom,
% 0.33/0.64 ! [X: num] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
% 0.33/0.64 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_numerals(8)
% 0.33/0.64 thf(fact_8037_or__numerals_I2_J,axiom,
% 0.33/0.64 ! [Y: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 0.33/0.64 = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_numerals(2)
% 0.33/0.64 thf(fact_8038_or__numerals_I2_J,axiom,
% 0.33/0.64 ! [Y: num] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.33/0.64 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_numerals(2)
% 0.33/0.64 thf(fact_8039_bit_Odisj__cancel__right,axiom,
% 0.33/0.64 ! [X: code_integer] :
% 0.33/0.64 ( ( bit_se1080825931792720795nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.disj_cancel_right
% 0.33/0.64 thf(fact_8040_bit_Odisj__cancel__right,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.disj_cancel_right
% 0.33/0.64 thf(fact_8041_bit_Odisj__cancel__left,axiom,
% 0.33/0.64 ! [X: code_integer] :
% 0.33/0.64 ( ( bit_se1080825931792720795nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.disj_cancel_left
% 0.33/0.64 thf(fact_8042_bit_Odisj__cancel__left,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.disj_cancel_left
% 0.33/0.64 thf(fact_8043_Re__divide__Reals,axiom,
% 0.33/0.64 ! [R: complex,Z2: complex] :
% 0.33/0.64 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( ( re @ ( divide1717551699836669952omplex @ Z2 @ R ) )
% 0.33/0.64 = ( divide_divide_real @ ( re @ Z2 ) @ ( re @ R ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Re_divide_Reals
% 0.33/0.64 thf(fact_8044_imaginary__eq__real__iff,axiom,
% 0.33/0.64 ! [Y: complex,X: complex] :
% 0.33/0.64 ( ( member_complex @ Y @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( ( member_complex @ X @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( ( ( times_times_complex @ imaginary_unit @ Y )
% 0.33/0.64 = X )
% 0.33/0.64 = ( ( X = zero_zero_complex )
% 0.33/0.64 & ( Y = zero_zero_complex ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % imaginary_eq_real_iff
% 0.33/0.64 thf(fact_8045_real__eq__imaginary__iff,axiom,
% 0.33/0.64 ! [Y: complex,X: complex] :
% 0.33/0.64 ( ( member_complex @ Y @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( ( member_complex @ X @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( ( X
% 0.33/0.64 = ( times_times_complex @ imaginary_unit @ Y ) )
% 0.33/0.64 = ( ( X = zero_zero_complex )
% 0.33/0.64 & ( Y = zero_zero_complex ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % real_eq_imaginary_iff
% 0.33/0.64 thf(fact_8046_or__numerals_I3_J,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 0.33/0.64 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_numerals(3)
% 0.33/0.64 thf(fact_8047_or__numerals_I3_J,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.33/0.64 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_numerals(3)
% 0.33/0.64 thf(fact_8048_or__numerals_I1_J,axiom,
% 0.33/0.64 ! [Y: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 0.33/0.64 = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_numerals(1)
% 0.33/0.64 thf(fact_8049_or__numerals_I1_J,axiom,
% 0.33/0.64 ! [Y: num] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.33/0.64 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_numerals(1)
% 0.33/0.64 thf(fact_8050_or__numerals_I5_J,axiom,
% 0.33/0.64 ! [X: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 0.33/0.64 = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_numerals(5)
% 0.33/0.64 thf(fact_8051_or__numerals_I5_J,axiom,
% 0.33/0.64 ! [X: num] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 0.33/0.64 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_numerals(5)
% 0.33/0.64 thf(fact_8052_or__minus__numerals_I6_J,axiom,
% 0.33/0.64 ! [N: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 0.33/0.64 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_minus_numerals(6)
% 0.33/0.64 thf(fact_8053_or__minus__numerals_I2_J,axiom,
% 0.33/0.64 ! [N: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.33/0.64 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_minus_numerals(2)
% 0.33/0.64 thf(fact_8054_Im__divide__Reals,axiom,
% 0.33/0.64 ! [R: complex,Z2: complex] :
% 0.33/0.64 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( ( im @ ( divide1717551699836669952omplex @ Z2 @ R ) )
% 0.33/0.64 = ( divide_divide_real @ ( im @ Z2 ) @ ( re @ R ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Im_divide_Reals
% 0.33/0.64 thf(fact_8055_or__minus__minus__numerals,axiom,
% 0.33/0.64 ! [M: num,N: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_minus_minus_numerals
% 0.33/0.64 thf(fact_8056_and__minus__minus__numerals,axiom,
% 0.33/0.64 ! [M: num,N: num] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_minus_minus_numerals
% 0.33/0.64 thf(fact_8057_or__numerals_I4_J,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 0.33/0.64 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_numerals(4)
% 0.33/0.64 thf(fact_8058_or__numerals_I4_J,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.33/0.64 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_numerals(4)
% 0.33/0.64 thf(fact_8059_or__numerals_I6_J,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 0.33/0.64 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_numerals(6)
% 0.33/0.64 thf(fact_8060_or__numerals_I6_J,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.33/0.64 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_numerals(6)
% 0.33/0.64 thf(fact_8061_or__numerals_I7_J,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 0.33/0.64 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_numerals(7)
% 0.33/0.64 thf(fact_8062_or__numerals_I7_J,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.33/0.64 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_numerals(7)
% 0.33/0.64 thf(fact_8063_of__nat__or__eq,axiom,
% 0.33/0.64 ! [M: nat,N: nat] :
% 0.33/0.64 ( ( semiri1314217659103216013at_int @ ( bit_se1412395901928357646or_nat @ M @ N ) )
% 0.33/0.64 = ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_nat_or_eq
% 0.33/0.64 thf(fact_8064_of__nat__or__eq,axiom,
% 0.33/0.64 ! [M: nat,N: nat] :
% 0.33/0.64 ( ( semiri1316708129612266289at_nat @ ( bit_se1412395901928357646or_nat @ M @ N ) )
% 0.33/0.64 = ( bit_se1412395901928357646or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_nat_or_eq
% 0.33/0.64 thf(fact_8065_Reals__numeral,axiom,
% 0.33/0.64 ! [W: num] : ( member_complex @ ( numera6690914467698888265omplex @ W ) @ real_V2521375963428798218omplex ) ).
% 0.33/0.64
% 0.33/0.64 % Reals_numeral
% 0.33/0.64 thf(fact_8066_Reals__numeral,axiom,
% 0.33/0.64 ! [W: num] : ( member_real @ ( numeral_numeral_real @ W ) @ real_V470468836141973256s_real ) ).
% 0.33/0.64
% 0.33/0.64 % Reals_numeral
% 0.33/0.64 thf(fact_8067_Reals__mult,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( member_real @ A @ real_V470468836141973256s_real )
% 0.33/0.64 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 0.33/0.64 => ( member_real @ ( times_times_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Reals_mult
% 0.33/0.64 thf(fact_8068_Reals__mult,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( member_complex @ ( times_times_complex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Reals_mult
% 0.33/0.64 thf(fact_8069_Reals__1,axiom,
% 0.33/0.64 member_real @ one_one_real @ real_V470468836141973256s_real ).
% 0.33/0.64
% 0.33/0.64 % Reals_1
% 0.33/0.64 thf(fact_8070_Reals__1,axiom,
% 0.33/0.64 member_complex @ one_one_complex @ real_V2521375963428798218omplex ).
% 0.33/0.64
% 0.33/0.64 % Reals_1
% 0.33/0.64 thf(fact_8071_Reals__add,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( member_real @ A @ real_V470468836141973256s_real )
% 0.33/0.64 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 0.33/0.64 => ( member_real @ ( plus_plus_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Reals_add
% 0.33/0.64 thf(fact_8072_Reals__add,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( member_complex @ ( plus_plus_complex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Reals_add
% 0.33/0.64 thf(fact_8073_Reals__diff,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( member_real @ A @ real_V470468836141973256s_real )
% 0.33/0.64 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 0.33/0.64 => ( member_real @ ( minus_minus_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Reals_diff
% 0.33/0.64 thf(fact_8074_Reals__diff,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( member_complex @ ( minus_minus_complex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Reals_diff
% 0.33/0.64 thf(fact_8075_Reals__divide,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( member_complex @ ( divide1717551699836669952omplex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Reals_divide
% 0.33/0.64 thf(fact_8076_Reals__divide,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( member_real @ A @ real_V470468836141973256s_real )
% 0.33/0.64 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 0.33/0.64 => ( member_real @ ( divide_divide_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Reals_divide
% 0.33/0.64 thf(fact_8077_bit__or__iff,axiom,
% 0.33/0.64 ! [A: int,B: int,N: nat] :
% 0.33/0.64 ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ N )
% 0.33/0.64 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.33/0.64 | ( bit_se1146084159140164899it_int @ B @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit_or_iff
% 0.33/0.64 thf(fact_8078_bit__or__iff,axiom,
% 0.33/0.64 ! [A: nat,B: nat,N: nat] :
% 0.33/0.64 ( ( bit_se1148574629649215175it_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ N )
% 0.33/0.64 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.33/0.64 | ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit_or_iff
% 0.33/0.64 thf(fact_8079_size__neq__size__imp__neq,axiom,
% 0.33/0.64 ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
% 0.33/0.64 ( ( ( size_s6755466524823107622T_VEBT @ X )
% 0.33/0.64 != ( size_s6755466524823107622T_VEBT @ Y ) )
% 0.33/0.64 => ( X != Y ) ) ).
% 0.33/0.64
% 0.33/0.64 % size_neq_size_imp_neq
% 0.33/0.64 thf(fact_8080_size__neq__size__imp__neq,axiom,
% 0.33/0.64 ! [X: list_o,Y: list_o] :
% 0.33/0.64 ( ( ( size_size_list_o @ X )
% 0.33/0.64 != ( size_size_list_o @ Y ) )
% 0.33/0.64 => ( X != Y ) ) ).
% 0.33/0.64
% 0.33/0.64 % size_neq_size_imp_neq
% 0.33/0.64 thf(fact_8081_size__neq__size__imp__neq,axiom,
% 0.33/0.64 ! [X: list_nat,Y: list_nat] :
% 0.33/0.64 ( ( ( size_size_list_nat @ X )
% 0.33/0.64 != ( size_size_list_nat @ Y ) )
% 0.33/0.64 => ( X != Y ) ) ).
% 0.33/0.64
% 0.33/0.64 % size_neq_size_imp_neq
% 0.33/0.64 thf(fact_8082_size__neq__size__imp__neq,axiom,
% 0.33/0.64 ! [X: list_int,Y: list_int] :
% 0.33/0.64 ( ( ( size_size_list_int @ X )
% 0.33/0.64 != ( size_size_list_int @ Y ) )
% 0.33/0.64 => ( X != Y ) ) ).
% 0.33/0.64
% 0.33/0.64 % size_neq_size_imp_neq
% 0.33/0.64 thf(fact_8083_size__neq__size__imp__neq,axiom,
% 0.33/0.64 ! [X: num,Y: num] :
% 0.33/0.64 ( ( ( size_size_num @ X )
% 0.33/0.64 != ( size_size_num @ Y ) )
% 0.33/0.64 => ( X != Y ) ) ).
% 0.33/0.64
% 0.33/0.64 % size_neq_size_imp_neq
% 0.33/0.64 thf(fact_8084_or_Oleft__commute,axiom,
% 0.33/0.64 ! [B: int,A: int,C: int] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ B @ ( bit_se1409905431419307370or_int @ A @ C ) )
% 0.33/0.64 = ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ B @ C ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or.left_commute
% 0.33/0.64 thf(fact_8085_or_Oleft__commute,axiom,
% 0.33/0.64 ! [B: nat,A: nat,C: nat] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ B @ ( bit_se1412395901928357646or_nat @ A @ C ) )
% 0.33/0.64 = ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ B @ C ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or.left_commute
% 0.33/0.64 thf(fact_8086_or_Ocommute,axiom,
% 0.33/0.64 ( bit_se1409905431419307370or_int
% 0.33/0.64 = ( ^ [A2: int,B2: int] : ( bit_se1409905431419307370or_int @ B2 @ A2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or.commute
% 0.33/0.64 thf(fact_8087_or_Ocommute,axiom,
% 0.33/0.64 ( bit_se1412395901928357646or_nat
% 0.33/0.64 = ( ^ [A2: nat,B2: nat] : ( bit_se1412395901928357646or_nat @ B2 @ A2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or.commute
% 0.33/0.64 thf(fact_8088_or_Oassoc,axiom,
% 0.33/0.64 ! [A: int,B: int,C: int] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ C )
% 0.33/0.64 = ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ B @ C ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or.assoc
% 0.33/0.64 thf(fact_8089_or_Oassoc,axiom,
% 0.33/0.64 ! [A: nat,B: nat,C: nat] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ C )
% 0.33/0.64 = ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ B @ C ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or.assoc
% 0.33/0.64 thf(fact_8090_bit_Odisj__zero__right,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ X @ zero_zero_int )
% 0.33/0.64 = X ) ).
% 0.33/0.64
% 0.33/0.64 % bit.disj_zero_right
% 0.33/0.64 thf(fact_8091_or__eq__0__iff,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( ( bit_se1409905431419307370or_int @ A @ B )
% 0.33/0.64 = zero_zero_int )
% 0.33/0.64 = ( ( A = zero_zero_int )
% 0.33/0.64 & ( B = zero_zero_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_eq_0_iff
% 0.33/0.64 thf(fact_8092_or__eq__0__iff,axiom,
% 0.33/0.64 ! [A: nat,B: nat] :
% 0.33/0.64 ( ( ( bit_se1412395901928357646or_nat @ A @ B )
% 0.33/0.64 = zero_zero_nat )
% 0.33/0.64 = ( ( A = zero_zero_nat )
% 0.33/0.64 & ( B = zero_zero_nat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_eq_0_iff
% 0.33/0.64 thf(fact_8093_of__int__or__eq,axiom,
% 0.33/0.64 ! [K: int,L: int] :
% 0.33/0.64 ( ( ring_1_of_int_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
% 0.33/0.64 = ( bit_se1409905431419307370or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_int_or_eq
% 0.33/0.64 thf(fact_8094_bit__or__int__iff,axiom,
% 0.33/0.64 ! [K: int,L: int,N: nat] :
% 0.33/0.64 ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ N )
% 0.33/0.64 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.33/0.64 | ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit_or_int_iff
% 0.33/0.64 thf(fact_8095_bit_Odisj__conj__distrib2,axiom,
% 0.33/0.64 ! [Y: int,Z2: int,X: int] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ Y @ Z2 ) @ X )
% 0.33/0.64 = ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ Y @ X ) @ ( bit_se1409905431419307370or_int @ Z2 @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.disj_conj_distrib2
% 0.33/0.64 thf(fact_8096_bit_Oconj__disj__distrib2,axiom,
% 0.33/0.64 ! [Y: int,Z2: int,X: int] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ Y @ Z2 ) @ X )
% 0.33/0.64 = ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ Y @ X ) @ ( bit_se725231765392027082nd_int @ Z2 @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.conj_disj_distrib2
% 0.33/0.64 thf(fact_8097_bit_Odisj__conj__distrib,axiom,
% 0.33/0.64 ! [X: int,Y: int,Z2: int] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ X @ ( bit_se725231765392027082nd_int @ Y @ Z2 ) )
% 0.33/0.64 = ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Z2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.disj_conj_distrib
% 0.33/0.64 thf(fact_8098_bit_Oconj__disj__distrib,axiom,
% 0.33/0.64 ! [X: int,Y: int,Z2: int] :
% 0.33/0.64 ( ( bit_se725231765392027082nd_int @ X @ ( bit_se1409905431419307370or_int @ Y @ Z2 ) )
% 0.33/0.64 = ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se725231765392027082nd_int @ X @ Z2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.conj_disj_distrib
% 0.33/0.64 thf(fact_8099_Reals__0,axiom,
% 0.33/0.64 member_real @ zero_zero_real @ real_V470468836141973256s_real ).
% 0.33/0.64
% 0.33/0.64 % Reals_0
% 0.33/0.64 thf(fact_8100_Reals__0,axiom,
% 0.33/0.64 member_complex @ zero_zero_complex @ real_V2521375963428798218omplex ).
% 0.33/0.64
% 0.33/0.64 % Reals_0
% 0.33/0.64 thf(fact_8101_Reals__power,axiom,
% 0.33/0.64 ! [A: real,N: nat] :
% 0.33/0.64 ( ( member_real @ A @ real_V470468836141973256s_real )
% 0.33/0.64 => ( member_real @ ( power_power_real @ A @ N ) @ real_V470468836141973256s_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % Reals_power
% 0.33/0.64 thf(fact_8102_Reals__power,axiom,
% 0.33/0.64 ! [A: complex,N: nat] :
% 0.33/0.64 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( member_complex @ ( power_power_complex @ A @ N ) @ real_V2521375963428798218omplex ) ) ).
% 0.33/0.64
% 0.33/0.64 % Reals_power
% 0.33/0.64 thf(fact_8103_Reals__of__nat,axiom,
% 0.33/0.64 ! [N: nat] : ( member_complex @ ( semiri8010041392384452111omplex @ N ) @ real_V2521375963428798218omplex ) ).
% 0.33/0.64
% 0.33/0.64 % Reals_of_nat
% 0.33/0.64 thf(fact_8104_Reals__of__nat,axiom,
% 0.33/0.64 ! [N: nat] : ( member_real @ ( semiri5074537144036343181t_real @ N ) @ real_V470468836141973256s_real ) ).
% 0.33/0.64
% 0.33/0.64 % Reals_of_nat
% 0.33/0.64 thf(fact_8105_or__greater__eq,axiom,
% 0.33/0.64 ! [L: int,K: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ zero_zero_int @ L )
% 0.33/0.64 => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_greater_eq
% 0.33/0.64 thf(fact_8106_OR__lower,axiom,
% 0.33/0.64 ! [X: int,Y: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.64 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.64 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % OR_lower
% 0.33/0.64 thf(fact_8107_disjunctive__add,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ! [N2: nat] :
% 0.33/0.64 ( ~ ( bit_se1146084159140164899it_int @ A @ N2 )
% 0.33/0.64 | ~ ( bit_se1146084159140164899it_int @ B @ N2 ) )
% 0.33/0.64 => ( ( plus_plus_int @ A @ B )
% 0.33/0.64 = ( bit_se1409905431419307370or_int @ A @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % disjunctive_add
% 0.33/0.64 thf(fact_8108_disjunctive__add,axiom,
% 0.33/0.64 ! [A: nat,B: nat] :
% 0.33/0.64 ( ! [N2: nat] :
% 0.33/0.64 ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 )
% 0.33/0.64 | ~ ( bit_se1148574629649215175it_nat @ B @ N2 ) )
% 0.33/0.64 => ( ( plus_plus_nat @ A @ B )
% 0.33/0.64 = ( bit_se1412395901928357646or_nat @ A @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % disjunctive_add
% 0.33/0.64 thf(fact_8109_plus__and__or,axiom,
% 0.33/0.64 ! [X: int,Y: int] :
% 0.33/0.64 ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Y ) )
% 0.33/0.64 = ( plus_plus_int @ X @ Y ) ) ).
% 0.33/0.64
% 0.33/0.64 % plus_and_or
% 0.33/0.64 thf(fact_8110_or__eq__not__not__and,axiom,
% 0.33/0.64 ( bit_se1409905431419307370or_int
% 0.33/0.64 = ( ^ [A2: int,B2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ A2 ) @ ( bit_ri7919022796975470100ot_int @ B2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_eq_not_not_and
% 0.33/0.64 thf(fact_8111_and__eq__not__not__or,axiom,
% 0.33/0.64 ( bit_se725231765392027082nd_int
% 0.33/0.64 = ( ^ [A2: int,B2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ A2 ) @ ( bit_ri7919022796975470100ot_int @ B2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % and_eq_not_not_or
% 0.33/0.64 thf(fact_8112_or__int__def,axiom,
% 0.33/0.64 ( bit_se1409905431419307370or_int
% 0.33/0.64 = ( ^ [K2: int,L2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ ( bit_ri7919022796975470100ot_int @ L2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_int_def
% 0.33/0.64 thf(fact_8113_nonzero__Reals__divide,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( ( B != zero_zero_complex )
% 0.33/0.64 => ( member_complex @ ( divide1717551699836669952omplex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % nonzero_Reals_divide
% 0.33/0.64 thf(fact_8114_nonzero__Reals__divide,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( member_real @ A @ real_V470468836141973256s_real )
% 0.33/0.64 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 0.33/0.64 => ( ( B != zero_zero_real )
% 0.33/0.64 => ( member_real @ ( divide_divide_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % nonzero_Reals_divide
% 0.33/0.64 thf(fact_8115_nonzero__Reals__inverse,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( member_real @ A @ real_V470468836141973256s_real )
% 0.33/0.64 => ( ( A != zero_zero_real )
% 0.33/0.64 => ( member_real @ ( inverse_inverse_real @ A ) @ real_V470468836141973256s_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % nonzero_Reals_inverse
% 0.33/0.64 thf(fact_8116_nonzero__Reals__inverse,axiom,
% 0.33/0.64 ! [A: complex] :
% 0.33/0.64 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 0.33/0.64 => ( ( A != zero_zero_complex )
% 0.33/0.64 => ( member_complex @ ( invers8013647133539491842omplex @ A ) @ real_V2521375963428798218omplex ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % nonzero_Reals_inverse
% 0.33/0.64 thf(fact_8117_or__not__numerals_I1_J,axiom,
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_not_numerals(1)
% 0.33/0.64 thf(fact_8118_bit_Oxor__def2,axiom,
% 0.33/0.64 ( bit_se6526347334894502574or_int
% 0.33/0.64 = ( ^ [X3: int,Y6: int] : ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ X3 @ Y6 ) @ ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ X3 ) @ ( bit_ri7919022796975470100ot_int @ Y6 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.xor_def2
% 0.33/0.64 thf(fact_8119_bit_Oxor__def,axiom,
% 0.33/0.64 ( bit_se6526347334894502574or_int
% 0.33/0.64 = ( ^ [X3: int,Y6: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ X3 @ ( bit_ri7919022796975470100ot_int @ Y6 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X3 ) @ Y6 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.xor_def
% 0.33/0.64 thf(fact_8120_set__bit__eq__or,axiom,
% 0.33/0.64 ( bit_se7882103937844011126it_nat
% 0.33/0.64 = ( ^ [N3: nat,A2: nat] : ( bit_se1412395901928357646or_nat @ A2 @ ( bit_se547839408752420682it_nat @ N3 @ one_one_nat ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % set_bit_eq_or
% 0.33/0.64 thf(fact_8121_set__bit__eq__or,axiom,
% 0.33/0.64 ( bit_se7879613467334960850it_int
% 0.33/0.64 = ( ^ [N3: nat,A2: int] : ( bit_se1409905431419307370or_int @ A2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % set_bit_eq_or
% 0.33/0.64 thf(fact_8122_set__bit__eq__or,axiom,
% 0.33/0.64 ( bit_se2793503036327961859nteger
% 0.33/0.64 = ( ^ [N3: nat,A2: code_integer] : ( bit_se1080825931792720795nteger @ A2 @ ( bit_se7788150548672797655nteger @ N3 @ one_one_Code_integer ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % set_bit_eq_or
% 0.33/0.64 thf(fact_8123_set__bit__eq__or,axiom,
% 0.33/0.64 ( bit_se1617098188084679374atural
% 0.33/0.64 = ( ^ [N3: nat,A2: code_natural] : ( bit_se9127793120404214118atural @ A2 @ ( bit_se6611745700429515170atural @ N3 @ one_one_Code_natural ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % set_bit_eq_or
% 0.33/0.64 thf(fact_8124_xor__int__def,axiom,
% 0.33/0.64 ( bit_se6526347334894502574or_int
% 0.33/0.64 = ( ^ [K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ L2 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ L2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_int_def
% 0.33/0.64 thf(fact_8125_concat__bit__def,axiom,
% 0.33/0.64 ( bit_concat_bit
% 0.33/0.64 = ( ^ [N3: nat,K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N3 @ K2 ) @ ( bit_se545348938243370406it_int @ N3 @ L2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % concat_bit_def
% 0.33/0.64 thf(fact_8126_set__bit__int__def,axiom,
% 0.33/0.64 ( bit_se7879613467334960850it_int
% 0.33/0.64 = ( ^ [N3: nat,K2: int] : ( bit_se1409905431419307370or_int @ K2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % set_bit_int_def
% 0.33/0.64 thf(fact_8127_even__or__iff,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ A @ B ) )
% 0.33/0.64 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.33/0.64 & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % even_or_iff
% 0.33/0.64 thf(fact_8128_even__or__iff,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 0.33/0.64 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.33/0.64 & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % even_or_iff
% 0.33/0.64 thf(fact_8129_even__or__iff,axiom,
% 0.33/0.64 ! [A: nat,B: nat] :
% 0.33/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 0.33/0.64 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.33/0.64 & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % even_or_iff
% 0.33/0.64 thf(fact_8130_bit_Ocomplement__unique,axiom,
% 0.33/0.64 ! [A: code_integer,X: code_integer,Y: code_integer] :
% 0.33/0.64 ( ( ( bit_se3949692690581998587nteger @ A @ X )
% 0.33/0.64 = zero_z3403309356797280102nteger )
% 0.33/0.64 => ( ( ( bit_se1080825931792720795nteger @ A @ X )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.64 => ( ( ( bit_se3949692690581998587nteger @ A @ Y )
% 0.33/0.64 = zero_z3403309356797280102nteger )
% 0.33/0.64 => ( ( ( bit_se1080825931792720795nteger @ A @ Y )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.64 => ( X = Y ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.complement_unique
% 0.33/0.64 thf(fact_8131_bit_Ocomplement__unique,axiom,
% 0.33/0.64 ! [A: int,X: int,Y: int] :
% 0.33/0.64 ( ( ( bit_se725231765392027082nd_int @ A @ X )
% 0.33/0.64 = zero_zero_int )
% 0.33/0.64 => ( ( ( bit_se1409905431419307370or_int @ A @ X )
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.64 => ( ( ( bit_se725231765392027082nd_int @ A @ Y )
% 0.33/0.64 = zero_zero_int )
% 0.33/0.64 => ( ( ( bit_se1409905431419307370or_int @ A @ Y )
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.64 => ( X = Y ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.complement_unique
% 0.33/0.64 thf(fact_8132_or__not__numerals_I2_J,axiom,
% 0.33/0.64 ! [N: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_not_numerals(2)
% 0.33/0.64 thf(fact_8133_or__not__numerals_I4_J,axiom,
% 0.33/0.64 ! [M: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_not_numerals(4)
% 0.33/0.64 thf(fact_8134_num_Osize_I4_J,axiom,
% 0.33/0.64 ( ( size_size_num @ one )
% 0.33/0.64 = zero_zero_nat ) ).
% 0.33/0.64
% 0.33/0.64 % num.size(4)
% 0.33/0.64 thf(fact_8135_bit_Ocompl__unique,axiom,
% 0.33/0.64 ! [X: code_integer,Y: code_integer] :
% 0.33/0.64 ( ( ( bit_se3949692690581998587nteger @ X @ Y )
% 0.33/0.64 = zero_z3403309356797280102nteger )
% 0.33/0.64 => ( ( ( bit_se1080825931792720795nteger @ X @ Y )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.64 => ( ( bit_ri7632146776885996613nteger @ X )
% 0.33/0.64 = Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.compl_unique
% 0.33/0.64 thf(fact_8136_bit_Ocompl__unique,axiom,
% 0.33/0.64 ! [X: int,Y: int] :
% 0.33/0.64 ( ( ( bit_se725231765392027082nd_int @ X @ Y )
% 0.33/0.64 = zero_zero_int )
% 0.33/0.64 => ( ( ( bit_se1409905431419307370or_int @ X @ Y )
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.64 => ( ( bit_ri7919022796975470100ot_int @ X )
% 0.33/0.64 = Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % bit.compl_unique
% 0.33/0.64 thf(fact_8137_or__not__numerals_I3_J,axiom,
% 0.33/0.64 ! [N: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_not_numerals(3)
% 0.33/0.64 thf(fact_8138_or__not__numerals_I7_J,axiom,
% 0.33/0.64 ! [M: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.33/0.64 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_not_numerals(7)
% 0.33/0.64 thf(fact_8139_signed__take__bit__eq__if__negative,axiom,
% 0.33/0.64 ! [A: int,N: nat] :
% 0.33/0.64 ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.33/0.64 => ( ( bit_ri631733984087533419it_int @ N @ A )
% 0.33/0.64 = ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % signed_take_bit_eq_if_negative
% 0.33/0.64 thf(fact_8140_mask__Suc__exp,axiom,
% 0.33/0.64 ! [N: nat] :
% 0.33/0.64 ( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
% 0.33/0.64 = ( bit_se1412395901928357646or_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mask_Suc_exp
% 0.33/0.64 thf(fact_8141_mask__Suc__exp,axiom,
% 0.33/0.64 ! [N: nat] :
% 0.33/0.64 ( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
% 0.33/0.64 = ( bit_se1409905431419307370or_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mask_Suc_exp
% 0.33/0.64 thf(fact_8142_or__not__numerals_I6_J,axiom,
% 0.33/0.64 ! [M: num,N: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.33/0.64 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_not_numerals(6)
% 0.33/0.64 thf(fact_8143_or__one__eq,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( bit_se1080825931792720795nteger @ A @ one_one_Code_integer )
% 0.33/0.64 = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_one_eq
% 0.33/0.64 thf(fact_8144_or__one__eq,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ A @ one_one_int )
% 0.33/0.64 = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_one_eq
% 0.33/0.64 thf(fact_8145_or__one__eq,axiom,
% 0.33/0.64 ! [A: nat] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ A @ one_one_nat )
% 0.33/0.64 = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_one_eq
% 0.33/0.64 thf(fact_8146_one__or__eq,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ A )
% 0.33/0.64 = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % one_or_eq
% 0.33/0.64 thf(fact_8147_one__or__eq,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ one_one_int @ A )
% 0.33/0.64 = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % one_or_eq
% 0.33/0.64 thf(fact_8148_one__or__eq,axiom,
% 0.33/0.64 ! [A: nat] :
% 0.33/0.64 ( ( bit_se1412395901928357646or_nat @ one_one_nat @ A )
% 0.33/0.64 = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % one_or_eq
% 0.33/0.64 thf(fact_8149_mask__Suc__double,axiom,
% 0.33/0.64 ! [N: nat] :
% 0.33/0.64 ( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
% 0.33/0.64 = ( bit_se1412395901928357646or_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mask_Suc_double
% 0.33/0.64 thf(fact_8150_mask__Suc__double,axiom,
% 0.33/0.64 ! [N: nat] :
% 0.33/0.64 ( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
% 0.33/0.64 = ( bit_se1409905431419307370or_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mask_Suc_double
% 0.33/0.64 thf(fact_8151_OR__upper,axiom,
% 0.33/0.64 ! [X: int,N: nat,Y: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.64 => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.64 => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.64 => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % OR_upper
% 0.33/0.64 thf(fact_8152_or__not__numerals_I5_J,axiom,
% 0.33/0.64 ! [M: num,N: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.33/0.64 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_not_numerals(5)
% 0.33/0.64 thf(fact_8153_signed__take__bit__def,axiom,
% 0.33/0.64 ( bit_ri6519982836138164636nteger
% 0.33/0.64 = ( ^ [N3: nat,A2: code_integer] : ( bit_se1080825931792720795nteger @ ( bit_se1745604003318907178nteger @ N3 @ A2 ) @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( bit_se9216721137139052372nteger @ A2 @ N3 ) ) @ ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N3 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % signed_take_bit_def
% 0.33/0.64 thf(fact_8154_signed__take__bit__def,axiom,
% 0.33/0.64 ( bit_ri631733984087533419it_int
% 0.33/0.64 = ( ^ [N3: nat,A2: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N3 @ A2 ) @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ A2 @ N3 ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N3 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % signed_take_bit_def
% 0.33/0.64 thf(fact_8155_or__not__numerals_I9_J,axiom,
% 0.33/0.64 ! [M: num,N: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.33/0.64 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_not_numerals(9)
% 0.33/0.64 thf(fact_8156_or__not__numerals_I8_J,axiom,
% 0.33/0.64 ! [M: num,N: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.33/0.64 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_not_numerals(8)
% 0.33/0.64 thf(fact_8157_or__int__unfold,axiom,
% 0.33/0.64 ( bit_se1409905431419307370or_int
% 0.33/0.64 = ( ^ [K2: int,L2: int] :
% 0.33/0.64 ( if_int
% 0.33/0.64 @ ( ( K2
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.64 | ( L2
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) ) )
% 0.33/0.64 @ ( uminus_uminus_int @ one_one_int )
% 0.33/0.64 @ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_int_unfold
% 0.33/0.64 thf(fact_8158_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
% 0.33/0.64 boolea2445317508997433345nteger @ bit_se3949692690581998587nteger @ bit_se1080825931792720795nteger @ bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ bit_se3222712562003087583nteger ).
% 0.33/0.64
% 0.33/0.64 % bit.abstract_boolean_algebra_sym_diff_axioms
% 0.33/0.64 thf(fact_8159_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
% 0.33/0.64 boolea8527374999097803216ff_int @ bit_se725231765392027082nd_int @ bit_se1409905431419307370or_int @ bit_ri7919022796975470100ot_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) @ bit_se6526347334894502574or_int ).
% 0.33/0.64
% 0.33/0.64 % bit.abstract_boolean_algebra_sym_diff_axioms
% 0.33/0.64 thf(fact_8160_or__minus__numerals_I4_J,axiom,
% 0.33/0.64 ! [M: num,N: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.33/0.64 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_minus_numerals(4)
% 0.33/0.64 thf(fact_8161_or__minus__numerals_I8_J,axiom,
% 0.33/0.64 ! [N: num,M: num] :
% 0.33/0.64 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.33/0.64 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % or_minus_numerals(8)
% 0.33/0.64 thf(fact_8162_max_Oidem,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( ord_max_int @ A @ A )
% 0.33/0.64 = A ) ).
% 0.33/0.64
% 0.33/0.64 % max.idem
% 0.33/0.64 thf(fact_8163_max_Oidem,axiom,
% 0.33/0.64 ! [A: nat] :
% 0.33/0.64 ( ( ord_max_nat @ A @ A )
% 0.33/0.64 = A ) ).
% 0.33/0.64
% 0.33/0.64 % max.idem
% 0.33/0.64 thf(fact_8164_max_Oleft__idem,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( ord_max_int @ A @ ( ord_max_int @ A @ B ) )
% 0.33/0.64 = ( ord_max_int @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.left_idem
% 0.33/0.64 thf(fact_8165_max_Oleft__idem,axiom,
% 0.33/0.64 ! [A: nat,B: nat] :
% 0.33/0.64 ( ( ord_max_nat @ A @ ( ord_max_nat @ A @ B ) )
% 0.33/0.64 = ( ord_max_nat @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.left_idem
% 0.33/0.64 thf(fact_8166_max_Oright__idem,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( ord_max_int @ ( ord_max_int @ A @ B ) @ B )
% 0.33/0.64 = ( ord_max_int @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.right_idem
% 0.33/0.64 thf(fact_8167_max_Oright__idem,axiom,
% 0.33/0.64 ! [A: nat,B: nat] :
% 0.33/0.64 ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ B )
% 0.33/0.64 = ( ord_max_nat @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.right_idem
% 0.33/0.64 thf(fact_8168_max_Obounded__iff,axiom,
% 0.33/0.64 ! [B: rat,C: rat,A: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 0.33/0.64 = ( ( ord_less_eq_rat @ B @ A )
% 0.33/0.64 & ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.bounded_iff
% 0.33/0.64 thf(fact_8169_max_Obounded__iff,axiom,
% 0.33/0.64 ! [B: num,C: num,A: num] :
% 0.33/0.64 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 0.33/0.64 = ( ( ord_less_eq_num @ B @ A )
% 0.33/0.64 & ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.bounded_iff
% 0.33/0.64 thf(fact_8170_max_Obounded__iff,axiom,
% 0.33/0.64 ! [B: nat,C: nat,A: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.33/0.64 = ( ( ord_less_eq_nat @ B @ A )
% 0.33/0.64 & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.bounded_iff
% 0.33/0.64 thf(fact_8171_max_Obounded__iff,axiom,
% 0.33/0.64 ! [B: int,C: int,A: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 0.33/0.64 = ( ( ord_less_eq_int @ B @ A )
% 0.33/0.64 & ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.bounded_iff
% 0.33/0.64 thf(fact_8172_max_Oabsorb2,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.64 => ( ( ord_max_rat @ A @ B )
% 0.33/0.64 = B ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb2
% 0.33/0.64 thf(fact_8173_max_Oabsorb2,axiom,
% 0.33/0.64 ! [A: num,B: num] :
% 0.33/0.64 ( ( ord_less_eq_num @ A @ B )
% 0.33/0.64 => ( ( ord_max_num @ A @ B )
% 0.33/0.64 = B ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb2
% 0.33/0.64 thf(fact_8174_max_Oabsorb2,axiom,
% 0.33/0.64 ! [A: nat,B: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.64 => ( ( ord_max_nat @ A @ B )
% 0.33/0.64 = B ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb2
% 0.33/0.64 thf(fact_8175_max_Oabsorb2,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ A @ B )
% 0.33/0.64 => ( ( ord_max_int @ A @ B )
% 0.33/0.64 = B ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb2
% 0.33/0.64 thf(fact_8176_max_Oabsorb1,axiom,
% 0.33/0.64 ! [B: rat,A: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ B @ A )
% 0.33/0.64 => ( ( ord_max_rat @ A @ B )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb1
% 0.33/0.64 thf(fact_8177_max_Oabsorb1,axiom,
% 0.33/0.64 ! [B: num,A: num] :
% 0.33/0.64 ( ( ord_less_eq_num @ B @ A )
% 0.33/0.64 => ( ( ord_max_num @ A @ B )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb1
% 0.33/0.64 thf(fact_8178_max_Oabsorb1,axiom,
% 0.33/0.64 ! [B: nat,A: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ B @ A )
% 0.33/0.64 => ( ( ord_max_nat @ A @ B )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb1
% 0.33/0.64 thf(fact_8179_max_Oabsorb1,axiom,
% 0.33/0.64 ! [B: int,A: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ B @ A )
% 0.33/0.64 => ( ( ord_max_int @ A @ B )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb1
% 0.33/0.64 thf(fact_8180_max__less__iff__conj,axiom,
% 0.33/0.64 ! [X: real,Y: real,Z2: real] :
% 0.33/0.64 ( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z2 )
% 0.33/0.64 = ( ( ord_less_real @ X @ Z2 )
% 0.33/0.64 & ( ord_less_real @ Y @ Z2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % max_less_iff_conj
% 0.33/0.64 thf(fact_8181_max__less__iff__conj,axiom,
% 0.33/0.64 ! [X: rat,Y: rat,Z2: rat] :
% 0.33/0.64 ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
% 0.33/0.64 = ( ( ord_less_rat @ X @ Z2 )
% 0.33/0.64 & ( ord_less_rat @ Y @ Z2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % max_less_iff_conj
% 0.33/0.64 thf(fact_8182_max__less__iff__conj,axiom,
% 0.33/0.64 ! [X: num,Y: num,Z2: num] :
% 0.33/0.64 ( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z2 )
% 0.33/0.64 = ( ( ord_less_num @ X @ Z2 )
% 0.33/0.64 & ( ord_less_num @ Y @ Z2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % max_less_iff_conj
% 0.33/0.64 thf(fact_8183_max__less__iff__conj,axiom,
% 0.33/0.64 ! [X: nat,Y: nat,Z2: nat] :
% 0.33/0.64 ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
% 0.33/0.64 = ( ( ord_less_nat @ X @ Z2 )
% 0.33/0.64 & ( ord_less_nat @ Y @ Z2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % max_less_iff_conj
% 0.33/0.64 thf(fact_8184_max__less__iff__conj,axiom,
% 0.33/0.64 ! [X: int,Y: int,Z2: int] :
% 0.33/0.64 ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z2 )
% 0.33/0.64 = ( ( ord_less_int @ X @ Z2 )
% 0.33/0.64 & ( ord_less_int @ Y @ Z2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % max_less_iff_conj
% 0.33/0.64 thf(fact_8185_max_Oabsorb4,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_real @ A @ B )
% 0.33/0.64 => ( ( ord_max_real @ A @ B )
% 0.33/0.64 = B ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb4
% 0.33/0.64 thf(fact_8186_max_Oabsorb4,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_rat @ A @ B )
% 0.33/0.64 => ( ( ord_max_rat @ A @ B )
% 0.33/0.64 = B ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb4
% 0.33/0.64 thf(fact_8187_max_Oabsorb4,axiom,
% 0.33/0.64 ! [A: num,B: num] :
% 0.33/0.64 ( ( ord_less_num @ A @ B )
% 0.33/0.64 => ( ( ord_max_num @ A @ B )
% 0.33/0.64 = B ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb4
% 0.33/0.64 thf(fact_8188_max_Oabsorb4,axiom,
% 0.33/0.64 ! [A: nat,B: nat] :
% 0.33/0.64 ( ( ord_less_nat @ A @ B )
% 0.33/0.64 => ( ( ord_max_nat @ A @ B )
% 0.33/0.64 = B ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb4
% 0.33/0.64 thf(fact_8189_max_Oabsorb4,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( ord_less_int @ A @ B )
% 0.33/0.64 => ( ( ord_max_int @ A @ B )
% 0.33/0.64 = B ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb4
% 0.33/0.64 thf(fact_8190_max_Oabsorb3,axiom,
% 0.33/0.64 ! [B: real,A: real] :
% 0.33/0.64 ( ( ord_less_real @ B @ A )
% 0.33/0.64 => ( ( ord_max_real @ A @ B )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb3
% 0.33/0.64 thf(fact_8191_max_Oabsorb3,axiom,
% 0.33/0.64 ! [B: rat,A: rat] :
% 0.33/0.64 ( ( ord_less_rat @ B @ A )
% 0.33/0.64 => ( ( ord_max_rat @ A @ B )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb3
% 0.33/0.64 thf(fact_8192_max_Oabsorb3,axiom,
% 0.33/0.64 ! [B: num,A: num] :
% 0.33/0.64 ( ( ord_less_num @ B @ A )
% 0.33/0.64 => ( ( ord_max_num @ A @ B )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb3
% 0.33/0.64 thf(fact_8193_max_Oabsorb3,axiom,
% 0.33/0.64 ! [B: nat,A: nat] :
% 0.33/0.64 ( ( ord_less_nat @ B @ A )
% 0.33/0.64 => ( ( ord_max_nat @ A @ B )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb3
% 0.33/0.64 thf(fact_8194_max_Oabsorb3,axiom,
% 0.33/0.64 ! [B: int,A: int] :
% 0.33/0.64 ( ( ord_less_int @ B @ A )
% 0.33/0.64 => ( ( ord_max_int @ A @ B )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % max.absorb3
% 0.33/0.64 thf(fact_8195_of__bool__or__iff,axiom,
% 0.33/0.64 ! [P: $o,Q: $o] :
% 0.33/0.64 ( ( zero_n2687167440665602831ol_nat
% 0.33/0.64 @ ( P
% 0.33/0.64 | Q ) )
% 0.33/0.64 = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_bool_or_iff
% 0.33/0.64 thf(fact_8196_of__bool__or__iff,axiom,
% 0.33/0.64 ! [P: $o,Q: $o] :
% 0.33/0.64 ( ( zero_n2684676970156552555ol_int
% 0.33/0.64 @ ( P
% 0.33/0.64 | Q ) )
% 0.33/0.64 = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_bool_or_iff
% 0.33/0.64 thf(fact_8197_of__bool__or__iff,axiom,
% 0.33/0.64 ! [P: $o,Q: $o] :
% 0.33/0.64 ( ( zero_n356916108424825756nteger
% 0.33/0.64 @ ( P
% 0.33/0.64 | Q ) )
% 0.33/0.64 = ( ord_max_Code_integer @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % of_bool_or_iff
% 0.33/0.64 thf(fact_8198_max__number__of_I1_J,axiom,
% 0.33/0.64 ! [U: num,V: num] :
% 0.33/0.64 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.33/0.64 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.33/0.64 = ( numeral_numeral_real @ V ) ) )
% 0.33/0.64 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.33/0.64 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.33/0.64 = ( numeral_numeral_real @ U ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_number_of(1)
% 0.33/0.65 thf(fact_8199_max__number__of_I1_J,axiom,
% 0.33/0.65 ! [U: num,V: num] :
% 0.33/0.65 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.33/0.65 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.33/0.65 = ( numeral_numeral_rat @ V ) ) )
% 0.33/0.65 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.33/0.65 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.33/0.65 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_number_of(1)
% 0.33/0.65 thf(fact_8200_max__number__of_I1_J,axiom,
% 0.33/0.65 ! [U: num,V: num] :
% 0.33/0.65 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.33/0.65 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.33/0.65 = ( numeral_numeral_nat @ V ) ) )
% 0.33/0.65 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.33/0.65 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.33/0.65 = ( numeral_numeral_nat @ U ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_number_of(1)
% 0.33/0.65 thf(fact_8201_max__number__of_I1_J,axiom,
% 0.33/0.65 ! [U: num,V: num] :
% 0.33/0.65 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.33/0.65 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.33/0.65 = ( numeral_numeral_int @ V ) ) )
% 0.33/0.65 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.33/0.65 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.33/0.65 = ( numeral_numeral_int @ U ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_number_of(1)
% 0.33/0.65 thf(fact_8202_max__0__1_I3_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
% 0.33/0.65 = ( numeral_numeral_real @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(3)
% 0.33/0.65 thf(fact_8203_max__0__1_I3_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
% 0.33/0.65 = ( numeral_numeral_rat @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(3)
% 0.33/0.65 thf(fact_8204_max__0__1_I3_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
% 0.33/0.65 = ( numeral_numeral_nat @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(3)
% 0.33/0.65 thf(fact_8205_max__0__1_I3_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
% 0.33/0.65 = ( numeral_numeral_int @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(3)
% 0.33/0.65 thf(fact_8206_max__0__1_I4_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
% 0.33/0.65 = ( numeral_numeral_real @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(4)
% 0.33/0.65 thf(fact_8207_max__0__1_I4_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
% 0.33/0.65 = ( numeral_numeral_rat @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(4)
% 0.33/0.65 thf(fact_8208_max__0__1_I4_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
% 0.33/0.65 = ( numeral_numeral_nat @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(4)
% 0.33/0.65 thf(fact_8209_max__0__1_I4_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
% 0.33/0.65 = ( numeral_numeral_int @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(4)
% 0.33/0.65 thf(fact_8210_max__0__1_I2_J,axiom,
% 0.33/0.65 ( ( ord_max_real @ one_one_real @ zero_zero_real )
% 0.33/0.65 = one_one_real ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(2)
% 0.33/0.65 thf(fact_8211_max__0__1_I2_J,axiom,
% 0.33/0.65 ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
% 0.33/0.65 = one_one_rat ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(2)
% 0.33/0.65 thf(fact_8212_max__0__1_I2_J,axiom,
% 0.33/0.65 ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 0.33/0.65 = one_one_int ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(2)
% 0.33/0.65 thf(fact_8213_max__0__1_I2_J,axiom,
% 0.33/0.65 ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 0.33/0.65 = one_one_nat ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(2)
% 0.33/0.65 thf(fact_8214_max__0__1_I1_J,axiom,
% 0.33/0.65 ( ( ord_max_real @ zero_zero_real @ one_one_real )
% 0.33/0.65 = one_one_real ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(1)
% 0.33/0.65 thf(fact_8215_max__0__1_I1_J,axiom,
% 0.33/0.65 ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
% 0.33/0.65 = one_one_rat ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(1)
% 0.33/0.65 thf(fact_8216_max__0__1_I1_J,axiom,
% 0.33/0.65 ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 0.33/0.65 = one_one_int ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(1)
% 0.33/0.65 thf(fact_8217_max__0__1_I1_J,axiom,
% 0.33/0.65 ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 0.33/0.65 = one_one_nat ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(1)
% 0.33/0.65 thf(fact_8218_max__0__1_I5_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 0.33/0.65 = ( numeral_numeral_real @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(5)
% 0.33/0.65 thf(fact_8219_max__0__1_I5_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 0.33/0.65 = ( numeral_numeral_rat @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(5)
% 0.33/0.65 thf(fact_8220_max__0__1_I5_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 0.33/0.65 = ( numeral_numeral_nat @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(5)
% 0.33/0.65 thf(fact_8221_max__0__1_I5_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 0.33/0.65 = ( numeral_numeral_int @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(5)
% 0.33/0.65 thf(fact_8222_max__0__1_I6_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
% 0.33/0.65 = ( numeral_numeral_real @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(6)
% 0.33/0.65 thf(fact_8223_max__0__1_I6_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
% 0.33/0.65 = ( numeral_numeral_rat @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(6)
% 0.33/0.65 thf(fact_8224_max__0__1_I6_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
% 0.33/0.65 = ( numeral_numeral_nat @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(6)
% 0.33/0.65 thf(fact_8225_max__0__1_I6_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
% 0.33/0.65 = ( numeral_numeral_int @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_0_1(6)
% 0.33/0.65 thf(fact_8226_max__number__of_I4_J,axiom,
% 0.33/0.65 ! [U: num,V: num] :
% 0.33/0.65 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.33/0.65 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.33/0.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 0.33/0.65 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.33/0.65 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.33/0.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_number_of(4)
% 0.33/0.65 thf(fact_8227_max__number__of_I4_J,axiom,
% 0.33/0.65 ! [U: num,V: num] :
% 0.33/0.65 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.33/0.65 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.33/0.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 0.33/0.65 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.33/0.65 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.33/0.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_number_of(4)
% 0.33/0.65 thf(fact_8228_max__number__of_I4_J,axiom,
% 0.33/0.65 ! [U: num,V: num] :
% 0.33/0.65 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.33/0.65 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.33/0.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 0.33/0.65 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.33/0.65 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.33/0.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_number_of(4)
% 0.33/0.65 thf(fact_8229_max__number__of_I4_J,axiom,
% 0.33/0.65 ! [U: num,V: num] :
% 0.33/0.65 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.33/0.65 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 0.33/0.65 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.33/0.65 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_number_of(4)
% 0.33/0.65 thf(fact_8230_max__number__of_I3_J,axiom,
% 0.33/0.65 ! [U: num,V: num] :
% 0.33/0.65 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.33/0.65 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.33/0.65 = ( numeral_numeral_real @ V ) ) )
% 0.33/0.65 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.33/0.65 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.33/0.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_number_of(3)
% 0.33/0.65 thf(fact_8231_max__number__of_I3_J,axiom,
% 0.33/0.65 ! [U: num,V: num] :
% 0.33/0.65 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.33/0.65 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.33/0.65 = ( numera6620942414471956472nteger @ V ) ) )
% 0.33/0.65 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.33/0.65 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.33/0.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_number_of(3)
% 0.33/0.65 thf(fact_8232_max__number__of_I3_J,axiom,
% 0.33/0.65 ! [U: num,V: num] :
% 0.33/0.65 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.33/0.65 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.33/0.65 = ( numeral_numeral_rat @ V ) ) )
% 0.33/0.65 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.33/0.65 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.33/0.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_number_of(3)
% 0.33/0.65 thf(fact_8233_max__number__of_I3_J,axiom,
% 0.33/0.65 ! [U: num,V: num] :
% 0.33/0.65 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.33/0.65 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.33/0.65 = ( numeral_numeral_int @ V ) ) )
% 0.33/0.65 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.33/0.65 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_number_of(3)
% 0.33/0.65 thf(fact_8234_max__number__of_I2_J,axiom,
% 0.33/0.65 ! [U: num,V: num] :
% 0.33/0.65 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.33/0.65 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.33/0.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 0.33/0.65 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.33/0.65 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.33/0.65 = ( numeral_numeral_real @ U ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_number_of(2)
% 0.33/0.65 thf(fact_8235_max__number__of_I2_J,axiom,
% 0.33/0.65 ! [U: num,V: num] :
% 0.33/0.65 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.33/0.65 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.33/0.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 0.33/0.65 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.33/0.65 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.33/0.65 = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_number_of(2)
% 0.33/0.65 thf(fact_8236_max__number__of_I2_J,axiom,
% 0.33/0.65 ! [U: num,V: num] :
% 0.33/0.65 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.33/0.65 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.33/0.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 0.33/0.65 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.33/0.65 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.33/0.65 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_number_of(2)
% 0.33/0.65 thf(fact_8237_max__number__of_I2_J,axiom,
% 0.33/0.65 ! [U: num,V: num] :
% 0.33/0.65 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.33/0.65 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 0.33/0.65 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.33/0.65 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.33/0.65 = ( numeral_numeral_int @ U ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_number_of(2)
% 0.33/0.65 thf(fact_8238_or__nat__numerals_I4_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.33/0.65 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_nat_numerals(4)
% 0.33/0.65 thf(fact_8239_or__nat__numerals_I2_J,axiom,
% 0.33/0.65 ! [Y: num] :
% 0.33/0.65 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.33/0.65 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_nat_numerals(2)
% 0.33/0.65 thf(fact_8240_or__nat__numerals_I3_J,axiom,
% 0.33/0.65 ! [X: num] :
% 0.33/0.65 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.33/0.65 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_nat_numerals(3)
% 0.33/0.65 thf(fact_8241_or__nat__numerals_I1_J,axiom,
% 0.33/0.65 ! [Y: num] :
% 0.33/0.65 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.33/0.65 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_nat_numerals(1)
% 0.33/0.65 thf(fact_8242_of__int__max,axiom,
% 0.33/0.65 ! [X: int,Y: int] :
% 0.33/0.65 ( ( ring_1_of_int_real @ ( ord_max_int @ X @ Y ) )
% 0.33/0.65 = ( ord_max_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_max
% 0.33/0.65 thf(fact_8243_of__int__max,axiom,
% 0.33/0.65 ! [X: int,Y: int] :
% 0.33/0.65 ( ( ring_1_of_int_rat @ ( ord_max_int @ X @ Y ) )
% 0.33/0.65 = ( ord_max_rat @ ( ring_1_of_int_rat @ X ) @ ( ring_1_of_int_rat @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_max
% 0.33/0.65 thf(fact_8244_of__int__max,axiom,
% 0.33/0.65 ! [X: int,Y: int] :
% 0.33/0.65 ( ( ring_1_of_int_int @ ( ord_max_int @ X @ Y ) )
% 0.33/0.65 = ( ord_max_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_max
% 0.33/0.65 thf(fact_8245_max_OcoboundedI2,axiom,
% 0.33/0.65 ! [C: rat,B: rat,A: rat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ C @ B )
% 0.33/0.65 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.coboundedI2
% 0.33/0.65 thf(fact_8246_max_OcoboundedI2,axiom,
% 0.33/0.65 ! [C: num,B: num,A: num] :
% 0.33/0.65 ( ( ord_less_eq_num @ C @ B )
% 0.33/0.65 => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.coboundedI2
% 0.33/0.65 thf(fact_8247_max_OcoboundedI2,axiom,
% 0.33/0.65 ! [C: nat,B: nat,A: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ C @ B )
% 0.33/0.65 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.coboundedI2
% 0.33/0.65 thf(fact_8248_max_OcoboundedI2,axiom,
% 0.33/0.65 ! [C: int,B: int,A: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ C @ B )
% 0.33/0.65 => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.coboundedI2
% 0.33/0.65 thf(fact_8249_max_OcoboundedI1,axiom,
% 0.33/0.65 ! [C: rat,A: rat,B: rat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ C @ A )
% 0.33/0.65 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.coboundedI1
% 0.33/0.65 thf(fact_8250_max_OcoboundedI1,axiom,
% 0.33/0.65 ! [C: num,A: num,B: num] :
% 0.33/0.65 ( ( ord_less_eq_num @ C @ A )
% 0.33/0.65 => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.coboundedI1
% 0.33/0.65 thf(fact_8251_max_OcoboundedI1,axiom,
% 0.33/0.65 ! [C: nat,A: nat,B: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ C @ A )
% 0.33/0.65 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.coboundedI1
% 0.33/0.65 thf(fact_8252_max_OcoboundedI1,axiom,
% 0.33/0.65 ! [C: int,A: int,B: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ C @ A )
% 0.33/0.65 => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.coboundedI1
% 0.33/0.65 thf(fact_8253_max_Oabsorb__iff2,axiom,
% 0.33/0.65 ( ord_less_eq_rat
% 0.33/0.65 = ( ^ [A2: rat,B2: rat] :
% 0.33/0.65 ( ( ord_max_rat @ A2 @ B2 )
% 0.33/0.65 = B2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.absorb_iff2
% 0.33/0.65 thf(fact_8254_max_Oabsorb__iff2,axiom,
% 0.33/0.65 ( ord_less_eq_num
% 0.33/0.65 = ( ^ [A2: num,B2: num] :
% 0.33/0.65 ( ( ord_max_num @ A2 @ B2 )
% 0.33/0.65 = B2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.absorb_iff2
% 0.33/0.65 thf(fact_8255_max_Oabsorb__iff2,axiom,
% 0.33/0.65 ( ord_less_eq_nat
% 0.33/0.65 = ( ^ [A2: nat,B2: nat] :
% 0.33/0.65 ( ( ord_max_nat @ A2 @ B2 )
% 0.33/0.65 = B2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.absorb_iff2
% 0.33/0.65 thf(fact_8256_max_Oabsorb__iff2,axiom,
% 0.33/0.65 ( ord_less_eq_int
% 0.33/0.65 = ( ^ [A2: int,B2: int] :
% 0.33/0.65 ( ( ord_max_int @ A2 @ B2 )
% 0.33/0.65 = B2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.absorb_iff2
% 0.33/0.65 thf(fact_8257_max_Oabsorb__iff1,axiom,
% 0.33/0.65 ( ord_less_eq_rat
% 0.33/0.65 = ( ^ [B2: rat,A2: rat] :
% 0.33/0.65 ( ( ord_max_rat @ A2 @ B2 )
% 0.33/0.65 = A2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.absorb_iff1
% 0.33/0.65 thf(fact_8258_max_Oabsorb__iff1,axiom,
% 0.33/0.65 ( ord_less_eq_num
% 0.33/0.65 = ( ^ [B2: num,A2: num] :
% 0.33/0.65 ( ( ord_max_num @ A2 @ B2 )
% 0.33/0.65 = A2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.absorb_iff1
% 0.33/0.65 thf(fact_8259_max_Oabsorb__iff1,axiom,
% 0.33/0.65 ( ord_less_eq_nat
% 0.33/0.65 = ( ^ [B2: nat,A2: nat] :
% 0.33/0.65 ( ( ord_max_nat @ A2 @ B2 )
% 0.33/0.65 = A2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.absorb_iff1
% 0.33/0.65 thf(fact_8260_max_Oabsorb__iff1,axiom,
% 0.33/0.65 ( ord_less_eq_int
% 0.33/0.65 = ( ^ [B2: int,A2: int] :
% 0.33/0.65 ( ( ord_max_int @ A2 @ B2 )
% 0.33/0.65 = A2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.absorb_iff1
% 0.33/0.65 thf(fact_8261_le__max__iff__disj,axiom,
% 0.33/0.65 ! [Z2: rat,X: rat,Y: rat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ Z2 @ ( ord_max_rat @ X @ Y ) )
% 0.33/0.65 = ( ( ord_less_eq_rat @ Z2 @ X )
% 0.33/0.65 | ( ord_less_eq_rat @ Z2 @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % le_max_iff_disj
% 0.33/0.65 thf(fact_8262_le__max__iff__disj,axiom,
% 0.33/0.65 ! [Z2: num,X: num,Y: num] :
% 0.33/0.65 ( ( ord_less_eq_num @ Z2 @ ( ord_max_num @ X @ Y ) )
% 0.33/0.65 = ( ( ord_less_eq_num @ Z2 @ X )
% 0.33/0.65 | ( ord_less_eq_num @ Z2 @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % le_max_iff_disj
% 0.33/0.65 thf(fact_8263_le__max__iff__disj,axiom,
% 0.33/0.65 ! [Z2: nat,X: nat,Y: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
% 0.33/0.65 = ( ( ord_less_eq_nat @ Z2 @ X )
% 0.33/0.65 | ( ord_less_eq_nat @ Z2 @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % le_max_iff_disj
% 0.33/0.65 thf(fact_8264_le__max__iff__disj,axiom,
% 0.33/0.65 ! [Z2: int,X: int,Y: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ Z2 @ ( ord_max_int @ X @ Y ) )
% 0.33/0.65 = ( ( ord_less_eq_int @ Z2 @ X )
% 0.33/0.65 | ( ord_less_eq_int @ Z2 @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % le_max_iff_disj
% 0.33/0.65 thf(fact_8265_max_Ocobounded2,axiom,
% 0.33/0.65 ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.cobounded2
% 0.33/0.65 thf(fact_8266_max_Ocobounded2,axiom,
% 0.33/0.65 ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.cobounded2
% 0.33/0.65 thf(fact_8267_max_Ocobounded2,axiom,
% 0.33/0.65 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.cobounded2
% 0.33/0.65 thf(fact_8268_max_Ocobounded2,axiom,
% 0.33/0.65 ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.cobounded2
% 0.33/0.65 thf(fact_8269_max_Ocobounded1,axiom,
% 0.33/0.65 ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.cobounded1
% 0.33/0.65 thf(fact_8270_max_Ocobounded1,axiom,
% 0.33/0.65 ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.cobounded1
% 0.33/0.65 thf(fact_8271_max_Ocobounded1,axiom,
% 0.33/0.65 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.cobounded1
% 0.33/0.65 thf(fact_8272_max_Ocobounded1,axiom,
% 0.33/0.65 ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.cobounded1
% 0.33/0.65 thf(fact_8273_max_Oorder__iff,axiom,
% 0.33/0.65 ( ord_less_eq_rat
% 0.33/0.65 = ( ^ [B2: rat,A2: rat] :
% 0.33/0.65 ( A2
% 0.33/0.65 = ( ord_max_rat @ A2 @ B2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.order_iff
% 0.33/0.65 thf(fact_8274_max_Oorder__iff,axiom,
% 0.33/0.65 ( ord_less_eq_num
% 0.33/0.65 = ( ^ [B2: num,A2: num] :
% 0.33/0.65 ( A2
% 0.33/0.65 = ( ord_max_num @ A2 @ B2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.order_iff
% 0.33/0.65 thf(fact_8275_max_Oorder__iff,axiom,
% 0.33/0.65 ( ord_less_eq_nat
% 0.33/0.65 = ( ^ [B2: nat,A2: nat] :
% 0.33/0.65 ( A2
% 0.33/0.65 = ( ord_max_nat @ A2 @ B2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.order_iff
% 0.33/0.65 thf(fact_8276_max_Oorder__iff,axiom,
% 0.33/0.65 ( ord_less_eq_int
% 0.33/0.65 = ( ^ [B2: int,A2: int] :
% 0.33/0.65 ( A2
% 0.33/0.65 = ( ord_max_int @ A2 @ B2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.order_iff
% 0.33/0.65 thf(fact_8277_max_OboundedI,axiom,
% 0.33/0.65 ! [B: rat,A: rat,C: rat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ B @ A )
% 0.33/0.65 => ( ( ord_less_eq_rat @ C @ A )
% 0.33/0.65 => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.boundedI
% 0.33/0.65 thf(fact_8278_max_OboundedI,axiom,
% 0.33/0.65 ! [B: num,A: num,C: num] :
% 0.33/0.65 ( ( ord_less_eq_num @ B @ A )
% 0.33/0.65 => ( ( ord_less_eq_num @ C @ A )
% 0.33/0.65 => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.boundedI
% 0.33/0.65 thf(fact_8279_max_OboundedI,axiom,
% 0.33/0.65 ! [B: nat,A: nat,C: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ B @ A )
% 0.33/0.65 => ( ( ord_less_eq_nat @ C @ A )
% 0.33/0.65 => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.boundedI
% 0.33/0.65 thf(fact_8280_max_OboundedI,axiom,
% 0.33/0.65 ! [B: int,A: int,C: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ B @ A )
% 0.33/0.65 => ( ( ord_less_eq_int @ C @ A )
% 0.33/0.65 => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.boundedI
% 0.33/0.65 thf(fact_8281_max_OboundedE,axiom,
% 0.33/0.65 ! [B: rat,C: rat,A: rat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 0.33/0.65 => ~ ( ( ord_less_eq_rat @ B @ A )
% 0.33/0.65 => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.boundedE
% 0.33/0.65 thf(fact_8282_max_OboundedE,axiom,
% 0.33/0.65 ! [B: num,C: num,A: num] :
% 0.33/0.65 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 0.33/0.65 => ~ ( ( ord_less_eq_num @ B @ A )
% 0.33/0.65 => ~ ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.boundedE
% 0.33/0.65 thf(fact_8283_max_OboundedE,axiom,
% 0.33/0.65 ! [B: nat,C: nat,A: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.33/0.65 => ~ ( ( ord_less_eq_nat @ B @ A )
% 0.33/0.65 => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.boundedE
% 0.33/0.65 thf(fact_8284_max_OboundedE,axiom,
% 0.33/0.65 ! [B: int,C: int,A: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 0.33/0.65 => ~ ( ( ord_less_eq_int @ B @ A )
% 0.33/0.65 => ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.boundedE
% 0.33/0.65 thf(fact_8285_max_OorderI,axiom,
% 0.33/0.65 ! [A: rat,B: rat] :
% 0.33/0.65 ( ( A
% 0.33/0.65 = ( ord_max_rat @ A @ B ) )
% 0.33/0.65 => ( ord_less_eq_rat @ B @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.orderI
% 0.33/0.65 thf(fact_8286_max_OorderI,axiom,
% 0.33/0.65 ! [A: num,B: num] :
% 0.33/0.65 ( ( A
% 0.33/0.65 = ( ord_max_num @ A @ B ) )
% 0.33/0.65 => ( ord_less_eq_num @ B @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.orderI
% 0.33/0.65 thf(fact_8287_max_OorderI,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( A
% 0.33/0.65 = ( ord_max_nat @ A @ B ) )
% 0.33/0.65 => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.orderI
% 0.33/0.65 thf(fact_8288_max_OorderI,axiom,
% 0.33/0.65 ! [A: int,B: int] :
% 0.33/0.65 ( ( A
% 0.33/0.65 = ( ord_max_int @ A @ B ) )
% 0.33/0.65 => ( ord_less_eq_int @ B @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.orderI
% 0.33/0.65 thf(fact_8289_max_OorderE,axiom,
% 0.33/0.65 ! [B: rat,A: rat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ B @ A )
% 0.33/0.65 => ( A
% 0.33/0.65 = ( ord_max_rat @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.orderE
% 0.33/0.65 thf(fact_8290_max_OorderE,axiom,
% 0.33/0.65 ! [B: num,A: num] :
% 0.33/0.65 ( ( ord_less_eq_num @ B @ A )
% 0.33/0.65 => ( A
% 0.33/0.65 = ( ord_max_num @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.orderE
% 0.33/0.65 thf(fact_8291_max_OorderE,axiom,
% 0.33/0.65 ! [B: nat,A: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ B @ A )
% 0.33/0.65 => ( A
% 0.33/0.65 = ( ord_max_nat @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.orderE
% 0.33/0.65 thf(fact_8292_max_OorderE,axiom,
% 0.33/0.65 ! [B: int,A: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ B @ A )
% 0.33/0.65 => ( A
% 0.33/0.65 = ( ord_max_int @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.orderE
% 0.33/0.65 thf(fact_8293_max_Omono,axiom,
% 0.33/0.65 ! [C: rat,A: rat,D: rat,B: rat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ C @ A )
% 0.33/0.65 => ( ( ord_less_eq_rat @ D @ B )
% 0.33/0.65 => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.mono
% 0.33/0.65 thf(fact_8294_max_Omono,axiom,
% 0.33/0.65 ! [C: num,A: num,D: num,B: num] :
% 0.33/0.65 ( ( ord_less_eq_num @ C @ A )
% 0.33/0.65 => ( ( ord_less_eq_num @ D @ B )
% 0.33/0.65 => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.mono
% 0.33/0.65 thf(fact_8295_max_Omono,axiom,
% 0.33/0.65 ! [C: nat,A: nat,D: nat,B: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ C @ A )
% 0.33/0.65 => ( ( ord_less_eq_nat @ D @ B )
% 0.33/0.65 => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.mono
% 0.33/0.65 thf(fact_8296_max_Omono,axiom,
% 0.33/0.65 ! [C: int,A: int,D: int,B: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ C @ A )
% 0.33/0.65 => ( ( ord_less_eq_int @ D @ B )
% 0.33/0.65 => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.mono
% 0.33/0.65 thf(fact_8297_max_Ostrict__coboundedI2,axiom,
% 0.33/0.65 ! [C: real,B: real,A: real] :
% 0.33/0.65 ( ( ord_less_real @ C @ B )
% 0.33/0.65 => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_coboundedI2
% 0.33/0.65 thf(fact_8298_max_Ostrict__coboundedI2,axiom,
% 0.33/0.65 ! [C: rat,B: rat,A: rat] :
% 0.33/0.65 ( ( ord_less_rat @ C @ B )
% 0.33/0.65 => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_coboundedI2
% 0.33/0.65 thf(fact_8299_max_Ostrict__coboundedI2,axiom,
% 0.33/0.65 ! [C: num,B: num,A: num] :
% 0.33/0.65 ( ( ord_less_num @ C @ B )
% 0.33/0.65 => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_coboundedI2
% 0.33/0.65 thf(fact_8300_max_Ostrict__coboundedI2,axiom,
% 0.33/0.65 ! [C: nat,B: nat,A: nat] :
% 0.33/0.65 ( ( ord_less_nat @ C @ B )
% 0.33/0.65 => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_coboundedI2
% 0.33/0.65 thf(fact_8301_max_Ostrict__coboundedI2,axiom,
% 0.33/0.65 ! [C: int,B: int,A: int] :
% 0.33/0.65 ( ( ord_less_int @ C @ B )
% 0.33/0.65 => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_coboundedI2
% 0.33/0.65 thf(fact_8302_max_Ostrict__coboundedI1,axiom,
% 0.33/0.65 ! [C: real,A: real,B: real] :
% 0.33/0.65 ( ( ord_less_real @ C @ A )
% 0.33/0.65 => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_coboundedI1
% 0.33/0.65 thf(fact_8303_max_Ostrict__coboundedI1,axiom,
% 0.33/0.65 ! [C: rat,A: rat,B: rat] :
% 0.33/0.65 ( ( ord_less_rat @ C @ A )
% 0.33/0.65 => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_coboundedI1
% 0.33/0.65 thf(fact_8304_max_Ostrict__coboundedI1,axiom,
% 0.33/0.65 ! [C: num,A: num,B: num] :
% 0.33/0.65 ( ( ord_less_num @ C @ A )
% 0.33/0.65 => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_coboundedI1
% 0.33/0.65 thf(fact_8305_max_Ostrict__coboundedI1,axiom,
% 0.33/0.65 ! [C: nat,A: nat,B: nat] :
% 0.33/0.65 ( ( ord_less_nat @ C @ A )
% 0.33/0.65 => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_coboundedI1
% 0.33/0.65 thf(fact_8306_max_Ostrict__coboundedI1,axiom,
% 0.33/0.65 ! [C: int,A: int,B: int] :
% 0.33/0.65 ( ( ord_less_int @ C @ A )
% 0.33/0.65 => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_coboundedI1
% 0.33/0.65 thf(fact_8307_max_Ostrict__order__iff,axiom,
% 0.33/0.65 ( ord_less_real
% 0.33/0.65 = ( ^ [B2: real,A2: real] :
% 0.33/0.65 ( ( A2
% 0.33/0.65 = ( ord_max_real @ A2 @ B2 ) )
% 0.33/0.65 & ( A2 != B2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_order_iff
% 0.33/0.65 thf(fact_8308_max_Ostrict__order__iff,axiom,
% 0.33/0.65 ( ord_less_rat
% 0.33/0.65 = ( ^ [B2: rat,A2: rat] :
% 0.33/0.65 ( ( A2
% 0.33/0.65 = ( ord_max_rat @ A2 @ B2 ) )
% 0.33/0.65 & ( A2 != B2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_order_iff
% 0.33/0.65 thf(fact_8309_max_Ostrict__order__iff,axiom,
% 0.33/0.65 ( ord_less_num
% 0.33/0.65 = ( ^ [B2: num,A2: num] :
% 0.33/0.65 ( ( A2
% 0.33/0.65 = ( ord_max_num @ A2 @ B2 ) )
% 0.33/0.65 & ( A2 != B2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_order_iff
% 0.33/0.65 thf(fact_8310_max_Ostrict__order__iff,axiom,
% 0.33/0.65 ( ord_less_nat
% 0.33/0.65 = ( ^ [B2: nat,A2: nat] :
% 0.33/0.65 ( ( A2
% 0.33/0.65 = ( ord_max_nat @ A2 @ B2 ) )
% 0.33/0.65 & ( A2 != B2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_order_iff
% 0.33/0.65 thf(fact_8311_max_Ostrict__order__iff,axiom,
% 0.33/0.65 ( ord_less_int
% 0.33/0.65 = ( ^ [B2: int,A2: int] :
% 0.33/0.65 ( ( A2
% 0.33/0.65 = ( ord_max_int @ A2 @ B2 ) )
% 0.33/0.65 & ( A2 != B2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_order_iff
% 0.33/0.65 thf(fact_8312_max_Ostrict__boundedE,axiom,
% 0.33/0.65 ! [B: real,C: real,A: real] :
% 0.33/0.65 ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
% 0.33/0.65 => ~ ( ( ord_less_real @ B @ A )
% 0.33/0.65 => ~ ( ord_less_real @ C @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_boundedE
% 0.33/0.65 thf(fact_8313_max_Ostrict__boundedE,axiom,
% 0.33/0.65 ! [B: rat,C: rat,A: rat] :
% 0.33/0.65 ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
% 0.33/0.65 => ~ ( ( ord_less_rat @ B @ A )
% 0.33/0.65 => ~ ( ord_less_rat @ C @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_boundedE
% 0.33/0.65 thf(fact_8314_max_Ostrict__boundedE,axiom,
% 0.33/0.65 ! [B: num,C: num,A: num] :
% 0.33/0.65 ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
% 0.33/0.65 => ~ ( ( ord_less_num @ B @ A )
% 0.33/0.65 => ~ ( ord_less_num @ C @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_boundedE
% 0.33/0.65 thf(fact_8315_max_Ostrict__boundedE,axiom,
% 0.33/0.65 ! [B: nat,C: nat,A: nat] :
% 0.33/0.65 ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.33/0.65 => ~ ( ( ord_less_nat @ B @ A )
% 0.33/0.65 => ~ ( ord_less_nat @ C @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_boundedE
% 0.33/0.65 thf(fact_8316_max_Ostrict__boundedE,axiom,
% 0.33/0.65 ! [B: int,C: int,A: int] :
% 0.33/0.65 ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
% 0.33/0.65 => ~ ( ( ord_less_int @ B @ A )
% 0.33/0.65 => ~ ( ord_less_int @ C @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.strict_boundedE
% 0.33/0.65 thf(fact_8317_less__max__iff__disj,axiom,
% 0.33/0.65 ! [Z2: real,X: real,Y: real] :
% 0.33/0.65 ( ( ord_less_real @ Z2 @ ( ord_max_real @ X @ Y ) )
% 0.33/0.65 = ( ( ord_less_real @ Z2 @ X )
% 0.33/0.65 | ( ord_less_real @ Z2 @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % less_max_iff_disj
% 0.33/0.65 thf(fact_8318_less__max__iff__disj,axiom,
% 0.33/0.65 ! [Z2: rat,X: rat,Y: rat] :
% 0.33/0.65 ( ( ord_less_rat @ Z2 @ ( ord_max_rat @ X @ Y ) )
% 0.33/0.65 = ( ( ord_less_rat @ Z2 @ X )
% 0.33/0.65 | ( ord_less_rat @ Z2 @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % less_max_iff_disj
% 0.33/0.65 thf(fact_8319_less__max__iff__disj,axiom,
% 0.33/0.65 ! [Z2: num,X: num,Y: num] :
% 0.33/0.65 ( ( ord_less_num @ Z2 @ ( ord_max_num @ X @ Y ) )
% 0.33/0.65 = ( ( ord_less_num @ Z2 @ X )
% 0.33/0.65 | ( ord_less_num @ Z2 @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % less_max_iff_disj
% 0.33/0.65 thf(fact_8320_less__max__iff__disj,axiom,
% 0.33/0.65 ! [Z2: nat,X: nat,Y: nat] :
% 0.33/0.65 ( ( ord_less_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
% 0.33/0.65 = ( ( ord_less_nat @ Z2 @ X )
% 0.33/0.65 | ( ord_less_nat @ Z2 @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % less_max_iff_disj
% 0.33/0.65 thf(fact_8321_less__max__iff__disj,axiom,
% 0.33/0.65 ! [Z2: int,X: int,Y: int] :
% 0.33/0.65 ( ( ord_less_int @ Z2 @ ( ord_max_int @ X @ Y ) )
% 0.33/0.65 = ( ( ord_less_int @ Z2 @ X )
% 0.33/0.65 | ( ord_less_int @ Z2 @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % less_max_iff_disj
% 0.33/0.65 thf(fact_8322_max__add__distrib__right,axiom,
% 0.33/0.65 ! [X: real,Y: real,Z2: real] :
% 0.33/0.65 ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z2 ) )
% 0.33/0.65 = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_add_distrib_right
% 0.33/0.65 thf(fact_8323_max__add__distrib__right,axiom,
% 0.33/0.65 ! [X: rat,Y: rat,Z2: rat] :
% 0.33/0.65 ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z2 ) )
% 0.33/0.65 = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_add_distrib_right
% 0.33/0.65 thf(fact_8324_max__add__distrib__right,axiom,
% 0.33/0.65 ! [X: int,Y: int,Z2: int] :
% 0.33/0.65 ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z2 ) )
% 0.33/0.65 = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_add_distrib_right
% 0.33/0.65 thf(fact_8325_max__add__distrib__right,axiom,
% 0.33/0.65 ! [X: nat,Y: nat,Z2: nat] :
% 0.33/0.65 ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z2 ) )
% 0.33/0.65 = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_add_distrib_right
% 0.33/0.65 thf(fact_8326_max__add__distrib__left,axiom,
% 0.33/0.65 ! [X: real,Y: real,Z2: real] :
% 0.33/0.65 ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
% 0.33/0.65 = ( ord_max_real @ ( plus_plus_real @ X @ Z2 ) @ ( plus_plus_real @ Y @ Z2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_add_distrib_left
% 0.33/0.65 thf(fact_8327_max__add__distrib__left,axiom,
% 0.33/0.65 ! [X: rat,Y: rat,Z2: rat] :
% 0.33/0.65 ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
% 0.33/0.65 = ( ord_max_rat @ ( plus_plus_rat @ X @ Z2 ) @ ( plus_plus_rat @ Y @ Z2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_add_distrib_left
% 0.33/0.65 thf(fact_8328_max__add__distrib__left,axiom,
% 0.33/0.65 ! [X: int,Y: int,Z2: int] :
% 0.33/0.65 ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z2 )
% 0.33/0.65 = ( ord_max_int @ ( plus_plus_int @ X @ Z2 ) @ ( plus_plus_int @ Y @ Z2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_add_distrib_left
% 0.33/0.65 thf(fact_8329_max__add__distrib__left,axiom,
% 0.33/0.65 ! [X: nat,Y: nat,Z2: nat] :
% 0.33/0.65 ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
% 0.33/0.65 = ( ord_max_nat @ ( plus_plus_nat @ X @ Z2 ) @ ( plus_plus_nat @ Y @ Z2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_add_distrib_left
% 0.33/0.65 thf(fact_8330_max__diff__distrib__left,axiom,
% 0.33/0.65 ! [X: real,Y: real,Z2: real] :
% 0.33/0.65 ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
% 0.33/0.65 = ( ord_max_real @ ( minus_minus_real @ X @ Z2 ) @ ( minus_minus_real @ Y @ Z2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_diff_distrib_left
% 0.33/0.65 thf(fact_8331_max__diff__distrib__left,axiom,
% 0.33/0.65 ! [X: rat,Y: rat,Z2: rat] :
% 0.33/0.65 ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
% 0.33/0.65 = ( ord_max_rat @ ( minus_minus_rat @ X @ Z2 ) @ ( minus_minus_rat @ Y @ Z2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_diff_distrib_left
% 0.33/0.65 thf(fact_8332_max__diff__distrib__left,axiom,
% 0.33/0.65 ! [X: int,Y: int,Z2: int] :
% 0.33/0.65 ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z2 )
% 0.33/0.65 = ( ord_max_int @ ( minus_minus_int @ X @ Z2 ) @ ( minus_minus_int @ Y @ Z2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_diff_distrib_left
% 0.33/0.65 thf(fact_8333_max_Oassoc,axiom,
% 0.33/0.65 ! [A: int,B: int,C: int] :
% 0.33/0.65 ( ( ord_max_int @ ( ord_max_int @ A @ B ) @ C )
% 0.33/0.65 = ( ord_max_int @ A @ ( ord_max_int @ B @ C ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.assoc
% 0.33/0.65 thf(fact_8334_max_Oassoc,axiom,
% 0.33/0.65 ! [A: nat,B: nat,C: nat] :
% 0.33/0.65 ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ C )
% 0.33/0.65 = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.assoc
% 0.33/0.65 thf(fact_8335_max_Ocommute,axiom,
% 0.33/0.65 ( ord_max_int
% 0.33/0.65 = ( ^ [A2: int,B2: int] : ( ord_max_int @ B2 @ A2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.commute
% 0.33/0.65 thf(fact_8336_max_Ocommute,axiom,
% 0.33/0.65 ( ord_max_nat
% 0.33/0.65 = ( ^ [A2: nat,B2: nat] : ( ord_max_nat @ B2 @ A2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.commute
% 0.33/0.65 thf(fact_8337_max_Oleft__commute,axiom,
% 0.33/0.65 ! [B: int,A: int,C: int] :
% 0.33/0.65 ( ( ord_max_int @ B @ ( ord_max_int @ A @ C ) )
% 0.33/0.65 = ( ord_max_int @ A @ ( ord_max_int @ B @ C ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.left_commute
% 0.33/0.65 thf(fact_8338_max_Oleft__commute,axiom,
% 0.33/0.65 ! [B: nat,A: nat,C: nat] :
% 0.33/0.65 ( ( ord_max_nat @ B @ ( ord_max_nat @ A @ C ) )
% 0.33/0.65 = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max.left_commute
% 0.33/0.65 thf(fact_8339_of__nat__max,axiom,
% 0.33/0.65 ! [X: nat,Y: nat] :
% 0.33/0.65 ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
% 0.33/0.65 = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_max
% 0.33/0.65 thf(fact_8340_of__nat__max,axiom,
% 0.33/0.65 ! [X: nat,Y: nat] :
% 0.33/0.65 ( ( semiri681578069525770553at_rat @ ( ord_max_nat @ X @ Y ) )
% 0.33/0.65 = ( ord_max_rat @ ( semiri681578069525770553at_rat @ X ) @ ( semiri681578069525770553at_rat @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_max
% 0.33/0.65 thf(fact_8341_of__nat__max,axiom,
% 0.33/0.65 ! [X: nat,Y: nat] :
% 0.33/0.65 ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
% 0.33/0.65 = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_max
% 0.33/0.65 thf(fact_8342_of__nat__max,axiom,
% 0.33/0.65 ! [X: nat,Y: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
% 0.33/0.65 = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_max
% 0.33/0.65 thf(fact_8343_or__not__num__neg_Osimps_I1_J,axiom,
% 0.33/0.65 ( ( bit_or_not_num_neg @ one @ one )
% 0.33/0.65 = one ) ).
% 0.33/0.65
% 0.33/0.65 % or_not_num_neg.simps(1)
% 0.33/0.65 thf(fact_8344_set__bit__nat__def,axiom,
% 0.33/0.65 ( bit_se7882103937844011126it_nat
% 0.33/0.65 = ( ^ [M3: nat,N3: nat] : ( bit_se1412395901928357646or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % set_bit_nat_def
% 0.33/0.65 thf(fact_8345_or__not__num__neg_Osimps_I4_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
% 0.33/0.65 = ( bit0 @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_not_num_neg.simps(4)
% 0.33/0.65 thf(fact_8346_or__not__num__neg_Osimps_I6_J,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
% 0.33/0.65 = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_not_num_neg.simps(6)
% 0.33/0.65 thf(fact_8347_or__not__num__neg_Osimps_I7_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
% 0.33/0.65 = one ) ).
% 0.33/0.65
% 0.33/0.65 % or_not_num_neg.simps(7)
% 0.33/0.65 thf(fact_8348_or__not__num__neg_Osimps_I3_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
% 0.33/0.65 = ( bit1 @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_not_num_neg.simps(3)
% 0.33/0.65 thf(fact_8349_or__nat__def,axiom,
% 0.33/0.65 ( bit_se1412395901928357646or_nat
% 0.33/0.65 = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_nat_def
% 0.33/0.65 thf(fact_8350_or__not__num__neg_Osimps_I2_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
% 0.33/0.65 = ( bit1 @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_not_num_neg.simps(2)
% 0.33/0.65 thf(fact_8351_int__numeral__or__not__num__neg,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_numeral_or_not_num_neg
% 0.33/0.65 thf(fact_8352_int__numeral__not__or__num__neg,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_numeral_not_or_num_neg
% 0.33/0.65 thf(fact_8353_numeral__or__not__num__eq,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_or_not_num_eq
% 0.33/0.65 thf(fact_8354_Suc__0__or__eq,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.33/0.65 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Suc_0_or_eq
% 0.33/0.65 thf(fact_8355_or__Suc__0__eq,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.33/0.65 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_Suc_0_eq
% 0.33/0.65 thf(fact_8356_or__nat__rec,axiom,
% 0.33/0.65 ( bit_se1412395901928357646or_nat
% 0.33/0.65 = ( ^ [M3: nat,N3: nat] :
% 0.33/0.65 ( plus_plus_nat
% 0.33/0.65 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.65 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 0.33/0.65 | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.33/0.65 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_nat_rec
% 0.33/0.65 thf(fact_8357_or__minus__numerals_I5_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_minus_numerals(5)
% 0.33/0.65 thf(fact_8358_or__minus__numerals_I1_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_minus_numerals(1)
% 0.33/0.65 thf(fact_8359_or__nat__unfold,axiom,
% 0.33/0.65 ( bit_se1412395901928357646or_nat
% 0.33/0.65 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_nat_unfold
% 0.33/0.65 thf(fact_8360_horner__sum__of__bool__2__less,axiom,
% 0.33/0.65 ! [Bs: list_o] : ( ord_less_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_size_list_o @ Bs ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % horner_sum_of_bool_2_less
% 0.33/0.65 thf(fact_8361_max__Suc__Suc,axiom,
% 0.33/0.65 ! [M: nat,N: nat] :
% 0.33/0.65 ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.33/0.65 = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_Suc_Suc
% 0.33/0.65 thf(fact_8362_max__0R,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( ord_max_nat @ N @ zero_zero_nat )
% 0.33/0.65 = N ) ).
% 0.33/0.65
% 0.33/0.65 % max_0R
% 0.33/0.65 thf(fact_8363_max__0L,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( ord_max_nat @ zero_zero_nat @ N )
% 0.33/0.65 = N ) ).
% 0.33/0.65
% 0.33/0.65 % max_0L
% 0.33/0.65 thf(fact_8364_max__nat_Oright__neutral,axiom,
% 0.33/0.65 ! [A: nat] :
% 0.33/0.65 ( ( ord_max_nat @ A @ zero_zero_nat )
% 0.33/0.65 = A ) ).
% 0.33/0.65
% 0.33/0.65 % max_nat.right_neutral
% 0.33/0.65 thf(fact_8365_max__nat_Oneutr__eq__iff,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( zero_zero_nat
% 0.33/0.65 = ( ord_max_nat @ A @ B ) )
% 0.33/0.65 = ( ( A = zero_zero_nat )
% 0.33/0.65 & ( B = zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_nat.neutr_eq_iff
% 0.33/0.65 thf(fact_8366_max__nat_Oleft__neutral,axiom,
% 0.33/0.65 ! [A: nat] :
% 0.33/0.65 ( ( ord_max_nat @ zero_zero_nat @ A )
% 0.33/0.65 = A ) ).
% 0.33/0.65
% 0.33/0.65 % max_nat.left_neutral
% 0.33/0.65 thf(fact_8367_max__nat_Oeq__neutr__iff,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( ( ord_max_nat @ A @ B )
% 0.33/0.65 = zero_zero_nat )
% 0.33/0.65 = ( ( A = zero_zero_nat )
% 0.33/0.65 & ( B = zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_nat.eq_neutr_iff
% 0.33/0.65 thf(fact_8368_dbl__dec__simps_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.33/0.65 = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(5)
% 0.33/0.65 thf(fact_8369_dbl__dec__simps_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) )
% 0.33/0.65 = ( numeral_numeral_real @ ( bitM @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(5)
% 0.33/0.65 thf(fact_8370_dbl__dec__simps_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) )
% 0.33/0.65 = ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(5)
% 0.33/0.65 thf(fact_8371_dbl__dec__simps_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
% 0.33/0.65 = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(5)
% 0.33/0.65 thf(fact_8372_pred__numeral__simps_I2_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( pred_numeral @ ( bit0 @ K ) )
% 0.33/0.65 = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % pred_numeral_simps(2)
% 0.33/0.65 thf(fact_8373_max__numeral__Suc,axiom,
% 0.33/0.65 ! [K: num,N: nat] :
% 0.33/0.65 ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.33/0.65 = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_numeral_Suc
% 0.33/0.65 thf(fact_8374_max__Suc__numeral,axiom,
% 0.33/0.65 ! [N: nat,K: num] :
% 0.33/0.65 ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.33/0.65 = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_Suc_numeral
% 0.33/0.65 thf(fact_8375_or__minus__numerals_I7_J,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_minus_numerals(7)
% 0.33/0.65 thf(fact_8376_or__minus__numerals_I3_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_minus_numerals(3)
% 0.33/0.65 thf(fact_8377_nat__mult__max__left,axiom,
% 0.33/0.65 ! [M: nat,N: nat,Q2: nat] :
% 0.33/0.65 ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 0.33/0.65 = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_mult_max_left
% 0.33/0.65 thf(fact_8378_nat__mult__max__right,axiom,
% 0.33/0.65 ! [M: nat,N: nat,Q2: nat] :
% 0.33/0.65 ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 0.33/0.65 = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_mult_max_right
% 0.33/0.65 thf(fact_8379_nat__add__max__left,axiom,
% 0.33/0.65 ! [M: nat,N: nat,Q2: nat] :
% 0.33/0.65 ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 0.33/0.65 = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_add_max_left
% 0.33/0.65 thf(fact_8380_nat__add__max__right,axiom,
% 0.33/0.65 ! [M: nat,N: nat,Q2: nat] :
% 0.33/0.65 ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 0.33/0.65 = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_add_max_right
% 0.33/0.65 thf(fact_8381_semiring__norm_I26_J,axiom,
% 0.33/0.65 ( ( bitM @ one )
% 0.33/0.65 = one ) ).
% 0.33/0.65
% 0.33/0.65 % semiring_norm(26)
% 0.33/0.65 thf(fact_8382_nat__minus__add__max,axiom,
% 0.33/0.65 ! [N: nat,M: nat] :
% 0.33/0.65 ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 0.33/0.65 = ( ord_max_nat @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_minus_add_max
% 0.33/0.65 thf(fact_8383_semiring__norm_I27_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( bitM @ ( bit0 @ N ) )
% 0.33/0.65 = ( bit1 @ ( bitM @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % semiring_norm(27)
% 0.33/0.65 thf(fact_8384_semiring__norm_I28_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( bitM @ ( bit1 @ N ) )
% 0.33/0.65 = ( bit1 @ ( bit0 @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % semiring_norm(28)
% 0.33/0.65 thf(fact_8385_or__not__num__neg_Osimps_I5_J,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
% 0.33/0.65 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_not_num_neg.simps(5)
% 0.33/0.65 thf(fact_8386_inc__BitM__eq,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( inc @ ( bitM @ N ) )
% 0.33/0.65 = ( bit0 @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % inc_BitM_eq
% 0.33/0.65 thf(fact_8387_or__not__num__neg_Osimps_I9_J,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
% 0.33/0.65 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_not_num_neg.simps(9)
% 0.33/0.65 thf(fact_8388_BitM__inc__eq,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( bitM @ ( inc @ N ) )
% 0.33/0.65 = ( bit1 @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % BitM_inc_eq
% 0.33/0.65 thf(fact_8389_eval__nat__numeral_I2_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.33/0.65 = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % eval_nat_numeral(2)
% 0.33/0.65 thf(fact_8390_one__plus__BitM,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( plus_plus_num @ one @ ( bitM @ N ) )
% 0.33/0.65 = ( bit0 @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % one_plus_BitM
% 0.33/0.65 thf(fact_8391_BitM__plus__one,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( plus_plus_num @ ( bitM @ N ) @ one )
% 0.33/0.65 = ( bit0 @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % BitM_plus_one
% 0.33/0.65 thf(fact_8392_or__not__num__neg_Osimps_I8_J,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
% 0.33/0.65 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_not_num_neg.simps(8)
% 0.33/0.65 thf(fact_8393_numeral__BitM,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( numera6690914467698888265omplex @ ( bitM @ N ) )
% 0.33/0.65 = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N ) ) @ one_one_complex ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_BitM
% 0.33/0.65 thf(fact_8394_numeral__BitM,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( numeral_numeral_real @ ( bitM @ N ) )
% 0.33/0.65 = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N ) ) @ one_one_real ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_BitM
% 0.33/0.65 thf(fact_8395_numeral__BitM,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( numeral_numeral_rat @ ( bitM @ N ) )
% 0.33/0.65 = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_BitM
% 0.33/0.65 thf(fact_8396_numeral__BitM,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( numeral_numeral_int @ ( bitM @ N ) )
% 0.33/0.65 = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_BitM
% 0.33/0.65 thf(fact_8397_odd__numeral__BitM,axiom,
% 0.33/0.65 ! [W: num] :
% 0.33/0.65 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % odd_numeral_BitM
% 0.33/0.65 thf(fact_8398_odd__numeral__BitM,axiom,
% 0.33/0.65 ! [W: num] :
% 0.33/0.65 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % odd_numeral_BitM
% 0.33/0.65 thf(fact_8399_odd__numeral__BitM,axiom,
% 0.33/0.65 ! [W: num] :
% 0.33/0.65 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % odd_numeral_BitM
% 0.33/0.65 thf(fact_8400_not__numeral__BitM__eq,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bitM @ N ) ) )
% 0.33/0.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % not_numeral_BitM_eq
% 0.33/0.65 thf(fact_8401_not__numeral__BitM__eq,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bitM @ N ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % not_numeral_BitM_eq
% 0.33/0.65 thf(fact_8402_or__not__num__neg_Oelims,axiom,
% 0.33/0.65 ! [X: num,Xa: num,Y: num] :
% 0.33/0.65 ( ( ( bit_or_not_num_neg @ X @ Xa )
% 0.33/0.65 = Y )
% 0.33/0.65 => ( ( ( X = one )
% 0.33/0.65 => ( ( Xa = one )
% 0.33/0.65 => ( Y != one ) ) )
% 0.33/0.65 => ( ( ( X = one )
% 0.33/0.65 => ! [M2: num] :
% 0.33/0.65 ( ( Xa
% 0.33/0.65 = ( bit0 @ M2 ) )
% 0.33/0.65 => ( Y
% 0.33/0.65 != ( bit1 @ M2 ) ) ) )
% 0.33/0.65 => ( ( ( X = one )
% 0.33/0.65 => ! [M2: num] :
% 0.33/0.65 ( ( Xa
% 0.33/0.65 = ( bit1 @ M2 ) )
% 0.33/0.65 => ( Y
% 0.33/0.65 != ( bit1 @ M2 ) ) ) )
% 0.33/0.65 => ( ( ? [N2: num] :
% 0.33/0.65 ( X
% 0.33/0.65 = ( bit0 @ N2 ) )
% 0.33/0.65 => ( ( Xa = one )
% 0.33/0.65 => ( Y
% 0.33/0.65 != ( bit0 @ one ) ) ) )
% 0.33/0.65 => ( ! [N2: num] :
% 0.33/0.65 ( ( X
% 0.33/0.65 = ( bit0 @ N2 ) )
% 0.33/0.65 => ! [M2: num] :
% 0.33/0.65 ( ( Xa
% 0.33/0.65 = ( bit0 @ M2 ) )
% 0.33/0.65 => ( Y
% 0.33/0.65 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M2 ) ) ) ) )
% 0.33/0.65 => ( ! [N2: num] :
% 0.33/0.65 ( ( X
% 0.33/0.65 = ( bit0 @ N2 ) )
% 0.33/0.65 => ! [M2: num] :
% 0.33/0.65 ( ( Xa
% 0.33/0.65 = ( bit1 @ M2 ) )
% 0.33/0.65 => ( Y
% 0.33/0.65 != ( bit0 @ ( bit_or_not_num_neg @ N2 @ M2 ) ) ) ) )
% 0.33/0.65 => ( ( ? [N2: num] :
% 0.33/0.65 ( X
% 0.33/0.65 = ( bit1 @ N2 ) )
% 0.33/0.65 => ( ( Xa = one )
% 0.33/0.65 => ( Y != one ) ) )
% 0.33/0.65 => ( ! [N2: num] :
% 0.33/0.65 ( ( X
% 0.33/0.65 = ( bit1 @ N2 ) )
% 0.33/0.65 => ! [M2: num] :
% 0.33/0.65 ( ( Xa
% 0.33/0.65 = ( bit0 @ M2 ) )
% 0.33/0.65 => ( Y
% 0.33/0.65 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M2 ) ) ) ) )
% 0.33/0.65 => ~ ! [N2: num] :
% 0.33/0.65 ( ( X
% 0.33/0.65 = ( bit1 @ N2 ) )
% 0.33/0.65 => ! [M2: num] :
% 0.33/0.65 ( ( Xa
% 0.33/0.65 = ( bit1 @ M2 ) )
% 0.33/0.65 => ( Y
% 0.33/0.65 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % or_not_num_neg.elims
% 0.33/0.65 thf(fact_8403_bit__horner__sum__bit__iff,axiom,
% 0.33/0.65 ! [Bs: list_o,N: nat] :
% 0.33/0.65 ( ( bit_se9216721137139052372nteger @ ( groups3417619833198082522nteger @ zero_n356916108424825756nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Bs ) @ N )
% 0.33/0.65 = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 0.33/0.65 & ( nth_o @ Bs @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bit_horner_sum_bit_iff
% 0.33/0.65 thf(fact_8404_bit__horner__sum__bit__iff,axiom,
% 0.33/0.65 ! [Bs: list_o,N: nat] :
% 0.33/0.65 ( ( bit_se1148574629649215175it_nat @ ( groups9119017779487936845_o_nat @ zero_n2687167440665602831ol_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Bs ) @ N )
% 0.33/0.65 = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 0.33/0.65 & ( nth_o @ Bs @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bit_horner_sum_bit_iff
% 0.33/0.65 thf(fact_8405_bit__horner__sum__bit__iff,axiom,
% 0.33/0.65 ! [Bs: list_o,N: nat] :
% 0.33/0.65 ( ( bit_se1146084159140164899it_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ N )
% 0.33/0.65 = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 0.33/0.65 & ( nth_o @ Bs @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bit_horner_sum_bit_iff
% 0.33/0.65 thf(fact_8406_length__subseqs,axiom,
% 0.33/0.65 ! [Xs: list_VEBT_VEBT] :
% 0.33/0.65 ( ( size_s8217280938318005548T_VEBT @ ( subseqs_VEBT_VEBT @ Xs ) )
% 0.33/0.65 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % length_subseqs
% 0.33/0.65 thf(fact_8407_length__subseqs,axiom,
% 0.33/0.65 ! [Xs: list_o] :
% 0.33/0.65 ( ( size_s2710708370519433104list_o @ ( subseqs_o @ Xs ) )
% 0.33/0.65 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_o @ Xs ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % length_subseqs
% 0.33/0.65 thf(fact_8408_length__subseqs,axiom,
% 0.33/0.65 ! [Xs: list_nat] :
% 0.33/0.65 ( ( size_s3023201423986296836st_nat @ ( subseqs_nat @ Xs ) )
% 0.33/0.65 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % length_subseqs
% 0.33/0.65 thf(fact_8409_length__subseqs,axiom,
% 0.33/0.65 ! [Xs: list_int] :
% 0.33/0.65 ( ( size_s533118279054570080st_int @ ( subseqs_int @ Xs ) )
% 0.33/0.65 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_int @ Xs ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % length_subseqs
% 0.33/0.65 thf(fact_8410_drop__bit__numeral__minus__bit1,axiom,
% 0.33/0.65 ! [L: num,K: num] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.33/0.65 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_numeral_minus_bit1
% 0.33/0.65 thf(fact_8411_drop__bit__rec,axiom,
% 0.33/0.65 ( bit_se8568078237143864401it_int
% 0.33/0.65 = ( ^ [N3: nat,A2: int] : ( if_int @ ( N3 = zero_zero_nat ) @ A2 @ ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_rec
% 0.33/0.65 thf(fact_8412_drop__bit__rec,axiom,
% 0.33/0.65 ( bit_se8570568707652914677it_nat
% 0.33/0.65 = ( ^ [N3: nat,A2: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ A2 @ ( bit_se8570568707652914677it_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_rec
% 0.33/0.65 thf(fact_8413_drop__bit__of__0,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ N @ zero_zero_int )
% 0.33/0.65 = zero_zero_int ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_of_0
% 0.33/0.65 thf(fact_8414_drop__bit__of__0,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ N @ zero_zero_nat )
% 0.33/0.65 = zero_zero_nat ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_of_0
% 0.33/0.65 thf(fact_8415_drop__bit__drop__bit,axiom,
% 0.33/0.65 ! [M: nat,N: nat,A: int] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se8568078237143864401it_int @ N @ A ) )
% 0.33/0.65 = ( bit_se8568078237143864401it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_drop_bit
% 0.33/0.65 thf(fact_8416_drop__bit__drop__bit,axiom,
% 0.33/0.65 ! [M: nat,N: nat,A: nat] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ M @ ( bit_se8570568707652914677it_nat @ N @ A ) )
% 0.33/0.65 = ( bit_se8570568707652914677it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_drop_bit
% 0.33/0.65 thf(fact_8417_drop__bit__and,axiom,
% 0.33/0.65 ! [N: nat,A: int,B: int] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 0.33/0.65 = ( bit_se725231765392027082nd_int @ ( bit_se8568078237143864401it_int @ N @ A ) @ ( bit_se8568078237143864401it_int @ N @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_and
% 0.33/0.65 thf(fact_8418_drop__bit__and,axiom,
% 0.33/0.65 ! [N: nat,A: nat,B: nat] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 0.33/0.65 = ( bit_se727722235901077358nd_nat @ ( bit_se8570568707652914677it_nat @ N @ A ) @ ( bit_se8570568707652914677it_nat @ N @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_and
% 0.33/0.65 thf(fact_8419_drop__bit__or,axiom,
% 0.33/0.65 ! [N: nat,A: int,B: int] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ N @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 0.33/0.65 = ( bit_se1409905431419307370or_int @ ( bit_se8568078237143864401it_int @ N @ A ) @ ( bit_se8568078237143864401it_int @ N @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_or
% 0.33/0.65 thf(fact_8420_drop__bit__or,axiom,
% 0.33/0.65 ! [N: nat,A: nat,B: nat] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ N @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 0.33/0.65 = ( bit_se1412395901928357646or_nat @ ( bit_se8570568707652914677it_nat @ N @ A ) @ ( bit_se8570568707652914677it_nat @ N @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_or
% 0.33/0.65 thf(fact_8421_drop__bit__xor,axiom,
% 0.33/0.65 ! [N: nat,A: int,B: int] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 0.33/0.65 = ( bit_se6526347334894502574or_int @ ( bit_se8568078237143864401it_int @ N @ A ) @ ( bit_se8568078237143864401it_int @ N @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_xor
% 0.33/0.65 thf(fact_8422_drop__bit__xor,axiom,
% 0.33/0.65 ! [N: nat,A: nat,B: nat] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 0.33/0.65 = ( bit_se6528837805403552850or_nat @ ( bit_se8570568707652914677it_nat @ N @ A ) @ ( bit_se8570568707652914677it_nat @ N @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_xor
% 0.33/0.65 thf(fact_8423_drop__bit__of__bool,axiom,
% 0.33/0.65 ! [N: nat,B: $o] :
% 0.33/0.65 ( ( bit_se3928097537394005634nteger @ N @ ( zero_n356916108424825756nteger @ B ) )
% 0.33/0.65 = ( zero_n356916108424825756nteger
% 0.33/0.65 @ ( ( N = zero_zero_nat )
% 0.33/0.65 & B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_of_bool
% 0.33/0.65 thf(fact_8424_drop__bit__of__bool,axiom,
% 0.33/0.65 ! [N: nat,B: $o] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ N @ ( zero_n2684676970156552555ol_int @ B ) )
% 0.33/0.65 = ( zero_n2684676970156552555ol_int
% 0.33/0.65 @ ( ( N = zero_zero_nat )
% 0.33/0.65 & B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_of_bool
% 0.33/0.65 thf(fact_8425_drop__bit__of__bool,axiom,
% 0.33/0.65 ! [N: nat,B: $o] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ N @ ( zero_n2687167440665602831ol_nat @ B ) )
% 0.33/0.65 = ( zero_n2687167440665602831ol_nat
% 0.33/0.65 @ ( ( N = zero_zero_nat )
% 0.33/0.65 & B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_of_bool
% 0.33/0.65 thf(fact_8426_drop__bit__of__Suc__0,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.33/0.65 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_of_Suc_0
% 0.33/0.65 thf(fact_8427_drop__bit__nonnegative__int__iff,axiom,
% 0.33/0.65 ! [N: nat,K: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
% 0.33/0.65 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_nonnegative_int_iff
% 0.33/0.65 thf(fact_8428_drop__bit__negative__int__iff,axiom,
% 0.33/0.65 ! [N: nat,K: int] :
% 0.33/0.65 ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 0.33/0.65 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_negative_int_iff
% 0.33/0.65 thf(fact_8429_drop__bit__minus__one,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.65 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_minus_one
% 0.33/0.65 thf(fact_8430_drop__bit__Suc__bit0,axiom,
% 0.33/0.65 ! [N: nat,K: num] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.33/0.65 = ( bit_se8568078237143864401it_int @ N @ ( numeral_numeral_int @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_Suc_bit0
% 0.33/0.65 thf(fact_8431_drop__bit__Suc__bit0,axiom,
% 0.33/0.65 ! [N: nat,K: num] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.33/0.65 = ( bit_se8570568707652914677it_nat @ N @ ( numeral_numeral_nat @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_Suc_bit0
% 0.33/0.65 thf(fact_8432_drop__bit__Suc__bit1,axiom,
% 0.33/0.65 ! [N: nat,K: num] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.33/0.65 = ( bit_se8568078237143864401it_int @ N @ ( numeral_numeral_int @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_Suc_bit1
% 0.33/0.65 thf(fact_8433_drop__bit__Suc__bit1,axiom,
% 0.33/0.65 ! [N: nat,K: num] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.33/0.65 = ( bit_se8570568707652914677it_nat @ N @ ( numeral_numeral_nat @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_Suc_bit1
% 0.33/0.65 thf(fact_8434_drop__bit__of__1,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( bit_se3928097537394005634nteger @ N @ one_one_Code_integer )
% 0.33/0.65 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_of_1
% 0.33/0.65 thf(fact_8435_drop__bit__of__1,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ N @ one_one_int )
% 0.33/0.65 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_of_1
% 0.33/0.65 thf(fact_8436_drop__bit__of__1,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ N @ one_one_nat )
% 0.33/0.65 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_of_1
% 0.33/0.65 thf(fact_8437_drop__bit__numeral__bit0,axiom,
% 0.33/0.65 ! [L: num,K: num] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.33/0.65 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_numeral_bit0
% 0.33/0.65 thf(fact_8438_drop__bit__numeral__bit0,axiom,
% 0.33/0.65 ! [L: num,K: num] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.33/0.65 = ( bit_se8570568707652914677it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_numeral_bit0
% 0.33/0.65 thf(fact_8439_drop__bit__numeral__bit1,axiom,
% 0.33/0.65 ! [L: num,K: num] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.33/0.65 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_numeral_bit1
% 0.33/0.65 thf(fact_8440_drop__bit__numeral__bit1,axiom,
% 0.33/0.65 ! [L: num,K: num] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.33/0.65 = ( bit_se8570568707652914677it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_numeral_bit1
% 0.33/0.65 thf(fact_8441_drop__bit__Suc__minus__bit0,axiom,
% 0.33/0.65 ! [N: nat,K: num] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.33/0.65 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_Suc_minus_bit0
% 0.33/0.65 thf(fact_8442_drop__bit__numeral__minus__bit0,axiom,
% 0.33/0.65 ! [L: num,K: num] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.33/0.65 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_numeral_minus_bit0
% 0.33/0.65 thf(fact_8443_drop__bit__Suc__minus__bit1,axiom,
% 0.33/0.65 ! [N: nat,K: num] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.33/0.65 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_Suc_minus_bit1
% 0.33/0.65 thf(fact_8444_drop__bit__nat__eq,axiom,
% 0.33/0.65 ! [N: nat,K: int] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
% 0.33/0.65 = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_nat_eq
% 0.33/0.65 thf(fact_8445_of__nat__drop__bit,axiom,
% 0.33/0.65 ! [M: nat,N: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( bit_se8570568707652914677it_nat @ M @ N ) )
% 0.33/0.65 = ( bit_se8568078237143864401it_int @ M @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_drop_bit
% 0.33/0.65 thf(fact_8446_of__nat__drop__bit,axiom,
% 0.33/0.65 ! [M: nat,N: nat] :
% 0.33/0.65 ( ( semiri1316708129612266289at_nat @ ( bit_se8570568707652914677it_nat @ M @ N ) )
% 0.33/0.65 = ( bit_se8570568707652914677it_nat @ M @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_drop_bit
% 0.33/0.65 thf(fact_8447_drop__bit__of__nat,axiom,
% 0.33/0.65 ! [N: nat,M: nat] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 0.33/0.65 = ( semiri1314217659103216013at_int @ ( bit_se8570568707652914677it_nat @ N @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_of_nat
% 0.33/0.65 thf(fact_8448_drop__bit__of__nat,axiom,
% 0.33/0.65 ! [N: nat,M: nat] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 0.33/0.65 = ( semiri1316708129612266289at_nat @ ( bit_se8570568707652914677it_nat @ N @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_of_nat
% 0.33/0.65 thf(fact_8449_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
% 0.33/0.65 ! [N: nat,A: int] :
% 0.33/0.65 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.33/0.65 = A )
% 0.33/0.65 = ( ( bit_se8568078237143864401it_int @ N @ A )
% 0.33/0.65 = zero_zero_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % take_bit_eq_self_iff_drop_bit_eq_0
% 0.33/0.65 thf(fact_8450_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
% 0.33/0.65 ! [N: nat,A: nat] :
% 0.33/0.65 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.33/0.65 = A )
% 0.33/0.65 = ( ( bit_se8570568707652914677it_nat @ N @ A )
% 0.33/0.65 = zero_zero_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % take_bit_eq_self_iff_drop_bit_eq_0
% 0.33/0.65 thf(fact_8451_drop__bit__push__bit__int,axiom,
% 0.33/0.65 ! [M: nat,N: nat,K: int] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
% 0.33/0.65 = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_push_bit_int
% 0.33/0.65 thf(fact_8452_take__bit__drop__bit,axiom,
% 0.33/0.65 ! [M: nat,N: nat,A: int] :
% 0.33/0.65 ( ( bit_se2923211474154528505it_int @ M @ ( bit_se8568078237143864401it_int @ N @ A ) )
% 0.33/0.65 = ( bit_se8568078237143864401it_int @ N @ ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % take_bit_drop_bit
% 0.33/0.65 thf(fact_8453_take__bit__drop__bit,axiom,
% 0.33/0.65 ! [M: nat,N: nat,A: nat] :
% 0.33/0.65 ( ( bit_se2925701944663578781it_nat @ M @ ( bit_se8570568707652914677it_nat @ N @ A ) )
% 0.33/0.65 = ( bit_se8570568707652914677it_nat @ N @ ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % take_bit_drop_bit
% 0.33/0.65 thf(fact_8454_drop__bit__take__bit,axiom,
% 0.33/0.65 ! [M: nat,N: nat,A: int] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.33/0.65 = ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N @ M ) @ ( bit_se8568078237143864401it_int @ M @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_take_bit
% 0.33/0.65 thf(fact_8455_drop__bit__take__bit,axiom,
% 0.33/0.65 ! [M: nat,N: nat,A: nat] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 0.33/0.65 = ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N @ M ) @ ( bit_se8570568707652914677it_nat @ M @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_take_bit
% 0.33/0.65 thf(fact_8456_div__push__bit__of__1__eq__drop__bit,axiom,
% 0.33/0.65 ! [A: nat,N: nat] :
% 0.33/0.65 ( ( divide_divide_nat @ A @ ( bit_se547839408752420682it_nat @ N @ one_one_nat ) )
% 0.33/0.65 = ( bit_se8570568707652914677it_nat @ N @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % div_push_bit_of_1_eq_drop_bit
% 0.33/0.65 thf(fact_8457_div__push__bit__of__1__eq__drop__bit,axiom,
% 0.33/0.65 ! [A: int,N: nat] :
% 0.33/0.65 ( ( divide_divide_int @ A @ ( bit_se545348938243370406it_int @ N @ one_one_int ) )
% 0.33/0.65 = ( bit_se8568078237143864401it_int @ N @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % div_push_bit_of_1_eq_drop_bit
% 0.33/0.65 thf(fact_8458_div__push__bit__of__1__eq__drop__bit,axiom,
% 0.33/0.65 ! [A: code_integer,N: nat] :
% 0.33/0.65 ( ( divide6298287555418463151nteger @ A @ ( bit_se7788150548672797655nteger @ N @ one_one_Code_integer ) )
% 0.33/0.65 = ( bit_se3928097537394005634nteger @ N @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % div_push_bit_of_1_eq_drop_bit
% 0.33/0.65 thf(fact_8459_div__push__bit__of__1__eq__drop__bit,axiom,
% 0.33/0.65 ! [A: code_natural,N: nat] :
% 0.33/0.65 ( ( divide5121882707175180666atural @ A @ ( bit_se6611745700429515170atural @ N @ one_one_Code_natural ) )
% 0.33/0.65 = ( bit_se2751692689150723149atural @ N @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % div_push_bit_of_1_eq_drop_bit
% 0.33/0.65 thf(fact_8460_bit__iff__and__drop__bit__eq__1,axiom,
% 0.33/0.65 ( bit_se1146084159140164899it_int
% 0.33/0.65 = ( ^ [A2: int,N3: nat] :
% 0.33/0.65 ( ( bit_se725231765392027082nd_int @ ( bit_se8568078237143864401it_int @ N3 @ A2 ) @ one_one_int )
% 0.33/0.65 = one_one_int ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bit_iff_and_drop_bit_eq_1
% 0.33/0.65 thf(fact_8461_bit__iff__and__drop__bit__eq__1,axiom,
% 0.33/0.65 ( bit_se1148574629649215175it_nat
% 0.33/0.65 = ( ^ [A2: nat,N3: nat] :
% 0.33/0.65 ( ( bit_se727722235901077358nd_nat @ ( bit_se8570568707652914677it_nat @ N3 @ A2 ) @ one_one_nat )
% 0.33/0.65 = one_one_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bit_iff_and_drop_bit_eq_1
% 0.33/0.65 thf(fact_8462_bits__ident,axiom,
% 0.33/0.65 ! [N: nat,A: code_integer] :
% 0.33/0.65 ( ( plus_p5714425477246183910nteger @ ( bit_se7788150548672797655nteger @ N @ ( bit_se3928097537394005634nteger @ N @ A ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
% 0.33/0.65 = A ) ).
% 0.33/0.65
% 0.33/0.65 % bits_ident
% 0.33/0.65 thf(fact_8463_bits__ident,axiom,
% 0.33/0.65 ! [N: nat,A: code_natural] :
% 0.33/0.65 ( ( plus_p4538020629002901425atural @ ( bit_se6611745700429515170atural @ N @ ( bit_se2751692689150723149atural @ N @ A ) ) @ ( bit_se569199155075624693atural @ N @ A ) )
% 0.33/0.65 = A ) ).
% 0.33/0.65
% 0.33/0.65 % bits_ident
% 0.33/0.65 thf(fact_8464_bits__ident,axiom,
% 0.33/0.65 ! [N: nat,A: int] :
% 0.33/0.65 ( ( plus_plus_int @ ( bit_se545348938243370406it_int @ N @ ( bit_se8568078237143864401it_int @ N @ A ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.33/0.65 = A ) ).
% 0.33/0.65
% 0.33/0.65 % bits_ident
% 0.33/0.65 thf(fact_8465_bits__ident,axiom,
% 0.33/0.65 ! [N: nat,A: nat] :
% 0.33/0.65 ( ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N @ ( bit_se8570568707652914677it_nat @ N @ A ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 0.33/0.65 = A ) ).
% 0.33/0.65
% 0.33/0.65 % bits_ident
% 0.33/0.65 thf(fact_8466_drop__bit__half,axiom,
% 0.33/0.65 ! [N: nat,A: int] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.33/0.65 = ( divide_divide_int @ ( bit_se8568078237143864401it_int @ N @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_half
% 0.33/0.65 thf(fact_8467_drop__bit__half,axiom,
% 0.33/0.65 ! [N: nat,A: nat] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.65 = ( divide_divide_nat @ ( bit_se8570568707652914677it_nat @ N @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_half
% 0.33/0.65 thf(fact_8468_stable__imp__drop__bit__eq,axiom,
% 0.33/0.65 ! [A: int,N: nat] :
% 0.33/0.65 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.65 = A )
% 0.33/0.65 => ( ( bit_se8568078237143864401it_int @ N @ A )
% 0.33/0.65 = A ) ) ).
% 0.33/0.65
% 0.33/0.65 % stable_imp_drop_bit_eq
% 0.33/0.65 thf(fact_8469_stable__imp__drop__bit__eq,axiom,
% 0.33/0.65 ! [A: nat,N: nat] :
% 0.33/0.65 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.65 = A )
% 0.33/0.65 => ( ( bit_se8570568707652914677it_nat @ N @ A )
% 0.33/0.65 = A ) ) ).
% 0.33/0.65
% 0.33/0.65 % stable_imp_drop_bit_eq
% 0.33/0.65 thf(fact_8470_drop__bit__int__def,axiom,
% 0.33/0.65 ( bit_se8568078237143864401it_int
% 0.33/0.65 = ( ^ [N3: nat,K2: int] : ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_int_def
% 0.33/0.65 thf(fact_8471_drop__bit__nat__def,axiom,
% 0.33/0.65 ( bit_se8570568707652914677it_nat
% 0.33/0.65 = ( ^ [N3: nat,M3: nat] : ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_nat_def
% 0.33/0.65 thf(fact_8472_drop__bit__Suc,axiom,
% 0.33/0.65 ! [N: nat,A: int] :
% 0.33/0.65 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ A )
% 0.33/0.65 = ( bit_se8568078237143864401it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_Suc
% 0.33/0.65 thf(fact_8473_drop__bit__Suc,axiom,
% 0.33/0.65 ! [N: nat,A: nat] :
% 0.33/0.65 ( ( bit_se8570568707652914677it_nat @ ( suc @ N ) @ A )
% 0.33/0.65 = ( bit_se8570568707652914677it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_Suc
% 0.33/0.65 thf(fact_8474_drop__bit__eq__div,axiom,
% 0.33/0.65 ( bit_se8568078237143864401it_int
% 0.33/0.65 = ( ^ [N3: nat,A2: int] : ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_eq_div
% 0.33/0.65 thf(fact_8475_drop__bit__eq__div,axiom,
% 0.33/0.65 ( bit_se8570568707652914677it_nat
% 0.33/0.65 = ( ^ [N3: nat,A2: nat] : ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % drop_bit_eq_div
% 0.33/0.65 thf(fact_8476_bit__iff__odd__drop__bit,axiom,
% 0.33/0.65 ( bit_se9216721137139052372nteger
% 0.33/0.65 = ( ^ [A2: code_integer,N3: nat] :
% 0.33/0.65 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3928097537394005634nteger @ N3 @ A2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bit_iff_odd_drop_bit
% 0.33/0.65 thf(fact_8477_bit__iff__odd__drop__bit,axiom,
% 0.33/0.65 ( bit_se1146084159140164899it_int
% 0.33/0.65 = ( ^ [A2: int,N3: nat] :
% 0.33/0.65 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se8568078237143864401it_int @ N3 @ A2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bit_iff_odd_drop_bit
% 0.33/0.65 thf(fact_8478_bit__iff__odd__drop__bit,axiom,
% 0.33/0.65 ( bit_se1148574629649215175it_nat
% 0.33/0.65 = ( ^ [A2: nat,N3: nat] :
% 0.33/0.65 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se8570568707652914677it_nat @ N3 @ A2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bit_iff_odd_drop_bit
% 0.33/0.65 thf(fact_8479_even__drop__bit__iff__not__bit,axiom,
% 0.33/0.65 ! [N: nat,A: code_integer] :
% 0.33/0.65 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3928097537394005634nteger @ N @ A ) )
% 0.33/0.65 = ( ~ ( bit_se9216721137139052372nteger @ A @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % even_drop_bit_iff_not_bit
% 0.33/0.65 thf(fact_8480_even__drop__bit__iff__not__bit,axiom,
% 0.33/0.65 ! [N: nat,A: int] :
% 0.33/0.65 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se8568078237143864401it_int @ N @ A ) )
% 0.33/0.65 = ( ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % even_drop_bit_iff_not_bit
% 0.33/0.65 thf(fact_8481_even__drop__bit__iff__not__bit,axiom,
% 0.33/0.65 ! [N: nat,A: nat] :
% 0.33/0.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se8570568707652914677it_nat @ N @ A ) )
% 0.33/0.65 = ( ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % even_drop_bit_iff_not_bit
% 0.33/0.65 thf(fact_8482_slice__eq__mask,axiom,
% 0.33/0.65 ! [N: nat,M: nat,A: code_integer] :
% 0.33/0.65 ( ( bit_se7788150548672797655nteger @ N @ ( bit_se1745604003318907178nteger @ M @ ( bit_se3928097537394005634nteger @ N @ A ) ) )
% 0.33/0.65 = ( bit_se3949692690581998587nteger @ A @ ( bit_se3949692690581998587nteger @ ( bit_se2119862282449309892nteger @ ( plus_plus_nat @ M @ N ) ) @ ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % slice_eq_mask
% 0.33/0.65 thf(fact_8483_slice__eq__mask,axiom,
% 0.33/0.65 ! [N: nat,M: nat,A: int] :
% 0.33/0.65 ( ( bit_se545348938243370406it_int @ N @ ( bit_se2923211474154528505it_int @ M @ ( bit_se8568078237143864401it_int @ N @ A ) ) )
% 0.33/0.65 = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ M @ N ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % slice_eq_mask
% 0.33/0.65 thf(fact_8484_inthall,axiom,
% 0.33/0.65 ! [Xs: list_real,P: real > $o,N: nat] :
% 0.33/0.65 ( ! [X4: real] :
% 0.33/0.65 ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 0.33/0.65 => ( P @ X4 ) )
% 0.33/0.65 => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 0.33/0.65 => ( P @ ( nth_real @ Xs @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % inthall
% 0.33/0.65 thf(fact_8485_inthall,axiom,
% 0.33/0.65 ! [Xs: list_complex,P: complex > $o,N: nat] :
% 0.33/0.65 ( ! [X4: complex] :
% 0.33/0.65 ( ( member_complex @ X4 @ ( set_complex2 @ Xs ) )
% 0.33/0.65 => ( P @ X4 ) )
% 0.33/0.65 => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
% 0.33/0.65 => ( P @ ( nth_complex @ Xs @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % inthall
% 0.33/0.65 thf(fact_8486_inthall,axiom,
% 0.33/0.65 ! [Xs: list_set_nat,P: set_nat > $o,N: nat] :
% 0.33/0.65 ( ! [X4: set_nat] :
% 0.33/0.65 ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
% 0.33/0.65 => ( P @ X4 ) )
% 0.33/0.65 => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.33/0.65 => ( P @ ( nth_set_nat @ Xs @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % inthall
% 0.33/0.65 thf(fact_8487_inthall,axiom,
% 0.33/0.65 ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 0.33/0.65 ( ! [X4: vEBT_VEBT] :
% 0.33/0.65 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.65 => ( P @ X4 ) )
% 0.33/0.65 => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.65 => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % inthall
% 0.33/0.65 thf(fact_8488_inthall,axiom,
% 0.33/0.65 ! [Xs: list_o,P: $o > $o,N: nat] :
% 0.33/0.65 ( ! [X4: $o] :
% 0.33/0.65 ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 0.33/0.65 => ( P @ X4 ) )
% 0.33/0.65 => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.33/0.65 => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % inthall
% 0.33/0.65 thf(fact_8489_inthall,axiom,
% 0.33/0.65 ! [Xs: list_nat,P: nat > $o,N: nat] :
% 0.33/0.65 ( ! [X4: nat] :
% 0.33/0.65 ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 0.33/0.65 => ( P @ X4 ) )
% 0.33/0.65 => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.33/0.65 => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % inthall
% 0.33/0.65 thf(fact_8490_inthall,axiom,
% 0.33/0.65 ! [Xs: list_int,P: int > $o,N: nat] :
% 0.33/0.65 ( ! [X4: int] :
% 0.33/0.65 ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 0.33/0.65 => ( P @ X4 ) )
% 0.33/0.65 => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.33/0.65 => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % inthall
% 0.33/0.65 thf(fact_8491_nth__rotate1,axiom,
% 0.33/0.65 ! [N: nat,Xs: list_VEBT_VEBT] :
% 0.33/0.65 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.65 => ( ( nth_VEBT_VEBT @ ( rotate1_VEBT_VEBT @ Xs ) @ N )
% 0.33/0.65 = ( nth_VEBT_VEBT @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nth_rotate1
% 0.33/0.65 thf(fact_8492_nth__rotate1,axiom,
% 0.33/0.65 ! [N: nat,Xs: list_o] :
% 0.33/0.65 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.33/0.65 => ( ( nth_o @ ( rotate1_o @ Xs ) @ N )
% 0.33/0.65 = ( nth_o @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_o @ Xs ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nth_rotate1
% 0.33/0.65 thf(fact_8493_nth__rotate1,axiom,
% 0.33/0.65 ! [N: nat,Xs: list_nat] :
% 0.33/0.65 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.33/0.65 => ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N )
% 0.33/0.65 = ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nth_rotate1
% 0.33/0.65 thf(fact_8494_nth__rotate1,axiom,
% 0.33/0.65 ! [N: nat,Xs: list_int] :
% 0.33/0.65 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.33/0.65 => ( ( nth_int @ ( rotate1_int @ Xs ) @ N )
% 0.33/0.65 = ( nth_int @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nth_rotate1
% 0.33/0.65 thf(fact_8495_root__powr__inverse,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ( root @ N @ X )
% 0.33/0.65 = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % root_powr_inverse
% 0.33/0.65 thf(fact_8496_xor__minus__numerals_I2_J,axiom,
% 0.33/0.65 ! [K: int,N: num] :
% 0.33/0.65 ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.65 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % xor_minus_numerals(2)
% 0.33/0.65 thf(fact_8497_real__root__zero,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( root @ N @ zero_zero_real )
% 0.33/0.65 = zero_zero_real ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_zero
% 0.33/0.65 thf(fact_8498_real__root__Suc__0,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( root @ ( suc @ zero_zero_nat ) @ X )
% 0.33/0.65 = X ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_Suc_0
% 0.33/0.65 thf(fact_8499_root__0,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( root @ zero_zero_nat @ X )
% 0.33/0.65 = zero_zero_real ) ).
% 0.33/0.65
% 0.33/0.65 % root_0
% 0.33/0.65 thf(fact_8500_real__root__eq__iff,axiom,
% 0.33/0.65 ! [N: nat,X: real,Y: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ( root @ N @ X )
% 0.33/0.65 = ( root @ N @ Y ) )
% 0.33/0.65 = ( X = Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_eq_iff
% 0.33/0.65 thf(fact_8501_sub__num__simps_I1_J,axiom,
% 0.33/0.65 ( ( neg_numeral_sub_real @ one @ one )
% 0.33/0.65 = zero_zero_real ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(1)
% 0.33/0.65 thf(fact_8502_sub__num__simps_I1_J,axiom,
% 0.33/0.65 ( ( neg_numeral_sub_rat @ one @ one )
% 0.33/0.65 = zero_zero_rat ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(1)
% 0.33/0.65 thf(fact_8503_sub__num__simps_I1_J,axiom,
% 0.33/0.65 ( ( neg_numeral_sub_int @ one @ one )
% 0.33/0.65 = zero_zero_int ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(1)
% 0.33/0.65 thf(fact_8504_diff__numeral__simps_I1_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 0.33/0.65 = ( neg_nu8416839295433526191omplex @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_simps(1)
% 0.33/0.65 thf(fact_8505_diff__numeral__simps_I1_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.33/0.65 = ( neg_numeral_sub_real @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_simps(1)
% 0.33/0.65 thf(fact_8506_diff__numeral__simps_I1_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.33/0.65 = ( neg_numeral_sub_rat @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_simps(1)
% 0.33/0.65 thf(fact_8507_diff__numeral__simps_I1_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.65 = ( neg_numeral_sub_int @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_simps(1)
% 0.33/0.65 thf(fact_8508_sub__num__simps_I6_J,axiom,
% 0.33/0.65 ! [K: num,L: num] :
% 0.33/0.65 ( ( neg_numeral_sub_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
% 0.33/0.65 = ( neg_numeral_dbl_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(6)
% 0.33/0.65 thf(fact_8509_sub__num__simps_I9_J,axiom,
% 0.33/0.65 ! [K: num,L: num] :
% 0.33/0.65 ( ( neg_numeral_sub_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
% 0.33/0.65 = ( neg_numeral_dbl_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(9)
% 0.33/0.65 thf(fact_8510_semiring__norm_I167_J,axiom,
% 0.33/0.65 ! [V: num,W: num,Y: real] :
% 0.33/0.65 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Y ) )
% 0.33/0.65 = ( plus_plus_real @ ( neg_numeral_sub_real @ W @ V ) @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % semiring_norm(167)
% 0.33/0.65 thf(fact_8511_semiring__norm_I167_J,axiom,
% 0.33/0.65 ! [V: num,W: num,Y: int] :
% 0.33/0.65 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Y ) )
% 0.33/0.65 = ( plus_plus_int @ ( neg_numeral_sub_int @ W @ V ) @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % semiring_norm(167)
% 0.33/0.65 thf(fact_8512_semiring__norm_I167_J,axiom,
% 0.33/0.65 ! [V: num,W: num,Y: complex] :
% 0.33/0.65 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Y ) )
% 0.33/0.65 = ( plus_plus_complex @ ( neg_nu8416839295433526191omplex @ W @ V ) @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % semiring_norm(167)
% 0.33/0.65 thf(fact_8513_semiring__norm_I167_J,axiom,
% 0.33/0.65 ! [V: num,W: num,Y: rat] :
% 0.33/0.65 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
% 0.33/0.65 = ( plus_plus_rat @ ( neg_numeral_sub_rat @ W @ V ) @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % semiring_norm(167)
% 0.33/0.65 thf(fact_8514_semiring__norm_I167_J,axiom,
% 0.33/0.65 ! [V: num,W: num,Y: code_integer] :
% 0.33/0.65 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
% 0.33/0.65 = ( plus_p5714425477246183910nteger @ ( neg_nu5755505904847501662nteger @ W @ V ) @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % semiring_norm(167)
% 0.33/0.65 thf(fact_8515_semiring__norm_I166_J,axiom,
% 0.33/0.65 ! [V: num,W: num,Y: real] :
% 0.33/0.65 ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 0.33/0.65 = ( plus_plus_real @ ( neg_numeral_sub_real @ V @ W ) @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % semiring_norm(166)
% 0.33/0.65 thf(fact_8516_semiring__norm_I166_J,axiom,
% 0.33/0.65 ! [V: num,W: num,Y: int] :
% 0.33/0.65 ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 0.33/0.65 = ( plus_plus_int @ ( neg_numeral_sub_int @ V @ W ) @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % semiring_norm(166)
% 0.33/0.65 thf(fact_8517_semiring__norm_I166_J,axiom,
% 0.33/0.65 ! [V: num,W: num,Y: complex] :
% 0.33/0.65 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 0.33/0.65 = ( plus_plus_complex @ ( neg_nu8416839295433526191omplex @ V @ W ) @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % semiring_norm(166)
% 0.33/0.65 thf(fact_8518_semiring__norm_I166_J,axiom,
% 0.33/0.65 ! [V: num,W: num,Y: rat] :
% 0.33/0.65 ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 0.33/0.65 = ( plus_plus_rat @ ( neg_numeral_sub_rat @ V @ W ) @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % semiring_norm(166)
% 0.33/0.65 thf(fact_8519_semiring__norm_I166_J,axiom,
% 0.33/0.65 ! [V: num,W: num,Y: code_integer] :
% 0.33/0.65 ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 0.33/0.65 = ( plus_p5714425477246183910nteger @ ( neg_nu5755505904847501662nteger @ V @ W ) @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % semiring_norm(166)
% 0.33/0.65 thf(fact_8520_add__neg__numeral__simps_I2_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 0.33/0.65 = ( neg_numeral_sub_real @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_simps(2)
% 0.33/0.65 thf(fact_8521_add__neg__numeral__simps_I2_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.65 = ( neg_numeral_sub_int @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_simps(2)
% 0.33/0.65 thf(fact_8522_add__neg__numeral__simps_I2_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 0.33/0.65 = ( neg_nu8416839295433526191omplex @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_simps(2)
% 0.33/0.65 thf(fact_8523_add__neg__numeral__simps_I2_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 0.33/0.65 = ( neg_numeral_sub_rat @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_simps(2)
% 0.33/0.65 thf(fact_8524_add__neg__numeral__simps_I2_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 0.33/0.65 = ( neg_nu5755505904847501662nteger @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_simps(2)
% 0.33/0.65 thf(fact_8525_add__neg__numeral__simps_I1_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.33/0.65 = ( neg_numeral_sub_real @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_simps(1)
% 0.33/0.65 thf(fact_8526_add__neg__numeral__simps_I1_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.65 = ( neg_numeral_sub_int @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_simps(1)
% 0.33/0.65 thf(fact_8527_add__neg__numeral__simps_I1_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.33/0.65 = ( neg_nu8416839295433526191omplex @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_simps(1)
% 0.33/0.65 thf(fact_8528_add__neg__numeral__simps_I1_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.33/0.65 = ( neg_numeral_sub_rat @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_simps(1)
% 0.33/0.65 thf(fact_8529_add__neg__numeral__simps_I1_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.33/0.65 = ( neg_nu5755505904847501662nteger @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_simps(1)
% 0.33/0.65 thf(fact_8530_real__root__eq__0__iff,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ( root @ N @ X )
% 0.33/0.65 = zero_zero_real )
% 0.33/0.65 = ( X = zero_zero_real ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_eq_0_iff
% 0.33/0.65 thf(fact_8531_real__root__less__iff,axiom,
% 0.33/0.65 ! [N: nat,X: real,Y: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
% 0.33/0.65 = ( ord_less_real @ X @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_less_iff
% 0.33/0.65 thf(fact_8532_diff__numeral__simps_I4_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.33/0.65 = ( neg_numeral_sub_real @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_simps(4)
% 0.33/0.65 thf(fact_8533_diff__numeral__simps_I4_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.65 = ( neg_numeral_sub_int @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_simps(4)
% 0.33/0.65 thf(fact_8534_diff__numeral__simps_I4_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.33/0.65 = ( neg_nu8416839295433526191omplex @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_simps(4)
% 0.33/0.65 thf(fact_8535_diff__numeral__simps_I4_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.33/0.65 = ( neg_numeral_sub_rat @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_simps(4)
% 0.33/0.65 thf(fact_8536_diff__numeral__simps_I4_J,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.33/0.65 = ( neg_nu5755505904847501662nteger @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_simps(4)
% 0.33/0.65 thf(fact_8537_real__root__le__iff,axiom,
% 0.33/0.65 ! [N: nat,X: real,Y: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
% 0.33/0.65 = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_le_iff
% 0.33/0.65 thf(fact_8538_real__root__eq__1__iff,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ( root @ N @ X )
% 0.33/0.65 = one_one_real )
% 0.33/0.65 = ( X = one_one_real ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_eq_1_iff
% 0.33/0.65 thf(fact_8539_real__root__one,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( root @ N @ one_one_real )
% 0.33/0.65 = one_one_real ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_one
% 0.33/0.65 thf(fact_8540_rotate1__length01,axiom,
% 0.33/0.65 ! [Xs: list_VEBT_VEBT] :
% 0.33/0.65 ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ one_one_nat )
% 0.33/0.65 => ( ( rotate1_VEBT_VEBT @ Xs )
% 0.33/0.65 = Xs ) ) ).
% 0.33/0.65
% 0.33/0.65 % rotate1_length01
% 0.33/0.65 thf(fact_8541_rotate1__length01,axiom,
% 0.33/0.65 ! [Xs: list_o] :
% 0.33/0.65 ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ one_one_nat )
% 0.33/0.65 => ( ( rotate1_o @ Xs )
% 0.33/0.65 = Xs ) ) ).
% 0.33/0.65
% 0.33/0.65 % rotate1_length01
% 0.33/0.65 thf(fact_8542_rotate1__length01,axiom,
% 0.33/0.65 ! [Xs: list_nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
% 0.33/0.65 => ( ( rotate1_nat @ Xs )
% 0.33/0.65 = Xs ) ) ).
% 0.33/0.65
% 0.33/0.65 % rotate1_length01
% 0.33/0.65 thf(fact_8543_rotate1__length01,axiom,
% 0.33/0.65 ! [Xs: list_int] :
% 0.33/0.65 ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
% 0.33/0.65 => ( ( rotate1_int @ Xs )
% 0.33/0.65 = Xs ) ) ).
% 0.33/0.65
% 0.33/0.65 % rotate1_length01
% 0.33/0.65 thf(fact_8544_sub__num__simps_I7_J,axiom,
% 0.33/0.65 ! [K: num,L: num] :
% 0.33/0.65 ( ( neg_numeral_sub_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
% 0.33/0.65 = ( neg_nu3811975205180677377ec_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(7)
% 0.33/0.65 thf(fact_8545_sub__num__simps_I8_J,axiom,
% 0.33/0.65 ! [K: num,L: num] :
% 0.33/0.65 ( ( neg_numeral_sub_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
% 0.33/0.65 = ( neg_nu5851722552734809277nc_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(8)
% 0.33/0.65 thf(fact_8546_diff__numeral__special_I1_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( minus_minus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
% 0.33/0.65 = ( neg_nu8416839295433526191omplex @ one @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(1)
% 0.33/0.65 thf(fact_8547_diff__numeral__special_I1_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( minus_minus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.33/0.65 = ( neg_numeral_sub_real @ one @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(1)
% 0.33/0.65 thf(fact_8548_diff__numeral__special_I1_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( minus_minus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.33/0.65 = ( neg_numeral_sub_rat @ one @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(1)
% 0.33/0.65 thf(fact_8549_diff__numeral__special_I1_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( minus_minus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.33/0.65 = ( neg_numeral_sub_int @ one @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(1)
% 0.33/0.65 thf(fact_8550_diff__numeral__special_I2_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ one_one_complex )
% 0.33/0.65 = ( neg_nu8416839295433526191omplex @ M @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(2)
% 0.33/0.65 thf(fact_8551_diff__numeral__special_I2_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ one_one_real )
% 0.33/0.65 = ( neg_numeral_sub_real @ M @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(2)
% 0.33/0.65 thf(fact_8552_diff__numeral__special_I2_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ one_one_rat )
% 0.33/0.65 = ( neg_numeral_sub_rat @ M @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(2)
% 0.33/0.65 thf(fact_8553_diff__numeral__special_I2_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int )
% 0.33/0.65 = ( neg_numeral_sub_int @ M @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(2)
% 0.33/0.65 thf(fact_8554_real__root__gt__0__iff,axiom,
% 0.33/0.65 ! [N: nat,Y: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
% 0.33/0.65 = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_gt_0_iff
% 0.33/0.65 thf(fact_8555_real__root__lt__0__iff,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
% 0.33/0.65 = ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_lt_0_iff
% 0.33/0.65 thf(fact_8556_real__root__ge__0__iff,axiom,
% 0.33/0.65 ! [N: nat,Y: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
% 0.33/0.65 = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_ge_0_iff
% 0.33/0.65 thf(fact_8557_real__root__le__0__iff,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
% 0.33/0.65 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_le_0_iff
% 0.33/0.65 thf(fact_8558_sub__num__simps_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu8416839295433526191omplex @ ( bit1 @ K ) @ one )
% 0.33/0.65 = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(5)
% 0.33/0.65 thf(fact_8559_sub__num__simps_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_numeral_sub_real @ ( bit1 @ K ) @ one )
% 0.33/0.65 = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(5)
% 0.33/0.65 thf(fact_8560_sub__num__simps_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_numeral_sub_rat @ ( bit1 @ K ) @ one )
% 0.33/0.65 = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(5)
% 0.33/0.65 thf(fact_8561_sub__num__simps_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_numeral_sub_int @ ( bit1 @ K ) @ one )
% 0.33/0.65 = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(5)
% 0.33/0.65 thf(fact_8562_real__root__gt__1__iff,axiom,
% 0.33/0.65 ! [N: nat,Y: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
% 0.33/0.65 = ( ord_less_real @ one_one_real @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_gt_1_iff
% 0.33/0.65 thf(fact_8563_real__root__lt__1__iff,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
% 0.33/0.65 = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_lt_1_iff
% 0.33/0.65 thf(fact_8564_real__root__ge__1__iff,axiom,
% 0.33/0.65 ! [N: nat,Y: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
% 0.33/0.65 = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_ge_1_iff
% 0.33/0.65 thf(fact_8565_real__root__le__1__iff,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
% 0.33/0.65 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_le_1_iff
% 0.33/0.65 thf(fact_8566_not__minus__numeral__eq,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.33/0.65 = ( neg_nu5755505904847501662nteger @ N @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % not_minus_numeral_eq
% 0.33/0.65 thf(fact_8567_not__minus__numeral__eq,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.65 = ( neg_numeral_sub_int @ N @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % not_minus_numeral_eq
% 0.33/0.65 thf(fact_8568_sub__num__simps_I4_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu8416839295433526191omplex @ ( bit0 @ K ) @ one )
% 0.33/0.65 = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(4)
% 0.33/0.65 thf(fact_8569_sub__num__simps_I4_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_numeral_sub_real @ ( bit0 @ K ) @ one )
% 0.33/0.65 = ( numeral_numeral_real @ ( bitM @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(4)
% 0.33/0.65 thf(fact_8570_sub__num__simps_I4_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_numeral_sub_rat @ ( bit0 @ K ) @ one )
% 0.33/0.65 = ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(4)
% 0.33/0.65 thf(fact_8571_sub__num__simps_I4_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_numeral_sub_int @ ( bit0 @ K ) @ one )
% 0.33/0.65 = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(4)
% 0.33/0.65 thf(fact_8572_add__neg__numeral__special_I4_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
% 0.33/0.65 = ( neg_numeral_sub_real @ N @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(4)
% 0.33/0.65 thf(fact_8573_add__neg__numeral__special_I4_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.65 = ( neg_numeral_sub_int @ N @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(4)
% 0.33/0.65 thf(fact_8574_add__neg__numeral__special_I4_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
% 0.33/0.65 = ( neg_nu8416839295433526191omplex @ N @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(4)
% 0.33/0.65 thf(fact_8575_add__neg__numeral__special_I4_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
% 0.33/0.65 = ( neg_numeral_sub_rat @ N @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(4)
% 0.33/0.65 thf(fact_8576_add__neg__numeral__special_I4_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
% 0.33/0.65 = ( neg_nu5755505904847501662nteger @ N @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(4)
% 0.33/0.65 thf(fact_8577_add__neg__numeral__special_I3_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.65 = ( neg_numeral_sub_real @ M @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(3)
% 0.33/0.65 thf(fact_8578_add__neg__numeral__special_I3_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.65 = ( neg_numeral_sub_int @ M @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(3)
% 0.33/0.65 thf(fact_8579_add__neg__numeral__special_I3_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.65 = ( neg_nu8416839295433526191omplex @ M @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(3)
% 0.33/0.65 thf(fact_8580_add__neg__numeral__special_I3_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.65 = ( neg_numeral_sub_rat @ M @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(3)
% 0.33/0.65 thf(fact_8581_add__neg__numeral__special_I3_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.65 = ( neg_nu5755505904847501662nteger @ M @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(3)
% 0.33/0.65 thf(fact_8582_add__neg__numeral__special_I2_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 0.33/0.65 = ( neg_numeral_sub_real @ one @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(2)
% 0.33/0.65 thf(fact_8583_add__neg__numeral__special_I2_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 0.33/0.65 = ( neg_numeral_sub_int @ one @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(2)
% 0.33/0.65 thf(fact_8584_add__neg__numeral__special_I2_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 0.33/0.65 = ( neg_nu8416839295433526191omplex @ one @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(2)
% 0.33/0.65 thf(fact_8585_add__neg__numeral__special_I2_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 0.33/0.65 = ( neg_numeral_sub_rat @ one @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(2)
% 0.33/0.65 thf(fact_8586_add__neg__numeral__special_I2_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 0.33/0.65 = ( neg_nu5755505904847501662nteger @ one @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(2)
% 0.33/0.65 thf(fact_8587_add__neg__numeral__special_I1_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) )
% 0.33/0.65 = ( neg_numeral_sub_real @ one @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(1)
% 0.33/0.65 thf(fact_8588_add__neg__numeral__special_I1_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) )
% 0.33/0.65 = ( neg_numeral_sub_int @ one @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(1)
% 0.33/0.65 thf(fact_8589_add__neg__numeral__special_I1_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) )
% 0.33/0.65 = ( neg_nu8416839295433526191omplex @ one @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(1)
% 0.33/0.65 thf(fact_8590_add__neg__numeral__special_I1_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) )
% 0.33/0.65 = ( neg_numeral_sub_rat @ one @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(1)
% 0.33/0.65 thf(fact_8591_add__neg__numeral__special_I1_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) )
% 0.33/0.65 = ( neg_nu5755505904847501662nteger @ one @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % add_neg_numeral_special(1)
% 0.33/0.65 thf(fact_8592_diff__numeral__special_I8_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.65 = ( neg_numeral_sub_real @ one @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(8)
% 0.33/0.65 thf(fact_8593_diff__numeral__special_I8_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.65 = ( neg_numeral_sub_int @ one @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(8)
% 0.33/0.65 thf(fact_8594_diff__numeral__special_I8_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.65 = ( neg_nu8416839295433526191omplex @ one @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(8)
% 0.33/0.65 thf(fact_8595_diff__numeral__special_I8_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.65 = ( neg_numeral_sub_rat @ one @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(8)
% 0.33/0.65 thf(fact_8596_diff__numeral__special_I8_J,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.65 = ( neg_nu5755505904847501662nteger @ one @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(8)
% 0.33/0.65 thf(fact_8597_diff__numeral__special_I7_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.33/0.65 = ( neg_numeral_sub_real @ N @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(7)
% 0.33/0.65 thf(fact_8598_diff__numeral__special_I7_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.65 = ( neg_numeral_sub_int @ N @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(7)
% 0.33/0.65 thf(fact_8599_diff__numeral__special_I7_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.33/0.65 = ( neg_nu8416839295433526191omplex @ N @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(7)
% 0.33/0.65 thf(fact_8600_diff__numeral__special_I7_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.33/0.65 = ( neg_numeral_sub_rat @ N @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(7)
% 0.33/0.65 thf(fact_8601_diff__numeral__special_I7_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.33/0.65 = ( neg_nu5755505904847501662nteger @ N @ one ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_special(7)
% 0.33/0.65 thf(fact_8602_minus__sub__one__diff__one,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( minus_minus_real @ ( uminus_uminus_real @ ( neg_numeral_sub_real @ M @ one ) ) @ one_one_real )
% 0.33/0.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % minus_sub_one_diff_one
% 0.33/0.65 thf(fact_8603_minus__sub__one__diff__one,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( minus_minus_int @ ( uminus_uminus_int @ ( neg_numeral_sub_int @ M @ one ) ) @ one_one_int )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % minus_sub_one_diff_one
% 0.33/0.65 thf(fact_8604_minus__sub__one__diff__one,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( neg_nu8416839295433526191omplex @ M @ one ) ) @ one_one_complex )
% 0.33/0.65 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % minus_sub_one_diff_one
% 0.33/0.65 thf(fact_8605_minus__sub__one__diff__one,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( neg_numeral_sub_rat @ M @ one ) ) @ one_one_rat )
% 0.33/0.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % minus_sub_one_diff_one
% 0.33/0.65 thf(fact_8606_minus__sub__one__diff__one,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( neg_nu5755505904847501662nteger @ M @ one ) ) @ one_one_Code_integer )
% 0.33/0.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % minus_sub_one_diff_one
% 0.33/0.65 thf(fact_8607_sub__num__simps_I3_J,axiom,
% 0.33/0.65 ! [L: num] :
% 0.33/0.65 ( ( neg_numeral_sub_real @ one @ ( bit1 @ L ) )
% 0.33/0.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ L ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(3)
% 0.33/0.65 thf(fact_8608_sub__num__simps_I3_J,axiom,
% 0.33/0.65 ! [L: num] :
% 0.33/0.65 ( ( neg_numeral_sub_int @ one @ ( bit1 @ L ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ L ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(3)
% 0.33/0.65 thf(fact_8609_sub__num__simps_I3_J,axiom,
% 0.33/0.65 ! [L: num] :
% 0.33/0.65 ( ( neg_nu8416839295433526191omplex @ one @ ( bit1 @ L ) )
% 0.33/0.65 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ L ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(3)
% 0.33/0.65 thf(fact_8610_sub__num__simps_I3_J,axiom,
% 0.33/0.65 ! [L: num] :
% 0.33/0.65 ( ( neg_numeral_sub_rat @ one @ ( bit1 @ L ) )
% 0.33/0.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ L ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(3)
% 0.33/0.65 thf(fact_8611_sub__num__simps_I3_J,axiom,
% 0.33/0.65 ! [L: num] :
% 0.33/0.65 ( ( neg_nu5755505904847501662nteger @ one @ ( bit1 @ L ) )
% 0.33/0.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ L ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(3)
% 0.33/0.65 thf(fact_8612_real__root__pow__pos2,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 0.33/0.65 = X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_pow_pos2
% 0.33/0.65 thf(fact_8613_sub__num__simps_I2_J,axiom,
% 0.33/0.65 ! [L: num] :
% 0.33/0.65 ( ( neg_numeral_sub_real @ one @ ( bit0 @ L ) )
% 0.33/0.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bitM @ L ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(2)
% 0.33/0.65 thf(fact_8614_sub__num__simps_I2_J,axiom,
% 0.33/0.65 ! [L: num] :
% 0.33/0.65 ( ( neg_numeral_sub_int @ one @ ( bit0 @ L ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bitM @ L ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(2)
% 0.33/0.65 thf(fact_8615_sub__num__simps_I2_J,axiom,
% 0.33/0.65 ! [L: num] :
% 0.33/0.65 ( ( neg_nu8416839295433526191omplex @ one @ ( bit0 @ L ) )
% 0.33/0.65 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bitM @ L ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(2)
% 0.33/0.65 thf(fact_8616_sub__num__simps_I2_J,axiom,
% 0.33/0.65 ! [L: num] :
% 0.33/0.65 ( ( neg_numeral_sub_rat @ one @ ( bit0 @ L ) )
% 0.33/0.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bitM @ L ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(2)
% 0.33/0.65 thf(fact_8617_sub__num__simps_I2_J,axiom,
% 0.33/0.65 ! [L: num] :
% 0.33/0.65 ( ( neg_nu5755505904847501662nteger @ one @ ( bit0 @ L ) )
% 0.33/0.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bitM @ L ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_num_simps(2)
% 0.33/0.65 thf(fact_8618_xor__minus__numerals_I1_J,axiom,
% 0.33/0.65 ! [N: num,K: int] :
% 0.33/0.65 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
% 0.33/0.65 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % xor_minus_numerals(1)
% 0.33/0.65 thf(fact_8619_real__root__inverse,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( root @ N @ ( inverse_inverse_real @ X ) )
% 0.33/0.65 = ( inverse_inverse_real @ ( root @ N @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_inverse
% 0.33/0.65 thf(fact_8620_real__root__mult__exp,axiom,
% 0.33/0.65 ! [M: nat,N: nat,X: real] :
% 0.33/0.65 ( ( root @ ( times_times_nat @ M @ N ) @ X )
% 0.33/0.65 = ( root @ M @ ( root @ N @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_mult_exp
% 0.33/0.65 thf(fact_8621_real__root__mult,axiom,
% 0.33/0.65 ! [N: nat,X: real,Y: real] :
% 0.33/0.65 ( ( root @ N @ ( times_times_real @ X @ Y ) )
% 0.33/0.65 = ( times_times_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_mult
% 0.33/0.65 thf(fact_8622_real__root__minus,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( root @ N @ ( uminus_uminus_real @ X ) )
% 0.33/0.65 = ( uminus_uminus_real @ ( root @ N @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_minus
% 0.33/0.65 thf(fact_8623_real__root__commute,axiom,
% 0.33/0.65 ! [M: nat,N: nat,X: real] :
% 0.33/0.65 ( ( root @ M @ ( root @ N @ X ) )
% 0.33/0.65 = ( root @ N @ ( root @ M @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_commute
% 0.33/0.65 thf(fact_8624_real__root__divide,axiom,
% 0.33/0.65 ! [N: nat,X: real,Y: real] :
% 0.33/0.65 ( ( root @ N @ ( divide_divide_real @ X @ Y ) )
% 0.33/0.65 = ( divide_divide_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_divide
% 0.33/0.65 thf(fact_8625_real__root__pos__pos__le,axiom,
% 0.33/0.65 ! [X: real,N: nat] :
% 0.33/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_pos_pos_le
% 0.33/0.65 thf(fact_8626_neg__numeral__class_Osub__def,axiom,
% 0.33/0.65 ( neg_nu8416839295433526191omplex
% 0.33/0.65 = ( ^ [K2: num,L2: num] : ( minus_minus_complex @ ( numera6690914467698888265omplex @ K2 ) @ ( numera6690914467698888265omplex @ L2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_class.sub_def
% 0.33/0.65 thf(fact_8627_neg__numeral__class_Osub__def,axiom,
% 0.33/0.65 ( neg_numeral_sub_real
% 0.33/0.65 = ( ^ [K2: num,L2: num] : ( minus_minus_real @ ( numeral_numeral_real @ K2 ) @ ( numeral_numeral_real @ L2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_class.sub_def
% 0.33/0.65 thf(fact_8628_neg__numeral__class_Osub__def,axiom,
% 0.33/0.65 ( neg_numeral_sub_rat
% 0.33/0.65 = ( ^ [K2: num,L2: num] : ( minus_minus_rat @ ( numeral_numeral_rat @ K2 ) @ ( numeral_numeral_rat @ L2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_class.sub_def
% 0.33/0.65 thf(fact_8629_neg__numeral__class_Osub__def,axiom,
% 0.33/0.65 ( neg_numeral_sub_int
% 0.33/0.65 = ( ^ [K2: num,L2: num] : ( minus_minus_int @ ( numeral_numeral_int @ K2 ) @ ( numeral_numeral_int @ L2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_class.sub_def
% 0.33/0.65 thf(fact_8630_real__root__less__mono,axiom,
% 0.33/0.65 ! [N: nat,X: real,Y: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_real @ X @ Y )
% 0.33/0.65 => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_less_mono
% 0.33/0.65 thf(fact_8631_real__root__le__mono,axiom,
% 0.33/0.65 ! [N: nat,X: real,Y: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.65 => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_le_mono
% 0.33/0.65 thf(fact_8632_real__root__power,axiom,
% 0.33/0.65 ! [N: nat,X: real,K: nat] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( root @ N @ ( power_power_real @ X @ K ) )
% 0.33/0.65 = ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_power
% 0.33/0.65 thf(fact_8633_real__root__abs,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( root @ N @ ( abs_abs_real @ X ) )
% 0.33/0.65 = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_abs
% 0.33/0.65 thf(fact_8634_length__pos__if__in__set,axiom,
% 0.33/0.65 ! [X: real,Xs: list_real] :
% 0.33/0.65 ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.33/0.65 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % length_pos_if_in_set
% 0.33/0.65 thf(fact_8635_length__pos__if__in__set,axiom,
% 0.33/0.65 ! [X: complex,Xs: list_complex] :
% 0.33/0.65 ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
% 0.33/0.65 => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % length_pos_if_in_set
% 0.33/0.65 thf(fact_8636_length__pos__if__in__set,axiom,
% 0.33/0.65 ! [X: set_nat,Xs: list_set_nat] :
% 0.33/0.65 ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.33/0.65 => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % length_pos_if_in_set
% 0.33/0.65 thf(fact_8637_length__pos__if__in__set,axiom,
% 0.33/0.65 ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.33/0.65 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.65 => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % length_pos_if_in_set
% 0.33/0.65 thf(fact_8638_length__pos__if__in__set,axiom,
% 0.33/0.65 ! [X: $o,Xs: list_o] :
% 0.33/0.65 ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.33/0.65 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % length_pos_if_in_set
% 0.33/0.65 thf(fact_8639_length__pos__if__in__set,axiom,
% 0.33/0.65 ! [X: nat,Xs: list_nat] :
% 0.33/0.65 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.33/0.65 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % length_pos_if_in_set
% 0.33/0.65 thf(fact_8640_length__pos__if__in__set,axiom,
% 0.33/0.65 ! [X: int,Xs: list_int] :
% 0.33/0.65 ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.33/0.65 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % length_pos_if_in_set
% 0.33/0.65 thf(fact_8641_sgn__root,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( sgn_sgn_real @ ( root @ N @ X ) )
% 0.33/0.65 = ( sgn_sgn_real @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sgn_root
% 0.33/0.65 thf(fact_8642_sub__non__negative,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( ord_less_eq_real @ zero_zero_real @ ( neg_numeral_sub_real @ N @ M ) )
% 0.33/0.65 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_non_negative
% 0.33/0.65 thf(fact_8643_sub__non__negative,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( ord_less_eq_rat @ zero_zero_rat @ ( neg_numeral_sub_rat @ N @ M ) )
% 0.33/0.65 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_non_negative
% 0.33/0.65 thf(fact_8644_sub__non__negative,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( ord_less_eq_int @ zero_zero_int @ ( neg_numeral_sub_int @ N @ M ) )
% 0.33/0.65 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_non_negative
% 0.33/0.65 thf(fact_8645_sub__non__positive,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( neg_numeral_sub_real @ N @ M ) @ zero_zero_real )
% 0.33/0.65 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_non_positive
% 0.33/0.65 thf(fact_8646_sub__non__positive,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( neg_numeral_sub_rat @ N @ M ) @ zero_zero_rat )
% 0.33/0.65 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_non_positive
% 0.33/0.65 thf(fact_8647_sub__non__positive,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( neg_numeral_sub_int @ N @ M ) @ zero_zero_int )
% 0.33/0.65 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_non_positive
% 0.33/0.65 thf(fact_8648_sub__negative,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( ord_less_real @ ( neg_numeral_sub_real @ N @ M ) @ zero_zero_real )
% 0.33/0.65 = ( ord_less_num @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_negative
% 0.33/0.65 thf(fact_8649_sub__negative,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( ord_less_rat @ ( neg_numeral_sub_rat @ N @ M ) @ zero_zero_rat )
% 0.33/0.65 = ( ord_less_num @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_negative
% 0.33/0.65 thf(fact_8650_sub__negative,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( ord_less_int @ ( neg_numeral_sub_int @ N @ M ) @ zero_zero_int )
% 0.33/0.65 = ( ord_less_num @ N @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_negative
% 0.33/0.65 thf(fact_8651_sub__positive,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( ord_less_real @ zero_zero_real @ ( neg_numeral_sub_real @ N @ M ) )
% 0.33/0.65 = ( ord_less_num @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_positive
% 0.33/0.65 thf(fact_8652_sub__positive,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( ord_less_rat @ zero_zero_rat @ ( neg_numeral_sub_rat @ N @ M ) )
% 0.33/0.65 = ( ord_less_num @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_positive
% 0.33/0.65 thf(fact_8653_sub__positive,axiom,
% 0.33/0.65 ! [N: num,M: num] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ ( neg_numeral_sub_int @ N @ M ) )
% 0.33/0.65 = ( ord_less_num @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_positive
% 0.33/0.65 thf(fact_8654_real__root__gt__zero,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_gt_zero
% 0.33/0.65 thf(fact_8655_real__root__strict__decreasing,axiom,
% 0.33/0.65 ! [N: nat,N5: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_nat @ N @ N5 )
% 0.33/0.65 => ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.65 => ( ord_less_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_strict_decreasing
% 0.33/0.65 thf(fact_8656_sqrt__def,axiom,
% 0.33/0.65 ( sqrt
% 0.33/0.65 = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sqrt_def
% 0.33/0.65 thf(fact_8657_root__abs__power,axiom,
% 0.33/0.65 ! [N: nat,Y: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
% 0.33/0.65 = ( abs_abs_real @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % root_abs_power
% 0.33/0.65 thf(fact_8658_real__root__pos__pos,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_pos_pos
% 0.33/0.65 thf(fact_8659_real__root__strict__increasing,axiom,
% 0.33/0.65 ! [N: nat,N5: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_nat @ N @ N5 )
% 0.33/0.65 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.65 => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_strict_increasing
% 0.33/0.65 thf(fact_8660_real__root__decreasing,axiom,
% 0.33/0.65 ! [N: nat,N5: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.33/0.65 => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.33/0.65 => ( ord_less_eq_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_decreasing
% 0.33/0.65 thf(fact_8661_real__root__pow__pos,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 0.33/0.65 = X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_pow_pos
% 0.33/0.65 thf(fact_8662_real__root__pos__unique,axiom,
% 0.33/0.65 ! [N: nat,Y: real,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.65 => ( ( ( power_power_real @ Y @ N )
% 0.33/0.65 = X )
% 0.33/0.65 => ( ( root @ N @ X )
% 0.33/0.65 = Y ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_pos_unique
% 0.33/0.65 thf(fact_8663_real__root__power__cancel,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 0.33/0.65 = X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_power_cancel
% 0.33/0.65 thf(fact_8664_odd__real__root__power__cancel,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.65 => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 0.33/0.65 = X ) ) ).
% 0.33/0.65
% 0.33/0.65 % odd_real_root_power_cancel
% 0.33/0.65 thf(fact_8665_odd__real__root__unique,axiom,
% 0.33/0.65 ! [N: nat,Y: real,X: real] :
% 0.33/0.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.65 => ( ( ( power_power_real @ Y @ N )
% 0.33/0.65 = X )
% 0.33/0.65 => ( ( root @ N @ X )
% 0.33/0.65 = Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % odd_real_root_unique
% 0.33/0.65 thf(fact_8666_odd__real__root__pow,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.65 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 0.33/0.65 = X ) ) ).
% 0.33/0.65
% 0.33/0.65 % odd_real_root_pow
% 0.33/0.65 thf(fact_8667_real__root__increasing,axiom,
% 0.33/0.65 ! [N: nat,N5: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.33/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.65 => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_root_increasing
% 0.33/0.65 thf(fact_8668_sub__BitM__One__eq,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
% 0.33/0.65 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sub_BitM_One_eq
% 0.33/0.65 thf(fact_8669_root__sgn__power,axiom,
% 0.33/0.65 ! [N: nat,Y: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
% 0.33/0.65 = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % root_sgn_power
% 0.33/0.65 thf(fact_8670_sgn__power__root,axiom,
% 0.33/0.65 ! [N: nat,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
% 0.33/0.65 = X ) ) ).
% 0.33/0.65
% 0.33/0.65 % sgn_power_root
% 0.33/0.65 thf(fact_8671_ln__root,axiom,
% 0.33/0.65 ! [N: nat,B: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.65 => ( ( ln_ln_real @ ( root @ N @ B ) )
% 0.33/0.65 = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % ln_root
% 0.33/0.65 thf(fact_8672_log__root,axiom,
% 0.33/0.65 ! [N: nat,A: real,B: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.65 => ( ( log2 @ B @ ( root @ N @ A ) )
% 0.33/0.65 = ( divide_divide_real @ ( log2 @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % log_root
% 0.33/0.65 thf(fact_8673_log__base__root,axiom,
% 0.33/0.65 ! [N: nat,B: real,X: real] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.65 => ( ( log2 @ ( root @ N @ B ) @ X )
% 0.33/0.65 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ B @ X ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % log_base_root
% 0.33/0.65 thf(fact_8674_split__root,axiom,
% 0.33/0.65 ! [P: real > $o,N: nat,X: real] :
% 0.33/0.65 ( ( P @ ( root @ N @ X ) )
% 0.33/0.65 = ( ( ( N = zero_zero_nat )
% 0.33/0.65 => ( P @ zero_zero_real ) )
% 0.33/0.65 & ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ! [Y6: real] :
% 0.33/0.65 ( ( ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N ) )
% 0.33/0.65 = X )
% 0.33/0.65 => ( P @ Y6 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % split_root
% 0.33/0.65 thf(fact_8675_num__of__nat_Osimps_I2_J,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( num_of_nat @ ( suc @ N ) )
% 0.33/0.65 = ( inc @ ( num_of_nat @ N ) ) ) )
% 0.33/0.65 & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( num_of_nat @ ( suc @ N ) )
% 0.33/0.65 = one ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % num_of_nat.simps(2)
% 0.33/0.65 thf(fact_8676_pow_Osimps_I3_J,axiom,
% 0.33/0.65 ! [X: num,Y: num] :
% 0.33/0.65 ( ( pow @ X @ ( bit1 @ Y ) )
% 0.33/0.65 = ( times_times_num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % pow.simps(3)
% 0.33/0.65 thf(fact_8677_num__of__nat__numeral__eq,axiom,
% 0.33/0.65 ! [Q2: num] :
% 0.33/0.65 ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
% 0.33/0.65 = Q2 ) ).
% 0.33/0.65
% 0.33/0.65 % num_of_nat_numeral_eq
% 0.33/0.65 thf(fact_8678_sqr_Osimps_I2_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( sqr @ ( bit0 @ N ) )
% 0.33/0.65 = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sqr.simps(2)
% 0.33/0.65 thf(fact_8679_sqr_Osimps_I1_J,axiom,
% 0.33/0.65 ( ( sqr @ one )
% 0.33/0.65 = one ) ).
% 0.33/0.65
% 0.33/0.65 % sqr.simps(1)
% 0.33/0.65 thf(fact_8680_sqr__conv__mult,axiom,
% 0.33/0.65 ( sqr
% 0.33/0.65 = ( ^ [X3: num] : ( times_times_num @ X3 @ X3 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sqr_conv_mult
% 0.33/0.65 thf(fact_8681_num__of__nat_Osimps_I1_J,axiom,
% 0.33/0.65 ( ( num_of_nat @ zero_zero_nat )
% 0.33/0.65 = one ) ).
% 0.33/0.65
% 0.33/0.65 % num_of_nat.simps(1)
% 0.33/0.65 thf(fact_8682_numeral__num__of__nat,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 0.33/0.65 = N ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_num_of_nat
% 0.33/0.65 thf(fact_8683_num__of__nat__One,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ N @ one_one_nat )
% 0.33/0.65 => ( ( num_of_nat @ N )
% 0.33/0.65 = one ) ) ).
% 0.33/0.65
% 0.33/0.65 % num_of_nat_One
% 0.33/0.65 thf(fact_8684_pow_Osimps_I2_J,axiom,
% 0.33/0.65 ! [X: num,Y: num] :
% 0.33/0.65 ( ( pow @ X @ ( bit0 @ Y ) )
% 0.33/0.65 = ( sqr @ ( pow @ X @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % pow.simps(2)
% 0.33/0.65 thf(fact_8685_num__of__nat__double,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 0.33/0.65 = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % num_of_nat_double
% 0.33/0.65 thf(fact_8686_num__of__nat__plus__distrib,axiom,
% 0.33/0.65 ! [M: nat,N: nat] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.65 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 0.33/0.65 = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % num_of_nat_plus_distrib
% 0.33/0.65 thf(fact_8687_sqr_Osimps_I3_J,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( sqr @ ( bit1 @ N ) )
% 0.33/0.65 = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sqr.simps(3)
% 0.33/0.65 thf(fact_8688_and__int_Oelims,axiom,
% 0.33/0.65 ! [X: int,Xa: int,Y: int] :
% 0.33/0.65 ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 0.33/0.65 = Y )
% 0.33/0.65 => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.33/0.65 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.33/0.65 => ( Y
% 0.33/0.65 = ( uminus_uminus_int
% 0.33/0.65 @ ( zero_n2684676970156552555ol_int
% 0.33/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.33/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 0.33/0.65 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.33/0.65 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.33/0.65 => ( Y
% 0.33/0.65 = ( plus_plus_int
% 0.33/0.65 @ ( zero_n2684676970156552555ol_int
% 0.33/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.33/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 0.33/0.65 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % and_int.elims
% 0.33/0.65 thf(fact_8689_and__int_Osimps,axiom,
% 0.33/0.65 ( bit_se725231765392027082nd_int
% 0.33/0.65 = ( ^ [K2: int,L2: int] :
% 0.33/0.65 ( if_int
% 0.33/0.65 @ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.33/0.65 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.33/0.65 @ ( uminus_uminus_int
% 0.33/0.65 @ ( zero_n2684676970156552555ol_int
% 0.33/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.33/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 0.33/0.65 @ ( plus_plus_int
% 0.33/0.65 @ ( zero_n2684676970156552555ol_int
% 0.33/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.33/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.33/0.65 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % and_int.simps
% 0.33/0.65 thf(fact_8690_min__Suc__Suc,axiom,
% 0.33/0.65 ! [M: nat,N: nat] :
% 0.33/0.65 ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.33/0.65 = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % min_Suc_Suc
% 0.33/0.65 thf(fact_8691_min__0R,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( ord_min_nat @ N @ zero_zero_nat )
% 0.33/0.65 = zero_zero_nat ) ).
% 0.33/0.65
% 0.33/0.65 % min_0R
% 0.33/0.65 thf(fact_8692_min__0L,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( ord_min_nat @ zero_zero_nat @ N )
% 0.33/0.65 = zero_zero_nat ) ).
% 0.33/0.65
% 0.33/0.65 % min_0L
% 0.33/0.65 thf(fact_8693_min__Suc__numeral,axiom,
% 0.33/0.65 ! [N: nat,K: num] :
% 0.33/0.65 ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.33/0.65 = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % min_Suc_numeral
% 0.33/0.65 thf(fact_8694_min__numeral__Suc,axiom,
% 0.33/0.65 ! [K: num,N: nat] :
% 0.33/0.65 ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.33/0.65 = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % min_numeral_Suc
% 0.33/0.65 thf(fact_8695_min__diff,axiom,
% 0.33/0.65 ! [M: nat,I: nat,N: nat] :
% 0.33/0.65 ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
% 0.33/0.65 = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% 0.33/0.65
% 0.33/0.65 % min_diff
% 0.33/0.65 thf(fact_8696_nat__mult__min__left,axiom,
% 0.33/0.65 ! [M: nat,N: nat,Q2: nat] :
% 0.33/0.65 ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
% 0.33/0.65 = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_mult_min_left
% 0.33/0.65 thf(fact_8697_nat__mult__min__right,axiom,
% 0.33/0.65 ! [M: nat,N: nat,Q2: nat] :
% 0.33/0.65 ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
% 0.33/0.65 = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_mult_min_right
% 0.33/0.65 thf(fact_8698_concat__bit__assoc__sym,axiom,
% 0.33/0.65 ! [M: nat,N: nat,K: int,L: int,R: int] :
% 0.33/0.65 ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L ) @ R )
% 0.33/0.65 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L @ R ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % concat_bit_assoc_sym
% 0.33/0.65 thf(fact_8699_take__bit__concat__bit__eq,axiom,
% 0.33/0.65 ! [M: nat,N: nat,K: int,L: int] :
% 0.33/0.65 ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L ) )
% 0.33/0.65 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % take_bit_concat_bit_eq
% 0.33/0.65 thf(fact_8700_set__decode__zero,axiom,
% 0.33/0.65 ( ( nat_set_decode @ zero_zero_nat )
% 0.33/0.65 = bot_bot_set_nat ) ).
% 0.33/0.65
% 0.33/0.65 % set_decode_zero
% 0.33/0.65 thf(fact_8701_set__decode__plus__power__2,axiom,
% 0.33/0.65 ! [N: nat,Z2: nat] :
% 0.33/0.65 ( ~ ( member_nat @ N @ ( nat_set_decode @ Z2 ) )
% 0.33/0.65 => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z2 ) )
% 0.33/0.65 = ( insert_nat @ N @ ( nat_set_decode @ Z2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % set_decode_plus_power_2
% 0.33/0.65 thf(fact_8702_and__int_Opsimps,axiom,
% 0.33/0.65 ! [K: int,L: int] :
% 0.33/0.65 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
% 0.33/0.65 => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.33/0.65 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.33/0.65 => ( ( bit_se725231765392027082nd_int @ K @ L )
% 0.33/0.65 = ( uminus_uminus_int
% 0.33/0.65 @ ( zero_n2684676970156552555ol_int
% 0.33/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.33/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 0.33/0.65 & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.33/0.65 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.33/0.65 => ( ( bit_se725231765392027082nd_int @ K @ L )
% 0.33/0.65 = ( plus_plus_int
% 0.33/0.65 @ ( zero_n2684676970156552555ol_int
% 0.33/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.33/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.33/0.65 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % and_int.psimps
% 0.33/0.65 thf(fact_8703_and__int_Opelims,axiom,
% 0.33/0.65 ! [X: int,Xa: int,Y: int] :
% 0.33/0.65 ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 0.33/0.65 = Y )
% 0.33/0.65 => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 0.33/0.65 => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.33/0.65 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.33/0.65 => ( Y
% 0.33/0.65 = ( uminus_uminus_int
% 0.33/0.65 @ ( zero_n2684676970156552555ol_int
% 0.33/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.33/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 0.33/0.65 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.33/0.65 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.33/0.65 => ( Y
% 0.33/0.65 = ( plus_plus_int
% 0.33/0.65 @ ( zero_n2684676970156552555ol_int
% 0.33/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.33/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 0.33/0.65 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.33/0.65 => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % and_int.pelims
% 0.33/0.65 thf(fact_8704_bot__nat__def,axiom,
% 0.33/0.65 bot_bot_nat = zero_zero_nat ).
% 0.33/0.65
% 0.33/0.65 % bot_nat_def
% 0.33/0.65 thf(fact_8705_and__int_Opinduct,axiom,
% 0.33/0.65 ! [A0: int,A1: int,P: int > int > $o] :
% 0.33/0.65 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 0.33/0.65 => ( ! [K3: int,L3: int] :
% 0.33/0.65 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L3 ) )
% 0.33/0.65 => ( ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.33/0.65 & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.33/0.65 => ( P @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.33/0.65 => ( P @ K3 @ L3 ) ) )
% 0.33/0.65 => ( P @ A0 @ A1 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % and_int.pinduct
% 0.33/0.65 thf(fact_8706_Divides_Oadjust__div__eq,axiom,
% 0.33/0.65 ! [Q2: int,R: int] :
% 0.33/0.65 ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.33/0.65 = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R != zero_zero_int ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Divides.adjust_div_eq
% 0.33/0.65 thf(fact_8707_neg__eucl__rel__int__mult__2,axiom,
% 0.33/0.65 ! [B: int,A: int,Q2: int,R: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.33/0.65 => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.33/0.65 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) @ one_one_int ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_eucl_rel_int_mult_2
% 0.33/0.65 thf(fact_8708_upto_Opinduct,axiom,
% 0.33/0.65 ! [A0: int,A1: int,P: int > int > $o] :
% 0.33/0.65 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 0.33/0.65 => ( ! [I2: int,J2: int] :
% 0.33/0.65 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
% 0.33/0.65 => ( ( ( ord_less_eq_int @ I2 @ J2 )
% 0.33/0.65 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
% 0.33/0.65 => ( P @ I2 @ J2 ) ) )
% 0.33/0.65 => ( P @ A0 @ A1 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % upto.pinduct
% 0.33/0.65 thf(fact_8709_unique__remainder,axiom,
% 0.33/0.65 ! [A: int,B: int,Q2: int,R: int,Q4: int,R4: int] :
% 0.33/0.65 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.33/0.65 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R4 ) )
% 0.33/0.65 => ( R = R4 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % unique_remainder
% 0.33/0.65 thf(fact_8710_unique__quotient,axiom,
% 0.33/0.65 ! [A: int,B: int,Q2: int,R: int,Q4: int,R4: int] :
% 0.33/0.65 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.33/0.65 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R4 ) )
% 0.33/0.65 => ( Q2 = Q4 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % unique_quotient
% 0.33/0.65 thf(fact_8711_xor__num_Ocases,axiom,
% 0.33/0.65 ! [X: product_prod_num_num] :
% 0.33/0.65 ( ( X
% 0.33/0.65 != ( product_Pair_num_num @ one @ one ) )
% 0.33/0.65 => ( ! [N2: num] :
% 0.33/0.65 ( X
% 0.33/0.65 != ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) )
% 0.33/0.65 => ( ! [N2: num] :
% 0.33/0.65 ( X
% 0.33/0.65 != ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) )
% 0.33/0.65 => ( ! [M2: num] :
% 0.33/0.65 ( X
% 0.33/0.65 != ( product_Pair_num_num @ ( bit0 @ M2 ) @ one ) )
% 0.33/0.65 => ( ! [M2: num,N2: num] :
% 0.33/0.65 ( X
% 0.33/0.65 != ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) ) )
% 0.33/0.65 => ( ! [M2: num,N2: num] :
% 0.33/0.65 ( X
% 0.33/0.65 != ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) ) )
% 0.33/0.65 => ( ! [M2: num] :
% 0.33/0.65 ( X
% 0.33/0.65 != ( product_Pair_num_num @ ( bit1 @ M2 ) @ one ) )
% 0.33/0.65 => ( ! [M2: num,N2: num] :
% 0.33/0.65 ( X
% 0.33/0.65 != ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) ) )
% 0.33/0.65 => ~ ! [M2: num,N2: num] :
% 0.33/0.65 ( X
% 0.33/0.65 != ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % xor_num.cases
% 0.33/0.65 thf(fact_8712_eucl__rel__int__by0,axiom,
% 0.33/0.65 ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% 0.33/0.65
% 0.33/0.65 % eucl_rel_int_by0
% 0.33/0.65 thf(fact_8713_div__int__unique,axiom,
% 0.33/0.65 ! [K: int,L: int,Q2: int,R: int] :
% 0.33/0.65 ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.33/0.65 => ( ( divide_divide_int @ K @ L )
% 0.33/0.65 = Q2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % div_int_unique
% 0.33/0.65 thf(fact_8714_mod__int__unique,axiom,
% 0.33/0.65 ! [K: int,L: int,Q2: int,R: int] :
% 0.33/0.65 ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.33/0.65 => ( ( modulo_modulo_int @ K @ L )
% 0.33/0.65 = R ) ) ).
% 0.33/0.65
% 0.33/0.65 % mod_int_unique
% 0.33/0.65 thf(fact_8715_eucl__rel__int__dividesI,axiom,
% 0.33/0.65 ! [L: int,K: int,Q2: int] :
% 0.33/0.65 ( ( L != zero_zero_int )
% 0.33/0.65 => ( ( K
% 0.33/0.65 = ( times_times_int @ Q2 @ L ) )
% 0.33/0.65 => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % eucl_rel_int_dividesI
% 0.33/0.65 thf(fact_8716_eucl__rel__int,axiom,
% 0.33/0.65 ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % eucl_rel_int
% 0.33/0.65 thf(fact_8717_zminus1__lemma,axiom,
% 0.33/0.65 ! [A: int,B: int,Q2: int,R: int] :
% 0.33/0.65 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.33/0.65 => ( ( B != zero_zero_int )
% 0.33/0.65 => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % zminus1_lemma
% 0.33/0.65 thf(fact_8718_eucl__rel__int__iff,axiom,
% 0.33/0.65 ! [K: int,L: int,Q2: int,R: int] :
% 0.33/0.65 ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.33/0.65 = ( ( K
% 0.33/0.65 = ( plus_plus_int @ ( times_times_int @ L @ Q2 ) @ R ) )
% 0.33/0.65 & ( ( ord_less_int @ zero_zero_int @ L )
% 0.33/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.33/0.65 & ( ord_less_int @ R @ L ) ) )
% 0.33/0.65 & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 0.33/0.65 => ( ( ( ord_less_int @ L @ zero_zero_int )
% 0.33/0.65 => ( ( ord_less_int @ L @ R )
% 0.33/0.65 & ( ord_less_eq_int @ R @ zero_zero_int ) ) )
% 0.33/0.65 & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 0.33/0.65 => ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % eucl_rel_int_iff
% 0.33/0.65 thf(fact_8719_eucl__rel__int__remainderI,axiom,
% 0.33/0.65 ! [R: int,L: int,K: int,Q2: int] :
% 0.33/0.65 ( ( ( sgn_sgn_int @ R )
% 0.33/0.65 = ( sgn_sgn_int @ L ) )
% 0.33/0.65 => ( ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ L ) )
% 0.33/0.65 => ( ( K
% 0.33/0.65 = ( plus_plus_int @ ( times_times_int @ Q2 @ L ) @ R ) )
% 0.33/0.65 => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % eucl_rel_int_remainderI
% 0.33/0.65 thf(fact_8720_eucl__rel__int_Ocases,axiom,
% 0.33/0.65 ! [A1: int,A22: int,A32: product_prod_int_int] :
% 0.33/0.65 ( ( eucl_rel_int @ A1 @ A22 @ A32 )
% 0.33/0.65 => ( ( ( A22 = zero_zero_int )
% 0.33/0.65 => ( A32
% 0.33/0.65 != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
% 0.33/0.65 => ( ! [Q6: int] :
% 0.33/0.65 ( ( A32
% 0.33/0.65 = ( product_Pair_int_int @ Q6 @ zero_zero_int ) )
% 0.33/0.65 => ( ( A22 != zero_zero_int )
% 0.33/0.65 => ( A1
% 0.33/0.65 != ( times_times_int @ Q6 @ A22 ) ) ) )
% 0.33/0.65 => ~ ! [R2: int,Q6: int] :
% 0.33/0.65 ( ( A32
% 0.33/0.65 = ( product_Pair_int_int @ Q6 @ R2 ) )
% 0.33/0.65 => ( ( ( sgn_sgn_int @ R2 )
% 0.33/0.65 = ( sgn_sgn_int @ A22 ) )
% 0.33/0.65 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ A22 ) )
% 0.33/0.65 => ( A1
% 0.33/0.65 != ( plus_plus_int @ ( times_times_int @ Q6 @ A22 ) @ R2 ) ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % eucl_rel_int.cases
% 0.33/0.65 thf(fact_8721_eucl__rel__int_Osimps,axiom,
% 0.33/0.65 ( eucl_rel_int
% 0.33/0.65 = ( ^ [A12: int,A23: int,A33: product_prod_int_int] :
% 0.33/0.65 ( ? [K2: int] :
% 0.33/0.65 ( ( A12 = K2 )
% 0.33/0.65 & ( A23 = zero_zero_int )
% 0.33/0.65 & ( A33
% 0.33/0.65 = ( product_Pair_int_int @ zero_zero_int @ K2 ) ) )
% 0.33/0.65 | ? [L2: int,K2: int,Q5: int] :
% 0.33/0.65 ( ( A12 = K2 )
% 0.33/0.65 & ( A23 = L2 )
% 0.33/0.65 & ( A33
% 0.33/0.65 = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
% 0.33/0.65 & ( L2 != zero_zero_int )
% 0.33/0.65 & ( K2
% 0.33/0.65 = ( times_times_int @ Q5 @ L2 ) ) )
% 0.33/0.65 | ? [R5: int,L2: int,K2: int,Q5: int] :
% 0.33/0.65 ( ( A12 = K2 )
% 0.33/0.65 & ( A23 = L2 )
% 0.33/0.65 & ( A33
% 0.33/0.65 = ( product_Pair_int_int @ Q5 @ R5 ) )
% 0.33/0.65 & ( ( sgn_sgn_int @ R5 )
% 0.33/0.65 = ( sgn_sgn_int @ L2 ) )
% 0.33/0.65 & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
% 0.33/0.65 & ( K2
% 0.33/0.65 = ( plus_plus_int @ ( times_times_int @ Q5 @ L2 ) @ R5 ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % eucl_rel_int.simps
% 0.33/0.65 thf(fact_8722_pos__eucl__rel__int__mult__2,axiom,
% 0.33/0.65 ! [B: int,A: int,Q2: int,R: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.65 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.33/0.65 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % pos_eucl_rel_int_mult_2
% 0.33/0.65 thf(fact_8723_minus__numeral__div__numeral,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % minus_numeral_div_numeral
% 0.33/0.65 thf(fact_8724_numeral__div__minus__numeral,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_div_minus_numeral
% 0.33/0.65 thf(fact_8725_minus__one__div__numeral,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % minus_one_div_numeral
% 0.33/0.65 thf(fact_8726_one__div__minus__numeral,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % one_div_minus_numeral
% 0.33/0.65 thf(fact_8727_divmod__BitM__2__eq,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 0.33/0.65 = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % divmod_BitM_2_eq
% 0.33/0.65 thf(fact_8728_divmod_H__nat__def,axiom,
% 0.33/0.65 ( unique5055182867167087721od_nat
% 0.33/0.65 = ( ^ [M3: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % divmod'_nat_def
% 0.33/0.65 thf(fact_8729_divmod__int__def,axiom,
% 0.33/0.65 ( unique5052692396658037445od_int
% 0.33/0.65 = ( ^ [M3: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % divmod_int_def
% 0.33/0.65 thf(fact_8730_bezw__0,axiom,
% 0.33/0.65 ! [X: nat] :
% 0.33/0.65 ( ( bezw @ X @ zero_zero_nat )
% 0.33/0.65 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % bezw_0
% 0.33/0.65 thf(fact_8731_rcis__inverse,axiom,
% 0.33/0.65 ! [R: real,A: real] :
% 0.33/0.65 ( ( invers8013647133539491842omplex @ ( rcis @ R @ A ) )
% 0.33/0.65 = ( rcis @ ( divide_divide_real @ one_one_real @ R ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % rcis_inverse
% 0.33/0.65 thf(fact_8732_abs__Gcd__eq,axiom,
% 0.33/0.65 ! [K5: set_int] :
% 0.33/0.65 ( ( abs_abs_int @ ( gcd_Gcd_int @ K5 ) )
% 0.33/0.65 = ( gcd_Gcd_int @ K5 ) ) ).
% 0.33/0.65
% 0.33/0.65 % abs_Gcd_eq
% 0.33/0.65 thf(fact_8733_Re__rcis,axiom,
% 0.33/0.65 ! [R: real,A: real] :
% 0.33/0.65 ( ( re @ ( rcis @ R @ A ) )
% 0.33/0.65 = ( times_times_real @ R @ ( cos_real @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Re_rcis
% 0.33/0.65 thf(fact_8734_Im__rcis,axiom,
% 0.33/0.65 ! [R: real,A: real] :
% 0.33/0.65 ( ( im @ ( rcis @ R @ A ) )
% 0.33/0.65 = ( times_times_real @ R @ ( sin_real @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Im_rcis
% 0.33/0.65 thf(fact_8735_prod__decode__aux_Ocases,axiom,
% 0.33/0.65 ! [X: product_prod_nat_nat] :
% 0.33/0.65 ~ ! [K3: nat,M2: nat] :
% 0.33/0.65 ( X
% 0.33/0.65 != ( product_Pair_nat_nat @ K3 @ M2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % prod_decode_aux.cases
% 0.33/0.65 thf(fact_8736_Gcd__greatest__int,axiom,
% 0.33/0.65 ! [A3: set_int,A: int] :
% 0.33/0.65 ( ! [B4: int] :
% 0.33/0.65 ( ( member_int @ B4 @ A3 )
% 0.33/0.65 => ( dvd_dvd_int @ A @ B4 ) )
% 0.33/0.65 => ( dvd_dvd_int @ A @ ( gcd_Gcd_int @ A3 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Gcd_greatest_int
% 0.33/0.65 thf(fact_8737_Gcd__dvd__int,axiom,
% 0.33/0.65 ! [A: int,A3: set_int] :
% 0.33/0.65 ( ( member_int @ A @ A3 )
% 0.33/0.65 => ( dvd_dvd_int @ ( gcd_Gcd_int @ A3 ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % Gcd_dvd_int
% 0.33/0.65 thf(fact_8738_Gcd__nat__eq__one,axiom,
% 0.33/0.65 ! [N5: set_nat] :
% 0.33/0.65 ( ( member_nat @ one_one_nat @ N5 )
% 0.33/0.65 => ( ( gcd_Gcd_nat @ N5 )
% 0.33/0.65 = one_one_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % Gcd_nat_eq_one
% 0.33/0.65 thf(fact_8739_Gcd__greatest__nat,axiom,
% 0.33/0.65 ! [A3: set_nat,A: nat] :
% 0.33/0.65 ( ! [B4: nat] :
% 0.33/0.65 ( ( member_nat @ B4 @ A3 )
% 0.33/0.65 => ( dvd_dvd_nat @ A @ B4 ) )
% 0.33/0.65 => ( dvd_dvd_nat @ A @ ( gcd_Gcd_nat @ A3 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Gcd_greatest_nat
% 0.33/0.65 thf(fact_8740_Gcd__dvd__nat,axiom,
% 0.33/0.65 ! [A: nat,A3: set_nat] :
% 0.33/0.65 ( ( member_nat @ A @ A3 )
% 0.33/0.65 => ( dvd_dvd_nat @ ( gcd_Gcd_nat @ A3 ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % Gcd_dvd_nat
% 0.33/0.65 thf(fact_8741_Gcd__int__greater__eq__0,axiom,
% 0.33/0.65 ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).
% 0.33/0.65
% 0.33/0.65 % Gcd_int_greater_eq_0
% 0.33/0.65 thf(fact_8742_cis__rcis__eq,axiom,
% 0.33/0.65 ( cis
% 0.33/0.65 = ( rcis @ one_one_real ) ) ).
% 0.33/0.65
% 0.33/0.65 % cis_rcis_eq
% 0.33/0.65 thf(fact_8743_rcis__mult,axiom,
% 0.33/0.65 ! [R1: real,A: real,R22: real,B: real] :
% 0.33/0.65 ( ( times_times_complex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B ) )
% 0.33/0.65 = ( rcis @ ( times_times_real @ R1 @ R22 ) @ ( plus_plus_real @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % rcis_mult
% 0.33/0.65 thf(fact_8744_rcis__divide,axiom,
% 0.33/0.65 ! [R1: real,A: real,R22: real,B: real] :
% 0.33/0.65 ( ( divide1717551699836669952omplex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B ) )
% 0.33/0.65 = ( rcis @ ( divide_divide_real @ R1 @ R22 ) @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % rcis_divide
% 0.33/0.65 thf(fact_8745_rcis__def,axiom,
% 0.33/0.65 ( rcis
% 0.33/0.65 = ( ^ [R5: real,A2: real] : ( times_times_complex @ ( real_V4546457046886955230omplex @ R5 ) @ ( cis @ A2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % rcis_def
% 0.33/0.65 thf(fact_8746_DeMoivre2,axiom,
% 0.33/0.65 ! [R: real,A: real,N: nat] :
% 0.33/0.65 ( ( power_power_complex @ ( rcis @ R @ A ) @ N )
% 0.33/0.65 = ( rcis @ ( power_power_real @ R @ N ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % DeMoivre2
% 0.33/0.65 thf(fact_8747_prod__decode__aux_Osimps,axiom,
% 0.33/0.65 ( nat_prod_decode_aux
% 0.33/0.65 = ( ^ [K2: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K2 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K2 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M3 @ ( suc @ K2 ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % prod_decode_aux.simps
% 0.33/0.65 thf(fact_8748_prod__decode__aux_Oelims,axiom,
% 0.33/0.65 ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
% 0.33/0.65 ( ( ( nat_prod_decode_aux @ X @ Xa )
% 0.33/0.65 = Y )
% 0.33/0.65 => ( ( ( ord_less_eq_nat @ Xa @ X )
% 0.33/0.65 => ( Y
% 0.33/0.65 = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
% 0.33/0.65 & ( ~ ( ord_less_eq_nat @ Xa @ X )
% 0.33/0.65 => ( Y
% 0.33/0.65 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % prod_decode_aux.elims
% 0.33/0.65 thf(fact_8749_Suc__0__div__numeral,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.33/0.65 = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Suc_0_div_numeral
% 0.33/0.65 thf(fact_8750_Suc__0__mod__numeral,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.33/0.65 = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Suc_0_mod_numeral
% 0.33/0.65 thf(fact_8751_snd__divmod__nat,axiom,
% 0.33/0.65 ! [M: nat,N: nat] :
% 0.33/0.65 ( ( product_snd_nat_nat @ ( divmod_nat @ M @ N ) )
% 0.33/0.65 = ( modulo_modulo_nat @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % snd_divmod_nat
% 0.33/0.65 thf(fact_8752_fst__divmod__nat,axiom,
% 0.33/0.65 ! [M: nat,N: nat] :
% 0.33/0.65 ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
% 0.33/0.65 = ( divide_divide_nat @ M @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % fst_divmod_nat
% 0.33/0.65 thf(fact_8753_bezw__non__0,axiom,
% 0.33/0.65 ! [Y: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ Y )
% 0.33/0.65 => ( ( bezw @ X @ Y )
% 0.33/0.65 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bezw_non_0
% 0.33/0.65 thf(fact_8754_bezw_Osimps,axiom,
% 0.33/0.65 ( bezw
% 0.33/0.65 = ( ^ [X3: nat,Y6: nat] : ( if_Pro3027730157355071871nt_int @ ( Y6 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y6 ) ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bezw.simps
% 0.33/0.65 thf(fact_8755_bezw_Oelims,axiom,
% 0.33/0.65 ! [X: nat,Xa: nat,Y: product_prod_int_int] :
% 0.33/0.65 ( ( ( bezw @ X @ Xa )
% 0.33/0.65 = Y )
% 0.33/0.65 => ( ( ( Xa = zero_zero_nat )
% 0.33/0.65 => ( Y
% 0.33/0.65 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.33/0.65 & ( ( Xa != zero_zero_nat )
% 0.33/0.65 => ( Y
% 0.33/0.65 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bezw.elims
% 0.33/0.65 thf(fact_8756_divmod__nat__def,axiom,
% 0.33/0.65 ( divmod_nat
% 0.33/0.65 = ( ^ [M3: nat,N3: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M3 @ N3 ) @ ( modulo_modulo_nat @ M3 @ N3 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % divmod_nat_def
% 0.33/0.65 thf(fact_8757_one__mod__minus__numeral,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % one_mod_minus_numeral
% 0.33/0.65 thf(fact_8758_minus__one__mod__numeral,axiom,
% 0.33/0.65 ! [N: num] :
% 0.33/0.65 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.65 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % minus_one_mod_numeral
% 0.33/0.65 thf(fact_8759_minus__numeral__mod__numeral,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.65 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % minus_numeral_mod_numeral
% 0.33/0.65 thf(fact_8760_numeral__mod__minus__numeral,axiom,
% 0.33/0.65 ! [M: num,N: num] :
% 0.33/0.65 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_mod_minus_numeral
% 0.33/0.65 thf(fact_8761_Divides_Oadjust__mod__def,axiom,
% 0.33/0.65 ( adjust_mod
% 0.33/0.65 = ( ^ [L2: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L2 @ R5 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Divides.adjust_mod_def
% 0.33/0.65 thf(fact_8762_bezw_Opelims,axiom,
% 0.33/0.65 ! [X: nat,Xa: nat,Y: product_prod_int_int] :
% 0.33/0.65 ( ( ( bezw @ X @ Xa )
% 0.33/0.65 = Y )
% 0.33/0.65 => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 0.33/0.65 => ~ ( ( ( ( Xa = zero_zero_nat )
% 0.33/0.65 => ( Y
% 0.33/0.65 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.33/0.65 & ( ( Xa != zero_zero_nat )
% 0.33/0.65 => ( Y
% 0.33/0.65 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) )
% 0.33/0.65 => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bezw.pelims
% 0.33/0.65 thf(fact_8763_prod__encode__prod__decode__aux,axiom,
% 0.33/0.65 ! [K: nat,M: nat] :
% 0.33/0.65 ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 0.33/0.65 = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % prod_encode_prod_decode_aux
% 0.33/0.65 thf(fact_8764_bezw__aux,axiom,
% 0.33/0.65 ! [X: nat,Y: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y ) )
% 0.33/0.65 = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bezw_aux
% 0.33/0.65 thf(fact_8765_gcd__idem__nat,axiom,
% 0.33/0.65 ! [X: nat] :
% 0.33/0.65 ( ( gcd_gcd_nat @ X @ X )
% 0.33/0.65 = X ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_idem_nat
% 0.33/0.65 thf(fact_8766_gcd__nat_Oright__idem,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( gcd_gcd_nat @ ( gcd_gcd_nat @ A @ B ) @ B )
% 0.33/0.65 = ( gcd_gcd_nat @ A @ B ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.right_idem
% 0.33/0.65 thf(fact_8767_gcd__nat_Oleft__idem,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( gcd_gcd_nat @ A @ ( gcd_gcd_nat @ A @ B ) )
% 0.33/0.65 = ( gcd_gcd_nat @ A @ B ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.left_idem
% 0.33/0.65 thf(fact_8768_gcd__nat_Oidem,axiom,
% 0.33/0.65 ! [A: nat] :
% 0.33/0.65 ( ( gcd_gcd_nat @ A @ A )
% 0.33/0.65 = A ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.idem
% 0.33/0.65 thf(fact_8769_prod__encode__eq,axiom,
% 0.33/0.65 ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 0.33/0.65 ( ( ( nat_prod_encode @ X )
% 0.33/0.65 = ( nat_prod_encode @ Y ) )
% 0.33/0.65 = ( X = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % prod_encode_eq
% 0.33/0.65 thf(fact_8770_gcd__0__left__nat,axiom,
% 0.33/0.65 ! [X: nat] :
% 0.33/0.65 ( ( gcd_gcd_nat @ zero_zero_nat @ X )
% 0.33/0.65 = X ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_0_left_nat
% 0.33/0.65 thf(fact_8771_gcd__0__nat,axiom,
% 0.33/0.65 ! [X: nat] :
% 0.33/0.65 ( ( gcd_gcd_nat @ X @ zero_zero_nat )
% 0.33/0.65 = X ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_0_nat
% 0.33/0.65 thf(fact_8772_gcd__nat_Oright__neutral,axiom,
% 0.33/0.65 ! [A: nat] :
% 0.33/0.65 ( ( gcd_gcd_nat @ A @ zero_zero_nat )
% 0.33/0.65 = A ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.right_neutral
% 0.33/0.65 thf(fact_8773_gcd__nat_Oneutr__eq__iff,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( zero_zero_nat
% 0.33/0.65 = ( gcd_gcd_nat @ A @ B ) )
% 0.33/0.65 = ( ( A = zero_zero_nat )
% 0.33/0.65 & ( B = zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.neutr_eq_iff
% 0.33/0.65 thf(fact_8774_gcd__nat_Oleft__neutral,axiom,
% 0.33/0.65 ! [A: nat] :
% 0.33/0.65 ( ( gcd_gcd_nat @ zero_zero_nat @ A )
% 0.33/0.65 = A ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.left_neutral
% 0.33/0.65 thf(fact_8775_gcd__nat_Oeq__neutr__iff,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( ( gcd_gcd_nat @ A @ B )
% 0.33/0.65 = zero_zero_nat )
% 0.33/0.65 = ( ( A = zero_zero_nat )
% 0.33/0.65 & ( B = zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.eq_neutr_iff
% 0.33/0.65 thf(fact_8776_gcd__1__nat,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ( ( gcd_gcd_nat @ M @ one_one_nat )
% 0.33/0.65 = one_one_nat ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_1_nat
% 0.33/0.65 thf(fact_8777_gcd__proj2__if__dvd__nat,axiom,
% 0.33/0.65 ! [Y: nat,X: nat] :
% 0.33/0.65 ( ( dvd_dvd_nat @ Y @ X )
% 0.33/0.65 => ( ( gcd_gcd_nat @ X @ Y )
% 0.33/0.65 = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_proj2_if_dvd_nat
% 0.33/0.65 thf(fact_8778_gcd__proj1__if__dvd__nat,axiom,
% 0.33/0.65 ! [X: nat,Y: nat] :
% 0.33/0.65 ( ( dvd_dvd_nat @ X @ Y )
% 0.33/0.65 => ( ( gcd_gcd_nat @ X @ Y )
% 0.33/0.65 = X ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_proj1_if_dvd_nat
% 0.33/0.65 thf(fact_8779_gcd__nat_Obounded__iff,axiom,
% 0.33/0.65 ! [A: nat,B: nat,C: nat] :
% 0.33/0.65 ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 0.33/0.65 = ( ( dvd_dvd_nat @ A @ B )
% 0.33/0.65 & ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.bounded_iff
% 0.33/0.65 thf(fact_8780_gcd__nat_Oabsorb2,axiom,
% 0.33/0.65 ! [B: nat,A: nat] :
% 0.33/0.65 ( ( dvd_dvd_nat @ B @ A )
% 0.33/0.65 => ( ( gcd_gcd_nat @ A @ B )
% 0.33/0.65 = B ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.absorb2
% 0.33/0.65 thf(fact_8781_gcd__nat_Oabsorb1,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( dvd_dvd_nat @ A @ B )
% 0.33/0.65 => ( ( gcd_gcd_nat @ A @ B )
% 0.33/0.65 = A ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.absorb1
% 0.33/0.65 thf(fact_8782_gcd__Suc__0,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.33/0.65 = ( suc @ zero_zero_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_Suc_0
% 0.33/0.65 thf(fact_8783_gcd__pos__nat,axiom,
% 0.33/0.65 ! [M: nat,N: nat] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 0.33/0.65 = ( ( M != zero_zero_nat )
% 0.33/0.65 | ( N != zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_pos_nat
% 0.33/0.65 thf(fact_8784_gcd__mult__distrib__nat,axiom,
% 0.33/0.65 ! [K: nat,M: nat,N: nat] :
% 0.33/0.65 ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
% 0.33/0.65 = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_mult_distrib_nat
% 0.33/0.65 thf(fact_8785_gcd__unique__nat,axiom,
% 0.33/0.65 ! [D: nat,A: nat,B: nat] :
% 0.33/0.65 ( ( ( dvd_dvd_nat @ D @ A )
% 0.33/0.65 & ( dvd_dvd_nat @ D @ B )
% 0.33/0.65 & ! [E3: nat] :
% 0.33/0.65 ( ( ( dvd_dvd_nat @ E3 @ A )
% 0.33/0.65 & ( dvd_dvd_nat @ E3 @ B ) )
% 0.33/0.65 => ( dvd_dvd_nat @ E3 @ D ) ) )
% 0.33/0.65 = ( D
% 0.33/0.65 = ( gcd_gcd_nat @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_unique_nat
% 0.33/0.65 thf(fact_8786_gcd__nat_Ostrict__coboundedI2,axiom,
% 0.33/0.65 ! [B: nat,C: nat,A: nat] :
% 0.33/0.65 ( ( ( dvd_dvd_nat @ B @ C )
% 0.33/0.65 & ( B != C ) )
% 0.33/0.65 => ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
% 0.33/0.65 & ( ( gcd_gcd_nat @ A @ B )
% 0.33/0.65 != C ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.strict_coboundedI2
% 0.33/0.65 thf(fact_8787_gcd__nat_Ostrict__coboundedI1,axiom,
% 0.33/0.65 ! [A: nat,C: nat,B: nat] :
% 0.33/0.65 ( ( ( dvd_dvd_nat @ A @ C )
% 0.33/0.65 & ( A != C ) )
% 0.33/0.65 => ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
% 0.33/0.65 & ( ( gcd_gcd_nat @ A @ B )
% 0.33/0.65 != C ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.strict_coboundedI1
% 0.33/0.65 thf(fact_8788_gcd__nat_Ostrict__order__iff,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( ( dvd_dvd_nat @ A @ B )
% 0.33/0.65 & ( A != B ) )
% 0.33/0.65 = ( ( A
% 0.33/0.65 = ( gcd_gcd_nat @ A @ B ) )
% 0.33/0.65 & ( A != B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.strict_order_iff
% 0.33/0.65 thf(fact_8789_gcd__nat_Ostrict__boundedE,axiom,
% 0.33/0.65 ! [A: nat,B: nat,C: nat] :
% 0.33/0.65 ( ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 0.33/0.65 & ( A
% 0.33/0.65 != ( gcd_gcd_nat @ B @ C ) ) )
% 0.33/0.65 => ~ ( ( ( dvd_dvd_nat @ A @ B )
% 0.33/0.65 & ( A != B ) )
% 0.33/0.65 => ~ ( ( dvd_dvd_nat @ A @ C )
% 0.33/0.65 & ( A != C ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.strict_boundedE
% 0.33/0.65 thf(fact_8790_gcd__nat_OcoboundedI2,axiom,
% 0.33/0.65 ! [B: nat,C: nat,A: nat] :
% 0.33/0.65 ( ( dvd_dvd_nat @ B @ C )
% 0.33/0.65 => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.coboundedI2
% 0.33/0.65 thf(fact_8791_gcd__nat_OcoboundedI1,axiom,
% 0.33/0.65 ! [A: nat,C: nat,B: nat] :
% 0.33/0.65 ( ( dvd_dvd_nat @ A @ C )
% 0.33/0.65 => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.coboundedI1
% 0.33/0.65 thf(fact_8792_gcd__nat_Oabsorb__iff2,axiom,
% 0.33/0.65 ( dvd_dvd_nat
% 0.33/0.65 = ( ^ [B2: nat,A2: nat] :
% 0.33/0.65 ( ( gcd_gcd_nat @ A2 @ B2 )
% 0.33/0.65 = B2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.absorb_iff2
% 0.33/0.65 thf(fact_8793_gcd__nat_Oabsorb__iff1,axiom,
% 0.33/0.65 ( dvd_dvd_nat
% 0.33/0.65 = ( ^ [A2: nat,B2: nat] :
% 0.33/0.65 ( ( gcd_gcd_nat @ A2 @ B2 )
% 0.33/0.65 = A2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.absorb_iff1
% 0.33/0.65 thf(fact_8794_gcd__nat_Ocobounded2,axiom,
% 0.33/0.65 ! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.cobounded2
% 0.33/0.65 thf(fact_8795_gcd__nat_Ocobounded1,axiom,
% 0.33/0.65 ! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.cobounded1
% 0.33/0.65 thf(fact_8796_gcd__nat_Oorder__iff,axiom,
% 0.33/0.65 ( dvd_dvd_nat
% 0.33/0.65 = ( ^ [A2: nat,B2: nat] :
% 0.33/0.65 ( A2
% 0.33/0.65 = ( gcd_gcd_nat @ A2 @ B2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.order_iff
% 0.33/0.65 thf(fact_8797_gcd__nat_OboundedI,axiom,
% 0.33/0.65 ! [A: nat,B: nat,C: nat] :
% 0.33/0.65 ( ( dvd_dvd_nat @ A @ B )
% 0.33/0.65 => ( ( dvd_dvd_nat @ A @ C )
% 0.33/0.65 => ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.boundedI
% 0.33/0.65 thf(fact_8798_gcd__nat_OboundedE,axiom,
% 0.33/0.65 ! [A: nat,B: nat,C: nat] :
% 0.33/0.65 ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 0.33/0.65 => ~ ( ( dvd_dvd_nat @ A @ B )
% 0.33/0.65 => ~ ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.boundedE
% 0.33/0.65 thf(fact_8799_gcd__nat_Oabsorb4,axiom,
% 0.33/0.65 ! [B: nat,A: nat] :
% 0.33/0.65 ( ( ( dvd_dvd_nat @ B @ A )
% 0.33/0.65 & ( B != A ) )
% 0.33/0.65 => ( ( gcd_gcd_nat @ A @ B )
% 0.33/0.65 = B ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.absorb4
% 0.33/0.65 thf(fact_8800_gcd__nat_Oabsorb3,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( ( dvd_dvd_nat @ A @ B )
% 0.33/0.65 & ( A != B ) )
% 0.33/0.65 => ( ( gcd_gcd_nat @ A @ B )
% 0.33/0.65 = A ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.absorb3
% 0.33/0.65 thf(fact_8801_gcd__nat_OorderI,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( A
% 0.33/0.65 = ( gcd_gcd_nat @ A @ B ) )
% 0.33/0.65 => ( dvd_dvd_nat @ A @ B ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.orderI
% 0.33/0.65 thf(fact_8802_gcd__nat_OorderE,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( dvd_dvd_nat @ A @ B )
% 0.33/0.65 => ( A
% 0.33/0.65 = ( gcd_gcd_nat @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.orderE
% 0.33/0.65 thf(fact_8803_gcd__nat_Omono,axiom,
% 0.33/0.65 ! [A: nat,C: nat,B: nat,D: nat] :
% 0.33/0.65 ( ( dvd_dvd_nat @ A @ C )
% 0.33/0.65 => ( ( dvd_dvd_nat @ B @ D )
% 0.33/0.65 => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ ( gcd_gcd_nat @ C @ D ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.mono
% 0.33/0.65 thf(fact_8804_gcd__red__nat,axiom,
% 0.33/0.65 ( gcd_gcd_nat
% 0.33/0.65 = ( ^ [X3: nat,Y6: nat] : ( gcd_gcd_nat @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_red_nat
% 0.33/0.65 thf(fact_8805_gcd__le2__nat,axiom,
% 0.33/0.65 ! [B: nat,A: nat] :
% 0.33/0.65 ( ( B != zero_zero_nat )
% 0.33/0.65 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_le2_nat
% 0.33/0.65 thf(fact_8806_gcd__le1__nat,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( A != zero_zero_nat )
% 0.33/0.65 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_le1_nat
% 0.33/0.65 thf(fact_8807_gcd__diff2__nat,axiom,
% 0.33/0.65 ! [M: nat,N: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.65 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 0.33/0.65 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_diff2_nat
% 0.33/0.65 thf(fact_8808_gcd__diff1__nat,axiom,
% 0.33/0.65 ! [N: nat,M: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.65 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 0.33/0.65 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_diff1_nat
% 0.33/0.65 thf(fact_8809_gcd__non__0__nat,axiom,
% 0.33/0.65 ! [Y: nat,X: nat] :
% 0.33/0.65 ( ( Y != zero_zero_nat )
% 0.33/0.65 => ( ( gcd_gcd_nat @ X @ Y )
% 0.33/0.65 = ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_non_0_nat
% 0.33/0.65 thf(fact_8810_gcd__nat_Osimps,axiom,
% 0.33/0.65 ( gcd_gcd_nat
% 0.33/0.65 = ( ^ [X3: nat,Y6: nat] : ( if_nat @ ( Y6 = zero_zero_nat ) @ X3 @ ( gcd_gcd_nat @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.simps
% 0.33/0.65 thf(fact_8811_gcd__nat_Oelims,axiom,
% 0.33/0.65 ! [X: nat,Xa: nat,Y: nat] :
% 0.33/0.65 ( ( ( gcd_gcd_nat @ X @ Xa )
% 0.33/0.65 = Y )
% 0.33/0.65 => ( ( ( Xa = zero_zero_nat )
% 0.33/0.65 => ( Y = X ) )
% 0.33/0.65 & ( ( Xa != zero_zero_nat )
% 0.33/0.65 => ( Y
% 0.33/0.65 = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.elims
% 0.33/0.65 thf(fact_8812_Gcd__in,axiom,
% 0.33/0.65 ! [A3: set_nat] :
% 0.33/0.65 ( ! [A4: nat,B4: nat] :
% 0.33/0.65 ( ( member_nat @ A4 @ A3 )
% 0.33/0.65 => ( ( member_nat @ B4 @ A3 )
% 0.33/0.65 => ( member_nat @ ( gcd_gcd_nat @ A4 @ B4 ) @ A3 ) ) )
% 0.33/0.65 => ( ( A3 != bot_bot_set_nat )
% 0.33/0.65 => ( member_nat @ ( gcd_Gcd_nat @ A3 ) @ A3 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Gcd_in
% 0.33/0.65 thf(fact_8813_bezout__nat,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( A != zero_zero_nat )
% 0.33/0.65 => ? [X4: nat,Y4: nat] :
% 0.33/0.65 ( ( times_times_nat @ A @ X4 )
% 0.33/0.65 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bezout_nat
% 0.33/0.65 thf(fact_8814_bezout__gcd__nat_H,axiom,
% 0.33/0.65 ! [B: nat,A: nat] :
% 0.33/0.65 ? [X4: nat,Y4: nat] :
% 0.33/0.65 ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y4 ) @ ( times_times_nat @ A @ X4 ) )
% 0.33/0.65 & ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y4 ) )
% 0.33/0.65 = ( gcd_gcd_nat @ A @ B ) ) )
% 0.33/0.65 | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B @ X4 ) )
% 0.33/0.65 & ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y4 ) )
% 0.33/0.65 = ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bezout_gcd_nat'
% 0.33/0.65 thf(fact_8815_le__prod__encode__1,axiom,
% 0.33/0.65 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % le_prod_encode_1
% 0.33/0.65 thf(fact_8816_le__prod__encode__2,axiom,
% 0.33/0.65 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % le_prod_encode_2
% 0.33/0.65 thf(fact_8817_prod__decode__aux_Opelims,axiom,
% 0.33/0.65 ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
% 0.33/0.65 ( ( ( nat_prod_decode_aux @ X @ Xa )
% 0.33/0.65 = Y )
% 0.33/0.65 => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 0.33/0.65 => ~ ( ( ( ( ord_less_eq_nat @ Xa @ X )
% 0.33/0.65 => ( Y
% 0.33/0.65 = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
% 0.33/0.65 & ( ~ ( ord_less_eq_nat @ Xa @ X )
% 0.33/0.65 => ( Y
% 0.33/0.65 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) )
% 0.33/0.65 => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % prod_decode_aux.pelims
% 0.33/0.65 thf(fact_8818_gcd__nat_Opelims,axiom,
% 0.33/0.65 ! [X: nat,Xa: nat,Y: nat] :
% 0.33/0.65 ( ( ( gcd_gcd_nat @ X @ Xa )
% 0.33/0.65 = Y )
% 0.33/0.65 => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 0.33/0.65 => ~ ( ( ( ( Xa = zero_zero_nat )
% 0.33/0.65 => ( Y = X ) )
% 0.33/0.65 & ( ( Xa != zero_zero_nat )
% 0.33/0.65 => ( Y
% 0.33/0.65 = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) )
% 0.33/0.65 => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat.pelims
% 0.33/0.65 thf(fact_8819_gcd__1__int,axiom,
% 0.33/0.65 ! [M: int] :
% 0.33/0.65 ( ( gcd_gcd_int @ M @ one_one_int )
% 0.33/0.65 = one_one_int ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_1_int
% 0.33/0.65 thf(fact_8820_gcd__neg1__int,axiom,
% 0.33/0.65 ! [X: int,Y: int] :
% 0.33/0.65 ( ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y )
% 0.33/0.65 = ( gcd_gcd_int @ X @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_neg1_int
% 0.33/0.65 thf(fact_8821_gcd__neg2__int,axiom,
% 0.33/0.65 ! [X: int,Y: int] :
% 0.33/0.65 ( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) )
% 0.33/0.65 = ( gcd_gcd_int @ X @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_neg2_int
% 0.33/0.65 thf(fact_8822_abs__gcd__int,axiom,
% 0.33/0.65 ! [X: int,Y: int] :
% 0.33/0.65 ( ( abs_abs_int @ ( gcd_gcd_int @ X @ Y ) )
% 0.33/0.65 = ( gcd_gcd_int @ X @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % abs_gcd_int
% 0.33/0.65 thf(fact_8823_gcd__abs1__int,axiom,
% 0.33/0.65 ! [X: int,Y: int] :
% 0.33/0.65 ( ( gcd_gcd_int @ ( abs_abs_int @ X ) @ Y )
% 0.33/0.65 = ( gcd_gcd_int @ X @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_abs1_int
% 0.33/0.65 thf(fact_8824_gcd__abs2__int,axiom,
% 0.33/0.65 ! [X: int,Y: int] :
% 0.33/0.65 ( ( gcd_gcd_int @ X @ ( abs_abs_int @ Y ) )
% 0.33/0.65 = ( gcd_gcd_int @ X @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_abs2_int
% 0.33/0.65 thf(fact_8825_gcd__pos__int,axiom,
% 0.33/0.65 ! [M: int,N: int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
% 0.33/0.65 = ( ( M != zero_zero_int )
% 0.33/0.65 | ( N != zero_zero_int ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_pos_int
% 0.33/0.65 thf(fact_8826_gcd__neg__numeral__2__int,axiom,
% 0.33/0.65 ! [X: int,N: num] :
% 0.33/0.65 ( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.65 = ( gcd_gcd_int @ X @ ( numeral_numeral_int @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_neg_numeral_2_int
% 0.33/0.65 thf(fact_8827_gcd__neg__numeral__1__int,axiom,
% 0.33/0.65 ! [N: num,X: int] :
% 0.33/0.65 ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X )
% 0.33/0.65 = ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_neg_numeral_1_int
% 0.33/0.65 thf(fact_8828_gcd__0__int,axiom,
% 0.33/0.65 ! [X: int] :
% 0.33/0.65 ( ( gcd_gcd_int @ X @ zero_zero_int )
% 0.33/0.65 = ( abs_abs_int @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_0_int
% 0.33/0.65 thf(fact_8829_gcd__0__left__int,axiom,
% 0.33/0.65 ! [X: int] :
% 0.33/0.65 ( ( gcd_gcd_int @ zero_zero_int @ X )
% 0.33/0.65 = ( abs_abs_int @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_0_left_int
% 0.33/0.65 thf(fact_8830_gcd__proj1__if__dvd__int,axiom,
% 0.33/0.65 ! [X: int,Y: int] :
% 0.33/0.65 ( ( dvd_dvd_int @ X @ Y )
% 0.33/0.65 => ( ( gcd_gcd_int @ X @ Y )
% 0.33/0.65 = ( abs_abs_int @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_proj1_if_dvd_int
% 0.33/0.65 thf(fact_8831_gcd__proj2__if__dvd__int,axiom,
% 0.33/0.65 ! [Y: int,X: int] :
% 0.33/0.65 ( ( dvd_dvd_int @ Y @ X )
% 0.33/0.65 => ( ( gcd_gcd_int @ X @ Y )
% 0.33/0.65 = ( abs_abs_int @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_proj2_if_dvd_int
% 0.33/0.65 thf(fact_8832_gcd__int__int__eq,axiom,
% 0.33/0.65 ! [M: nat,N: nat] :
% 0.33/0.65 ( ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.33/0.65 = ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_int_int_eq
% 0.33/0.65 thf(fact_8833_gcd__nat__abs__right__eq,axiom,
% 0.33/0.65 ! [N: nat,K: int] :
% 0.33/0.65 ( ( gcd_gcd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 0.33/0.65 = ( nat2 @ ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat_abs_right_eq
% 0.33/0.65 thf(fact_8834_gcd__nat__abs__left__eq,axiom,
% 0.33/0.65 ! [K: int,N: nat] :
% 0.33/0.65 ( ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 0.33/0.65 = ( nat2 @ ( gcd_gcd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_nat_abs_left_eq
% 0.33/0.65 thf(fact_8835_gcd__red__int,axiom,
% 0.33/0.65 ( gcd_gcd_int
% 0.33/0.65 = ( ^ [X3: int,Y6: int] : ( gcd_gcd_int @ Y6 @ ( modulo_modulo_int @ X3 @ Y6 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_red_int
% 0.33/0.65 thf(fact_8836_gcd__idem__int,axiom,
% 0.33/0.65 ! [X: int] :
% 0.33/0.65 ( ( gcd_gcd_int @ X @ X )
% 0.33/0.65 = ( abs_abs_int @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_idem_int
% 0.33/0.65 thf(fact_8837_gcd__ge__0__int,axiom,
% 0.33/0.65 ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_ge_0_int
% 0.33/0.65 thf(fact_8838_bezout__int,axiom,
% 0.33/0.65 ! [X: int,Y: int] :
% 0.33/0.65 ? [U2: int,V3: int] :
% 0.33/0.65 ( ( plus_plus_int @ ( times_times_int @ U2 @ X ) @ ( times_times_int @ V3 @ Y ) )
% 0.33/0.65 = ( gcd_gcd_int @ X @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % bezout_int
% 0.33/0.65 thf(fact_8839_gcd__mult__distrib__int,axiom,
% 0.33/0.65 ! [K: int,M: int,N: int] :
% 0.33/0.65 ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
% 0.33/0.65 = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_mult_distrib_int
% 0.33/0.65 thf(fact_8840_gcd__le1__int,axiom,
% 0.33/0.65 ! [A: int,B: int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.65 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_le1_int
% 0.33/0.65 thf(fact_8841_gcd__le2__int,axiom,
% 0.33/0.65 ! [B: int,A: int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.65 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_le2_int
% 0.33/0.65 thf(fact_8842_gcd__cases__int,axiom,
% 0.33/0.65 ! [X: int,Y: int,P: int > $o] :
% 0.33/0.65 ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.65 => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) )
% 0.33/0.65 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.65 => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 0.33/0.65 => ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
% 0.33/0.65 => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.33/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.65 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
% 0.33/0.65 => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.33/0.65 => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 0.33/0.65 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
% 0.33/0.65 => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_cases_int
% 0.33/0.65 thf(fact_8843_gcd__unique__int,axiom,
% 0.33/0.65 ! [D: int,A: int,B: int] :
% 0.33/0.65 ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 0.33/0.65 & ( dvd_dvd_int @ D @ A )
% 0.33/0.65 & ( dvd_dvd_int @ D @ B )
% 0.33/0.65 & ! [E3: int] :
% 0.33/0.65 ( ( ( dvd_dvd_int @ E3 @ A )
% 0.33/0.65 & ( dvd_dvd_int @ E3 @ B ) )
% 0.33/0.65 => ( dvd_dvd_int @ E3 @ D ) ) )
% 0.33/0.65 = ( D
% 0.33/0.65 = ( gcd_gcd_int @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_unique_int
% 0.33/0.65 thf(fact_8844_gcd__non__0__int,axiom,
% 0.33/0.65 ! [Y: int,X: int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ Y )
% 0.33/0.65 => ( ( gcd_gcd_int @ X @ Y )
% 0.33/0.65 = ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_non_0_int
% 0.33/0.65 thf(fact_8845_gcd__code__int,axiom,
% 0.33/0.65 ( gcd_gcd_int
% 0.33/0.65 = ( ^ [K2: int,L2: int] : ( abs_abs_int @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( gcd_gcd_int @ L2 @ ( modulo_modulo_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_code_int
% 0.33/0.65 thf(fact_8846_gcd__int__def,axiom,
% 0.33/0.65 ( gcd_gcd_int
% 0.33/0.65 = ( ^ [X3: int,Y6: int] : ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ X3 ) ) @ ( nat2 @ ( abs_abs_int @ Y6 ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gcd_int_def
% 0.33/0.65 thf(fact_8847_upto__aux__rec,axiom,
% 0.33/0.65 ( upto_aux
% 0.33/0.65 = ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % upto_aux_rec
% 0.33/0.65 thf(fact_8848_powr__real__of__int_H,axiom,
% 0.33/0.65 ! [X: real,N: int] :
% 0.33/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ( ( X != zero_zero_real )
% 0.33/0.65 | ( ord_less_int @ zero_zero_int @ N ) )
% 0.33/0.65 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 0.33/0.65 = ( power_int_real @ X @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % powr_real_of_int'
% 0.33/0.65 thf(fact_8849_complex__is__Nat__iff,axiom,
% 0.33/0.65 ! [Z2: complex] :
% 0.33/0.65 ( ( member_complex @ Z2 @ semiri3842193898606819883omplex )
% 0.33/0.65 = ( ( ( im @ Z2 )
% 0.33/0.65 = zero_zero_real )
% 0.33/0.65 & ? [I4: nat] :
% 0.33/0.65 ( ( re @ Z2 )
% 0.33/0.65 = ( semiri5074537144036343181t_real @ I4 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % complex_is_Nat_iff
% 0.33/0.65 thf(fact_8850_cpmi,axiom,
% 0.33/0.65 ! [D3: int,P: int > $o,P4: int > $o,B5: set_int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.33/0.65 => ( ? [Z3: int] :
% 0.33/0.65 ! [X4: int] :
% 0.33/0.65 ( ( ord_less_int @ X4 @ Z3 )
% 0.33/0.65 => ( ( P @ X4 )
% 0.33/0.65 = ( P4 @ X4 ) ) )
% 0.33/0.65 => ( ! [X4: int] :
% 0.33/0.65 ( ! [Xa2: int] :
% 0.33/0.65 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb2: int] :
% 0.33/0.65 ( ( member_int @ Xb2 @ B5 )
% 0.33/0.65 => ( X4
% 0.33/0.65 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.33/0.65 => ( ( P @ X4 )
% 0.33/0.65 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 0.33/0.65 => ( ! [X4: int,K3: int] :
% 0.33/0.65 ( ( P4 @ X4 )
% 0.33/0.65 = ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
% 0.33/0.65 => ( ( ? [X8: int] : ( P @ X8 ) )
% 0.33/0.65 = ( ? [X3: int] :
% 0.33/0.65 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 & ( P4 @ X3 ) )
% 0.33/0.65 | ? [X3: int] :
% 0.33/0.65 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 & ? [Y6: int] :
% 0.33/0.65 ( ( member_int @ Y6 @ B5 )
% 0.33/0.65 & ( P @ ( plus_plus_int @ Y6 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % cpmi
% 0.33/0.65 thf(fact_8851_cppi,axiom,
% 0.33/0.65 ! [D3: int,P: int > $o,P4: int > $o,A3: set_int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.33/0.65 => ( ? [Z3: int] :
% 0.33/0.65 ! [X4: int] :
% 0.33/0.65 ( ( ord_less_int @ Z3 @ X4 )
% 0.33/0.65 => ( ( P @ X4 )
% 0.33/0.65 = ( P4 @ X4 ) ) )
% 0.33/0.65 => ( ! [X4: int] :
% 0.33/0.65 ( ! [Xa2: int] :
% 0.33/0.65 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb2: int] :
% 0.33/0.65 ( ( member_int @ Xb2 @ A3 )
% 0.33/0.65 => ( X4
% 0.33/0.65 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.33/0.65 => ( ( P @ X4 )
% 0.33/0.65 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 0.33/0.65 => ( ! [X4: int,K3: int] :
% 0.33/0.65 ( ( P4 @ X4 )
% 0.33/0.65 = ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
% 0.33/0.65 => ( ( ? [X8: int] : ( P @ X8 ) )
% 0.33/0.65 = ( ? [X3: int] :
% 0.33/0.65 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 & ( P4 @ X3 ) )
% 0.33/0.65 | ? [X3: int] :
% 0.33/0.65 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 & ? [Y6: int] :
% 0.33/0.65 ( ( member_int @ Y6 @ A3 )
% 0.33/0.65 & ( P @ ( minus_minus_int @ Y6 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % cppi
% 0.33/0.65 thf(fact_8852_aset_I2_J,axiom,
% 0.33/0.65 ! [D3: int,A3: set_int,P: int > $o,Q: int > $o] :
% 0.33/0.65 ( ! [X4: int] :
% 0.33/0.65 ( ! [Xa2: int] :
% 0.33/0.65 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb2: int] :
% 0.33/0.65 ( ( member_int @ Xb2 @ A3 )
% 0.33/0.65 => ( X4
% 0.33/0.65 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.33/0.65 => ( ( P @ X4 )
% 0.33/0.65 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 0.33/0.65 => ( ! [X4: int] :
% 0.33/0.65 ( ! [Xa2: int] :
% 0.33/0.65 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb2: int] :
% 0.33/0.65 ( ( member_int @ Xb2 @ A3 )
% 0.33/0.65 => ( X4
% 0.33/0.65 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.33/0.65 => ( ( Q @ X4 )
% 0.33/0.65 => ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ A3 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( ( P @ X5 )
% 0.33/0.65 | ( Q @ X5 ) )
% 0.33/0.65 => ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
% 0.33/0.65 | ( Q @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % aset(2)
% 0.33/0.65 thf(fact_8853_aset_I1_J,axiom,
% 0.33/0.65 ! [D3: int,A3: set_int,P: int > $o,Q: int > $o] :
% 0.33/0.65 ( ! [X4: int] :
% 0.33/0.65 ( ! [Xa2: int] :
% 0.33/0.65 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb2: int] :
% 0.33/0.65 ( ( member_int @ Xb2 @ A3 )
% 0.33/0.65 => ( X4
% 0.33/0.65 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.33/0.65 => ( ( P @ X4 )
% 0.33/0.65 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 0.33/0.65 => ( ! [X4: int] :
% 0.33/0.65 ( ! [Xa2: int] :
% 0.33/0.65 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb2: int] :
% 0.33/0.65 ( ( member_int @ Xb2 @ A3 )
% 0.33/0.65 => ( X4
% 0.33/0.65 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.33/0.65 => ( ( Q @ X4 )
% 0.33/0.65 => ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ A3 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( ( P @ X5 )
% 0.33/0.65 & ( Q @ X5 ) )
% 0.33/0.65 => ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
% 0.33/0.65 & ( Q @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % aset(1)
% 0.33/0.65 thf(fact_8854_bset_I2_J,axiom,
% 0.33/0.65 ! [D3: int,B5: set_int,P: int > $o,Q: int > $o] :
% 0.33/0.65 ( ! [X4: int] :
% 0.33/0.65 ( ! [Xa2: int] :
% 0.33/0.65 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb2: int] :
% 0.33/0.65 ( ( member_int @ Xb2 @ B5 )
% 0.33/0.65 => ( X4
% 0.33/0.65 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.33/0.65 => ( ( P @ X4 )
% 0.33/0.65 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 0.33/0.65 => ( ! [X4: int] :
% 0.33/0.65 ( ! [Xa2: int] :
% 0.33/0.65 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb2: int] :
% 0.33/0.65 ( ( member_int @ Xb2 @ B5 )
% 0.33/0.65 => ( X4
% 0.33/0.65 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.33/0.65 => ( ( Q @ X4 )
% 0.33/0.65 => ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ B5 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( ( P @ X5 )
% 0.33/0.65 | ( Q @ X5 ) )
% 0.33/0.65 => ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
% 0.33/0.65 | ( Q @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bset(2)
% 0.33/0.65 thf(fact_8855_bset_I1_J,axiom,
% 0.33/0.65 ! [D3: int,B5: set_int,P: int > $o,Q: int > $o] :
% 0.33/0.65 ( ! [X4: int] :
% 0.33/0.65 ( ! [Xa2: int] :
% 0.33/0.65 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb2: int] :
% 0.33/0.65 ( ( member_int @ Xb2 @ B5 )
% 0.33/0.65 => ( X4
% 0.33/0.65 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.33/0.65 => ( ( P @ X4 )
% 0.33/0.65 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 0.33/0.65 => ( ! [X4: int] :
% 0.33/0.65 ( ! [Xa2: int] :
% 0.33/0.65 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb2: int] :
% 0.33/0.65 ( ( member_int @ Xb2 @ B5 )
% 0.33/0.65 => ( X4
% 0.33/0.65 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.33/0.65 => ( ( Q @ X4 )
% 0.33/0.65 => ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ B5 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( ( P @ X5 )
% 0.33/0.65 & ( Q @ X5 ) )
% 0.33/0.65 => ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
% 0.33/0.65 & ( Q @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bset(1)
% 0.33/0.65 thf(fact_8856_aset_I10_J,axiom,
% 0.33/0.65 ! [D: int,D3: int,A3: set_int,T: int] :
% 0.33/0.65 ( ( dvd_dvd_int @ D @ D3 )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ A3 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.33/0.65 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % aset(10)
% 0.33/0.65 thf(fact_8857_aset_I9_J,axiom,
% 0.33/0.65 ! [D: int,D3: int,A3: set_int,T: int] :
% 0.33/0.65 ( ( dvd_dvd_int @ D @ D3 )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ A3 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.33/0.65 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % aset(9)
% 0.33/0.65 thf(fact_8858_bset_I10_J,axiom,
% 0.33/0.65 ! [D: int,D3: int,B5: set_int,T: int] :
% 0.33/0.65 ( ( dvd_dvd_int @ D @ D3 )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ B5 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.33/0.65 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bset(10)
% 0.33/0.65 thf(fact_8859_bset_I9_J,axiom,
% 0.33/0.65 ! [D: int,D3: int,B5: set_int,T: int] :
% 0.33/0.65 ( ( dvd_dvd_int @ D @ D3 )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ B5 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.33/0.65 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bset(9)
% 0.33/0.65 thf(fact_8860_atLeastAtMostPlus1__int__conv,axiom,
% 0.33/0.65 ! [M: int,N: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 0.33/0.65 => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 0.33/0.65 = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % atLeastAtMostPlus1_int_conv
% 0.33/0.65 thf(fact_8861_periodic__finite__ex,axiom,
% 0.33/0.65 ! [D: int,P: int > $o] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D )
% 0.33/0.65 => ( ! [X4: int,K3: int] :
% 0.33/0.65 ( ( P @ X4 )
% 0.33/0.65 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) )
% 0.33/0.65 => ( ( ? [X8: int] : ( P @ X8 ) )
% 0.33/0.65 = ( ? [X3: int] :
% 0.33/0.65 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 0.33/0.65 & ( P @ X3 ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % periodic_finite_ex
% 0.33/0.65 thf(fact_8862_bset_I3_J,axiom,
% 0.33/0.65 ! [D3: int,T: int,B5: set_int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.33/0.65 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ B5 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( X5 = T )
% 0.33/0.65 => ( ( minus_minus_int @ X5 @ D3 )
% 0.33/0.65 = T ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bset(3)
% 0.33/0.65 thf(fact_8863_bset_I4_J,axiom,
% 0.33/0.65 ! [D3: int,T: int,B5: set_int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.33/0.65 => ( ( member_int @ T @ B5 )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ B5 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( X5 != T )
% 0.33/0.65 => ( ( minus_minus_int @ X5 @ D3 )
% 0.33/0.65 != T ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bset(4)
% 0.33/0.65 thf(fact_8864_bset_I5_J,axiom,
% 0.33/0.65 ! [D3: int,B5: set_int,T: int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ B5 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( ord_less_int @ X5 @ T )
% 0.33/0.65 => ( ord_less_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bset(5)
% 0.33/0.65 thf(fact_8865_bset_I7_J,axiom,
% 0.33/0.65 ! [D3: int,T: int,B5: set_int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.33/0.65 => ( ( member_int @ T @ B5 )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ B5 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( ord_less_int @ T @ X5 )
% 0.33/0.65 => ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bset(7)
% 0.33/0.65 thf(fact_8866_aset_I3_J,axiom,
% 0.33/0.65 ! [D3: int,T: int,A3: set_int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.33/0.65 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ A3 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( X5 = T )
% 0.33/0.65 => ( ( plus_plus_int @ X5 @ D3 )
% 0.33/0.65 = T ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % aset(3)
% 0.33/0.65 thf(fact_8867_aset_I4_J,axiom,
% 0.33/0.65 ! [D3: int,T: int,A3: set_int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.33/0.65 => ( ( member_int @ T @ A3 )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ A3 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( X5 != T )
% 0.33/0.65 => ( ( plus_plus_int @ X5 @ D3 )
% 0.33/0.65 != T ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % aset(4)
% 0.33/0.65 thf(fact_8868_aset_I5_J,axiom,
% 0.33/0.65 ! [D3: int,T: int,A3: set_int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.33/0.65 => ( ( member_int @ T @ A3 )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ A3 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( ord_less_int @ X5 @ T )
% 0.33/0.65 => ( ord_less_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % aset(5)
% 0.33/0.65 thf(fact_8869_aset_I7_J,axiom,
% 0.33/0.65 ! [D3: int,A3: set_int,T: int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ A3 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( ord_less_int @ T @ X5 )
% 0.33/0.65 => ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % aset(7)
% 0.33/0.65 thf(fact_8870_simp__from__to,axiom,
% 0.33/0.65 ( set_or1266510415728281911st_int
% 0.33/0.65 = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % simp_from_to
% 0.33/0.65 thf(fact_8871_bset_I6_J,axiom,
% 0.33/0.65 ! [D3: int,B5: set_int,T: int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ B5 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( ord_less_eq_int @ X5 @ T )
% 0.33/0.65 => ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bset(6)
% 0.33/0.65 thf(fact_8872_bset_I8_J,axiom,
% 0.33/0.65 ! [D3: int,T: int,B5: set_int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.33/0.65 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ B5 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( ord_less_eq_int @ T @ X5 )
% 0.33/0.65 => ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bset(8)
% 0.33/0.65 thf(fact_8873_aset_I6_J,axiom,
% 0.33/0.65 ! [D3: int,T: int,A3: set_int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.33/0.65 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ A3 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( ord_less_eq_int @ X5 @ T )
% 0.33/0.65 => ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % aset(6)
% 0.33/0.65 thf(fact_8874_aset_I8_J,axiom,
% 0.33/0.65 ! [D3: int,A3: set_int,T: int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.33/0.65 => ! [X5: int] :
% 0.33/0.65 ( ! [Xa3: int] :
% 0.33/0.65 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.33/0.65 => ! [Xb3: int] :
% 0.33/0.65 ( ( member_int @ Xb3 @ A3 )
% 0.33/0.65 => ( X5
% 0.33/0.65 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.33/0.65 => ( ( ord_less_eq_int @ T @ X5 )
% 0.33/0.65 => ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % aset(8)
% 0.33/0.65 thf(fact_8875_card__atLeastAtMost__int,axiom,
% 0.33/0.65 ! [L: int,U: int] :
% 0.33/0.65 ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
% 0.33/0.65 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % card_atLeastAtMost_int
% 0.33/0.65 thf(fact_8876_dist__real__def,axiom,
% 0.33/0.65 ( real_V975177566351809787t_real
% 0.33/0.65 = ( ^ [X3: real,Y6: real] : ( abs_abs_real @ ( minus_minus_real @ X3 @ Y6 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dist_real_def
% 0.33/0.65 thf(fact_8877_dist__complex__def,axiom,
% 0.33/0.65 ( real_V3694042436643373181omplex
% 0.33/0.65 = ( ^ [X3: complex,Y6: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y6 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dist_complex_def
% 0.33/0.65 thf(fact_8878_normalize__negative,axiom,
% 0.33/0.65 ! [Q2: int,P2: int] :
% 0.33/0.65 ( ( ord_less_int @ Q2 @ zero_zero_int )
% 0.33/0.65 => ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
% 0.33/0.65 = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P2 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % normalize_negative
% 0.33/0.65 thf(fact_8879_card__greaterThanLessThan__int,axiom,
% 0.33/0.65 ! [L: int,U: int] :
% 0.33/0.65 ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
% 0.33/0.65 = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % card_greaterThanLessThan_int
% 0.33/0.65 thf(fact_8880_normalize__denom__zero,axiom,
% 0.33/0.65 ! [P2: int] :
% 0.33/0.65 ( ( normalize @ ( product_Pair_int_int @ P2 @ zero_zero_int ) )
% 0.33/0.65 = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % normalize_denom_zero
% 0.33/0.65 thf(fact_8881_quotient__of__number_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.33/0.65 = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % quotient_of_number(5)
% 0.33/0.65 thf(fact_8882_rat__one__code,axiom,
% 0.33/0.65 ( ( quotient_of @ one_one_rat )
% 0.33/0.65 = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % rat_one_code
% 0.33/0.65 thf(fact_8883_rat__zero__code,axiom,
% 0.33/0.65 ( ( quotient_of @ zero_zero_rat )
% 0.33/0.65 = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % rat_zero_code
% 0.33/0.65 thf(fact_8884_quotient__of__number_I4_J,axiom,
% 0.33/0.65 ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.65 = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % quotient_of_number(4)
% 0.33/0.65 thf(fact_8885_quotient__of__number_I3_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 0.33/0.65 = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % quotient_of_number(3)
% 0.33/0.65 thf(fact_8886_diff__rat__def,axiom,
% 0.33/0.65 ( minus_minus_rat
% 0.33/0.65 = ( ^ [Q5: rat,R5: rat] : ( plus_plus_rat @ Q5 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_rat_def
% 0.33/0.65 thf(fact_8887_sgn__rat__def,axiom,
% 0.33/0.65 ( sgn_sgn_rat
% 0.33/0.65 = ( ^ [A2: rat] : ( if_rat @ ( A2 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A2 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sgn_rat_def
% 0.33/0.65 thf(fact_8888_quotient__of__denom__pos,axiom,
% 0.33/0.65 ! [R: rat,P2: int,Q2: int] :
% 0.33/0.65 ( ( ( quotient_of @ R )
% 0.33/0.65 = ( product_Pair_int_int @ P2 @ Q2 ) )
% 0.33/0.65 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % quotient_of_denom_pos
% 0.33/0.65 thf(fact_8889_quotient__of__denom__pos_H,axiom,
% 0.33/0.65 ! [R: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % quotient_of_denom_pos'
% 0.33/0.65 thf(fact_8890_normalize__denom__pos,axiom,
% 0.33/0.65 ! [R: product_prod_int_int,P2: int,Q2: int] :
% 0.33/0.65 ( ( ( normalize @ R )
% 0.33/0.65 = ( product_Pair_int_int @ P2 @ Q2 ) )
% 0.33/0.65 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % normalize_denom_pos
% 0.33/0.65 thf(fact_8891_normalize__crossproduct,axiom,
% 0.33/0.65 ! [Q2: int,S: int,P2: int,R: int] :
% 0.33/0.65 ( ( Q2 != zero_zero_int )
% 0.33/0.65 => ( ( S != zero_zero_int )
% 0.33/0.65 => ( ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
% 0.33/0.65 = ( normalize @ ( product_Pair_int_int @ R @ S ) ) )
% 0.33/0.65 => ( ( times_times_int @ P2 @ S )
% 0.33/0.65 = ( times_times_int @ R @ Q2 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % normalize_crossproduct
% 0.33/0.65 thf(fact_8892_rat__sgn__code,axiom,
% 0.33/0.65 ! [P2: rat] :
% 0.33/0.65 ( ( quotient_of @ ( sgn_sgn_rat @ P2 ) )
% 0.33/0.65 = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P2 ) ) ) @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % rat_sgn_code
% 0.33/0.65 thf(fact_8893_quotient__of__int,axiom,
% 0.33/0.65 ! [A: int] :
% 0.33/0.65 ( ( quotient_of @ ( of_int @ A ) )
% 0.33/0.65 = ( product_Pair_int_int @ A @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % quotient_of_int
% 0.33/0.65 thf(fact_8894_Frct__code__post_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.65 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Frct_code_post(5)
% 0.33/0.65 thf(fact_8895_Frct__code__post_I4_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 0.33/0.65 = ( numeral_numeral_rat @ K ) ) ).
% 0.33/0.65
% 0.33/0.65 % Frct_code_post(4)
% 0.33/0.65 thf(fact_8896_Frct__code__post_I6_J,axiom,
% 0.33/0.65 ! [K: num,L: num] :
% 0.33/0.65 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
% 0.33/0.65 = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Frct_code_post(6)
% 0.33/0.65 thf(fact_8897_card__atLeastAtMost,axiom,
% 0.33/0.65 ! [L: nat,U: nat] :
% 0.33/0.65 ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 0.33/0.65 = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% 0.33/0.65
% 0.33/0.65 % card_atLeastAtMost
% 0.33/0.65 thf(fact_8898_card__greaterThanLessThan,axiom,
% 0.33/0.65 ! [L: nat,U: nat] :
% 0.33/0.65 ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
% 0.33/0.65 = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % card_greaterThanLessThan
% 0.33/0.65 thf(fact_8899_divide__rat__def,axiom,
% 0.33/0.65 ( divide_divide_rat
% 0.33/0.65 = ( ^ [Q5: rat,R5: rat] : ( times_times_rat @ Q5 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % divide_rat_def
% 0.33/0.65 thf(fact_8900_obtain__pos__sum,axiom,
% 0.33/0.65 ! [R: rat] :
% 0.33/0.65 ( ( ord_less_rat @ zero_zero_rat @ R )
% 0.33/0.65 => ~ ! [S3: rat] :
% 0.33/0.65 ( ( ord_less_rat @ zero_zero_rat @ S3 )
% 0.33/0.65 => ! [T3: rat] :
% 0.33/0.65 ( ( ord_less_rat @ zero_zero_rat @ T3 )
% 0.33/0.65 => ( R
% 0.33/0.65 != ( plus_plus_rat @ S3 @ T3 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % obtain_pos_sum
% 0.33/0.65 thf(fact_8901_all__nat__less,axiom,
% 0.33/0.65 ! [N: nat,P: nat > $o] :
% 0.33/0.65 ( ( ! [M3: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ M3 @ N )
% 0.33/0.65 => ( P @ M3 ) ) )
% 0.33/0.65 = ( ! [X3: nat] :
% 0.33/0.65 ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.33/0.65 => ( P @ X3 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % all_nat_less
% 0.33/0.65 thf(fact_8902_ex__nat__less,axiom,
% 0.33/0.65 ! [N: nat,P: nat > $o] :
% 0.33/0.65 ( ( ? [M3: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ M3 @ N )
% 0.33/0.65 & ( P @ M3 ) ) )
% 0.33/0.65 = ( ? [X3: nat] :
% 0.33/0.65 ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.33/0.65 & ( P @ X3 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % ex_nat_less
% 0.33/0.65 thf(fact_8903_atLeast0__atMost__Suc,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.33/0.65 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % atLeast0_atMost_Suc
% 0.33/0.65 thf(fact_8904_Icc__eq__insert__lb__nat,axiom,
% 0.33/0.65 ! [M: nat,N: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.65 => ( ( set_or1269000886237332187st_nat @ M @ N )
% 0.33/0.65 = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Icc_eq_insert_lb_nat
% 0.33/0.65 thf(fact_8905_atLeastAtMostSuc__conv,axiom,
% 0.33/0.65 ! [M: nat,N: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.33/0.65 => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 0.33/0.65 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % atLeastAtMostSuc_conv
% 0.33/0.65 thf(fact_8906_atLeastAtMost__insertL,axiom,
% 0.33/0.65 ! [M: nat,N: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.65 => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.33/0.65 = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % atLeastAtMost_insertL
% 0.33/0.65 thf(fact_8907_tanh__real__bounds,axiom,
% 0.33/0.65 ! [X: real] : ( member_real @ ( tanh_real @ X ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).
% 0.33/0.65
% 0.33/0.65 % tanh_real_bounds
% 0.33/0.65 thf(fact_8908_Frct__code__post_I2_J,axiom,
% 0.33/0.65 ! [A: int] :
% 0.33/0.65 ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
% 0.33/0.65 = zero_zero_rat ) ).
% 0.33/0.65
% 0.33/0.65 % Frct_code_post(2)
% 0.33/0.65 thf(fact_8909_Frct__code__post_I1_J,axiom,
% 0.33/0.65 ! [A: int] :
% 0.33/0.65 ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
% 0.33/0.65 = zero_zero_rat ) ).
% 0.33/0.65
% 0.33/0.65 % Frct_code_post(1)
% 0.33/0.65 thf(fact_8910_Frct__code__post_I3_J,axiom,
% 0.33/0.65 ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
% 0.33/0.65 = one_one_rat ) ).
% 0.33/0.65
% 0.33/0.65 % Frct_code_post(3)
% 0.33/0.65 thf(fact_8911_both__member__options__def,axiom,
% 0.33/0.65 ( vEBT_V8194947554948674370ptions
% 0.33/0.65 = ( ^ [T2: vEBT_VEBT,X3: nat] :
% 0.33/0.65 ( ( vEBT_V5719532721284313246member @ T2 @ X3 )
% 0.33/0.65 | ( vEBT_VEBT_membermima @ T2 @ X3 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % both_member_options_def
% 0.33/0.65 thf(fact_8912_image__Suc__atLeastAtMost,axiom,
% 0.33/0.65 ! [I: nat,J: nat] :
% 0.33/0.65 ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 0.33/0.65 = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % image_Suc_atLeastAtMost
% 0.33/0.65 thf(fact_8913_Gcd__abs__eq,axiom,
% 0.33/0.65 ! [K5: set_int] :
% 0.33/0.65 ( ( gcd_Gcd_int @ ( image_int_int @ abs_abs_int @ K5 ) )
% 0.33/0.65 = ( gcd_Gcd_int @ K5 ) ) ).
% 0.33/0.65
% 0.33/0.65 % Gcd_abs_eq
% 0.33/0.65 thf(fact_8914_card__Collect__le__nat,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( finite_card_nat
% 0.33/0.65 @ ( collect_nat
% 0.33/0.65 @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
% 0.33/0.65 = ( suc @ N ) ) ).
% 0.33/0.65
% 0.33/0.65 % card_Collect_le_nat
% 0.33/0.65 thf(fact_8915_Gcd__int__eq,axiom,
% 0.33/0.65 ! [N5: set_nat] :
% 0.33/0.65 ( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N5 ) )
% 0.33/0.65 = ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N5 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Gcd_int_eq
% 0.33/0.65 thf(fact_8916_Gcd__nat__abs__eq,axiom,
% 0.33/0.65 ! [K5: set_int] :
% 0.33/0.65 ( ( gcd_Gcd_nat
% 0.33/0.65 @ ( image_int_nat
% 0.33/0.65 @ ^ [K2: int] : ( nat2 @ ( abs_abs_int @ K2 ) )
% 0.33/0.65 @ K5 ) )
% 0.33/0.65 = ( nat2 @ ( gcd_Gcd_int @ K5 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Gcd_nat_abs_eq
% 0.33/0.65 thf(fact_8917_nat__less__as__int,axiom,
% 0.33/0.65 ( ord_less_nat
% 0.33/0.65 = ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_less_as_int
% 0.33/0.65 thf(fact_8918_card__less,axiom,
% 0.33/0.65 ! [M10: set_nat,I: nat] :
% 0.33/0.65 ( ( member_nat @ zero_zero_nat @ M10 )
% 0.33/0.65 => ( ( finite_card_nat
% 0.33/0.65 @ ( collect_nat
% 0.33/0.65 @ ^ [K2: nat] :
% 0.33/0.65 ( ( member_nat @ K2 @ M10 )
% 0.33/0.65 & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
% 0.33/0.65 != zero_zero_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % card_less
% 0.33/0.65 thf(fact_8919_card__less__Suc,axiom,
% 0.33/0.65 ! [M10: set_nat,I: nat] :
% 0.33/0.65 ( ( member_nat @ zero_zero_nat @ M10 )
% 0.33/0.65 => ( ( suc
% 0.33/0.65 @ ( finite_card_nat
% 0.33/0.65 @ ( collect_nat
% 0.33/0.65 @ ^ [K2: nat] :
% 0.33/0.65 ( ( member_nat @ ( suc @ K2 ) @ M10 )
% 0.33/0.65 & ( ord_less_nat @ K2 @ I ) ) ) ) )
% 0.33/0.65 = ( finite_card_nat
% 0.33/0.65 @ ( collect_nat
% 0.33/0.65 @ ^ [K2: nat] :
% 0.33/0.65 ( ( member_nat @ K2 @ M10 )
% 0.33/0.65 & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % card_less_Suc
% 0.33/0.65 thf(fact_8920_card__less__Suc2,axiom,
% 0.33/0.65 ! [M10: set_nat,I: nat] :
% 0.33/0.65 ( ~ ( member_nat @ zero_zero_nat @ M10 )
% 0.33/0.65 => ( ( finite_card_nat
% 0.33/0.65 @ ( collect_nat
% 0.33/0.65 @ ^ [K2: nat] :
% 0.33/0.65 ( ( member_nat @ ( suc @ K2 ) @ M10 )
% 0.33/0.65 & ( ord_less_nat @ K2 @ I ) ) ) )
% 0.33/0.65 = ( finite_card_nat
% 0.33/0.65 @ ( collect_nat
% 0.33/0.65 @ ^ [K2: nat] :
% 0.33/0.65 ( ( member_nat @ K2 @ M10 )
% 0.33/0.65 & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % card_less_Suc2
% 0.33/0.65 thf(fact_8921_nat__leq__as__int,axiom,
% 0.33/0.65 ( ord_less_eq_nat
% 0.33/0.65 = ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_leq_as_int
% 0.33/0.65 thf(fact_8922_nat__plus__as__int,axiom,
% 0.33/0.65 ( plus_plus_nat
% 0.33/0.65 = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_plus_as_int
% 0.33/0.65 thf(fact_8923_nat__times__as__int,axiom,
% 0.33/0.65 ( times_times_nat
% 0.33/0.65 = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_times_as_int
% 0.33/0.65 thf(fact_8924_nat__minus__as__int,axiom,
% 0.33/0.65 ( minus_minus_nat
% 0.33/0.65 = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_minus_as_int
% 0.33/0.65 thf(fact_8925_nat__div__as__int,axiom,
% 0.33/0.65 ( divide_divide_nat
% 0.33/0.65 = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_div_as_int
% 0.33/0.65 thf(fact_8926_nat__mod__as__int,axiom,
% 0.33/0.65 ( modulo_modulo_nat
% 0.33/0.65 = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_mod_as_int
% 0.33/0.65 thf(fact_8927_card__nth__roots,axiom,
% 0.33/0.65 ! [C: complex,N: nat] :
% 0.33/0.65 ( ( C != zero_zero_complex )
% 0.33/0.65 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( finite_card_complex
% 0.33/0.65 @ ( collect_complex
% 0.33/0.65 @ ^ [Z5: complex] :
% 0.33/0.65 ( ( power_power_complex @ Z5 @ N )
% 0.33/0.65 = C ) ) )
% 0.33/0.65 = N ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % card_nth_roots
% 0.33/0.65 thf(fact_8928_card__roots__unity__eq,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( ( finite_card_complex
% 0.33/0.65 @ ( collect_complex
% 0.33/0.65 @ ^ [Z5: complex] :
% 0.33/0.65 ( ( power_power_complex @ Z5 @ N )
% 0.33/0.65 = one_one_complex ) ) )
% 0.33/0.65 = N ) ) ).
% 0.33/0.65
% 0.33/0.65 % card_roots_unity_eq
% 0.33/0.65 thf(fact_8929_zero__notin__Suc__image,axiom,
% 0.33/0.65 ! [A3: set_nat] :
% 0.33/0.65 ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A3 ) ) ).
% 0.33/0.65
% 0.33/0.65 % zero_notin_Suc_image
% 0.33/0.65 thf(fact_8930_image__int__atLeastAtMost,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.33/0.65 = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % image_int_atLeastAtMost
% 0.33/0.65 thf(fact_8931_diff__nat__eq__if,axiom,
% 0.33/0.65 ! [Z6: int,Z2: int] :
% 0.33/0.65 ( ( ( ord_less_int @ Z6 @ zero_zero_int )
% 0.33/0.65 => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) )
% 0.33/0.65 = ( nat2 @ Z2 ) ) )
% 0.33/0.65 & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
% 0.33/0.65 => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) )
% 0.33/0.65 = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z2 @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z2 @ Z6 ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_nat_eq_if
% 0.33/0.65 thf(fact_8932_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.33/0.65 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % atLeast0_atMost_Suc_eq_insert_0
% 0.33/0.65 thf(fact_8933_set__decode__def,axiom,
% 0.33/0.65 ( nat_set_decode
% 0.33/0.65 = ( ^ [X3: nat] :
% 0.33/0.65 ( collect_nat
% 0.33/0.65 @ ^ [N3: nat] :
% 0.33/0.65 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % set_decode_def
% 0.33/0.65 thf(fact_8934_normalize__def,axiom,
% 0.33/0.65 ( normalize
% 0.33/0.65 = ( ^ [P3: product_prod_int_int] :
% 0.33/0.65 ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P3 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P3 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P3 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) )
% 0.33/0.65 @ ( if_Pro3027730157355071871nt_int
% 0.33/0.65 @ ( ( product_snd_int_int @ P3 )
% 0.33/0.65 = zero_zero_int )
% 0.33/0.65 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.33/0.65 @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P3 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P3 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % normalize_def
% 0.33/0.65 thf(fact_8935_in__children__def,axiom,
% 0.33/0.65 ( vEBT_V5917875025757280293ildren
% 0.33/0.65 = ( ^ [N3: nat,TreeList: list_VEBT_VEBT,X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ N3 ) ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % in_children_def
% 0.33/0.65 thf(fact_8936_monoseq__arctan__series,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.65 => ( topolo6980174941875973593q_real
% 0.33/0.65 @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % monoseq_arctan_series
% 0.33/0.65 thf(fact_8937_fact__eq__fact__times,axiom,
% 0.33/0.65 ! [N: nat,M: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.65 => ( ( semiri1408675320244567234ct_nat @ M )
% 0.33/0.65 = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 0.33/0.65 @ ( groups708209901874060359at_nat
% 0.33/0.65 @ ^ [X3: nat] : X3
% 0.33/0.65 @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % fact_eq_fact_times
% 0.33/0.65 thf(fact_8938_monoseq__realpow,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.65 => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % monoseq_realpow
% 0.33/0.65 thf(fact_8939_fact__div__fact,axiom,
% 0.33/0.65 ! [N: nat,M: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.65 => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.33/0.65 = ( groups708209901874060359at_nat
% 0.33/0.65 @ ^ [X3: nat] : X3
% 0.33/0.65 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % fact_div_fact
% 0.33/0.65 thf(fact_8940_prod__int__eq,axiom,
% 0.33/0.65 ! [I: nat,J: nat] :
% 0.33/0.65 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 0.33/0.65 = ( groups1705073143266064639nt_int
% 0.33/0.65 @ ^ [X3: int] : X3
% 0.33/0.65 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % prod_int_eq
% 0.33/0.65 thf(fact_8941_prod__int__plus__eq,axiom,
% 0.33/0.65 ! [I: nat,J: nat] :
% 0.33/0.65 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
% 0.33/0.65 = ( groups1705073143266064639nt_int
% 0.33/0.65 @ ^ [X3: int] : X3
% 0.33/0.65 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % prod_int_plus_eq
% 0.33/0.65 thf(fact_8942_ln__series,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.65 => ( ( ln_ln_real @ X )
% 0.33/0.65 = ( suminf_real
% 0.33/0.65 @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N3 ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % ln_series
% 0.33/0.65 thf(fact_8943_arctan__series,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.65 => ( ( arctan @ X )
% 0.33/0.65 = ( suminf_real
% 0.33/0.65 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % arctan_series
% 0.33/0.65 thf(fact_8944_bij__betw__nth__root__unity,axiom,
% 0.33/0.65 ! [C: complex,N: nat] :
% 0.33/0.65 ( ( C != zero_zero_complex )
% 0.33/0.65 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 0.33/0.65 @ ( collect_complex
% 0.33/0.65 @ ^ [Z5: complex] :
% 0.33/0.65 ( ( power_power_complex @ Z5 @ N )
% 0.33/0.65 = one_one_complex ) )
% 0.33/0.65 @ ( collect_complex
% 0.33/0.65 @ ^ [Z5: complex] :
% 0.33/0.65 ( ( power_power_complex @ Z5 @ N )
% 0.33/0.65 = C ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bij_betw_nth_root_unity
% 0.33/0.65 thf(fact_8945_pi__series,axiom,
% 0.33/0.65 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.65 = ( suminf_real
% 0.33/0.65 @ ^ [K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % pi_series
% 0.33/0.65 thf(fact_8946_bij__betw__Suc,axiom,
% 0.33/0.65 ! [M10: set_nat,N5: set_nat] :
% 0.33/0.65 ( ( bij_betw_nat_nat @ suc @ M10 @ N5 )
% 0.33/0.65 = ( ( image_nat_nat @ suc @ M10 )
% 0.33/0.65 = N5 ) ) ).
% 0.33/0.65
% 0.33/0.65 % bij_betw_Suc
% 0.33/0.65 thf(fact_8947_summable__arctan__series,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.65 => ( summable_real
% 0.33/0.65 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % summable_arctan_series
% 0.33/0.65 thf(fact_8948_bij__betw__roots__unity,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.65 => ( bij_betw_nat_complex
% 0.33/0.65 @ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N )
% 0.33/0.65 @ ( collect_complex
% 0.33/0.65 @ ^ [Z5: complex] :
% 0.33/0.65 ( ( power_power_complex @ Z5 @ N )
% 0.33/0.65 = one_one_complex ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % bij_betw_roots_unity
% 0.33/0.65 thf(fact_8949_sin__paired,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( sums_real
% 0.33/0.65 @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 0.33/0.65 @ ( sin_real @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % sin_paired
% 0.33/0.65 thf(fact_8950_lessThan__0,axiom,
% 0.33/0.65 ( ( set_ord_lessThan_nat @ zero_zero_nat )
% 0.33/0.65 = bot_bot_set_nat ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_0
% 0.33/0.65 thf(fact_8951_lessThan__Suc,axiom,
% 0.33/0.65 ! [K: nat] :
% 0.33/0.65 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 0.33/0.65 = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_Suc
% 0.33/0.65 thf(fact_8952_lessThan__empty__iff,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( ( set_ord_lessThan_nat @ N )
% 0.33/0.65 = bot_bot_set_nat )
% 0.33/0.65 = ( N = zero_zero_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_empty_iff
% 0.33/0.65 thf(fact_8953_lessThan__nat__numeral,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 0.33/0.65 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_nat_numeral
% 0.33/0.65 thf(fact_8954_image__Suc__lessThan,axiom,
% 0.33/0.65 ! [N: nat] :
% 0.33/0.65 ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.65 = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % image_Suc_lessThan
% 0.33/0.66 thf(fact_8955_lessThan__Suc__eq__insert__0,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( set_ord_lessThan_nat @ ( suc @ N ) )
% 0.33/0.66 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % lessThan_Suc_eq_insert_0
% 0.33/0.66 thf(fact_8956_summable__power__series,axiom,
% 0.33/0.66 ! [F: nat > real,Z2: real] :
% 0.33/0.66 ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
% 0.33/0.66 => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 0.33/0.66 => ( ( ord_less_eq_real @ zero_zero_real @ Z2 )
% 0.33/0.66 => ( ( ord_less_real @ Z2 @ one_one_real )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z2 @ I4 ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_power_series
% 0.33/0.66 thf(fact_8957_power__half__series,axiom,
% 0.33/0.66 ( sums_real
% 0.33/0.66 @ ^ [N3: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N3 ) )
% 0.33/0.66 @ one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % power_half_series
% 0.33/0.66 thf(fact_8958_sums__if_H,axiom,
% 0.33/0.66 ! [G: nat > real,X: real] :
% 0.33/0.66 ( ( sums_real @ G @ X )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.66 @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_if'
% 0.33/0.66 thf(fact_8959_sums__if,axiom,
% 0.33/0.66 ! [G: nat > real,X: real,F: nat > real,Y: real] :
% 0.33/0.66 ( ( sums_real @ G @ X )
% 0.33/0.66 => ( ( sums_real @ F @ Y )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( F @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.66 @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_if
% 0.33/0.66 thf(fact_8960_cos__paired,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( sums_real
% 0.33/0.66 @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.33/0.66 @ ( cos_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_paired
% 0.33/0.66 thf(fact_8961_Maclaurin__minus__cos__expansion,axiom,
% 0.33/0.66 ! [N: nat,X: real] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.66 => ? [T3: real] :
% 0.33/0.66 ( ( ord_less_real @ X @ T3 )
% 0.33/0.66 & ( ord_less_real @ T3 @ zero_zero_real )
% 0.33/0.66 & ( ( cos_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_minus_cos_expansion
% 0.33/0.66 thf(fact_8962_Maclaurin__cos__expansion2,axiom,
% 0.33/0.66 ! [X: real,N: nat] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ? [T3: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ T3 )
% 0.33/0.66 & ( ord_less_real @ T3 @ X )
% 0.33/0.66 & ( ( cos_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_cos_expansion2
% 0.33/0.66 thf(fact_8963_Maclaurin__sin__expansion3,axiom,
% 0.33/0.66 ! [N: nat,X: real] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ? [T3: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ T3 )
% 0.33/0.66 & ( ord_less_real @ T3 @ X )
% 0.33/0.66 & ( ( sin_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_sin_expansion3
% 0.33/0.66 thf(fact_8964_sumr__cos__zero__one,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ zero_zero_real @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % sumr_cos_zero_one
% 0.33/0.66 thf(fact_8965_Maclaurin__lemma,axiom,
% 0.33/0.66 ! [H: real,F: real > real,J: nat > real,N: nat] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ H )
% 0.33/0.66 => ? [B7: real] :
% 0.33/0.66 ( ( F @ H )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ B7 @ ( divide_divide_real @ ( power_power_real @ H @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_lemma
% 0.33/0.66 thf(fact_8966_sum__split__even__odd,axiom,
% 0.33/0.66 ! [F: nat > real,G: nat > real,N: nat] :
% 0.33/0.66 ( ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( F @ I4 ) @ ( G @ I4 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ one_one_nat ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_split_even_odd
% 0.33/0.66 thf(fact_8967_Maclaurin__exp__le,axiom,
% 0.33/0.66 ! [X: real,N: nat] :
% 0.33/0.66 ? [T3: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.33/0.66 & ( ( exp_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_exp_le
% 0.33/0.66 thf(fact_8968_Maclaurin__sin__bound,axiom,
% 0.33/0.66 ! [X: real,N: nat] :
% 0.33/0.66 ( ord_less_eq_real
% 0.33/0.66 @ ( abs_abs_real
% 0.33/0.66 @ ( minus_minus_real @ ( sin_real @ X )
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) ) ) )
% 0.33/0.66 @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_sin_bound
% 0.33/0.66 thf(fact_8969_sum__pos__lt__pair,axiom,
% 0.33/0.66 ! [F: nat > real,K: nat] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ! [D2: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D2 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D2 ) @ one_one_nat ) ) ) ) )
% 0.33/0.66 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_pos_lt_pair
% 0.33/0.66 thf(fact_8970_Maclaurin__exp__lt,axiom,
% 0.33/0.66 ! [X: real,N: nat] :
% 0.33/0.66 ( ( X != zero_zero_real )
% 0.33/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ? [T3: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
% 0.33/0.66 & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.33/0.66 & ( ( exp_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_exp_lt
% 0.33/0.66 thf(fact_8971_Maclaurin__sin__expansion,axiom,
% 0.33/0.66 ! [X: real,N: nat] :
% 0.33/0.66 ? [T3: real] :
% 0.33/0.66 ( ( sin_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_sin_expansion
% 0.33/0.66 thf(fact_8972_Maclaurin__sin__expansion2,axiom,
% 0.33/0.66 ! [X: real,N: nat] :
% 0.33/0.66 ? [T3: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.33/0.66 & ( ( sin_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_sin_expansion2
% 0.33/0.66 thf(fact_8973_Maclaurin__cos__expansion,axiom,
% 0.33/0.66 ! [X: real,N: nat] :
% 0.33/0.66 ? [T3: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.33/0.66 & ( ( cos_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_cos_expansion
% 0.33/0.66 thf(fact_8974_Maclaurin__sin__expansion4,axiom,
% 0.33/0.66 ! [X: real,N: nat] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ? [T3: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ T3 )
% 0.33/0.66 & ( ord_less_eq_real @ T3 @ X )
% 0.33/0.66 & ( ( sin_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_sin_expansion4
% 0.33/0.66 thf(fact_8975_card__atMost,axiom,
% 0.33/0.66 ! [U: nat] :
% 0.33/0.66 ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
% 0.33/0.66 = ( suc @ U ) ) ).
% 0.33/0.66
% 0.33/0.66 % card_atMost
% 0.33/0.66 thf(fact_8976_atMost__0,axiom,
% 0.33/0.66 ( ( set_ord_atMost_nat @ zero_zero_nat )
% 0.33/0.66 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % atMost_0
% 0.33/0.66 thf(fact_8977_sum__choose__upper,axiom,
% 0.33/0.66 ! [M: nat,N: nat] :
% 0.33/0.66 ( ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [K2: nat] : ( binomial @ K2 @ M )
% 0.33/0.66 @ ( set_ord_atMost_nat @ N ) )
% 0.33/0.66 = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_choose_upper
% 0.33/0.66 thf(fact_8978_choose__rising__sum_I2_J,axiom,
% 0.33/0.66 ! [N: nat,M: nat] :
% 0.33/0.66 ( ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 0.33/0.66 @ ( set_ord_atMost_nat @ M ) )
% 0.33/0.66 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).
% 0.33/0.66
% 0.33/0.66 % choose_rising_sum(2)
% 0.33/0.66 thf(fact_8979_choose__rising__sum_I1_J,axiom,
% 0.33/0.66 ! [N: nat,M: nat] :
% 0.33/0.66 ( ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 0.33/0.66 @ ( set_ord_atMost_nat @ M ) )
% 0.33/0.66 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % choose_rising_sum(1)
% 0.33/0.66 thf(fact_8980_sum__choose__lower,axiom,
% 0.33/0.66 ! [R: nat,N: nat] :
% 0.33/0.66 ( ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R @ K2 ) @ K2 )
% 0.33/0.66 @ ( set_ord_atMost_nat @ N ) )
% 0.33/0.66 = ( binomial @ ( suc @ ( plus_plus_nat @ R @ N ) ) @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_choose_lower
% 0.33/0.66 thf(fact_8981_atMost__atLeast0,axiom,
% 0.33/0.66 ( set_ord_atMost_nat
% 0.33/0.66 = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % atMost_atLeast0
% 0.33/0.66 thf(fact_8982_lessThan__Suc__atMost,axiom,
% 0.33/0.66 ! [K: nat] :
% 0.33/0.66 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 0.33/0.66 = ( set_ord_atMost_nat @ K ) ) ).
% 0.33/0.66
% 0.33/0.66 % lessThan_Suc_atMost
% 0.33/0.66 thf(fact_8983_sum__choose__diagonal,axiom,
% 0.33/0.66 ! [M: nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.66 => ( ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M @ K2 ) )
% 0.33/0.66 @ ( set_ord_atMost_nat @ M ) )
% 0.33/0.66 = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_choose_diagonal
% 0.33/0.66 thf(fact_8984_vandermonde,axiom,
% 0.33/0.66 ! [M: nat,N: nat,R: nat] :
% 0.33/0.66 ( ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M @ K2 ) @ ( binomial @ N @ ( minus_minus_nat @ R @ K2 ) ) )
% 0.33/0.66 @ ( set_ord_atMost_nat @ R ) )
% 0.33/0.66 = ( binomial @ ( plus_plus_nat @ M @ N ) @ R ) ) ).
% 0.33/0.66
% 0.33/0.66 % vandermonde
% 0.33/0.66 thf(fact_8985_atMost__Suc,axiom,
% 0.33/0.66 ! [K: nat] :
% 0.33/0.66 ( ( set_ord_atMost_nat @ ( suc @ K ) )
% 0.33/0.66 = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % atMost_Suc
% 0.33/0.66 thf(fact_8986_choose__row__sum,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
% 0.33/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % choose_row_sum
% 0.33/0.66 thf(fact_8987_binomial,axiom,
% 0.33/0.66 ! [A: nat,B: nat,N: nat] :
% 0.33/0.66 ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 0.33/0.66 = ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K2 ) ) )
% 0.33/0.66 @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % binomial
% 0.33/0.66 thf(fact_8988_polynomial__product__nat,axiom,
% 0.33/0.66 ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
% 0.33/0.66 ( ! [I2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ M @ I2 )
% 0.33/0.66 => ( ( A @ I2 )
% 0.33/0.66 = zero_zero_nat ) )
% 0.33/0.66 => ( ! [J2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ N @ J2 )
% 0.33/0.66 => ( ( B @ J2 )
% 0.33/0.66 = zero_zero_nat ) )
% 0.33/0.66 => ( ( times_times_nat
% 0.33/0.66 @ ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( power_power_nat @ X @ I4 ) )
% 0.33/0.66 @ ( set_ord_atMost_nat @ M ) )
% 0.33/0.66 @ ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
% 0.33/0.66 @ ( set_ord_atMost_nat @ N ) ) )
% 0.33/0.66 = ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [R5: nat] :
% 0.33/0.66 ( times_times_nat
% 0.33/0.66 @ ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 0.33/0.66 @ ( set_ord_atMost_nat @ R5 ) )
% 0.33/0.66 @ ( power_power_nat @ X @ R5 ) )
% 0.33/0.66 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % polynomial_product_nat
% 0.33/0.66 thf(fact_8989_choose__square__sum,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [K2: nat] : ( power_power_nat @ ( binomial @ N @ K2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.66 @ ( set_ord_atMost_nat @ N ) )
% 0.33/0.66 = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % choose_square_sum
% 0.33/0.66 thf(fact_8990_atMost__nat__numeral,axiom,
% 0.33/0.66 ! [K: num] :
% 0.33/0.66 ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 0.33/0.66 = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % atMost_nat_numeral
% 0.33/0.66 thf(fact_8991_binomial__r__part__sum,axiom,
% 0.33/0.66 ! [M: nat] :
% 0.33/0.66 ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 0.33/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % binomial_r_part_sum
% 0.33/0.66 thf(fact_8992_choose__linear__sum,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N @ I4 ) )
% 0.33/0.66 @ ( set_ord_atMost_nat @ N ) )
% 0.33/0.66 = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % choose_linear_sum
% 0.33/0.66 thf(fact_8993_image__Suc__atMost,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
% 0.33/0.66 = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % image_Suc_atMost
% 0.33/0.66 thf(fact_8994_atMost__Suc__eq__insert__0,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( set_ord_atMost_nat @ ( suc @ N ) )
% 0.33/0.66 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % atMost_Suc_eq_insert_0
% 0.33/0.66 thf(fact_8995_sum__nth__roots,axiom,
% 0.33/0.66 ! [N: nat,C: complex] :
% 0.33/0.66 ( ( ord_less_nat @ one_one_nat @ N )
% 0.33/0.66 => ( ( groups7754918857620584856omplex
% 0.33/0.66 @ ^ [X3: complex] : X3
% 0.33/0.66 @ ( collect_complex
% 0.33/0.66 @ ^ [Z5: complex] :
% 0.33/0.66 ( ( power_power_complex @ Z5 @ N )
% 0.33/0.66 = C ) ) )
% 0.33/0.66 = zero_zero_complex ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_nth_roots
% 0.33/0.66 thf(fact_8996_atLeast1__atMost__eq__remove0,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.33/0.66 = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % atLeast1_atMost_eq_remove0
% 0.33/0.66 thf(fact_8997_sum__roots__unity,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( ord_less_nat @ one_one_nat @ N )
% 0.33/0.66 => ( ( groups7754918857620584856omplex
% 0.33/0.66 @ ^ [X3: complex] : X3
% 0.33/0.66 @ ( collect_complex
% 0.33/0.66 @ ^ [Z5: complex] :
% 0.33/0.66 ( ( power_power_complex @ Z5 @ N )
% 0.33/0.66 = one_one_complex ) ) )
% 0.33/0.66 = zero_zero_complex ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_roots_unity
% 0.33/0.66 thf(fact_8998_mask__eq__sum__exp__nat,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 0.33/0.66 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.66 @ ( collect_nat
% 0.33/0.66 @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % mask_eq_sum_exp_nat
% 0.33/0.66 thf(fact_8999_gauss__sum__nat,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [X3: nat] : X3
% 0.33/0.66 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.33/0.66 = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % gauss_sum_nat
% 0.33/0.66 thf(fact_9000_arith__series__nat,axiom,
% 0.33/0.66 ! [A: nat,D: nat,N: nat] :
% 0.33/0.66 ( ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I4 @ D ) )
% 0.33/0.66 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.33/0.66 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % arith_series_nat
% 0.33/0.66 thf(fact_9001_Sum__Icc__nat,axiom,
% 0.33/0.66 ! [M: nat,N: nat] :
% 0.33/0.66 ( ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [X3: nat] : X3
% 0.33/0.66 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.33/0.66 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Sum_Icc_nat
% 0.33/0.66 thf(fact_9002_Sum__Icc__int,axiom,
% 0.33/0.66 ! [M: int,N: int] :
% 0.33/0.66 ( ( ord_less_eq_int @ M @ N )
% 0.33/0.66 => ( ( groups4538972089207619220nt_int
% 0.33/0.66 @ ^ [X3: int] : X3
% 0.33/0.66 @ ( set_or1266510415728281911st_int @ M @ N ) )
% 0.33/0.66 = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Sum_Icc_int
% 0.33/0.66 thf(fact_9003_set__vebt__def,axiom,
% 0.33/0.66 ( vEBT_set_vebt
% 0.33/0.66 = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % set_vebt_def
% 0.33/0.66 thf(fact_9004_rat__times__code,axiom,
% 0.33/0.66 ! [P2: rat,Q2: rat] :
% 0.33/0.66 ( ( quotient_of @ ( times_times_rat @ P2 @ Q2 ) )
% 0.33/0.66 = ( produc4245557441103728435nt_int
% 0.33/0.66 @ ^ [A2: int,C4: int] :
% 0.33/0.66 ( produc4245557441103728435nt_int
% 0.33/0.66 @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ C4 @ D4 ) ) )
% 0.33/0.66 @ ( quotient_of @ Q2 ) )
% 0.33/0.66 @ ( quotient_of @ P2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % rat_times_code
% 0.33/0.66 thf(fact_9005_rat__divide__code,axiom,
% 0.33/0.66 ! [P2: rat,Q2: rat] :
% 0.33/0.66 ( ( quotient_of @ ( divide_divide_rat @ P2 @ Q2 ) )
% 0.33/0.66 = ( produc4245557441103728435nt_int
% 0.33/0.66 @ ^ [A2: int,C4: int] :
% 0.33/0.66 ( produc4245557441103728435nt_int
% 0.33/0.66 @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A2 @ D4 ) @ ( times_times_int @ C4 @ B2 ) ) )
% 0.33/0.66 @ ( quotient_of @ Q2 ) )
% 0.33/0.66 @ ( quotient_of @ P2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % rat_divide_code
% 0.33/0.66 thf(fact_9006_divmod__nat__if,axiom,
% 0.33/0.66 ( divmod_nat
% 0.33/0.66 = ( ^ [M3: nat,N3: nat] :
% 0.33/0.66 ( if_Pro6206227464963214023at_nat
% 0.33/0.66 @ ( ( N3 = zero_zero_nat )
% 0.33/0.66 | ( ord_less_nat @ M3 @ N3 ) )
% 0.33/0.66 @ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
% 0.33/0.66 @ ( produc2626176000494625587at_nat
% 0.33/0.66 @ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
% 0.33/0.66 @ ( divmod_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % divmod_nat_if
% 0.33/0.66 thf(fact_9007_rat__plus__code,axiom,
% 0.33/0.66 ! [P2: rat,Q2: rat] :
% 0.33/0.66 ( ( quotient_of @ ( plus_plus_rat @ P2 @ Q2 ) )
% 0.33/0.66 = ( produc4245557441103728435nt_int
% 0.33/0.66 @ ^ [A2: int,C4: int] :
% 0.33/0.66 ( produc4245557441103728435nt_int
% 0.33/0.66 @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A2 @ D4 ) @ ( times_times_int @ B2 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
% 0.33/0.66 @ ( quotient_of @ Q2 ) )
% 0.33/0.66 @ ( quotient_of @ P2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % rat_plus_code
% 0.33/0.66 thf(fact_9008_rat__minus__code,axiom,
% 0.33/0.66 ! [P2: rat,Q2: rat] :
% 0.33/0.66 ( ( quotient_of @ ( minus_minus_rat @ P2 @ Q2 ) )
% 0.33/0.66 = ( produc4245557441103728435nt_int
% 0.33/0.66 @ ^ [A2: int,C4: int] :
% 0.33/0.66 ( produc4245557441103728435nt_int
% 0.33/0.66 @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A2 @ D4 ) @ ( times_times_int @ B2 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
% 0.33/0.66 @ ( quotient_of @ Q2 ) )
% 0.33/0.66 @ ( quotient_of @ P2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % rat_minus_code
% 0.33/0.66 thf(fact_9009_divmod__step__nat__def,axiom,
% 0.33/0.66 ( unique5026877609467782581ep_nat
% 0.33/0.66 = ( ^ [L2: num] :
% 0.33/0.66 ( produc2626176000494625587at_nat
% 0.33/0.66 @ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % divmod_step_nat_def
% 0.33/0.66 thf(fact_9010_rat__inverse__code,axiom,
% 0.33/0.66 ! [P2: rat] :
% 0.33/0.66 ( ( quotient_of @ ( inverse_inverse_rat @ P2 ) )
% 0.33/0.66 = ( produc4245557441103728435nt_int
% 0.33/0.66 @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( A2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A2 ) @ B2 ) @ ( abs_abs_int @ A2 ) ) )
% 0.33/0.66 @ ( quotient_of @ P2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % rat_inverse_code
% 0.33/0.66 thf(fact_9011_divmod__step__int__def,axiom,
% 0.33/0.66 ( unique5024387138958732305ep_int
% 0.33/0.66 = ( ^ [L2: num] :
% 0.33/0.66 ( produc4245557441103728435nt_int
% 0.33/0.66 @ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % divmod_step_int_def
% 0.33/0.66 thf(fact_9012_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 0.33/0.66 ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
% 0.33/0.66 ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X )
% 0.33/0.66 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.33/0.66 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.naive_member.simps(3)
% 0.33/0.66 thf(fact_9013_arctan__def,axiom,
% 0.33/0.66 ( arctan
% 0.33/0.66 = ( ^ [Y6: real] :
% 0.33/0.66 ( the_real
% 0.33/0.66 @ ^ [X3: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.33/0.66 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 & ( ( tan_real @ X3 )
% 0.33/0.66 = Y6 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % arctan_def
% 0.33/0.66 thf(fact_9014_VEBT_Oinject_I1_J,axiom,
% 0.33/0.66 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
% 0.33/0.66 ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 0.33/0.66 = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
% 0.33/0.66 = ( ( X11 = Y11 )
% 0.33/0.66 & ( X12 = Y12 )
% 0.33/0.66 & ( X13 = Y13 )
% 0.33/0.66 & ( X14 = Y14 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT.inject(1)
% 0.33/0.66 thf(fact_9015_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 0.33/0.66 ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 0.33/0.66 ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.naive_member.simps(2)
% 0.33/0.66 thf(fact_9016_pi__half,axiom,
% 0.33/0.66 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.66 = ( the_real
% 0.33/0.66 @ ^ [X3: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.33/0.66 & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.66 & ( ( cos_real @ X3 )
% 0.33/0.66 = zero_zero_real ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pi_half
% 0.33/0.66 thf(fact_9017_pi__def,axiom,
% 0.33/0.66 ( pi
% 0.33/0.66 = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.33/0.66 @ ( the_real
% 0.33/0.66 @ ^ [X3: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.33/0.66 & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.66 & ( ( cos_real @ X3 )
% 0.33/0.66 = zero_zero_real ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pi_def
% 0.33/0.66 thf(fact_9018_arcsin__def,axiom,
% 0.33/0.66 ( arcsin
% 0.33/0.66 = ( ^ [Y6: real] :
% 0.33/0.66 ( the_real
% 0.33/0.66 @ ^ [X3: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.33/0.66 & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 & ( ( sin_real @ X3 )
% 0.33/0.66 = Y6 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % arcsin_def
% 0.33/0.66 thf(fact_9019_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 0.33/0.66 ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 0.33/0.66 ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.33/0.66 = Y )
% 0.33/0.66 => ( ! [A4: $o,B4: $o] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 = ( ~ ( ( ( Xa = zero_zero_nat )
% 0.33/0.66 => A4 )
% 0.33/0.66 & ( ( Xa != zero_zero_nat )
% 0.33/0.66 => ( ( ( Xa = one_one_nat )
% 0.33/0.66 => B4 )
% 0.33/0.66 & ( Xa = one_one_nat ) ) ) ) ) ) )
% 0.33/0.66 => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.33/0.66 => Y )
% 0.33/0.66 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.33/0.66 ( ? [S3: vEBT_VEBT] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.naive_member.elims(1)
% 0.33/0.66 thf(fact_9020_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 0.33/0.66 ! [X: vEBT_VEBT,Xa: nat] :
% 0.33/0.66 ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.33/0.66 => ( ! [A4: $o,B4: $o] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.33/0.66 => ~ ( ( ( Xa = zero_zero_nat )
% 0.33/0.66 => A4 )
% 0.33/0.66 & ( ( Xa != zero_zero_nat )
% 0.33/0.66 => ( ( ( Xa = one_one_nat )
% 0.33/0.66 => B4 )
% 0.33/0.66 & ( Xa = one_one_nat ) ) ) ) )
% 0.33/0.66 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.33/0.66 ( ? [S3: vEBT_VEBT] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.33/0.66 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.naive_member.elims(2)
% 0.33/0.66 thf(fact_9021_VEBT_Oinject_I2_J,axiom,
% 0.33/0.66 ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
% 0.33/0.66 ( ( ( vEBT_Leaf @ X21 @ X222 )
% 0.33/0.66 = ( vEBT_Leaf @ Y21 @ Y22 ) )
% 0.33/0.66 = ( ( X21 = Y21 )
% 0.33/0.66 & ( X222 = Y22 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT.inject(2)
% 0.33/0.66 thf(fact_9022_VEBT_Osize_I4_J,axiom,
% 0.33/0.66 ! [X21: $o,X222: $o] :
% 0.33/0.66 ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 0.33/0.66 = zero_zero_nat ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT.size(4)
% 0.33/0.66 thf(fact_9023_VEBT_Oexhaust,axiom,
% 0.33/0.66 ! [Y: vEBT_VEBT] :
% 0.33/0.66 ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
% 0.33/0.66 ( Y
% 0.33/0.66 != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
% 0.33/0.66 => ~ ! [X212: $o,X223: $o] :
% 0.33/0.66 ( Y
% 0.33/0.66 != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT.exhaust
% 0.33/0.66 thf(fact_9024_VEBT_Odistinct_I1_J,axiom,
% 0.33/0.66 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
% 0.33/0.66 ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 0.33/0.66 != ( vEBT_Leaf @ X21 @ X222 ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT.distinct(1)
% 0.33/0.66 thf(fact_9025_VEBT__internal_Ovalid_H_Ocases,axiom,
% 0.33/0.66 ! [X: produc9072475918466114483BT_nat] :
% 0.33/0.66 ( ! [Uu2: $o,Uv2: $o,D2: nat] :
% 0.33/0.66 ( X
% 0.33/0.66 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D2 ) )
% 0.33/0.66 => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg2: nat] :
% 0.33/0.66 ( X
% 0.33/0.66 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) @ Deg2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.valid'.cases
% 0.33/0.66 thf(fact_9026_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
% 0.33/0.66 ! [Uu: $o,Uv: $o,Uw: nat] :
% 0.33/0.66 ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.membermima.simps(1)
% 0.33/0.66 thf(fact_9027_vebt__buildup_Osimps_I1_J,axiom,
% 0.33/0.66 ( ( vEBT_vebt_buildup @ zero_zero_nat )
% 0.33/0.66 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.33/0.66
% 0.33/0.66 % vebt_buildup.simps(1)
% 0.33/0.66 thf(fact_9028_VEBT__internal_Onaive__member_Ocases,axiom,
% 0.33/0.66 ! [X: produc9072475918466114483BT_nat] :
% 0.33/0.66 ( ! [A4: $o,B4: $o,X4: nat] :
% 0.33/0.66 ( X
% 0.33/0.66 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X4 ) )
% 0.33/0.66 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 0.33/0.66 ( X
% 0.33/0.66 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 0.33/0.66 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
% 0.33/0.66 ( X
% 0.33/0.66 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ X4 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.naive_member.cases
% 0.33/0.66 thf(fact_9029_vebt__buildup_Osimps_I2_J,axiom,
% 0.33/0.66 ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 0.33/0.66 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.33/0.66
% 0.33/0.66 % vebt_buildup.simps(2)
% 0.33/0.66 thf(fact_9030_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 0.33/0.66 ! [A: $o,B: $o,X: nat] :
% 0.33/0.66 ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.33/0.66 = ( ( ( X = zero_zero_nat )
% 0.33/0.66 => A )
% 0.33/0.66 & ( ( X != zero_zero_nat )
% 0.33/0.66 => ( ( ( X = one_one_nat )
% 0.33/0.66 => B )
% 0.33/0.66 & ( X = one_one_nat ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.naive_member.simps(1)
% 0.33/0.66 thf(fact_9031_prod__encode__def,axiom,
% 0.33/0.66 ( nat_prod_encode
% 0.33/0.66 = ( produc6842872674320459806at_nat
% 0.33/0.66 @ ^ [M3: nat,N3: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M3 @ N3 ) ) @ M3 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % prod_encode_def
% 0.33/0.66 thf(fact_9032_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 0.33/0.66 ! [X: vEBT_VEBT,Xa: nat] :
% 0.33/0.66 ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.33/0.66 => ( ! [A4: $o,B4: $o] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.33/0.66 => ( ( ( Xa = zero_zero_nat )
% 0.33/0.66 => A4 )
% 0.33/0.66 & ( ( Xa != zero_zero_nat )
% 0.33/0.66 => ( ( ( Xa = one_one_nat )
% 0.33/0.66 => B4 )
% 0.33/0.66 & ( Xa = one_one_nat ) ) ) ) )
% 0.33/0.66 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.33/0.66 ( X
% 0.33/0.66 != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.33/0.66 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.33/0.66 ( ? [S3: vEBT_VEBT] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.33/0.66 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.naive_member.elims(3)
% 0.33/0.66 thf(fact_9033_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 0.33/0.66 ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
% 0.33/0.66 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X )
% 0.33/0.66 = ( ( X = Mi )
% 0.33/0.66 | ( X = Ma )
% 0.33/0.66 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.33/0.66 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.membermima.simps(4)
% 0.33/0.66 thf(fact_9034_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 0.33/0.66 ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
% 0.33/0.66 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X )
% 0.33/0.66 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.33/0.66 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.membermima.simps(5)
% 0.33/0.66 thf(fact_9035_finite__interval__int1,axiom,
% 0.33/0.66 ! [A: int,B: int] :
% 0.33/0.66 ( finite_finite_int
% 0.33/0.66 @ ( collect_int
% 0.33/0.66 @ ^ [I4: int] :
% 0.33/0.66 ( ( ord_less_eq_int @ A @ I4 )
% 0.33/0.66 & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % finite_interval_int1
% 0.33/0.66 thf(fact_9036_finite__interval__int4,axiom,
% 0.33/0.66 ! [A: int,B: int] :
% 0.33/0.66 ( finite_finite_int
% 0.33/0.66 @ ( collect_int
% 0.33/0.66 @ ^ [I4: int] :
% 0.33/0.66 ( ( ord_less_int @ A @ I4 )
% 0.33/0.66 & ( ord_less_int @ I4 @ B ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % finite_interval_int4
% 0.33/0.66 thf(fact_9037_finite__interval__int2,axiom,
% 0.33/0.66 ! [A: int,B: int] :
% 0.33/0.66 ( finite_finite_int
% 0.33/0.66 @ ( collect_int
% 0.33/0.66 @ ^ [I4: int] :
% 0.33/0.66 ( ( ord_less_eq_int @ A @ I4 )
% 0.33/0.66 & ( ord_less_int @ I4 @ B ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % finite_interval_int2
% 0.33/0.66 thf(fact_9038_finite__interval__int3,axiom,
% 0.33/0.66 ! [A: int,B: int] :
% 0.33/0.66 ( finite_finite_int
% 0.33/0.66 @ ( collect_int
% 0.33/0.66 @ ^ [I4: int] :
% 0.33/0.66 ( ( ord_less_int @ A @ I4 )
% 0.33/0.66 & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % finite_interval_int3
% 0.33/0.66 thf(fact_9039_finite__nth__roots,axiom,
% 0.33/0.66 ! [N: nat,C: complex] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ( finite3207457112153483333omplex
% 0.33/0.66 @ ( collect_complex
% 0.33/0.66 @ ^ [Z5: complex] :
% 0.33/0.66 ( ( power_power_complex @ Z5 @ N )
% 0.33/0.66 = C ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % finite_nth_roots
% 0.33/0.66 thf(fact_9040_finite__set__decode,axiom,
% 0.33/0.66 ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % finite_set_decode
% 0.33/0.66 thf(fact_9041_VEBT__internal_Omembermima_Ocases,axiom,
% 0.33/0.66 ! [X: produc9072475918466114483BT_nat] :
% 0.33/0.66 ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 0.33/0.66 ( X
% 0.33/0.66 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 0.33/0.66 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz: nat] :
% 0.33/0.66 ( X
% 0.33/0.66 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz ) )
% 0.33/0.66 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X4: nat] :
% 0.33/0.66 ( X
% 0.33/0.66 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) @ X4 ) )
% 0.33/0.66 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
% 0.33/0.66 ( X
% 0.33/0.66 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ X4 ) )
% 0.33/0.66 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] :
% 0.33/0.66 ( X
% 0.33/0.66 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ X4 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.membermima.cases
% 0.33/0.66 thf(fact_9042_finite__divisors__int,axiom,
% 0.33/0.66 ! [I: int] :
% 0.33/0.66 ( ( I != zero_zero_int )
% 0.33/0.66 => ( finite_finite_int
% 0.33/0.66 @ ( collect_int
% 0.33/0.66 @ ^ [D4: int] : ( dvd_dvd_int @ D4 @ I ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % finite_divisors_int
% 0.33/0.66 thf(fact_9043_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 0.33/0.66 ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz2: nat] :
% 0.33/0.66 ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz2 ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.membermima.simps(2)
% 0.33/0.66 thf(fact_9044_finite__divisors__nat,axiom,
% 0.33/0.66 ! [M: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.66 => ( finite_finite_nat
% 0.33/0.66 @ ( collect_nat
% 0.33/0.66 @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % finite_divisors_nat
% 0.33/0.66 thf(fact_9045_subset__eq__atLeast0__atMost__finite,axiom,
% 0.33/0.66 ! [N5: set_nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_set_nat @ N5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.33/0.66 => ( finite_finite_nat @ N5 ) ) ).
% 0.33/0.66
% 0.33/0.66 % subset_eq_atLeast0_atMost_finite
% 0.33/0.66 thf(fact_9046_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 0.33/0.66 ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X: nat] :
% 0.33/0.66 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X )
% 0.33/0.66 = ( ( X = Mi )
% 0.33/0.66 | ( X = Ma ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.membermima.simps(3)
% 0.33/0.66 thf(fact_9047_Gcd__remove0__nat,axiom,
% 0.33/0.66 ! [M10: set_nat] :
% 0.33/0.66 ( ( finite_finite_nat @ M10 )
% 0.33/0.66 => ( ( gcd_Gcd_nat @ M10 )
% 0.33/0.66 = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M10 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Gcd_remove0_nat
% 0.33/0.66 thf(fact_9048_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 0.33/0.66 ! [X: vEBT_VEBT,Xa: nat] :
% 0.33/0.66 ( ( vEBT_VEBT_membermima @ X @ Xa )
% 0.33/0.66 => ( ! [Mi2: nat,Ma2: nat] :
% 0.33/0.66 ( ? [Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
% 0.33/0.66 => ~ ( ( Xa = Mi2 )
% 0.33/0.66 | ( Xa = Ma2 ) ) )
% 0.33/0.66 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.33/0.66 ( ? [Vc2: vEBT_VEBT] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 0.33/0.66 => ~ ( ( Xa = Mi2 )
% 0.33/0.66 | ( Xa = Ma2 )
% 0.33/0.66 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.33/0.66 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.33/0.66 ( ? [Vd2: vEBT_VEBT] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 0.33/0.66 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.membermima.elims(2)
% 0.33/0.66 thf(fact_9049_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 0.33/0.66 ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 0.33/0.66 ( ( ( vEBT_VEBT_membermima @ X @ Xa )
% 0.33/0.66 = Y )
% 0.33/0.66 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.33/0.66 => Y )
% 0.33/0.66 => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.33/0.66 => Y )
% 0.33/0.66 => ( ! [Mi2: nat,Ma2: nat] :
% 0.33/0.66 ( ? [Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 = ( ~ ( ( Xa = Mi2 )
% 0.33/0.66 | ( Xa = Ma2 ) ) ) ) )
% 0.33/0.66 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.33/0.66 ( ? [Vc2: vEBT_VEBT] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 = ( ~ ( ( Xa = Mi2 )
% 0.33/0.66 | ( Xa = Ma2 )
% 0.33/0.66 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
% 0.33/0.66 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.33/0.66 ( ? [Vd2: vEBT_VEBT] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.membermima.elims(1)
% 0.33/0.66 thf(fact_9050_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 0.33/0.66 ! [X: vEBT_VEBT,Xa: nat] :
% 0.33/0.66 ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
% 0.33/0.66 => ( ! [Uu2: $o,Uv2: $o] :
% 0.33/0.66 ( X
% 0.33/0.66 != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.33/0.66 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.33/0.66 ( X
% 0.33/0.66 != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.33/0.66 => ( ! [Mi2: nat,Ma2: nat] :
% 0.33/0.66 ( ? [Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
% 0.33/0.66 => ( ( Xa = Mi2 )
% 0.33/0.66 | ( Xa = Ma2 ) ) )
% 0.33/0.66 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.33/0.66 ( ? [Vc2: vEBT_VEBT] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 0.33/0.66 => ( ( Xa = Mi2 )
% 0.33/0.66 | ( Xa = Ma2 )
% 0.33/0.66 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.33/0.66 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 0.33/0.66 ( ? [Vd2: vEBT_VEBT] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 0.33/0.66 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.membermima.elims(3)
% 0.33/0.66 thf(fact_9051_invar__vebt_Osimps,axiom,
% 0.33/0.66 ( vEBT_invar_vebt
% 0.33/0.66 = ( ^ [A12: vEBT_VEBT,A23: nat] :
% 0.33/0.66 ( ( ? [A2: $o,B2: $o] :
% 0.33/0.66 ( A12
% 0.33/0.66 = ( vEBT_Leaf @ A2 @ B2 ) )
% 0.33/0.66 & ( A23
% 0.33/0.66 = ( suc @ zero_zero_nat ) ) )
% 0.33/0.66 | ? [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT] :
% 0.33/0.66 ( ( A12
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList @ Summary2 ) )
% 0.33/0.66 & ! [X3: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.66 => ( vEBT_invar_vebt @ X3 @ N3 ) )
% 0.33/0.66 & ( vEBT_invar_vebt @ Summary2 @ N3 )
% 0.33/0.66 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.33/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 0.33/0.66 & ( A23
% 0.33/0.66 = ( plus_plus_nat @ N3 @ N3 ) )
% 0.33/0.66 & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 0.33/0.66 & ! [X3: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.66 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.66 | ? [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT] :
% 0.33/0.66 ( ( A12
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList @ Summary2 ) )
% 0.33/0.66 & ! [X3: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.66 => ( vEBT_invar_vebt @ X3 @ N3 ) )
% 0.33/0.66 & ( vEBT_invar_vebt @ Summary2 @ ( suc @ N3 ) )
% 0.33/0.66 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.33/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 0.33/0.66 & ( A23
% 0.33/0.66 = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
% 0.33/0.66 & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 0.33/0.66 & ! [X3: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.66 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.66 | ? [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.33/0.66 ( ( A12
% 0.33/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary2 ) )
% 0.33/0.66 & ! [X3: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.66 => ( vEBT_invar_vebt @ X3 @ N3 ) )
% 0.33/0.66 & ( vEBT_invar_vebt @ Summary2 @ N3 )
% 0.33/0.66 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.33/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 0.33/0.66 & ( A23
% 0.33/0.66 = ( plus_plus_nat @ N3 @ N3 ) )
% 0.33/0.66 & ! [I4: nat] :
% 0.33/0.66 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 0.33/0.66 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
% 0.33/0.66 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.33/0.66 & ( ( Mi3 = Ma3 )
% 0.33/0.66 => ! [X3: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.66 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.66 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.33/0.66 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 0.33/0.66 & ( ( Mi3 != Ma3 )
% 0.33/0.66 => ! [I4: nat] :
% 0.33/0.66 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 0.33/0.66 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
% 0.33/0.66 = I4 )
% 0.33/0.66 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
% 0.33/0.66 & ! [X3: nat] :
% 0.33/0.66 ( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
% 0.33/0.66 = I4 )
% 0.33/0.66 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
% 0.33/0.66 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.33/0.66 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) )
% 0.33/0.66 | ? [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.33/0.66 ( ( A12
% 0.33/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary2 ) )
% 0.33/0.66 & ! [X3: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.66 => ( vEBT_invar_vebt @ X3 @ N3 ) )
% 0.33/0.66 & ( vEBT_invar_vebt @ Summary2 @ ( suc @ N3 ) )
% 0.33/0.66 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.33/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 0.33/0.66 & ( A23
% 0.33/0.66 = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
% 0.33/0.66 & ! [I4: nat] :
% 0.33/0.66 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 0.33/0.66 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
% 0.33/0.66 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.33/0.66 & ( ( Mi3 = Ma3 )
% 0.33/0.66 => ! [X3: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.66 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.66 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.33/0.66 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 0.33/0.66 & ( ( Mi3 != Ma3 )
% 0.33/0.66 => ! [I4: nat] :
% 0.33/0.66 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 0.33/0.66 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
% 0.33/0.66 = I4 )
% 0.33/0.66 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
% 0.33/0.66 & ! [X3: nat] :
% 0.33/0.66 ( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
% 0.33/0.66 = I4 )
% 0.33/0.66 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
% 0.33/0.66 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.33/0.66 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % invar_vebt.simps
% 0.33/0.66 thf(fact_9052_invar__vebt_Ocases,axiom,
% 0.33/0.66 ! [A1: vEBT_VEBT,A22: nat] :
% 0.33/0.66 ( ( vEBT_invar_vebt @ A1 @ A22 )
% 0.33/0.66 => ( ( ? [A4: $o,B4: $o] :
% 0.33/0.66 ( A1
% 0.33/0.66 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.33/0.66 => ( A22
% 0.33/0.66 != ( suc @ zero_zero_nat ) ) )
% 0.33/0.66 => ( ! [TreeList3: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
% 0.33/0.66 ( ( A1
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) )
% 0.33/0.66 => ( ( A22 = Deg )
% 0.33/0.66 => ( ! [X5: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_invar_vebt @ X5 @ N2 ) )
% 0.33/0.66 => ( ( vEBT_invar_vebt @ Summary @ M2 )
% 0.33/0.66 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.33/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.33/0.66 => ( ( M2 = N2 )
% 0.33/0.66 => ( ( Deg
% 0.33/0.66 = ( plus_plus_nat @ N2 @ M2 ) )
% 0.33/0.66 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.33/0.66 => ~ ! [X5: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.66 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) ) ) ) ) ) ) )
% 0.33/0.66 => ( ! [TreeList3: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
% 0.33/0.66 ( ( A1
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) )
% 0.33/0.66 => ( ( A22 = Deg )
% 0.33/0.66 => ( ! [X5: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_invar_vebt @ X5 @ N2 ) )
% 0.33/0.66 => ( ( vEBT_invar_vebt @ Summary @ M2 )
% 0.33/0.66 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.33/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.33/0.66 => ( ( M2
% 0.33/0.66 = ( suc @ N2 ) )
% 0.33/0.66 => ( ( Deg
% 0.33/0.66 = ( plus_plus_nat @ N2 @ M2 ) )
% 0.33/0.66 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.33/0.66 => ~ ! [X5: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.66 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) ) ) ) ) ) ) )
% 0.33/0.66 => ( ! [TreeList3: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi2: nat,Ma2: nat] :
% 0.33/0.66 ( ( A1
% 0.33/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList3 @ Summary ) )
% 0.33/0.66 => ( ( A22 = Deg )
% 0.33/0.66 => ( ! [X5: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_invar_vebt @ X5 @ N2 ) )
% 0.33/0.66 => ( ( vEBT_invar_vebt @ Summary @ M2 )
% 0.33/0.66 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.33/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.33/0.66 => ( ( M2 = N2 )
% 0.33/0.66 => ( ( Deg
% 0.33/0.66 = ( plus_plus_nat @ N2 @ M2 ) )
% 0.33/0.66 => ( ! [I3: nat] :
% 0.33/0.66 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.33/0.66 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X8 ) )
% 0.33/0.66 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 0.33/0.66 => ( ( ( Mi2 = Ma2 )
% 0.33/0.66 => ! [X5: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.66 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
% 0.33/0.66 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.33/0.66 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.66 => ~ ( ( Mi2 != Ma2 )
% 0.33/0.66 => ! [I3: nat] :
% 0.33/0.66 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.33/0.66 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 0.33/0.66 = I3 )
% 0.33/0.66 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 0.33/0.66 & ! [X5: nat] :
% 0.33/0.66 ( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
% 0.33/0.66 = I3 )
% 0.33/0.66 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
% 0.33/0.66 => ( ( ord_less_nat @ Mi2 @ X5 )
% 0.33/0.66 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.33/0.66 => ~ ! [TreeList3: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi2: nat,Ma2: nat] :
% 0.33/0.66 ( ( A1
% 0.33/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList3 @ Summary ) )
% 0.33/0.66 => ( ( A22 = Deg )
% 0.33/0.66 => ( ! [X5: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_invar_vebt @ X5 @ N2 ) )
% 0.33/0.66 => ( ( vEBT_invar_vebt @ Summary @ M2 )
% 0.33/0.66 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.33/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.33/0.66 => ( ( M2
% 0.33/0.66 = ( suc @ N2 ) )
% 0.33/0.66 => ( ( Deg
% 0.33/0.66 = ( plus_plus_nat @ N2 @ M2 ) )
% 0.33/0.66 => ( ! [I3: nat] :
% 0.33/0.66 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.33/0.66 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X8 ) )
% 0.33/0.66 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 0.33/0.66 => ( ( ( Mi2 = Ma2 )
% 0.33/0.66 => ! [X5: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.66 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
% 0.33/0.66 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.33/0.66 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.66 => ~ ( ( Mi2 != Ma2 )
% 0.33/0.66 => ! [I3: nat] :
% 0.33/0.66 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.33/0.66 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 0.33/0.66 = I3 )
% 0.33/0.66 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 0.33/0.66 & ! [X5: nat] :
% 0.33/0.66 ( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
% 0.33/0.66 = I3 )
% 0.33/0.66 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
% 0.33/0.66 => ( ( ord_less_nat @ Mi2 @ X5 )
% 0.33/0.66 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % invar_vebt.cases
% 0.33/0.66 thf(fact_9053_invar__vebt_Ointros_I1_J,axiom,
% 0.33/0.66 ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % invar_vebt.intros(1)
% 0.33/0.66 thf(fact_9054_invar__vebt_Ointros_I2_J,axiom,
% 0.33/0.66 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,M: nat,Deg3: nat] :
% 0.33/0.66 ( ! [X4: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.33/0.66 => ( vEBT_invar_vebt @ X4 @ N ) )
% 0.33/0.66 => ( ( vEBT_invar_vebt @ Summary3 @ M )
% 0.33/0.66 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.33/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.66 => ( ( M = N )
% 0.33/0.66 => ( ( Deg3
% 0.33/0.66 = ( plus_plus_nat @ N @ M ) )
% 0.33/0.66 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_12 )
% 0.33/0.66 => ( ! [X4: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.33/0.66 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) )
% 0.33/0.66 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg3 @ TreeList2 @ Summary3 ) @ Deg3 ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % invar_vebt.intros(2)
% 0.33/0.66 thf(fact_9055_invar__vebt_Ointros_I3_J,axiom,
% 0.33/0.66 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,M: nat,Deg3: nat] :
% 0.33/0.66 ( ! [X4: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.33/0.66 => ( vEBT_invar_vebt @ X4 @ N ) )
% 0.33/0.66 => ( ( vEBT_invar_vebt @ Summary3 @ M )
% 0.33/0.66 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.33/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.66 => ( ( M
% 0.33/0.66 = ( suc @ N ) )
% 0.33/0.66 => ( ( Deg3
% 0.33/0.66 = ( plus_plus_nat @ N @ M ) )
% 0.33/0.66 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_12 )
% 0.33/0.66 => ( ! [X4: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.33/0.66 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) )
% 0.33/0.66 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg3 @ TreeList2 @ Summary3 ) @ Deg3 ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % invar_vebt.intros(3)
% 0.33/0.66 thf(fact_9056_invar__vebt_Ointros_I4_J,axiom,
% 0.33/0.66 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,M: nat,Deg3: nat,Mi: nat,Ma: nat] :
% 0.33/0.66 ( ! [X4: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.33/0.66 => ( vEBT_invar_vebt @ X4 @ N ) )
% 0.33/0.66 => ( ( vEBT_invar_vebt @ Summary3 @ M )
% 0.33/0.66 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.33/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.66 => ( ( M = N )
% 0.33/0.66 => ( ( Deg3
% 0.33/0.66 = ( plus_plus_nat @ N @ M ) )
% 0.33/0.66 => ( ! [I2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.66 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 0.33/0.66 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
% 0.33/0.66 => ( ( ( Mi = Ma )
% 0.33/0.66 => ! [X4: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.33/0.66 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
% 0.33/0.66 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.33/0.66 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg3 ) )
% 0.33/0.66 => ( ( ( Mi != Ma )
% 0.33/0.66 => ! [I2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.66 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.33/0.66 = I2 )
% 0.33/0.66 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.33/0.66 & ! [X4: nat] :
% 0.33/0.66 ( ( ( ( vEBT_VEBT_high @ X4 @ N )
% 0.33/0.66 = I2 )
% 0.33/0.66 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
% 0.33/0.66 => ( ( ord_less_nat @ Mi @ X4 )
% 0.33/0.66 & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 0.33/0.66 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg3 @ TreeList2 @ Summary3 ) @ Deg3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % invar_vebt.intros(4)
% 0.33/0.66 thf(fact_9057_invar__vebt_Ointros_I5_J,axiom,
% 0.33/0.66 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,M: nat,Deg3: nat,Mi: nat,Ma: nat] :
% 0.33/0.66 ( ! [X4: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.33/0.66 => ( vEBT_invar_vebt @ X4 @ N ) )
% 0.33/0.66 => ( ( vEBT_invar_vebt @ Summary3 @ M )
% 0.33/0.66 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.33/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.66 => ( ( M
% 0.33/0.66 = ( suc @ N ) )
% 0.33/0.66 => ( ( Deg3
% 0.33/0.66 = ( plus_plus_nat @ N @ M ) )
% 0.33/0.66 => ( ! [I2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.66 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 0.33/0.66 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
% 0.33/0.66 => ( ( ( Mi = Ma )
% 0.33/0.66 => ! [X4: vEBT_VEBT] :
% 0.33/0.66 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.33/0.66 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
% 0.33/0.66 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.33/0.66 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg3 ) )
% 0.33/0.66 => ( ( ( Mi != Ma )
% 0.33/0.66 => ! [I2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.66 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.33/0.66 = I2 )
% 0.33/0.66 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.33/0.66 & ! [X4: nat] :
% 0.33/0.66 ( ( ( ( vEBT_VEBT_high @ X4 @ N )
% 0.33/0.66 = I2 )
% 0.33/0.66 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
% 0.33/0.66 => ( ( ord_less_nat @ Mi @ X4 )
% 0.33/0.66 & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 0.33/0.66 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg3 @ TreeList2 @ Summary3 ) @ Deg3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % invar_vebt.intros(5)
% 0.33/0.66 thf(fact_9058_set__encode__insert,axiom,
% 0.33/0.66 ! [A3: set_nat,N: nat] :
% 0.33/0.66 ( ( finite_finite_nat @ A3 )
% 0.33/0.66 => ( ~ ( member_nat @ N @ A3 )
% 0.33/0.66 => ( ( nat_set_encode @ ( insert_nat @ N @ A3 ) )
% 0.33/0.66 = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % set_encode_insert
% 0.33/0.66 thf(fact_9059_vebt__buildup_Oelims,axiom,
% 0.33/0.66 ! [X: nat,Y: vEBT_VEBT] :
% 0.33/0.66 ( ( ( vEBT_vebt_buildup @ X )
% 0.33/0.66 = Y )
% 0.33/0.66 => ( ( ( X = zero_zero_nat )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.33/0.66 => ( ( ( X
% 0.33/0.66 = ( suc @ zero_zero_nat ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.33/0.66 => ~ ! [Va: nat] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( suc @ ( suc @ Va ) ) )
% 0.33/0.66 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.33/0.66 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % vebt_buildup.elims
% 0.33/0.66 thf(fact_9060_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 0.33/0.66 ! [X: vEBT_VEBT,Xa: nat] :
% 0.33/0.66 ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
% 0.33/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.33/0.66 => ( ! [Uu2: $o,Uv2: $o] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.33/0.66 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
% 0.33/0.66 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.33/0.66 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) )
% 0.33/0.66 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
% 0.33/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) @ Xa ) )
% 0.33/0.66 => ( ( Xa = Mi2 )
% 0.33/0.66 | ( Xa = Ma2 ) ) ) )
% 0.33/0.66 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 0.33/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa ) )
% 0.33/0.66 => ( ( Xa = Mi2 )
% 0.33/0.66 | ( Xa = Ma2 )
% 0.33/0.66 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 0.33/0.66 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 0.33/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ Xa ) )
% 0.33/0.66 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.membermima.pelims(3)
% 0.33/0.66 thf(fact_9061_set__decode__inverse,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( nat_set_encode @ ( nat_set_decode @ N ) )
% 0.33/0.66 = N ) ).
% 0.33/0.66
% 0.33/0.66 % set_decode_inverse
% 0.33/0.66 thf(fact_9062_set__encode__empty,axiom,
% 0.33/0.66 ( ( nat_set_encode @ bot_bot_set_nat )
% 0.33/0.66 = zero_zero_nat ) ).
% 0.33/0.66
% 0.33/0.66 % set_encode_empty
% 0.33/0.66 thf(fact_9063_set__encode__inverse,axiom,
% 0.33/0.66 ! [A3: set_nat] :
% 0.33/0.66 ( ( finite_finite_nat @ A3 )
% 0.33/0.66 => ( ( nat_set_decode @ ( nat_set_encode @ A3 ) )
% 0.33/0.66 = A3 ) ) ).
% 0.33/0.66
% 0.33/0.66 % set_encode_inverse
% 0.33/0.66 thf(fact_9064_set__encode__eq,axiom,
% 0.33/0.66 ! [A3: set_nat,B5: set_nat] :
% 0.33/0.66 ( ( finite_finite_nat @ A3 )
% 0.33/0.66 => ( ( finite_finite_nat @ B5 )
% 0.33/0.66 => ( ( ( nat_set_encode @ A3 )
% 0.33/0.66 = ( nat_set_encode @ B5 ) )
% 0.33/0.66 = ( A3 = B5 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % set_encode_eq
% 0.33/0.66 thf(fact_9065_set__encode__inf,axiom,
% 0.33/0.66 ! [A3: set_nat] :
% 0.33/0.66 ( ~ ( finite_finite_nat @ A3 )
% 0.33/0.66 => ( ( nat_set_encode @ A3 )
% 0.33/0.66 = zero_zero_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % set_encode_inf
% 0.33/0.66 thf(fact_9066_set__encode__def,axiom,
% 0.33/0.66 ( nat_set_encode
% 0.33/0.66 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % set_encode_def
% 0.33/0.66 thf(fact_9067_even__set__encode__iff,axiom,
% 0.33/0.66 ! [A3: set_nat] :
% 0.33/0.66 ( ( finite_finite_nat @ A3 )
% 0.33/0.66 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A3 ) )
% 0.33/0.66 = ( ~ ( member_nat @ zero_zero_nat @ A3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % even_set_encode_iff
% 0.33/0.66 thf(fact_9068_vebt__buildup_Osimps_I3_J,axiom,
% 0.33/0.66 ! [Va3: nat] :
% 0.33/0.66 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 0.33/0.66 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va3 ) ) )
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.33/0.66 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 0.33/0.66 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va3 ) ) )
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % vebt_buildup.simps(3)
% 0.33/0.66 thf(fact_9069_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 0.33/0.66 ! [X: vEBT_VEBT,Xa: nat] :
% 0.33/0.66 ( ( vEBT_VEBT_membermima @ X @ Xa )
% 0.33/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.33/0.66 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
% 0.33/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) @ Xa ) )
% 0.33/0.66 => ~ ( ( Xa = Mi2 )
% 0.33/0.66 | ( Xa = Ma2 ) ) ) )
% 0.33/0.66 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 0.33/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa ) )
% 0.33/0.66 => ~ ( ( Xa = Mi2 )
% 0.33/0.66 | ( Xa = Ma2 )
% 0.33/0.66 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 0.33/0.66 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 0.33/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ Xa ) )
% 0.33/0.66 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.membermima.pelims(2)
% 0.33/0.66 thf(fact_9070_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 0.33/0.66 ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 0.33/0.66 ( ( ( vEBT_VEBT_membermima @ X @ Xa )
% 0.33/0.66 = Y )
% 0.33/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.33/0.66 => ( ! [Uu2: $o,Uv2: $o] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.33/0.66 => ( ~ Y
% 0.33/0.66 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
% 0.33/0.66 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.33/0.66 => ( ~ Y
% 0.33/0.66 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) ) )
% 0.33/0.66 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
% 0.33/0.66 => ( ( Y
% 0.33/0.66 = ( ( Xa = Mi2 )
% 0.33/0.66 | ( Xa = Ma2 ) ) )
% 0.33/0.66 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) @ Xa ) ) ) )
% 0.33/0.66 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 0.33/0.66 => ( ( Y
% 0.33/0.66 = ( ( Xa = Mi2 )
% 0.33/0.66 | ( Xa = Ma2 )
% 0.33/0.66 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.33/0.66 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa ) ) ) )
% 0.33/0.66 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 0.33/0.66 => ( ( Y
% 0.33/0.66 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 0.33/0.66 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.membermima.pelims(1)
% 0.33/0.66 thf(fact_9071_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 0.33/0.66 ! [X: vEBT_VEBT,Xa: nat] :
% 0.33/0.66 ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.33/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.33/0.66 => ( ! [A4: $o,B4: $o] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.33/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) )
% 0.33/0.66 => ( ( ( Xa = zero_zero_nat )
% 0.33/0.66 => A4 )
% 0.33/0.66 & ( ( Xa != zero_zero_nat )
% 0.33/0.66 => ( ( ( Xa = one_one_nat )
% 0.33/0.66 => B4 )
% 0.33/0.66 & ( Xa = one_one_nat ) ) ) ) ) )
% 0.33/0.66 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.33/0.66 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) )
% 0.33/0.66 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.33/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) )
% 0.33/0.66 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.naive_member.pelims(3)
% 0.33/0.66 thf(fact_9072_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 0.33/0.66 ! [X: vEBT_VEBT,Xa: nat] :
% 0.33/0.66 ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.33/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.33/0.66 => ( ! [A4: $o,B4: $o] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.33/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) )
% 0.33/0.66 => ~ ( ( ( Xa = zero_zero_nat )
% 0.33/0.66 => A4 )
% 0.33/0.66 & ( ( Xa != zero_zero_nat )
% 0.33/0.66 => ( ( ( Xa = one_one_nat )
% 0.33/0.66 => B4 )
% 0.33/0.66 & ( Xa = one_one_nat ) ) ) ) ) )
% 0.33/0.66 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.33/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) )
% 0.33/0.66 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.naive_member.pelims(2)
% 0.33/0.66 thf(fact_9073_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 0.33/0.66 ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 0.33/0.66 ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.33/0.66 = Y )
% 0.33/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.33/0.66 => ( ! [A4: $o,B4: $o] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.33/0.66 => ( ( Y
% 0.33/0.66 = ( ( ( Xa = zero_zero_nat )
% 0.33/0.66 => A4 )
% 0.33/0.66 & ( ( Xa != zero_zero_nat )
% 0.33/0.66 => ( ( ( Xa = one_one_nat )
% 0.33/0.66 => B4 )
% 0.33/0.66 & ( Xa = one_one_nat ) ) ) ) )
% 0.33/0.66 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) ) ) )
% 0.33/0.66 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.33/0.66 => ( ~ Y
% 0.33/0.66 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) ) )
% 0.33/0.66 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 0.33/0.66 => ( ( Y
% 0.33/0.66 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.33/0.66 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 0.33/0.66 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.naive_member.pelims(1)
% 0.33/0.66 thf(fact_9074_vebt__buildup_Opelims,axiom,
% 0.33/0.66 ! [X: nat,Y: vEBT_VEBT] :
% 0.33/0.66 ( ( ( vEBT_vebt_buildup @ X )
% 0.33/0.66 = Y )
% 0.33/0.66 => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
% 0.33/0.66 => ( ( ( X = zero_zero_nat )
% 0.33/0.66 => ( ( Y
% 0.33/0.66 = ( vEBT_Leaf @ $false @ $false ) )
% 0.33/0.66 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 0.33/0.66 => ( ( ( X
% 0.33/0.66 = ( suc @ zero_zero_nat ) )
% 0.33/0.66 => ( ( Y
% 0.33/0.66 = ( vEBT_Leaf @ $false @ $false ) )
% 0.33/0.66 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 0.33/0.66 => ~ ! [Va: nat] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( suc @ ( suc @ Va ) ) )
% 0.33/0.66 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.33/0.66 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.33/0.66 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % vebt_buildup.pelims
% 0.33/0.66 thf(fact_9075_valid__eq2,axiom,
% 0.33/0.66 ! [T: vEBT_VEBT,D: nat] :
% 0.33/0.66 ( ( vEBT_VEBT_valid @ T @ D )
% 0.33/0.66 => ( vEBT_invar_vebt @ T @ D ) ) ).
% 0.33/0.66
% 0.33/0.66 % valid_eq2
% 0.33/0.66 thf(fact_9076_valid__eq1,axiom,
% 0.33/0.66 ! [T: vEBT_VEBT,D: nat] :
% 0.33/0.66 ( ( vEBT_invar_vebt @ T @ D )
% 0.33/0.66 => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 0.33/0.66
% 0.33/0.66 % valid_eq1
% 0.33/0.66 thf(fact_9077_valid__eq,axiom,
% 0.33/0.66 vEBT_VEBT_valid = vEBT_invar_vebt ).
% 0.33/0.66
% 0.33/0.66 % valid_eq
% 0.33/0.66 thf(fact_9078_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 0.33/0.66 ! [Uu: $o,Uv: $o,D: nat] :
% 0.33/0.66 ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 0.33/0.66 = ( D = one_one_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT_internal.valid'.simps(1)
% 0.33/0.66 thf(fact_9079_VEBT_Osize_I3_J,axiom,
% 0.33/0.66 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.33/0.66 ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.33/0.66 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT.size(3)
% 0.33/0.66 thf(fact_9080_Sum__Ico__nat,axiom,
% 0.33/0.66 ! [M: nat,N: nat] :
% 0.33/0.66 ( ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [X3: nat] : X3
% 0.33/0.66 @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 0.33/0.66 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Sum_Ico_nat
% 0.33/0.66 thf(fact_9081_image__Suc__atLeastLessThan,axiom,
% 0.33/0.66 ! [I: nat,J: nat] :
% 0.33/0.66 ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
% 0.33/0.66 = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % image_Suc_atLeastLessThan
% 0.33/0.66 thf(fact_9082_card__atLeastLessThan,axiom,
% 0.33/0.66 ! [L: nat,U: nat] :
% 0.33/0.66 ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
% 0.33/0.66 = ( minus_minus_nat @ U @ L ) ) ).
% 0.33/0.66
% 0.33/0.66 % card_atLeastLessThan
% 0.33/0.66 thf(fact_9083_atLeastLessThan__singleton,axiom,
% 0.33/0.66 ! [M: nat] :
% 0.33/0.66 ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 0.33/0.66 = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % atLeastLessThan_singleton
% 0.33/0.66 thf(fact_9084_ex__nat__less__eq,axiom,
% 0.33/0.66 ! [N: nat,P: nat > $o] :
% 0.33/0.66 ( ( ? [M3: nat] :
% 0.33/0.66 ( ( ord_less_nat @ M3 @ N )
% 0.33/0.66 & ( P @ M3 ) ) )
% 0.33/0.66 = ( ? [X3: nat] :
% 0.33/0.66 ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.33/0.66 & ( P @ X3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % ex_nat_less_eq
% 0.33/0.66 thf(fact_9085_all__nat__less__eq,axiom,
% 0.33/0.66 ! [N: nat,P: nat > $o] :
% 0.33/0.66 ( ( ! [M3: nat] :
% 0.33/0.66 ( ( ord_less_nat @ M3 @ N )
% 0.33/0.66 => ( P @ M3 ) ) )
% 0.33/0.66 = ( ! [X3: nat] :
% 0.33/0.66 ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.33/0.66 => ( P @ X3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % all_nat_less_eq
% 0.33/0.66 thf(fact_9086_atLeastLessThanSuc__atLeastAtMost,axiom,
% 0.33/0.66 ! [L: nat,U: nat] :
% 0.33/0.66 ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
% 0.33/0.66 = ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 0.33/0.66
% 0.33/0.66 % atLeastLessThanSuc_atLeastAtMost
% 0.33/0.66 thf(fact_9087_lessThan__atLeast0,axiom,
% 0.33/0.66 ( set_ord_lessThan_nat
% 0.33/0.66 = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % lessThan_atLeast0
% 0.33/0.66 thf(fact_9088_atLeastLessThan0,axiom,
% 0.33/0.66 ! [M: nat] :
% 0.33/0.66 ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
% 0.33/0.66 = bot_bot_set_nat ) ).
% 0.33/0.66
% 0.33/0.66 % atLeastLessThan0
% 0.33/0.66 thf(fact_9089_atLeastSucLessThan__greaterThanLessThan,axiom,
% 0.33/0.66 ! [L: nat,U: nat] :
% 0.33/0.66 ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
% 0.33/0.66 = ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 0.33/0.66
% 0.33/0.66 % atLeastSucLessThan_greaterThanLessThan
% 0.33/0.66 thf(fact_9090_atLeast0__lessThan__Suc,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.33/0.66 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % atLeast0_lessThan_Suc
% 0.33/0.66 thf(fact_9091_subset__eq__atLeast0__lessThan__finite,axiom,
% 0.33/0.66 ! [N5: set_nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.33/0.66 => ( finite_finite_nat @ N5 ) ) ).
% 0.33/0.66
% 0.33/0.66 % subset_eq_atLeast0_lessThan_finite
% 0.33/0.66 thf(fact_9092_subset__card__intvl__is__intvl,axiom,
% 0.33/0.66 ! [A3: set_nat,K: nat] :
% 0.33/0.66 ( ( ord_less_eq_set_nat @ A3 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) )
% 0.33/0.66 => ( A3
% 0.33/0.66 = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % subset_card_intvl_is_intvl
% 0.33/0.66 thf(fact_9093_atLeastLessThanSuc,axiom,
% 0.33/0.66 ! [M: nat,N: nat] :
% 0.33/0.66 ( ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.66 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.33/0.66 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 0.33/0.66 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.33/0.66 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.33/0.66 = bot_bot_set_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % atLeastLessThanSuc
% 0.33/0.66 thf(fact_9094_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.33/0.66 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % atLeast0_lessThan_Suc_eq_insert_0
% 0.33/0.66 thf(fact_9095_prod__Suc__fact,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.33/0.66 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % prod_Suc_fact
% 0.33/0.66 thf(fact_9096_prod__Suc__Suc__fact,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.33/0.66 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % prod_Suc_Suc_fact
% 0.33/0.66 thf(fact_9097_subset__eq__atLeast0__lessThan__card,axiom,
% 0.33/0.66 ! [N5: set_nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.33/0.66 => ( ord_less_eq_nat @ ( finite_card_nat @ N5 ) @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % subset_eq_atLeast0_lessThan_card
% 0.33/0.66 thf(fact_9098_card__sum__le__nat__sum,axiom,
% 0.33/0.66 ! [S2: set_nat] :
% 0.33/0.66 ( ord_less_eq_nat
% 0.33/0.66 @ ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [X3: nat] : X3
% 0.33/0.66 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
% 0.33/0.66 @ ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [X3: nat] : X3
% 0.33/0.66 @ S2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % card_sum_le_nat_sum
% 0.33/0.66 thf(fact_9099_atLeastLessThan__nat__numeral,axiom,
% 0.33/0.66 ! [M: nat,K: num] :
% 0.33/0.66 ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.33/0.66 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.33/0.66 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 0.33/0.66 & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.33/0.66 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.33/0.66 = bot_bot_set_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % atLeastLessThan_nat_numeral
% 0.33/0.66 thf(fact_9100_image__minus__const__atLeastLessThan__nat,axiom,
% 0.33/0.66 ! [C: nat,Y: nat,X: nat] :
% 0.33/0.66 ( ( ( ord_less_nat @ C @ Y )
% 0.33/0.66 => ( ( image_nat_nat
% 0.33/0.66 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.33/0.66 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 0.33/0.66 = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
% 0.33/0.66 & ( ~ ( ord_less_nat @ C @ Y )
% 0.33/0.66 => ( ( ( ord_less_nat @ X @ Y )
% 0.33/0.66 => ( ( image_nat_nat
% 0.33/0.66 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.33/0.66 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 0.33/0.66 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 0.33/0.66 & ( ~ ( ord_less_nat @ X @ Y )
% 0.33/0.66 => ( ( image_nat_nat
% 0.33/0.66 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.33/0.66 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 0.33/0.66 = bot_bot_set_nat ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % image_minus_const_atLeastLessThan_nat
% 0.33/0.66 thf(fact_9101_atLeast1__lessThan__eq__remove0,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.33/0.66 = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % atLeast1_lessThan_eq_remove0
% 0.33/0.66 thf(fact_9102_sum__power2,axiom,
% 0.33/0.66 ! [K: nat] :
% 0.33/0.66 ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 0.33/0.66 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_power2
% 0.33/0.66 thf(fact_9103_Chebyshev__sum__upper__nat,axiom,
% 0.33/0.66 ! [N: nat,A: nat > nat,B: nat > nat] :
% 0.33/0.66 ( ! [I2: nat,J2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ I2 @ J2 )
% 0.33/0.66 => ( ( ord_less_nat @ J2 @ N )
% 0.33/0.66 => ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J2 ) ) ) )
% 0.33/0.66 => ( ! [I2: nat,J2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ I2 @ J2 )
% 0.33/0.66 => ( ( ord_less_nat @ J2 @ N )
% 0.33/0.66 => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I2 ) ) ) )
% 0.33/0.66 => ( ord_less_eq_nat
% 0.33/0.66 @ ( times_times_nat @ N
% 0.33/0.66 @ ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( B @ I4 ) )
% 0.33/0.66 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 0.33/0.66 @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Chebyshev_sum_upper_nat
% 0.33/0.66 thf(fact_9104_VEBT_Osize__gen_I1_J,axiom,
% 0.33/0.66 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.33/0.66 ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.33/0.66 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT.size_gen(1)
% 0.33/0.66 thf(fact_9105_card__atLeastLessThan__int,axiom,
% 0.33/0.66 ! [L: int,U: int] :
% 0.33/0.66 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
% 0.33/0.66 = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % card_atLeastLessThan_int
% 0.33/0.66 thf(fact_9106_finite__atLeastZeroLessThan__int,axiom,
% 0.33/0.66 ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% 0.33/0.66
% 0.33/0.66 % finite_atLeastZeroLessThan_int
% 0.33/0.66 thf(fact_9107_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 0.33/0.66 ! [L: int,U: int] :
% 0.33/0.66 ( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
% 0.33/0.66 = ( set_or1266510415728281911st_int @ L @ U ) ) ).
% 0.33/0.66
% 0.33/0.66 % atLeastLessThanPlusOne_atLeastAtMost_int
% 0.33/0.66 thf(fact_9108_image__int__atLeastLessThan,axiom,
% 0.33/0.66 ! [A: nat,B: nat] :
% 0.33/0.66 ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
% 0.33/0.66 = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % image_int_atLeastLessThan
% 0.33/0.66 thf(fact_9109_card__atLeastZeroLessThan__int,axiom,
% 0.33/0.66 ! [U: int] :
% 0.33/0.66 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
% 0.33/0.66 = ( nat2 @ U ) ) ).
% 0.33/0.66
% 0.33/0.66 % card_atLeastZeroLessThan_int
% 0.33/0.66 thf(fact_9110_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 0.33/0.66 ! [L: int,U: int] :
% 0.33/0.66 ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
% 0.33/0.66 = ( set_or5832277885323065728an_int @ L @ U ) ) ).
% 0.33/0.66
% 0.33/0.66 % atLeastPlusOneLessThan_greaterThanLessThan_int
% 0.33/0.66 thf(fact_9111_image__add__int__atLeastLessThan,axiom,
% 0.33/0.66 ! [L: int,U: int] :
% 0.33/0.66 ( ( image_int_int
% 0.33/0.66 @ ^ [X3: int] : ( plus_plus_int @ X3 @ L )
% 0.33/0.66 @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
% 0.33/0.66 = ( set_or4662586982721622107an_int @ L @ U ) ) ).
% 0.33/0.66
% 0.33/0.66 % image_add_int_atLeastLessThan
% 0.33/0.66 thf(fact_9112_VEBT_Osize__gen_I2_J,axiom,
% 0.33/0.66 ! [X21: $o,X222: $o] :
% 0.33/0.66 ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 0.33/0.66 = zero_zero_nat ) ).
% 0.33/0.66
% 0.33/0.66 % VEBT.size_gen(2)
% 0.33/0.66 thf(fact_9113_image__atLeastZeroLessThan__int,axiom,
% 0.33/0.66 ! [U: int] :
% 0.33/0.66 ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.33/0.66 => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
% 0.33/0.66 = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % image_atLeastZeroLessThan_int
% 0.33/0.66 thf(fact_9114_int__of__nat__def,axiom,
% 0.33/0.66 code_T6385005292777649522of_nat = semiri1314217659103216013at_int ).
% 0.33/0.66
% 0.33/0.66 % int_of_nat_def
% 0.33/0.66 thf(fact_9115_Code__Target__Int_Opositive__def,axiom,
% 0.33/0.66 code_Target_positive = numeral_numeral_int ).
% 0.33/0.66
% 0.33/0.66 % Code_Target_Int.positive_def
% 0.33/0.66 thf(fact_9116_divmod__step__integer__def,axiom,
% 0.33/0.66 ( unique4921790084139445826nteger
% 0.33/0.66 = ( ^ [L2: num] :
% 0.33/0.66 ( produc6916734918728496179nteger
% 0.33/0.66 @ ^ [Q5: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % divmod_step_integer_def
% 0.33/0.66 thf(fact_9117_minus__integer__code_I1_J,axiom,
% 0.33/0.66 ! [K: code_integer] :
% 0.33/0.66 ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
% 0.33/0.66 = K ) ).
% 0.33/0.66
% 0.33/0.66 % minus_integer_code(1)
% 0.33/0.66 thf(fact_9118_times__integer__code_I2_J,axiom,
% 0.33/0.66 ! [L: code_integer] :
% 0.33/0.66 ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
% 0.33/0.66 = zero_z3403309356797280102nteger ) ).
% 0.33/0.66
% 0.33/0.66 % times_integer_code(2)
% 0.33/0.66 thf(fact_9119_times__integer__code_I1_J,axiom,
% 0.33/0.66 ! [K: code_integer] :
% 0.33/0.66 ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
% 0.33/0.66 = zero_z3403309356797280102nteger ) ).
% 0.33/0.66
% 0.33/0.66 % times_integer_code(1)
% 0.33/0.66 thf(fact_9120_plus__integer__code_I2_J,axiom,
% 0.33/0.66 ! [L: code_integer] :
% 0.33/0.66 ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
% 0.33/0.66 = L ) ).
% 0.33/0.66
% 0.33/0.66 % plus_integer_code(2)
% 0.33/0.66 thf(fact_9121_plus__integer__code_I1_J,axiom,
% 0.33/0.66 ! [K: code_integer] :
% 0.33/0.66 ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
% 0.33/0.66 = K ) ).
% 0.33/0.66
% 0.33/0.66 % plus_integer_code(1)
% 0.33/0.66 thf(fact_9122_sgn__integer__code,axiom,
% 0.33/0.66 ( sgn_sgn_Code_integer
% 0.33/0.66 = ( ^ [K2: code_integer] : ( if_Code_integer @ ( K2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sgn_integer_code
% 0.33/0.66 thf(fact_9123_minus__integer__code_I2_J,axiom,
% 0.33/0.66 ! [L: code_integer] :
% 0.33/0.66 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
% 0.33/0.66 = ( uminus1351360451143612070nteger @ L ) ) ).
% 0.33/0.66
% 0.33/0.66 % minus_integer_code(2)
% 0.33/0.66 thf(fact_9124_nat_Odisc__eq__case_I2_J,axiom,
% 0.33/0.66 ! [Nat: nat] :
% 0.33/0.66 ( ( Nat != zero_zero_nat )
% 0.33/0.66 = ( case_nat_o @ $false
% 0.33/0.66 @ ^ [Uu3: nat] : $true
% 0.33/0.66 @ Nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % nat.disc_eq_case(2)
% 0.33/0.66 thf(fact_9125_nat_Odisc__eq__case_I1_J,axiom,
% 0.33/0.66 ! [Nat: nat] :
% 0.33/0.66 ( ( Nat = zero_zero_nat )
% 0.33/0.66 = ( case_nat_o @ $true
% 0.33/0.66 @ ^ [Uu3: nat] : $false
% 0.33/0.66 @ Nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % nat.disc_eq_case(1)
% 0.33/0.66 thf(fact_9126_zero__natural_Orsp,axiom,
% 0.33/0.66 zero_zero_nat = zero_zero_nat ).
% 0.33/0.66
% 0.33/0.66 % zero_natural.rsp
% 0.33/0.66 thf(fact_9127_zero__integer_Orsp,axiom,
% 0.33/0.66 zero_zero_int = zero_zero_int ).
% 0.33/0.66
% 0.33/0.66 % zero_integer.rsp
% 0.33/0.66 thf(fact_9128_one__integer_Orsp,axiom,
% 0.33/0.66 one_one_int = one_one_int ).
% 0.33/0.66
% 0.33/0.66 % one_integer.rsp
% 0.33/0.66 thf(fact_9129_one__natural_Orsp,axiom,
% 0.33/0.66 one_one_nat = one_one_nat ).
% 0.33/0.66
% 0.33/0.66 % one_natural.rsp
% 0.33/0.66 thf(fact_9130_less__eq__nat_Osimps_I2_J,axiom,
% 0.33/0.66 ! [M: nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.33/0.66 = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % less_eq_nat.simps(2)
% 0.33/0.66 thf(fact_9131_max__Suc2,axiom,
% 0.33/0.66 ! [M: nat,N: nat] :
% 0.33/0.66 ( ( ord_max_nat @ M @ ( suc @ N ) )
% 0.33/0.66 = ( case_nat_nat @ ( suc @ N )
% 0.33/0.66 @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ M5 @ N ) )
% 0.33/0.66 @ M ) ) ).
% 0.33/0.66
% 0.33/0.66 % max_Suc2
% 0.33/0.66 thf(fact_9132_max__Suc1,axiom,
% 0.33/0.66 ! [N: nat,M: nat] :
% 0.33/0.66 ( ( ord_max_nat @ ( suc @ N ) @ M )
% 0.33/0.66 = ( case_nat_nat @ ( suc @ N )
% 0.33/0.66 @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ N @ M5 ) )
% 0.33/0.66 @ M ) ) ).
% 0.33/0.66
% 0.33/0.66 % max_Suc1
% 0.33/0.66 thf(fact_9133_diff__Suc,axiom,
% 0.33/0.66 ! [M: nat,N: nat] :
% 0.33/0.66 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.33/0.66 = ( case_nat_nat @ zero_zero_nat
% 0.33/0.66 @ ^ [K2: nat] : K2
% 0.33/0.66 @ ( minus_minus_nat @ M @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % diff_Suc
% 0.33/0.66 thf(fact_9134_min__Suc2,axiom,
% 0.33/0.66 ! [M: nat,N: nat] :
% 0.33/0.66 ( ( ord_min_nat @ M @ ( suc @ N ) )
% 0.33/0.66 = ( case_nat_nat @ zero_zero_nat
% 0.33/0.66 @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ M5 @ N ) )
% 0.33/0.66 @ M ) ) ).
% 0.33/0.66
% 0.33/0.66 % min_Suc2
% 0.33/0.66 thf(fact_9135_min__Suc1,axiom,
% 0.33/0.66 ! [N: nat,M: nat] :
% 0.33/0.66 ( ( ord_min_nat @ ( suc @ N ) @ M )
% 0.33/0.66 = ( case_nat_nat @ zero_zero_nat
% 0.33/0.66 @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ N @ M5 ) )
% 0.33/0.66 @ M ) ) ).
% 0.33/0.66
% 0.33/0.66 % min_Suc1
% 0.33/0.66 thf(fact_9136_integer__of__int__code,axiom,
% 0.33/0.66 ( code_integer_of_int
% 0.33/0.66 = ( ^ [K2: int] :
% 0.33/0.66 ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K2 ) ) )
% 0.33/0.66 @ ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.33/0.66 @ ( if_Code_integer
% 0.33/0.66 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.66 = zero_zero_int )
% 0.33/0.66 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.33/0.66 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % integer_of_int_code
% 0.33/0.66 thf(fact_9137_integer__of__num_I3_J,axiom,
% 0.33/0.66 ! [N: num] :
% 0.33/0.66 ( ( code_integer_of_num @ ( bit1 @ N ) )
% 0.33/0.66 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% 0.33/0.66
% 0.33/0.66 % integer_of_num(3)
% 0.33/0.66 thf(fact_9138_push__bit__integer_Oabs__eq,axiom,
% 0.33/0.66 ! [Xa: nat,X: int] :
% 0.33/0.66 ( ( bit_se7788150548672797655nteger @ Xa @ ( code_integer_of_int @ X ) )
% 0.33/0.66 = ( code_integer_of_int @ ( bit_se545348938243370406it_int @ Xa @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % push_bit_integer.abs_eq
% 0.33/0.66 thf(fact_9139_zero__integer__def,axiom,
% 0.33/0.66 ( zero_z3403309356797280102nteger
% 0.33/0.66 = ( code_integer_of_int @ zero_zero_int ) ) ).
% 0.33/0.66
% 0.33/0.66 % zero_integer_def
% 0.33/0.66 thf(fact_9140_divide__integer_Oabs__eq,axiom,
% 0.33/0.66 ! [Xa: int,X: int] :
% 0.33/0.66 ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.33/0.66 = ( code_integer_of_int @ ( divide_divide_int @ Xa @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % divide_integer.abs_eq
% 0.33/0.66 thf(fact_9141_gcd__code__integer,axiom,
% 0.33/0.66 ( gcd_gcd_Code_integer
% 0.33/0.66 = ( ^ [K2: code_integer,L2: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L2 = zero_z3403309356797280102nteger ) @ K2 @ ( gcd_gcd_Code_integer @ L2 @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % gcd_code_integer
% 0.33/0.66 thf(fact_9142_gcd__integer_Oabs__eq,axiom,
% 0.33/0.66 ! [Xa: int,X: int] :
% 0.33/0.66 ( ( gcd_gcd_Code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.33/0.66 = ( code_integer_of_int @ ( gcd_gcd_int @ Xa @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % gcd_integer.abs_eq
% 0.33/0.66 thf(fact_9143_plus__integer_Oabs__eq,axiom,
% 0.33/0.66 ! [Xa: int,X: int] :
% 0.33/0.66 ( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.33/0.66 = ( code_integer_of_int @ ( plus_plus_int @ Xa @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % plus_integer.abs_eq
% 0.33/0.66 thf(fact_9144_times__integer_Oabs__eq,axiom,
% 0.33/0.66 ! [Xa: int,X: int] :
% 0.33/0.66 ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.33/0.66 = ( code_integer_of_int @ ( times_times_int @ Xa @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % times_integer.abs_eq
% 0.33/0.66 thf(fact_9145_one__integer__def,axiom,
% 0.33/0.66 ( one_one_Code_integer
% 0.33/0.66 = ( code_integer_of_int @ one_one_int ) ) ).
% 0.33/0.66
% 0.33/0.66 % one_integer_def
% 0.33/0.66 thf(fact_9146_minus__integer_Oabs__eq,axiom,
% 0.33/0.66 ! [Xa: int,X: int] :
% 0.33/0.66 ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.33/0.66 = ( code_integer_of_int @ ( minus_minus_int @ Xa @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % minus_integer.abs_eq
% 0.33/0.66 thf(fact_9147_floor__rat__def,axiom,
% 0.33/0.66 ( archim3151403230148437115or_rat
% 0.33/0.66 = ( ^ [X3: rat] :
% 0.33/0.66 ( the_int
% 0.33/0.66 @ ^ [Z5: int] :
% 0.33/0.66 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X3 )
% 0.33/0.66 & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % floor_rat_def
% 0.33/0.66 thf(fact_9148_integer__of__num__triv_I1_J,axiom,
% 0.33/0.66 ( ( code_integer_of_num @ one )
% 0.33/0.66 = one_one_Code_integer ) ).
% 0.33/0.66
% 0.33/0.66 % integer_of_num_triv(1)
% 0.33/0.66 thf(fact_9149_integer__of__num_I2_J,axiom,
% 0.33/0.66 ! [N: num] :
% 0.33/0.66 ( ( code_integer_of_num @ ( bit0 @ N ) )
% 0.33/0.66 = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % integer_of_num(2)
% 0.33/0.66 thf(fact_9150_integer__of__num__triv_I2_J,axiom,
% 0.33/0.66 ( ( code_integer_of_num @ ( bit0 @ one ) )
% 0.33/0.66 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % integer_of_num_triv(2)
% 0.33/0.66 thf(fact_9151_floor__real__def,axiom,
% 0.33/0.66 ( archim6058952711729229775r_real
% 0.33/0.66 = ( ^ [X3: real] :
% 0.33/0.66 ( the_int
% 0.33/0.66 @ ^ [Z5: int] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X3 )
% 0.33/0.66 & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % floor_real_def
% 0.33/0.66 thf(fact_9152_bit__cut__integer__code,axiom,
% 0.33/0.66 ( code_bit_cut_integer
% 0.33/0.66 = ( ^ [K2: code_integer] :
% 0.33/0.66 ( if_Pro5737122678794959658eger_o @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 0.33/0.66 @ ( produc9125791028180074456eger_o
% 0.33/0.66 @ ^ [R5: code_integer,S4: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S4 ) ) @ ( S4 = one_one_Code_integer ) )
% 0.33/0.66 @ ( code_divmod_abs @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % bit_cut_integer_code
% 0.33/0.66 thf(fact_9153_bit__cut__integer__def,axiom,
% 0.33/0.66 ( code_bit_cut_integer
% 0.33/0.66 = ( ^ [K2: code_integer] :
% 0.33/0.66 ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.66 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K2 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % bit_cut_integer_def
% 0.33/0.66 thf(fact_9154_pred__def,axiom,
% 0.33/0.66 ( pred
% 0.33/0.66 = ( case_nat_nat @ zero_zero_nat
% 0.33/0.66 @ ^ [X23: nat] : X23 ) ) ).
% 0.33/0.66
% 0.33/0.66 % pred_def
% 0.33/0.66 thf(fact_9155_divmod__integer__code,axiom,
% 0.33/0.66 ( code_divmod_integer
% 0.33/0.66 = ( ^ [K2: code_integer,L2: code_integer] :
% 0.33/0.66 ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.33/0.66 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
% 0.33/0.66 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ ( code_divmod_abs @ K2 @ L2 )
% 0.33/0.66 @ ( produc6916734918728496179nteger
% 0.33/0.66 @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S4 ) ) )
% 0.33/0.66 @ ( code_divmod_abs @ K2 @ L2 ) ) )
% 0.33/0.66 @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 0.33/0.66 @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 0.33/0.66 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K2 @ L2 )
% 0.33/0.66 @ ( produc6916734918728496179nteger
% 0.33/0.66 @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S4 ) ) )
% 0.33/0.66 @ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % divmod_integer_code
% 0.33/0.66 thf(fact_9156_Gcd__int__def,axiom,
% 0.33/0.66 ( gcd_Gcd_int
% 0.33/0.66 = ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Gcd_int_def
% 0.33/0.66 thf(fact_9157_divmod__integer__eq__cases,axiom,
% 0.33/0.66 ( code_divmod_integer
% 0.33/0.66 = ( ^ [K2: code_integer,L2: code_integer] :
% 0.33/0.66 ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.33/0.66 @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 0.33/0.66 @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
% 0.33/0.66 @ ( if_Pro6119634080678213985nteger
% 0.33/0.66 @ ( ( sgn_sgn_Code_integer @ K2 )
% 0.33/0.66 = ( sgn_sgn_Code_integer @ L2 ) )
% 0.33/0.66 @ ( code_divmod_abs @ K2 @ L2 )
% 0.33/0.66 @ ( produc6916734918728496179nteger
% 0.33/0.66 @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S4 ) ) )
% 0.33/0.66 @ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % divmod_integer_eq_cases
% 0.33/0.66 thf(fact_9158_nat__of__integer__non__positive,axiom,
% 0.33/0.66 ! [K: code_integer] :
% 0.33/0.66 ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 0.33/0.66 => ( ( code_nat_of_integer @ K )
% 0.33/0.66 = zero_zero_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % nat_of_integer_non_positive
% 0.33/0.66 thf(fact_9159_Suc__funpow,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( compow_nat_nat @ N @ suc )
% 0.33/0.66 = ( plus_plus_nat @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % Suc_funpow
% 0.33/0.66 thf(fact_9160_card_Ocomp__fun__commute__on,axiom,
% 0.33/0.66 ( ( comp_nat_nat_nat @ suc @ suc )
% 0.33/0.66 = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 0.33/0.66
% 0.33/0.66 % card.comp_fun_commute_on
% 0.33/0.66 thf(fact_9161_nat__of__integer__code__post_I1_J,axiom,
% 0.33/0.66 ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
% 0.33/0.66 = zero_zero_nat ) ).
% 0.33/0.66
% 0.33/0.66 % nat_of_integer_code_post(1)
% 0.33/0.66 thf(fact_9162_nat__of__integer__code__post_I3_J,axiom,
% 0.33/0.66 ! [K: num] :
% 0.33/0.66 ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 0.33/0.66 = ( numeral_numeral_nat @ K ) ) ).
% 0.33/0.66
% 0.33/0.66 % nat_of_integer_code_post(3)
% 0.33/0.66 thf(fact_9163_nat__of__integer__code__post_I2_J,axiom,
% 0.33/0.66 ( ( code_nat_of_integer @ one_one_Code_integer )
% 0.33/0.66 = one_one_nat ) ).
% 0.33/0.66
% 0.33/0.66 % nat_of_integer_code_post(2)
% 0.33/0.66 thf(fact_9164_Code__Target__Int_Onegative__def,axiom,
% 0.33/0.66 ( code_Target_negative
% 0.33/0.66 = ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).
% 0.33/0.66
% 0.33/0.66 % Code_Target_Int.negative_def
% 0.33/0.66 thf(fact_9165_nat__of__integer__code,axiom,
% 0.33/0.66 ( code_nat_of_integer
% 0.33/0.66 = ( ^ [K2: code_integer] :
% 0.33/0.66 ( if_nat @ ( ord_le3102999989581377725nteger @ K2 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 0.33/0.66 @ ( produc1555791787009142072er_nat
% 0.33/0.66 @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
% 0.33/0.66 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % nat_of_integer_code
% 0.33/0.66 thf(fact_9166_measure__function__int,axiom,
% 0.33/0.66 fun_is_measure_int @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) ).
% 0.33/0.66
% 0.33/0.66 % measure_function_int
% 0.33/0.66 thf(fact_9167_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.33/0.66 ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 0.33/0.66 @ ^ [X3: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X3 )
% 0.33/0.66 @ ^ [X3: nat,Y6: nat] : ( ord_less_nat @ Y6 @ X3 ) ) ).
% 0.33/0.66
% 0.33/0.66 % max_nat.semilattice_neutr_order_axioms
% 0.33/0.66 thf(fact_9168_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.33/0.66 ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
% 0.33/0.66 @ ^ [M3: nat,N3: nat] :
% 0.33/0.66 ( ( dvd_dvd_nat @ M3 @ N3 )
% 0.33/0.66 & ( M3 != N3 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % gcd_nat.semilattice_neutr_order_axioms
% 0.33/0.66 thf(fact_9169_int__of__integer__code,axiom,
% 0.33/0.66 ( code_int_of_integer
% 0.33/0.66 = ( ^ [K2: code_integer] :
% 0.33/0.66 ( if_int @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K2 ) ) )
% 0.33/0.66 @ ( if_int @ ( K2 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 0.33/0.66 @ ( produc1553301316500091796er_int
% 0.33/0.66 @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
% 0.33/0.66 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % int_of_integer_code
% 0.33/0.66 thf(fact_9170_times__int_Oabs__eq,axiom,
% 0.33/0.66 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.33/0.66 ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.33/0.66 = ( abs_Integ
% 0.33/0.66 @ ( produc27273713700761075at_nat
% 0.33/0.66 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.66 ( produc2626176000494625587at_nat
% 0.33/0.66 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) )
% 0.33/0.66 @ Xa
% 0.33/0.66 @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % times_int.abs_eq
% 0.33/0.66 thf(fact_9171_int__of__integer__of__nat,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( code_int_of_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.33/0.66 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % int_of_integer_of_nat
% 0.33/0.66 thf(fact_9172_zero__integer_Orep__eq,axiom,
% 0.33/0.66 ( ( code_int_of_integer @ zero_z3403309356797280102nteger )
% 0.33/0.66 = zero_zero_int ) ).
% 0.33/0.66
% 0.33/0.66 % zero_integer.rep_eq
% 0.33/0.66 thf(fact_9173_int__of__integer__numeral,axiom,
% 0.33/0.66 ! [K: num] :
% 0.33/0.66 ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 0.33/0.66 = ( numeral_numeral_int @ K ) ) ).
% 0.33/0.66
% 0.33/0.66 % int_of_integer_numeral
% 0.33/0.66 thf(fact_9174_plus__integer_Orep__eq,axiom,
% 0.33/0.66 ! [X: code_integer,Xa: code_integer] :
% 0.33/0.66 ( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X @ Xa ) )
% 0.33/0.66 = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % plus_integer.rep_eq
% 0.33/0.66 thf(fact_9175_times__integer_Orep__eq,axiom,
% 0.33/0.66 ! [X: code_integer,Xa: code_integer] :
% 0.33/0.66 ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa ) )
% 0.33/0.66 = ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % times_integer.rep_eq
% 0.33/0.66 thf(fact_9176_one__integer_Orep__eq,axiom,
% 0.33/0.66 ( ( code_int_of_integer @ one_one_Code_integer )
% 0.33/0.66 = one_one_int ) ).
% 0.33/0.66
% 0.33/0.66 % one_integer.rep_eq
% 0.33/0.66 thf(fact_9177_minus__integer_Orep__eq,axiom,
% 0.33/0.66 ! [X: code_integer,Xa: code_integer] :
% 0.33/0.66 ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X @ Xa ) )
% 0.33/0.66 = ( minus_minus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % minus_integer.rep_eq
% 0.33/0.66 thf(fact_9178_divide__integer_Orep__eq,axiom,
% 0.33/0.66 ! [X: code_integer,Xa: code_integer] :
% 0.33/0.66 ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa ) )
% 0.33/0.66 = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % divide_integer.rep_eq
% 0.33/0.66 thf(fact_9179_eq__Abs__Integ,axiom,
% 0.33/0.66 ! [Z2: int] :
% 0.33/0.66 ~ ! [X4: nat,Y4: nat] :
% 0.33/0.66 ( Z2
% 0.33/0.66 != ( abs_Integ @ ( product_Pair_nat_nat @ X4 @ Y4 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % eq_Abs_Integ
% 0.33/0.66 thf(fact_9180_int_Oabs__induct,axiom,
% 0.33/0.66 ! [P: int > $o,X: int] :
% 0.33/0.66 ( ! [Y4: product_prod_nat_nat] : ( P @ ( abs_Integ @ Y4 ) )
% 0.33/0.66 => ( P @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % int.abs_induct
% 0.33/0.66 thf(fact_9181_gcd__integer_Orep__eq,axiom,
% 0.33/0.66 ! [X: code_integer,Xa: code_integer] :
% 0.33/0.66 ( ( code_int_of_integer @ ( gcd_gcd_Code_integer @ X @ Xa ) )
% 0.33/0.66 = ( gcd_gcd_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % gcd_integer.rep_eq
% 0.33/0.66 thf(fact_9182_push__bit__integer_Orep__eq,axiom,
% 0.33/0.66 ! [X: nat,Xa: code_integer] :
% 0.33/0.66 ( ( code_int_of_integer @ ( bit_se7788150548672797655nteger @ X @ Xa ) )
% 0.33/0.66 = ( bit_se545348938243370406it_int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % push_bit_integer.rep_eq
% 0.33/0.66 thf(fact_9183_zero__int__def,axiom,
% 0.33/0.66 ( zero_zero_int
% 0.33/0.66 = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % zero_int_def
% 0.33/0.66 thf(fact_9184_int__def,axiom,
% 0.33/0.66 ( semiri1314217659103216013at_int
% 0.33/0.66 = ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N3 @ zero_zero_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % int_def
% 0.33/0.66 thf(fact_9185_nat_Oabs__eq,axiom,
% 0.33/0.66 ! [X: product_prod_nat_nat] :
% 0.33/0.66 ( ( nat2 @ ( abs_Integ @ X ) )
% 0.33/0.66 = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % nat.abs_eq
% 0.33/0.66 thf(fact_9186_uminus__int_Oabs__eq,axiom,
% 0.33/0.66 ! [X: product_prod_nat_nat] :
% 0.33/0.66 ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
% 0.33/0.66 = ( abs_Integ
% 0.33/0.66 @ ( produc2626176000494625587at_nat
% 0.33/0.66 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 )
% 0.33/0.66 @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % uminus_int.abs_eq
% 0.33/0.66 thf(fact_9187_one__int__def,axiom,
% 0.33/0.66 ( one_one_int
% 0.33/0.66 = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % one_int_def
% 0.33/0.66 thf(fact_9188_less__int_Oabs__eq,axiom,
% 0.33/0.66 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.33/0.66 ( ( ord_less_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.33/0.66 = ( produc8739625826339149834_nat_o
% 0.33/0.66 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.66 ( produc6081775807080527818_nat_o
% 0.33/0.66 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) )
% 0.33/0.66 @ Xa
% 0.33/0.66 @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % less_int.abs_eq
% 0.33/0.66 thf(fact_9189_less__eq__int_Oabs__eq,axiom,
% 0.33/0.66 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.33/0.66 ( ( ord_less_eq_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.33/0.66 = ( produc8739625826339149834_nat_o
% 0.33/0.66 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.66 ( produc6081775807080527818_nat_o
% 0.33/0.66 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) )
% 0.33/0.66 @ Xa
% 0.33/0.66 @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % less_eq_int.abs_eq
% 0.33/0.66 thf(fact_9190_plus__int_Oabs__eq,axiom,
% 0.33/0.66 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.33/0.66 ( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.33/0.66 = ( abs_Integ
% 0.33/0.66 @ ( produc27273713700761075at_nat
% 0.33/0.66 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.66 ( produc2626176000494625587at_nat
% 0.33/0.66 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) )
% 0.33/0.66 @ Xa
% 0.33/0.66 @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % plus_int.abs_eq
% 0.33/0.66 thf(fact_9191_minus__int_Oabs__eq,axiom,
% 0.33/0.66 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.33/0.66 ( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.33/0.66 = ( abs_Integ
% 0.33/0.66 @ ( produc27273713700761075at_nat
% 0.33/0.66 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.66 ( produc2626176000494625587at_nat
% 0.33/0.66 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) )
% 0.33/0.66 @ Xa
% 0.33/0.66 @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % minus_int.abs_eq
% 0.33/0.66 thf(fact_9192_rat__floor__lemma,axiom,
% 0.33/0.66 ! [A: int,B: int] :
% 0.33/0.66 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
% 0.33/0.66 & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % rat_floor_lemma
% 0.33/0.66 thf(fact_9193_mult__rat,axiom,
% 0.33/0.66 ! [A: int,B: int,C: int,D: int] :
% 0.33/0.66 ( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.33/0.66 = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % mult_rat
% 0.33/0.66 thf(fact_9194_divide__rat,axiom,
% 0.33/0.66 ! [A: int,B: int,C: int,D: int] :
% 0.33/0.66 ( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.33/0.66 = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % divide_rat
% 0.33/0.66 thf(fact_9195_floor__Fract,axiom,
% 0.33/0.66 ! [A: int,B: int] :
% 0.33/0.66 ( ( archim3151403230148437115or_rat @ ( fract @ A @ B ) )
% 0.33/0.66 = ( divide_divide_int @ A @ B ) ) ).
% 0.33/0.66
% 0.33/0.66 % floor_Fract
% 0.33/0.66 thf(fact_9196_less__rat,axiom,
% 0.33/0.66 ! [B: int,D: int,A: int,C: int] :
% 0.33/0.66 ( ( B != zero_zero_int )
% 0.33/0.66 => ( ( D != zero_zero_int )
% 0.33/0.66 => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.33/0.66 = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % less_rat
% 0.33/0.66 thf(fact_9197_add__rat,axiom,
% 0.33/0.66 ! [B: int,D: int,A: int,C: int] :
% 0.33/0.66 ( ( B != zero_zero_int )
% 0.33/0.66 => ( ( D != zero_zero_int )
% 0.33/0.66 => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.33/0.66 = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % add_rat
% 0.33/0.66 thf(fact_9198_le__rat,axiom,
% 0.33/0.66 ! [B: int,D: int,A: int,C: int] :
% 0.33/0.66 ( ( B != zero_zero_int )
% 0.33/0.66 => ( ( D != zero_zero_int )
% 0.33/0.66 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.33/0.66 = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % le_rat
% 0.33/0.66 thf(fact_9199_diff__rat,axiom,
% 0.33/0.66 ! [B: int,D: int,A: int,C: int] :
% 0.33/0.66 ( ( B != zero_zero_int )
% 0.33/0.66 => ( ( D != zero_zero_int )
% 0.33/0.66 => ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.33/0.66 = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % diff_rat
% 0.33/0.66 thf(fact_9200_sgn__rat,axiom,
% 0.33/0.66 ! [A: int,B: int] :
% 0.33/0.66 ( ( sgn_sgn_rat @ ( fract @ A @ B ) )
% 0.33/0.66 = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sgn_rat
% 0.33/0.66 thf(fact_9201_eq__rat_I3_J,axiom,
% 0.33/0.66 ! [A: int,C: int] :
% 0.33/0.66 ( ( fract @ zero_zero_int @ A )
% 0.33/0.66 = ( fract @ zero_zero_int @ C ) ) ).
% 0.33/0.66
% 0.33/0.66 % eq_rat(3)
% 0.33/0.66 thf(fact_9202_eq__rat_I2_J,axiom,
% 0.33/0.66 ! [A: int] :
% 0.33/0.66 ( ( fract @ A @ zero_zero_int )
% 0.33/0.66 = ( fract @ zero_zero_int @ one_one_int ) ) ).
% 0.33/0.66
% 0.33/0.66 % eq_rat(2)
% 0.33/0.66 thf(fact_9203_Rat__induct__pos,axiom,
% 0.33/0.66 ! [P: rat > $o,Q2: rat] :
% 0.33/0.66 ( ! [A4: int,B4: int] :
% 0.33/0.66 ( ( ord_less_int @ zero_zero_int @ B4 )
% 0.33/0.66 => ( P @ ( fract @ A4 @ B4 ) ) )
% 0.33/0.66 => ( P @ Q2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % Rat_induct_pos
% 0.33/0.66 thf(fact_9204_mult__rat__cancel,axiom,
% 0.33/0.66 ! [C: int,A: int,B: int] :
% 0.33/0.66 ( ( C != zero_zero_int )
% 0.33/0.66 => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.33/0.66 = ( fract @ A @ B ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % mult_rat_cancel
% 0.33/0.66 thf(fact_9205_eq__rat_I1_J,axiom,
% 0.33/0.66 ! [B: int,D: int,A: int,C: int] :
% 0.33/0.66 ( ( B != zero_zero_int )
% 0.33/0.66 => ( ( D != zero_zero_int )
% 0.33/0.66 => ( ( ( fract @ A @ B )
% 0.33/0.66 = ( fract @ C @ D ) )
% 0.33/0.66 = ( ( times_times_int @ A @ D )
% 0.33/0.66 = ( times_times_int @ C @ B ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % eq_rat(1)
% 0.33/0.66 thf(fact_9206_Fract__of__nat__eq,axiom,
% 0.33/0.66 ! [K: nat] :
% 0.33/0.66 ( ( fract @ ( semiri1314217659103216013at_int @ K ) @ one_one_int )
% 0.33/0.66 = ( semiri681578069525770553at_rat @ K ) ) ).
% 0.33/0.66
% 0.33/0.66 % Fract_of_nat_eq
% 0.33/0.66 thf(fact_9207_rat__number__collapse_I6_J,axiom,
% 0.33/0.66 ! [K: int] :
% 0.33/0.66 ( ( fract @ K @ zero_zero_int )
% 0.33/0.66 = zero_zero_rat ) ).
% 0.33/0.66
% 0.33/0.66 % rat_number_collapse(6)
% 0.33/0.66 thf(fact_9208_rat__number__collapse_I1_J,axiom,
% 0.33/0.66 ! [K: int] :
% 0.33/0.66 ( ( fract @ zero_zero_int @ K )
% 0.33/0.66 = zero_zero_rat ) ).
% 0.33/0.66
% 0.33/0.66 % rat_number_collapse(1)
% 0.33/0.66 thf(fact_9209_Fract__coprime,axiom,
% 0.33/0.66 ! [A: int,B: int] :
% 0.33/0.66 ( ( fract @ ( divide_divide_int @ A @ ( gcd_gcd_int @ A @ B ) ) @ ( divide_divide_int @ B @ ( gcd_gcd_int @ A @ B ) ) )
% 0.33/0.66 = ( fract @ A @ B ) ) ).
% 0.33/0.66
% 0.33/0.66 % Fract_coprime
% 0.33/0.66 thf(fact_9210_One__rat__def,axiom,
% 0.33/0.66 ( one_one_rat
% 0.33/0.66 = ( fract @ one_one_int @ one_one_int ) ) ).
% 0.33/0.66
% 0.33/0.66 % One_rat_def
% 0.33/0.66 thf(fact_9211_Fract__of__int__eq,axiom,
% 0.33/0.66 ! [K: int] :
% 0.33/0.66 ( ( fract @ K @ one_one_int )
% 0.33/0.66 = ( ring_1_of_int_rat @ K ) ) ).
% 0.33/0.66
% 0.33/0.66 % Fract_of_int_eq
% 0.33/0.66 thf(fact_9212_Zero__rat__def,axiom,
% 0.33/0.66 ( zero_zero_rat
% 0.33/0.66 = ( fract @ zero_zero_int @ one_one_int ) ) ).
% 0.33/0.66
% 0.33/0.66 % Zero_rat_def
% 0.33/0.66 thf(fact_9213_rat__number__collapse_I3_J,axiom,
% 0.33/0.66 ! [W: num] :
% 0.33/0.66 ( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
% 0.33/0.66 = ( numeral_numeral_rat @ W ) ) ).
% 0.33/0.66
% 0.33/0.66 % rat_number_collapse(3)
% 0.33/0.66 thf(fact_9214_rat__number__expand_I3_J,axiom,
% 0.33/0.66 ( numeral_numeral_rat
% 0.33/0.66 = ( ^ [K2: num] : ( fract @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % rat_number_expand(3)
% 0.33/0.66 thf(fact_9215_sorted__list__of__set__greaterThanLessThan,axiom,
% 0.33/0.66 ! [I: nat,J: nat] :
% 0.33/0.66 ( ( ord_less_nat @ ( suc @ I ) @ J )
% 0.33/0.66 => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 0.33/0.66 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sorted_list_of_set_greaterThanLessThan
% 0.33/0.66 thf(fact_9216_zero__less__Fract__iff,axiom,
% 0.33/0.66 ! [B: int,A: int] :
% 0.33/0.66 ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.66 => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 0.33/0.66 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % zero_less_Fract_iff
% 0.33/0.66 thf(fact_9217_Fract__less__zero__iff,axiom,
% 0.33/0.66 ! [B: int,A: int] :
% 0.33/0.66 ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.66 => ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 0.33/0.66 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Fract_less_zero_iff
% 0.33/0.66 thf(fact_9218_Fract__less__one__iff,axiom,
% 0.33/0.66 ! [B: int,A: int] :
% 0.33/0.66 ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.66 => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
% 0.33/0.66 = ( ord_less_int @ A @ B ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Fract_less_one_iff
% 0.33/0.66 thf(fact_9219_one__less__Fract__iff,axiom,
% 0.33/0.66 ! [B: int,A: int] :
% 0.33/0.66 ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.66 => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
% 0.33/0.66 = ( ord_less_int @ B @ A ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % one_less_Fract_iff
% 0.33/0.66 thf(fact_9220_rat__number__collapse_I5_J,axiom,
% 0.33/0.66 ( ( fract @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.33/0.66 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.66
% 0.33/0.66 % rat_number_collapse(5)
% 0.33/0.66 thf(fact_9221_Fract__add__one,axiom,
% 0.33/0.66 ! [N: int,M: int] :
% 0.33/0.66 ( ( N != zero_zero_int )
% 0.33/0.66 => ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
% 0.33/0.66 = ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Fract_add_one
% 0.33/0.66 thf(fact_9222_zero__le__Fract__iff,axiom,
% 0.33/0.66 ! [B: int,A: int] :
% 0.33/0.66 ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.66 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 0.33/0.66 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % zero_le_Fract_iff
% 0.33/0.66 thf(fact_9223_Fract__le__zero__iff,axiom,
% 0.33/0.66 ! [B: int,A: int] :
% 0.33/0.66 ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.66 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 0.33/0.66 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Fract_le_zero_iff
% 0.33/0.66 thf(fact_9224_Fract__le__one__iff,axiom,
% 0.33/0.66 ! [B: int,A: int] :
% 0.33/0.66 ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.66 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
% 0.33/0.66 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Fract_le_one_iff
% 0.33/0.66 thf(fact_9225_one__le__Fract__iff,axiom,
% 0.33/0.66 ! [B: int,A: int] :
% 0.33/0.66 ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.66 => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
% 0.33/0.66 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % one_le_Fract_iff
% 0.33/0.66 thf(fact_9226_rat__number__collapse_I4_J,axiom,
% 0.33/0.66 ! [W: num] :
% 0.33/0.66 ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
% 0.33/0.66 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % rat_number_collapse(4)
% 0.33/0.66 thf(fact_9227_rat__number__expand_I5_J,axiom,
% 0.33/0.66 ! [K: num] :
% 0.33/0.66 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
% 0.33/0.66 = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 0.33/0.66
% 0.33/0.66 % rat_number_expand(5)
% 0.33/0.66 thf(fact_9228_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 0.33/0.66 ! [N: nat,J: nat,I: nat] :
% 0.33/0.66 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 0.33/0.66 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
% 0.33/0.66 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % nth_sorted_list_of_set_greaterThanLessThan
% 0.33/0.66 thf(fact_9229_less__eq__int_Orep__eq,axiom,
% 0.33/0.66 ( ord_less_eq_int
% 0.33/0.66 = ( ^ [X3: int,Xa4: int] :
% 0.33/0.66 ( produc8739625826339149834_nat_o
% 0.33/0.66 @ ^ [Y6: nat,Z5: nat] :
% 0.33/0.66 ( produc6081775807080527818_nat_o
% 0.33/0.66 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y6 @ V4 ) @ ( plus_plus_nat @ U3 @ Z5 ) ) )
% 0.33/0.66 @ ( rep_Integ @ X3 )
% 0.33/0.66 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % less_eq_int.rep_eq
% 0.33/0.66 thf(fact_9230_less__int_Orep__eq,axiom,
% 0.33/0.66 ( ord_less_int
% 0.33/0.66 = ( ^ [X3: int,Xa4: int] :
% 0.33/0.66 ( produc8739625826339149834_nat_o
% 0.33/0.66 @ ^ [Y6: nat,Z5: nat] :
% 0.33/0.66 ( produc6081775807080527818_nat_o
% 0.33/0.66 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y6 @ V4 ) @ ( plus_plus_nat @ U3 @ Z5 ) ) )
% 0.33/0.66 @ ( rep_Integ @ X3 )
% 0.33/0.66 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % less_int.rep_eq
% 0.33/0.66 thf(fact_9231_inf__nat__def,axiom,
% 0.33/0.66 inf_inf_nat = ord_min_nat ).
% 0.33/0.66
% 0.33/0.66 % inf_nat_def
% 0.33/0.66 thf(fact_9232_inf__int__def,axiom,
% 0.33/0.66 inf_inf_int = ord_min_int ).
% 0.33/0.66
% 0.33/0.66 % inf_int_def
% 0.33/0.66 thf(fact_9233_nat_Orep__eq,axiom,
% 0.33/0.66 ( nat2
% 0.33/0.66 = ( ^ [X3: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % nat.rep_eq
% 0.33/0.66 thf(fact_9234_list__encode_Osimps_I2_J,axiom,
% 0.33/0.66 ! [X: nat,Xs: list_nat] :
% 0.33/0.66 ( ( nat_list_encode @ ( cons_nat @ X @ Xs ) )
% 0.33/0.66 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % list_encode.simps(2)
% 0.33/0.66 thf(fact_9235_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 0.33/0.66 ! [N: nat,J: nat,I: nat] :
% 0.33/0.66 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
% 0.33/0.66 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
% 0.33/0.66 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % nth_sorted_list_of_set_greaterThanAtMost
% 0.33/0.66 thf(fact_9236_card__greaterThanAtMost,axiom,
% 0.33/0.66 ! [L: nat,U: nat] :
% 0.33/0.66 ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
% 0.33/0.66 = ( minus_minus_nat @ U @ L ) ) ).
% 0.33/0.66
% 0.33/0.66 % card_greaterThanAtMost
% 0.33/0.66 thf(fact_9237_list__encode__eq,axiom,
% 0.33/0.66 ! [X: list_nat,Y: list_nat] :
% 0.33/0.66 ( ( ( nat_list_encode @ X )
% 0.33/0.66 = ( nat_list_encode @ Y ) )
% 0.33/0.66 = ( X = Y ) ) ).
% 0.33/0.66
% 0.33/0.66 % list_encode_eq
% 0.33/0.66 thf(fact_9238_atLeastSucAtMost__greaterThanAtMost,axiom,
% 0.33/0.66 ! [L: nat,U: nat] :
% 0.33/0.66 ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
% 0.33/0.66 = ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 0.33/0.66
% 0.33/0.66 % atLeastSucAtMost_greaterThanAtMost
% 0.33/0.66 thf(fact_9239_sorted__list__of__set__greaterThanAtMost,axiom,
% 0.33/0.66 ! [I: nat,J: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 0.33/0.66 => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 0.33/0.66 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sorted_list_of_set_greaterThanAtMost
% 0.33/0.66 thf(fact_9240_list__encode_Oelims,axiom,
% 0.33/0.66 ! [X: list_nat,Y: nat] :
% 0.33/0.66 ( ( ( nat_list_encode @ X )
% 0.33/0.66 = Y )
% 0.33/0.66 => ( ( ( X = nil_nat )
% 0.33/0.66 => ( Y != zero_zero_nat ) )
% 0.33/0.66 => ~ ! [X4: nat,Xs2: list_nat] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( cons_nat @ X4 @ Xs2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % list_encode.elims
% 0.33/0.66 thf(fact_9241_uminus__int__def,axiom,
% 0.33/0.66 ( uminus_uminus_int
% 0.33/0.66 = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
% 0.33/0.66 @ ( produc2626176000494625587at_nat
% 0.33/0.66 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % uminus_int_def
% 0.33/0.66 thf(fact_9242_take__bit__numeral__minus__numeral__int,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.66 = ( case_option_int_num @ zero_zero_int
% 0.33/0.66 @ ^ [Q5: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q5 ) ) )
% 0.33/0.66 @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % take_bit_numeral_minus_numeral_int
% 0.33/0.66 thf(fact_9243_take__bit__num__simps_I1_J,axiom,
% 0.33/0.66 ! [M: num] :
% 0.33/0.66 ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 0.33/0.66 = none_num ) ).
% 0.33/0.66
% 0.33/0.66 % take_bit_num_simps(1)
% 0.33/0.66 thf(fact_9244_take__bit__num__simps_I2_J,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( bit_take_bit_num @ ( suc @ N ) @ one )
% 0.33/0.66 = ( some_num @ one ) ) ).
% 0.33/0.66
% 0.33/0.66 % take_bit_num_simps(2)
% 0.33/0.66 thf(fact_9245_take__bit__num__simps_I5_J,axiom,
% 0.33/0.66 ! [R: num] :
% 0.33/0.66 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ one )
% 0.33/0.66 = ( some_num @ one ) ) ).
% 0.33/0.66
% 0.33/0.66 % take_bit_num_simps(5)
% 0.33/0.66 thf(fact_9246_card__greaterThanAtMost__int,axiom,
% 0.33/0.66 ! [L: int,U: int] :
% 0.33/0.66 ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
% 0.33/0.66 = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % card_greaterThanAtMost_int
% 0.33/0.66 thf(fact_9247_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( bit_take_bit_num @ N @ one )
% 0.33/0.66 = ( case_nat_option_num @ none_num
% 0.33/0.66 @ ^ [N3: nat] : ( some_num @ one )
% 0.33/0.66 @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % Code_Abstract_Nat.take_bit_num_code(1)
% 0.33/0.66 thf(fact_9248_list__encode_Ocases,axiom,
% 0.33/0.66 ! [X: list_nat] :
% 0.33/0.66 ( ( X != nil_nat )
% 0.33/0.66 => ~ ! [X4: nat,Xs2: list_nat] :
% 0.33/0.66 ( X
% 0.33/0.66 != ( cons_nat @ X4 @ Xs2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % list_encode.cases
% 0.33/0.66 thf(fact_9249_list__encode_Osimps_I1_J,axiom,
% 0.33/0.66 ( ( nat_list_encode @ nil_nat )
% 0.33/0.66 = zero_zero_nat ) ).
% 0.33/0.66
% 0.33/0.66 % list_encode.simps(1)
% 0.33/0.66 thf(fact_9250_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 0.33/0.66 ! [L: int,U: int] :
% 0.33/0.66 ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
% 0.33/0.66 = ( set_or6656581121297822940st_int @ L @ U ) ) ).
% 0.33/0.66
% 0.33/0.66 % atLeastPlusOneAtMost_greaterThanAtMost_int
% 0.33/0.66 thf(fact_9251_take__bit__num__def,axiom,
% 0.33/0.66 ( bit_take_bit_num
% 0.33/0.66 = ( ^ [N3: nat,M3: num] :
% 0.33/0.66 ( if_option_num
% 0.33/0.66 @ ( ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M3 ) )
% 0.33/0.66 = zero_zero_nat )
% 0.33/0.66 @ none_num
% 0.33/0.66 @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M3 ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % take_bit_num_def
% 0.33/0.66 thf(fact_9252_and__minus__numerals_I3_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.33/0.66 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_minus_numerals(3)
% 0.33/0.66 thf(fact_9253_and__minus__numerals_I7_J,axiom,
% 0.33/0.66 ! [N: num,M: num] :
% 0.33/0.66 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.33/0.66 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_minus_numerals(7)
% 0.33/0.66 thf(fact_9254_and__minus__numerals_I4_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.33/0.66 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_minus_numerals(4)
% 0.33/0.66 thf(fact_9255_take__bit__num__simps_I4_J,axiom,
% 0.33/0.66 ! [N: nat,M: num] :
% 0.33/0.66 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 0.33/0.66 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % take_bit_num_simps(4)
% 0.33/0.66 thf(fact_9256_take__bit__num__simps_I3_J,axiom,
% 0.33/0.66 ! [N: nat,M: num] :
% 0.33/0.66 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 0.33/0.66 = ( case_o6005452278849405969um_num @ none_num
% 0.33/0.66 @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 0.33/0.66 @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % take_bit_num_simps(3)
% 0.33/0.66 thf(fact_9257_take__bit__num__simps_I7_J,axiom,
% 0.33/0.66 ! [R: num,M: num] :
% 0.33/0.66 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit1 @ M ) )
% 0.33/0.66 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % take_bit_num_simps(7)
% 0.33/0.66 thf(fact_9258_take__bit__num__simps_I6_J,axiom,
% 0.33/0.66 ! [R: num,M: num] :
% 0.33/0.66 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit0 @ M ) )
% 0.33/0.66 = ( case_o6005452278849405969um_num @ none_num
% 0.33/0.66 @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 0.33/0.66 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % take_bit_num_simps(6)
% 0.33/0.66 thf(fact_9259_and__minus__numerals_I8_J,axiom,
% 0.33/0.66 ! [N: num,M: num] :
% 0.33/0.66 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.33/0.66 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_minus_numerals(8)
% 0.33/0.66 thf(fact_9260_and__not__num_Osimps_I8_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.33/0.66 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.33/0.66 @ ^ [N7: num] : ( some_num @ ( bit1 @ N7 ) )
% 0.33/0.66 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_not_num.simps(8)
% 0.33/0.66 thf(fact_9261_and__not__num_Osimps_I1_J,axiom,
% 0.33/0.66 ( ( bit_and_not_num @ one @ one )
% 0.33/0.66 = none_num ) ).
% 0.33/0.66
% 0.33/0.66 % and_not_num.simps(1)
% 0.33/0.66 thf(fact_9262_and__not__num_Osimps_I2_J,axiom,
% 0.33/0.66 ! [N: num] :
% 0.33/0.66 ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
% 0.33/0.66 = ( some_num @ one ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_not_num.simps(2)
% 0.33/0.66 thf(fact_9263_and__not__num_Osimps_I4_J,axiom,
% 0.33/0.66 ! [M: num] :
% 0.33/0.66 ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 0.33/0.66 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_not_num.simps(4)
% 0.33/0.66 thf(fact_9264_and__not__num_Osimps_I3_J,axiom,
% 0.33/0.66 ! [N: num] :
% 0.33/0.66 ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
% 0.33/0.66 = none_num ) ).
% 0.33/0.66
% 0.33/0.66 % and_not_num.simps(3)
% 0.33/0.66 thf(fact_9265_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 0.33/0.66 ! [N: nat,M: num] :
% 0.33/0.66 ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
% 0.33/0.66 = ( case_nat_option_num @ none_num
% 0.33/0.66 @ ^ [N3: nat] :
% 0.33/0.66 ( case_o6005452278849405969um_num @ none_num
% 0.33/0.66 @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 0.33/0.66 @ ( bit_take_bit_num @ N3 @ M ) )
% 0.33/0.66 @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % Code_Abstract_Nat.take_bit_num_code(2)
% 0.33/0.66 thf(fact_9266_and__not__num_Osimps_I7_J,axiom,
% 0.33/0.66 ! [M: num] :
% 0.33/0.66 ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 0.33/0.66 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_not_num.simps(7)
% 0.33/0.66 thf(fact_9267_and__not__num__eq__Some__iff,axiom,
% 0.33/0.66 ! [M: num,N: num,Q2: num] :
% 0.33/0.66 ( ( ( bit_and_not_num @ M @ N )
% 0.33/0.66 = ( some_num @ Q2 ) )
% 0.33/0.66 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.66 = ( numeral_numeral_int @ Q2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_not_num_eq_Some_iff
% 0.33/0.66 thf(fact_9268_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 0.33/0.66 ! [N: nat,M: num] :
% 0.33/0.66 ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
% 0.33/0.66 = ( case_nat_option_num @ none_num
% 0.33/0.66 @ ^ [N3: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N3 @ M ) ) )
% 0.33/0.66 @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % Code_Abstract_Nat.take_bit_num_code(3)
% 0.33/0.66 thf(fact_9269_and__not__num__eq__None__iff,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( ( bit_and_not_num @ M @ N )
% 0.33/0.66 = none_num )
% 0.33/0.66 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.66 = zero_zero_int ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_not_num_eq_None_iff
% 0.33/0.66 thf(fact_9270_int__numeral__not__and__num,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.66 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % int_numeral_not_and_num
% 0.33/0.66 thf(fact_9271_int__numeral__and__not__num,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.66 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % int_numeral_and_not_num
% 0.33/0.66 thf(fact_9272_times__int__def,axiom,
% 0.33/0.66 ( times_times_int
% 0.33/0.66 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.33/0.66 @ ( produc27273713700761075at_nat
% 0.33/0.66 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.66 ( produc2626176000494625587at_nat
% 0.33/0.66 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % times_int_def
% 0.33/0.66 thf(fact_9273_minus__int__def,axiom,
% 0.33/0.66 ( minus_minus_int
% 0.33/0.66 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.33/0.66 @ ( produc27273713700761075at_nat
% 0.33/0.66 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.66 ( produc2626176000494625587at_nat
% 0.33/0.66 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % minus_int_def
% 0.33/0.66 thf(fact_9274_plus__int__def,axiom,
% 0.33/0.66 ( plus_plus_int
% 0.33/0.66 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.33/0.66 @ ( produc27273713700761075at_nat
% 0.33/0.66 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.66 ( produc2626176000494625587at_nat
% 0.33/0.66 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % plus_int_def
% 0.33/0.66 thf(fact_9275_upto_Opelims,axiom,
% 0.33/0.66 ! [X: int,Xa: int,Y: list_int] :
% 0.33/0.66 ( ( ( upto @ X @ Xa )
% 0.33/0.66 = Y )
% 0.33/0.66 => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 0.33/0.66 => ~ ( ( ( ( ord_less_eq_int @ X @ Xa )
% 0.33/0.66 => ( Y
% 0.33/0.66 = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
% 0.33/0.66 & ( ~ ( ord_less_eq_int @ X @ Xa )
% 0.33/0.66 => ( Y = nil_int ) ) )
% 0.33/0.66 => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upto.pelims
% 0.33/0.66 thf(fact_9276_upto_Opsimps,axiom,
% 0.33/0.66 ! [I: int,J: int] :
% 0.33/0.66 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 0.33/0.66 => ( ( ( ord_less_eq_int @ I @ J )
% 0.33/0.66 => ( ( upto @ I @ J )
% 0.33/0.66 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 0.33/0.66 & ( ~ ( ord_less_eq_int @ I @ J )
% 0.33/0.66 => ( ( upto @ I @ J )
% 0.33/0.66 = nil_int ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upto.psimps
% 0.33/0.66 thf(fact_9277_nth__upto,axiom,
% 0.33/0.66 ! [I: int,K: nat,J: int] :
% 0.33/0.66 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 0.33/0.66 => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 0.33/0.66 = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % nth_upto
% 0.33/0.66 thf(fact_9278_length__upto,axiom,
% 0.33/0.66 ! [I: int,J: int] :
% 0.33/0.66 ( ( size_size_list_int @ ( upto @ I @ J ) )
% 0.33/0.66 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % length_upto
% 0.33/0.66 thf(fact_9279_sorted__list__of__set__lessThan__Suc,axiom,
% 0.33/0.66 ! [K: nat] :
% 0.33/0.66 ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
% 0.33/0.66 = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sorted_list_of_set_lessThan_Suc
% 0.33/0.66 thf(fact_9280_sorted__list__of__set__atMost__Suc,axiom,
% 0.33/0.66 ! [K: nat] :
% 0.33/0.66 ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
% 0.33/0.66 = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sorted_list_of_set_atMost_Suc
% 0.33/0.66 thf(fact_9281_upto__rec__numeral_I1_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.66 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.66 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 0.33/0.66 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.66 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.66 = nil_int ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upto_rec_numeral(1)
% 0.33/0.66 thf(fact_9282_upto__rec__numeral_I2_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.66 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.66 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 0.33/0.66 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.66 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.66 = nil_int ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upto_rec_numeral(2)
% 0.33/0.66 thf(fact_9283_upto__rec__numeral_I3_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.66 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.66 = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 0.33/0.66 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.66 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.66 = nil_int ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upto_rec_numeral(3)
% 0.33/0.66 thf(fact_9284_upto__rec__numeral_I4_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.66 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.66 = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 0.33/0.66 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.66 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.33/0.66 = nil_int ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upto_rec_numeral(4)
% 0.33/0.66 thf(fact_9285_upto__split1,axiom,
% 0.33/0.66 ! [I: int,J: int,K: int] :
% 0.33/0.66 ( ( ord_less_eq_int @ I @ J )
% 0.33/0.66 => ( ( ord_less_eq_int @ J @ K )
% 0.33/0.66 => ( ( upto @ I @ K )
% 0.33/0.66 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upto_split1
% 0.33/0.66 thf(fact_9286_upto__split2,axiom,
% 0.33/0.66 ! [I: int,J: int,K: int] :
% 0.33/0.66 ( ( ord_less_eq_int @ I @ J )
% 0.33/0.66 => ( ( ord_less_eq_int @ J @ K )
% 0.33/0.66 => ( ( upto @ I @ K )
% 0.33/0.66 = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upto_split2
% 0.33/0.66 thf(fact_9287_upto__rec2,axiom,
% 0.33/0.66 ! [I: int,J: int] :
% 0.33/0.66 ( ( ord_less_eq_int @ I @ J )
% 0.33/0.66 => ( ( upto @ I @ J )
% 0.33/0.66 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upto_rec2
% 0.33/0.66 thf(fact_9288_upto__split3,axiom,
% 0.33/0.66 ! [I: int,J: int,K: int] :
% 0.33/0.66 ( ( ord_less_eq_int @ I @ J )
% 0.33/0.66 => ( ( ord_less_eq_int @ J @ K )
% 0.33/0.66 => ( ( upto @ I @ K )
% 0.33/0.66 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upto_split3
% 0.33/0.66 thf(fact_9289_atLeastLessThan__upto,axiom,
% 0.33/0.66 ( set_or4662586982721622107an_int
% 0.33/0.66 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % atLeastLessThan_upto
% 0.33/0.66 thf(fact_9290_greaterThanAtMost__upto,axiom,
% 0.33/0.66 ( set_or6656581121297822940st_int
% 0.33/0.66 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % greaterThanAtMost_upto
% 0.33/0.66 thf(fact_9291_upto_Osimps,axiom,
% 0.33/0.66 ( upto
% 0.33/0.66 = ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upto.simps
% 0.33/0.66 thf(fact_9292_upto_Oelims,axiom,
% 0.33/0.66 ! [X: int,Xa: int,Y: list_int] :
% 0.33/0.66 ( ( ( upto @ X @ Xa )
% 0.33/0.66 = Y )
% 0.33/0.66 => ( ( ( ord_less_eq_int @ X @ Xa )
% 0.33/0.66 => ( Y
% 0.33/0.66 = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
% 0.33/0.66 & ( ~ ( ord_less_eq_int @ X @ Xa )
% 0.33/0.66 => ( Y = nil_int ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upto.elims
% 0.33/0.66 thf(fact_9293_upto__rec1,axiom,
% 0.33/0.66 ! [I: int,J: int] :
% 0.33/0.66 ( ( ord_less_eq_int @ I @ J )
% 0.33/0.66 => ( ( upto @ I @ J )
% 0.33/0.66 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upto_rec1
% 0.33/0.66 thf(fact_9294_greaterThanLessThan__upto,axiom,
% 0.33/0.66 ( set_or5832277885323065728an_int
% 0.33/0.66 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % greaterThanLessThan_upto
% 0.33/0.66 thf(fact_9295_and__not__num_Oelims,axiom,
% 0.33/0.66 ! [X: num,Xa: num,Y: option_num] :
% 0.33/0.66 ( ( ( bit_and_not_num @ X @ Xa )
% 0.33/0.66 = Y )
% 0.33/0.66 => ( ( ( X = one )
% 0.33/0.66 => ( ( Xa = one )
% 0.33/0.66 => ( Y != none_num ) ) )
% 0.33/0.66 => ( ( ( X = one )
% 0.33/0.66 => ( ? [N2: num] :
% 0.33/0.66 ( Xa
% 0.33/0.66 = ( bit0 @ N2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( some_num @ one ) ) ) )
% 0.33/0.66 => ( ( ( X = one )
% 0.33/0.66 => ( ? [N2: num] :
% 0.33/0.66 ( Xa
% 0.33/0.66 = ( bit1 @ N2 ) )
% 0.33/0.66 => ( Y != none_num ) ) )
% 0.33/0.66 => ( ! [M2: num] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( bit0 @ M2 ) )
% 0.33/0.66 => ( ( Xa = one )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( some_num @ ( bit0 @ M2 ) ) ) ) )
% 0.33/0.66 => ( ! [M2: num] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( bit0 @ M2 ) )
% 0.33/0.66 => ! [N2: num] :
% 0.33/0.66 ( ( Xa
% 0.33/0.66 = ( bit0 @ N2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N2 ) ) ) ) )
% 0.33/0.66 => ( ! [M2: num] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( bit0 @ M2 ) )
% 0.33/0.66 => ! [N2: num] :
% 0.33/0.66 ( ( Xa
% 0.33/0.66 = ( bit1 @ N2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N2 ) ) ) ) )
% 0.33/0.66 => ( ! [M2: num] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( bit1 @ M2 ) )
% 0.33/0.66 => ( ( Xa = one )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( some_num @ ( bit0 @ M2 ) ) ) ) )
% 0.33/0.66 => ( ! [M2: num] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( bit1 @ M2 ) )
% 0.33/0.66 => ! [N2: num] :
% 0.33/0.66 ( ( Xa
% 0.33/0.66 = ( bit0 @ N2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.33/0.66 @ ^ [N7: num] : ( some_num @ ( bit1 @ N7 ) )
% 0.33/0.66 @ ( bit_and_not_num @ M2 @ N2 ) ) ) ) )
% 0.33/0.66 => ~ ! [M2: num] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( bit1 @ M2 ) )
% 0.33/0.66 => ! [N2: num] :
% 0.33/0.66 ( ( Xa
% 0.33/0.66 = ( bit1 @ N2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_not_num.elims
% 0.33/0.66 thf(fact_9296_and__not__num_Osimps_I5_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.33/0.66 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_not_num.simps(5)
% 0.33/0.66 thf(fact_9297_and__not__num_Osimps_I6_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.33/0.66 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_not_num.simps(6)
% 0.33/0.66 thf(fact_9298_and__not__num_Osimps_I9_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.33/0.66 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_not_num.simps(9)
% 0.33/0.66 thf(fact_9299_and__num_Oelims,axiom,
% 0.33/0.66 ! [X: num,Xa: num,Y: option_num] :
% 0.33/0.66 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 0.33/0.66 = Y )
% 0.33/0.66 => ( ( ( X = one )
% 0.33/0.66 => ( ( Xa = one )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( some_num @ one ) ) ) )
% 0.33/0.66 => ( ( ( X = one )
% 0.33/0.66 => ( ? [N2: num] :
% 0.33/0.66 ( Xa
% 0.33/0.66 = ( bit0 @ N2 ) )
% 0.33/0.66 => ( Y != none_num ) ) )
% 0.33/0.66 => ( ( ( X = one )
% 0.33/0.66 => ( ? [N2: num] :
% 0.33/0.66 ( Xa
% 0.33/0.66 = ( bit1 @ N2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( some_num @ one ) ) ) )
% 0.33/0.66 => ( ( ? [M2: num] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( bit0 @ M2 ) )
% 0.33/0.66 => ( ( Xa = one )
% 0.33/0.66 => ( Y != none_num ) ) )
% 0.33/0.66 => ( ! [M2: num] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( bit0 @ M2 ) )
% 0.33/0.66 => ! [N2: num] :
% 0.33/0.66 ( ( Xa
% 0.33/0.66 = ( bit0 @ N2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) ) ) )
% 0.33/0.66 => ( ! [M2: num] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( bit0 @ M2 ) )
% 0.33/0.66 => ! [N2: num] :
% 0.33/0.66 ( ( Xa
% 0.33/0.66 = ( bit1 @ N2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) ) ) )
% 0.33/0.66 => ( ( ? [M2: num] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( bit1 @ M2 ) )
% 0.33/0.66 => ( ( Xa = one )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( some_num @ one ) ) ) )
% 0.33/0.66 => ( ! [M2: num] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( bit1 @ M2 ) )
% 0.33/0.66 => ! [N2: num] :
% 0.33/0.66 ( ( Xa
% 0.33/0.66 = ( bit0 @ N2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) ) ) )
% 0.33/0.66 => ~ ! [M2: num] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( bit1 @ M2 ) )
% 0.33/0.66 => ! [N2: num] :
% 0.33/0.66 ( ( Xa
% 0.33/0.66 = ( bit1 @ N2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.33/0.66 @ ^ [N7: num] : ( some_num @ ( bit1 @ N7 ) )
% 0.33/0.66 @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_num.elims
% 0.33/0.66 thf(fact_9300_xor__num_Oelims,axiom,
% 0.33/0.66 ! [X: num,Xa: num,Y: option_num] :
% 0.33/0.66 ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 0.33/0.66 = Y )
% 0.33/0.66 => ( ( ( X = one )
% 0.33/0.66 => ( ( Xa = one )
% 0.33/0.66 => ( Y != none_num ) ) )
% 0.33/0.66 => ( ( ( X = one )
% 0.33/0.66 => ! [N2: num] :
% 0.33/0.66 ( ( Xa
% 0.33/0.66 = ( bit0 @ N2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( some_num @ ( bit1 @ N2 ) ) ) ) )
% 0.33/0.66 => ( ( ( X = one )
% 0.33/0.66 => ! [N2: num] :
% 0.33/0.66 ( ( Xa
% 0.33/0.66 = ( bit1 @ N2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( some_num @ ( bit0 @ N2 ) ) ) ) )
% 0.33/0.66 => ( ! [M2: num] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( bit0 @ M2 ) )
% 0.33/0.66 => ( ( Xa = one )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( some_num @ ( bit1 @ M2 ) ) ) ) )
% 0.33/0.66 => ( ! [M2: num] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( bit0 @ M2 ) )
% 0.33/0.66 => ! [N2: num] :
% 0.33/0.66 ( ( Xa
% 0.33/0.66 = ( bit0 @ N2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) ) )
% 0.33/0.66 => ( ! [M2: num] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( bit0 @ M2 ) )
% 0.33/0.66 => ! [N2: num] :
% 0.33/0.66 ( ( Xa
% 0.33/0.66 = ( bit1 @ N2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) ) ) )
% 0.33/0.66 => ( ! [M2: num] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( bit1 @ M2 ) )
% 0.33/0.66 => ( ( Xa = one )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( some_num @ ( bit0 @ M2 ) ) ) ) )
% 0.33/0.66 => ( ! [M2: num] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( bit1 @ M2 ) )
% 0.33/0.66 => ! [N2: num] :
% 0.33/0.66 ( ( Xa
% 0.33/0.66 = ( bit0 @ N2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) ) ) )
% 0.33/0.66 => ~ ! [M2: num] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( bit1 @ M2 ) )
% 0.33/0.66 => ! [N2: num] :
% 0.33/0.66 ( ( Xa
% 0.33/0.66 = ( bit1 @ N2 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % xor_num.elims
% 0.33/0.66 thf(fact_9301_and__num_Osimps_I1_J,axiom,
% 0.33/0.66 ( ( bit_un7362597486090784418nd_num @ one @ one )
% 0.33/0.66 = ( some_num @ one ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_num.simps(1)
% 0.33/0.66 thf(fact_9302_xor__num_Osimps_I1_J,axiom,
% 0.33/0.66 ( ( bit_un2480387367778600638or_num @ one @ one )
% 0.33/0.66 = none_num ) ).
% 0.33/0.66
% 0.33/0.66 % xor_num.simps(1)
% 0.33/0.66 thf(fact_9303_xor__num_Osimps_I5_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.33/0.66 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % xor_num.simps(5)
% 0.33/0.66 thf(fact_9304_and__num_Osimps_I5_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.33/0.66 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_num.simps(5)
% 0.33/0.66 thf(fact_9305_and__num_Osimps_I3_J,axiom,
% 0.33/0.66 ! [N: num] :
% 0.33/0.66 ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
% 0.33/0.66 = ( some_num @ one ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_num.simps(3)
% 0.33/0.66 thf(fact_9306_and__num_Osimps_I7_J,axiom,
% 0.33/0.66 ! [M: num] :
% 0.33/0.66 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 0.33/0.66 = ( some_num @ one ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_num.simps(7)
% 0.33/0.66 thf(fact_9307_and__num_Osimps_I2_J,axiom,
% 0.33/0.66 ! [N: num] :
% 0.33/0.66 ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
% 0.33/0.66 = none_num ) ).
% 0.33/0.66
% 0.33/0.66 % and_num.simps(2)
% 0.33/0.66 thf(fact_9308_and__num_Osimps_I4_J,axiom,
% 0.33/0.66 ! [M: num] :
% 0.33/0.66 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 0.33/0.66 = none_num ) ).
% 0.33/0.66
% 0.33/0.66 % and_num.simps(4)
% 0.33/0.66 thf(fact_9309_and__num_Osimps_I8_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.33/0.66 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_num.simps(8)
% 0.33/0.66 thf(fact_9310_and__num_Osimps_I6_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.33/0.66 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_num.simps(6)
% 0.33/0.66 thf(fact_9311_xor__num_Osimps_I9_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.33/0.66 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % xor_num.simps(9)
% 0.33/0.66 thf(fact_9312_xor__num_Osimps_I7_J,axiom,
% 0.33/0.66 ! [M: num] :
% 0.33/0.66 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 0.33/0.66 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % xor_num.simps(7)
% 0.33/0.66 thf(fact_9313_xor__num_Osimps_I4_J,axiom,
% 0.33/0.66 ! [M: num] :
% 0.33/0.66 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 0.33/0.66 = ( some_num @ ( bit1 @ M ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % xor_num.simps(4)
% 0.33/0.66 thf(fact_9314_xor__num_Osimps_I3_J,axiom,
% 0.33/0.66 ! [N: num] :
% 0.33/0.66 ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
% 0.33/0.66 = ( some_num @ ( bit0 @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % xor_num.simps(3)
% 0.33/0.66 thf(fact_9315_xor__num_Osimps_I2_J,axiom,
% 0.33/0.66 ! [N: num] :
% 0.33/0.66 ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
% 0.33/0.66 = ( some_num @ ( bit1 @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % xor_num.simps(2)
% 0.33/0.66 thf(fact_9316_and__num_Osimps_I9_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.33/0.66 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.33/0.66 @ ^ [N7: num] : ( some_num @ ( bit1 @ N7 ) )
% 0.33/0.66 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % and_num.simps(9)
% 0.33/0.66 thf(fact_9317_xor__num_Osimps_I6_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.33/0.66 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % xor_num.simps(6)
% 0.33/0.66 thf(fact_9318_xor__num_Osimps_I8_J,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.33/0.66 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % xor_num.simps(8)
% 0.33/0.66 thf(fact_9319_xor__num__dict,axiom,
% 0.33/0.66 bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% 0.33/0.66
% 0.33/0.66 % xor_num_dict
% 0.33/0.66 thf(fact_9320_and__num__dict,axiom,
% 0.33/0.66 bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% 0.33/0.66
% 0.33/0.66 % and_num_dict
% 0.33/0.66 thf(fact_9321_Bit__Operations_Otake__bit__num__code,axiom,
% 0.33/0.66 ( bit_take_bit_num
% 0.33/0.66 = ( ^ [N3: nat,M3: num] :
% 0.33/0.66 ( produc478579273971653890on_num
% 0.33/0.66 @ ^ [A2: nat,X3: num] :
% 0.33/0.66 ( case_nat_option_num @ none_num
% 0.33/0.66 @ ^ [O: nat] :
% 0.33/0.66 ( case_num_option_num @ ( some_num @ one )
% 0.33/0.66 @ ^ [P3: num] :
% 0.33/0.66 ( case_o6005452278849405969um_num @ none_num
% 0.33/0.66 @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 0.33/0.66 @ ( bit_take_bit_num @ O @ P3 ) )
% 0.33/0.66 @ ^ [P3: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P3 ) ) )
% 0.33/0.66 @ X3 )
% 0.33/0.66 @ A2 )
% 0.33/0.66 @ ( product_Pair_nat_num @ N3 @ M3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Bit_Operations.take_bit_num_code
% 0.33/0.66 thf(fact_9322_of__nat__eq__id,axiom,
% 0.33/0.66 semiri1316708129612266289at_nat = id_nat ).
% 0.33/0.66
% 0.33/0.66 % of_nat_eq_id
% 0.33/0.66 thf(fact_9323_less__int__def,axiom,
% 0.33/0.66 ( ord_less_int
% 0.33/0.66 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 0.33/0.66 @ ( produc8739625826339149834_nat_o
% 0.33/0.66 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.66 ( produc6081775807080527818_nat_o
% 0.33/0.66 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % less_int_def
% 0.33/0.66 thf(fact_9324_less__eq__int__def,axiom,
% 0.33/0.66 ( ord_less_eq_int
% 0.33/0.66 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 0.33/0.66 @ ( produc8739625826339149834_nat_o
% 0.33/0.66 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.66 ( produc6081775807080527818_nat_o
% 0.33/0.66 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % less_eq_int_def
% 0.33/0.66 thf(fact_9325_nat__def,axiom,
% 0.33/0.66 ( nat2
% 0.33/0.66 = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % nat_def
% 0.33/0.66 thf(fact_9326_positive__rat,axiom,
% 0.33/0.66 ! [A: int,B: int] :
% 0.33/0.66 ( ( positive @ ( fract @ A @ B ) )
% 0.33/0.66 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % positive_rat
% 0.33/0.66 thf(fact_9327_list__encode_Opelims,axiom,
% 0.33/0.66 ! [X: list_nat,Y: nat] :
% 0.33/0.66 ( ( ( nat_list_encode @ X )
% 0.33/0.66 = Y )
% 0.33/0.66 => ( ( accp_list_nat @ nat_list_encode_rel @ X )
% 0.33/0.66 => ( ( ( X = nil_nat )
% 0.33/0.66 => ( ( Y = zero_zero_nat )
% 0.33/0.66 => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
% 0.33/0.66 => ~ ! [X4: nat,Xs2: list_nat] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( cons_nat @ X4 @ Xs2 ) )
% 0.33/0.66 => ( ( Y
% 0.33/0.66 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) )
% 0.33/0.66 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % list_encode.pelims
% 0.33/0.66 thf(fact_9328_num__of__integer__code,axiom,
% 0.33/0.66 ( code_num_of_integer
% 0.33/0.66 = ( ^ [K2: code_integer] :
% 0.33/0.66 ( if_num @ ( ord_le3102999989581377725nteger @ K2 @ one_one_Code_integer ) @ one
% 0.33/0.66 @ ( produc7336495610019696514er_num
% 0.33/0.66 @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
% 0.33/0.66 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % num_of_integer_code
% 0.33/0.66 thf(fact_9329_Rat_Opositive__add,axiom,
% 0.33/0.66 ! [X: rat,Y: rat] :
% 0.33/0.66 ( ( positive @ X )
% 0.33/0.66 => ( ( positive @ Y )
% 0.33/0.66 => ( positive @ ( plus_plus_rat @ X @ Y ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Rat.positive_add
% 0.33/0.66 thf(fact_9330_Rat_Opositive__mult,axiom,
% 0.33/0.66 ! [X: rat,Y: rat] :
% 0.33/0.66 ( ( positive @ X )
% 0.33/0.66 => ( ( positive @ Y )
% 0.33/0.66 => ( positive @ ( times_times_rat @ X @ Y ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Rat.positive_mult
% 0.33/0.66 thf(fact_9331_less__rat__def,axiom,
% 0.33/0.66 ( ord_less_rat
% 0.33/0.66 = ( ^ [X3: rat,Y6: rat] : ( positive @ ( minus_minus_rat @ Y6 @ X3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % less_rat_def
% 0.33/0.66 thf(fact_9332_Rat_Opositive_Orep__eq,axiom,
% 0.33/0.66 ( positive
% 0.33/0.66 = ( ^ [X3: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X3 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X3 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Rat.positive.rep_eq
% 0.33/0.66 thf(fact_9333_Rat_Opositive__def,axiom,
% 0.33/0.66 ( positive
% 0.33/0.66 = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
% 0.33/0.66 @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Rat.positive_def
% 0.33/0.66 thf(fact_9334_upt__rec__numeral,axiom,
% 0.33/0.66 ! [M: num,N: num] :
% 0.33/0.66 ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.33/0.66 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.33/0.66 = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
% 0.33/0.66 & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.33/0.66 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.33/0.66 = nil_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upt_rec_numeral
% 0.33/0.66 thf(fact_9335_tl__upt,axiom,
% 0.33/0.66 ! [M: nat,N: nat] :
% 0.33/0.66 ( ( tl_nat @ ( upt @ M @ N ) )
% 0.33/0.66 = ( upt @ ( suc @ M ) @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % tl_upt
% 0.33/0.66 thf(fact_9336_drop__upt,axiom,
% 0.33/0.66 ! [M: nat,I: nat,J: nat] :
% 0.33/0.66 ( ( drop_nat @ M @ ( upt @ I @ J ) )
% 0.33/0.66 = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% 0.33/0.66
% 0.33/0.66 % drop_upt
% 0.33/0.66 thf(fact_9337_length__upt,axiom,
% 0.33/0.66 ! [I: nat,J: nat] :
% 0.33/0.66 ( ( size_size_list_nat @ ( upt @ I @ J ) )
% 0.33/0.66 = ( minus_minus_nat @ J @ I ) ) ).
% 0.33/0.66
% 0.33/0.66 % length_upt
% 0.33/0.66 thf(fact_9338_take__upt,axiom,
% 0.33/0.66 ! [I: nat,M: nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
% 0.33/0.66 => ( ( take_nat @ M @ ( upt @ I @ N ) )
% 0.33/0.66 = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % take_upt
% 0.33/0.66 thf(fact_9339_upt__eq__Nil__conv,axiom,
% 0.33/0.66 ! [I: nat,J: nat] :
% 0.33/0.66 ( ( ( upt @ I @ J )
% 0.33/0.66 = nil_nat )
% 0.33/0.66 = ( ( J = zero_zero_nat )
% 0.33/0.66 | ( ord_less_eq_nat @ J @ I ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upt_eq_Nil_conv
% 0.33/0.66 thf(fact_9340_nth__upt,axiom,
% 0.33/0.66 ! [I: nat,K: nat,J: nat] :
% 0.33/0.66 ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
% 0.33/0.66 => ( ( nth_nat @ ( upt @ I @ J ) @ K )
% 0.33/0.66 = ( plus_plus_nat @ I @ K ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % nth_upt
% 0.33/0.66 thf(fact_9341_mono__Suc,axiom,
% 0.33/0.66 order_mono_nat_nat @ suc ).
% 0.33/0.66
% 0.33/0.66 % mono_Suc
% 0.33/0.66 thf(fact_9342_atLeastAtMost__upt,axiom,
% 0.33/0.66 ( set_or1269000886237332187st_nat
% 0.33/0.66 = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ N3 @ ( suc @ M3 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % atLeastAtMost_upt
% 0.33/0.66 thf(fact_9343_atLeast__upt,axiom,
% 0.33/0.66 ( set_ord_lessThan_nat
% 0.33/0.66 = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % atLeast_upt
% 0.33/0.66 thf(fact_9344_upt__conv__Cons__Cons,axiom,
% 0.33/0.66 ! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
% 0.33/0.66 ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
% 0.33/0.66 = ( upt @ M @ Q2 ) )
% 0.33/0.66 = ( ( cons_nat @ N @ Ns )
% 0.33/0.66 = ( upt @ ( suc @ M ) @ Q2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upt_conv_Cons_Cons
% 0.33/0.66 thf(fact_9345_upt__0,axiom,
% 0.33/0.66 ! [I: nat] :
% 0.33/0.66 ( ( upt @ I @ zero_zero_nat )
% 0.33/0.66 = nil_nat ) ).
% 0.33/0.66
% 0.33/0.66 % upt_0
% 0.33/0.66 thf(fact_9346_greaterThanAtMost__upt,axiom,
% 0.33/0.66 ( set_or6659071591806873216st_nat
% 0.33/0.66 = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ ( suc @ M3 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % greaterThanAtMost_upt
% 0.33/0.66 thf(fact_9347_greaterThanLessThan__upt,axiom,
% 0.33/0.66 ( set_or5834768355832116004an_nat
% 0.33/0.66 = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ M3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % greaterThanLessThan_upt
% 0.33/0.66 thf(fact_9348_mono__times__nat,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % mono_times_nat
% 0.33/0.66 thf(fact_9349_atMost__upto,axiom,
% 0.33/0.66 ( set_ord_atMost_nat
% 0.33/0.66 = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N3 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % atMost_upto
% 0.33/0.66 thf(fact_9350_upt__conv__Cons,axiom,
% 0.33/0.66 ! [I: nat,J: nat] :
% 0.33/0.66 ( ( ord_less_nat @ I @ J )
% 0.33/0.66 => ( ( upt @ I @ J )
% 0.33/0.66 = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upt_conv_Cons
% 0.33/0.66 thf(fact_9351_upt__add__eq__append,axiom,
% 0.33/0.66 ! [I: nat,J: nat,K: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.66 => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
% 0.33/0.66 = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upt_add_eq_append
% 0.33/0.66 thf(fact_9352_upt__eq__Cons__conv,axiom,
% 0.33/0.66 ! [I: nat,J: nat,X: nat,Xs: list_nat] :
% 0.33/0.66 ( ( ( upt @ I @ J )
% 0.33/0.66 = ( cons_nat @ X @ Xs ) )
% 0.33/0.66 = ( ( ord_less_nat @ I @ J )
% 0.33/0.66 & ( I = X )
% 0.33/0.66 & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
% 0.33/0.66 = Xs ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upt_eq_Cons_conv
% 0.33/0.66 thf(fact_9353_upt__rec,axiom,
% 0.33/0.66 ( upt
% 0.33/0.66 = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upt_rec
% 0.33/0.66 thf(fact_9354_mono__ge2__power__minus__self,axiom,
% 0.33/0.66 ! [K: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.33/0.66 => ( order_mono_nat_nat
% 0.33/0.66 @ ^ [M3: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M3 ) @ M3 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % mono_ge2_power_minus_self
% 0.33/0.66 thf(fact_9355_upt__Suc__append,axiom,
% 0.33/0.66 ! [I: nat,J: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.66 => ( ( upt @ I @ ( suc @ J ) )
% 0.33/0.66 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upt_Suc_append
% 0.33/0.66 thf(fact_9356_upt__Suc,axiom,
% 0.33/0.66 ! [I: nat,J: nat] :
% 0.33/0.66 ( ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.66 => ( ( upt @ I @ ( suc @ J ) )
% 0.33/0.66 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 0.33/0.66 & ( ~ ( ord_less_eq_nat @ I @ J )
% 0.33/0.66 => ( ( upt @ I @ ( suc @ J ) )
% 0.33/0.66 = nil_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % upt_Suc
% 0.33/0.66 thf(fact_9357_of__rat__dense,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_real @ X @ Y )
% 0.33/0.66 => ? [Q6: rat] :
% 0.33/0.66 ( ( ord_less_real @ X @ ( field_7254667332652039916t_real @ Q6 ) )
% 0.33/0.66 & ( ord_less_real @ ( field_7254667332652039916t_real @ Q6 ) @ Y ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % of_rat_dense
% 0.33/0.66 thf(fact_9358_range__mod,axiom,
% 0.33/0.66 ! [N: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ( ( image_nat_nat
% 0.33/0.66 @ ^ [M3: nat] : ( modulo_modulo_nat @ M3 @ N )
% 0.33/0.66 @ top_top_set_nat )
% 0.33/0.66 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % range_mod
% 0.33/0.66 thf(fact_9359_UNIV__nat__eq,axiom,
% 0.33/0.66 ( top_top_set_nat
% 0.33/0.66 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % UNIV_nat_eq
% 0.33/0.66 thf(fact_9360_plus__rat__def,axiom,
% 0.33/0.66 ( plus_plus_rat
% 0.33/0.66 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 0.33/0.66 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % plus_rat_def
% 0.33/0.66 thf(fact_9361_card__UNIV__unit,axiom,
% 0.33/0.66 ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 0.33/0.66 = one_one_nat ) ).
% 0.33/0.66
% 0.33/0.66 % card_UNIV_unit
% 0.33/0.66 thf(fact_9362_range__mult,axiom,
% 0.33/0.66 ! [A: real] :
% 0.33/0.66 ( ( ( A = zero_zero_real )
% 0.33/0.66 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.33/0.66 = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 0.33/0.66 & ( ( A != zero_zero_real )
% 0.33/0.66 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.33/0.66 = top_top_set_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % range_mult
% 0.33/0.66 thf(fact_9363_range__abs__Nats,axiom,
% 0.33/0.66 ( ( image_int_int @ abs_abs_int @ top_top_set_int )
% 0.33/0.66 = semiring_1_Nats_int ) ).
% 0.33/0.66
% 0.33/0.66 % range_abs_Nats
% 0.33/0.66 thf(fact_9364_card__UNIV__bool,axiom,
% 0.33/0.66 ( ( finite_card_o @ top_top_set_o )
% 0.33/0.66 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % card_UNIV_bool
% 0.33/0.66 thf(fact_9365_map__Suc__upt,axiom,
% 0.33/0.66 ! [M: nat,N: nat] :
% 0.33/0.66 ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
% 0.33/0.66 = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % map_Suc_upt
% 0.33/0.66 thf(fact_9366_infinite__UNIV__int,axiom,
% 0.33/0.66 ~ ( finite_finite_int @ top_top_set_int ) ).
% 0.33/0.66
% 0.33/0.66 % infinite_UNIV_int
% 0.33/0.66 thf(fact_9367_surj__list__encode,axiom,
% 0.33/0.66 ( ( image_list_nat_nat @ nat_list_encode @ top_top_set_list_nat )
% 0.33/0.66 = top_top_set_nat ) ).
% 0.33/0.66
% 0.33/0.66 % surj_list_encode
% 0.33/0.66 thf(fact_9368_bij__list__encode,axiom,
% 0.33/0.66 bij_be8532844293280997160at_nat @ nat_list_encode @ top_top_set_list_nat @ top_top_set_nat ).
% 0.33/0.66
% 0.33/0.66 % bij_list_encode
% 0.33/0.66 thf(fact_9369_surj__prod__encode,axiom,
% 0.33/0.66 ( ( image_2486076414777270412at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat )
% 0.33/0.66 = top_top_set_nat ) ).
% 0.33/0.66
% 0.33/0.66 % surj_prod_encode
% 0.33/0.66 thf(fact_9370_bij__prod__encode,axiom,
% 0.33/0.66 bij_be5333170631980326235at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat @ top_top_set_nat ).
% 0.33/0.66
% 0.33/0.66 % bij_prod_encode
% 0.33/0.66 thf(fact_9371_map__add__upt,axiom,
% 0.33/0.66 ! [N: nat,M: nat] :
% 0.33/0.66 ( ( map_nat_nat
% 0.33/0.66 @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
% 0.33/0.66 @ ( upt @ zero_zero_nat @ M ) )
% 0.33/0.66 = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % map_add_upt
% 0.33/0.66 thf(fact_9372_int__in__range__abs,axiom,
% 0.33/0.66 ! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% 0.33/0.66
% 0.33/0.66 % int_in_range_abs
% 0.33/0.66 thf(fact_9373_map__decr__upt,axiom,
% 0.33/0.66 ! [M: nat,N: nat] :
% 0.33/0.66 ( ( map_nat_nat
% 0.33/0.66 @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
% 0.33/0.66 @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.33/0.66 = ( upt @ M @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % map_decr_upt
% 0.33/0.66 thf(fact_9374_one__rat__def,axiom,
% 0.33/0.66 ( one_one_rat
% 0.33/0.66 = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % one_rat_def
% 0.33/0.66 thf(fact_9375_Fract_Oabs__eq,axiom,
% 0.33/0.66 ( fract
% 0.33/0.66 = ( ^ [Xa4: int,X3: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X3 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Fract.abs_eq
% 0.33/0.66 thf(fact_9376_zero__rat__def,axiom,
% 0.33/0.66 ( zero_zero_rat
% 0.33/0.66 = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % zero_rat_def
% 0.33/0.66 thf(fact_9377_times__rat__def,axiom,
% 0.33/0.66 ( times_times_rat
% 0.33/0.66 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 0.33/0.66 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % times_rat_def
% 0.33/0.66 thf(fact_9378_inverse__rat__def,axiom,
% 0.33/0.66 ( inverse_inverse_rat
% 0.33/0.66 = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
% 0.33/0.66 @ ^ [X3: product_prod_int_int] :
% 0.33/0.66 ( if_Pro3027730157355071871nt_int
% 0.33/0.66 @ ( ( product_fst_int_int @ X3 )
% 0.33/0.66 = zero_zero_int )
% 0.33/0.66 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.33/0.66 @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % inverse_rat_def
% 0.33/0.66 thf(fact_9379_root__def,axiom,
% 0.33/0.66 ( root
% 0.33/0.66 = ( ^ [N3: nat,X3: real] :
% 0.33/0.66 ( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
% 0.33/0.66 @ ( the_in5290026491893676941l_real @ top_top_set_real
% 0.33/0.66 @ ^ [Y6: real] : ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N3 ) )
% 0.33/0.66 @ X3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % root_def
% 0.33/0.66 thf(fact_9380_card__UNIV__char,axiom,
% 0.33/0.66 ( ( finite_card_char @ top_top_set_char )
% 0.33/0.66 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % card_UNIV_char
% 0.33/0.66 thf(fact_9381_plus__rat_Oabs__eq,axiom,
% 0.33/0.66 ! [Xa: product_prod_int_int,X: product_prod_int_int] :
% 0.33/0.66 ( ( ratrel @ Xa @ Xa )
% 0.33/0.66 => ( ( ratrel @ X @ X )
% 0.33/0.66 => ( ( plus_plus_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 0.33/0.66 = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % plus_rat.abs_eq
% 0.33/0.66 thf(fact_9382_ratrel__iff,axiom,
% 0.33/0.66 ( ratrel
% 0.33/0.66 = ( ^ [X3: product_prod_int_int,Y6: product_prod_int_int] :
% 0.33/0.66 ( ( ( product_snd_int_int @ X3 )
% 0.33/0.66 != zero_zero_int )
% 0.33/0.66 & ( ( product_snd_int_int @ Y6 )
% 0.33/0.66 != zero_zero_int )
% 0.33/0.66 & ( ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) )
% 0.33/0.66 = ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % ratrel_iff
% 0.33/0.66 thf(fact_9383_one__rat_Orsp,axiom,
% 0.33/0.66 ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).
% 0.33/0.66
% 0.33/0.66 % one_rat.rsp
% 0.33/0.66 thf(fact_9384_zero__rat_Orsp,axiom,
% 0.33/0.66 ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).
% 0.33/0.66
% 0.33/0.66 % zero_rat.rsp
% 0.33/0.66 thf(fact_9385_ratrel__def,axiom,
% 0.33/0.66 ( ratrel
% 0.33/0.66 = ( ^ [X3: product_prod_int_int,Y6: product_prod_int_int] :
% 0.33/0.66 ( ( ( product_snd_int_int @ X3 )
% 0.33/0.66 != zero_zero_int )
% 0.33/0.66 & ( ( product_snd_int_int @ Y6 )
% 0.33/0.66 != zero_zero_int )
% 0.33/0.66 & ( ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) )
% 0.33/0.66 = ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % ratrel_def
% 0.33/0.66 thf(fact_9386_times__rat_Oabs__eq,axiom,
% 0.33/0.66 ! [Xa: product_prod_int_int,X: product_prod_int_int] :
% 0.33/0.66 ( ( ratrel @ Xa @ Xa )
% 0.33/0.66 => ( ( ratrel @ X @ X )
% 0.33/0.66 => ( ( times_times_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 0.33/0.66 = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % times_rat.abs_eq
% 0.33/0.66 thf(fact_9387_Rat_Opositive_Oabs__eq,axiom,
% 0.33/0.66 ! [X: product_prod_int_int] :
% 0.33/0.66 ( ( ratrel @ X @ X )
% 0.33/0.66 => ( ( positive @ ( abs_Rat @ X ) )
% 0.33/0.66 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Rat.positive.abs_eq
% 0.33/0.66 thf(fact_9388_inverse__rat_Oabs__eq,axiom,
% 0.33/0.66 ! [X: product_prod_int_int] :
% 0.33/0.66 ( ( ratrel @ X @ X )
% 0.33/0.66 => ( ( inverse_inverse_rat @ ( abs_Rat @ X ) )
% 0.33/0.66 = ( abs_Rat
% 0.33/0.66 @ ( if_Pro3027730157355071871nt_int
% 0.33/0.66 @ ( ( product_fst_int_int @ X )
% 0.33/0.66 = zero_zero_int )
% 0.33/0.66 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.33/0.66 @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % inverse_rat.abs_eq
% 0.33/0.66 thf(fact_9389_UNIV__char__of__nat,axiom,
% 0.33/0.66 ( top_top_set_char
% 0.33/0.66 = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % UNIV_char_of_nat
% 0.33/0.66 thf(fact_9390_char_Osize_I2_J,axiom,
% 0.33/0.66 ! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
% 0.33/0.66 ( ( size_size_char @ ( char2 @ X1 @ X2 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
% 0.33/0.66 = zero_zero_nat ) ).
% 0.33/0.66
% 0.33/0.66 % char.size(2)
% 0.33/0.66 thf(fact_9391_nat__of__char__less__256,axiom,
% 0.33/0.66 ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % nat_of_char_less_256
% 0.33/0.66 thf(fact_9392_range__nat__of__char,axiom,
% 0.33/0.66 ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 0.33/0.66 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % range_nat_of_char
% 0.33/0.66 thf(fact_9393_integer__of__char__code,axiom,
% 0.33/0.66 ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 0.33/0.66 ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 0.33/0.66 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % integer_of_char_code
% 0.33/0.66 thf(fact_9394_char_Osize__gen,axiom,
% 0.33/0.66 ! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
% 0.33/0.66 ( ( size_char @ ( char2 @ X1 @ X2 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
% 0.33/0.66 = zero_zero_nat ) ).
% 0.33/0.66
% 0.33/0.66 % char.size_gen
% 0.33/0.66 thf(fact_9395_String_Ochar__of__ascii__of,axiom,
% 0.33/0.66 ! [C: char] :
% 0.33/0.66 ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
% 0.33/0.66 = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % String.char_of_ascii_of
% 0.33/0.66 thf(fact_9396_DERIV__real__root__generic,axiom,
% 0.33/0.66 ! [N: nat,X: real,D3: real] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ( ( X != zero_zero_real )
% 0.33/0.66 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( D3
% 0.33/0.66 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 0.33/0.66 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.66 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.66 => ( D3
% 0.33/0.66 = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 0.33/0.66 => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.66 => ( D3
% 0.33/0.66 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_real_root_generic
% 0.33/0.66 thf(fact_9397_has__real__derivative__neg__dec__left,axiom,
% 0.33/0.66 ! [F: real > real,L: real,X: real,S2: set_real] :
% 0.33/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 0.33/0.66 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.33/0.66 => ? [D2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.33/0.66 & ! [H2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.33/0.66 => ( ( member_real @ ( minus_minus_real @ X @ H2 ) @ S2 )
% 0.33/0.66 => ( ( ord_less_real @ H2 @ D2 )
% 0.33/0.66 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H2 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % has_real_derivative_neg_dec_left
% 0.33/0.66 thf(fact_9398_has__real__derivative__pos__inc__left,axiom,
% 0.33/0.66 ! [F: real > real,L: real,X: real,S2: set_real] :
% 0.33/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.33/0.66 => ? [D2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.33/0.66 & ! [H2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.33/0.66 => ( ( member_real @ ( minus_minus_real @ X @ H2 ) @ S2 )
% 0.33/0.66 => ( ( ord_less_real @ H2 @ D2 )
% 0.33/0.66 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % has_real_derivative_pos_inc_left
% 0.33/0.66 thf(fact_9399_has__real__derivative__pos__inc__right,axiom,
% 0.33/0.66 ! [F: real > real,L: real,X: real,S2: set_real] :
% 0.33/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.33/0.66 => ? [D2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.33/0.66 & ! [H2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.33/0.66 => ( ( member_real @ ( plus_plus_real @ X @ H2 ) @ S2 )
% 0.33/0.66 => ( ( ord_less_real @ H2 @ D2 )
% 0.33/0.66 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H2 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % has_real_derivative_pos_inc_right
% 0.33/0.66 thf(fact_9400_has__real__derivative__neg__dec__right,axiom,
% 0.33/0.66 ! [F: real > real,L: real,X: real,S2: set_real] :
% 0.33/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 0.33/0.66 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.33/0.66 => ? [D2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.33/0.66 & ! [H2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.33/0.66 => ( ( member_real @ ( plus_plus_real @ X @ H2 ) @ S2 )
% 0.33/0.66 => ( ( ord_less_real @ H2 @ D2 )
% 0.33/0.66 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % has_real_derivative_neg_dec_right
% 0.33/0.66 thf(fact_9401_DERIV__const__ratio__const2,axiom,
% 0.33/0.66 ! [A: real,B: real,F: real > real,K: real] :
% 0.33/0.66 ( ( A != B )
% 0.33/0.66 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.66 => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
% 0.33/0.66 = K ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_const_ratio_const2
% 0.33/0.66 thf(fact_9402_DERIV__const__ratio__const,axiom,
% 0.33/0.66 ! [A: real,B: real,F: real > real,K: real] :
% 0.33/0.66 ( ( A != B )
% 0.33/0.66 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.66 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.33/0.66 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_const_ratio_const
% 0.33/0.66 thf(fact_9403_DERIV__neg__dec__right,axiom,
% 0.33/0.66 ! [F: real > real,L: real,X: real] :
% 0.33/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.66 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.33/0.66 => ? [D2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.33/0.66 & ! [H2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.33/0.66 => ( ( ord_less_real @ H2 @ D2 )
% 0.33/0.66 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_neg_dec_right
% 0.33/0.66 thf(fact_9404_DERIV__pos__inc__right,axiom,
% 0.33/0.66 ! [F: real > real,L: real,X: real] :
% 0.33/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.33/0.66 => ? [D2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.33/0.66 & ! [H2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.33/0.66 => ( ( ord_less_real @ H2 @ D2 )
% 0.33/0.66 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H2 ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_pos_inc_right
% 0.33/0.66 thf(fact_9405_DERIV__neg__dec__left,axiom,
% 0.33/0.66 ! [F: real > real,L: real,X: real] :
% 0.33/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.66 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.33/0.66 => ? [D2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.33/0.66 & ! [H2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.33/0.66 => ( ( ord_less_real @ H2 @ D2 )
% 0.33/0.66 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H2 ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_neg_dec_left
% 0.33/0.66 thf(fact_9406_DERIV__pos__inc__left,axiom,
% 0.33/0.66 ! [F: real > real,L: real,X: real] :
% 0.33/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.33/0.66 => ? [D2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.33/0.66 & ! [H2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.33/0.66 => ( ( ord_less_real @ H2 @ D2 )
% 0.33/0.66 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_pos_inc_left
% 0.33/0.66 thf(fact_9407_MVT2,axiom,
% 0.33/0.66 ! [A: real,B: real,F: real > real,F3: real > real] :
% 0.33/0.66 ( ( ord_less_real @ A @ B )
% 0.33/0.66 => ( ! [X4: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ A @ X4 )
% 0.33/0.66 => ( ( ord_less_eq_real @ X4 @ B )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.33/0.66 => ? [Z4: real] :
% 0.33/0.66 ( ( ord_less_real @ A @ Z4 )
% 0.33/0.66 & ( ord_less_real @ Z4 @ B )
% 0.33/0.66 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.33/0.66 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F3 @ Z4 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % MVT2
% 0.33/0.66 thf(fact_9408_DERIV__local__const,axiom,
% 0.33/0.66 ! [F: real > real,L: real,X: real,D: real] :
% 0.33/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.33/0.66 => ( ! [Y4: real] :
% 0.33/0.66 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 0.33/0.66 => ( ( F @ X )
% 0.33/0.66 = ( F @ Y4 ) ) )
% 0.33/0.66 => ( L = zero_zero_real ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_local_const
% 0.33/0.66 thf(fact_9409_DERIV__const__average,axiom,
% 0.33/0.66 ! [A: real,B: real,V: real > real,K: real] :
% 0.33/0.66 ( ( A != B )
% 0.33/0.66 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.66 => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_const_average
% 0.33/0.66 thf(fact_9410_DERIV__local__max,axiom,
% 0.33/0.66 ! [F: real > real,L: real,X: real,D: real] :
% 0.33/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.33/0.66 => ( ! [Y4: real] :
% 0.33/0.66 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 0.33/0.66 => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X ) ) )
% 0.33/0.66 => ( L = zero_zero_real ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_local_max
% 0.33/0.66 thf(fact_9411_DERIV__local__min,axiom,
% 0.33/0.66 ! [F: real > real,L: real,X: real,D: real] :
% 0.33/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.33/0.66 => ( ! [Y4: real] :
% 0.33/0.66 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 0.33/0.66 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.33/0.66 => ( L = zero_zero_real ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_local_min
% 0.33/0.66 thf(fact_9412_DERIV__ln__divide,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_ln_divide
% 0.33/0.66 thf(fact_9413_DERIV__pow,axiom,
% 0.33/0.66 ! [N: nat,X: real,S: set_real] :
% 0.33/0.66 ( has_fi5821293074295781190e_real
% 0.33/0.66 @ ^ [X3: real] : ( power_power_real @ X3 @ N )
% 0.33/0.66 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ X @ S ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_pow
% 0.33/0.66 thf(fact_9414_DERIV__fun__pow,axiom,
% 0.33/0.66 ! [G: real > real,M: real,X: real,N: nat] :
% 0.33/0.66 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.66 => ( has_fi5821293074295781190e_real
% 0.33/0.66 @ ^ [X3: real] : ( power_power_real @ ( G @ X3 ) @ N )
% 0.33/0.66 @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_fun_pow
% 0.33/0.66 thf(fact_9415_has__real__derivative__powr,axiom,
% 0.33/0.66 ! [Z2: real,R: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ Z2 )
% 0.33/0.66 => ( has_fi5821293074295781190e_real
% 0.33/0.66 @ ^ [Z5: real] : ( powr_real @ Z5 @ R )
% 0.33/0.66 @ ( times_times_real @ R @ ( powr_real @ Z2 @ ( minus_minus_real @ R @ one_one_real ) ) )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % has_real_derivative_powr
% 0.33/0.66 thf(fact_9416_DERIV__log,axiom,
% 0.33/0.66 ! [X: real,B: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ ( log2 @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_log
% 0.33/0.66 thf(fact_9417_DERIV__fun__powr,axiom,
% 0.33/0.66 ! [G: real > real,M: real,X: real,R: real] :
% 0.33/0.66 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 0.33/0.66 => ( has_fi5821293074295781190e_real
% 0.33/0.66 @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ R )
% 0.33/0.66 @ ( times_times_real @ ( times_times_real @ R @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_fun_powr
% 0.33/0.66 thf(fact_9418_DERIV__powr,axiom,
% 0.33/0.66 ! [G: real > real,M: real,X: real,F: real > real,R: real] :
% 0.33/0.66 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 0.33/0.66 => ( ( has_fi5821293074295781190e_real @ F @ R @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.66 => ( has_fi5821293074295781190e_real
% 0.33/0.66 @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) )
% 0.33/0.66 @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_powr
% 0.33/0.66 thf(fact_9419_DERIV__real__sqrt,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_real_sqrt
% 0.33/0.66 thf(fact_9420_DERIV__series_H,axiom,
% 0.33/0.66 ! [F: real > nat > real,F3: real > nat > real,X0: real,A: real,B: real,L4: nat > real] :
% 0.33/0.66 ( ! [N2: nat] :
% 0.33/0.66 ( has_fi5821293074295781190e_real
% 0.33/0.66 @ ^ [X3: real] : ( F @ X3 @ N2 )
% 0.33/0.66 @ ( F3 @ X0 @ N2 )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 0.33/0.66 => ( ! [X4: real] :
% 0.33/0.66 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.33/0.66 => ( summable_real @ ( F @ X4 ) ) )
% 0.33/0.66 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.33/0.66 => ( ( summable_real @ ( F3 @ X0 ) )
% 0.33/0.66 => ( ( summable_real @ L4 )
% 0.33/0.66 => ( ! [N2: nat,X4: real,Y4: real] :
% 0.33/0.66 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.33/0.66 => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.33/0.66 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X4 @ N2 ) @ ( F @ Y4 @ N2 ) ) ) @ ( times_times_real @ ( L4 @ N2 ) @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y4 ) ) ) ) ) )
% 0.33/0.66 => ( has_fi5821293074295781190e_real
% 0.33/0.66 @ ^ [X3: real] : ( suminf_real @ ( F @ X3 ) )
% 0.33/0.66 @ ( suminf_real @ ( F3 @ X0 ) )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_series'
% 0.33/0.66 thf(fact_9421_DERIV__arctan,axiom,
% 0.33/0.66 ! [X: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_arctan
% 0.33/0.66 thf(fact_9422_arsinh__real__has__field__derivative,axiom,
% 0.33/0.66 ! [X: real,A3: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A3 ) ) ).
% 0.33/0.66
% 0.33/0.66 % arsinh_real_has_field_derivative
% 0.33/0.66 thf(fact_9423_DERIV__real__sqrt__generic,axiom,
% 0.33/0.66 ! [X: real,D3: real] :
% 0.33/0.66 ( ( X != zero_zero_real )
% 0.33/0.66 => ( ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( D3
% 0.33/0.66 = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.33/0.66 => ( ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.66 => ( D3
% 0.33/0.66 = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ sqrt @ D3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_real_sqrt_generic
% 0.33/0.66 thf(fact_9424_arcosh__real__has__field__derivative,axiom,
% 0.33/0.66 ! [X: real,A3: set_real] :
% 0.33/0.66 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A3 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % arcosh_real_has_field_derivative
% 0.33/0.66 thf(fact_9425_artanh__real__has__field__derivative,axiom,
% 0.33/0.66 ! [X: real,A3: set_real] :
% 0.33/0.66 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A3 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % artanh_real_has_field_derivative
% 0.33/0.66 thf(fact_9426_DERIV__power__series_H,axiom,
% 0.33/0.66 ! [R3: real,F: nat > real,X0: real] :
% 0.33/0.66 ( ! [X4: real] :
% 0.33/0.66 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R3 ) @ R3 ) )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X4 @ N3 ) ) ) )
% 0.33/0.66 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R3 ) @ R3 ) )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.33/0.66 => ( has_fi5821293074295781190e_real
% 0.33/0.66 @ ^ [X3: real] :
% 0.33/0.66 ( suminf_real
% 0.33/0.66 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X3 @ ( suc @ N3 ) ) ) )
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X0 @ N3 ) ) )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_power_series'
% 0.33/0.66 thf(fact_9427_DERIV__real__root,axiom,
% 0.33/0.66 ! [N: nat,X: real] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_real_root
% 0.33/0.66 thf(fact_9428_DERIV__arccos,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_arccos
% 0.33/0.66 thf(fact_9429_DERIV__arcsin,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_arcsin
% 0.33/0.66 thf(fact_9430_Maclaurin__all__le__objl,axiom,
% 0.33/0.66 ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 0.33/0.66 ( ( ( ( Diff @ zero_zero_nat )
% 0.33/0.66 = F )
% 0.33/0.66 & ! [M2: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.33/0.66 => ? [T3: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.33/0.66 & ( ( F @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_all_le_objl
% 0.33/0.66 thf(fact_9431_Maclaurin__all__le,axiom,
% 0.33/0.66 ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 0.33/0.66 ( ( ( Diff @ zero_zero_nat )
% 0.33/0.66 = F )
% 0.33/0.66 => ( ! [M2: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.66 => ? [T3: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.33/0.66 & ( ( F @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_all_le
% 0.33/0.66 thf(fact_9432_DERIV__odd__real__root,axiom,
% 0.33/0.66 ! [N: nat,X: real] :
% 0.33/0.66 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.66 => ( ( X != zero_zero_real )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_odd_real_root
% 0.33/0.66 thf(fact_9433_Maclaurin__minus,axiom,
% 0.33/0.66 ! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.33/0.66 ( ( ord_less_real @ H @ zero_zero_real )
% 0.33/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ( ( ( Diff @ zero_zero_nat )
% 0.33/0.66 = F )
% 0.33/0.66 => ( ! [M2: nat,T3: real] :
% 0.33/0.66 ( ( ( ord_less_nat @ M2 @ N )
% 0.33/0.66 & ( ord_less_eq_real @ H @ T3 )
% 0.33/0.66 & ( ord_less_eq_real @ T3 @ zero_zero_real ) )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.33/0.66 => ? [T3: real] :
% 0.33/0.66 ( ( ord_less_real @ H @ T3 )
% 0.33/0.66 & ( ord_less_real @ T3 @ zero_zero_real )
% 0.33/0.66 & ( ( F @ H )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_minus
% 0.33/0.66 thf(fact_9434_Maclaurin2,axiom,
% 0.33/0.66 ! [H: real,Diff: nat > real > real,F: real > real,N: nat] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ H )
% 0.33/0.66 => ( ( ( Diff @ zero_zero_nat )
% 0.33/0.66 = F )
% 0.33/0.66 => ( ! [M2: nat,T3: real] :
% 0.33/0.66 ( ( ( ord_less_nat @ M2 @ N )
% 0.33/0.66 & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.33/0.66 & ( ord_less_eq_real @ T3 @ H ) )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.33/0.66 => ? [T3: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ T3 )
% 0.33/0.66 & ( ord_less_eq_real @ T3 @ H )
% 0.33/0.66 & ( ( F @ H )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin2
% 0.33/0.66 thf(fact_9435_Maclaurin,axiom,
% 0.33/0.66 ! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ H )
% 0.33/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ( ( ( Diff @ zero_zero_nat )
% 0.33/0.66 = F )
% 0.33/0.66 => ( ! [M2: nat,T3: real] :
% 0.33/0.66 ( ( ( ord_less_nat @ M2 @ N )
% 0.33/0.66 & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.33/0.66 & ( ord_less_eq_real @ T3 @ H ) )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.33/0.66 => ? [T3: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ T3 )
% 0.33/0.66 & ( ord_less_real @ T3 @ H )
% 0.33/0.66 & ( ( F @ H )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin
% 0.33/0.66 thf(fact_9436_Maclaurin__all__lt,axiom,
% 0.33/0.66 ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 0.33/0.66 ( ( ( Diff @ zero_zero_nat )
% 0.33/0.66 = F )
% 0.33/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ( ( X != zero_zero_real )
% 0.33/0.66 => ( ! [M2: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.66 => ? [T3: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
% 0.33/0.66 & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.33/0.66 & ( ( F @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_all_lt
% 0.33/0.66 thf(fact_9437_Maclaurin__bi__le,axiom,
% 0.33/0.66 ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 0.33/0.66 ( ( ( Diff @ zero_zero_nat )
% 0.33/0.66 = F )
% 0.33/0.66 => ( ! [M2: nat,T3: real] :
% 0.33/0.66 ( ( ( ord_less_nat @ M2 @ N )
% 0.33/0.66 & ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) ) )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.33/0.66 => ? [T3: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 0.33/0.66 & ( ( F @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_bi_le
% 0.33/0.66 thf(fact_9438_Taylor__down,axiom,
% 0.33/0.66 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ( ( ( Diff @ zero_zero_nat )
% 0.33/0.66 = F )
% 0.33/0.66 => ( ! [M2: nat,T3: real] :
% 0.33/0.66 ( ( ( ord_less_nat @ M2 @ N )
% 0.33/0.66 & ( ord_less_eq_real @ A @ T3 )
% 0.33/0.66 & ( ord_less_eq_real @ T3 @ B ) )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.33/0.66 => ( ( ord_less_real @ A @ C )
% 0.33/0.66 => ( ( ord_less_eq_real @ C @ B )
% 0.33/0.66 => ? [T3: real] :
% 0.33/0.66 ( ( ord_less_real @ A @ T3 )
% 0.33/0.66 & ( ord_less_real @ T3 @ C )
% 0.33/0.66 & ( ( F @ A )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Taylor_down
% 0.33/0.66 thf(fact_9439_Taylor__up,axiom,
% 0.33/0.66 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ( ( ( Diff @ zero_zero_nat )
% 0.33/0.66 = F )
% 0.33/0.66 => ( ! [M2: nat,T3: real] :
% 0.33/0.66 ( ( ( ord_less_nat @ M2 @ N )
% 0.33/0.66 & ( ord_less_eq_real @ A @ T3 )
% 0.33/0.66 & ( ord_less_eq_real @ T3 @ B ) )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.33/0.66 => ( ( ord_less_eq_real @ A @ C )
% 0.33/0.66 => ( ( ord_less_real @ C @ B )
% 0.33/0.66 => ? [T3: real] :
% 0.33/0.66 ( ( ord_less_real @ C @ T3 )
% 0.33/0.66 & ( ord_less_real @ T3 @ B )
% 0.33/0.66 & ( ( F @ B )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Taylor_up
% 0.33/0.66 thf(fact_9440_Taylor,axiom,
% 0.33/0.66 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ( ( ( Diff @ zero_zero_nat )
% 0.33/0.66 = F )
% 0.33/0.66 => ( ! [M2: nat,T3: real] :
% 0.33/0.66 ( ( ( ord_less_nat @ M2 @ N )
% 0.33/0.66 & ( ord_less_eq_real @ A @ T3 )
% 0.33/0.66 & ( ord_less_eq_real @ T3 @ B ) )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.33/0.66 => ( ( ord_less_eq_real @ A @ C )
% 0.33/0.66 => ( ( ord_less_eq_real @ C @ B )
% 0.33/0.66 => ( ( ord_less_eq_real @ A @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ X @ B )
% 0.33/0.66 => ( ( X != C )
% 0.33/0.66 => ? [T3: real] :
% 0.33/0.66 ( ( ( ord_less_real @ X @ C )
% 0.33/0.66 => ( ( ord_less_real @ X @ T3 )
% 0.33/0.66 & ( ord_less_real @ T3 @ C ) ) )
% 0.33/0.66 & ( ~ ( ord_less_real @ X @ C )
% 0.33/0.66 => ( ( ord_less_real @ C @ T3 )
% 0.33/0.66 & ( ord_less_real @ T3 @ X ) ) )
% 0.33/0.66 & ( ( F @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Taylor
% 0.33/0.66 thf(fact_9441_Maclaurin__lemma2,axiom,
% 0.33/0.66 ! [N: nat,H: real,Diff: nat > real > real,K: nat,B5: real] :
% 0.33/0.66 ( ! [M2: nat,T3: real] :
% 0.33/0.66 ( ( ( ord_less_nat @ M2 @ N )
% 0.33/0.66 & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 0.33/0.66 & ( ord_less_eq_real @ T3 @ H ) )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 0.33/0.66 => ( ( N
% 0.33/0.66 = ( suc @ K ) )
% 0.33/0.66 => ! [M4: nat,T4: real] :
% 0.33/0.66 ( ( ( ord_less_nat @ M4 @ N )
% 0.33/0.66 & ( ord_less_eq_real @ zero_zero_real @ T4 )
% 0.33/0.66 & ( ord_less_eq_real @ T4 @ H ) )
% 0.33/0.66 => ( has_fi5821293074295781190e_real
% 0.33/0.66 @ ^ [U3: real] :
% 0.33/0.66 ( minus_minus_real @ ( Diff @ M4 @ U3 )
% 0.33/0.66 @ ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [P3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M4 @ P3 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P3 ) ) @ ( power_power_real @ U3 @ P3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M4 ) ) )
% 0.33/0.66 @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ U3 @ ( minus_minus_nat @ N @ M4 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M4 ) ) ) ) ) )
% 0.33/0.66 @ ( minus_minus_real @ ( Diff @ ( suc @ M4 ) @ T4 )
% 0.33/0.66 @ ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [P3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M4 ) @ P3 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P3 ) ) @ ( power_power_real @ T4 @ P3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M4 ) ) ) )
% 0.33/0.66 @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ T4 @ ( minus_minus_nat @ N @ ( suc @ M4 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M4 ) ) ) ) ) ) )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_lemma2
% 0.33/0.66 thf(fact_9442_DERIV__arctan__series,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.66 => ( has_fi5821293074295781190e_real
% 0.33/0.66 @ ^ [X10: real] :
% 0.33/0.66 ( suminf_real
% 0.33/0.66 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X10 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_arctan_series
% 0.33/0.66 thf(fact_9443_DERIV__even__real__root,axiom,
% 0.33/0.66 ! [N: nat,X: real] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.66 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % DERIV_even_real_root
% 0.33/0.66 thf(fact_9444_LIM__fun__gt__zero,axiom,
% 0.33/0.66 ! [F: real > real,L: real,C: real] :
% 0.33/0.66 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.33/0.66 => ? [R2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.33/0.66 & ! [X5: real] :
% 0.33/0.66 ( ( ( X5 != C )
% 0.33/0.66 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R2 ) )
% 0.33/0.66 => ( ord_less_real @ zero_zero_real @ ( F @ X5 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % LIM_fun_gt_zero
% 0.33/0.66 thf(fact_9445_LIM__fun__not__zero,axiom,
% 0.33/0.66 ! [F: real > real,L: real,C: real] :
% 0.33/0.66 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.33/0.66 => ( ( L != zero_zero_real )
% 0.33/0.66 => ? [R2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.33/0.66 & ! [X5: real] :
% 0.33/0.66 ( ( ( X5 != C )
% 0.33/0.66 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R2 ) )
% 0.33/0.66 => ( ( F @ X5 )
% 0.33/0.66 != zero_zero_real ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % LIM_fun_not_zero
% 0.33/0.66 thf(fact_9446_LIM__fun__less__zero,axiom,
% 0.33/0.66 ! [F: real > real,L: real,C: real] :
% 0.33/0.66 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.33/0.66 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.33/0.66 => ? [R2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.33/0.66 & ! [X5: real] :
% 0.33/0.66 ( ( ( X5 != C )
% 0.33/0.66 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R2 ) )
% 0.33/0.66 => ( ord_less_real @ ( F @ X5 ) @ zero_zero_real ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % LIM_fun_less_zero
% 0.33/0.66 thf(fact_9447_isCont__real__sqrt,axiom,
% 0.33/0.66 ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ sqrt ) ).
% 0.33/0.66
% 0.33/0.66 % isCont_real_sqrt
% 0.33/0.66 thf(fact_9448_isCont__real__root,axiom,
% 0.33/0.66 ! [X: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( root @ N ) ) ).
% 0.33/0.66
% 0.33/0.66 % isCont_real_root
% 0.33/0.66 thf(fact_9449_isCont__arcosh,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % isCont_arcosh
% 0.33/0.66 thf(fact_9450_LIM__cos__div__sin,axiom,
% 0.33/0.66 ( filterlim_real_real
% 0.33/0.66 @ ^ [X3: real] : ( divide_divide_real @ ( cos_real @ X3 ) @ ( sin_real @ X3 ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % LIM_cos_div_sin
% 0.33/0.66 thf(fact_9451_isCont__arccos,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % isCont_arccos
% 0.33/0.66 thf(fact_9452_isCont__arcsin,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % isCont_arcsin
% 0.33/0.66 thf(fact_9453_isCont__artanh,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % isCont_artanh
% 0.33/0.66 thf(fact_9454_isCont__inverse__function,axiom,
% 0.33/0.66 ! [D: real,X: real,G: real > real,F: real > real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ D )
% 0.33/0.66 => ( ! [Z4: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X ) ) @ D )
% 0.33/0.66 => ( ( G @ ( F @ Z4 ) )
% 0.33/0.66 = Z4 ) )
% 0.33/0.66 => ( ! [Z4: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X ) ) @ D )
% 0.33/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) )
% 0.33/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % isCont_inverse_function
% 0.33/0.66 thf(fact_9455_GMVT_H,axiom,
% 0.33/0.66 ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F3: real > real] :
% 0.33/0.66 ( ( ord_less_real @ A @ B )
% 0.33/0.66 => ( ! [Z4: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ A @ Z4 )
% 0.33/0.66 => ( ( ord_less_eq_real @ Z4 @ B )
% 0.33/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
% 0.33/0.66 => ( ! [Z4: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ A @ Z4 )
% 0.33/0.66 => ( ( ord_less_eq_real @ Z4 @ B )
% 0.33/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ G ) ) )
% 0.33/0.66 => ( ! [Z4: real] :
% 0.33/0.66 ( ( ord_less_real @ A @ Z4 )
% 0.33/0.66 => ( ( ord_less_real @ Z4 @ B )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
% 0.33/0.66 => ( ! [Z4: real] :
% 0.33/0.66 ( ( ord_less_real @ A @ Z4 )
% 0.33/0.66 => ( ( ord_less_real @ Z4 @ B )
% 0.33/0.66 => ( has_fi5821293074295781190e_real @ F @ ( F3 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
% 0.33/0.66 => ? [C3: real] :
% 0.33/0.66 ( ( ord_less_real @ A @ C3 )
% 0.33/0.66 & ( ord_less_real @ C3 @ B )
% 0.33/0.66 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
% 0.33/0.66 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F3 @ C3 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % GMVT'
% 0.33/0.66 thf(fact_9456_summable__Leibniz_I3_J,axiom,
% 0.33/0.66 ! [A: nat > real] :
% 0.33/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.66 => ( ( topolo6980174941875973593q_real @ A )
% 0.33/0.66 => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 0.33/0.66 => ! [N6: nat] :
% 0.33/0.66 ( member_real
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 0.33/0.66 @ ( set_or1222579329274155063t_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) )
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_Leibniz(3)
% 0.33/0.66 thf(fact_9457_summable__Leibniz_I2_J,axiom,
% 0.33/0.66 ! [A: nat > real] :
% 0.33/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.66 => ( ( topolo6980174941875973593q_real @ A )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 0.33/0.66 => ! [N6: nat] :
% 0.33/0.66 ( member_real
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 0.33/0.66 @ ( set_or1222579329274155063t_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_Leibniz(2)
% 0.33/0.66 thf(fact_9458_mult__nat__right__at__top,axiom,
% 0.33/0.66 ! [C: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.33/0.66 => ( filterlim_nat_nat
% 0.33/0.66 @ ^ [X3: nat] : ( times_times_nat @ X3 @ C )
% 0.33/0.66 @ at_top_nat
% 0.33/0.66 @ at_top_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % mult_nat_right_at_top
% 0.33/0.66 thf(fact_9459_mult__nat__left__at__top,axiom,
% 0.33/0.66 ! [C: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.33/0.66 => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % mult_nat_left_at_top
% 0.33/0.66 thf(fact_9460_LIMSEQ__root,axiom,
% 0.33/0.66 ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] : ( root @ N3 @ ( semiri5074537144036343181t_real @ N3 ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.33/0.66 @ at_top_nat ) ).
% 0.33/0.66
% 0.33/0.66 % LIMSEQ_root
% 0.33/0.66 thf(fact_9461_nested__sequence__unique,axiom,
% 0.33/0.66 ! [F: nat > real,G: nat > real] :
% 0.33/0.66 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.33/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
% 0.33/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.33/0.66 => ( ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.66 @ at_top_nat )
% 0.33/0.66 => ? [L3: real] :
% 0.33/0.66 ( ! [N6: nat] : ( ord_less_eq_real @ ( F @ N6 ) @ L3 )
% 0.33/0.66 & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat )
% 0.33/0.66 & ! [N6: nat] : ( ord_less_eq_real @ L3 @ ( G @ N6 ) )
% 0.33/0.66 & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % nested_sequence_unique
% 0.33/0.66 thf(fact_9462_lim__inverse__n_H,axiom,
% 0.33/0.66 ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.66 @ at_top_nat ) ).
% 0.33/0.66
% 0.33/0.66 % lim_inverse_n'
% 0.33/0.66 thf(fact_9463_LIMSEQ__root__const,axiom,
% 0.33/0.66 ! [C: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.66 => ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] : ( root @ N3 @ C )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.33/0.66 @ at_top_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % LIMSEQ_root_const
% 0.33/0.66 thf(fact_9464_LIMSEQ__inverse__real__of__nat,axiom,
% 0.33/0.66 ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.66 @ at_top_nat ) ).
% 0.33/0.66
% 0.33/0.66 % LIMSEQ_inverse_real_of_nat
% 0.33/0.66 thf(fact_9465_LIMSEQ__inverse__real__of__nat__add,axiom,
% 0.33/0.66 ! [R: real] :
% 0.33/0.66 ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] : ( plus_plus_real @ R @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ R )
% 0.33/0.66 @ at_top_nat ) ).
% 0.33/0.66
% 0.33/0.66 % LIMSEQ_inverse_real_of_nat_add
% 0.33/0.66 thf(fact_9466_increasing__LIMSEQ,axiom,
% 0.33/0.66 ! [F: nat > real,L: real] :
% 0.33/0.66 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.33/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ L )
% 0.33/0.66 => ( ! [E2: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.33/0.66 => ? [N6: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N6 ) @ E2 ) ) )
% 0.33/0.66 => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % increasing_LIMSEQ
% 0.33/0.66 thf(fact_9467_LIMSEQ__realpow__zero,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.66 => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % LIMSEQ_realpow_zero
% 0.33/0.66 thf(fact_9468_LIMSEQ__divide__realpow__zero,axiom,
% 0.33/0.66 ! [X: real,A: real] :
% 0.33/0.66 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.66 => ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N3 ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.66 @ at_top_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % LIMSEQ_divide_realpow_zero
% 0.33/0.66 thf(fact_9469_LIMSEQ__abs__realpow__zero2,axiom,
% 0.33/0.66 ! [C: real] :
% 0.33/0.66 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.33/0.66 => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % LIMSEQ_abs_realpow_zero2
% 0.33/0.66 thf(fact_9470_LIMSEQ__abs__realpow__zero,axiom,
% 0.33/0.66 ! [C: real] :
% 0.33/0.66 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.33/0.66 => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % LIMSEQ_abs_realpow_zero
% 0.33/0.66 thf(fact_9471_LIMSEQ__inverse__realpow__zero,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.66 => ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N3 ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.66 @ at_top_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % LIMSEQ_inverse_realpow_zero
% 0.33/0.66 thf(fact_9472_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
% 0.33/0.66 ! [R: real] :
% 0.33/0.66 ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] : ( plus_plus_real @ R @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ R )
% 0.33/0.66 @ at_top_nat ) ).
% 0.33/0.66
% 0.33/0.66 % LIMSEQ_inverse_real_of_nat_add_minus
% 0.33/0.66 thf(fact_9473_tendsto__exp__limit__sequentially,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ) @ N3 )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 0.33/0.66 @ at_top_nat ) ).
% 0.33/0.66
% 0.33/0.66 % tendsto_exp_limit_sequentially
% 0.33/0.66 thf(fact_9474_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 0.33/0.66 ! [R: real] :
% 0.33/0.66 ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] : ( times_times_real @ R @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ R )
% 0.33/0.66 @ at_top_nat ) ).
% 0.33/0.66
% 0.33/0.66 % LIMSEQ_inverse_real_of_nat_add_minus_mult
% 0.33/0.66 thf(fact_9475_summable__Leibniz_I1_J,axiom,
% 0.33/0.66 ! [A: nat > real] :
% 0.33/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.66 => ( ( topolo6980174941875973593q_real @ A )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_Leibniz(1)
% 0.33/0.66 thf(fact_9476_summable,axiom,
% 0.33/0.66 ! [A: nat > real] :
% 0.33/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.33/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable
% 0.33/0.66 thf(fact_9477_cos__diff__limit__1,axiom,
% 0.33/0.66 ! [Theta: nat > real,Theta2: real] :
% 0.33/0.66 ( ( filterlim_nat_real
% 0.33/0.66 @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.33/0.66 @ at_top_nat )
% 0.33/0.66 => ~ ! [K3: nat > int] :
% 0.33/0.66 ~ ( filterlim_nat_real
% 0.33/0.66 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ Theta2 )
% 0.33/0.66 @ at_top_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_diff_limit_1
% 0.33/0.66 thf(fact_9478_cos__limit__1,axiom,
% 0.33/0.66 ! [Theta: nat > real] :
% 0.33/0.66 ( ( filterlim_nat_real
% 0.33/0.66 @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.33/0.66 @ at_top_nat )
% 0.33/0.66 => ? [K3: nat > int] :
% 0.33/0.66 ( filterlim_nat_real
% 0.33/0.66 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.66 @ at_top_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_limit_1
% 0.33/0.66 thf(fact_9479_summable__Leibniz_I4_J,axiom,
% 0.33/0.66 ! [A: nat > real] :
% 0.33/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.66 => ( ( topolo6980174941875973593q_real @ A )
% 0.33/0.66 => ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] :
% 0.33/0.66 ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 0.33/0.66 @ at_top_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_Leibniz(4)
% 0.33/0.66 thf(fact_9480_zeroseq__arctan__series,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.66 => ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.66 @ at_top_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % zeroseq_arctan_series
% 0.33/0.66 thf(fact_9481_summable__Leibniz_H_I2_J,axiom,
% 0.33/0.66 ! [A: nat > real,N: nat] :
% 0.33/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.33/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.33/0.66 => ( ord_less_eq_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_Leibniz'(2)
% 0.33/0.66 thf(fact_9482_summable__Leibniz_H_I3_J,axiom,
% 0.33/0.66 ! [A: nat > real] :
% 0.33/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.33/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.33/0.66 => ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] :
% 0.33/0.66 ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 0.33/0.66 @ at_top_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_Leibniz'(3)
% 0.33/0.66 thf(fact_9483_sums__alternating__upper__lower,axiom,
% 0.33/0.66 ! [A: nat > real] :
% 0.33/0.66 ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.33/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.33/0.66 => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.66 => ? [L3: real] :
% 0.33/0.66 ( ! [N6: nat] :
% 0.33/0.66 ( ord_less_eq_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
% 0.33/0.66 @ L3 )
% 0.33/0.66 & ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] :
% 0.33/0.66 ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ L3 )
% 0.33/0.66 @ at_top_nat )
% 0.33/0.66 & ! [N6: nat] :
% 0.33/0.66 ( ord_less_eq_real @ L3
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) )
% 0.33/0.66 & ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] :
% 0.33/0.66 ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ L3 )
% 0.33/0.66 @ at_top_nat ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_alternating_upper_lower
% 0.33/0.66 thf(fact_9484_summable__Leibniz_I5_J,axiom,
% 0.33/0.66 ! [A: nat > real] :
% 0.33/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.66 => ( ( topolo6980174941875973593q_real @ A )
% 0.33/0.66 => ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] :
% 0.33/0.66 ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 0.33/0.66 @ at_top_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_Leibniz(5)
% 0.33/0.66 thf(fact_9485_summable__Leibniz_H_I5_J,axiom,
% 0.33/0.66 ! [A: nat > real] :
% 0.33/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.33/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.33/0.66 => ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] :
% 0.33/0.66 ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 0.33/0.66 @ at_top_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_Leibniz'(5)
% 0.33/0.66 thf(fact_9486_summable__Leibniz_H_I4_J,axiom,
% 0.33/0.66 ! [A: nat > real,N: nat] :
% 0.33/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.33/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.33/0.66 => ( ord_less_eq_real
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_Leibniz'(4)
% 0.33/0.66 thf(fact_9487_real__bounded__linear,axiom,
% 0.33/0.66 ( real_V5970128139526366754l_real
% 0.33/0.66 = ( ^ [F2: real > real] :
% 0.33/0.66 ? [C4: real] :
% 0.33/0.66 ( F2
% 0.33/0.66 = ( ^ [X3: real] : ( times_times_real @ X3 @ C4 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_bounded_linear
% 0.33/0.66 thf(fact_9488_filterlim__Suc,axiom,
% 0.33/0.66 filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% 0.33/0.66
% 0.33/0.66 % filterlim_Suc
% 0.33/0.66 thf(fact_9489_tendsto__arctan__at__bot,axiom,
% 0.33/0.66 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% 0.33/0.66
% 0.33/0.66 % tendsto_arctan_at_bot
% 0.33/0.66 thf(fact_9490_tendsto__exp__limit__at__right,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( filterlim_real_real
% 0.33/0.66 @ ^ [Y6: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y6 ) ) @ ( divide_divide_real @ one_one_real @ Y6 ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tendsto_exp_limit_at_right
% 0.33/0.66 thf(fact_9491_artanh__real__at__right__1,axiom,
% 0.33/0.66 filterlim_real_real @ artanh_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ one_one_real ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % artanh_real_at_right_1
% 0.33/0.66 thf(fact_9492_filterlim__tan__at__right,axiom,
% 0.33/0.66 filterlim_real_real @ tan_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % filterlim_tan_at_right
% 0.33/0.66 thf(fact_9493_tanh__real__at__bot,axiom,
% 0.33/0.66 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).
% 0.33/0.66
% 0.33/0.66 % tanh_real_at_bot
% 0.33/0.66 thf(fact_9494_tendsto__arcosh__at__left__1,axiom,
% 0.33/0.66 filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % tendsto_arcosh_at_left_1
% 0.33/0.66 thf(fact_9495_filterlim__pow__at__bot__odd,axiom,
% 0.33/0.66 ! [N: nat,F: real > real,F4: filter_real] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
% 0.33/0.66 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.66 => ( filterlim_real_real
% 0.33/0.66 @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
% 0.33/0.66 @ at_bot_real
% 0.33/0.66 @ F4 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % filterlim_pow_at_bot_odd
% 0.33/0.66 thf(fact_9496_filterlim__pow__at__bot__even,axiom,
% 0.33/0.66 ! [N: nat,F: real > real,F4: filter_real] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.66 => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
% 0.33/0.66 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.66 => ( filterlim_real_real
% 0.33/0.66 @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
% 0.33/0.66 @ at_top_real
% 0.33/0.66 @ F4 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % filterlim_pow_at_bot_even
% 0.33/0.66 thf(fact_9497_sqrt__at__top,axiom,
% 0.33/0.66 filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).
% 0.33/0.66
% 0.33/0.66 % sqrt_at_top
% 0.33/0.66 thf(fact_9498_filterlim__real__sequentially,axiom,
% 0.33/0.66 filterlim_nat_real @ semiri5074537144036343181t_real @ at_top_real @ at_top_nat ).
% 0.33/0.66
% 0.33/0.66 % filterlim_real_sequentially
% 0.33/0.66 thf(fact_9499_filterlim__real__at__infinity__sequentially,axiom,
% 0.33/0.66 filterlim_nat_real @ semiri5074537144036343181t_real @ at_infinity_real @ at_top_nat ).
% 0.33/0.66
% 0.33/0.66 % filterlim_real_at_infinity_sequentially
% 0.33/0.66 thf(fact_9500_greaterThan__0,axiom,
% 0.33/0.66 ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 0.33/0.66 = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % greaterThan_0
% 0.33/0.66 thf(fact_9501_tanh__real__at__top,axiom,
% 0.33/0.66 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).
% 0.33/0.66
% 0.33/0.66 % tanh_real_at_top
% 0.33/0.66 thf(fact_9502_artanh__real__at__left__1,axiom,
% 0.33/0.66 filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % artanh_real_at_left_1
% 0.33/0.66 thf(fact_9503_greaterThan__Suc,axiom,
% 0.33/0.66 ! [K: nat] :
% 0.33/0.66 ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 0.33/0.66 = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % greaterThan_Suc
% 0.33/0.66 thf(fact_9504_ln__x__over__x__tendsto__0,axiom,
% 0.33/0.66 ( filterlim_real_real
% 0.33/0.66 @ ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ X3 ) @ X3 )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.66 @ at_top_real ) ).
% 0.33/0.66
% 0.33/0.66 % ln_x_over_x_tendsto_0
% 0.33/0.66 thf(fact_9505_tendsto__at__topI__sequentially__real,axiom,
% 0.33/0.66 ! [F: real > real,Y: real] :
% 0.33/0.66 ( ( order_mono_real_real @ F )
% 0.33/0.66 => ( ( filterlim_nat_real
% 0.33/0.66 @ ^ [N3: nat] : ( F @ ( semiri5074537144036343181t_real @ N3 ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ Y )
% 0.33/0.66 @ at_top_nat )
% 0.33/0.66 => ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Y ) @ at_top_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tendsto_at_topI_sequentially_real
% 0.33/0.66 thf(fact_9506_tendsto__power__div__exp__0,axiom,
% 0.33/0.66 ! [K: nat] :
% 0.33/0.66 ( filterlim_real_real
% 0.33/0.66 @ ^ [X3: real] : ( divide_divide_real @ ( power_power_real @ X3 @ K ) @ ( exp_real @ X3 ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.66 @ at_top_real ) ).
% 0.33/0.66
% 0.33/0.66 % tendsto_power_div_exp_0
% 0.33/0.66 thf(fact_9507_tendsto__exp__limit__at__top,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( filterlim_real_real
% 0.33/0.66 @ ^ [Y6: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y6 ) ) @ Y6 )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 0.33/0.66 @ at_top_real ) ).
% 0.33/0.66
% 0.33/0.66 % tendsto_exp_limit_at_top
% 0.33/0.66 thf(fact_9508_filterlim__tan__at__left,axiom,
% 0.33/0.66 filterlim_real_real @ tan_real @ at_top_real @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( set_or5984915006950818249n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % filterlim_tan_at_left
% 0.33/0.66 thf(fact_9509_tendsto__arctan__at__top,axiom,
% 0.33/0.66 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% 0.33/0.66
% 0.33/0.66 % tendsto_arctan_at_top
% 0.33/0.66 thf(fact_9510_lhopital__right__at__top,axiom,
% 0.33/0.66 ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
% 0.33/0.66 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.66 => ( ( eventually_real
% 0.33/0.66 @ ^ [X3: real] :
% 0.33/0.66 ( ( G2 @ X3 )
% 0.33/0.66 != zero_zero_real )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.66 => ( ( eventually_real
% 0.33/0.66 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.66 => ( ( eventually_real
% 0.33/0.66 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.66 => ( ( filterlim_real_real
% 0.33/0.66 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ Y )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.66 => ( filterlim_real_real
% 0.33/0.66 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.33/0.66 @ ( topolo2815343760600316023s_real @ Y )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % lhopital_right_at_top
% 0.33/0.66 thf(fact_9511_eventually__sequentially__Suc,axiom,
% 0.33/0.66 ! [P: nat > $o] :
% 0.33/0.66 ( ( eventually_nat
% 0.33/0.66 @ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
% 0.33/0.66 @ at_top_nat )
% 0.33/0.66 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % eventually_sequentially_Suc
% 0.33/0.66 thf(fact_9512_eventually__sequentially__seg,axiom,
% 0.33/0.66 ! [P: nat > $o,K: nat] :
% 0.33/0.66 ( ( eventually_nat
% 0.33/0.66 @ ^ [N3: nat] : ( P @ ( plus_plus_nat @ N3 @ K ) )
% 0.33/0.66 @ at_top_nat )
% 0.33/0.66 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % eventually_sequentially_seg
% 0.33/0.66 thf(fact_9513_sequentially__offset,axiom,
% 0.33/0.66 ! [P: nat > $o,K: nat] :
% 0.33/0.66 ( ( eventually_nat @ P @ at_top_nat )
% 0.33/0.66 => ( eventually_nat
% 0.33/0.66 @ ^ [I4: nat] : ( P @ ( plus_plus_nat @ I4 @ K ) )
% 0.33/0.66 @ at_top_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % sequentially_offset
% 0.33/0.66 thf(fact_9514_eventually__at__right__to__0,axiom,
% 0.33/0.66 ! [P: real > $o,A: real] :
% 0.33/0.66 ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.66 = ( eventually_real
% 0.33/0.66 @ ^ [X3: real] : ( P @ ( plus_plus_real @ X3 @ A ) )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % eventually_at_right_to_0
% 0.33/0.66 thf(fact_9515_lhopital__left__at__top__at__top,axiom,
% 0.33/0.66 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.33/0.66 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.66 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.66 => ( ( eventually_real
% 0.33/0.66 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.66 => ( ( eventually_real
% 0.33/0.66 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.66 => ( ( filterlim_real_real
% 0.33/0.66 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 0.33/0.66 @ at_top_real
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.66 => ( filterlim_real_real
% 0.33/0.66 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.33/0.66 @ at_top_real
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % lhopital_left_at_top_at_top
% 0.33/0.66 thf(fact_9516_lhopital__at__top__at__top,axiom,
% 0.33/0.66 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.33/0.66 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.66 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.66 => ( ( eventually_real
% 0.33/0.66 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.66 => ( ( eventually_real
% 0.33/0.66 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.66 => ( ( filterlim_real_real
% 0.33/0.66 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 0.33/0.66 @ at_top_real
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.66 => ( filterlim_real_real
% 0.33/0.66 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.33/0.66 @ at_top_real
% 0.33/0.66 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lhopital_at_top_at_top
% 0.33/0.67 thf(fact_9517_lhopital,axiom,
% 0.33/0.67 ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 0.33/0.67 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.67 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] :
% 0.33/0.67 ( ( G @ X3 )
% 0.33/0.67 != zero_zero_real )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] :
% 0.33/0.67 ( ( G2 @ X3 )
% 0.33/0.67 != zero_zero_real )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.67 => ( ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 0.33/0.67 @ F4
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.67 => ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.33/0.67 @ F4
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lhopital
% 0.33/0.67 thf(fact_9518_lhopital__left,axiom,
% 0.33/0.67 ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 0.33/0.67 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.67 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] :
% 0.33/0.67 ( ( G @ X3 )
% 0.33/0.67 != zero_zero_real )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] :
% 0.33/0.67 ( ( G2 @ X3 )
% 0.33/0.67 != zero_zero_real )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.67 => ( ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 0.33/0.67 @ F4
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.67 => ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.33/0.67 @ F4
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lhopital_left
% 0.33/0.67 thf(fact_9519_lhopital__right__at__top__at__top,axiom,
% 0.33/0.67 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.33/0.67 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.67 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.67 => ( ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 0.33/0.67 @ at_top_real
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.67 => ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.33/0.67 @ at_top_real
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lhopital_right_at_top_at_top
% 0.33/0.67 thf(fact_9520_lhopital__left__at__top__at__bot,axiom,
% 0.33/0.67 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.33/0.67 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.67 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.67 => ( ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 0.33/0.67 @ at_bot_real
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.67 => ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.33/0.67 @ at_bot_real
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lhopital_left_at_top_at_bot
% 0.33/0.67 thf(fact_9521_lhopital__at__top__at__bot,axiom,
% 0.33/0.67 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.33/0.67 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.67 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.67 => ( ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 0.33/0.67 @ at_bot_real
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.67 => ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.33/0.67 @ at_bot_real
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lhopital_at_top_at_bot
% 0.33/0.67 thf(fact_9522_lhopital__at__top,axiom,
% 0.33/0.67 ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
% 0.33/0.67 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] :
% 0.33/0.67 ( ( G2 @ X3 )
% 0.33/0.67 != zero_zero_real )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.67 => ( ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 0.33/0.67 @ ( topolo2815343760600316023s_real @ Y )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.67 => ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.33/0.67 @ ( topolo2815343760600316023s_real @ Y )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lhopital_at_top
% 0.33/0.67 thf(fact_9523_lhospital__at__top__at__top,axiom,
% 0.33/0.67 ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
% 0.33/0.67 ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] :
% 0.33/0.67 ( ( G2 @ X3 )
% 0.33/0.67 != zero_zero_real )
% 0.33/0.67 @ at_top_real )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ at_top_real )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ at_top_real )
% 0.33/0.67 => ( ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 0.33/0.67 @ ( topolo2815343760600316023s_real @ X )
% 0.33/0.67 @ at_top_real )
% 0.33/0.67 => ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.33/0.67 @ ( topolo2815343760600316023s_real @ X )
% 0.33/0.67 @ at_top_real ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lhospital_at_top_at_top
% 0.33/0.67 thf(fact_9524_lhopital__left__at__top,axiom,
% 0.33/0.67 ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
% 0.33/0.67 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] :
% 0.33/0.67 ( ( G2 @ X3 )
% 0.33/0.67 != zero_zero_real )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.67 => ( ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 0.33/0.67 @ ( topolo2815343760600316023s_real @ Y )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.67 => ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.33/0.67 @ ( topolo2815343760600316023s_real @ Y )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lhopital_left_at_top
% 0.33/0.67 thf(fact_9525_lhopital__right,axiom,
% 0.33/0.67 ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 0.33/0.67 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.67 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] :
% 0.33/0.67 ( ( G @ X3 )
% 0.33/0.67 != zero_zero_real )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] :
% 0.33/0.67 ( ( G2 @ X3 )
% 0.33/0.67 != zero_zero_real )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.67 => ( ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 0.33/0.67 @ F4
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.67 => ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.33/0.67 @ F4
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lhopital_right
% 0.33/0.67 thf(fact_9526_lhopital__right__0,axiom,
% 0.33/0.67 ! [F0: real > real,G0: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 0.33/0.67 ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.67 => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] :
% 0.33/0.67 ( ( G0 @ X3 )
% 0.33/0.67 != zero_zero_real )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] :
% 0.33/0.67 ( ( G2 @ X3 )
% 0.33/0.67 != zero_zero_real )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.67 => ( ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 0.33/0.67 @ F4
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.67 => ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F0 @ X3 ) @ ( G0 @ X3 ) )
% 0.33/0.67 @ F4
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lhopital_right_0
% 0.33/0.67 thf(fact_9527_lhopital__right__at__top__at__bot,axiom,
% 0.33/0.67 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.33/0.67 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.67 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.67 => ( ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 0.33/0.67 @ at_bot_real
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.67 => ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.33/0.67 @ at_bot_real
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lhopital_right_at_top_at_bot
% 0.33/0.67 thf(fact_9528_lhopital__right__0__at__top,axiom,
% 0.33/0.67 ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
% 0.33/0.67 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] :
% 0.33/0.67 ( ( G2 @ X3 )
% 0.33/0.67 != zero_zero_real )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.67 => ( ( eventually_real
% 0.33/0.67 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.67 => ( ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 0.33/0.67 @ ( topolo2815343760600316023s_real @ X )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.67 => ( filterlim_real_real
% 0.33/0.67 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.33/0.67 @ ( topolo2815343760600316023s_real @ X )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lhopital_right_0_at_top
% 0.33/0.67 thf(fact_9529_GMVT,axiom,
% 0.33/0.67 ! [A: real,B: real,F: real > real,G: real > real] :
% 0.33/0.67 ( ( ord_less_real @ A @ B )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( ( ord_less_eq_real @ A @ X4 )
% 0.33/0.67 & ( ord_less_eq_real @ X4 @ B ) )
% 0.33/0.67 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( ( ord_less_real @ A @ X4 )
% 0.33/0.67 & ( ord_less_real @ X4 @ B ) )
% 0.33/0.67 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( ( ord_less_eq_real @ A @ X4 )
% 0.33/0.67 & ( ord_less_eq_real @ X4 @ B ) )
% 0.33/0.67 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G ) )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( ( ord_less_real @ A @ X4 )
% 0.33/0.67 & ( ord_less_real @ X4 @ B ) )
% 0.33/0.67 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.33/0.67 => ? [G_c: real,F_c: real,C3: real] :
% 0.33/0.67 ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 0.33/0.67 & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 0.33/0.67 & ( ord_less_real @ A @ C3 )
% 0.33/0.67 & ( ord_less_real @ C3 @ B )
% 0.33/0.67 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
% 0.33/0.67 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % GMVT
% 0.33/0.67 thf(fact_9530_Bseq__realpow,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.67 => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Bseq_realpow
% 0.33/0.67 thf(fact_9531_MVT,axiom,
% 0.33/0.67 ! [A: real,B: real,F: real > real] :
% 0.33/0.67 ( ( ord_less_real @ A @ B )
% 0.33/0.67 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( ord_less_real @ A @ X4 )
% 0.33/0.67 => ( ( ord_less_real @ X4 @ B )
% 0.33/0.67 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.33/0.67 => ? [L3: real,Z4: real] :
% 0.33/0.67 ( ( ord_less_real @ A @ Z4 )
% 0.33/0.67 & ( ord_less_real @ Z4 @ B )
% 0.33/0.67 & ( has_fi5821293074295781190e_real @ F @ L3 @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) )
% 0.33/0.67 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.33/0.67 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % MVT
% 0.33/0.67 thf(fact_9532_continuous__on__arcosh_H,axiom,
% 0.33/0.67 ! [A3: set_real,F: real > real] :
% 0.33/0.67 ( ( topolo5044208981011980120l_real @ A3 @ F )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ A3 )
% 0.33/0.67 => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 0.33/0.67 => ( topolo5044208981011980120l_real @ A3
% 0.33/0.67 @ ^ [X3: real] : ( arcosh_real @ ( F @ X3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % continuous_on_arcosh'
% 0.33/0.67 thf(fact_9533_continuous__on__arccos_H,axiom,
% 0.33/0.67 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).
% 0.33/0.67
% 0.33/0.67 % continuous_on_arccos'
% 0.33/0.67 thf(fact_9534_continuous__on__arcsin_H,axiom,
% 0.33/0.67 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).
% 0.33/0.67
% 0.33/0.67 % continuous_on_arcsin'
% 0.33/0.67 thf(fact_9535_continuous__on__artanh,axiom,
% 0.33/0.67 ! [A3: set_real] :
% 0.33/0.67 ( ( ord_less_eq_set_real @ A3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 0.33/0.67 => ( topolo5044208981011980120l_real @ A3 @ artanh_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % continuous_on_artanh
% 0.33/0.67 thf(fact_9536_continuous__on__artanh_H,axiom,
% 0.33/0.67 ! [A3: set_real,F: real > real] :
% 0.33/0.67 ( ( topolo5044208981011980120l_real @ A3 @ F )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ A3 )
% 0.33/0.67 => ( member_real @ ( F @ X4 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
% 0.33/0.67 => ( topolo5044208981011980120l_real @ A3
% 0.33/0.67 @ ^ [X3: real] : ( artanh_real @ ( F @ X3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % continuous_on_artanh'
% 0.33/0.67 thf(fact_9537_mvt,axiom,
% 0.33/0.67 ! [A: real,B: real,F: real > real,F3: real > real > real] :
% 0.33/0.67 ( ( ord_less_real @ A @ B )
% 0.33/0.67 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( ord_less_real @ A @ X4 )
% 0.33/0.67 => ( ( ord_less_real @ X4 @ B )
% 0.33/0.67 => ( has_de1759254742604945161l_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.33/0.67 => ~ ! [Xi: real] :
% 0.33/0.67 ( ( ord_less_real @ A @ Xi )
% 0.33/0.67 => ( ( ord_less_real @ Xi @ B )
% 0.33/0.67 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.33/0.67 != ( F3 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % mvt
% 0.33/0.67 thf(fact_9538_card__length__sum__list__rec,axiom,
% 0.33/0.67 ! [M: nat,N5: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.33/0.67 => ( ( finite_card_list_nat
% 0.33/0.67 @ ( collect_list_nat
% 0.33/0.67 @ ^ [L2: list_nat] :
% 0.33/0.67 ( ( ( size_size_list_nat @ L2 )
% 0.33/0.67 = M )
% 0.33/0.67 & ( ( groups4561878855575611511st_nat @ L2 )
% 0.33/0.67 = N5 ) ) ) )
% 0.33/0.67 = ( plus_plus_nat
% 0.33/0.67 @ ( finite_card_list_nat
% 0.33/0.67 @ ( collect_list_nat
% 0.33/0.67 @ ^ [L2: list_nat] :
% 0.33/0.67 ( ( ( size_size_list_nat @ L2 )
% 0.33/0.67 = ( minus_minus_nat @ M @ one_one_nat ) )
% 0.33/0.67 & ( ( groups4561878855575611511st_nat @ L2 )
% 0.33/0.67 = N5 ) ) ) )
% 0.33/0.67 @ ( finite_card_list_nat
% 0.33/0.67 @ ( collect_list_nat
% 0.33/0.67 @ ^ [L2: list_nat] :
% 0.33/0.67 ( ( ( size_size_list_nat @ L2 )
% 0.33/0.67 = M )
% 0.33/0.67 & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
% 0.33/0.67 = N5 ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % card_length_sum_list_rec
% 0.33/0.67 thf(fact_9539_card__length__sum__list,axiom,
% 0.33/0.67 ! [M: nat,N5: nat] :
% 0.33/0.67 ( ( finite_card_list_nat
% 0.33/0.67 @ ( collect_list_nat
% 0.33/0.67 @ ^ [L2: list_nat] :
% 0.33/0.67 ( ( ( size_size_list_nat @ L2 )
% 0.33/0.67 = M )
% 0.33/0.67 & ( ( groups4561878855575611511st_nat @ L2 )
% 0.33/0.67 = N5 ) ) ) )
% 0.33/0.67 = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N5 @ M ) @ one_one_nat ) @ N5 ) ) ).
% 0.33/0.67
% 0.33/0.67 % card_length_sum_list
% 0.33/0.67 thf(fact_9540_surj__int__encode,axiom,
% 0.33/0.67 ( ( image_int_nat @ nat_int_encode @ top_top_set_int )
% 0.33/0.67 = top_top_set_nat ) ).
% 0.33/0.67
% 0.33/0.67 % surj_int_encode
% 0.33/0.67 thf(fact_9541_int__encode__eq,axiom,
% 0.33/0.67 ! [X: int,Y: int] :
% 0.33/0.67 ( ( ( nat_int_encode @ X )
% 0.33/0.67 = ( nat_int_encode @ Y ) )
% 0.33/0.67 = ( X = Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % int_encode_eq
% 0.33/0.67 thf(fact_9542_bij__int__encode,axiom,
% 0.33/0.67 bij_betw_int_nat @ nat_int_encode @ top_top_set_int @ top_top_set_nat ).
% 0.33/0.67
% 0.33/0.67 % bij_int_encode
% 0.33/0.67 thf(fact_9543_Sup__nat__empty,axiom,
% 0.33/0.67 ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
% 0.33/0.67 = zero_zero_nat ) ).
% 0.33/0.67
% 0.33/0.67 % Sup_nat_empty
% 0.33/0.67 thf(fact_9544_Inf__real__def,axiom,
% 0.33/0.67 ( comple4887499456419720421f_real
% 0.33/0.67 = ( ^ [X8: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X8 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Inf_real_def
% 0.33/0.67 thf(fact_9545_binomial__def,axiom,
% 0.33/0.67 ( binomial
% 0.33/0.67 = ( ^ [N3: nat,K2: nat] :
% 0.33/0.67 ( finite_card_set_nat
% 0.33/0.67 @ ( collect_set_nat
% 0.33/0.67 @ ^ [K7: set_nat] :
% 0.33/0.67 ( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) )
% 0.33/0.67 & ( ( finite_card_nat @ K7 )
% 0.33/0.67 = K2 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % binomial_def
% 0.33/0.67 thf(fact_9546_inj__sgn__power,axiom,
% 0.33/0.67 ! [N: nat] :
% 0.33/0.67 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.67 => ( inj_on_real_real
% 0.33/0.67 @ ^ [Y6: real] : ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N ) )
% 0.33/0.67 @ top_top_set_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % inj_sgn_power
% 0.33/0.67 thf(fact_9547_surj__int__decode,axiom,
% 0.33/0.67 ( ( image_nat_int @ nat_int_decode @ top_top_set_nat )
% 0.33/0.67 = top_top_set_int ) ).
% 0.33/0.67
% 0.33/0.67 % surj_int_decode
% 0.33/0.67 thf(fact_9548_int__encode__inverse,axiom,
% 0.33/0.67 ! [X: int] :
% 0.33/0.67 ( ( nat_int_decode @ ( nat_int_encode @ X ) )
% 0.33/0.67 = X ) ).
% 0.33/0.67
% 0.33/0.67 % int_encode_inverse
% 0.33/0.67 thf(fact_9549_int__decode__inverse,axiom,
% 0.33/0.67 ! [N: nat] :
% 0.33/0.67 ( ( nat_int_encode @ ( nat_int_decode @ N ) )
% 0.33/0.67 = N ) ).
% 0.33/0.67
% 0.33/0.67 % int_decode_inverse
% 0.33/0.67 thf(fact_9550_int__decode__eq,axiom,
% 0.33/0.67 ! [X: nat,Y: nat] :
% 0.33/0.67 ( ( ( nat_int_decode @ X )
% 0.33/0.67 = ( nat_int_decode @ Y ) )
% 0.33/0.67 = ( X = Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % int_decode_eq
% 0.33/0.67 thf(fact_9551_bij__int__decode,axiom,
% 0.33/0.67 bij_betw_nat_int @ nat_int_decode @ top_top_set_nat @ top_top_set_int ).
% 0.33/0.67
% 0.33/0.67 % bij_int_decode
% 0.33/0.67 thf(fact_9552_log__inj,axiom,
% 0.33/0.67 ! [B: real] :
% 0.33/0.67 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.67 => ( inj_on_real_real @ ( log2 @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % log_inj
% 0.33/0.67 thf(fact_9553_inj__int__decode,axiom,
% 0.33/0.67 ! [A3: set_nat] : ( inj_on_nat_int @ nat_int_decode @ A3 ) ).
% 0.33/0.67
% 0.33/0.67 % inj_int_decode
% 0.33/0.67 thf(fact_9554_inj__Suc,axiom,
% 0.33/0.67 ! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).
% 0.33/0.67
% 0.33/0.67 % inj_Suc
% 0.33/0.67 thf(fact_9555_inj__prod__encode,axiom,
% 0.33/0.67 ! [A3: set_Pr1261947904930325089at_nat] : ( inj_on2178005380612969504at_nat @ nat_prod_encode @ A3 ) ).
% 0.33/0.67
% 0.33/0.67 % inj_prod_encode
% 0.33/0.67 thf(fact_9556_inj__list__encode,axiom,
% 0.33/0.67 ! [A3: set_list_nat] : ( inj_on_list_nat_nat @ nat_list_encode @ A3 ) ).
% 0.33/0.67
% 0.33/0.67 % inj_list_encode
% 0.33/0.67 thf(fact_9557_inj__int__encode,axiom,
% 0.33/0.67 ! [A3: set_int] : ( inj_on_int_nat @ nat_int_encode @ A3 ) ).
% 0.33/0.67
% 0.33/0.67 % inj_int_encode
% 0.33/0.67 thf(fact_9558_inj__on__set__encode,axiom,
% 0.33/0.67 inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).
% 0.33/0.67
% 0.33/0.67 % inj_on_set_encode
% 0.33/0.67 thf(fact_9559_inj__on__diff__nat,axiom,
% 0.33/0.67 ! [N5: set_nat,K: nat] :
% 0.33/0.67 ( ! [N2: nat] :
% 0.33/0.67 ( ( member_nat @ N2 @ N5 )
% 0.33/0.67 => ( ord_less_eq_nat @ K @ N2 ) )
% 0.33/0.67 => ( inj_on_nat_nat
% 0.33/0.67 @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K )
% 0.33/0.67 @ N5 ) ) ).
% 0.33/0.67
% 0.33/0.67 % inj_on_diff_nat
% 0.33/0.67 thf(fact_9560_inj__on__char__of__nat,axiom,
% 0.33/0.67 inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % inj_on_char_of_nat
% 0.33/0.67 thf(fact_9561_Max__divisors__self__nat,axiom,
% 0.33/0.67 ! [N: nat] :
% 0.33/0.67 ( ( N != zero_zero_nat )
% 0.33/0.67 => ( ( lattic8265883725875713057ax_nat
% 0.33/0.67 @ ( collect_nat
% 0.33/0.67 @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ N ) ) )
% 0.33/0.67 = N ) ) ).
% 0.33/0.67
% 0.33/0.67 % Max_divisors_self_nat
% 0.33/0.67 thf(fact_9562_atLeastLessThan__add__Un,axiom,
% 0.33/0.67 ! [I: nat,J: nat,K: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.67 => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 0.33/0.67 = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % atLeastLessThan_add_Un
% 0.33/0.67 thf(fact_9563_card__le__Suc__Max,axiom,
% 0.33/0.67 ! [S2: set_nat] :
% 0.33/0.67 ( ( finite_finite_nat @ S2 )
% 0.33/0.67 => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % card_le_Suc_Max
% 0.33/0.67 thf(fact_9564_Sup__nat__def,axiom,
% 0.33/0.67 ( complete_Sup_Sup_nat
% 0.33/0.67 = ( ^ [X8: set_nat] : ( if_nat @ ( X8 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X8 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Sup_nat_def
% 0.33/0.67 thf(fact_9565_divide__nat__def,axiom,
% 0.33/0.67 ( divide_divide_nat
% 0.33/0.67 = ( ^ [M3: nat,N3: nat] :
% 0.33/0.67 ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
% 0.33/0.67 @ ( lattic8265883725875713057ax_nat
% 0.33/0.67 @ ( collect_nat
% 0.33/0.67 @ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N3 ) @ M3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % divide_nat_def
% 0.33/0.67 thf(fact_9566_gcd__is__Max__divisors__nat,axiom,
% 0.33/0.67 ! [N: nat,M: nat] :
% 0.33/0.67 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.67 => ( ( gcd_gcd_nat @ M @ N )
% 0.33/0.67 = ( lattic8265883725875713057ax_nat
% 0.33/0.67 @ ( collect_nat
% 0.33/0.67 @ ^ [D4: nat] :
% 0.33/0.67 ( ( dvd_dvd_nat @ D4 @ M )
% 0.33/0.67 & ( dvd_dvd_nat @ D4 @ N ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % gcd_is_Max_divisors_nat
% 0.33/0.67 thf(fact_9567_Gcd__eq__Max,axiom,
% 0.33/0.67 ! [M10: set_nat] :
% 0.33/0.67 ( ( finite_finite_nat @ M10 )
% 0.33/0.67 => ( ( M10 != bot_bot_set_nat )
% 0.33/0.67 => ( ~ ( member_nat @ zero_zero_nat @ M10 )
% 0.33/0.67 => ( ( gcd_Gcd_nat @ M10 )
% 0.33/0.67 = ( lattic8265883725875713057ax_nat
% 0.33/0.67 @ ( comple7806235888213564991et_nat
% 0.33/0.67 @ ( image_nat_set_nat
% 0.33/0.67 @ ^ [M3: nat] :
% 0.33/0.67 ( collect_nat
% 0.33/0.67 @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M3 ) )
% 0.33/0.67 @ M10 ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Gcd_eq_Max
% 0.33/0.67 thf(fact_9568_quotient__of__def,axiom,
% 0.33/0.67 ( quotient_of
% 0.33/0.67 = ( ^ [X3: rat] :
% 0.33/0.67 ( the_Pr4378521158711661632nt_int
% 0.33/0.67 @ ^ [Pair: product_prod_int_int] :
% 0.33/0.67 ( ( X3
% 0.33/0.67 = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
% 0.33/0.67 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
% 0.33/0.67 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % quotient_of_def
% 0.33/0.67 thf(fact_9569_coprime__abs__left__iff,axiom,
% 0.33/0.67 ! [K: int,L: int] :
% 0.33/0.67 ( ( algebr932160517623751201me_int @ ( abs_abs_int @ K ) @ L )
% 0.33/0.67 = ( algebr932160517623751201me_int @ K @ L ) ) ).
% 0.33/0.67
% 0.33/0.67 % coprime_abs_left_iff
% 0.33/0.67 thf(fact_9570_coprime__abs__right__iff,axiom,
% 0.33/0.67 ! [K: int,L: int] :
% 0.33/0.67 ( ( algebr932160517623751201me_int @ K @ ( abs_abs_int @ L ) )
% 0.33/0.67 = ( algebr932160517623751201me_int @ K @ L ) ) ).
% 0.33/0.67
% 0.33/0.67 % coprime_abs_right_iff
% 0.33/0.67 thf(fact_9571_Max__divisors__self__int,axiom,
% 0.33/0.67 ! [N: int] :
% 0.33/0.67 ( ( N != zero_zero_int )
% 0.33/0.67 => ( ( lattic8263393255366662781ax_int
% 0.33/0.67 @ ( collect_int
% 0.33/0.67 @ ^ [D4: int] : ( dvd_dvd_int @ D4 @ N ) ) )
% 0.33/0.67 = ( abs_abs_int @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Max_divisors_self_int
% 0.33/0.67 thf(fact_9572_normalize__stable,axiom,
% 0.33/0.67 ! [Q2: int,P2: int] :
% 0.33/0.67 ( ( ord_less_int @ zero_zero_int @ Q2 )
% 0.33/0.67 => ( ( algebr932160517623751201me_int @ P2 @ Q2 )
% 0.33/0.67 => ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
% 0.33/0.67 = ( product_Pair_int_int @ P2 @ Q2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % normalize_stable
% 0.33/0.67 thf(fact_9573_sup__nat__def,axiom,
% 0.33/0.67 sup_sup_nat = ord_max_nat ).
% 0.33/0.67
% 0.33/0.67 % sup_nat_def
% 0.33/0.67 thf(fact_9574_sup__int__def,axiom,
% 0.33/0.67 sup_sup_int = ord_max_int ).
% 0.33/0.67
% 0.33/0.67 % sup_int_def
% 0.33/0.67 thf(fact_9575_coprime__crossproduct__int,axiom,
% 0.33/0.67 ! [A: int,D: int,B: int,C: int] :
% 0.33/0.67 ( ( algebr932160517623751201me_int @ A @ D )
% 0.33/0.67 => ( ( algebr932160517623751201me_int @ B @ C )
% 0.33/0.67 => ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
% 0.33/0.67 = ( times_times_int @ ( abs_abs_int @ B ) @ ( abs_abs_int @ D ) ) )
% 0.33/0.67 = ( ( ( abs_abs_int @ A )
% 0.33/0.67 = ( abs_abs_int @ B ) )
% 0.33/0.67 & ( ( abs_abs_int @ C )
% 0.33/0.67 = ( abs_abs_int @ D ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % coprime_crossproduct_int
% 0.33/0.67 thf(fact_9576_Rat__cases,axiom,
% 0.33/0.67 ! [Q2: rat] :
% 0.33/0.67 ~ ! [A4: int,B4: int] :
% 0.33/0.67 ( ( Q2
% 0.33/0.67 = ( fract @ A4 @ B4 ) )
% 0.33/0.67 => ( ( ord_less_int @ zero_zero_int @ B4 )
% 0.33/0.67 => ~ ( algebr932160517623751201me_int @ A4 @ B4 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Rat_cases
% 0.33/0.67 thf(fact_9577_Rat__induct,axiom,
% 0.33/0.67 ! [P: rat > $o,Q2: rat] :
% 0.33/0.67 ( ! [A4: int,B4: int] :
% 0.33/0.67 ( ( ord_less_int @ zero_zero_int @ B4 )
% 0.33/0.67 => ( ( algebr932160517623751201me_int @ A4 @ B4 )
% 0.33/0.67 => ( P @ ( fract @ A4 @ B4 ) ) ) )
% 0.33/0.67 => ( P @ Q2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % Rat_induct
% 0.33/0.67 thf(fact_9578_coprime__common__divisor__int,axiom,
% 0.33/0.67 ! [A: int,B: int,X: int] :
% 0.33/0.67 ( ( algebr932160517623751201me_int @ A @ B )
% 0.33/0.67 => ( ( dvd_dvd_int @ X @ A )
% 0.33/0.67 => ( ( dvd_dvd_int @ X @ B )
% 0.33/0.67 => ( ( abs_abs_int @ X )
% 0.33/0.67 = one_one_int ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % coprime_common_divisor_int
% 0.33/0.67 thf(fact_9579_gcd__is__Max__divisors__int,axiom,
% 0.33/0.67 ! [N: int,M: int] :
% 0.33/0.67 ( ( N != zero_zero_int )
% 0.33/0.67 => ( ( gcd_gcd_int @ M @ N )
% 0.33/0.67 = ( lattic8263393255366662781ax_int
% 0.33/0.67 @ ( collect_int
% 0.33/0.67 @ ^ [D4: int] :
% 0.33/0.67 ( ( dvd_dvd_int @ D4 @ M )
% 0.33/0.67 & ( dvd_dvd_int @ D4 @ N ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % gcd_is_Max_divisors_int
% 0.33/0.67 thf(fact_9580_Rat__cases__nonzero,axiom,
% 0.33/0.67 ! [Q2: rat] :
% 0.33/0.67 ( ! [A4: int,B4: int] :
% 0.33/0.67 ( ( Q2
% 0.33/0.67 = ( fract @ A4 @ B4 ) )
% 0.33/0.67 => ( ( ord_less_int @ zero_zero_int @ B4 )
% 0.33/0.67 => ( ( A4 != zero_zero_int )
% 0.33/0.67 => ~ ( algebr932160517623751201me_int @ A4 @ B4 ) ) ) )
% 0.33/0.67 => ( Q2 = zero_zero_rat ) ) ).
% 0.33/0.67
% 0.33/0.67 % Rat_cases_nonzero
% 0.33/0.67 thf(fact_9581_quotient__of__unique,axiom,
% 0.33/0.67 ! [R: rat] :
% 0.33/0.67 ? [X4: product_prod_int_int] :
% 0.33/0.67 ( ( R
% 0.33/0.67 = ( fract @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) )
% 0.33/0.67 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X4 ) )
% 0.33/0.67 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) )
% 0.33/0.67 & ! [Y5: product_prod_int_int] :
% 0.33/0.67 ( ( ( R
% 0.33/0.67 = ( fract @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 0.33/0.67 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y5 ) )
% 0.33/0.67 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 0.33/0.67 => ( Y5 = X4 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % quotient_of_unique
% 0.33/0.67 thf(fact_9582_Rat_Opositive_Orsp,axiom,
% 0.33/0.67 ( bNF_re8699439704749558557nt_o_o @ ratrel
% 0.33/0.67 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.33/0.67 @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
% 0.33/0.67 @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Rat.positive.rsp
% 0.33/0.67 thf(fact_9583_Rats__abs__iff,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( member_real @ ( abs_abs_real @ X ) @ field_5140801741446780682s_real )
% 0.33/0.67 = ( member_real @ X @ field_5140801741446780682s_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % Rats_abs_iff
% 0.33/0.67 thf(fact_9584_coprime__int__iff,axiom,
% 0.33/0.67 ! [M: nat,N: nat] :
% 0.33/0.67 ( ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.33/0.67 = ( algebr934650988132801477me_nat @ M @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % coprime_int_iff
% 0.33/0.67 thf(fact_9585_coprime__nat__abs__right__iff,axiom,
% 0.33/0.67 ! [N: nat,K: int] :
% 0.33/0.67 ( ( algebr934650988132801477me_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 0.33/0.67 = ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% 0.33/0.67
% 0.33/0.67 % coprime_nat_abs_right_iff
% 0.33/0.67 thf(fact_9586_coprime__nat__abs__left__iff,axiom,
% 0.33/0.67 ! [K: int,N: nat] :
% 0.33/0.67 ( ( algebr934650988132801477me_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 0.33/0.67 = ( algebr932160517623751201me_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % coprime_nat_abs_left_iff
% 0.33/0.67 thf(fact_9587_coprime__crossproduct__nat,axiom,
% 0.33/0.67 ! [A: nat,D: nat,B: nat,C: nat] :
% 0.33/0.67 ( ( algebr934650988132801477me_nat @ A @ D )
% 0.33/0.67 => ( ( algebr934650988132801477me_nat @ B @ C )
% 0.33/0.67 => ( ( ( times_times_nat @ A @ C )
% 0.33/0.67 = ( times_times_nat @ B @ D ) )
% 0.33/0.67 = ( ( A = B )
% 0.33/0.67 & ( C = D ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % coprime_crossproduct_nat
% 0.33/0.67 thf(fact_9588_coprime__Suc__right__nat,axiom,
% 0.33/0.67 ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % coprime_Suc_right_nat
% 0.33/0.67 thf(fact_9589_coprime__Suc__left__nat,axiom,
% 0.33/0.67 ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ N ) @ N ) ).
% 0.33/0.67
% 0.33/0.67 % coprime_Suc_left_nat
% 0.33/0.67 thf(fact_9590_coprime__Suc__0__right,axiom,
% 0.33/0.67 ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % coprime_Suc_0_right
% 0.33/0.67 thf(fact_9591_coprime__Suc__0__left,axiom,
% 0.33/0.67 ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% 0.33/0.67
% 0.33/0.67 % coprime_Suc_0_left
% 0.33/0.67 thf(fact_9592_coprime__common__divisor__nat,axiom,
% 0.33/0.67 ! [A: nat,B: nat,X: nat] :
% 0.33/0.67 ( ( algebr934650988132801477me_nat @ A @ B )
% 0.33/0.67 => ( ( dvd_dvd_nat @ X @ A )
% 0.33/0.67 => ( ( dvd_dvd_nat @ X @ B )
% 0.33/0.67 => ( X = one_one_nat ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % coprime_common_divisor_nat
% 0.33/0.67 thf(fact_9593_Rats__abs__nat__div__natE,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( member_real @ X @ field_5140801741446780682s_real )
% 0.33/0.67 => ~ ! [M2: nat,N2: nat] :
% 0.33/0.67 ( ( N2 != zero_zero_nat )
% 0.33/0.67 => ( ( ( abs_abs_real @ X )
% 0.33/0.67 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.33/0.67 => ~ ( algebr934650988132801477me_nat @ M2 @ N2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Rats_abs_nat_div_natE
% 0.33/0.67 thf(fact_9594_Rats__no__bot__less,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ? [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 0.33/0.67 & ( ord_less_real @ X4 @ X ) ) ).
% 0.33/0.67
% 0.33/0.67 % Rats_no_bot_less
% 0.33/0.67 thf(fact_9595_Rats__dense__in__real,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_real @ X @ Y )
% 0.33/0.67 => ? [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 0.33/0.67 & ( ord_less_real @ X @ X4 )
% 0.33/0.67 & ( ord_less_real @ X4 @ Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Rats_dense_in_real
% 0.33/0.67 thf(fact_9596_Rats__no__top__le,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ? [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 0.33/0.67 & ( ord_less_eq_real @ X @ X4 ) ) ).
% 0.33/0.67
% 0.33/0.67 % Rats_no_top_le
% 0.33/0.67 thf(fact_9597_coprime__diff__one__right__nat,axiom,
% 0.33/0.67 ! [N: nat] :
% 0.33/0.67 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.67 => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % coprime_diff_one_right_nat
% 0.33/0.67 thf(fact_9598_coprime__diff__one__left__nat,axiom,
% 0.33/0.67 ! [N: nat] :
% 0.33/0.67 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.67 => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % coprime_diff_one_left_nat
% 0.33/0.67 thf(fact_9599_and__not__num_Opelims,axiom,
% 0.33/0.67 ! [X: num,Xa: num,Y: option_num] :
% 0.33/0.67 ( ( ( bit_and_not_num @ X @ Xa )
% 0.33/0.67 = Y )
% 0.33/0.67 => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( ( Y = none_num )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( some_num @ one ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ( ( Y = none_num )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( some_num @ ( bit0 @ M2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ one ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( some_num @ ( bit0 @ M2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ one ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.33/0.67 @ ^ [N7: num] : ( some_num @ ( bit1 @ N7 ) )
% 0.33/0.67 @ ( bit_and_not_num @ M2 @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ~ ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % and_not_num.pelims
% 0.33/0.67 thf(fact_9600_times__rat_Orsp,axiom,
% 0.33/0.67 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 0.33/0.67 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) )
% 0.33/0.67 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_rat.rsp
% 0.33/0.67 thf(fact_9601_integer__of__natural_Orsp,axiom,
% 0.33/0.67 ( bNF_re6650684261131312217nt_int
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ semiri1314217659103216013at_int
% 0.33/0.67 @ semiri1314217659103216013at_int ) ).
% 0.33/0.67
% 0.33/0.67 % integer_of_natural.rsp
% 0.33/0.67 thf(fact_9602_sub_Orsp,axiom,
% 0.33/0.67 ( bNF_re8402795839162346335um_int
% 0.33/0.67 @ ^ [Y3: num,Z: num] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re1822329894187522285nt_int
% 0.33/0.67 @ ^ [Y3: num,Z: num] : ( Y3 = Z )
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 0.33/0.67 @ ^ [M3: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) )
% 0.33/0.67 @ ^ [M3: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sub.rsp
% 0.33/0.67 thf(fact_9603_dup_Orsp,axiom,
% 0.33/0.67 ( bNF_re4712519889275205905nt_int
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ^ [K2: int] : ( plus_plus_int @ K2 @ K2 )
% 0.33/0.67 @ ^ [K2: int] : ( plus_plus_int @ K2 @ K2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % dup.rsp
% 0.33/0.67 thf(fact_9604_plus__integer_Orsp,axiom,
% 0.33/0.67 ( bNF_re711492959462206631nt_int
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re4712519889275205905nt_int
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 0.33/0.67 @ plus_plus_int
% 0.33/0.67 @ plus_plus_int ) ).
% 0.33/0.67
% 0.33/0.67 % plus_integer.rsp
% 0.33/0.67 thf(fact_9605_plus__natural_Orsp,axiom,
% 0.33/0.67 ( bNF_re1345281282404953727at_nat
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re5653821019739307937at_nat
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 0.33/0.67 @ plus_plus_nat
% 0.33/0.67 @ plus_plus_nat ) ).
% 0.33/0.67
% 0.33/0.67 % plus_natural.rsp
% 0.33/0.67 thf(fact_9606_Suc_Orsp,axiom,
% 0.33/0.67 ( bNF_re5653821019739307937at_nat
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ suc
% 0.33/0.67 @ suc ) ).
% 0.33/0.67
% 0.33/0.67 % Suc.rsp
% 0.33/0.67 thf(fact_9607_times__integer_Orsp,axiom,
% 0.33/0.67 ( bNF_re711492959462206631nt_int
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re4712519889275205905nt_int
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 0.33/0.67 @ times_times_int
% 0.33/0.67 @ times_times_int ) ).
% 0.33/0.67
% 0.33/0.67 % times_integer.rsp
% 0.33/0.67 thf(fact_9608_times__natural_Orsp,axiom,
% 0.33/0.67 ( bNF_re1345281282404953727at_nat
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re5653821019739307937at_nat
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 0.33/0.67 @ times_times_nat
% 0.33/0.67 @ times_times_nat ) ).
% 0.33/0.67
% 0.33/0.67 % times_natural.rsp
% 0.33/0.67 thf(fact_9609_minus__natural_Orsp,axiom,
% 0.33/0.67 ( bNF_re1345281282404953727at_nat
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re5653821019739307937at_nat
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 0.33/0.67 @ minus_minus_nat
% 0.33/0.67 @ minus_minus_nat ) ).
% 0.33/0.67
% 0.33/0.67 % minus_natural.rsp
% 0.33/0.67 thf(fact_9610_minus__integer_Orsp,axiom,
% 0.33/0.67 ( bNF_re711492959462206631nt_int
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re4712519889275205905nt_int
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 0.33/0.67 @ minus_minus_int
% 0.33/0.67 @ minus_minus_int ) ).
% 0.33/0.67
% 0.33/0.67 % minus_integer.rsp
% 0.33/0.67 thf(fact_9611_push__bit__natural_Orsp,axiom,
% 0.33/0.67 ( bNF_re1345281282404953727at_nat
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re5653821019739307937at_nat
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 0.33/0.67 @ bit_se547839408752420682it_nat
% 0.33/0.67 @ bit_se547839408752420682it_nat ) ).
% 0.33/0.67
% 0.33/0.67 % push_bit_natural.rsp
% 0.33/0.67 thf(fact_9612_push__bit__integer_Orsp,axiom,
% 0.33/0.67 ( bNF_re4785983289428654063nt_int
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re4712519889275205905nt_int
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 0.33/0.67 @ bit_se545348938243370406it_int
% 0.33/0.67 @ bit_se545348938243370406it_int ) ).
% 0.33/0.67
% 0.33/0.67 % push_bit_integer.rsp
% 0.33/0.67 thf(fact_9613_divide__natural_Orsp,axiom,
% 0.33/0.67 ( bNF_re1345281282404953727at_nat
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re5653821019739307937at_nat
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 0.33/0.67 @ divide_divide_nat
% 0.33/0.67 @ divide_divide_nat ) ).
% 0.33/0.67
% 0.33/0.67 % divide_natural.rsp
% 0.33/0.67 thf(fact_9614_divide__integer_Orsp,axiom,
% 0.33/0.67 ( bNF_re711492959462206631nt_int
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re4712519889275205905nt_int
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 0.33/0.67 @ divide_divide_int
% 0.33/0.67 @ divide_divide_int ) ).
% 0.33/0.67
% 0.33/0.67 % divide_integer.rsp
% 0.33/0.67 thf(fact_9615_gcd__integer_Orsp,axiom,
% 0.33/0.67 ( bNF_re711492959462206631nt_int
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re4712519889275205905nt_int
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 0.33/0.67 @ gcd_gcd_int
% 0.33/0.67 @ gcd_gcd_int ) ).
% 0.33/0.67
% 0.33/0.67 % gcd_integer.rsp
% 0.33/0.67 thf(fact_9616_Fract_Orsp,axiom,
% 0.33/0.67 ( bNF_re157797125943740599nt_int
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re6250860962936578807nt_int
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ratrel )
% 0.33/0.67 @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) )
% 0.33/0.67 @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Fract.rsp
% 0.33/0.67 thf(fact_9617_plus__rat_Orsp,axiom,
% 0.33/0.67 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 0.33/0.67 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) )
% 0.33/0.67 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_rat.rsp
% 0.33/0.67 thf(fact_9618_inverse__rat_Orsp,axiom,
% 0.33/0.67 ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
% 0.33/0.67 @ ^ [X3: product_prod_int_int] :
% 0.33/0.67 ( if_Pro3027730157355071871nt_int
% 0.33/0.67 @ ( ( product_fst_int_int @ X3 )
% 0.33/0.67 = zero_zero_int )
% 0.33/0.67 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.33/0.67 @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
% 0.33/0.67 @ ^ [X3: product_prod_int_int] :
% 0.33/0.67 ( if_Pro3027730157355071871nt_int
% 0.33/0.67 @ ( ( product_fst_int_int @ X3 )
% 0.33/0.67 = zero_zero_int )
% 0.33/0.67 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.33/0.67 @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % inverse_rat.rsp
% 0.33/0.67 thf(fact_9619_and__num_Opelims,axiom,
% 0.33/0.67 ! [X: num,Xa: num,Y: option_num] :
% 0.33/0.67 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 0.33/0.67 = Y )
% 0.33/0.67 => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( some_num @ one ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ( ( Y = none_num )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( some_num @ one ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( ( Y = none_num )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ one ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( some_num @ one ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ one ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ~ ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.33/0.67 @ ^ [N7: num] : ( some_num @ ( bit1 @ N7 ) )
% 0.33/0.67 @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % and_num.pelims
% 0.33/0.67 thf(fact_9620_xor__num_Opelims,axiom,
% 0.33/0.67 ! [X: num,Xa: num,Y: option_num] :
% 0.33/0.67 ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 0.33/0.67 = Y )
% 0.33/0.67 => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( ( Y = none_num )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( some_num @ ( bit1 @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( some_num @ ( bit0 @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( some_num @ ( bit1 @ M2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ one ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( some_num @ ( bit0 @ M2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ one ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ~ ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % xor_num.pelims
% 0.33/0.67 thf(fact_9621_or__not__num__neg_Opelims,axiom,
% 0.33/0.67 ! [X: num,Xa: num,Y: num] :
% 0.33/0.67 ( ( ( bit_or_not_num_neg @ X @ Xa )
% 0.33/0.67 = Y )
% 0.33/0.67 => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( ( Y = one )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ! [M2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M2 ) ) ) ) ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ! [M2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M2 ) ) ) ) ) )
% 0.33/0.67 => ( ! [N2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( bit0 @ one ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ one ) ) ) ) )
% 0.33/0.67 => ( ! [N2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ! [M2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit0 @ M2 ) ) ) ) ) )
% 0.33/0.67 => ( ! [N2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ! [M2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( bit0 @ ( bit_or_not_num_neg @ N2 @ M2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit1 @ M2 ) ) ) ) ) )
% 0.33/0.67 => ( ! [N2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( ( Y = one )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ one ) ) ) ) )
% 0.33/0.67 => ( ! [N2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ! [M2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit0 @ M2 ) ) ) ) ) )
% 0.33/0.67 => ~ ! [N2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ! [M2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit1 @ M2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % or_not_num_neg.pelims
% 0.33/0.67 thf(fact_9622_xor__num__rel__dict,axiom,
% 0.33/0.67 bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).
% 0.33/0.67
% 0.33/0.67 % xor_num_rel_dict
% 0.33/0.67 thf(fact_9623_and__num__rel__dict,axiom,
% 0.33/0.67 bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).
% 0.33/0.67
% 0.33/0.67 % and_num_rel_dict
% 0.33/0.67 thf(fact_9624_plus__rat_Otransfer,axiom,
% 0.33/0.67 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 0.33/0.67 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) )
% 0.33/0.67 @ plus_plus_rat ) ).
% 0.33/0.67
% 0.33/0.67 % plus_rat.transfer
% 0.33/0.67 thf(fact_9625_one__rat_Otransfer,axiom,
% 0.33/0.67 pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).
% 0.33/0.67
% 0.33/0.67 % one_rat.transfer
% 0.33/0.67 thf(fact_9626_zero__rat_Otransfer,axiom,
% 0.33/0.67 pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).
% 0.33/0.67
% 0.33/0.67 % zero_rat.transfer
% 0.33/0.67 thf(fact_9627_Fract_Otransfer,axiom,
% 0.33/0.67 ( bNF_re3461391660133120880nt_rat
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re2214769303045360666nt_rat
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ pcr_rat )
% 0.33/0.67 @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) )
% 0.33/0.67 @ fract ) ).
% 0.33/0.67
% 0.33/0.67 % Fract.transfer
% 0.33/0.67 thf(fact_9628_times__rat_Otransfer,axiom,
% 0.33/0.67 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 0.33/0.67 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) )
% 0.33/0.67 @ times_times_rat ) ).
% 0.33/0.67
% 0.33/0.67 % times_rat.transfer
% 0.33/0.67 thf(fact_9629_Rat_Opositive_Otransfer,axiom,
% 0.33/0.67 ( bNF_re1494630372529172596at_o_o @ pcr_rat
% 0.33/0.67 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.33/0.67 @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
% 0.33/0.67 @ positive ) ).
% 0.33/0.67
% 0.33/0.67 % Rat.positive.transfer
% 0.33/0.67 thf(fact_9630_inverse__rat_Otransfer,axiom,
% 0.33/0.67 ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
% 0.33/0.67 @ ^ [X3: product_prod_int_int] :
% 0.33/0.67 ( if_Pro3027730157355071871nt_int
% 0.33/0.67 @ ( ( product_fst_int_int @ X3 )
% 0.33/0.67 = zero_zero_int )
% 0.33/0.67 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.33/0.67 @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
% 0.33/0.67 @ inverse_inverse_rat ) ).
% 0.33/0.67
% 0.33/0.67 % inverse_rat.transfer
% 0.33/0.67 thf(fact_9631_times__int_Otransfer,axiom,
% 0.33/0.67 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.33/0.67 @ ( produc27273713700761075at_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.67 ( produc2626176000494625587at_nat
% 0.33/0.67 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) )
% 0.33/0.67 @ times_times_int ) ).
% 0.33/0.67
% 0.33/0.67 % times_int.transfer
% 0.33/0.67 thf(fact_9632_zero__int_Otransfer,axiom,
% 0.33/0.67 pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).
% 0.33/0.67
% 0.33/0.67 % zero_int.transfer
% 0.33/0.67 thf(fact_9633_int__transfer,axiom,
% 0.33/0.67 ( bNF_re6830278522597306478at_int
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ pcr_int
% 0.33/0.67 @ ^ [N3: nat] : ( product_Pair_nat_nat @ N3 @ zero_zero_nat )
% 0.33/0.67 @ semiri1314217659103216013at_int ) ).
% 0.33/0.67
% 0.33/0.67 % int_transfer
% 0.33/0.67 thf(fact_9634_uminus__int_Otransfer,axiom,
% 0.33/0.67 ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
% 0.33/0.67 @ ( produc2626176000494625587at_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 ) )
% 0.33/0.67 @ uminus_uminus_int ) ).
% 0.33/0.67
% 0.33/0.67 % uminus_int.transfer
% 0.33/0.67 thf(fact_9635_nat_Otransfer,axiom,
% 0.33/0.67 ( bNF_re4555766996558763186at_nat @ pcr_int
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 0.33/0.67 @ nat2 ) ).
% 0.33/0.67
% 0.33/0.67 % nat.transfer
% 0.33/0.67 thf(fact_9636_one__int_Otransfer,axiom,
% 0.33/0.67 pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% 0.33/0.67
% 0.33/0.67 % one_int.transfer
% 0.33/0.67 thf(fact_9637_less__int_Otransfer,axiom,
% 0.33/0.67 ( bNF_re717283939379294677_int_o @ pcr_int
% 0.33/0.67 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 0.33/0.67 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.33/0.67 @ ( produc8739625826339149834_nat_o
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.67 ( produc6081775807080527818_nat_o
% 0.33/0.67 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
% 0.33/0.67 @ ord_less_int ) ).
% 0.33/0.67
% 0.33/0.67 % less_int.transfer
% 0.33/0.67 thf(fact_9638_less__eq__int_Otransfer,axiom,
% 0.33/0.67 ( bNF_re717283939379294677_int_o @ pcr_int
% 0.33/0.67 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 0.33/0.67 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.33/0.67 @ ( produc8739625826339149834_nat_o
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.67 ( produc6081775807080527818_nat_o
% 0.33/0.67 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
% 0.33/0.67 @ ord_less_eq_int ) ).
% 0.33/0.67
% 0.33/0.67 % less_eq_int.transfer
% 0.33/0.67 thf(fact_9639_plus__int_Otransfer,axiom,
% 0.33/0.67 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.33/0.67 @ ( produc27273713700761075at_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.67 ( produc2626176000494625587at_nat
% 0.33/0.67 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) )
% 0.33/0.67 @ plus_plus_int ) ).
% 0.33/0.67
% 0.33/0.67 % plus_int.transfer
% 0.33/0.67 thf(fact_9640_minus__int_Otransfer,axiom,
% 0.33/0.67 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.33/0.67 @ ( produc27273713700761075at_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.67 ( produc2626176000494625587at_nat
% 0.33/0.67 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) )
% 0.33/0.67 @ minus_minus_int ) ).
% 0.33/0.67
% 0.33/0.67 % minus_int.transfer
% 0.33/0.67 thf(fact_9641_times__int_Orsp,axiom,
% 0.33/0.67 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.33/0.67 @ ( produc27273713700761075at_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.67 ( produc2626176000494625587at_nat
% 0.33/0.67 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) )
% 0.33/0.67 @ ( produc27273713700761075at_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.67 ( produc2626176000494625587at_nat
% 0.33/0.67 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_int.rsp
% 0.33/0.67 thf(fact_9642_intrel__iff,axiom,
% 0.33/0.67 ! [X: nat,Y: nat,U: nat,V: nat] :
% 0.33/0.67 ( ( intrel @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ U @ V ) )
% 0.33/0.67 = ( ( plus_plus_nat @ X @ V )
% 0.33/0.67 = ( plus_plus_nat @ U @ Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % intrel_iff
% 0.33/0.67 thf(fact_9643_int_Orel__eq__transfer,axiom,
% 0.33/0.67 ( bNF_re717283939379294677_int_o @ pcr_int
% 0.33/0.67 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 0.33/0.67 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.33/0.67 @ intrel
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) ) ).
% 0.33/0.67
% 0.33/0.67 % int.rel_eq_transfer
% 0.33/0.67 thf(fact_9644_int_Oabs__eq__iff,axiom,
% 0.33/0.67 ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 0.33/0.67 ( ( ( abs_Integ @ X )
% 0.33/0.67 = ( abs_Integ @ Y ) )
% 0.33/0.67 = ( intrel @ X @ Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % int.abs_eq_iff
% 0.33/0.67 thf(fact_9645_zero__int_Orsp,axiom,
% 0.33/0.67 intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.33/0.67
% 0.33/0.67 % zero_int.rsp
% 0.33/0.67 thf(fact_9646_uminus__int_Orsp,axiom,
% 0.33/0.67 ( bNF_re2241393799969408733at_nat @ intrel @ intrel
% 0.33/0.67 @ ( produc2626176000494625587at_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 ) )
% 0.33/0.67 @ ( produc2626176000494625587at_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % uminus_int.rsp
% 0.33/0.67 thf(fact_9647_nat_Orsp,axiom,
% 0.33/0.67 ( bNF_re8246922863344978751at_nat @ intrel
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 0.33/0.67 @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % nat.rsp
% 0.33/0.67 thf(fact_9648_one__int_Orsp,axiom,
% 0.33/0.67 intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.33/0.67
% 0.33/0.67 % one_int.rsp
% 0.33/0.67 thf(fact_9649_product__atMost__eq__Un,axiom,
% 0.33/0.67 ! [A3: set_nat,M: nat] :
% 0.33/0.67 ( ( produc457027306803732586at_nat @ A3
% 0.33/0.67 @ ^ [Uu3: nat] : ( set_ord_atMost_nat @ M ) )
% 0.33/0.67 = ( sup_su6327502436637775413at_nat
% 0.33/0.67 @ ( produc457027306803732586at_nat @ A3
% 0.33/0.67 @ ^ [I4: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ I4 ) ) )
% 0.33/0.67 @ ( produc457027306803732586at_nat @ A3
% 0.33/0.67 @ ^ [I4: nat] : ( set_or6659071591806873216st_nat @ ( minus_minus_nat @ M @ I4 ) @ M ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % product_atMost_eq_Un
% 0.33/0.67 thf(fact_9650_intrel__def,axiom,
% 0.33/0.67 ( intrel
% 0.33/0.67 = ( produc8739625826339149834_nat_o
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.67 ( produc6081775807080527818_nat_o
% 0.33/0.67 @ ^ [U3: nat,V4: nat] :
% 0.33/0.67 ( ( plus_plus_nat @ X3 @ V4 )
% 0.33/0.67 = ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % intrel_def
% 0.33/0.67 thf(fact_9651_pairs__le__eq__Sigma,axiom,
% 0.33/0.67 ! [M: nat] :
% 0.33/0.67 ( ( collec3392354462482085612at_nat
% 0.33/0.67 @ ( produc6081775807080527818_nat_o
% 0.33/0.67 @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ M ) ) )
% 0.33/0.67 = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
% 0.33/0.67 @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pairs_le_eq_Sigma
% 0.33/0.67 thf(fact_9652_less__int_Orsp,axiom,
% 0.33/0.67 ( bNF_re4202695980764964119_nat_o @ intrel
% 0.33/0.67 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 0.33/0.67 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.33/0.67 @ ( produc8739625826339149834_nat_o
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.67 ( produc6081775807080527818_nat_o
% 0.33/0.67 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
% 0.33/0.67 @ ( produc8739625826339149834_nat_o
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.67 ( produc6081775807080527818_nat_o
% 0.33/0.67 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_int.rsp
% 0.33/0.67 thf(fact_9653_less__eq__int_Orsp,axiom,
% 0.33/0.67 ( bNF_re4202695980764964119_nat_o @ intrel
% 0.33/0.67 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 0.33/0.67 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.33/0.67 @ ( produc8739625826339149834_nat_o
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.67 ( produc6081775807080527818_nat_o
% 0.33/0.67 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
% 0.33/0.67 @ ( produc8739625826339149834_nat_o
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.67 ( produc6081775807080527818_nat_o
% 0.33/0.67 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_eq_int.rsp
% 0.33/0.67 thf(fact_9654_minus__int_Orsp,axiom,
% 0.33/0.67 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.33/0.67 @ ( produc27273713700761075at_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.67 ( produc2626176000494625587at_nat
% 0.33/0.67 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) )
% 0.33/0.67 @ ( produc27273713700761075at_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.67 ( produc2626176000494625587at_nat
% 0.33/0.67 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % minus_int.rsp
% 0.33/0.67 thf(fact_9655_plus__int_Orsp,axiom,
% 0.33/0.67 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.33/0.67 @ ( produc27273713700761075at_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.67 ( produc2626176000494625587at_nat
% 0.33/0.67 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) )
% 0.33/0.67 @ ( produc27273713700761075at_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] :
% 0.33/0.67 ( produc2626176000494625587at_nat
% 0.33/0.67 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_int.rsp
% 0.33/0.67 thf(fact_9656_Rats__eq__int__div__nat,axiom,
% 0.33/0.67 ( field_5140801741446780682s_real
% 0.33/0.67 = ( collect_real
% 0.33/0.67 @ ^ [Uu3: real] :
% 0.33/0.67 ? [I4: int,N3: nat] :
% 0.33/0.67 ( ( Uu3
% 0.33/0.67 = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 0.33/0.67 & ( N3 != zero_zero_nat ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Rats_eq_int_div_nat
% 0.33/0.67 thf(fact_9657_rat__less__code,axiom,
% 0.33/0.67 ( ord_less_rat
% 0.33/0.67 = ( ^ [P3: rat,Q5: rat] :
% 0.33/0.67 ( produc4947309494688390418_int_o
% 0.33/0.67 @ ^ [A2: int,C4: int] :
% 0.33/0.67 ( produc4947309494688390418_int_o
% 0.33/0.67 @ ^ [B2: int,D4: int] : ( ord_less_int @ ( times_times_int @ A2 @ D4 ) @ ( times_times_int @ C4 @ B2 ) )
% 0.33/0.67 @ ( quotient_of @ Q5 ) )
% 0.33/0.67 @ ( quotient_of @ P3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % rat_less_code
% 0.33/0.67 thf(fact_9658_rat__floor__code,axiom,
% 0.33/0.67 ( archim3151403230148437115or_rat
% 0.33/0.67 = ( ^ [P3: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % rat_floor_code
% 0.33/0.67 thf(fact_9659_rat__less__eq__code,axiom,
% 0.33/0.67 ( ord_less_eq_rat
% 0.33/0.67 = ( ^ [P3: rat,Q5: rat] :
% 0.33/0.67 ( produc4947309494688390418_int_o
% 0.33/0.67 @ ^ [A2: int,C4: int] :
% 0.33/0.67 ( produc4947309494688390418_int_o
% 0.33/0.67 @ ^ [B2: int,D4: int] : ( ord_less_eq_int @ ( times_times_int @ A2 @ D4 ) @ ( times_times_int @ C4 @ B2 ) )
% 0.33/0.67 @ ( quotient_of @ Q5 ) )
% 0.33/0.67 @ ( quotient_of @ P3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % rat_less_eq_code
% 0.33/0.67 thf(fact_9660_Rats__eq__int__div__int,axiom,
% 0.33/0.67 ( field_5140801741446780682s_real
% 0.33/0.67 = ( collect_real
% 0.33/0.67 @ ^ [Uu3: real] :
% 0.33/0.67 ? [I4: int,J3: int] :
% 0.33/0.67 ( ( Uu3
% 0.33/0.67 = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( ring_1_of_int_real @ J3 ) ) )
% 0.33/0.67 & ( J3 != zero_zero_int ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Rats_eq_int_div_int
% 0.33/0.67 thf(fact_9661_Divides_Oadjust__div__def,axiom,
% 0.33/0.67 ( adjust_div
% 0.33/0.67 = ( produc8211389475949308722nt_int
% 0.33/0.67 @ ^ [Q5: int,R5: int] : ( plus_plus_int @ Q5 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Divides.adjust_div_def
% 0.33/0.67 thf(fact_9662_int__ge__less__than__def,axiom,
% 0.33/0.67 ( int_ge_less_than
% 0.33/0.67 = ( ^ [D4: int] :
% 0.33/0.67 ( collec213857154873943460nt_int
% 0.33/0.67 @ ( produc4947309494688390418_int_o
% 0.33/0.67 @ ^ [Z7: int,Z5: int] :
% 0.33/0.67 ( ( ord_less_eq_int @ D4 @ Z7 )
% 0.33/0.67 & ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % int_ge_less_than_def
% 0.33/0.67 thf(fact_9663_int__ge__less__than2__def,axiom,
% 0.33/0.67 ( int_ge_less_than2
% 0.33/0.67 = ( ^ [D4: int] :
% 0.33/0.67 ( collec213857154873943460nt_int
% 0.33/0.67 @ ( produc4947309494688390418_int_o
% 0.33/0.67 @ ^ [Z7: int,Z5: int] :
% 0.33/0.67 ( ( ord_less_eq_int @ D4 @ Z5 )
% 0.33/0.67 & ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % int_ge_less_than2_def
% 0.33/0.67 thf(fact_9664_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 0.33/0.67 ! [X: vEBT_VEBT,Xa: nat] :
% 0.33/0.67 ( ~ ( vEBT_VEBT_valid @ X @ Xa )
% 0.33/0.67 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.33/0.67 ( X
% 0.33/0.67 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.33/0.67 => ( Xa = one_one_nat ) )
% 0.33/0.67 => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) )
% 0.33/0.67 => ( ( Deg = Xa )
% 0.33/0.67 & ! [X4: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.33/0.67 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.67 & ( case_o184042715313410164at_nat
% 0.33/0.67 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
% 0.33/0.67 & ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.67 @ ( produc6081775807080527818_nat_o
% 0.33/0.67 @ ^ [Mi3: nat,Ma3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.33/0.67 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.67 & ! [I4: nat] :
% 0.33/0.67 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.67 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
% 0.33/0.67 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 0.33/0.67 & ( ( Mi3 = Ma3 )
% 0.33/0.67 => ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.67 & ( ( Mi3 != Ma3 )
% 0.33/0.67 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.33/0.67 & ! [X3: nat] :
% 0.33/0.67 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.67 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 0.33/0.67 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.33/0.67 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 0.33/0.67 @ Mima ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % VEBT_internal.valid'.elims(3)
% 0.33/0.67 thf(fact_9665_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 0.33/0.67 ! [X: vEBT_VEBT,Xa: nat] :
% 0.33/0.67 ( ( vEBT_VEBT_valid @ X @ Xa )
% 0.33/0.67 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.33/0.67 ( X
% 0.33/0.67 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.33/0.67 => ( Xa != one_one_nat ) )
% 0.33/0.67 => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) )
% 0.33/0.67 => ~ ( ( Deg = Xa )
% 0.33/0.67 & ! [X5: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.33/0.67 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.67 & ( case_o184042715313410164at_nat
% 0.33/0.67 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
% 0.33/0.67 & ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.67 @ ( produc6081775807080527818_nat_o
% 0.33/0.67 @ ^ [Mi3: nat,Ma3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.33/0.67 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.67 & ! [I4: nat] :
% 0.33/0.67 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.67 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
% 0.33/0.67 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 0.33/0.67 & ( ( Mi3 = Ma3 )
% 0.33/0.67 => ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.67 & ( ( Mi3 != Ma3 )
% 0.33/0.67 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.33/0.67 & ! [X3: nat] :
% 0.33/0.67 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.67 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 0.33/0.67 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.33/0.67 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 0.33/0.67 @ Mima ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % VEBT_internal.valid'.elims(2)
% 0.33/0.67 thf(fact_9666_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 0.33/0.67 ! [Mima2: option4927543243414619207at_nat,Deg3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,Deg4: nat] :
% 0.33/0.67 ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg3 @ TreeList2 @ Summary3 ) @ Deg4 )
% 0.33/0.67 = ( ( Deg3 = Deg4 )
% 0.33/0.67 & ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.33/0.67 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg3 @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.33/0.67 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg3 @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.67 & ( case_o184042715313410164at_nat
% 0.33/0.67 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
% 0.33/0.67 & ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.33/0.67 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.67 @ ( produc6081775807080527818_nat_o
% 0.33/0.67 @ ^ [Mi3: nat,Ma3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.33/0.67 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg3 ) )
% 0.33/0.67 & ! [I4: nat] :
% 0.33/0.67 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg3 @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.67 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
% 0.33/0.67 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 0.33/0.67 & ( ( Mi3 = Ma3 )
% 0.33/0.67 => ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.33/0.67 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.67 & ( ( Mi3 != Ma3 )
% 0.33/0.67 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.33/0.67 & ! [X3: nat] :
% 0.33/0.67 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg3 ) )
% 0.33/0.67 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
% 0.33/0.67 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.33/0.67 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 0.33/0.67 @ Mima2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % VEBT_internal.valid'.simps(2)
% 0.33/0.67 thf(fact_9667_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 0.33/0.67 ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 0.33/0.67 ( ( ( vEBT_VEBT_valid @ X @ Xa )
% 0.33/0.67 = Y )
% 0.33/0.67 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.33/0.67 ( X
% 0.33/0.67 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.33/0.67 => ( Y
% 0.33/0.67 = ( Xa != one_one_nat ) ) )
% 0.33/0.67 => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) )
% 0.33/0.67 => ( Y
% 0.33/0.67 = ( ~ ( ( Deg = Xa )
% 0.33/0.67 & ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.33/0.67 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.67 & ( case_o184042715313410164at_nat
% 0.33/0.67 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
% 0.33/0.67 & ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.67 @ ( produc6081775807080527818_nat_o
% 0.33/0.67 @ ^ [Mi3: nat,Ma3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.33/0.67 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.67 & ! [I4: nat] :
% 0.33/0.67 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.67 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
% 0.33/0.67 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 0.33/0.67 & ( ( Mi3 = Ma3 )
% 0.33/0.67 => ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.67 & ( ( Mi3 != Ma3 )
% 0.33/0.67 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.33/0.67 & ! [X3: nat] :
% 0.33/0.67 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.67 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 0.33/0.67 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.33/0.67 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 0.33/0.67 @ Mima ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % VEBT_internal.valid'.elims(1)
% 0.33/0.67 thf(fact_9668_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 0.33/0.67 ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 0.33/0.67 ( ( ( vEBT_VEBT_valid @ X @ Xa )
% 0.33/0.67 = Y )
% 0.33/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.33/0.67 => ( ! [Uu2: $o,Uv2: $o] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( Xa = one_one_nat ) )
% 0.33/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
% 0.33/0.67 => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( ( Deg = Xa )
% 0.33/0.67 & ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.33/0.67 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.67 & ( case_o184042715313410164at_nat
% 0.33/0.67 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
% 0.33/0.67 & ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.67 @ ( produc6081775807080527818_nat_o
% 0.33/0.67 @ ^ [Mi3: nat,Ma3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.33/0.67 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.67 & ! [I4: nat] :
% 0.33/0.67 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.67 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
% 0.33/0.67 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 0.33/0.67 & ( ( Mi3 = Ma3 )
% 0.33/0.67 => ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.67 & ( ( Mi3 != Ma3 )
% 0.33/0.67 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.33/0.67 & ! [X3: nat] :
% 0.33/0.67 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.67 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 0.33/0.67 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.33/0.67 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 0.33/0.67 @ Mima ) ) )
% 0.33/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % VEBT_internal.valid'.pelims(1)
% 0.33/0.67 thf(fact_9669_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 0.33/0.67 ! [X: vEBT_VEBT,Xa: nat] :
% 0.33/0.67 ( ( vEBT_VEBT_valid @ X @ Xa )
% 0.33/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.33/0.67 => ( ! [Uu2: $o,Uv2: $o] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.33/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
% 0.33/0.67 => ( Xa != one_one_nat ) ) )
% 0.33/0.67 => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) )
% 0.33/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) @ Xa ) )
% 0.33/0.67 => ~ ( ( Deg = Xa )
% 0.33/0.67 & ! [X5: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.33/0.67 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.67 & ( case_o184042715313410164at_nat
% 0.33/0.67 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
% 0.33/0.67 & ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.67 @ ( produc6081775807080527818_nat_o
% 0.33/0.67 @ ^ [Mi3: nat,Ma3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.33/0.67 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.67 & ! [I4: nat] :
% 0.33/0.67 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.67 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
% 0.33/0.67 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 0.33/0.67 & ( ( Mi3 = Ma3 )
% 0.33/0.67 => ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.67 & ( ( Mi3 != Ma3 )
% 0.33/0.67 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.33/0.67 & ! [X3: nat] :
% 0.33/0.67 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.67 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 0.33/0.67 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.33/0.67 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 0.33/0.67 @ Mima ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % VEBT_internal.valid'.pelims(2)
% 0.33/0.67 thf(fact_9670_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 0.33/0.67 ! [X: vEBT_VEBT,Xa: nat] :
% 0.33/0.67 ( ~ ( vEBT_VEBT_valid @ X @ Xa )
% 0.33/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.33/0.67 => ( ! [Uu2: $o,Uv2: $o] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.33/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
% 0.33/0.67 => ( Xa = one_one_nat ) ) )
% 0.33/0.67 => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) )
% 0.33/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) @ Xa ) )
% 0.33/0.67 => ( ( Deg = Xa )
% 0.33/0.67 & ! [X4: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.33/0.67 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.67 & ( case_o184042715313410164at_nat
% 0.33/0.67 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
% 0.33/0.67 & ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.67 @ ( produc6081775807080527818_nat_o
% 0.33/0.67 @ ^ [Mi3: nat,Ma3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.33/0.67 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.67 & ! [I4: nat] :
% 0.33/0.67 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.67 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
% 0.33/0.67 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 0.33/0.67 & ( ( Mi3 = Ma3 )
% 0.33/0.67 => ! [X3: vEBT_VEBT] :
% 0.33/0.67 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.67 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 0.33/0.67 & ( ( Mi3 != Ma3 )
% 0.33/0.67 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.33/0.67 & ! [X3: nat] :
% 0.33/0.67 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.67 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 0.33/0.67 => ( ( ord_less_nat @ Mi3 @ X3 )
% 0.33/0.67 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 0.33/0.67 @ Mima ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % VEBT_internal.valid'.pelims(3)
% 0.33/0.67 thf(fact_9671_Real_Opositive_Orsp,axiom,
% 0.33/0.67 ( bNF_re728719798268516973at_o_o @ realrel
% 0.33/0.67 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.33/0.67 @ ^ [X8: nat > rat] :
% 0.33/0.67 ? [R5: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.33/0.67 & ? [K2: nat] :
% 0.33/0.67 ! [N3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K2 @ N3 )
% 0.33/0.67 => ( ord_less_rat @ R5 @ ( X8 @ N3 ) ) ) )
% 0.33/0.67 @ ^ [X8: nat > rat] :
% 0.33/0.67 ? [R5: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.33/0.67 & ? [K2: nat] :
% 0.33/0.67 ! [N3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K2 @ N3 )
% 0.33/0.67 => ( ord_less_rat @ R5 @ ( X8 @ N3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Real.positive.rsp
% 0.33/0.67 thf(fact_9672_zero__real_Orsp,axiom,
% 0.33/0.67 ( realrel
% 0.33/0.67 @ ^ [N3: nat] : zero_zero_rat
% 0.33/0.67 @ ^ [N3: nat] : zero_zero_rat ) ).
% 0.33/0.67
% 0.33/0.67 % zero_real.rsp
% 0.33/0.67 thf(fact_9673_one__real_Orsp,axiom,
% 0.33/0.67 ( realrel
% 0.33/0.67 @ ^ [N3: nat] : one_one_rat
% 0.33/0.67 @ ^ [N3: nat] : one_one_rat ) ).
% 0.33/0.67
% 0.33/0.67 % one_real.rsp
% 0.33/0.67 thf(fact_9674_plus__real_Orsp,axiom,
% 0.33/0.67 ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
% 0.33/0.67 @ ^ [X8: nat > rat,Y7: nat > rat,N3: nat] : ( plus_plus_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) )
% 0.33/0.67 @ ^ [X8: nat > rat,Y7: nat > rat,N3: nat] : ( plus_plus_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_real.rsp
% 0.33/0.67 thf(fact_9675_times__real_Orsp,axiom,
% 0.33/0.67 ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
% 0.33/0.67 @ ^ [X8: nat > rat,Y7: nat > rat,N3: nat] : ( times_times_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) )
% 0.33/0.67 @ ^ [X8: nat > rat,Y7: nat > rat,N3: nat] : ( times_times_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_real.rsp
% 0.33/0.67 thf(fact_9676_uminus__real_Orsp,axiom,
% 0.33/0.67 ( bNF_re895249473297799549at_rat @ realrel @ realrel
% 0.33/0.67 @ ^ [X8: nat > rat,N3: nat] : ( uminus_uminus_rat @ ( X8 @ N3 ) )
% 0.33/0.67 @ ^ [X8: nat > rat,N3: nat] : ( uminus_uminus_rat @ ( X8 @ N3 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % uminus_real.rsp
% 0.33/0.67 thf(fact_9677_continuous__on__arcosh,axiom,
% 0.33/0.67 ! [A3: set_real] :
% 0.33/0.67 ( ( ord_less_eq_set_real @ A3 @ ( set_ord_atLeast_real @ one_one_real ) )
% 0.33/0.67 => ( topolo5044208981011980120l_real @ A3 @ arcosh_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % continuous_on_arcosh
% 0.33/0.67 thf(fact_9678_inverse__real_Orsp,axiom,
% 0.33/0.67 ( bNF_re895249473297799549at_rat @ realrel @ realrel
% 0.33/0.67 @ ^ [X8: nat > rat] :
% 0.33/0.67 ( if_nat_rat @ ( vanishes @ X8 )
% 0.33/0.67 @ ^ [N3: nat] : zero_zero_rat
% 0.33/0.67 @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X8 @ N3 ) ) )
% 0.33/0.67 @ ^ [X8: nat > rat] :
% 0.33/0.67 ( if_nat_rat @ ( vanishes @ X8 )
% 0.33/0.67 @ ^ [N3: nat] : zero_zero_rat
% 0.33/0.67 @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X8 @ N3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % inverse_real.rsp
% 0.33/0.67 thf(fact_9679_atLeast__0,axiom,
% 0.33/0.67 ( ( set_ord_atLeast_nat @ zero_zero_nat )
% 0.33/0.67 = top_top_set_nat ) ).
% 0.33/0.67
% 0.33/0.67 % atLeast_0
% 0.33/0.67 thf(fact_9680_vanishes__const,axiom,
% 0.33/0.67 ! [C: rat] :
% 0.33/0.67 ( ( vanishes
% 0.33/0.67 @ ^ [N3: nat] : C )
% 0.33/0.67 = ( C = zero_zero_rat ) ) ).
% 0.33/0.67
% 0.33/0.67 % vanishes_const
% 0.33/0.67 thf(fact_9681_last__upt,axiom,
% 0.33/0.67 ! [I: nat,J: nat] :
% 0.33/0.67 ( ( ord_less_nat @ I @ J )
% 0.33/0.67 => ( ( last_nat @ ( upt @ I @ J ) )
% 0.33/0.67 = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % last_upt
% 0.33/0.67 thf(fact_9682_atLeast__Suc__greaterThan,axiom,
% 0.33/0.67 ! [K: nat] :
% 0.33/0.67 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.33/0.67 = ( set_or1210151606488870762an_nat @ K ) ) ).
% 0.33/0.67
% 0.33/0.67 % atLeast_Suc_greaterThan
% 0.33/0.67 thf(fact_9683_vanishes__minus,axiom,
% 0.33/0.67 ! [X9: nat > rat] :
% 0.33/0.67 ( ( vanishes @ X9 )
% 0.33/0.67 => ( vanishes
% 0.33/0.67 @ ^ [N3: nat] : ( uminus_uminus_rat @ ( X9 @ N3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % vanishes_minus
% 0.33/0.67 thf(fact_9684_vanishes__add,axiom,
% 0.33/0.67 ! [X9: nat > rat,Y8: nat > rat] :
% 0.33/0.67 ( ( vanishes @ X9 )
% 0.33/0.67 => ( ( vanishes @ Y8 )
% 0.33/0.67 => ( vanishes
% 0.33/0.67 @ ^ [N3: nat] : ( plus_plus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % vanishes_add
% 0.33/0.67 thf(fact_9685_vanishes__diff,axiom,
% 0.33/0.67 ! [X9: nat > rat,Y8: nat > rat] :
% 0.33/0.67 ( ( vanishes @ X9 )
% 0.33/0.67 => ( ( vanishes @ Y8 )
% 0.33/0.67 => ( vanishes
% 0.33/0.67 @ ^ [N3: nat] : ( minus_minus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % vanishes_diff
% 0.33/0.67 thf(fact_9686_vanishes__def,axiom,
% 0.33/0.67 ( vanishes
% 0.33/0.67 = ( ^ [X8: nat > rat] :
% 0.33/0.67 ! [R5: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.33/0.67 => ? [K2: nat] :
% 0.33/0.67 ! [N3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K2 @ N3 )
% 0.33/0.67 => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N3 ) ) @ R5 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % vanishes_def
% 0.33/0.67 thf(fact_9687_vanishesI,axiom,
% 0.33/0.67 ! [X9: nat > rat] :
% 0.33/0.67 ( ! [R2: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.33/0.67 => ? [K4: nat] :
% 0.33/0.67 ! [N2: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K4 @ N2 )
% 0.33/0.67 => ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N2 ) ) @ R2 ) ) )
% 0.33/0.67 => ( vanishes @ X9 ) ) ).
% 0.33/0.67
% 0.33/0.67 % vanishesI
% 0.33/0.67 thf(fact_9688_vanishesD,axiom,
% 0.33/0.67 ! [X9: nat > rat,R: rat] :
% 0.33/0.67 ( ( vanishes @ X9 )
% 0.33/0.67 => ( ( ord_less_rat @ zero_zero_rat @ R )
% 0.33/0.67 => ? [K3: nat] :
% 0.33/0.67 ! [N6: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K3 @ N6 )
% 0.33/0.67 => ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N6 ) ) @ R ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % vanishesD
% 0.33/0.67 thf(fact_9689_vanishes__mult__bounded,axiom,
% 0.33/0.67 ! [X9: nat > rat,Y8: nat > rat] :
% 0.33/0.67 ( ? [A6: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ A6 )
% 0.33/0.67 & ! [N2: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N2 ) ) @ A6 ) )
% 0.33/0.67 => ( ( vanishes @ Y8 )
% 0.33/0.67 => ( vanishes
% 0.33/0.67 @ ^ [N3: nat] : ( times_times_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % vanishes_mult_bounded
% 0.33/0.67 thf(fact_9690_atLeast__Suc,axiom,
% 0.33/0.67 ! [K: nat] :
% 0.33/0.67 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.33/0.67 = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % atLeast_Suc
% 0.33/0.67 thf(fact_9691_inverse__real_Otransfer,axiom,
% 0.33/0.67 ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
% 0.33/0.67 @ ^ [X8: nat > rat] :
% 0.33/0.67 ( if_nat_rat @ ( vanishes @ X8 )
% 0.33/0.67 @ ^ [N3: nat] : zero_zero_rat
% 0.33/0.67 @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X8 @ N3 ) ) )
% 0.33/0.67 @ inverse_inverse_real ) ).
% 0.33/0.67
% 0.33/0.67 % inverse_real.transfer
% 0.33/0.67 thf(fact_9692_inverse__real_Oabs__eq,axiom,
% 0.33/0.67 ! [X: nat > rat] :
% 0.33/0.67 ( ( realrel @ X @ X )
% 0.33/0.67 => ( ( inverse_inverse_real @ ( real2 @ X ) )
% 0.33/0.67 = ( real2
% 0.33/0.67 @ ( if_nat_rat @ ( vanishes @ X )
% 0.33/0.67 @ ^ [N3: nat] : zero_zero_rat
% 0.33/0.67 @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X @ N3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % inverse_real.abs_eq
% 0.33/0.67 thf(fact_9693_one__real__def,axiom,
% 0.33/0.67 ( one_one_real
% 0.33/0.67 = ( real2
% 0.33/0.67 @ ^ [N3: nat] : one_one_rat ) ) ).
% 0.33/0.67
% 0.33/0.67 % one_real_def
% 0.33/0.67 thf(fact_9694_real_Oabs__induct,axiom,
% 0.33/0.67 ! [P: real > $o,X: real] :
% 0.33/0.67 ( ! [Y4: nat > rat] :
% 0.33/0.67 ( ( realrel @ Y4 @ Y4 )
% 0.33/0.67 => ( P @ ( real2 @ Y4 ) ) )
% 0.33/0.67 => ( P @ X ) ) ).
% 0.33/0.67
% 0.33/0.67 % real.abs_induct
% 0.33/0.67 thf(fact_9695_of__rat__Real,axiom,
% 0.33/0.67 ( field_7254667332652039916t_real
% 0.33/0.67 = ( ^ [X3: rat] :
% 0.33/0.67 ( real2
% 0.33/0.67 @ ^ [N3: nat] : X3 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % of_rat_Real
% 0.33/0.67 thf(fact_9696_zero__real__def,axiom,
% 0.33/0.67 ( zero_zero_real
% 0.33/0.67 = ( real2
% 0.33/0.67 @ ^ [N3: nat] : zero_zero_rat ) ) ).
% 0.33/0.67
% 0.33/0.67 % zero_real_def
% 0.33/0.67 thf(fact_9697_real_Orel__eq__transfer,axiom,
% 0.33/0.67 ( bNF_re4521903465945308077real_o @ pcr_real
% 0.33/0.67 @ ( bNF_re4297313714947099218al_o_o @ pcr_real
% 0.33/0.67 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.33/0.67 @ realrel
% 0.33/0.67 @ ^ [Y3: real,Z: real] : ( Y3 = Z ) ) ).
% 0.33/0.67
% 0.33/0.67 % real.rel_eq_transfer
% 0.33/0.67 thf(fact_9698_of__int__Real,axiom,
% 0.33/0.67 ( ring_1_of_int_real
% 0.33/0.67 = ( ^ [X3: int] :
% 0.33/0.67 ( real2
% 0.33/0.67 @ ^ [N3: nat] : ( ring_1_of_int_rat @ X3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % of_int_Real
% 0.33/0.67 thf(fact_9699_zero__real_Otransfer,axiom,
% 0.33/0.67 ( pcr_real
% 0.33/0.67 @ ^ [N3: nat] : zero_zero_rat
% 0.33/0.67 @ zero_zero_real ) ).
% 0.33/0.67
% 0.33/0.67 % zero_real.transfer
% 0.33/0.67 thf(fact_9700_one__real_Otransfer,axiom,
% 0.33/0.67 ( pcr_real
% 0.33/0.67 @ ^ [N3: nat] : one_one_rat
% 0.33/0.67 @ one_one_real ) ).
% 0.33/0.67
% 0.33/0.67 % one_real.transfer
% 0.33/0.67 thf(fact_9701_of__nat__Real,axiom,
% 0.33/0.67 ( semiri5074537144036343181t_real
% 0.33/0.67 = ( ^ [X3: nat] :
% 0.33/0.67 ( real2
% 0.33/0.67 @ ^ [N3: nat] : ( semiri681578069525770553at_rat @ X3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % of_nat_Real
% 0.33/0.67 thf(fact_9702_uminus__real_Otransfer,axiom,
% 0.33/0.67 ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
% 0.33/0.67 @ ^ [X8: nat > rat,N3: nat] : ( uminus_uminus_rat @ ( X8 @ N3 ) )
% 0.33/0.67 @ uminus_uminus_real ) ).
% 0.33/0.67
% 0.33/0.67 % uminus_real.transfer
% 0.33/0.67 thf(fact_9703_plus__real_Otransfer,axiom,
% 0.33/0.67 ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
% 0.33/0.67 @ ^ [X8: nat > rat,Y7: nat > rat,N3: nat] : ( plus_plus_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) )
% 0.33/0.67 @ plus_plus_real ) ).
% 0.33/0.67
% 0.33/0.67 % plus_real.transfer
% 0.33/0.67 thf(fact_9704_uminus__real_Oabs__eq,axiom,
% 0.33/0.67 ! [X: nat > rat] :
% 0.33/0.67 ( ( realrel @ X @ X )
% 0.33/0.67 => ( ( uminus_uminus_real @ ( real2 @ X ) )
% 0.33/0.67 = ( real2
% 0.33/0.67 @ ^ [N3: nat] : ( uminus_uminus_rat @ ( X @ N3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % uminus_real.abs_eq
% 0.33/0.67 thf(fact_9705_times__real_Otransfer,axiom,
% 0.33/0.67 ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
% 0.33/0.67 @ ^ [X8: nat > rat,Y7: nat > rat,N3: nat] : ( times_times_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) )
% 0.33/0.67 @ times_times_real ) ).
% 0.33/0.67
% 0.33/0.67 % times_real.transfer
% 0.33/0.67 thf(fact_9706_plus__real_Oabs__eq,axiom,
% 0.33/0.67 ! [Xa: nat > rat,X: nat > rat] :
% 0.33/0.67 ( ( realrel @ Xa @ Xa )
% 0.33/0.67 => ( ( realrel @ X @ X )
% 0.33/0.67 => ( ( plus_plus_real @ ( real2 @ Xa ) @ ( real2 @ X ) )
% 0.33/0.67 = ( real2
% 0.33/0.67 @ ^ [N3: nat] : ( plus_plus_rat @ ( Xa @ N3 ) @ ( X @ N3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_real.abs_eq
% 0.33/0.67 thf(fact_9707_times__real_Oabs__eq,axiom,
% 0.33/0.67 ! [Xa: nat > rat,X: nat > rat] :
% 0.33/0.67 ( ( realrel @ Xa @ Xa )
% 0.33/0.67 => ( ( realrel @ X @ X )
% 0.33/0.67 => ( ( times_times_real @ ( real2 @ Xa ) @ ( real2 @ X ) )
% 0.33/0.67 = ( real2
% 0.33/0.67 @ ^ [N3: nat] : ( times_times_rat @ ( Xa @ N3 ) @ ( X @ N3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_real.abs_eq
% 0.33/0.67 thf(fact_9708_Real_Opositive_Otransfer,axiom,
% 0.33/0.67 ( bNF_re4297313714947099218al_o_o @ pcr_real
% 0.33/0.67 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.33/0.67 @ ^ [X8: nat > rat] :
% 0.33/0.67 ? [R5: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.33/0.67 & ? [K2: nat] :
% 0.33/0.67 ! [N3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K2 @ N3 )
% 0.33/0.67 => ( ord_less_rat @ R5 @ ( X8 @ N3 ) ) ) )
% 0.33/0.67 @ positive2 ) ).
% 0.33/0.67
% 0.33/0.67 % Real.positive.transfer
% 0.33/0.67 thf(fact_9709_Real_Opositive_Oabs__eq,axiom,
% 0.33/0.67 ! [X: nat > rat] :
% 0.33/0.67 ( ( realrel @ X @ X )
% 0.33/0.67 => ( ( positive2 @ ( real2 @ X ) )
% 0.33/0.67 = ( ? [R5: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.33/0.67 & ? [K2: nat] :
% 0.33/0.67 ! [N3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K2 @ N3 )
% 0.33/0.67 => ( ord_less_rat @ R5 @ ( X @ N3 ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Real.positive.abs_eq
% 0.33/0.67 thf(fact_9710_Real_Opositive__mult,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( positive2 @ X )
% 0.33/0.67 => ( ( positive2 @ Y )
% 0.33/0.67 => ( positive2 @ ( times_times_real @ X @ Y ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Real.positive_mult
% 0.33/0.67 thf(fact_9711_Real_Opositive__add,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( positive2 @ X )
% 0.33/0.67 => ( ( positive2 @ Y )
% 0.33/0.67 => ( positive2 @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Real.positive_add
% 0.33/0.67 thf(fact_9712_Real_Opositive__zero,axiom,
% 0.33/0.67 ~ ( positive2 @ zero_zero_real ) ).
% 0.33/0.67
% 0.33/0.67 % Real.positive_zero
% 0.33/0.67 thf(fact_9713_Real_Opositive__minus,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ~ ( positive2 @ X )
% 0.33/0.67 => ( ( X != zero_zero_real )
% 0.33/0.67 => ( positive2 @ ( uminus_uminus_real @ X ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Real.positive_minus
% 0.33/0.67 thf(fact_9714_less__real__def,axiom,
% 0.33/0.67 ( ord_less_real
% 0.33/0.67 = ( ^ [X3: real,Y6: real] : ( positive2 @ ( minus_minus_real @ Y6 @ X3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_real_def
% 0.33/0.67 thf(fact_9715_le__Real,axiom,
% 0.33/0.67 ! [X9: nat > rat,Y8: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ( cauchy @ Y8 )
% 0.33/0.67 => ( ( ord_less_eq_real @ ( real2 @ X9 ) @ ( real2 @ Y8 ) )
% 0.33/0.67 = ( ! [R5: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.33/0.67 => ? [K2: nat] :
% 0.33/0.67 ! [N3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K2 @ N3 )
% 0.33/0.67 => ( ord_less_eq_rat @ ( X9 @ N3 ) @ ( plus_plus_rat @ ( Y8 @ N3 ) @ R5 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % le_Real
% 0.33/0.67 thf(fact_9716_Real_Opositive_Orep__eq,axiom,
% 0.33/0.67 ( positive2
% 0.33/0.67 = ( ^ [X3: real] :
% 0.33/0.67 ? [R5: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.33/0.67 & ? [K2: nat] :
% 0.33/0.67 ! [N3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K2 @ N3 )
% 0.33/0.67 => ( ord_less_rat @ R5 @ ( rep_real @ X3 @ N3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Real.positive.rep_eq
% 0.33/0.67 thf(fact_9717_realrel__refl,axiom,
% 0.33/0.67 ! [X9: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( realrel @ X9 @ X9 ) ) ).
% 0.33/0.67
% 0.33/0.67 % realrel_refl
% 0.33/0.67 thf(fact_9718_cauchy__diff,axiom,
% 0.33/0.67 ! [X9: nat > rat,Y8: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ( cauchy @ Y8 )
% 0.33/0.67 => ( cauchy
% 0.33/0.67 @ ^ [N3: nat] : ( minus_minus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cauchy_diff
% 0.33/0.67 thf(fact_9719_cauchy__mult,axiom,
% 0.33/0.67 ! [X9: nat > rat,Y8: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ( cauchy @ Y8 )
% 0.33/0.67 => ( cauchy
% 0.33/0.67 @ ^ [N3: nat] : ( times_times_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cauchy_mult
% 0.33/0.67 thf(fact_9720_cauchy__add,axiom,
% 0.33/0.67 ! [X9: nat > rat,Y8: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ( cauchy @ Y8 )
% 0.33/0.67 => ( cauchy
% 0.33/0.67 @ ^ [N3: nat] : ( plus_plus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cauchy_add
% 0.33/0.67 thf(fact_9721_cauchy__minus,axiom,
% 0.33/0.67 ! [X9: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( cauchy
% 0.33/0.67 @ ^ [N3: nat] : ( uminus_uminus_rat @ ( X9 @ N3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cauchy_minus
% 0.33/0.67 thf(fact_9722_cauchy__const,axiom,
% 0.33/0.67 ! [X: rat] :
% 0.33/0.67 ( cauchy
% 0.33/0.67 @ ^ [N3: nat] : X ) ).
% 0.33/0.67
% 0.33/0.67 % cauchy_const
% 0.33/0.67 thf(fact_9723_Real__induct,axiom,
% 0.33/0.67 ! [P: real > $o,X: real] :
% 0.33/0.67 ( ! [X15: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X15 )
% 0.33/0.67 => ( P @ ( real2 @ X15 ) ) )
% 0.33/0.67 => ( P @ X ) ) ).
% 0.33/0.67
% 0.33/0.67 % Real_induct
% 0.33/0.67 thf(fact_9724_cr__real__eq,axiom,
% 0.33/0.67 ( pcr_real
% 0.33/0.67 = ( ^ [X3: nat > rat,Y6: real] :
% 0.33/0.67 ( ( cauchy @ X3 )
% 0.33/0.67 & ( ( real2 @ X3 )
% 0.33/0.67 = Y6 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cr_real_eq
% 0.33/0.67 thf(fact_9725_cauchy__inverse,axiom,
% 0.33/0.67 ! [X9: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ~ ( vanishes @ X9 )
% 0.33/0.67 => ( cauchy
% 0.33/0.67 @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X9 @ N3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cauchy_inverse
% 0.33/0.67 thf(fact_9726_cauchy__imp__bounded,axiom,
% 0.33/0.67 ! [X9: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ? [B4: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ B4 )
% 0.33/0.67 & ! [N6: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N6 ) ) @ B4 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cauchy_imp_bounded
% 0.33/0.67 thf(fact_9727_less__RealD,axiom,
% 0.33/0.67 ! [Y8: nat > rat,X: real] :
% 0.33/0.67 ( ( cauchy @ Y8 )
% 0.33/0.67 => ( ( ord_less_real @ X @ ( real2 @ Y8 ) )
% 0.33/0.67 => ? [N2: nat] : ( ord_less_real @ X @ ( field_7254667332652039916t_real @ ( Y8 @ N2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_RealD
% 0.33/0.67 thf(fact_9728_le__RealI,axiom,
% 0.33/0.67 ! [Y8: nat > rat,X: real] :
% 0.33/0.67 ( ( cauchy @ Y8 )
% 0.33/0.67 => ( ! [N2: nat] : ( ord_less_eq_real @ X @ ( field_7254667332652039916t_real @ ( Y8 @ N2 ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ X @ ( real2 @ Y8 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % le_RealI
% 0.33/0.67 thf(fact_9729_Real__leI,axiom,
% 0.33/0.67 ! [X9: nat > rat,Y: real] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ! [N2: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X9 @ N2 ) ) @ Y )
% 0.33/0.67 => ( ord_less_eq_real @ ( real2 @ X9 ) @ Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Real_leI
% 0.33/0.67 thf(fact_9730_minus__Real,axiom,
% 0.33/0.67 ! [X9: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ( uminus_uminus_real @ ( real2 @ X9 ) )
% 0.33/0.67 = ( real2
% 0.33/0.67 @ ^ [N3: nat] : ( uminus_uminus_rat @ ( X9 @ N3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % minus_Real
% 0.33/0.67 thf(fact_9731_add__Real,axiom,
% 0.33/0.67 ! [X9: nat > rat,Y8: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ( cauchy @ Y8 )
% 0.33/0.67 => ( ( plus_plus_real @ ( real2 @ X9 ) @ ( real2 @ Y8 ) )
% 0.33/0.67 = ( real2
% 0.33/0.67 @ ^ [N3: nat] : ( plus_plus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % add_Real
% 0.33/0.67 thf(fact_9732_mult__Real,axiom,
% 0.33/0.67 ! [X9: nat > rat,Y8: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ( cauchy @ Y8 )
% 0.33/0.67 => ( ( times_times_real @ ( real2 @ X9 ) @ ( real2 @ Y8 ) )
% 0.33/0.67 = ( real2
% 0.33/0.67 @ ^ [N3: nat] : ( times_times_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % mult_Real
% 0.33/0.67 thf(fact_9733_diff__Real,axiom,
% 0.33/0.67 ! [X9: nat > rat,Y8: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ( cauchy @ Y8 )
% 0.33/0.67 => ( ( minus_minus_real @ ( real2 @ X9 ) @ ( real2 @ Y8 ) )
% 0.33/0.67 = ( real2
% 0.33/0.67 @ ^ [N3: nat] : ( minus_minus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % diff_Real
% 0.33/0.67 thf(fact_9734_realrelI,axiom,
% 0.33/0.67 ! [X9: nat > rat,Y8: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ( cauchy @ Y8 )
% 0.33/0.67 => ( ( vanishes
% 0.33/0.67 @ ^ [N3: nat] : ( minus_minus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) )
% 0.33/0.67 => ( realrel @ X9 @ Y8 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % realrelI
% 0.33/0.67 thf(fact_9735_eq__Real,axiom,
% 0.33/0.67 ! [X9: nat > rat,Y8: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ( cauchy @ Y8 )
% 0.33/0.67 => ( ( ( real2 @ X9 )
% 0.33/0.67 = ( real2 @ Y8 ) )
% 0.33/0.67 = ( vanishes
% 0.33/0.67 @ ^ [N3: nat] : ( minus_minus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % eq_Real
% 0.33/0.67 thf(fact_9736_vanishes__diff__inverse,axiom,
% 0.33/0.67 ! [X9: nat > rat,Y8: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ~ ( vanishes @ X9 )
% 0.33/0.67 => ( ( cauchy @ Y8 )
% 0.33/0.67 => ( ~ ( vanishes @ Y8 )
% 0.33/0.67 => ( ( vanishes
% 0.33/0.67 @ ^ [N3: nat] : ( minus_minus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) )
% 0.33/0.67 => ( vanishes
% 0.33/0.67 @ ^ [N3: nat] : ( minus_minus_rat @ ( inverse_inverse_rat @ ( X9 @ N3 ) ) @ ( inverse_inverse_rat @ ( Y8 @ N3 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % vanishes_diff_inverse
% 0.33/0.67 thf(fact_9737_realrel__def,axiom,
% 0.33/0.67 ( realrel
% 0.33/0.67 = ( ^ [X8: nat > rat,Y7: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X8 )
% 0.33/0.67 & ( cauchy @ Y7 )
% 0.33/0.67 & ( vanishes
% 0.33/0.67 @ ^ [N3: nat] : ( minus_minus_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % realrel_def
% 0.33/0.67 thf(fact_9738_cauchy__not__vanishes__cases,axiom,
% 0.33/0.67 ! [X9: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ~ ( vanishes @ X9 )
% 0.33/0.67 => ? [B4: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ B4 )
% 0.33/0.67 & ? [K3: nat] :
% 0.33/0.67 ( ! [N6: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K3 @ N6 )
% 0.33/0.67 => ( ord_less_rat @ B4 @ ( uminus_uminus_rat @ ( X9 @ N6 ) ) ) )
% 0.33/0.67 | ! [N6: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K3 @ N6 )
% 0.33/0.67 => ( ord_less_rat @ B4 @ ( X9 @ N6 ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cauchy_not_vanishes_cases
% 0.33/0.67 thf(fact_9739_positive__Real,axiom,
% 0.33/0.67 ! [X9: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ( positive2 @ ( real2 @ X9 ) )
% 0.33/0.67 = ( ? [R5: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.33/0.67 & ? [K2: nat] :
% 0.33/0.67 ! [N3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K2 @ N3 )
% 0.33/0.67 => ( ord_less_rat @ R5 @ ( X9 @ N3 ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % positive_Real
% 0.33/0.67 thf(fact_9740_cauchy__not__vanishes,axiom,
% 0.33/0.67 ! [X9: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ~ ( vanishes @ X9 )
% 0.33/0.67 => ? [B4: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ B4 )
% 0.33/0.67 & ? [K3: nat] :
% 0.33/0.67 ! [N6: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K3 @ N6 )
% 0.33/0.67 => ( ord_less_rat @ B4 @ ( abs_abs_rat @ ( X9 @ N6 ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cauchy_not_vanishes
% 0.33/0.67 thf(fact_9741_cauchyD,axiom,
% 0.33/0.67 ! [X9: nat > rat,R: rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ( ord_less_rat @ zero_zero_rat @ R )
% 0.33/0.67 => ? [K3: nat] :
% 0.33/0.67 ! [M4: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K3 @ M4 )
% 0.33/0.67 => ! [N6: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K3 @ N6 )
% 0.33/0.67 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X9 @ M4 ) @ ( X9 @ N6 ) ) ) @ R ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cauchyD
% 0.33/0.67 thf(fact_9742_cauchyI,axiom,
% 0.33/0.67 ! [X9: nat > rat] :
% 0.33/0.67 ( ! [R2: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.33/0.67 => ? [K4: nat] :
% 0.33/0.67 ! [M2: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K4 @ M2 )
% 0.33/0.67 => ! [N2: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K4 @ N2 )
% 0.33/0.67 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X9 @ M2 ) @ ( X9 @ N2 ) ) ) @ R2 ) ) ) )
% 0.33/0.67 => ( cauchy @ X9 ) ) ).
% 0.33/0.67
% 0.33/0.67 % cauchyI
% 0.33/0.67 thf(fact_9743_cauchy__def,axiom,
% 0.33/0.67 ( cauchy
% 0.33/0.67 = ( ^ [X8: nat > rat] :
% 0.33/0.67 ! [R5: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.33/0.67 => ? [K2: nat] :
% 0.33/0.67 ! [M3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K2 @ M3 )
% 0.33/0.67 => ! [N3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K2 @ N3 )
% 0.33/0.67 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) ) @ R5 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cauchy_def
% 0.33/0.67 thf(fact_9744_inverse__Real,axiom,
% 0.33/0.67 ! [X9: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ( ( vanishes @ X9 )
% 0.33/0.67 => ( ( inverse_inverse_real @ ( real2 @ X9 ) )
% 0.33/0.67 = zero_zero_real ) )
% 0.33/0.67 & ( ~ ( vanishes @ X9 )
% 0.33/0.67 => ( ( inverse_inverse_real @ ( real2 @ X9 ) )
% 0.33/0.67 = ( real2
% 0.33/0.67 @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X9 @ N3 ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % inverse_Real
% 0.33/0.67 thf(fact_9745_not__positive__Real,axiom,
% 0.33/0.67 ! [X9: nat > rat] :
% 0.33/0.67 ( ( cauchy @ X9 )
% 0.33/0.67 => ( ( ~ ( positive2 @ ( real2 @ X9 ) ) )
% 0.33/0.67 = ( ! [R5: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.33/0.67 => ? [K2: nat] :
% 0.33/0.67 ! [N3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K2 @ N3 )
% 0.33/0.67 => ( ord_less_eq_rat @ ( X9 @ N3 ) @ R5 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % not_positive_Real
% 0.33/0.67 thf(fact_9746_Real_Opositive__def,axiom,
% 0.33/0.67 ( positive2
% 0.33/0.67 = ( map_fu1856342031159181835at_o_o @ rep_real @ id_o
% 0.33/0.67 @ ^ [X8: nat > rat] :
% 0.33/0.67 ? [R5: rat] :
% 0.33/0.67 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.33/0.67 & ? [K2: nat] :
% 0.33/0.67 ! [N3: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K2 @ N3 )
% 0.33/0.67 => ( ord_less_rat @ R5 @ ( X8 @ N3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Real.positive_def
% 0.33/0.67 thf(fact_9747_inverse__real__def,axiom,
% 0.33/0.67 ( inverse_inverse_real
% 0.33/0.67 = ( map_fu7146612038024189824t_real @ rep_real @ real2
% 0.33/0.67 @ ^ [X8: nat > rat] :
% 0.33/0.67 ( if_nat_rat @ ( vanishes @ X8 )
% 0.33/0.67 @ ^ [N3: nat] : zero_zero_rat
% 0.33/0.67 @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X8 @ N3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % inverse_real_def
% 0.33/0.67 thf(fact_9748_uminus__real__def,axiom,
% 0.33/0.67 ( uminus_uminus_real
% 0.33/0.67 = ( map_fu7146612038024189824t_real @ rep_real @ real2
% 0.33/0.67 @ ^ [X8: nat > rat,N3: nat] : ( uminus_uminus_rat @ ( X8 @ N3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % uminus_real_def
% 0.33/0.67 thf(fact_9749_times__real__def,axiom,
% 0.33/0.67 ( times_times_real
% 0.33/0.67 = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
% 0.33/0.67 @ ^ [X8: nat > rat,Y7: nat > rat,N3: nat] : ( times_times_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_real_def
% 0.33/0.67 thf(fact_9750_plus__real__def,axiom,
% 0.33/0.67 ( plus_plus_real
% 0.33/0.67 = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
% 0.33/0.67 @ ^ [X8: nat > rat,Y7: nat > rat,N3: nat] : ( plus_plus_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_real_def
% 0.33/0.67 thf(fact_9751_cr__real__def,axiom,
% 0.33/0.67 ( cr_real
% 0.33/0.67 = ( ^ [X3: nat > rat,Y6: real] :
% 0.33/0.67 ( ( realrel @ X3 @ X3 )
% 0.33/0.67 & ( ( real2 @ X3 )
% 0.33/0.67 = Y6 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cr_real_def
% 0.33/0.67 thf(fact_9752_real_Opcr__cr__eq,axiom,
% 0.33/0.67 pcr_real = cr_real ).
% 0.33/0.67
% 0.33/0.67 % real.pcr_cr_eq
% 0.33/0.67 thf(fact_9753_Bseq__monoseq__convergent_H__dec,axiom,
% 0.33/0.67 ! [F: nat > real,M10: nat] :
% 0.33/0.67 ( ( bfun_nat_real
% 0.33/0.67 @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ M10 ) )
% 0.33/0.67 @ at_top_nat )
% 0.33/0.67 => ( ! [M2: nat,N2: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M10 @ M2 )
% 0.33/0.67 => ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.33/0.67 => ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ M2 ) ) ) )
% 0.33/0.67 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Bseq_monoseq_convergent'_dec
% 0.33/0.67 thf(fact_9754_Bseq__monoseq__convergent_H__inc,axiom,
% 0.33/0.67 ! [F: nat > real,M10: nat] :
% 0.33/0.67 ( ( bfun_nat_real
% 0.33/0.67 @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ M10 ) )
% 0.33/0.67 @ at_top_nat )
% 0.33/0.67 => ( ! [M2: nat,N2: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M10 @ M2 )
% 0.33/0.67 => ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.33/0.67 => ( ord_less_eq_real @ ( F @ M2 ) @ ( F @ N2 ) ) ) )
% 0.33/0.67 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Bseq_monoseq_convergent'_inc
% 0.33/0.67 thf(fact_9755_convergent__realpow,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.67 => ( topolo7531315842566124627t_real @ ( power_power_real @ X ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % convergent_realpow
% 0.33/0.67 thf(fact_9756_set__encode__vimage__Suc,axiom,
% 0.33/0.67 ! [A3: set_nat] :
% 0.33/0.67 ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A3 ) )
% 0.33/0.67 = ( divide_divide_nat @ ( nat_set_encode @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % set_encode_vimage_Suc
% 0.33/0.67 thf(fact_9757_vimage__Suc__insert__0,axiom,
% 0.33/0.67 ! [A3: set_nat] :
% 0.33/0.67 ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A3 ) )
% 0.33/0.67 = ( vimage_nat_nat @ suc @ A3 ) ) ).
% 0.33/0.67
% 0.33/0.67 % vimage_Suc_insert_0
% 0.33/0.67 thf(fact_9758_vimage__Suc__insert__Suc,axiom,
% 0.33/0.67 ! [N: nat,A3: set_nat] :
% 0.33/0.67 ( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N ) @ A3 ) )
% 0.33/0.67 = ( insert_nat @ N @ ( vimage_nat_nat @ suc @ A3 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % vimage_Suc_insert_Suc
% 0.33/0.67 thf(fact_9759_finite__vimage__Suc__iff,axiom,
% 0.33/0.67 ! [F4: set_nat] :
% 0.33/0.67 ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F4 ) )
% 0.33/0.67 = ( finite_finite_nat @ F4 ) ) ).
% 0.33/0.67
% 0.33/0.67 % finite_vimage_Suc_iff
% 0.33/0.67 thf(fact_9760_set__decode__div__2,axiom,
% 0.33/0.67 ! [X: nat] :
% 0.33/0.67 ( ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.67 = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % set_decode_div_2
% 0.33/0.67 thf(fact_9761_euclidean__size__nat__def,axiom,
% 0.33/0.67 euclid4777050414544973029ze_nat = id_nat ).
% 0.33/0.67
% 0.33/0.67 % euclidean_size_nat_def
% 0.33/0.67 thf(fact_9762_euclidean__size__int__def,axiom,
% 0.33/0.67 ( euclid4774559944035922753ze_int
% 0.33/0.67 = ( comp_int_nat_int @ nat2 @ abs_abs_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % euclidean_size_int_def
% 0.33/0.67 thf(fact_9763_abs__division__segment,axiom,
% 0.33/0.67 ! [K: int] :
% 0.33/0.67 ( ( abs_abs_int @ ( euclid3395696857347342551nt_int @ K ) )
% 0.33/0.67 = one_one_int ) ).
% 0.33/0.67
% 0.33/0.67 % abs_division_segment
% 0.33/0.67 thf(fact_9764_division__segment__eq__sgn,axiom,
% 0.33/0.67 ! [K: int] :
% 0.33/0.67 ( ( K != zero_zero_int )
% 0.33/0.67 => ( ( euclid3395696857347342551nt_int @ K )
% 0.33/0.67 = ( sgn_sgn_int @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % division_segment_eq_sgn
% 0.33/0.67 thf(fact_9765_division__segment__nat__def,axiom,
% 0.33/0.67 ( euclid3398187327856392827nt_nat
% 0.33/0.67 = ( ^ [N3: nat] : one_one_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % division_segment_nat_def
% 0.33/0.67 thf(fact_9766_division__segment__int__def,axiom,
% 0.33/0.67 ( euclid3395696857347342551nt_int
% 0.33/0.67 = ( ^ [K2: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % division_segment_int_def
% 0.33/0.67 thf(fact_9767_transp__realrel,axiom,
% 0.33/0.67 transp_nat_rat @ realrel ).
% 0.33/0.67
% 0.33/0.67 % transp_realrel
% 0.33/0.67 thf(fact_9768_bot__nat__0_Oordering__top__axioms,axiom,
% 0.33/0.67 ( ordering_top_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X3 )
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] : ( ord_less_nat @ Y6 @ X3 )
% 0.33/0.67 @ zero_zero_nat ) ).
% 0.33/0.67
% 0.33/0.67 % bot_nat_0.ordering_top_axioms
% 0.33/0.67 thf(fact_9769_strict__mono__imp__increasing,axiom,
% 0.33/0.67 ! [F: nat > nat,N: nat] :
% 0.33/0.67 ( ( order_5726023648592871131at_nat @ F )
% 0.33/0.67 => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % strict_mono_imp_increasing
% 0.33/0.67 thf(fact_9770_gcd__nat_Oordering__top__axioms,axiom,
% 0.33/0.67 ( ordering_top_nat @ dvd_dvd_nat
% 0.33/0.67 @ ^ [M3: nat,N3: nat] :
% 0.33/0.67 ( ( dvd_dvd_nat @ M3 @ N3 )
% 0.33/0.67 & ( M3 != N3 ) )
% 0.33/0.67 @ zero_zero_nat ) ).
% 0.33/0.67
% 0.33/0.67 % gcd_nat.ordering_top_axioms
% 0.33/0.67 thf(fact_9771_prod__decode__triangle__add,axiom,
% 0.33/0.67 ! [K: nat,M: nat] :
% 0.33/0.67 ( ( nat_prod_decode @ ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) )
% 0.33/0.67 = ( nat_prod_decode_aux @ K @ M ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_decode_triangle_add
% 0.33/0.67 thf(fact_9772_prod__decode__eq,axiom,
% 0.33/0.67 ! [X: nat,Y: nat] :
% 0.33/0.67 ( ( ( nat_prod_decode @ X )
% 0.33/0.67 = ( nat_prod_decode @ Y ) )
% 0.33/0.67 = ( X = Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_decode_eq
% 0.33/0.67 thf(fact_9773_prod__encode__inverse,axiom,
% 0.33/0.67 ! [X: product_prod_nat_nat] :
% 0.33/0.67 ( ( nat_prod_decode @ ( nat_prod_encode @ X ) )
% 0.33/0.67 = X ) ).
% 0.33/0.67
% 0.33/0.67 % prod_encode_inverse
% 0.33/0.67 thf(fact_9774_prod__decode__inverse,axiom,
% 0.33/0.67 ! [N: nat] :
% 0.33/0.67 ( ( nat_prod_encode @ ( nat_prod_decode @ N ) )
% 0.33/0.67 = N ) ).
% 0.33/0.67
% 0.33/0.67 % prod_decode_inverse
% 0.33/0.67 thf(fact_9775_inj__prod__decode,axiom,
% 0.33/0.67 ! [A3: set_nat] : ( inj_on5538052773655684606at_nat @ nat_prod_decode @ A3 ) ).
% 0.33/0.67
% 0.33/0.67 % inj_prod_decode
% 0.33/0.67 thf(fact_9776_prod__decode__def,axiom,
% 0.33/0.67 ( nat_prod_decode
% 0.33/0.67 = ( nat_prod_decode_aux @ zero_zero_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_decode_def
% 0.33/0.67 thf(fact_9777_surj__prod__decode,axiom,
% 0.33/0.67 ( ( image_5846123807819985514at_nat @ nat_prod_decode @ top_top_set_nat )
% 0.33/0.67 = top_to4669805908274784177at_nat ) ).
% 0.33/0.67
% 0.33/0.67 % surj_prod_decode
% 0.33/0.67 thf(fact_9778_bij__prod__decode,axiom,
% 0.33/0.67 bij_be8693218025023041337at_nat @ nat_prod_decode @ top_top_set_nat @ top_to4669805908274784177at_nat ).
% 0.33/0.67
% 0.33/0.67 % bij_prod_decode
% 0.33/0.67 thf(fact_9779_list__decode_Opinduct,axiom,
% 0.33/0.67 ! [A0: nat,P: nat > $o] :
% 0.33/0.67 ( ( accp_nat @ nat_list_decode_rel @ A0 )
% 0.33/0.67 => ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
% 0.33/0.67 => ( P @ zero_zero_nat ) )
% 0.33/0.67 => ( ! [N2: nat] :
% 0.33/0.67 ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N2 ) )
% 0.33/0.67 => ( ! [X5: nat,Y5: nat] :
% 0.33/0.67 ( ( ( product_Pair_nat_nat @ X5 @ Y5 )
% 0.33/0.67 = ( nat_prod_decode @ N2 ) )
% 0.33/0.67 => ( P @ Y5 ) )
% 0.33/0.67 => ( P @ ( suc @ N2 ) ) ) )
% 0.33/0.67 => ( P @ A0 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % list_decode.pinduct
% 0.33/0.67 thf(fact_9780_list__decode_Oelims,axiom,
% 0.33/0.67 ! [X: nat,Y: list_nat] :
% 0.33/0.67 ( ( ( nat_list_decode @ X )
% 0.33/0.67 = Y )
% 0.33/0.67 => ( ( ( X = zero_zero_nat )
% 0.33/0.67 => ( Y != nil_nat ) )
% 0.33/0.67 => ~ ! [N2: nat] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( suc @ N2 ) )
% 0.33/0.67 => ( Y
% 0.33/0.67 != ( produc2761476792215241774st_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y6 ) )
% 0.33/0.67 @ ( nat_prod_decode @ N2 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % list_decode.elims
% 0.33/0.67 thf(fact_9781_list__encode__inverse,axiom,
% 0.33/0.67 ! [X: list_nat] :
% 0.33/0.67 ( ( nat_list_decode @ ( nat_list_encode @ X ) )
% 0.33/0.67 = X ) ).
% 0.33/0.67
% 0.33/0.67 % list_encode_inverse
% 0.33/0.67 thf(fact_9782_list__decode__inverse,axiom,
% 0.33/0.67 ! [N: nat] :
% 0.33/0.67 ( ( nat_list_encode @ ( nat_list_decode @ N ) )
% 0.33/0.67 = N ) ).
% 0.33/0.67
% 0.33/0.67 % list_decode_inverse
% 0.33/0.67 thf(fact_9783_list__decode__eq,axiom,
% 0.33/0.67 ! [X: nat,Y: nat] :
% 0.33/0.67 ( ( ( nat_list_decode @ X )
% 0.33/0.67 = ( nat_list_decode @ Y ) )
% 0.33/0.67 = ( X = Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % list_decode_eq
% 0.33/0.67 thf(fact_9784_inj__list__decode,axiom,
% 0.33/0.67 ! [A3: set_nat] : ( inj_on_nat_list_nat @ nat_list_decode @ A3 ) ).
% 0.33/0.67
% 0.33/0.67 % inj_list_decode
% 0.33/0.67 thf(fact_9785_list__decode_Opsimps_I1_J,axiom,
% 0.33/0.67 ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
% 0.33/0.67 => ( ( nat_list_decode @ zero_zero_nat )
% 0.33/0.67 = nil_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % list_decode.psimps(1)
% 0.33/0.67 thf(fact_9786_list__decode_Osimps_I1_J,axiom,
% 0.33/0.67 ( ( nat_list_decode @ zero_zero_nat )
% 0.33/0.67 = nil_nat ) ).
% 0.33/0.67
% 0.33/0.67 % list_decode.simps(1)
% 0.33/0.67 thf(fact_9787_list__decode_Opsimps_I2_J,axiom,
% 0.33/0.67 ! [N: nat] :
% 0.33/0.67 ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N ) )
% 0.33/0.67 => ( ( nat_list_decode @ ( suc @ N ) )
% 0.33/0.67 = ( produc2761476792215241774st_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y6 ) )
% 0.33/0.67 @ ( nat_prod_decode @ N ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % list_decode.psimps(2)
% 0.33/0.67 thf(fact_9788_bij__list__decode,axiom,
% 0.33/0.67 bij_be6293887246118711976st_nat @ nat_list_decode @ top_top_set_nat @ top_top_set_list_nat ).
% 0.33/0.67
% 0.33/0.67 % bij_list_decode
% 0.33/0.67 thf(fact_9789_surj__list__decode,axiom,
% 0.33/0.67 ( ( image_nat_list_nat @ nat_list_decode @ top_top_set_nat )
% 0.33/0.67 = top_top_set_list_nat ) ).
% 0.33/0.67
% 0.33/0.67 % surj_list_decode
% 0.33/0.67 thf(fact_9790_list__decode_Osimps_I2_J,axiom,
% 0.33/0.67 ! [N: nat] :
% 0.33/0.67 ( ( nat_list_decode @ ( suc @ N ) )
% 0.33/0.67 = ( produc2761476792215241774st_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y6 ) )
% 0.33/0.67 @ ( nat_prod_decode @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % list_decode.simps(2)
% 0.33/0.67 thf(fact_9791_list__decode_Opelims,axiom,
% 0.33/0.67 ! [X: nat,Y: list_nat] :
% 0.33/0.67 ( ( ( nat_list_decode @ X )
% 0.33/0.67 = Y )
% 0.33/0.67 => ( ( accp_nat @ nat_list_decode_rel @ X )
% 0.33/0.67 => ( ( ( X = zero_zero_nat )
% 0.33/0.67 => ( ( Y = nil_nat )
% 0.33/0.67 => ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
% 0.33/0.67 => ~ ! [N2: nat] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( suc @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( produc2761476792215241774st_nat
% 0.33/0.67 @ ^ [X3: nat,Y6: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y6 ) )
% 0.33/0.67 @ ( nat_prod_decode @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N2 ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % list_decode.pelims
% 0.33/0.67 thf(fact_9792_compute__powr__real,axiom,
% 0.33/0.67 ( powr_real2
% 0.33/0.67 = ( ^ [B2: real,I4: real] :
% 0.33/0.67 ( if_real @ ( ord_less_eq_real @ B2 @ zero_zero_real )
% 0.33/0.67 @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.33/0.67 @ ^ [Uu3: product_unit] : ( powr_real2 @ B2 @ I4 ) )
% 0.33/0.67 @ ( if_real
% 0.33/0.67 @ ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ I4 ) )
% 0.33/0.67 = I4 )
% 0.33/0.67 @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ I4 ) @ ( power_power_real @ B2 @ ( nat2 @ ( archim6058952711729229775r_real @ I4 ) ) ) @ ( divide_divide_real @ one_one_real @ ( power_power_real @ B2 @ ( nat2 @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ I4 ) ) ) ) ) )
% 0.33/0.67 @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.33/0.67 @ ^ [Uu3: product_unit] : ( powr_real2 @ B2 @ I4 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % compute_powr_real
% 0.33/0.67 thf(fact_9793_int__of__integer__integer__of__nat,axiom,
% 0.33/0.67 ! [N: nat] :
% 0.33/0.67 ( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
% 0.33/0.67 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % int_of_integer_integer_of_nat
% 0.33/0.67 thf(fact_9794_integer__of__nat_Orep__eq,axiom,
% 0.33/0.67 ! [X: nat] :
% 0.33/0.67 ( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
% 0.33/0.67 = ( semiri1314217659103216013at_int @ X ) ) ).
% 0.33/0.67
% 0.33/0.67 % integer_of_nat.rep_eq
% 0.33/0.67 thf(fact_9795_integer__of__nat__0,axiom,
% 0.33/0.67 ( ( code_integer_of_nat @ zero_zero_nat )
% 0.33/0.67 = zero_z3403309356797280102nteger ) ).
% 0.33/0.67
% 0.33/0.67 % integer_of_nat_0
% 0.33/0.67 thf(fact_9796_integer__of__nat_Oabs__eq,axiom,
% 0.33/0.67 ( code_integer_of_nat
% 0.33/0.67 = ( ^ [X3: nat] : ( code_integer_of_int @ ( semiri1314217659103216013at_int @ X3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % integer_of_nat.abs_eq
% 0.33/0.67 thf(fact_9797_integer__of__nat__numeral,axiom,
% 0.33/0.67 ! [N: num] :
% 0.33/0.67 ( ( code_integer_of_nat @ ( numeral_numeral_nat @ N ) )
% 0.33/0.67 = ( numera6620942414471956472nteger @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % integer_of_nat_numeral
% 0.33/0.67 thf(fact_9798_integer__of__nat__1,axiom,
% 0.33/0.67 ( ( code_integer_of_nat @ one_one_nat )
% 0.33/0.67 = one_one_Code_integer ) ).
% 0.33/0.67
% 0.33/0.67 % integer_of_nat_1
% 0.33/0.67 thf(fact_9799_integer__of__nat__def,axiom,
% 0.33/0.67 ( code_integer_of_nat
% 0.33/0.67 = ( map_fu6290471996055670595nteger @ id_nat @ code_integer_of_int @ semiri1314217659103216013at_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % integer_of_nat_def
% 0.33/0.67 thf(fact_9800_Gcd__int__set__eq__fold,axiom,
% 0.33/0.67 ! [Xs: list_int] :
% 0.33/0.67 ( ( gcd_Gcd_int @ ( set_int2 @ Xs ) )
% 0.33/0.67 = ( fold_int_int @ gcd_gcd_int @ Xs @ zero_zero_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % Gcd_int_set_eq_fold
% 0.33/0.67 thf(fact_9801_numeral__le__enat__iff,axiom,
% 0.33/0.67 ! [M: num,N: nat] :
% 0.33/0.67 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
% 0.33/0.67 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % numeral_le_enat_iff
% 0.33/0.67 thf(fact_9802_plus__enat__simps_I1_J,axiom,
% 0.33/0.67 ! [M: nat,N: nat] :
% 0.33/0.67 ( ( plus_p3455044024723400733d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.33/0.67 = ( extended_enat2 @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_enat_simps(1)
% 0.33/0.67 thf(fact_9803_idiff__enat__0,axiom,
% 0.33/0.67 ! [N: extended_enat] :
% 0.33/0.67 ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ zero_zero_nat ) @ N )
% 0.33/0.67 = ( extended_enat2 @ zero_zero_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % idiff_enat_0
% 0.33/0.67 thf(fact_9804_idiff__enat__0__right,axiom,
% 0.33/0.67 ! [N: extended_enat] :
% 0.33/0.67 ( ( minus_3235023915231533773d_enat @ N @ ( extended_enat2 @ zero_zero_nat ) )
% 0.33/0.67 = N ) ).
% 0.33/0.67
% 0.33/0.67 % idiff_enat_0_right
% 0.33/0.67 thf(fact_9805_idiff__enat__enat,axiom,
% 0.33/0.67 ! [A: nat,B: nat] :
% 0.33/0.67 ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ A ) @ ( extended_enat2 @ B ) )
% 0.33/0.67 = ( extended_enat2 @ ( minus_minus_nat @ A @ B ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % idiff_enat_enat
% 0.33/0.67 thf(fact_9806_times__enat__simps_I1_J,axiom,
% 0.33/0.67 ! [M: nat,N: nat] :
% 0.33/0.67 ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.33/0.67 = ( extended_enat2 @ ( times_times_nat @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_enat_simps(1)
% 0.33/0.67 thf(fact_9807_numeral__less__enat__iff,axiom,
% 0.33/0.67 ! [M: num,N: nat] :
% 0.33/0.67 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
% 0.33/0.67 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % numeral_less_enat_iff
% 0.33/0.67 thf(fact_9808_enat__0__iff_I2_J,axiom,
% 0.33/0.67 ! [X: nat] :
% 0.33/0.67 ( ( zero_z5237406670263579293d_enat
% 0.33/0.67 = ( extended_enat2 @ X ) )
% 0.33/0.67 = ( X = zero_zero_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % enat_0_iff(2)
% 0.33/0.67 thf(fact_9809_enat__0__iff_I1_J,axiom,
% 0.33/0.67 ! [X: nat] :
% 0.33/0.67 ( ( ( extended_enat2 @ X )
% 0.33/0.67 = zero_z5237406670263579293d_enat )
% 0.33/0.67 = ( X = zero_zero_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % enat_0_iff(1)
% 0.33/0.67 thf(fact_9810_zero__enat__def,axiom,
% 0.33/0.67 ( zero_z5237406670263579293d_enat
% 0.33/0.67 = ( extended_enat2 @ zero_zero_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % zero_enat_def
% 0.33/0.67 thf(fact_9811_numeral__eq__enat,axiom,
% 0.33/0.67 ( numera1916890842035813515d_enat
% 0.33/0.67 = ( ^ [K2: num] : ( extended_enat2 @ ( numeral_numeral_nat @ K2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % numeral_eq_enat
% 0.33/0.67 thf(fact_9812_one__enat__def,axiom,
% 0.33/0.67 ( one_on7984719198319812577d_enat
% 0.33/0.67 = ( extended_enat2 @ one_one_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % one_enat_def
% 0.33/0.67 thf(fact_9813_enat__1__iff_I1_J,axiom,
% 0.33/0.67 ! [X: nat] :
% 0.33/0.67 ( ( ( extended_enat2 @ X )
% 0.33/0.67 = one_on7984719198319812577d_enat )
% 0.33/0.67 = ( X = one_one_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % enat_1_iff(1)
% 0.33/0.67 thf(fact_9814_enat__1__iff_I2_J,axiom,
% 0.33/0.67 ! [X: nat] :
% 0.33/0.67 ( ( one_on7984719198319812577d_enat
% 0.33/0.67 = ( extended_enat2 @ X ) )
% 0.33/0.67 = ( X = one_one_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % enat_1_iff(2)
% 0.33/0.67 thf(fact_9815_Suc__ile__eq,axiom,
% 0.33/0.67 ! [M: nat,N: extended_enat] :
% 0.33/0.67 ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N )
% 0.33/0.67 = ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % Suc_ile_eq
% 0.33/0.67 thf(fact_9816_filtermap__at__right__shift,axiom,
% 0.33/0.67 ! [D: real,A: real] :
% 0.33/0.67 ( ( filtermap_real_real
% 0.33/0.67 @ ^ [X3: real] : ( minus_minus_real @ X3 @ D )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.67 = ( topolo2177554685111907308n_real @ ( minus_minus_real @ A @ D ) @ ( set_or5849166863359141190n_real @ ( minus_minus_real @ A @ D ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % filtermap_at_right_shift
% 0.33/0.67 thf(fact_9817_iadd__le__enat__iff,axiom,
% 0.33/0.67 ! [X: extended_enat,Y: extended_enat,N: nat] :
% 0.33/0.67 ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ ( extended_enat2 @ N ) )
% 0.33/0.67 = ( ? [Y9: nat,X10: nat] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( extended_enat2 @ X10 ) )
% 0.33/0.67 & ( Y
% 0.33/0.67 = ( extended_enat2 @ Y9 ) )
% 0.33/0.67 & ( ord_less_eq_nat @ ( plus_plus_nat @ X10 @ Y9 ) @ N ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % iadd_le_enat_iff
% 0.33/0.67 thf(fact_9818_at__right__to__0,axiom,
% 0.33/0.67 ! [A: real] :
% 0.33/0.67 ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
% 0.33/0.67 = ( filtermap_real_real
% 0.33/0.67 @ ^ [X3: real] : ( plus_plus_real @ X3 @ A )
% 0.33/0.67 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % at_right_to_0
% 0.33/0.67 thf(fact_9819_Gcd__nat__set__eq__fold,axiom,
% 0.33/0.67 ! [Xs: list_nat] :
% 0.33/0.67 ( ( gcd_Gcd_nat @ ( set_nat2 @ Xs ) )
% 0.33/0.67 = ( fold_nat_nat @ gcd_gcd_nat @ Xs @ zero_zero_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % Gcd_nat_set_eq_fold
% 0.33/0.67 thf(fact_9820_times__enat__simps_I4_J,axiom,
% 0.33/0.67 ! [M: nat] :
% 0.33/0.67 ( ( ( M = zero_zero_nat )
% 0.33/0.67 => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
% 0.33/0.67 = zero_z5237406670263579293d_enat ) )
% 0.33/0.67 & ( ( M != zero_zero_nat )
% 0.33/0.67 => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
% 0.33/0.67 = extend5688581933313929465d_enat ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_enat_simps(4)
% 0.33/0.67 thf(fact_9821_times__enat__simps_I3_J,axiom,
% 0.33/0.67 ! [N: nat] :
% 0.33/0.67 ( ( ( N = zero_zero_nat )
% 0.33/0.67 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
% 0.33/0.67 = zero_z5237406670263579293d_enat ) )
% 0.33/0.67 & ( ( N != zero_zero_nat )
% 0.33/0.67 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
% 0.33/0.67 = extend5688581933313929465d_enat ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_enat_simps(3)
% 0.33/0.67 thf(fact_9822_plus__enat__simps_I2_J,axiom,
% 0.33/0.67 ! [Q2: extended_enat] :
% 0.33/0.67 ( ( plus_p3455044024723400733d_enat @ extend5688581933313929465d_enat @ Q2 )
% 0.33/0.67 = extend5688581933313929465d_enat ) ).
% 0.33/0.67
% 0.33/0.67 % plus_enat_simps(2)
% 0.33/0.67 thf(fact_9823_plus__enat__simps_I3_J,axiom,
% 0.33/0.67 ! [Q2: extended_enat] :
% 0.33/0.67 ( ( plus_p3455044024723400733d_enat @ Q2 @ extend5688581933313929465d_enat )
% 0.33/0.67 = extend5688581933313929465d_enat ) ).
% 0.33/0.67
% 0.33/0.67 % plus_enat_simps(3)
% 0.33/0.67 thf(fact_9824_idiff__infinity,axiom,
% 0.33/0.67 ! [N: extended_enat] :
% 0.33/0.67 ( ( minus_3235023915231533773d_enat @ extend5688581933313929465d_enat @ N )
% 0.33/0.67 = extend5688581933313929465d_enat ) ).
% 0.33/0.67
% 0.33/0.67 % idiff_infinity
% 0.33/0.67 thf(fact_9825_times__enat__simps_I2_J,axiom,
% 0.33/0.67 ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ extend5688581933313929465d_enat )
% 0.33/0.67 = extend5688581933313929465d_enat ) ).
% 0.33/0.67
% 0.33/0.67 % times_enat_simps(2)
% 0.33/0.67 thf(fact_9826_idiff__self,axiom,
% 0.33/0.67 ! [N: extended_enat] :
% 0.33/0.67 ( ( N != extend5688581933313929465d_enat )
% 0.33/0.67 => ( ( minus_3235023915231533773d_enat @ N @ N )
% 0.33/0.67 = zero_z5237406670263579293d_enat ) ) ).
% 0.33/0.67
% 0.33/0.67 % idiff_self
% 0.33/0.67 thf(fact_9827_add__diff__cancel__enat,axiom,
% 0.33/0.67 ! [X: extended_enat,Y: extended_enat] :
% 0.33/0.67 ( ( X != extend5688581933313929465d_enat )
% 0.33/0.67 => ( ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ X )
% 0.33/0.67 = Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % add_diff_cancel_enat
% 0.33/0.67 thf(fact_9828_idiff__infinity__right,axiom,
% 0.33/0.67 ! [A: nat] :
% 0.33/0.67 ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ A ) @ extend5688581933313929465d_enat )
% 0.33/0.67 = zero_z5237406670263579293d_enat ) ).
% 0.33/0.67
% 0.33/0.67 % idiff_infinity_right
% 0.33/0.67 thf(fact_9829_enat__add__left__cancel__le,axiom,
% 0.33/0.67 ! [A: extended_enat,B: extended_enat,C: extended_enat] :
% 0.33/0.67 ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( plus_p3455044024723400733d_enat @ A @ C ) )
% 0.33/0.67 = ( ( A = extend5688581933313929465d_enat )
% 0.33/0.67 | ( ord_le2932123472753598470d_enat @ B @ C ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % enat_add_left_cancel_le
% 0.33/0.67 thf(fact_9830_enat__add__left__cancel__less,axiom,
% 0.33/0.67 ! [A: extended_enat,B: extended_enat,C: extended_enat] :
% 0.33/0.67 ( ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( plus_p3455044024723400733d_enat @ A @ C ) )
% 0.33/0.67 = ( ( A != extend5688581933313929465d_enat )
% 0.33/0.67 & ( ord_le72135733267957522d_enat @ B @ C ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % enat_add_left_cancel_less
% 0.33/0.67 thf(fact_9831_infinity__ne__i1,axiom,
% 0.33/0.67 extend5688581933313929465d_enat != one_on7984719198319812577d_enat ).
% 0.33/0.67
% 0.33/0.67 % infinity_ne_i1
% 0.33/0.67 thf(fact_9832_enat__add__left__cancel,axiom,
% 0.33/0.67 ! [A: extended_enat,B: extended_enat,C: extended_enat] :
% 0.33/0.67 ( ( ( plus_p3455044024723400733d_enat @ A @ B )
% 0.33/0.67 = ( plus_p3455044024723400733d_enat @ A @ C ) )
% 0.33/0.67 = ( ( A = extend5688581933313929465d_enat )
% 0.33/0.67 | ( B = C ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % enat_add_left_cancel
% 0.33/0.67 thf(fact_9833_plus__eq__infty__iff__enat,axiom,
% 0.33/0.67 ! [M: extended_enat,N: extended_enat] :
% 0.33/0.67 ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 0.33/0.67 = extend5688581933313929465d_enat )
% 0.33/0.67 = ( ( M = extend5688581933313929465d_enat )
% 0.33/0.67 | ( N = extend5688581933313929465d_enat ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_eq_infty_iff_enat
% 0.33/0.67 thf(fact_9834_imult__is__infinity,axiom,
% 0.33/0.67 ! [A: extended_enat,B: extended_enat] :
% 0.33/0.67 ( ( ( times_7803423173614009249d_enat @ A @ B )
% 0.33/0.67 = extend5688581933313929465d_enat )
% 0.33/0.67 = ( ( ( A = extend5688581933313929465d_enat )
% 0.33/0.67 & ( B != zero_z5237406670263579293d_enat ) )
% 0.33/0.67 | ( ( B = extend5688581933313929465d_enat )
% 0.33/0.67 & ( A != zero_z5237406670263579293d_enat ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % imult_is_infinity
% 0.33/0.67 thf(fact_9835_imult__infinity__right,axiom,
% 0.33/0.67 ! [N: extended_enat] :
% 0.33/0.67 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.33/0.67 => ( ( times_7803423173614009249d_enat @ N @ extend5688581933313929465d_enat )
% 0.33/0.67 = extend5688581933313929465d_enat ) ) ).
% 0.33/0.67
% 0.33/0.67 % imult_infinity_right
% 0.33/0.67 thf(fact_9836_imult__infinity,axiom,
% 0.33/0.67 ! [N: extended_enat] :
% 0.33/0.67 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.33/0.67 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ N )
% 0.33/0.67 = extend5688581933313929465d_enat ) ) ).
% 0.33/0.67
% 0.33/0.67 % imult_infinity
% 0.33/0.67 thf(fact_9837_times__enat__def,axiom,
% 0.33/0.67 ( times_7803423173614009249d_enat
% 0.33/0.67 = ( ^ [M3: extended_enat,N3: extended_enat] :
% 0.33/0.67 ( extend3600170679010898289d_enat
% 0.33/0.67 @ ^ [O: nat] :
% 0.33/0.67 ( extend3600170679010898289d_enat
% 0.33/0.67 @ ^ [P3: nat] : ( extended_enat2 @ ( times_times_nat @ O @ P3 ) )
% 0.33/0.67 @ ( if_Extended_enat @ ( O = zero_zero_nat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
% 0.33/0.67 @ N3 )
% 0.33/0.67 @ ( if_Extended_enat @ ( N3 = zero_z5237406670263579293d_enat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
% 0.33/0.67 @ M3 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_enat_def
% 0.33/0.67 thf(fact_9838_diff__enat__def,axiom,
% 0.33/0.67 ( minus_3235023915231533773d_enat
% 0.33/0.67 = ( ^ [A2: extended_enat,B2: extended_enat] :
% 0.33/0.67 ( extend3600170679010898289d_enat
% 0.33/0.67 @ ^ [X3: nat] :
% 0.33/0.67 ( extend3600170679010898289d_enat
% 0.33/0.67 @ ^ [Y6: nat] : ( extended_enat2 @ ( minus_minus_nat @ X3 @ Y6 ) )
% 0.33/0.67 @ zero_z5237406670263579293d_enat
% 0.33/0.67 @ B2 )
% 0.33/0.67 @ extend5688581933313929465d_enat
% 0.33/0.67 @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % diff_enat_def
% 0.33/0.67 thf(fact_9839_plus__enat__def,axiom,
% 0.33/0.67 ( plus_p3455044024723400733d_enat
% 0.33/0.67 = ( ^ [M3: extended_enat,N3: extended_enat] :
% 0.33/0.67 ( extend3600170679010898289d_enat
% 0.33/0.67 @ ^ [O: nat] :
% 0.33/0.67 ( extend3600170679010898289d_enat
% 0.33/0.67 @ ^ [P3: nat] : ( extended_enat2 @ ( plus_plus_nat @ O @ P3 ) )
% 0.33/0.67 @ extend5688581933313929465d_enat
% 0.33/0.67 @ N3 )
% 0.33/0.67 @ extend5688581933313929465d_enat
% 0.33/0.67 @ M3 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_enat_def
% 0.33/0.67 thf(fact_9840_eSuc__def,axiom,
% 0.33/0.67 ( extended_eSuc
% 0.33/0.67 = ( extend3600170679010898289d_enat
% 0.33/0.67 @ ^ [N3: nat] : ( extended_enat2 @ ( suc @ N3 ) )
% 0.33/0.67 @ extend5688581933313929465d_enat ) ) ).
% 0.33/0.67
% 0.33/0.67 % eSuc_def
% 0.33/0.67 thf(fact_9841_eSuc__numeral,axiom,
% 0.33/0.67 ! [K: num] :
% 0.33/0.67 ( ( extended_eSuc @ ( numera1916890842035813515d_enat @ K ) )
% 0.33/0.67 = ( numera1916890842035813515d_enat @ ( plus_plus_num @ K @ one ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % eSuc_numeral
% 0.33/0.67 thf(fact_9842_eSuc__minus__eSuc,axiom,
% 0.33/0.67 ! [N: extended_enat,M: extended_enat] :
% 0.33/0.67 ( ( minus_3235023915231533773d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 0.33/0.67 = ( minus_3235023915231533773d_enat @ N @ M ) ) ).
% 0.33/0.67
% 0.33/0.67 % eSuc_minus_eSuc
% 0.33/0.67 thf(fact_9843_eSuc__minus__1,axiom,
% 0.33/0.67 ! [N: extended_enat] :
% 0.33/0.67 ( ( minus_3235023915231533773d_enat @ ( extended_eSuc @ N ) @ one_on7984719198319812577d_enat )
% 0.33/0.67 = N ) ).
% 0.33/0.67
% 0.33/0.67 % eSuc_minus_1
% 0.33/0.67 thf(fact_9844_eSuc__enat,axiom,
% 0.33/0.67 ! [N: nat] :
% 0.33/0.67 ( ( extended_eSuc @ ( extended_enat2 @ N ) )
% 0.33/0.67 = ( extended_enat2 @ ( suc @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % eSuc_enat
% 0.33/0.67 thf(fact_9845_eSuc__enat__iff,axiom,
% 0.33/0.67 ! [X: extended_enat,Y: nat] :
% 0.33/0.67 ( ( ( extended_eSuc @ X )
% 0.33/0.67 = ( extended_enat2 @ Y ) )
% 0.33/0.67 = ( ? [N3: nat] :
% 0.33/0.67 ( ( Y
% 0.33/0.67 = ( suc @ N3 ) )
% 0.33/0.67 & ( X
% 0.33/0.67 = ( extended_enat2 @ N3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % eSuc_enat_iff
% 0.33/0.67 thf(fact_9846_enat__eSuc__iff,axiom,
% 0.33/0.67 ! [Y: nat,X: extended_enat] :
% 0.33/0.67 ( ( ( extended_enat2 @ Y )
% 0.33/0.67 = ( extended_eSuc @ X ) )
% 0.33/0.67 = ( ? [N3: nat] :
% 0.33/0.67 ( ( Y
% 0.33/0.67 = ( suc @ N3 ) )
% 0.33/0.67 & ( ( extended_enat2 @ N3 )
% 0.33/0.67 = X ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % enat_eSuc_iff
% 0.33/0.67 thf(fact_9847_iadd__Suc,axiom,
% 0.33/0.67 ! [M: extended_enat,N: extended_enat] :
% 0.33/0.67 ( ( plus_p3455044024723400733d_enat @ ( extended_eSuc @ M ) @ N )
% 0.33/0.67 = ( extended_eSuc @ ( plus_p3455044024723400733d_enat @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % iadd_Suc
% 0.33/0.67 thf(fact_9848_iadd__Suc__right,axiom,
% 0.33/0.67 ! [M: extended_enat,N: extended_enat] :
% 0.33/0.67 ( ( plus_p3455044024723400733d_enat @ M @ ( extended_eSuc @ N ) )
% 0.33/0.67 = ( extended_eSuc @ ( plus_p3455044024723400733d_enat @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % iadd_Suc_right
% 0.33/0.67 thf(fact_9849_one__eSuc,axiom,
% 0.33/0.67 ( one_on7984719198319812577d_enat
% 0.33/0.67 = ( extended_eSuc @ zero_z5237406670263579293d_enat ) ) ).
% 0.33/0.67
% 0.33/0.67 % one_eSuc
% 0.33/0.67 thf(fact_9850_mult__eSuc,axiom,
% 0.33/0.67 ! [M: extended_enat,N: extended_enat] :
% 0.33/0.67 ( ( times_7803423173614009249d_enat @ ( extended_eSuc @ M ) @ N )
% 0.33/0.67 = ( plus_p3455044024723400733d_enat @ N @ ( times_7803423173614009249d_enat @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % mult_eSuc
% 0.33/0.67 thf(fact_9851_mult__eSuc__right,axiom,
% 0.33/0.67 ! [M: extended_enat,N: extended_enat] :
% 0.33/0.67 ( ( times_7803423173614009249d_enat @ M @ ( extended_eSuc @ N ) )
% 0.33/0.67 = ( plus_p3455044024723400733d_enat @ M @ ( times_7803423173614009249d_enat @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % mult_eSuc_right
% 0.33/0.67 thf(fact_9852_eSuc__plus__1,axiom,
% 0.33/0.67 ( extended_eSuc
% 0.33/0.67 = ( ^ [N3: extended_enat] : ( plus_p3455044024723400733d_enat @ N3 @ one_on7984719198319812577d_enat ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % eSuc_plus_1
% 0.33/0.67 thf(fact_9853_plus__1__eSuc_I1_J,axiom,
% 0.33/0.67 ! [Q2: extended_enat] :
% 0.33/0.67 ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ Q2 )
% 0.33/0.67 = ( extended_eSuc @ Q2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_1_eSuc(1)
% 0.33/0.67 thf(fact_9854_plus__1__eSuc_I2_J,axiom,
% 0.33/0.67 ! [Q2: extended_enat] :
% 0.33/0.67 ( ( plus_p3455044024723400733d_enat @ Q2 @ one_on7984719198319812577d_enat )
% 0.33/0.67 = ( extended_eSuc @ Q2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_1_eSuc(2)
% 0.33/0.67 thf(fact_9855_Quotient__real,axiom,
% 0.33/0.67 quotie3684837364556693515t_real @ realrel @ real2 @ rep_real @ cr_real ).
% 0.33/0.67
% 0.33/0.67 % Quotient_real
% 0.33/0.67 thf(fact_9856_has__vector__derivative__id,axiom,
% 0.33/0.67 ! [Net: filter_real] :
% 0.33/0.67 ( has_ve631408500373753343e_real
% 0.33/0.67 @ ^ [X3: real] : X3
% 0.33/0.67 @ one_one_real
% 0.33/0.67 @ Net ) ).
% 0.33/0.67
% 0.33/0.67 % has_vector_derivative_id
% 0.33/0.67 thf(fact_9857_lcm__code__integer,axiom,
% 0.33/0.67 ( gcd_lcm_Code_integer
% 0.33/0.67 = ( ^ [A2: code_integer,B2: code_integer] : ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( gcd_gcd_Code_integer @ A2 @ B2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_code_integer
% 0.33/0.67 thf(fact_9858_lcm__0__iff__nat,axiom,
% 0.33/0.67 ! [M: nat,N: nat] :
% 0.33/0.67 ( ( ( gcd_lcm_nat @ M @ N )
% 0.33/0.67 = zero_zero_nat )
% 0.33/0.67 = ( ( M = zero_zero_nat )
% 0.33/0.67 | ( N = zero_zero_nat ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_0_iff_nat
% 0.33/0.67 thf(fact_9859_lcm__0__iff__int,axiom,
% 0.33/0.67 ! [M: int,N: int] :
% 0.33/0.67 ( ( ( gcd_lcm_int @ M @ N )
% 0.33/0.67 = zero_zero_int )
% 0.33/0.67 = ( ( M = zero_zero_int )
% 0.33/0.67 | ( N = zero_zero_int ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_0_iff_int
% 0.33/0.67 thf(fact_9860_lcm__proj1__iff__nat,axiom,
% 0.33/0.67 ! [M: nat,N: nat] :
% 0.33/0.67 ( ( ( gcd_lcm_nat @ M @ N )
% 0.33/0.67 = M )
% 0.33/0.67 = ( dvd_dvd_nat @ N @ M ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_proj1_iff_nat
% 0.33/0.67 thf(fact_9861_lcm__proj2__iff__nat,axiom,
% 0.33/0.67 ! [M: nat,N: nat] :
% 0.33/0.67 ( ( ( gcd_lcm_nat @ M @ N )
% 0.33/0.67 = N )
% 0.33/0.67 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_proj2_iff_nat
% 0.33/0.67 thf(fact_9862_lcm__proj1__if__dvd__nat,axiom,
% 0.33/0.67 ! [X: nat,Y: nat] :
% 0.33/0.67 ( ( dvd_dvd_nat @ X @ Y )
% 0.33/0.67 => ( ( gcd_lcm_nat @ Y @ X )
% 0.33/0.67 = Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_proj1_if_dvd_nat
% 0.33/0.67 thf(fact_9863_lcm__proj2__if__dvd__nat,axiom,
% 0.33/0.67 ! [X: nat,Y: nat] :
% 0.33/0.67 ( ( dvd_dvd_nat @ X @ Y )
% 0.33/0.67 => ( ( gcd_lcm_nat @ X @ Y )
% 0.33/0.67 = Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_proj2_if_dvd_nat
% 0.33/0.67 thf(fact_9864_lcm__int__int__eq,axiom,
% 0.33/0.67 ! [M: nat,N: nat] :
% 0.33/0.67 ( ( gcd_lcm_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.33/0.67 = ( semiri1314217659103216013at_int @ ( gcd_lcm_nat @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_int_int_eq
% 0.33/0.67 thf(fact_9865_abs__lcm__int,axiom,
% 0.33/0.67 ! [I: int,J: int] :
% 0.33/0.67 ( ( abs_abs_int @ ( gcd_lcm_int @ I @ J ) )
% 0.33/0.67 = ( gcd_lcm_int @ I @ J ) ) ).
% 0.33/0.67
% 0.33/0.67 % abs_lcm_int
% 0.33/0.67 thf(fact_9866_lcm__abs1__int,axiom,
% 0.33/0.67 ! [X: int,Y: int] :
% 0.33/0.67 ( ( gcd_lcm_int @ ( abs_abs_int @ X ) @ Y )
% 0.33/0.67 = ( gcd_lcm_int @ X @ Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_abs1_int
% 0.33/0.67 thf(fact_9867_lcm__abs2__int,axiom,
% 0.33/0.67 ! [X: int,Y: int] :
% 0.33/0.67 ( ( gcd_lcm_int @ X @ ( abs_abs_int @ Y ) )
% 0.33/0.67 = ( gcd_lcm_int @ X @ Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_abs2_int
% 0.33/0.67 thf(fact_9868_lcm__1__iff__nat,axiom,
% 0.33/0.67 ! [M: nat,N: nat] :
% 0.33/0.67 ( ( ( gcd_lcm_nat @ M @ N )
% 0.33/0.67 = ( suc @ zero_zero_nat ) )
% 0.33/0.67 = ( ( M
% 0.33/0.67 = ( suc @ zero_zero_nat ) )
% 0.33/0.67 & ( N
% 0.33/0.67 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_1_iff_nat
% 0.33/0.67 thf(fact_9869_lcm__1__iff__int,axiom,
% 0.33/0.67 ! [M: int,N: int] :
% 0.33/0.67 ( ( ( gcd_lcm_int @ M @ N )
% 0.33/0.67 = one_one_int )
% 0.33/0.67 = ( ( ( M = one_one_int )
% 0.33/0.67 | ( M
% 0.33/0.67 = ( uminus_uminus_int @ one_one_int ) ) )
% 0.33/0.67 & ( ( N = one_one_int )
% 0.33/0.67 | ( N
% 0.33/0.67 = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_1_iff_int
% 0.33/0.67 thf(fact_9870_lcm__proj2__if__dvd__int,axiom,
% 0.33/0.67 ! [X: int,Y: int] :
% 0.33/0.67 ( ( dvd_dvd_int @ X @ Y )
% 0.33/0.67 => ( ( gcd_lcm_int @ X @ Y )
% 0.33/0.67 = ( abs_abs_int @ Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_proj2_if_dvd_int
% 0.33/0.67 thf(fact_9871_lcm__proj1__if__dvd__int,axiom,
% 0.33/0.67 ! [X: int,Y: int] :
% 0.33/0.67 ( ( dvd_dvd_int @ X @ Y )
% 0.33/0.67 => ( ( gcd_lcm_int @ Y @ X )
% 0.33/0.67 = ( abs_abs_int @ Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_proj1_if_dvd_int
% 0.33/0.67 thf(fact_9872_lcm__proj2__iff__int,axiom,
% 0.33/0.67 ! [M: int,N: int] :
% 0.33/0.67 ( ( ( gcd_lcm_int @ M @ N )
% 0.33/0.67 = ( abs_abs_int @ N ) )
% 0.33/0.67 = ( dvd_dvd_int @ M @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_proj2_iff_int
% 0.33/0.67 thf(fact_9873_lcm__proj1__iff__int,axiom,
% 0.33/0.67 ! [M: int,N: int] :
% 0.33/0.67 ( ( ( gcd_lcm_int @ M @ N )
% 0.33/0.67 = ( abs_abs_int @ M ) )
% 0.33/0.67 = ( dvd_dvd_int @ N @ M ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_proj1_iff_int
% 0.33/0.67 thf(fact_9874_lcm__nat__abs__left__eq,axiom,
% 0.33/0.67 ! [K: int,N: nat] :
% 0.33/0.67 ( ( gcd_lcm_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 0.33/0.67 = ( nat2 @ ( gcd_lcm_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_nat_abs_left_eq
% 0.33/0.67 thf(fact_9875_lcm__nat__abs__right__eq,axiom,
% 0.33/0.67 ! [N: nat,K: int] :
% 0.33/0.67 ( ( gcd_lcm_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 0.33/0.67 = ( nat2 @ ( gcd_lcm_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_nat_abs_right_eq
% 0.33/0.67 thf(fact_9876_lcm__unique__int,axiom,
% 0.33/0.67 ! [D: int,A: int,B: int] :
% 0.33/0.67 ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 0.33/0.67 & ( dvd_dvd_int @ A @ D )
% 0.33/0.67 & ( dvd_dvd_int @ B @ D )
% 0.33/0.67 & ! [E3: int] :
% 0.33/0.67 ( ( ( dvd_dvd_int @ A @ E3 )
% 0.33/0.67 & ( dvd_dvd_int @ B @ E3 ) )
% 0.33/0.67 => ( dvd_dvd_int @ D @ E3 ) ) )
% 0.33/0.67 = ( D
% 0.33/0.67 = ( gcd_lcm_int @ A @ B ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_unique_int
% 0.33/0.67 thf(fact_9877_prod__gcd__lcm__nat,axiom,
% 0.33/0.67 ( times_times_nat
% 0.33/0.67 = ( ^ [M3: nat,N3: nat] : ( times_times_nat @ ( gcd_gcd_nat @ M3 @ N3 ) @ ( gcd_lcm_nat @ M3 @ N3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_gcd_lcm_nat
% 0.33/0.67 thf(fact_9878_lcm__integer_Orep__eq,axiom,
% 0.33/0.67 ! [X: code_integer,Xa: code_integer] :
% 0.33/0.67 ( ( code_int_of_integer @ ( gcd_lcm_Code_integer @ X @ Xa ) )
% 0.33/0.67 = ( gcd_lcm_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_integer.rep_eq
% 0.33/0.67 thf(fact_9879_dvd__lcm__I1__nat,axiom,
% 0.33/0.67 ! [K: nat,M: nat,N: nat] :
% 0.33/0.67 ( ( dvd_dvd_nat @ K @ M )
% 0.33/0.67 => ( dvd_dvd_nat @ K @ ( gcd_lcm_nat @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % dvd_lcm_I1_nat
% 0.33/0.67 thf(fact_9880_dvd__lcm__I2__nat,axiom,
% 0.33/0.67 ! [K: nat,N: nat,M: nat] :
% 0.33/0.67 ( ( dvd_dvd_nat @ K @ N )
% 0.33/0.67 => ( dvd_dvd_nat @ K @ ( gcd_lcm_nat @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % dvd_lcm_I2_nat
% 0.33/0.67 thf(fact_9881_lcm__unique__nat,axiom,
% 0.33/0.67 ! [A: nat,D: nat,B: nat] :
% 0.33/0.67 ( ( ( dvd_dvd_nat @ A @ D )
% 0.33/0.67 & ( dvd_dvd_nat @ B @ D )
% 0.33/0.67 & ! [E3: nat] :
% 0.33/0.67 ( ( ( dvd_dvd_nat @ A @ E3 )
% 0.33/0.67 & ( dvd_dvd_nat @ B @ E3 ) )
% 0.33/0.67 => ( dvd_dvd_nat @ D @ E3 ) ) )
% 0.33/0.67 = ( D
% 0.33/0.67 = ( gcd_lcm_nat @ A @ B ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_unique_nat
% 0.33/0.67 thf(fact_9882_dvd__lcm__I1__int,axiom,
% 0.33/0.67 ! [I: int,M: int,N: int] :
% 0.33/0.67 ( ( dvd_dvd_int @ I @ M )
% 0.33/0.67 => ( dvd_dvd_int @ I @ ( gcd_lcm_int @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % dvd_lcm_I1_int
% 0.33/0.67 thf(fact_9883_dvd__lcm__I2__int,axiom,
% 0.33/0.67 ! [I: int,N: int,M: int] :
% 0.33/0.67 ( ( dvd_dvd_int @ I @ N )
% 0.33/0.67 => ( dvd_dvd_int @ I @ ( gcd_lcm_int @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % dvd_lcm_I2_int
% 0.33/0.67 thf(fact_9884_lcm__neg2__int,axiom,
% 0.33/0.67 ! [X: int,Y: int] :
% 0.33/0.67 ( ( gcd_lcm_int @ X @ ( uminus_uminus_int @ Y ) )
% 0.33/0.67 = ( gcd_lcm_int @ X @ Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_neg2_int
% 0.33/0.67 thf(fact_9885_lcm__neg1__int,axiom,
% 0.33/0.67 ! [X: int,Y: int] :
% 0.33/0.67 ( ( gcd_lcm_int @ ( uminus_uminus_int @ X ) @ Y )
% 0.33/0.67 = ( gcd_lcm_int @ X @ Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_neg1_int
% 0.33/0.67 thf(fact_9886_lcm__integer_Oabs__eq,axiom,
% 0.33/0.67 ! [Xa: int,X: int] :
% 0.33/0.67 ( ( gcd_lcm_Code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.33/0.67 = ( code_integer_of_int @ ( gcd_lcm_int @ Xa @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_integer.abs_eq
% 0.33/0.67 thf(fact_9887_lcm__int__def,axiom,
% 0.33/0.67 ( gcd_lcm_int
% 0.33/0.67 = ( ^ [X3: int,Y6: int] : ( semiri1314217659103216013at_int @ ( gcd_lcm_nat @ ( nat2 @ ( abs_abs_int @ X3 ) ) @ ( nat2 @ ( abs_abs_int @ Y6 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_int_def
% 0.33/0.67 thf(fact_9888_lcm__integer_Orsp,axiom,
% 0.33/0.67 ( bNF_re711492959462206631nt_int
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re4712519889275205905nt_int
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 0.33/0.67 @ gcd_lcm_int
% 0.33/0.67 @ gcd_lcm_int ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_integer.rsp
% 0.33/0.67 thf(fact_9889_lcm__pos__int,axiom,
% 0.33/0.67 ! [M: int,N: int] :
% 0.33/0.67 ( ( M != zero_zero_int )
% 0.33/0.67 => ( ( N != zero_zero_int )
% 0.33/0.67 => ( ord_less_int @ zero_zero_int @ ( gcd_lcm_int @ M @ N ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_pos_int
% 0.33/0.67 thf(fact_9890_lcm__pos__nat,axiom,
% 0.33/0.67 ! [M: nat,N: nat] :
% 0.33/0.67 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.67 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.67 => ( ord_less_nat @ zero_zero_nat @ ( gcd_lcm_nat @ M @ N ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_pos_nat
% 0.33/0.67 thf(fact_9891_lcm__ge__0__int,axiom,
% 0.33/0.67 ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_lcm_int @ X @ Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_ge_0_int
% 0.33/0.67 thf(fact_9892_lcm__cases__int,axiom,
% 0.33/0.67 ! [X: int,Y: int,P: int > $o] :
% 0.33/0.67 ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.67 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.67 => ( P @ ( gcd_lcm_int @ X @ Y ) ) ) )
% 0.33/0.67 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.67 => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 0.33/0.67 => ( P @ ( gcd_lcm_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
% 0.33/0.67 => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.33/0.67 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.67 => ( P @ ( gcd_lcm_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
% 0.33/0.67 => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.33/0.67 => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 0.33/0.67 => ( P @ ( gcd_lcm_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
% 0.33/0.67 => ( P @ ( gcd_lcm_int @ X @ Y ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_cases_int
% 0.33/0.67 thf(fact_9893_lcm__nat__def,axiom,
% 0.33/0.67 ( gcd_lcm_nat
% 0.33/0.67 = ( ^ [X3: nat,Y6: nat] : ( divide_divide_nat @ ( times_times_nat @ X3 @ Y6 ) @ ( gcd_gcd_nat @ X3 @ Y6 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_nat_def
% 0.33/0.67 thf(fact_9894_prod__gcd__lcm__int,axiom,
% 0.33/0.67 ! [M: int,N: int] :
% 0.33/0.67 ( ( times_times_int @ ( abs_abs_int @ M ) @ ( abs_abs_int @ N ) )
% 0.33/0.67 = ( times_times_int @ ( gcd_gcd_int @ M @ N ) @ ( gcd_lcm_int @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_gcd_lcm_int
% 0.33/0.67 thf(fact_9895_lcm__altdef__int,axiom,
% 0.33/0.67 ( gcd_lcm_int
% 0.33/0.67 = ( ^ [A2: int,B2: int] : ( divide_divide_int @ ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( gcd_gcd_int @ A2 @ B2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_altdef_int
% 0.33/0.67 thf(fact_9896_Lcm__eq__Max__nat,axiom,
% 0.33/0.67 ! [M10: set_nat] :
% 0.33/0.67 ( ( finite_finite_nat @ M10 )
% 0.33/0.67 => ( ( M10 != bot_bot_set_nat )
% 0.33/0.67 => ( ~ ( member_nat @ zero_zero_nat @ M10 )
% 0.33/0.67 => ( ! [M2: nat,N2: nat] :
% 0.33/0.67 ( ( member_nat @ M2 @ M10 )
% 0.33/0.67 => ( ( member_nat @ N2 @ M10 )
% 0.33/0.67 => ( member_nat @ ( gcd_lcm_nat @ M2 @ N2 ) @ M10 ) ) )
% 0.33/0.67 => ( ( gcd_Lcm_nat @ M10 )
% 0.33/0.67 = ( lattic8265883725875713057ax_nat @ M10 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_eq_Max_nat
% 0.33/0.67 thf(fact_9897_Lcm__int__set__eq__fold,axiom,
% 0.33/0.67 ! [Xs: list_int] :
% 0.33/0.67 ( ( gcd_Lcm_int @ ( set_int2 @ Xs ) )
% 0.33/0.67 = ( fold_int_int @ gcd_lcm_int @ Xs @ one_one_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_int_set_eq_fold
% 0.33/0.67 thf(fact_9898_Lcm__eq__0__I__nat,axiom,
% 0.33/0.67 ! [A3: set_nat] :
% 0.33/0.67 ( ( member_nat @ zero_zero_nat @ A3 )
% 0.33/0.67 => ( ( gcd_Lcm_nat @ A3 )
% 0.33/0.67 = zero_zero_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_eq_0_I_nat
% 0.33/0.67 thf(fact_9899_abs__Lcm__eq,axiom,
% 0.33/0.67 ! [K5: set_int] :
% 0.33/0.67 ( ( abs_abs_int @ ( gcd_Lcm_int @ K5 ) )
% 0.33/0.67 = ( gcd_Lcm_int @ K5 ) ) ).
% 0.33/0.67
% 0.33/0.67 % abs_Lcm_eq
% 0.33/0.67 thf(fact_9900_Lcm__0__iff__nat,axiom,
% 0.33/0.67 ! [A3: set_nat] :
% 0.33/0.67 ( ( finite_finite_nat @ A3 )
% 0.33/0.67 => ( ( ( gcd_Lcm_nat @ A3 )
% 0.33/0.67 = zero_zero_nat )
% 0.33/0.67 = ( member_nat @ zero_zero_nat @ A3 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_0_iff_nat
% 0.33/0.67 thf(fact_9901_Lcm__abs__eq,axiom,
% 0.33/0.67 ! [K5: set_int] :
% 0.33/0.67 ( ( gcd_Lcm_int @ ( image_int_int @ abs_abs_int @ K5 ) )
% 0.33/0.67 = ( gcd_Lcm_int @ K5 ) ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_abs_eq
% 0.33/0.67 thf(fact_9902_Lcm__int__eq,axiom,
% 0.33/0.67 ! [N5: set_nat] :
% 0.33/0.67 ( ( gcd_Lcm_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N5 ) )
% 0.33/0.67 = ( semiri1314217659103216013at_int @ ( gcd_Lcm_nat @ N5 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_int_eq
% 0.33/0.67 thf(fact_9903_Lcm__nat__abs__eq,axiom,
% 0.33/0.67 ! [K5: set_int] :
% 0.33/0.67 ( ( gcd_Lcm_nat
% 0.33/0.67 @ ( image_int_nat
% 0.33/0.67 @ ^ [K2: int] : ( nat2 @ ( abs_abs_int @ K2 ) )
% 0.33/0.67 @ K5 ) )
% 0.33/0.67 = ( nat2 @ ( gcd_Lcm_int @ K5 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_nat_abs_eq
% 0.33/0.67 thf(fact_9904_Lcm__in__lcm__closed__set__nat,axiom,
% 0.33/0.67 ! [M10: set_nat] :
% 0.33/0.67 ( ( finite_finite_nat @ M10 )
% 0.33/0.67 => ( ( M10 != bot_bot_set_nat )
% 0.33/0.67 => ( ! [M2: nat,N2: nat] :
% 0.33/0.67 ( ( member_nat @ M2 @ M10 )
% 0.33/0.67 => ( ( member_nat @ N2 @ M10 )
% 0.33/0.67 => ( member_nat @ ( gcd_lcm_nat @ M2 @ N2 ) @ M10 ) ) )
% 0.33/0.67 => ( member_nat @ ( gcd_Lcm_nat @ M10 ) @ M10 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_in_lcm_closed_set_nat
% 0.33/0.67 thf(fact_9905_Lcm__nat__insert,axiom,
% 0.33/0.67 ! [N: nat,M10: set_nat] :
% 0.33/0.67 ( ( gcd_Lcm_nat @ ( insert_nat @ N @ M10 ) )
% 0.33/0.67 = ( gcd_lcm_nat @ N @ ( gcd_Lcm_nat @ M10 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_nat_insert
% 0.33/0.67 thf(fact_9906_Lcm__nat__empty,axiom,
% 0.33/0.67 ( ( gcd_Lcm_nat @ bot_bot_set_nat )
% 0.33/0.67 = one_one_nat ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_nat_empty
% 0.33/0.67 thf(fact_9907_Lcm__dvd__nat,axiom,
% 0.33/0.67 ! [M10: set_nat,N: nat] :
% 0.33/0.67 ( ! [X4: nat] :
% 0.33/0.67 ( ( member_nat @ X4 @ M10 )
% 0.33/0.67 => ( dvd_dvd_nat @ X4 @ N ) )
% 0.33/0.67 => ( dvd_dvd_nat @ ( gcd_Lcm_nat @ M10 ) @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_dvd_nat
% 0.33/0.67 thf(fact_9908_dvd__Lcm__nat,axiom,
% 0.33/0.67 ! [M: nat,M10: set_nat] :
% 0.33/0.67 ( ( member_nat @ M @ M10 )
% 0.33/0.67 => ( dvd_dvd_nat @ M @ ( gcd_Lcm_nat @ M10 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % dvd_Lcm_nat
% 0.33/0.67 thf(fact_9909_dvd__Lcm__int,axiom,
% 0.33/0.67 ! [M: int,M10: set_int] :
% 0.33/0.67 ( ( member_int @ M @ M10 )
% 0.33/0.67 => ( dvd_dvd_int @ M @ ( gcd_Lcm_int @ M10 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % dvd_Lcm_int
% 0.33/0.67 thf(fact_9910_Lcm__least__int,axiom,
% 0.33/0.67 ! [A3: set_int,A: int] :
% 0.33/0.67 ( ! [B4: int] :
% 0.33/0.67 ( ( member_int @ B4 @ A3 )
% 0.33/0.67 => ( dvd_dvd_int @ B4 @ A ) )
% 0.33/0.67 => ( dvd_dvd_int @ ( gcd_Lcm_int @ A3 ) @ A ) ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_least_int
% 0.33/0.67 thf(fact_9911_Lcm__nat__infinite,axiom,
% 0.33/0.67 ! [M10: set_nat] :
% 0.33/0.67 ( ~ ( finite_finite_nat @ M10 )
% 0.33/0.67 => ( ( gcd_Lcm_nat @ M10 )
% 0.33/0.67 = zero_zero_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_nat_infinite
% 0.33/0.67 thf(fact_9912_Lcm__int__greater__eq__0,axiom,
% 0.33/0.67 ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Lcm_int @ K5 ) ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_int_greater_eq_0
% 0.33/0.67 thf(fact_9913_Gcd__nat__def,axiom,
% 0.33/0.67 ( gcd_Gcd_nat
% 0.33/0.67 = ( ^ [M7: set_nat] :
% 0.33/0.67 ( gcd_Lcm_nat
% 0.33/0.67 @ ( collect_nat
% 0.33/0.67 @ ^ [D4: nat] :
% 0.33/0.67 ! [X3: nat] :
% 0.33/0.67 ( ( member_nat @ X3 @ M7 )
% 0.33/0.67 => ( dvd_dvd_nat @ D4 @ X3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Gcd_nat_def
% 0.33/0.67 thf(fact_9914_Lcm__int__def,axiom,
% 0.33/0.67 ( gcd_Lcm_int
% 0.33/0.67 = ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Lcm_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_int_def
% 0.33/0.67 thf(fact_9915_Lcm__nat__set__eq__fold,axiom,
% 0.33/0.67 ! [Xs: list_nat] :
% 0.33/0.67 ( ( gcd_Lcm_nat @ ( set_nat2 @ Xs ) )
% 0.33/0.67 = ( fold_nat_nat @ gcd_lcm_nat @ Xs @ one_one_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_nat_set_eq_fold
% 0.33/0.67 thf(fact_9916_Lcm__nat__def,axiom,
% 0.33/0.67 ( gcd_Lcm_nat
% 0.33/0.67 = ( ^ [M7: set_nat] : ( if_nat @ ( finite_finite_nat @ M7 ) @ ( lattic7826324295020591184_F_nat @ gcd_lcm_nat @ one_one_nat @ M7 ) @ zero_zero_nat ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Lcm_nat_def
% 0.33/0.67 thf(fact_9917_normalize__int__def,axiom,
% 0.33/0.67 normal571682499473568032ze_int = abs_abs_int ).
% 0.33/0.67
% 0.33/0.67 % normalize_int_def
% 0.33/0.67 thf(fact_9918_normalize__nat__def,axiom,
% 0.33/0.67 normal574172969982618308ze_nat = id_nat ).
% 0.33/0.67
% 0.33/0.67 % normalize_nat_def
% 0.33/0.67 thf(fact_9919_cr__int__def,axiom,
% 0.33/0.67 ( cr_int
% 0.33/0.67 = ( ^ [X3: product_prod_nat_nat] :
% 0.33/0.67 ( ^ [Y3: int,Z: int] : ( Y3 = Z )
% 0.33/0.67 @ ( abs_Integ @ X3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cr_int_def
% 0.33/0.67 thf(fact_9920_int_Opcr__cr__eq,axiom,
% 0.33/0.67 pcr_int = cr_int ).
% 0.33/0.67
% 0.33/0.67 % int.pcr_cr_eq
% 0.33/0.67 thf(fact_9921_Quotient__int,axiom,
% 0.33/0.67 quotie1194848508323700631at_int @ intrel @ abs_Integ @ rep_Integ @ cr_int ).
% 0.33/0.67
% 0.33/0.67 % Quotient_int
% 0.33/0.67 thf(fact_9922_gcd__nat_Osemilattice__neutr__axioms,axiom,
% 0.33/0.67 semila9081495762789891438tr_nat @ gcd_gcd_nat @ zero_zero_nat ).
% 0.33/0.67
% 0.33/0.67 % gcd_nat.semilattice_neutr_axioms
% 0.33/0.67 thf(fact_9923_max__nat_Osemilattice__neutr__axioms,axiom,
% 0.33/0.67 semila9081495762789891438tr_nat @ ord_max_nat @ zero_zero_nat ).
% 0.33/0.67
% 0.33/0.67 % max_nat.semilattice_neutr_axioms
% 0.33/0.67 thf(fact_9924_linear__scale__real,axiom,
% 0.33/0.67 ! [F: real > real,R: real,B: real] :
% 0.33/0.67 ( ( real_V4572627801940501177l_real @ F )
% 0.33/0.67 => ( ( F @ ( times_times_real @ R @ B ) )
% 0.33/0.67 = ( times_times_real @ R @ ( F @ B ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % linear_scale_real
% 0.33/0.67 thf(fact_9925_real__linearD,axiom,
% 0.33/0.67 ! [F: real > real] :
% 0.33/0.67 ( ( real_V4572627801940501177l_real @ F )
% 0.33/0.67 => ~ ! [C3: real] :
% 0.33/0.67 ( F
% 0.33/0.67 != ( times_times_real @ C3 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_linearD
% 0.33/0.67 thf(fact_9926_filterlim__int__sequentially,axiom,
% 0.33/0.67 filterlim_nat_int @ semiri1314217659103216013at_int @ at_top_int @ at_top_nat ).
% 0.33/0.67
% 0.33/0.67 % filterlim_int_sequentially
% 0.33/0.67 thf(fact_9927_sub_Oabs__eq,axiom,
% 0.33/0.67 ( code_sub
% 0.33/0.67 = ( ^ [Xa4: num,X3: num] : ( code_integer_of_int @ ( minus_minus_int @ ( numeral_numeral_int @ Xa4 ) @ ( numeral_numeral_int @ X3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sub.abs_eq
% 0.33/0.67 thf(fact_9928_sub_Orep__eq,axiom,
% 0.33/0.67 ! [X: num,Xa: num] :
% 0.33/0.67 ( ( code_int_of_integer @ ( code_sub @ X @ Xa ) )
% 0.33/0.67 = ( minus_minus_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Xa ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sub.rep_eq
% 0.33/0.67 thf(fact_9929_Code__Numeral_Osub__code_I1_J,axiom,
% 0.33/0.67 ( ( code_sub @ one @ one )
% 0.33/0.67 = zero_z3403309356797280102nteger ) ).
% 0.33/0.67
% 0.33/0.67 % Code_Numeral.sub_code(1)
% 0.33/0.67 thf(fact_9930_Code__Numeral_Osub__code_I9_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( code_sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.33/0.67 = ( minus_8373710615458151222nteger @ ( code_dup @ ( code_sub @ M @ N ) ) @ one_one_Code_integer ) ) ).
% 0.33/0.67
% 0.33/0.67 % Code_Numeral.sub_code(9)
% 0.33/0.67 thf(fact_9931_Code__Numeral_Osub__code_I8_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( code_sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.33/0.67 = ( plus_p5714425477246183910nteger @ ( code_dup @ ( code_sub @ M @ N ) ) @ one_one_Code_integer ) ) ).
% 0.33/0.67
% 0.33/0.67 % Code_Numeral.sub_code(8)
% 0.33/0.67 thf(fact_9932_dup_Orep__eq,axiom,
% 0.33/0.67 ! [X: code_integer] :
% 0.33/0.67 ( ( code_int_of_integer @ ( code_dup @ X ) )
% 0.33/0.67 = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % dup.rep_eq
% 0.33/0.67 thf(fact_9933_dup_Oabs__eq,axiom,
% 0.33/0.67 ! [X: int] :
% 0.33/0.67 ( ( code_dup @ ( code_integer_of_int @ X ) )
% 0.33/0.67 = ( code_integer_of_int @ ( plus_plus_int @ X @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % dup.abs_eq
% 0.33/0.67 thf(fact_9934_Code__Numeral_Osub__code_I6_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( code_sub @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.33/0.67 = ( code_dup @ ( code_sub @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Code_Numeral.sub_code(6)
% 0.33/0.67 thf(fact_9935_Code__Numeral_Osub__code_I4_J,axiom,
% 0.33/0.67 ! [N: num] :
% 0.33/0.67 ( ( code_sub @ one @ ( bit0 @ N ) )
% 0.33/0.67 = ( code_Neg @ ( bitM @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Code_Numeral.sub_code(4)
% 0.33/0.67 thf(fact_9936_Code__Numeral_Odup__def,axiom,
% 0.33/0.67 ( code_dup
% 0.33/0.67 = ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int
% 0.33/0.67 @ ^ [K2: int] : ( plus_plus_int @ K2 @ K2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Code_Numeral.dup_def
% 0.33/0.67 thf(fact_9937_Code__Numeral_Odup__code_I3_J,axiom,
% 0.33/0.67 ! [N: num] :
% 0.33/0.67 ( ( code_dup @ ( code_Neg @ N ) )
% 0.33/0.67 = ( code_Neg @ ( bit0 @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Code_Numeral.dup_code(3)
% 0.33/0.67 thf(fact_9938_plus__integer__code_I6_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( plus_p5714425477246183910nteger @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
% 0.33/0.67 = ( code_Neg @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_integer_code(6)
% 0.33/0.67 thf(fact_9939_minus__integer__code_I6_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( minus_8373710615458151222nteger @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
% 0.33/0.67 = ( code_sub @ N @ M ) ) ).
% 0.33/0.67
% 0.33/0.67 % minus_integer_code(6)
% 0.33/0.67 thf(fact_9940_Code__Numeral_Osub__code_I5_J,axiom,
% 0.33/0.67 ! [N: num] :
% 0.33/0.67 ( ( code_sub @ one @ ( bit1 @ N ) )
% 0.33/0.67 = ( code_Neg @ ( bit0 @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Code_Numeral.sub_code(5)
% 0.33/0.67 thf(fact_9941_push__bit__integer__def,axiom,
% 0.33/0.67 ( bit_se7788150548672797655nteger
% 0.33/0.67 = ( map_fu8224359078269604234nteger @ id_nat @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se545348938243370406it_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % push_bit_integer_def
% 0.33/0.67 thf(fact_9942_lcm__integer__def,axiom,
% 0.33/0.67 ( gcd_lcm_Code_integer
% 0.33/0.67 = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ gcd_lcm_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % lcm_integer_def
% 0.33/0.67 thf(fact_9943_plus__integer__def,axiom,
% 0.33/0.67 ( plus_p5714425477246183910nteger
% 0.33/0.67 = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ plus_plus_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_integer_def
% 0.33/0.67 thf(fact_9944_times__integer__def,axiom,
% 0.33/0.67 ( times_3573771949741848930nteger
% 0.33/0.67 = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ times_times_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_integer_def
% 0.33/0.67 thf(fact_9945_minus__integer__def,axiom,
% 0.33/0.67 ( minus_8373710615458151222nteger
% 0.33/0.67 = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ minus_minus_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % minus_integer_def
% 0.33/0.67 thf(fact_9946_divide__integer__def,axiom,
% 0.33/0.67 ( divide6298287555418463151nteger
% 0.33/0.67 = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ divide_divide_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % divide_integer_def
% 0.33/0.67 thf(fact_9947_gcd__integer__def,axiom,
% 0.33/0.67 ( gcd_gcd_Code_integer
% 0.33/0.67 = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ gcd_gcd_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % gcd_integer_def
% 0.33/0.67 thf(fact_9948_Code__Numeral_Osub__code_I2_J,axiom,
% 0.33/0.67 ! [M: num] :
% 0.33/0.67 ( ( code_sub @ ( bit0 @ M ) @ one )
% 0.33/0.67 = ( code_Pos @ ( bitM @ M ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Code_Numeral.sub_code(2)
% 0.33/0.67 thf(fact_9949_Code__Numeral_Osub__code_I3_J,axiom,
% 0.33/0.67 ! [M: num] :
% 0.33/0.67 ( ( code_sub @ ( bit1 @ M ) @ one )
% 0.33/0.67 = ( code_Pos @ ( bit0 @ M ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Code_Numeral.sub_code(3)
% 0.33/0.67 thf(fact_9950_Code__Numeral_Odup__code_I2_J,axiom,
% 0.33/0.67 ! [N: num] :
% 0.33/0.67 ( ( code_dup @ ( code_Pos @ N ) )
% 0.33/0.67 = ( code_Pos @ ( bit0 @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Code_Numeral.dup_code(2)
% 0.33/0.67 thf(fact_9951_times__integer__code_I3_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( times_3573771949741848930nteger @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
% 0.33/0.67 = ( code_Pos @ ( times_times_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_integer_code(3)
% 0.33/0.67 thf(fact_9952_one__integer__code,axiom,
% 0.33/0.67 ( one_one_Code_integer
% 0.33/0.67 = ( code_Pos @ one ) ) ).
% 0.33/0.67
% 0.33/0.67 % one_integer_code
% 0.33/0.67 thf(fact_9953_Pos__fold_I1_J,axiom,
% 0.33/0.67 ( ( numera6620942414471956472nteger @ one )
% 0.33/0.67 = ( code_Pos @ one ) ) ).
% 0.33/0.67
% 0.33/0.67 % Pos_fold(1)
% 0.33/0.67 thf(fact_9954_Pos__fold_I2_J,axiom,
% 0.33/0.67 ! [K: num] :
% 0.33/0.67 ( ( numera6620942414471956472nteger @ ( bit0 @ K ) )
% 0.33/0.67 = ( code_Pos @ ( bit0 @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Pos_fold(2)
% 0.33/0.67 thf(fact_9955_plus__integer__code_I3_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( plus_p5714425477246183910nteger @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
% 0.33/0.67 = ( code_Pos @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_integer_code(3)
% 0.33/0.67 thf(fact_9956_minus__integer__code_I3_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( minus_8373710615458151222nteger @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
% 0.33/0.67 = ( code_sub @ M @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % minus_integer_code(3)
% 0.33/0.67 thf(fact_9957_minus__integer__code_I4_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( minus_8373710615458151222nteger @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
% 0.33/0.67 = ( code_Pos @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % minus_integer_code(4)
% 0.33/0.67 thf(fact_9958_minus__integer__code_I5_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( minus_8373710615458151222nteger @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
% 0.33/0.67 = ( code_Neg @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % minus_integer_code(5)
% 0.33/0.67 thf(fact_9959_times__integer__code_I6_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( times_3573771949741848930nteger @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
% 0.33/0.67 = ( code_Pos @ ( times_times_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_integer_code(6)
% 0.33/0.67 thf(fact_9960_times__integer__code_I5_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( times_3573771949741848930nteger @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
% 0.33/0.67 = ( code_Neg @ ( times_times_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_integer_code(5)
% 0.33/0.67 thf(fact_9961_times__integer__code_I4_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( times_3573771949741848930nteger @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
% 0.33/0.67 = ( code_Neg @ ( times_times_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_integer_code(4)
% 0.33/0.67 thf(fact_9962_plus__integer__code_I4_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( plus_p5714425477246183910nteger @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
% 0.33/0.67 = ( code_sub @ M @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_integer_code(4)
% 0.33/0.67 thf(fact_9963_plus__integer__code_I5_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( plus_p5714425477246183910nteger @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
% 0.33/0.67 = ( code_sub @ N @ M ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_integer_code(5)
% 0.33/0.67 thf(fact_9964_max__nat_Omonoid__axioms,axiom,
% 0.33/0.67 monoid_nat @ ord_max_nat @ zero_zero_nat ).
% 0.33/0.67
% 0.33/0.67 % max_nat.monoid_axioms
% 0.33/0.67 thf(fact_9965_gcd__nat_Omonoid__axioms,axiom,
% 0.33/0.67 monoid_nat @ gcd_gcd_nat @ zero_zero_nat ).
% 0.33/0.67
% 0.33/0.67 % gcd_nat.monoid_axioms
% 0.33/0.67 thf(fact_9966_Sup__real__def,axiom,
% 0.33/0.67 ( comple1385675409528146559p_real
% 0.33/0.67 = ( ^ [X8: set_real] :
% 0.33/0.67 ( ord_Least_real
% 0.33/0.67 @ ^ [Z5: real] :
% 0.33/0.67 ! [X3: real] :
% 0.33/0.67 ( ( member_real @ X3 @ X8 )
% 0.33/0.67 => ( ord_less_eq_real @ X3 @ Z5 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Sup_real_def
% 0.33/0.67 thf(fact_9967_Least__eq__0,axiom,
% 0.33/0.67 ! [P: nat > $o] :
% 0.33/0.67 ( ( P @ zero_zero_nat )
% 0.33/0.67 => ( ( ord_Least_nat @ P )
% 0.33/0.67 = zero_zero_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % Least_eq_0
% 0.33/0.67 thf(fact_9968_Least__Suc,axiom,
% 0.33/0.67 ! [P: nat > $o,N: nat] :
% 0.33/0.67 ( ( P @ N )
% 0.33/0.67 => ( ~ ( P @ zero_zero_nat )
% 0.33/0.67 => ( ( ord_Least_nat @ P )
% 0.33/0.67 = ( suc
% 0.33/0.67 @ ( ord_Least_nat
% 0.33/0.67 @ ^ [M3: nat] : ( P @ ( suc @ M3 ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Least_Suc
% 0.33/0.67 thf(fact_9969_Least__Suc2,axiom,
% 0.33/0.67 ! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
% 0.33/0.67 ( ( P @ N )
% 0.33/0.67 => ( ( Q @ M )
% 0.33/0.67 => ( ~ ( P @ zero_zero_nat )
% 0.33/0.67 => ( ! [K3: nat] :
% 0.33/0.67 ( ( P @ ( suc @ K3 ) )
% 0.33/0.67 = ( Q @ K3 ) )
% 0.33/0.67 => ( ( ord_Least_nat @ P )
% 0.33/0.67 = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Least_Suc2
% 0.33/0.67 thf(fact_9970_unit__factor__simps_I1_J,axiom,
% 0.33/0.67 ( ( unit_f2748546683901255202or_nat @ zero_zero_nat )
% 0.33/0.67 = zero_zero_nat ) ).
% 0.33/0.67
% 0.33/0.67 % unit_factor_simps(1)
% 0.33/0.67 thf(fact_9971_unit__factor__simps_I2_J,axiom,
% 0.33/0.67 ! [N: nat] :
% 0.33/0.67 ( ( unit_f2748546683901255202or_nat @ ( suc @ N ) )
% 0.33/0.67 = one_one_nat ) ).
% 0.33/0.67
% 0.33/0.67 % unit_factor_simps(2)
% 0.33/0.67 thf(fact_9972_unit__factor__int__def,axiom,
% 0.33/0.67 unit_f2746056213392204926or_int = sgn_sgn_int ).
% 0.33/0.67
% 0.33/0.67 % unit_factor_int_def
% 0.33/0.67 thf(fact_9973_unit__factor__nat__def,axiom,
% 0.33/0.67 ( unit_f2748546683901255202or_nat
% 0.33/0.67 = ( ^ [N3: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % unit_factor_nat_def
% 0.33/0.67 thf(fact_9974_gcd__nat_Ocomm__monoid__axioms,axiom,
% 0.33/0.67 comm_monoid_nat @ gcd_gcd_nat @ zero_zero_nat ).
% 0.33/0.67
% 0.33/0.67 % gcd_nat.comm_monoid_axioms
% 0.33/0.67 thf(fact_9975_max__nat_Ocomm__monoid__axioms,axiom,
% 0.33/0.67 comm_monoid_nat @ ord_max_nat @ zero_zero_nat ).
% 0.33/0.67
% 0.33/0.67 % max_nat.comm_monoid_axioms
% 0.33/0.67 thf(fact_9976_times__num__def,axiom,
% 0.33/0.67 ( times_times_num
% 0.33/0.67 = ( ^ [M3: num,N3: num] : ( num_of_nat @ ( times_times_nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_num_def
% 0.33/0.67 thf(fact_9977_or__num_Osimps_I4_J,axiom,
% 0.33/0.67 ! [M: num] :
% 0.33/0.67 ( ( bit_un6697907153464112080or_num @ ( bit0 @ M ) @ one )
% 0.33/0.67 = ( bit1 @ M ) ) ).
% 0.33/0.67
% 0.33/0.67 % or_num.simps(4)
% 0.33/0.67 thf(fact_9978_nat__of__num__sqr,axiom,
% 0.33/0.67 ! [X: num] :
% 0.33/0.67 ( ( nat_of_num @ ( sqr @ X ) )
% 0.33/0.67 = ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % nat_of_num_sqr
% 0.33/0.67 thf(fact_9979_nat__of__num__code_I2_J,axiom,
% 0.33/0.67 ! [N: num] :
% 0.33/0.67 ( ( nat_of_num @ ( bit0 @ N ) )
% 0.33/0.67 = ( plus_plus_nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % nat_of_num_code(2)
% 0.33/0.67 thf(fact_9980_or__num_Osimps_I8_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( bit_un6697907153464112080or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.33/0.67 = ( bit1 @ ( bit_un6697907153464112080or_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % or_num.simps(8)
% 0.33/0.67 thf(fact_9981_or__num_Osimps_I6_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( bit_un6697907153464112080or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.33/0.67 = ( bit1 @ ( bit_un6697907153464112080or_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % or_num.simps(6)
% 0.33/0.67 thf(fact_9982_or__num_Osimps_I7_J,axiom,
% 0.33/0.67 ! [M: num] :
% 0.33/0.67 ( ( bit_un6697907153464112080or_num @ ( bit1 @ M ) @ one )
% 0.33/0.67 = ( bit1 @ M ) ) ).
% 0.33/0.67
% 0.33/0.67 % or_num.simps(7)
% 0.33/0.67 thf(fact_9983_or__num_Osimps_I3_J,axiom,
% 0.33/0.67 ! [N: num] :
% 0.33/0.67 ( ( bit_un6697907153464112080or_num @ one @ ( bit1 @ N ) )
% 0.33/0.67 = ( bit1 @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % or_num.simps(3)
% 0.33/0.67 thf(fact_9984_less__eq__num__def,axiom,
% 0.33/0.67 ( ord_less_eq_num
% 0.33/0.67 = ( ^ [M3: num,N3: num] : ( ord_less_eq_nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_eq_num_def
% 0.33/0.67 thf(fact_9985_nat__of__num__add,axiom,
% 0.33/0.67 ! [X: num,Y: num] :
% 0.33/0.67 ( ( nat_of_num @ ( plus_plus_num @ X @ Y ) )
% 0.33/0.67 = ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % nat_of_num_add
% 0.33/0.67 thf(fact_9986_nat__of__num_Osimps_I2_J,axiom,
% 0.33/0.67 ! [X: num] :
% 0.33/0.67 ( ( nat_of_num @ ( bit0 @ X ) )
% 0.33/0.67 = ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % nat_of_num.simps(2)
% 0.33/0.67 thf(fact_9987_nat__of__num__inverse,axiom,
% 0.33/0.67 ! [X: num] :
% 0.33/0.67 ( ( num_of_nat @ ( nat_of_num @ X ) )
% 0.33/0.67 = X ) ).
% 0.33/0.67
% 0.33/0.67 % nat_of_num_inverse
% 0.33/0.67 thf(fact_9988_or__num_Osimps_I9_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( bit_un6697907153464112080or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.33/0.67 = ( bit1 @ ( bit_un6697907153464112080or_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % or_num.simps(9)
% 0.33/0.67 thf(fact_9989_or__num_Osimps_I5_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( bit_un6697907153464112080or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.33/0.67 = ( bit0 @ ( bit_un6697907153464112080or_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % or_num.simps(5)
% 0.33/0.67 thf(fact_9990_num__eq__iff,axiom,
% 0.33/0.67 ( ( ^ [Y3: num,Z: num] : ( Y3 = Z ) )
% 0.33/0.67 = ( ^ [X3: num,Y6: num] :
% 0.33/0.67 ( ( nat_of_num @ X3 )
% 0.33/0.67 = ( nat_of_num @ Y6 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % num_eq_iff
% 0.33/0.67 thf(fact_9991_or__num_Osimps_I1_J,axiom,
% 0.33/0.67 ( ( bit_un6697907153464112080or_num @ one @ one )
% 0.33/0.67 = one ) ).
% 0.33/0.67
% 0.33/0.67 % or_num.simps(1)
% 0.33/0.67 thf(fact_9992_nat__of__num__numeral,axiom,
% 0.33/0.67 nat_of_num = numeral_numeral_nat ).
% 0.33/0.67
% 0.33/0.67 % nat_of_num_numeral
% 0.33/0.67 thf(fact_9993_nat__of__num__code_I1_J,axiom,
% 0.33/0.67 ( ( nat_of_num @ one )
% 0.33/0.67 = one_one_nat ) ).
% 0.33/0.67
% 0.33/0.67 % nat_of_num_code(1)
% 0.33/0.67 thf(fact_9994_less__num__def,axiom,
% 0.33/0.67 ( ord_less_num
% 0.33/0.67 = ( ^ [M3: num,N3: num] : ( ord_less_nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_num_def
% 0.33/0.67 thf(fact_9995_nat__of__num__mult,axiom,
% 0.33/0.67 ! [X: num,Y: num] :
% 0.33/0.67 ( ( nat_of_num @ ( times_times_num @ X @ Y ) )
% 0.33/0.67 = ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % nat_of_num_mult
% 0.33/0.67 thf(fact_9996_nat__of__num__inc,axiom,
% 0.33/0.67 ! [X: num] :
% 0.33/0.67 ( ( nat_of_num @ ( inc @ X ) )
% 0.33/0.67 = ( suc @ ( nat_of_num @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % nat_of_num_inc
% 0.33/0.67 thf(fact_9997_nat__of__num__pos,axiom,
% 0.33/0.67 ! [X: num] : ( ord_less_nat @ zero_zero_nat @ ( nat_of_num @ X ) ) ).
% 0.33/0.67
% 0.33/0.67 % nat_of_num_pos
% 0.33/0.67 thf(fact_9998_nat__of__num__neq__0,axiom,
% 0.33/0.67 ! [X: num] :
% 0.33/0.67 ( ( nat_of_num @ X )
% 0.33/0.67 != zero_zero_nat ) ).
% 0.33/0.67
% 0.33/0.67 % nat_of_num_neq_0
% 0.33/0.67 thf(fact_9999_nat__of__num_Osimps_I1_J,axiom,
% 0.33/0.67 ( ( nat_of_num @ one )
% 0.33/0.67 = ( suc @ zero_zero_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % nat_of_num.simps(1)
% 0.33/0.67 thf(fact_10000_nat__of__num_Osimps_I3_J,axiom,
% 0.33/0.67 ! [X: num] :
% 0.33/0.67 ( ( nat_of_num @ ( bit1 @ X ) )
% 0.33/0.67 = ( suc @ ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % nat_of_num.simps(3)
% 0.33/0.67 thf(fact_10001_num__of__nat__inverse,axiom,
% 0.33/0.67 ! [N: nat] :
% 0.33/0.67 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.67 => ( ( nat_of_num @ ( num_of_nat @ N ) )
% 0.33/0.67 = N ) ) ).
% 0.33/0.67
% 0.33/0.67 % num_of_nat_inverse
% 0.33/0.67 thf(fact_10002_nat__of__num__code_I3_J,axiom,
% 0.33/0.67 ! [N: num] :
% 0.33/0.67 ( ( nat_of_num @ ( bit1 @ N ) )
% 0.33/0.67 = ( suc @ ( plus_plus_nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % nat_of_num_code(3)
% 0.33/0.67 thf(fact_10003_plus__num__def,axiom,
% 0.33/0.67 ( plus_plus_num
% 0.33/0.67 = ( ^ [M3: num,N3: num] : ( num_of_nat @ ( plus_plus_nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_num_def
% 0.33/0.67 thf(fact_10004_or__num_Oelims,axiom,
% 0.33/0.67 ! [X: num,Xa: num,Y: num] :
% 0.33/0.67 ( ( ( bit_un6697907153464112080or_num @ X @ Xa )
% 0.33/0.67 = Y )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( Y != one ) ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ( Y
% 0.33/0.67 != ( bit1 @ N2 ) ) ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ( Y
% 0.33/0.67 != ( bit1 @ N2 ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( Y
% 0.33/0.67 != ( bit1 @ M2 ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ( Y
% 0.33/0.67 != ( bit0 @ ( bit_un6697907153464112080or_num @ M2 @ N2 ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ( Y
% 0.33/0.67 != ( bit1 @ ( bit_un6697907153464112080or_num @ M2 @ N2 ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( Y
% 0.33/0.67 != ( bit1 @ M2 ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ( Y
% 0.33/0.67 != ( bit1 @ ( bit_un6697907153464112080or_num @ M2 @ N2 ) ) ) ) )
% 0.33/0.67 => ~ ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ( Y
% 0.33/0.67 != ( bit1 @ ( bit_un6697907153464112080or_num @ M2 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % or_num.elims
% 0.33/0.67 thf(fact_10005_or__num_Osimps_I2_J,axiom,
% 0.33/0.67 ! [N: num] :
% 0.33/0.67 ( ( bit_un6697907153464112080or_num @ one @ ( bit0 @ N ) )
% 0.33/0.67 = ( bit1 @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % or_num.simps(2)
% 0.33/0.67 thf(fact_10006_or__num_Opelims,axiom,
% 0.33/0.67 ! [X: num,Xa: num,Y: num] :
% 0.33/0.67 ( ( ( bit_un6697907153464112080or_num @ X @ Xa )
% 0.33/0.67 = Y )
% 0.33/0.67 => ( ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( ( Y = one )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ( ( ( X = one )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ one ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( bit0 @ ( bit_un6697907153464112080or_num @ M2 @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit0 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( bit1 @ ( bit_un6697907153464112080or_num @ M2 @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ( ( Xa = one )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ one ) ) ) ) )
% 0.33/0.67 => ( ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit0 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( bit1 @ ( bit_un6697907153464112080or_num @ M2 @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.33/0.67 => ~ ! [M2: num] :
% 0.33/0.67 ( ( X
% 0.33/0.67 = ( bit1 @ M2 ) )
% 0.33/0.67 => ! [N2: num] :
% 0.33/0.67 ( ( Xa
% 0.33/0.67 = ( bit1 @ N2 ) )
% 0.33/0.67 => ( ( Y
% 0.33/0.67 = ( bit1 @ ( bit_un6697907153464112080or_num @ M2 @ N2 ) ) )
% 0.33/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % or_num.pelims
% 0.33/0.67 thf(fact_10007_or__num__dict,axiom,
% 0.33/0.67 bit_un6697907153464112080or_num = bit_un2785000775030745342or_num ).
% 0.33/0.67
% 0.33/0.67 % or_num_dict
% 0.33/0.67 thf(fact_10008_or__num__rel__dict,axiom,
% 0.33/0.67 bit_un4773296044027857193um_rel = bit_un6909899581280750971um_rel ).
% 0.33/0.67
% 0.33/0.67 % or_num_rel_dict
% 0.33/0.67 thf(fact_10009_real__floor__code,axiom,
% 0.33/0.67 ! [X: rat] :
% 0.33/0.67 ( ( archim6058952711729229775r_real @ ( ratreal @ X ) )
% 0.33/0.67 = ( archim3151403230148437115or_rat @ X ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_floor_code
% 0.33/0.67 thf(fact_10010_real__minus__code,axiom,
% 0.33/0.67 ! [X: rat,Y: rat] :
% 0.33/0.67 ( ( minus_minus_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
% 0.33/0.67 = ( ratreal @ ( minus_minus_rat @ X @ Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_minus_code
% 0.33/0.67 thf(fact_10011_real__inverse__code,axiom,
% 0.33/0.67 ! [X: rat] :
% 0.33/0.67 ( ( inverse_inverse_real @ ( ratreal @ X ) )
% 0.33/0.67 = ( ratreal @ ( inverse_inverse_rat @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_inverse_code
% 0.33/0.67 thf(fact_10012_real__plus__code,axiom,
% 0.33/0.67 ! [X: rat,Y: rat] :
% 0.33/0.67 ( ( plus_plus_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
% 0.33/0.67 = ( ratreal @ ( plus_plus_rat @ X @ Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_plus_code
% 0.33/0.67 thf(fact_10013_one__real__code,axiom,
% 0.33/0.67 ( one_one_real
% 0.33/0.67 = ( ratreal @ one_one_rat ) ) ).
% 0.33/0.67
% 0.33/0.67 % one_real_code
% 0.33/0.67 thf(fact_10014_zero__real__code,axiom,
% 0.33/0.67 ( zero_zero_real
% 0.33/0.67 = ( ratreal @ zero_zero_rat ) ) ).
% 0.33/0.67
% 0.33/0.67 % zero_real_code
% 0.33/0.67 thf(fact_10015_real__less__code,axiom,
% 0.33/0.67 ! [X: rat,Y: rat] :
% 0.33/0.67 ( ( ord_less_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
% 0.33/0.67 = ( ord_less_rat @ X @ Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_less_code
% 0.33/0.67 thf(fact_10016_Ratreal__def,axiom,
% 0.33/0.67 ratreal = field_7254667332652039916t_real ).
% 0.33/0.67
% 0.33/0.67 % Ratreal_def
% 0.33/0.67 thf(fact_10017_real__uminus__code,axiom,
% 0.33/0.67 ! [X: rat] :
% 0.33/0.67 ( ( uminus_uminus_real @ ( ratreal @ X ) )
% 0.33/0.67 = ( ratreal @ ( uminus_uminus_rat @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_uminus_code
% 0.33/0.67 thf(fact_10018_real__less__eq__code,axiom,
% 0.33/0.67 ! [X: rat,Y: rat] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
% 0.33/0.67 = ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_less_eq_code
% 0.33/0.67 thf(fact_10019_real__times__code,axiom,
% 0.33/0.67 ! [X: rat,Y: rat] :
% 0.33/0.67 ( ( times_times_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
% 0.33/0.67 = ( ratreal @ ( times_times_rat @ X @ Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_times_code
% 0.33/0.67 thf(fact_10020_real__divide__code,axiom,
% 0.33/0.67 ! [X: rat,Y: rat] :
% 0.33/0.67 ( ( divide_divide_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
% 0.33/0.67 = ( ratreal @ ( divide_divide_rat @ X @ Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_divide_code
% 0.33/0.67 thf(fact_10021_gcd__nat_Osemilattice__order__axioms,axiom,
% 0.33/0.67 ( semila1248733672344298208er_nat @ gcd_gcd_nat @ dvd_dvd_nat
% 0.33/0.67 @ ^ [M3: nat,N3: nat] :
% 0.33/0.67 ( ( dvd_dvd_nat @ M3 @ N3 )
% 0.33/0.67 & ( M3 != N3 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % gcd_nat.semilattice_order_axioms
% 0.33/0.67 thf(fact_10022_natural__decr,axiom,
% 0.33/0.67 ! [N: code_natural] :
% 0.33/0.67 ( ( N != zero_z2226904508553997617atural )
% 0.33/0.67 => ( ord_less_nat @ ( minus_minus_nat @ ( code_nat_of_natural @ N ) @ ( suc @ zero_zero_nat ) ) @ ( code_nat_of_natural @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % natural_decr
% 0.33/0.67 thf(fact_10023_nat__of__natural__numeral,axiom,
% 0.33/0.67 ! [K: num] :
% 0.33/0.67 ( ( code_nat_of_natural @ ( numera5444537566228673987atural @ K ) )
% 0.33/0.67 = ( numeral_numeral_nat @ K ) ) ).
% 0.33/0.67
% 0.33/0.67 % nat_of_natural_numeral
% 0.33/0.67 thf(fact_10024_plus__natural_Orep__eq,axiom,
% 0.33/0.67 ! [X: code_natural,Xa: code_natural] :
% 0.33/0.67 ( ( code_nat_of_natural @ ( plus_p4538020629002901425atural @ X @ Xa ) )
% 0.33/0.67 = ( plus_plus_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_natural.rep_eq
% 0.33/0.67 thf(fact_10025_minus__natural_Orep__eq,axiom,
% 0.33/0.67 ! [X: code_natural,Xa: code_natural] :
% 0.33/0.67 ( ( code_nat_of_natural @ ( minus_7197305767214868737atural @ X @ Xa ) )
% 0.33/0.67 = ( minus_minus_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % minus_natural.rep_eq
% 0.33/0.67 thf(fact_10026_times__natural_Orep__eq,axiom,
% 0.33/0.67 ! [X: code_natural,Xa: code_natural] :
% 0.33/0.67 ( ( code_nat_of_natural @ ( times_2397367101498566445atural @ X @ Xa ) )
% 0.33/0.67 = ( times_times_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_natural.rep_eq
% 0.33/0.67 thf(fact_10027_one__natural_Orep__eq,axiom,
% 0.33/0.67 ( ( code_nat_of_natural @ one_one_Code_natural )
% 0.33/0.67 = one_one_nat ) ).
% 0.33/0.67
% 0.33/0.67 % one_natural.rep_eq
% 0.33/0.67 thf(fact_10028_divide__natural_Orep__eq,axiom,
% 0.33/0.67 ! [X: code_natural,Xa: code_natural] :
% 0.33/0.67 ( ( code_nat_of_natural @ ( divide5121882707175180666atural @ X @ Xa ) )
% 0.33/0.67 = ( divide_divide_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % divide_natural.rep_eq
% 0.33/0.67 thf(fact_10029_zero__natural_Orep__eq,axiom,
% 0.33/0.67 ( ( code_nat_of_natural @ zero_z2226904508553997617atural )
% 0.33/0.67 = zero_zero_nat ) ).
% 0.33/0.67
% 0.33/0.67 % zero_natural.rep_eq
% 0.33/0.67 thf(fact_10030_push__bit__natural_Orep__eq,axiom,
% 0.33/0.67 ! [X: nat,Xa: code_natural] :
% 0.33/0.67 ( ( code_nat_of_natural @ ( bit_se6611745700429515170atural @ X @ Xa ) )
% 0.33/0.67 = ( bit_se547839408752420682it_nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % push_bit_natural.rep_eq
% 0.33/0.67 thf(fact_10031_natural__zero__minus__one,axiom,
% 0.33/0.67 ( ( minus_7197305767214868737atural @ zero_z2226904508553997617atural @ one_one_Code_natural )
% 0.33/0.67 = zero_z2226904508553997617atural ) ).
% 0.33/0.67
% 0.33/0.67 % natural_zero_minus_one
% 0.33/0.67 thf(fact_10032_integer__of__natural_Orep__eq,axiom,
% 0.33/0.67 ! [X: code_natural] :
% 0.33/0.67 ( ( code_int_of_integer @ ( code_i5400310926305786745atural @ X ) )
% 0.33/0.67 = ( semiri1314217659103216013at_int @ ( code_nat_of_natural @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % integer_of_natural.rep_eq
% 0.33/0.67 thf(fact_10033_int__of__integer__of__natural,axiom,
% 0.33/0.67 ! [N: code_natural] :
% 0.33/0.67 ( ( code_int_of_integer @ ( code_i5400310926305786745atural @ N ) )
% 0.33/0.67 = ( semiri1314217659103216013at_int @ ( code_nat_of_natural @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % int_of_integer_of_natural
% 0.33/0.67 thf(fact_10034_next_Osimps,axiom,
% 0.33/0.67 ! [V: code_natural,W: code_natural] :
% 0.33/0.67 ( ( next @ ( produc3574140220909816553atural @ V @ W ) )
% 0.33/0.67 = ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ one_one_Code_natural ) ) @ one_one_Code_natural ) @ ( produc3574140220909816553atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % next.simps
% 0.33/0.67 thf(fact_10035_log_Osimps,axiom,
% 0.33/0.67 ( log
% 0.33/0.67 = ( ^ [B2: code_natural,I4: code_natural] :
% 0.33/0.67 ( if_Code_natural
% 0.33/0.67 @ ( ( ord_le1926595141338095240atural @ B2 @ one_one_Code_natural )
% 0.33/0.67 | ( ord_le5570908160329646204atural @ I4 @ B2 ) )
% 0.33/0.67 @ one_one_Code_natural
% 0.33/0.67 @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ B2 @ ( divide5121882707175180666atural @ I4 @ B2 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % log.simps
% 0.33/0.67 thf(fact_10036_minus__shift__def,axiom,
% 0.33/0.67 ( minus_shift
% 0.33/0.67 = ( ^ [R5: code_natural,K2: code_natural,L2: code_natural] : ( if_Code_natural @ ( ord_le5570908160329646204atural @ K2 @ L2 ) @ ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ R5 @ K2 ) @ L2 ) @ ( minus_7197305767214868737atural @ K2 @ L2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % minus_shift_def
% 0.33/0.67 thf(fact_10037_log_Oelims,axiom,
% 0.33/0.67 ! [X: code_natural,Xa: code_natural,Y: code_natural] :
% 0.33/0.67 ( ( ( log @ X @ Xa )
% 0.33/0.67 = Y )
% 0.33/0.67 => ( ( ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
% 0.33/0.67 | ( ord_le5570908160329646204atural @ Xa @ X ) )
% 0.33/0.67 => ( Y = one_one_Code_natural ) )
% 0.33/0.67 & ( ~ ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
% 0.33/0.67 | ( ord_le5570908160329646204atural @ Xa @ X ) )
% 0.33/0.67 => ( Y
% 0.33/0.67 = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ X @ ( divide5121882707175180666atural @ Xa @ X ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % log.elims
% 0.33/0.67 thf(fact_10038_split__seed__def,axiom,
% 0.33/0.67 ( split_seed
% 0.33/0.67 = ( ^ [S4: produc7822875418678951345atural] :
% 0.33/0.67 ( produc8282080750456430313atural
% 0.33/0.67 @ ^ [V4: code_natural,W2: code_natural] :
% 0.33/0.67 ( produc8282080750456430313atural
% 0.33/0.67 @ ^ [V5: code_natural,W3: code_natural] : ( produc4480994950612372183atural @ ( produc3574140220909816553atural @ ( inc_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ V4 ) @ W3 ) @ ( produc3574140220909816553atural @ V5 @ ( inc_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ W2 ) ) )
% 0.33/0.67 @ ( produc6591912806276919810atural @ ( next @ S4 ) ) )
% 0.33/0.67 @ S4 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % split_seed_def
% 0.33/0.67 thf(fact_10039_Random_Orange__def,axiom,
% 0.33/0.67 ( range
% 0.33/0.67 = ( ^ [K2: code_natural] :
% 0.33/0.67 ( produc5538323210962509403atural
% 0.33/0.67 @ ( iterat8892046348760725948atural @ ( log @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K2 )
% 0.33/0.67 @ ^ [L2: code_natural] :
% 0.33/0.67 ( produc5538323210962509403atural @ next
% 0.33/0.67 @ ^ [V4: code_natural] : ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ V4 @ ( times_2397367101498566445atural @ L2 @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.33/0.67 @ one_one_Code_natural )
% 0.33/0.67 @ ^ [V4: code_natural] : ( produc6639722614265839536atural @ ( modulo8411746178871703098atural @ V4 @ K2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Random.range_def
% 0.33/0.67 thf(fact_10040_inc__shift__def,axiom,
% 0.33/0.67 ( inc_shift
% 0.33/0.67 = ( ^ [V4: code_natural,K2: code_natural] : ( if_Code_natural @ ( V4 = K2 ) @ one_one_Code_natural @ ( plus_p4538020629002901425atural @ K2 @ one_one_Code_natural ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % inc_shift_def
% 0.33/0.67 thf(fact_10041_log_Opelims,axiom,
% 0.33/0.67 ! [X: code_natural,Xa: code_natural,Y: code_natural] :
% 0.33/0.67 ( ( ( log @ X @ Xa )
% 0.33/0.67 = Y )
% 0.33/0.67 => ( ( accp_P8126237942716283194atural @ log_rel @ ( produc3574140220909816553atural @ X @ Xa ) )
% 0.33/0.67 => ~ ( ( ( ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
% 0.33/0.67 | ( ord_le5570908160329646204atural @ Xa @ X ) )
% 0.33/0.67 => ( Y = one_one_Code_natural ) )
% 0.33/0.67 & ( ~ ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
% 0.33/0.67 | ( ord_le5570908160329646204atural @ Xa @ X ) )
% 0.33/0.67 => ( Y
% 0.33/0.67 = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ X @ ( divide5121882707175180666atural @ Xa @ X ) ) ) ) ) )
% 0.33/0.67 => ~ ( accp_P8126237942716283194atural @ log_rel @ ( produc3574140220909816553atural @ X @ Xa ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % log.pelims
% 0.33/0.67 thf(fact_10042_divide__natural_Oabs__eq,axiom,
% 0.33/0.67 ! [Xa: nat,X: nat] :
% 0.33/0.67 ( ( divide5121882707175180666atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 0.33/0.67 = ( code_natural_of_nat @ ( divide_divide_nat @ Xa @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % divide_natural.abs_eq
% 0.33/0.67 thf(fact_10043_push__bit__natural_Oabs__eq,axiom,
% 0.33/0.67 ! [Xa: nat,X: nat] :
% 0.33/0.67 ( ( bit_se6611745700429515170atural @ Xa @ ( code_natural_of_nat @ X ) )
% 0.33/0.67 = ( code_natural_of_nat @ ( bit_se547839408752420682it_nat @ Xa @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % push_bit_natural.abs_eq
% 0.33/0.67 thf(fact_10044_times__natural_Oabs__eq,axiom,
% 0.33/0.67 ! [Xa: nat,X: nat] :
% 0.33/0.67 ( ( times_2397367101498566445atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 0.33/0.67 = ( code_natural_of_nat @ ( times_times_nat @ Xa @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_natural.abs_eq
% 0.33/0.67 thf(fact_10045_one__natural__def,axiom,
% 0.33/0.67 ( one_one_Code_natural
% 0.33/0.67 = ( code_natural_of_nat @ one_one_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % one_natural_def
% 0.33/0.67 thf(fact_10046_plus__natural_Oabs__eq,axiom,
% 0.33/0.67 ! [Xa: nat,X: nat] :
% 0.33/0.67 ( ( plus_p4538020629002901425atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 0.33/0.67 = ( code_natural_of_nat @ ( plus_plus_nat @ Xa @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_natural.abs_eq
% 0.33/0.67 thf(fact_10047_minus__natural_Oabs__eq,axiom,
% 0.33/0.67 ! [Xa: nat,X: nat] :
% 0.33/0.67 ( ( minus_7197305767214868737atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 0.33/0.67 = ( code_natural_of_nat @ ( minus_minus_nat @ Xa @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % minus_natural.abs_eq
% 0.33/0.67 thf(fact_10048_zero__natural__def,axiom,
% 0.33/0.67 ( zero_z2226904508553997617atural
% 0.33/0.67 = ( code_natural_of_nat @ zero_zero_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % zero_natural_def
% 0.33/0.67 thf(fact_10049_integer__of__natural_Oabs__eq,axiom,
% 0.33/0.67 ! [X: nat] :
% 0.33/0.67 ( ( code_i5400310926305786745atural @ ( code_natural_of_nat @ X ) )
% 0.33/0.67 = ( code_integer_of_int @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % integer_of_natural.abs_eq
% 0.33/0.67 thf(fact_10050_integer__of__natural__def,axiom,
% 0.33/0.67 ( code_i5400310926305786745atural
% 0.33/0.67 = ( map_fu2787874002554666395nteger @ code_nat_of_natural @ code_integer_of_int @ semiri1314217659103216013at_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % integer_of_natural_def
% 0.33/0.67 thf(fact_10051_Suc_Orep__eq,axiom,
% 0.33/0.67 ! [X: code_natural] :
% 0.33/0.67 ( ( code_nat_of_natural @ ( code_Suc @ X ) )
% 0.33/0.67 = ( suc @ ( code_nat_of_natural @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Suc.rep_eq
% 0.33/0.67 thf(fact_10052_Suc__natural__minus__one,axiom,
% 0.33/0.67 ! [N: code_natural] :
% 0.33/0.67 ( ( minus_7197305767214868737atural @ ( code_Suc @ N ) @ one_one_Code_natural )
% 0.33/0.67 = N ) ).
% 0.33/0.67
% 0.33/0.67 % Suc_natural_minus_one
% 0.33/0.67 thf(fact_10053_Suc_Oabs__eq,axiom,
% 0.33/0.67 ! [X: nat] :
% 0.33/0.67 ( ( code_Suc @ ( code_natural_of_nat @ X ) )
% 0.33/0.67 = ( code_natural_of_nat @ ( suc @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Suc.abs_eq
% 0.33/0.67 thf(fact_10054_Code__Numeral_OSuc__def,axiom,
% 0.33/0.67 ( code_Suc
% 0.33/0.67 = ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat @ suc ) ) ).
% 0.33/0.67
% 0.33/0.67 % Code_Numeral.Suc_def
% 0.33/0.67 thf(fact_10055_Quotient3__int,axiom,
% 0.33/0.67 quotie6776551016481293500at_int @ intrel @ abs_Integ @ rep_Integ ).
% 0.33/0.67
% 0.33/0.67 % Quotient3_int
% 0.33/0.67 thf(fact_10056_push__bit__natural__def,axiom,
% 0.33/0.67 ( bit_se6611745700429515170atural
% 0.33/0.67 = ( map_fu6981160461238011112atural @ id_nat @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se547839408752420682it_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % push_bit_natural_def
% 0.33/0.67 thf(fact_10057_divide__natural__def,axiom,
% 0.33/0.67 ( divide5121882707175180666atural
% 0.33/0.67 = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ divide_divide_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % divide_natural_def
% 0.33/0.67 thf(fact_10058_plus__natural__def,axiom,
% 0.33/0.67 ( plus_p4538020629002901425atural
% 0.33/0.67 = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ plus_plus_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_natural_def
% 0.33/0.67 thf(fact_10059_minus__natural__def,axiom,
% 0.33/0.67 ( minus_7197305767214868737atural
% 0.33/0.67 = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ minus_minus_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % minus_natural_def
% 0.33/0.67 thf(fact_10060_times__natural__def,axiom,
% 0.33/0.67 ( times_2397367101498566445atural
% 0.33/0.67 = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ times_times_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_natural_def
% 0.33/0.67 thf(fact_10061_sub_Otransfer,axiom,
% 0.33/0.67 ( bNF_re7876454716742015248nteger
% 0.33/0.67 @ ^ [Y3: num,Z: num] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re6501075790457514782nteger
% 0.33/0.67 @ ^ [Y3: num,Z: num] : ( Y3 = Z )
% 0.33/0.67 @ code_pcr_integer )
% 0.33/0.67 @ ^ [M3: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) )
% 0.33/0.67 @ code_sub ) ).
% 0.33/0.67
% 0.33/0.67 % sub.transfer
% 0.33/0.67 thf(fact_10062_push__bit__integer_Otransfer,axiom,
% 0.33/0.67 ( bNF_re4935368626670024657nteger
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
% 0.33/0.67 @ bit_se545348938243370406it_int
% 0.33/0.67 @ bit_se7788150548672797655nteger ) ).
% 0.33/0.67
% 0.33/0.67 % push_bit_integer.transfer
% 0.33/0.67 thf(fact_10063_divide__integer_Otransfer,axiom,
% 0.33/0.67 bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ divide_divide_int @ divide6298287555418463151nteger ).
% 0.33/0.67
% 0.33/0.67 % divide_integer.transfer
% 0.33/0.67 thf(fact_10064_minus__integer_Otransfer,axiom,
% 0.33/0.67 bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ minus_minus_int @ minus_8373710615458151222nteger ).
% 0.33/0.67
% 0.33/0.67 % minus_integer.transfer
% 0.33/0.67 thf(fact_10065_times__integer_Otransfer,axiom,
% 0.33/0.67 bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ times_times_int @ times_3573771949741848930nteger ).
% 0.33/0.67
% 0.33/0.67 % times_integer.transfer
% 0.33/0.67 thf(fact_10066_plus__integer_Otransfer,axiom,
% 0.33/0.67 bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ plus_plus_int @ plus_p5714425477246183910nteger ).
% 0.33/0.67
% 0.33/0.67 % plus_integer.transfer
% 0.33/0.67 thf(fact_10067_one__integer_Otransfer,axiom,
% 0.33/0.67 code_pcr_integer @ one_one_int @ one_one_Code_integer ).
% 0.33/0.67
% 0.33/0.67 % one_integer.transfer
% 0.33/0.67 thf(fact_10068_dup_Otransfer,axiom,
% 0.33/0.67 ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer
% 0.33/0.67 @ ^ [K2: int] : ( plus_plus_int @ K2 @ K2 )
% 0.33/0.67 @ code_dup ) ).
% 0.33/0.67
% 0.33/0.67 % dup.transfer
% 0.33/0.67 thf(fact_10069_integer__of__nat_Otransfer,axiom,
% 0.33/0.67 ( bNF_re4153400068438556298nteger
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ code_pcr_integer
% 0.33/0.67 @ semiri1314217659103216013at_int
% 0.33/0.67 @ code_integer_of_nat ) ).
% 0.33/0.67
% 0.33/0.67 % integer_of_nat.transfer
% 0.33/0.67 thf(fact_10070_gcd__integer_Otransfer,axiom,
% 0.33/0.67 bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ gcd_gcd_int @ gcd_gcd_Code_integer ).
% 0.33/0.67
% 0.33/0.67 % gcd_integer.transfer
% 0.33/0.67 thf(fact_10071_lcm__integer_Otransfer,axiom,
% 0.33/0.67 bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ gcd_lcm_int @ gcd_lcm_Code_integer ).
% 0.33/0.67
% 0.33/0.67 % lcm_integer.transfer
% 0.33/0.67 thf(fact_10072_zero__integer_Otransfer,axiom,
% 0.33/0.67 code_pcr_integer @ zero_zero_int @ zero_z3403309356797280102nteger ).
% 0.33/0.67
% 0.33/0.67 % zero_integer.transfer
% 0.33/0.67 thf(fact_10073_integer__of__natural_Otransfer,axiom,
% 0.33/0.67 bNF_re5252274238750452962nteger @ code_pcr_natural @ code_pcr_integer @ semiri1314217659103216013at_int @ code_i5400310926305786745atural ).
% 0.33/0.67
% 0.33/0.67 % integer_of_natural.transfer
% 0.33/0.67 thf(fact_10074_Suc_Otransfer,axiom,
% 0.33/0.67 bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural @ suc @ code_Suc ).
% 0.33/0.67
% 0.33/0.67 % Suc.transfer
% 0.33/0.67 thf(fact_10075_divide__natural_Otransfer,axiom,
% 0.33/0.67 bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ divide_divide_nat @ divide5121882707175180666atural ).
% 0.33/0.67
% 0.33/0.67 % divide_natural.transfer
% 0.33/0.67 thf(fact_10076_push__bit__natural_Otransfer,axiom,
% 0.33/0.67 ( bNF_re3692170009638431535atural
% 0.33/0.67 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 0.33/0.67 @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural )
% 0.33/0.67 @ bit_se547839408752420682it_nat
% 0.33/0.67 @ bit_se6611745700429515170atural ) ).
% 0.33/0.67
% 0.33/0.67 % push_bit_natural.transfer
% 0.33/0.67 thf(fact_10077_minus__natural_Otransfer,axiom,
% 0.33/0.67 bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ minus_minus_nat @ minus_7197305767214868737atural ).
% 0.33/0.67
% 0.33/0.67 % minus_natural.transfer
% 0.33/0.67 thf(fact_10078_times__natural_Otransfer,axiom,
% 0.33/0.67 bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ times_times_nat @ times_2397367101498566445atural ).
% 0.33/0.67
% 0.33/0.67 % times_natural.transfer
% 0.33/0.67 thf(fact_10079_plus__natural_Otransfer,axiom,
% 0.33/0.67 bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ plus_plus_nat @ plus_p4538020629002901425atural ).
% 0.33/0.67
% 0.33/0.67 % plus_natural.transfer
% 0.33/0.67 thf(fact_10080_less__eq__int__code_I9_J,axiom,
% 0.33/0.67 ! [K: num,L: num] :
% 0.33/0.67 ( ( ord_less_eq_int @ ( neg @ K ) @ ( neg @ L ) )
% 0.33/0.67 = ( ord_less_eq_num @ L @ K ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_eq_int_code(9)
% 0.33/0.67 thf(fact_10081_less__eq__int__code_I3_J,axiom,
% 0.33/0.67 ! [L: num] :
% 0.33/0.67 ~ ( ord_less_eq_int @ zero_zero_int @ ( neg @ L ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_eq_int_code(3)
% 0.33/0.67 thf(fact_10082_less__eq__int__code_I7_J,axiom,
% 0.33/0.67 ! [K: num] : ( ord_less_eq_int @ ( neg @ K ) @ zero_zero_int ) ).
% 0.33/0.67
% 0.33/0.67 % less_eq_int_code(7)
% 0.33/0.67 thf(fact_10083_zero__natural_Otransfer,axiom,
% 0.33/0.67 code_pcr_natural @ zero_zero_nat @ zero_z2226904508553997617atural ).
% 0.33/0.67
% 0.33/0.67 % zero_natural.transfer
% 0.33/0.67 thf(fact_10084_nat__code_I1_J,axiom,
% 0.33/0.67 ! [K: num] :
% 0.33/0.67 ( ( nat2 @ ( neg @ K ) )
% 0.33/0.67 = zero_zero_nat ) ).
% 0.33/0.67
% 0.33/0.67 % nat_code(1)
% 0.33/0.67 thf(fact_10085_less__int__code_I9_J,axiom,
% 0.33/0.67 ! [K: num,L: num] :
% 0.33/0.67 ( ( ord_less_int @ ( neg @ K ) @ ( neg @ L ) )
% 0.33/0.67 = ( ord_less_num @ L @ K ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_int_code(9)
% 0.33/0.67 thf(fact_10086_one__natural_Otransfer,axiom,
% 0.33/0.67 code_pcr_natural @ one_one_nat @ one_one_Code_natural ).
% 0.33/0.67
% 0.33/0.67 % one_natural.transfer
% 0.33/0.67 thf(fact_10087_plus__int__code_I6_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( plus_plus_int @ ( neg @ M ) @ ( neg @ N ) )
% 0.33/0.67 = ( neg @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_int_code(6)
% 0.33/0.67 thf(fact_10088_less__int__code_I3_J,axiom,
% 0.33/0.67 ! [L: num] :
% 0.33/0.67 ~ ( ord_less_int @ zero_zero_int @ ( neg @ L ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_int_code(3)
% 0.33/0.67 thf(fact_10089_less__int__code_I7_J,axiom,
% 0.33/0.67 ! [K: num] : ( ord_less_int @ ( neg @ K ) @ zero_zero_int ) ).
% 0.33/0.67
% 0.33/0.67 % less_int_code(7)
% 0.33/0.67 thf(fact_10090_Int_Osub__code_I4_J,axiom,
% 0.33/0.67 ! [N: num] :
% 0.33/0.67 ( ( sub @ one @ ( bit0 @ N ) )
% 0.33/0.67 = ( neg @ ( bitM @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Int.sub_code(4)
% 0.33/0.67 thf(fact_10091_Int_Osub__code_I5_J,axiom,
% 0.33/0.67 ! [N: num] :
% 0.33/0.67 ( ( sub @ one @ ( bit1 @ N ) )
% 0.33/0.67 = ( neg @ ( bit0 @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Int.sub_code(5)
% 0.33/0.67 thf(fact_10092_Int_Osub__def,axiom,
% 0.33/0.67 ( sub
% 0.33/0.67 = ( ^ [M3: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Int.sub_def
% 0.33/0.67 thf(fact_10093_Int_Osub__code_I1_J,axiom,
% 0.33/0.67 ( ( sub @ one @ one )
% 0.33/0.67 = zero_zero_int ) ).
% 0.33/0.67
% 0.33/0.67 % Int.sub_code(1)
% 0.33/0.67 thf(fact_10094_minus__int__code_I6_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( minus_minus_int @ ( neg @ M ) @ ( neg @ N ) )
% 0.33/0.67 = ( sub @ N @ M ) ) ).
% 0.33/0.67
% 0.33/0.67 % minus_int_code(6)
% 0.33/0.67 thf(fact_10095_Int_Osub__code_I9_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.33/0.67 = ( minus_minus_int @ ( dup @ ( sub @ M @ N ) ) @ one_one_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % Int.sub_code(9)
% 0.33/0.67 thf(fact_10096_Int_Osub__code_I8_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.33/0.67 = ( plus_plus_int @ ( dup @ ( sub @ M @ N ) ) @ one_one_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % Int.sub_code(8)
% 0.33/0.67 thf(fact_10097_Int_Odup__code_I1_J,axiom,
% 0.33/0.67 ( ( dup @ zero_zero_int )
% 0.33/0.67 = zero_zero_int ) ).
% 0.33/0.67
% 0.33/0.67 % Int.dup_code(1)
% 0.33/0.67 thf(fact_10098_Int_Odup__def,axiom,
% 0.33/0.67 ( dup
% 0.33/0.67 = ( ^ [K2: int] : ( plus_plus_int @ K2 @ K2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Int.dup_def
% 0.33/0.67 thf(fact_10099_Int_Odup__code_I3_J,axiom,
% 0.33/0.67 ! [N: num] :
% 0.33/0.67 ( ( dup @ ( neg @ N ) )
% 0.33/0.67 = ( neg @ ( bit0 @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Int.dup_code(3)
% 0.33/0.67 thf(fact_10100_Int_Osub__code_I6_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( sub @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.33/0.67 = ( dup @ ( sub @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Int.sub_code(6)
% 0.33/0.67 thf(fact_10101_Int_Osub__code_I7_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( sub @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.33/0.67 = ( dup @ ( sub @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Int.sub_code(7)
% 0.33/0.67 thf(fact_10102_Int_Osub__code_I2_J,axiom,
% 0.33/0.67 ! [M: num] :
% 0.33/0.67 ( ( sub @ ( bit0 @ M ) @ one )
% 0.33/0.67 = ( pos @ ( bitM @ M ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Int.sub_code(2)
% 0.33/0.67 thf(fact_10103_Int_Osub__code_I3_J,axiom,
% 0.33/0.67 ! [M: num] :
% 0.33/0.67 ( ( sub @ ( bit1 @ M ) @ one )
% 0.33/0.67 = ( pos @ ( bit0 @ M ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Int.sub_code(3)
% 0.33/0.67 thf(fact_10104_Int_Odup__code_I2_J,axiom,
% 0.33/0.67 ! [N: num] :
% 0.33/0.67 ( ( dup @ ( pos @ N ) )
% 0.33/0.67 = ( pos @ ( bit0 @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Int.dup_code(2)
% 0.33/0.67 thf(fact_10105_less__eq__int__code_I6_J,axiom,
% 0.33/0.67 ! [K: num,L: num] :
% 0.33/0.67 ~ ( ord_less_eq_int @ ( pos @ K ) @ ( neg @ L ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_eq_int_code(6)
% 0.33/0.67 thf(fact_10106_less__eq__int__code_I8_J,axiom,
% 0.33/0.67 ! [K: num,L: num] : ( ord_less_eq_int @ ( neg @ K ) @ ( pos @ L ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_eq_int_code(8)
% 0.33/0.67 thf(fact_10107_Int_ONeg__def,axiom,
% 0.33/0.67 ( neg
% 0.33/0.67 = ( ^ [N3: num] : ( uminus_uminus_int @ ( pos @ N3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Int.Neg_def
% 0.33/0.67 thf(fact_10108_uminus__int__code_I2_J,axiom,
% 0.33/0.67 ! [M: num] :
% 0.33/0.67 ( ( uminus_uminus_int @ ( pos @ M ) )
% 0.33/0.67 = ( neg @ M ) ) ).
% 0.33/0.67
% 0.33/0.67 % uminus_int_code(2)
% 0.33/0.67 thf(fact_10109_uminus__int__code_I3_J,axiom,
% 0.33/0.67 ! [M: num] :
% 0.33/0.67 ( ( uminus_uminus_int @ ( neg @ M ) )
% 0.33/0.67 = ( pos @ M ) ) ).
% 0.33/0.67
% 0.33/0.67 % uminus_int_code(3)
% 0.33/0.67 thf(fact_10110_less__int__code_I8_J,axiom,
% 0.33/0.67 ! [K: num,L: num] : ( ord_less_int @ ( neg @ K ) @ ( pos @ L ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_int_code(8)
% 0.33/0.67 thf(fact_10111_less__int__code_I6_J,axiom,
% 0.33/0.67 ! [K: num,L: num] :
% 0.33/0.67 ~ ( ord_less_int @ ( pos @ K ) @ ( neg @ L ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_int_code(6)
% 0.33/0.67 thf(fact_10112_less__int__code_I4_J,axiom,
% 0.33/0.67 ! [K: num] :
% 0.33/0.67 ~ ( ord_less_int @ ( pos @ K ) @ zero_zero_int ) ).
% 0.33/0.67
% 0.33/0.67 % less_int_code(4)
% 0.33/0.67 thf(fact_10113_less__int__code_I2_J,axiom,
% 0.33/0.67 ! [L: num] : ( ord_less_int @ zero_zero_int @ ( pos @ L ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_int_code(2)
% 0.33/0.67 thf(fact_10114_Int_OPos__def,axiom,
% 0.33/0.67 pos = numeral_numeral_int ).
% 0.33/0.67
% 0.33/0.67 % Int.Pos_def
% 0.33/0.67 thf(fact_10115_one__int__code,axiom,
% 0.33/0.67 ( one_one_int
% 0.33/0.67 = ( pos @ one ) ) ).
% 0.33/0.67
% 0.33/0.67 % one_int_code
% 0.33/0.67 thf(fact_10116_plus__int__code_I3_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( plus_plus_int @ ( pos @ M ) @ ( pos @ N ) )
% 0.33/0.67 = ( pos @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_int_code(3)
% 0.33/0.67 thf(fact_10117_times__int__code_I3_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( times_times_int @ ( pos @ M ) @ ( pos @ N ) )
% 0.33/0.67 = ( pos @ ( times_times_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_int_code(3)
% 0.33/0.67 thf(fact_10118_less__int__code_I5_J,axiom,
% 0.33/0.67 ! [K: num,L: num] :
% 0.33/0.67 ( ( ord_less_int @ ( pos @ K ) @ ( pos @ L ) )
% 0.33/0.67 = ( ord_less_num @ K @ L ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_int_code(5)
% 0.33/0.67 thf(fact_10119_nat__code_I3_J,axiom,
% 0.33/0.67 ! [K: num] :
% 0.33/0.67 ( ( nat2 @ ( pos @ K ) )
% 0.33/0.67 = ( nat_of_num @ K ) ) ).
% 0.33/0.67
% 0.33/0.67 % nat_code(3)
% 0.33/0.67 thf(fact_10120_less__eq__int__code_I2_J,axiom,
% 0.33/0.67 ! [L: num] : ( ord_less_eq_int @ zero_zero_int @ ( pos @ L ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_eq_int_code(2)
% 0.33/0.67 thf(fact_10121_less__eq__int__code_I4_J,axiom,
% 0.33/0.67 ! [K: num] :
% 0.33/0.67 ~ ( ord_less_eq_int @ ( pos @ K ) @ zero_zero_int ) ).
% 0.33/0.67
% 0.33/0.67 % less_eq_int_code(4)
% 0.33/0.67 thf(fact_10122_less__eq__int__code_I5_J,axiom,
% 0.33/0.67 ! [K: num,L: num] :
% 0.33/0.67 ( ( ord_less_eq_int @ ( pos @ K ) @ ( pos @ L ) )
% 0.33/0.67 = ( ord_less_eq_num @ K @ L ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_eq_int_code(5)
% 0.33/0.67 thf(fact_10123_minus__int__code_I3_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( minus_minus_int @ ( pos @ M ) @ ( pos @ N ) )
% 0.33/0.67 = ( sub @ M @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % minus_int_code(3)
% 0.33/0.67 thf(fact_10124_minus__int__code_I5_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( minus_minus_int @ ( neg @ M ) @ ( pos @ N ) )
% 0.33/0.67 = ( neg @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % minus_int_code(5)
% 0.33/0.67 thf(fact_10125_minus__int__code_I4_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( minus_minus_int @ ( pos @ M ) @ ( neg @ N ) )
% 0.33/0.67 = ( pos @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % minus_int_code(4)
% 0.33/0.67 thf(fact_10126_times__int__code_I6_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( times_times_int @ ( neg @ M ) @ ( neg @ N ) )
% 0.33/0.67 = ( pos @ ( times_times_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_int_code(6)
% 0.33/0.67 thf(fact_10127_times__int__code_I5_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( times_times_int @ ( neg @ M ) @ ( pos @ N ) )
% 0.33/0.67 = ( neg @ ( times_times_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_int_code(5)
% 0.33/0.67 thf(fact_10128_times__int__code_I4_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( times_times_int @ ( pos @ M ) @ ( neg @ N ) )
% 0.33/0.67 = ( neg @ ( times_times_num @ M @ N ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % times_int_code(4)
% 0.33/0.67 thf(fact_10129_plus__int__code_I4_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( plus_plus_int @ ( pos @ M ) @ ( neg @ N ) )
% 0.33/0.67 = ( sub @ M @ N ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_int_code(4)
% 0.33/0.67 thf(fact_10130_plus__int__code_I5_J,axiom,
% 0.33/0.67 ! [M: num,N: num] :
% 0.33/0.67 ( ( plus_plus_int @ ( neg @ M ) @ ( pos @ N ) )
% 0.33/0.67 = ( sub @ N @ M ) ) ).
% 0.33/0.67
% 0.33/0.67 % plus_int_code(5)
% 0.33/0.67 thf(fact_10131_sum__encode__def,axiom,
% 0.33/0.67 ( nat_sum_encode
% 0.33/0.67 = ( sum_ca6763686470577984908at_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.67 @ ^ [B2: nat] : ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sum_encode_def
% 0.33/0.67 thf(fact_10132_Quotient3__real,axiom,
% 0.33/0.67 quotie8700032322157300518t_real @ realrel @ real2 @ rep_real ).
% 0.33/0.67
% 0.33/0.67 % Quotient3_real
% 0.33/0.67 thf(fact_10133_sum__encode__eq,axiom,
% 0.33/0.67 ! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] :
% 0.33/0.67 ( ( ( nat_sum_encode @ X )
% 0.33/0.67 = ( nat_sum_encode @ Y ) )
% 0.33/0.67 = ( X = Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % sum_encode_eq
% 0.33/0.67 thf(fact_10134_bij__sum__encode,axiom,
% 0.33/0.67 bij_be5432664580149595207at_nat @ nat_sum_encode @ top_to6661820994512907621at_nat @ top_top_set_nat ).
% 0.33/0.67
% 0.33/0.67 % bij_sum_encode
% 0.33/0.67 thf(fact_10135_surj__sum__encode,axiom,
% 0.33/0.67 ( ( image_1320371278474632150at_nat @ nat_sum_encode @ top_to6661820994512907621at_nat )
% 0.33/0.67 = top_top_set_nat ) ).
% 0.33/0.67
% 0.33/0.67 % surj_sum_encode
% 0.33/0.67 thf(fact_10136_inj__sum__encode,axiom,
% 0.33/0.67 ! [A3: set_Sum_sum_nat_nat] : ( inj_on6343450744447823682at_nat @ nat_sum_encode @ A3 ) ).
% 0.33/0.67
% 0.33/0.67 % inj_sum_encode
% 0.33/0.67 thf(fact_10137_int__encode__def,axiom,
% 0.33/0.67 ( nat_int_encode
% 0.33/0.67 = ( ^ [I4: int] : ( nat_sum_encode @ ( if_Sum_sum_nat_nat @ ( ord_less_eq_int @ zero_zero_int @ I4 ) @ ( sum_Inl_nat_nat @ ( nat2 @ I4 ) ) @ ( sum_Inr_nat_nat @ ( nat2 @ ( minus_minus_int @ ( uminus_uminus_int @ I4 ) @ one_one_int ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % int_encode_def
% 0.33/0.67 thf(fact_10138_sum__decode__def,axiom,
% 0.33/0.67 ( nat_sum_decode
% 0.33/0.67 = ( ^ [N3: nat] : ( if_Sum_sum_nat_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( sum_Inl_nat_nat @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( sum_Inr_nat_nat @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sum_decode_def
% 0.33/0.67 thf(fact_10139_sum__encode__inverse,axiom,
% 0.33/0.67 ! [X: sum_sum_nat_nat] :
% 0.33/0.67 ( ( nat_sum_decode @ ( nat_sum_encode @ X ) )
% 0.33/0.67 = X ) ).
% 0.33/0.67
% 0.33/0.67 % sum_encode_inverse
% 0.33/0.67 thf(fact_10140_sum__decode__inverse,axiom,
% 0.33/0.67 ! [N: nat] :
% 0.33/0.67 ( ( nat_sum_encode @ ( nat_sum_decode @ N ) )
% 0.33/0.67 = N ) ).
% 0.33/0.67
% 0.33/0.67 % sum_decode_inverse
% 0.33/0.67 thf(fact_10141_inj__sum__decode,axiom,
% 0.33/0.67 ! [A3: set_nat] : ( inj_on5701776251185195458at_nat @ nat_sum_decode @ A3 ) ).
% 0.33/0.67
% 0.33/0.67 % inj_sum_decode
% 0.33/0.67 thf(fact_10142_surj__sum__decode,axiom,
% 0.33/0.67 ( ( image_678696785212003926at_nat @ nat_sum_decode @ top_top_set_nat )
% 0.33/0.67 = top_to6661820994512907621at_nat ) ).
% 0.33/0.67
% 0.33/0.67 % surj_sum_decode
% 0.33/0.67 thf(fact_10143_bij__sum__decode,axiom,
% 0.33/0.67 bij_be4790990086886966983at_nat @ nat_sum_decode @ top_top_set_nat @ top_to6661820994512907621at_nat ).
% 0.33/0.67
% 0.33/0.67 % bij_sum_decode
% 0.33/0.67 thf(fact_10144_sum__decode__eq,axiom,
% 0.33/0.67 ! [X: nat,Y: nat] :
% 0.33/0.67 ( ( ( nat_sum_decode @ X )
% 0.33/0.67 = ( nat_sum_decode @ Y ) )
% 0.33/0.67 = ( X = Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % sum_decode_eq
% 0.33/0.67 thf(fact_10145_nth__item_Opinduct,axiom,
% 0.33/0.67 ! [A0: nat,P: nat > $o] :
% 0.33/0.67 ( ( accp_nat @ nth_item_rel @ A0 )
% 0.33/0.67 => ( ( ( accp_nat @ nth_item_rel @ zero_zero_nat )
% 0.33/0.67 => ( P @ zero_zero_nat ) )
% 0.33/0.67 => ( ! [N2: nat] :
% 0.33/0.67 ( ( accp_nat @ nth_item_rel @ ( suc @ N2 ) )
% 0.33/0.67 => ( ! [A6: nat,Aa: nat] :
% 0.33/0.67 ( ( ( nat_sum_decode @ N2 )
% 0.33/0.67 = ( sum_Inl_nat_nat @ A6 ) )
% 0.33/0.67 => ( ( ( nat_sum_decode @ A6 )
% 0.33/0.67 = ( sum_Inl_nat_nat @ Aa ) )
% 0.33/0.67 => ( P @ Aa ) ) )
% 0.33/0.67 => ( ! [A6: nat,B8: nat] :
% 0.33/0.67 ( ( ( nat_sum_decode @ N2 )
% 0.33/0.67 = ( sum_Inl_nat_nat @ A6 ) )
% 0.33/0.67 => ( ( ( nat_sum_decode @ A6 )
% 0.33/0.67 = ( sum_Inr_nat_nat @ B8 ) )
% 0.33/0.67 => ( P @ B8 ) ) )
% 0.33/0.67 => ( ! [B8: nat,Ba: nat,X5: nat,Y5: nat] :
% 0.33/0.67 ( ( ( nat_sum_decode @ N2 )
% 0.33/0.67 = ( sum_Inr_nat_nat @ B8 ) )
% 0.33/0.67 => ( ( ( nat_sum_decode @ B8 )
% 0.33/0.67 = ( sum_Inr_nat_nat @ Ba ) )
% 0.33/0.67 => ( ( ( product_Pair_nat_nat @ X5 @ Y5 )
% 0.33/0.67 = ( nat_prod_decode @ Ba ) )
% 0.33/0.67 => ( P @ X5 ) ) ) )
% 0.33/0.67 => ( ! [B8: nat,Ba: nat,X5: nat,Y5: nat] :
% 0.33/0.67 ( ( ( nat_sum_decode @ N2 )
% 0.33/0.67 = ( sum_Inr_nat_nat @ B8 ) )
% 0.33/0.67 => ( ( ( nat_sum_decode @ B8 )
% 0.33/0.67 = ( sum_Inr_nat_nat @ Ba ) )
% 0.33/0.67 => ( ( ( product_Pair_nat_nat @ X5 @ Y5 )
% 0.33/0.67 = ( nat_prod_decode @ Ba ) )
% 0.33/0.67 => ( P @ Y5 ) ) ) )
% 0.33/0.67 => ( P @ ( suc @ N2 ) ) ) ) ) ) )
% 0.33/0.67 => ( P @ A0 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % nth_item.pinduct
% 0.33/0.67 thf(fact_10146_int__decode__def,axiom,
% 0.33/0.67 ( nat_int_decode
% 0.33/0.67 = ( ^ [N3: nat] :
% 0.33/0.67 ( sum_ca7763040182479039464nt_nat @ semiri1314217659103216013at_int
% 0.33/0.67 @ ^ [B2: nat] : ( minus_minus_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ B2 ) ) @ one_one_int )
% 0.33/0.67 @ ( nat_sum_decode @ N3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % int_decode_def
% 0.33/0.67
% 0.33/0.67 % Helper facts (45)
% 0.33/0.67 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 0.33/0.67 ! [X: int,Y: int] :
% 0.33/0.67 ( ( if_int @ $false @ X @ Y )
% 0.33/0.67 = Y ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 0.33/0.67 ! [X: int,Y: int] :
% 0.33/0.67 ( ( if_int @ $true @ X @ Y )
% 0.33/0.67 = X ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.33/0.67 ! [X: nat,Y: nat] :
% 0.33/0.67 ( ( if_nat @ $false @ X @ Y )
% 0.33/0.67 = Y ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.33/0.67 ! [X: nat,Y: nat] :
% 0.33/0.67 ( ( if_nat @ $true @ X @ Y )
% 0.33/0.67 = X ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 0.33/0.67 ! [X: num,Y: num] :
% 0.33/0.67 ( ( if_num @ $false @ X @ Y )
% 0.33/0.67 = Y ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 0.33/0.67 ! [X: num,Y: num] :
% 0.33/0.67 ( ( if_num @ $true @ X @ Y )
% 0.33/0.67 = X ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 0.33/0.67 ! [X: rat,Y: rat] :
% 0.33/0.67 ( ( if_rat @ $false @ X @ Y )
% 0.33/0.67 = Y ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 0.33/0.67 ! [X: rat,Y: rat] :
% 0.33/0.67 ( ( if_rat @ $true @ X @ Y )
% 0.33/0.67 = X ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( if_real @ $false @ X @ Y )
% 0.33/0.67 = Y ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( if_real @ $true @ X @ Y )
% 0.33/0.67 = X ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.33/0.67 ! [X: complex,Y: complex] :
% 0.33/0.67 ( ( if_complex @ $false @ X @ Y )
% 0.33/0.67 = Y ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.33/0.67 ! [X: complex,Y: complex] :
% 0.33/0.67 ( ( if_complex @ $true @ X @ Y )
% 0.33/0.67 = X ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 0.33/0.67 ! [X: extended_enat,Y: extended_enat] :
% 0.33/0.67 ( ( if_Extended_enat @ $false @ X @ Y )
% 0.33/0.67 = Y ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 0.33/0.67 ! [X: extended_enat,Y: extended_enat] :
% 0.33/0.67 ( ( if_Extended_enat @ $true @ X @ Y )
% 0.33/0.67 = X ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.33/0.67 ! [X: code_integer,Y: code_integer] :
% 0.33/0.67 ( ( if_Code_integer @ $false @ X @ Y )
% 0.33/0.67 = Y ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.33/0.67 ! [X: code_integer,Y: code_integer] :
% 0.33/0.67 ( ( if_Code_integer @ $true @ X @ Y )
% 0.33/0.67 = X ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_2_1_If_001t__Code____Numeral__Onatural_T,axiom,
% 0.33/0.67 ! [X: code_natural,Y: code_natural] :
% 0.33/0.67 ( ( if_Code_natural @ $false @ X @ Y )
% 0.33/0.67 = Y ) ).
% 0.33/0.67
% 0.33/0.67 thf(help_If_1_1_If_001t__Code____Numeral__Onatural_T,axiom,
% 0.33/0.68 ! [X: code_natural,Y: code_natural] :
% 0.33/0.68 ( ( if_Code_natural @ $true @ X @ Y )
% 0.33/0.68 = X ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.33/0.68 ! [X: set_int,Y: set_int] :
% 0.33/0.68 ( ( if_set_int @ $false @ X @ Y )
% 0.33/0.68 = Y ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.33/0.68 ! [X: set_int,Y: set_int] :
% 0.33/0.68 ( ( if_set_int @ $true @ X @ Y )
% 0.33/0.68 = X ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.33/0.68 ! [X: list_int,Y: list_int] :
% 0.33/0.68 ( ( if_list_int @ $false @ X @ Y )
% 0.33/0.68 = Y ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.33/0.68 ! [X: list_int,Y: list_int] :
% 0.33/0.68 ( ( if_list_int @ $true @ X @ Y )
% 0.33/0.68 = X ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.33/0.68 ! [X: list_nat,Y: list_nat] :
% 0.33/0.68 ( ( if_list_nat @ $false @ X @ Y )
% 0.33/0.68 = Y ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.33/0.68 ! [X: list_nat,Y: list_nat] :
% 0.33/0.68 ( ( if_list_nat @ $true @ X @ Y )
% 0.33/0.68 = X ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.33/0.68 ! [X: int > int,Y: int > int] :
% 0.33/0.68 ( ( if_int_int @ $false @ X @ Y )
% 0.33/0.68 = Y ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.33/0.68 ! [X: int > int,Y: int > int] :
% 0.33/0.68 ( ( if_int_int @ $true @ X @ Y )
% 0.33/0.68 = X ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_2_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
% 0.33/0.68 ! [X: nat > rat,Y: nat > rat] :
% 0.33/0.68 ( ( if_nat_rat @ $false @ X @ Y )
% 0.33/0.68 = Y ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_1_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
% 0.33/0.68 ! [X: nat > rat,Y: nat > rat] :
% 0.33/0.68 ( ( if_nat_rat @ $true @ X @ Y )
% 0.33/0.68 = X ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.33/0.68 ! [X: option_num,Y: option_num] :
% 0.33/0.68 ( ( if_option_num @ $false @ X @ Y )
% 0.33/0.68 = Y ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.33/0.68 ! [X: option_num,Y: option_num] :
% 0.33/0.68 ( ( if_option_num @ $true @ X @ Y )
% 0.33/0.68 = X ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_2_1_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.33/0.68 ! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] :
% 0.33/0.68 ( ( if_Sum_sum_nat_nat @ $false @ X @ Y )
% 0.33/0.68 = Y ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_1_1_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.33/0.68 ! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] :
% 0.33/0.68 ( ( if_Sum_sum_nat_nat @ $true @ X @ Y )
% 0.33/0.68 = X ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.33/0.68 ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 0.33/0.68 ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
% 0.33/0.68 = Y ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.33/0.68 ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 0.33/0.68 ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
% 0.33/0.68 = X ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.33/0.68 ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 0.33/0.68 ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
% 0.33/0.68 = Y ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.33/0.68 ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 0.33/0.68 ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
% 0.33/0.68 = X ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
% 0.33/0.68 ! [X: nat > int > int,Y: nat > int > int] :
% 0.33/0.68 ( ( if_nat_int_int @ $false @ X @ Y )
% 0.33/0.68 = Y ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
% 0.33/0.68 ! [X: nat > int > int,Y: nat > int > int] :
% 0.33/0.68 ( ( if_nat_int_int @ $true @ X @ Y )
% 0.33/0.68 = X ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
% 0.33/0.68 ! [X: nat > nat > nat,Y: nat > nat > nat] :
% 0.33/0.68 ( ( if_nat_nat_nat @ $false @ X @ Y )
% 0.33/0.68 = Y ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
% 0.33/0.68 ! [X: nat > nat > nat,Y: nat > nat > nat] :
% 0.33/0.68 ( ( if_nat_nat_nat @ $true @ X @ Y )
% 0.33/0.68 = X ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.33/0.68 ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
% 0.33/0.68 ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
% 0.33/0.68 = Y ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.33/0.68 ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
% 0.33/0.68 ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
% 0.33/0.68 = X ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.33/0.68 ! [P: $o] :
% 0.33/0.68 ( ( P = $true )
% 0.33/0.68 | ( P = $false ) ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.33/0.68 ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 0.33/0.68 ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
% 0.33/0.68 = Y ) ).
% 0.33/0.68
% 0.33/0.68 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.33/0.68 ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 0.33/0.68 ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
% 0.33/0.68 = X ) ).
% 0.33/0.68
% 0.33/0.68 % Conjectures (1)
% 0.33/0.68 thf(conj_0,conjecture,
% 0.33/0.68 ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ y ) ).
% 0.33/0.68
% 0.33/0.68 %------------------------------------------------------------------------------
% 0.33/0.68 ------------------------
% 6.25/2.42 ---- Embedded in THF ---
% 6.25/2.42 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 6.25/2.42 %%% Generated on Thu Feb 26 06:50:36 EST 2026
% 6.25/2.42 %%% using '$$dhol' embedding, version 1.3.0.
% 6.25/2.42 %%% Logic specification used:
% 6.25/2.42 %%% thf(spec, logic, $$dhol).
% 6.25/2.42
% 6.25/2.42 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.iMRIUser6P/DTF2DTF27261.p
% See solution above
% 6.25/2.47 ------------------------
% 8.55/4.72 ---- Cleaned THF ---
% 8.55/4.72 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
% 8.55/4.72 produc6756470014200349799atural: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
% 8.55/4.72 produc5835291356934675326atural: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
% 8.55/4.72 produc7822875418678951345atural: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 8.55/4.72 produc8923325533196201883nteger: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 8.55/4.72 option4927543243414619207at_nat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 8.55/4.72 produc9072475918466114483BT_nat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 8.55/4.72 set_Pr1261947904930325089at_nat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 8.55/4.72 set_Pr958786334691620121nt_int: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 8.55/4.72 set_Sum_sum_nat_nat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__List__Olist_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
% 8.55/4.72 list_list_VEBT_VEBT: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 8.55/4.72 produc6271795597528267376eger_o: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 8.55/4.72 product_prod_num_num: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 8.55/4.72 product_prod_nat_num: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.55/4.72 product_prod_nat_nat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 8.55/4.72 product_prod_int_int: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.55/4.72 sum_sum_nat_nat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 8.55/4.72 list_list_nat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
% 8.55/4.72 list_list_int: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 8.55/4.72 list_VEBT_VEBT: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 8.55/4.72 set_list_nat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 8.55/4.72 list_set_nat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
% 8.55/4.72 set_VEBT_VEBT: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 8.55/4.72 set_set_nat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 8.55/4.72 set_Code_integer: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
% 8.55/4.72 set_Product_unit: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__List__Olist_It__List__Olist_I_Eo_J_J,type,
% 8.55/4.72 list_list_o: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
% 8.55/4.72 list_complex: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
% 8.55/4.72 set_complex: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
% 8.55/4.72 filter_real: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
% 8.55/4.72 option_num: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
% 8.55/4.72 filter_nat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Filter__Ofilter_It__Int__Oint_J,type,
% 8.55/4.72 filter_int: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
% 8.55/4.72 set_char: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
% 8.55/4.72 list_real: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 8.55/4.72 set_real: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 8.55/4.72 list_nat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
% 8.55/4.72 list_int: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__VEBT____Definitions__OVEBT,type,
% 8.55/4.72 vEBT_VEBT: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
% 8.55/4.72 set_rat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 8.55/4.72 set_nat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
% 8.55/4.72 set_int: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Code____Numeral__Onatural,type,
% 8.55/4.72 code_natural: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Code____Numeral__Ointeger,type,
% 8.55/4.72 code_integer: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Product____Type__Ounit,type,
% 8.55/4.72 product_unit: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Extended____Nat__Oenat,type,
% 8.55/4.72 extended_enat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__List__Olist_I_Eo_J,type,
% 8.55/4.72 list_o: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Complex__Ocomplex,type,
% 8.55/4.72 complex: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__String__Oliteral,type,
% 8.55/4.72 literal: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Set__Oset_I_Eo_J,type,
% 8.55/4.72 set_o: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__String__Ochar,type,
% 8.55/4.72 char: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Real__Oreal,type,
% 8.55/4.72 real: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Rat__Orat,type,
% 8.55/4.72 rat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Num__Onum,type,
% 8.55/4.72 num: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Nat__Onat,type,
% 8.55/4.72 nat: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(ty_n_t__Int__Oint,type,
% 8.55/4.72 int: $tType ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
% 8.55/4.72 archim2889992004027027881ng_rat: rat > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
% 8.55/4.72 archim7802044766580827645g_real: real > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
% 8.55/4.72 archim3151403230148437115or_rat: rat > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
% 8.55/4.72 archim6058952711729229775r_real: real > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
% 8.55/4.72 archimedean_frac_rat: rat > rat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
% 8.55/4.72 archim2898591450579166408c_real: real > real ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
% 8.55/4.72 archim7778729529865785530nd_rat: rat > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
% 8.55/4.72 archim8280529875227126926d_real: real > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
% 8.55/4.72 bNF_re1962705104956426057at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
% 8.55/4.72 bNF_re895249473297799549at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
% 8.55/4.72 bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 8.55/4.72 bNF_re4695409256820837752l_real: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( real > real > real ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_Eo_J_001_062_It__Real__Oreal_M_Eo_J,type,
% 8.55/4.72 bNF_re4521903465945308077real_o: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > $o ) > ( real > $o ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( real > real > $o ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
% 8.55/4.72 bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_Eo_001_Eo,type,
% 8.55/4.72 bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 8.55/4.72 bNF_re398004352372739002nteger: ( int > code_integer > $o ) > ( ( int > int ) > ( code_integer > code_integer ) > $o ) > ( int > int > int ) > ( code_integer > code_integer > code_integer ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
% 8.55/4.72 bNF_re3379532845092657523nteger: ( int > code_integer > $o ) > ( int > code_integer > $o ) > ( int > int ) > ( code_integer > code_integer ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 8.55/4.72 bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 8.55/4.72 bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Rat__Orat_J,type,
% 8.55/4.72 bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 8.55/4.72 bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 8.55/4.72 bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 8.55/4.72 bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
% 8.55/4.72 bNF_re88643428490162567atural: ( nat > code_natural > $o ) > ( ( nat > nat ) > ( code_natural > code_natural ) > $o ) > ( nat > nat > nat ) > ( code_natural > code_natural > code_natural ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
% 8.55/4.72 bNF_re5252274238750452962nteger: ( nat > code_natural > $o ) > ( int > code_integer > $o ) > ( nat > int ) > ( code_natural > code_integer ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
% 8.55/4.72 bNF_re3704215830270325841atural: ( nat > code_natural > $o ) > ( nat > code_natural > $o ) > ( nat > nat ) > ( code_natural > code_natural ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 8.55/4.72 bNF_re4935368626670024657nteger: ( nat > nat > $o ) > ( ( int > int ) > ( code_integer > code_integer ) > $o ) > ( nat > int > int ) > ( nat > code_integer > code_integer ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 8.55/4.72 bNF_re4785983289428654063nt_int: ( nat > nat > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( nat > int > int ) > ( nat > int > int ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
% 8.55/4.72 bNF_re3692170009638431535atural: ( nat > nat > $o ) > ( ( nat > nat ) > ( code_natural > code_natural ) > $o ) > ( nat > nat > nat ) > ( nat > code_natural > code_natural ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.55/4.72 bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
% 8.55/4.72 bNF_re4153400068438556298nteger: ( nat > nat > $o ) > ( int > code_integer > $o ) > ( nat > int ) > ( nat > code_integer ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Int__Oint,type,
% 8.55/4.72 bNF_re6650684261131312217nt_int: ( nat > nat > $o ) > ( int > int > $o ) > ( nat > int ) > ( nat > int ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.55/4.72 bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 8.55/4.72 bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J,type,
% 8.55/4.72 bNF_re7876454716742015248nteger: ( num > num > $o ) > ( ( num > int ) > ( num > code_integer ) > $o ) > ( num > num > int ) > ( num > num > code_integer ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Int__Oint_J,type,
% 8.55/4.72 bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
% 8.55/4.72 bNF_re6501075790457514782nteger: ( num > num > $o ) > ( int > code_integer > $o ) > ( num > int ) > ( num > code_integer ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
% 8.55/4.72 bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 8.55/4.72 bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
% 8.55/4.72 bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 8.55/4.72 bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
% 8.55/4.72 bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
% 8.55/4.72 bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 8.55/4.72 bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
% 8.55/4.72 bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 8.55/4.72 bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
% 8.55/4.72 bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.55/4.72 bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 8.55/4.72 bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 8.55/4.72 bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 8.55/4.72 bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
% 8.55/4.72 bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.55/4.72 bNF_re8246922863344978751at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.55/4.72 bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Binomial_Obinomial,type,
% 8.55/4.72 binomial: nat > nat > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
% 8.55/4.72 gbinomial_complex: complex > nat > complex ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
% 8.55/4.72 gbinomial_int: int > nat > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
% 8.55/4.72 gbinomial_nat: nat > nat > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
% 8.55/4.72 gbinomial_rat: rat > nat > rat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
% 8.55/4.72 gbinomial_real: real > nat > real ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Bit__Operations_Oand__int__rel,type,
% 8.55/4.72 bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Bit__Operations_Oand__not__num,type,
% 8.55/4.72 bit_and_not_num: num > num > option_num ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
% 8.55/4.72 bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Bit__Operations_Oconcat__bit,type,
% 8.55/4.72 bit_concat_bit: nat > int > int > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
% 8.55/4.72 bit_or_not_num_neg: num > num > num ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
% 8.55/4.72 bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,type,
% 8.55/4.72 bit_ri7632146776885996613nteger: code_integer > code_integer ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
% 8.55/4.72 bit_ri7919022796975470100ot_int: int > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
% 8.55/4.72 bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
% 8.55/4.72 bit_ri631733984087533419it_int: nat > int > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
% 8.55/4.72 bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Onatural,type,
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% 8.55/4.72 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Extended__Nat_OeSuc,type,
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% 8.55/4.72 thf(sy_c_Extended__Nat_Oenat,type,
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% 8.55/4.72
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Filter_Oat__top_001t__Int__Oint,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
% 8.55/4.72 eventually_nat: ( nat > $o ) > filter_nat > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Int__Oint,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Filter_Ofiltermap_001t__Real__Oreal_001t__Real__Oreal,type,
% 8.55/4.72 filtermap_real_real: ( real > real ) > filter_real > filter_real ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Finite__Set_Ocard_001_Eo,type,
% 8.55/4.72 finite_card_o: set_o > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
% 8.55/4.72 finite_card_list_nat: set_list_nat > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
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% 8.55/4.72 thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
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% 8.55/4.72
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% 8.55/4.72
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% 8.55/4.72
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_If_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J,type,
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% 8.55/4.72
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
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% 8.55/4.72
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% 8.55/4.72
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% 8.55/4.72
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_If_001t__Int__Oint,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_If_001t__Nat__Onat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_If_001t__Num__Onum,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_If_001t__Option__Ooption_It__Num__Onum_J,type,
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% 8.55/4.72
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% 8.55/4.72
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% 8.55/4.72
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% 8.55/4.72
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_If_001t__Rat__Orat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_If_001t__Real__Oreal,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_OAbs__Integ,type,
% 8.55/4.72 abs_Integ: product_prod_nat_nat > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_ONeg,type,
% 8.55/4.72 neg: num > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_OPos,type,
% 8.55/4.72 pos: num > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_ORep__Integ,type,
% 8.55/4.72 rep_Integ: int > product_prod_nat_nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Ocr__int,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Odup,type,
% 8.55/4.72 dup: int > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Oint__ge__less__than,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Oint__ge__less__than2,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Ointrel,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Onat,type,
% 8.55/4.72 nat2: int > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Opcr__int,type,
% 8.55/4.72 pcr_int: product_prod_nat_nat > int > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Opower__int_001t__Real__Oreal,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Oring__1__class_OInts_001t__Code____Numeral__Ointeger,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Oring__1__class_OInts_001t__Complex__Ocomplex,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
% 8.55/4.72 ring_1_Ints_int: set_int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Oring__1__class_OInts_001t__Rat__Orat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
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% 8.55/4.72
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
% 8.55/4.72 ring_17405671764205052669omplex: int > complex ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 8.55/4.72 ring_1_of_int_real: int > real ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Int_Osub,type,
% 8.55/4.72 sub: num > num > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
% 8.55/4.72 inf_inf_nat: nat > nat > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Lattices_Osemilattice__neutr_001t__Nat__Onat,type,
% 8.55/4.72 semila9081495762789891438tr_nat: ( nat > nat > nat ) > nat > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
% 8.55/4.72 semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Lattices_Osemilattice__order_001t__Nat__Onat,type,
% 8.55/4.72 semila1248733672344298208er_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
% 8.55/4.72 sup_sup_set_nat: set_nat > set_nat > set_nat ).
% 8.55/4.72
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
% 8.55/4.72 lattic8265883725875713057ax_nat: set_nat > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF_001t__Nat__Onat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Lifting_OQuotient_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Lifting_OQuotient_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 8.55/4.72 quotie1194848508323700631at_int: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > int ) > ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int > $o ) > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Oappend_001t__Int__Oint,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Oappend_001t__Nat__Onat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Odrop_001t__Nat__Onat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Ofold_001t__Int__Oint_001t__Int__Oint,type,
% 8.55/4.72 fold_int_int: ( int > int > int ) > list_int > int > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.55/4.72 fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Olast_001t__Nat__Onat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
% 8.55/4.72 cons_int: int > list_int > list_int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Olist_Oset_001_Eo,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
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% 8.55/4.72
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% 8.55/4.72
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Onth_001_Eo,type,
% 8.55/4.72 nth_o: list_o > nat > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Onth_001t__Int__Oint,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Onth_001t__Nat__Onat,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Onth_001t__Real__Oreal,type,
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% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
% 8.55/4.72 nth_set_nat: list_set_nat > nat > set_nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 8.55/4.72 nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 8.55/4.72 replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Orotate1_001_Eo,type,
% 8.55/4.72 rotate1_o: list_o > list_o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Orotate1_001t__Int__Oint,type,
% 8.55/4.72 rotate1_int: list_int > list_int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
% 8.55/4.72 rotate1_nat: list_nat > list_nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Orotate1_001t__VEBT____Definitions__OVEBT,type,
% 8.55/4.72 rotate1_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Osubseqs_001_Eo,type,
% 8.55/4.72 subseqs_o: list_o > list_list_o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Osubseqs_001t__Int__Oint,type,
% 8.55/4.72 subseqs_int: list_int > list_list_int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
% 8.55/4.72 subseqs_nat: list_nat > list_list_nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Osubseqs_001t__VEBT____Definitions__OVEBT,type,
% 8.55/4.72 subseqs_VEBT_VEBT: list_VEBT_VEBT > list_list_VEBT_VEBT ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 8.55/4.72 take_nat: nat > list_nat > list_nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Oupt,type,
% 8.55/4.72 upt: nat > nat > list_nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Oupto,type,
% 8.55/4.72 upto: int > int > list_int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Oupto__aux,type,
% 8.55/4.72 upto_aux: int > int > list_int > list_int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_List_Oupto__rel,type,
% 8.55/4.72 upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_OSuc,type,
% 8.55/4.72 suc: nat > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.55/4.72 compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 8.55/4.72 case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 8.55/4.72 case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 8.55/4.72 case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Onat_Opred,type,
% 8.55/4.72 pred: nat > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Complex__Ocomplex,type,
% 8.55/4.72 semiri3842193898606819883omplex: set_complex ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
% 8.55/4.72 semiring_1_Nats_int: set_int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 8.55/4.72 semiri4939895301339042750nteger: nat > code_integer ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Onatural,type,
% 8.55/4.72 semiri3763490453095760265atural: nat > code_natural ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 8.55/4.72 semiri8010041392384452111omplex: nat > complex ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 8.55/4.72 semiri1314217659103216013at_int: nat > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 8.55/4.72 semiri1316708129612266289at_nat: nat > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
% 8.55/4.72 semiri681578069525770553at_rat: nat > rat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 8.55/4.72 semiri5074537144036343181t_real: nat > real ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 8.55/4.72 size_size_list_o: list_o > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 8.55/4.72 size_s3451745648224563538omplex: list_complex > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 8.55/4.72 size_size_list_int: list_int > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
% 8.55/4.72 size_s2710708370519433104list_o: list_list_o > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
% 8.55/4.72 size_s533118279054570080st_int: list_list_int > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 8.55/4.72 size_s3023201423986296836st_nat: list_list_nat > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
% 8.55/4.72 size_s8217280938318005548T_VEBT: list_list_VEBT_VEBT > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 8.55/4.72 size_size_list_nat: list_nat > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 8.55/4.72 size_size_list_real: list_real > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 8.55/4.72 size_s3254054031482475050et_nat: list_set_nat > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 8.55/4.72 size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 8.55/4.72 size_size_num: num > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
% 8.55/4.72 size_size_char: char > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
% 8.55/4.72 size_size_VEBT_VEBT: vEBT_VEBT > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat__Bijection_Oint__decode,type,
% 8.55/4.72 nat_int_decode: nat > int ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat__Bijection_Oint__encode,type,
% 8.55/4.72 nat_int_encode: int > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat__Bijection_Olist__decode,type,
% 8.55/4.72 nat_list_decode: nat > list_nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
% 8.55/4.72 nat_list_decode_rel: nat > nat > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat__Bijection_Olist__encode,type,
% 8.55/4.72 nat_list_encode: list_nat > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 8.55/4.72 nat_list_encode_rel: list_nat > list_nat > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat__Bijection_Oprod__decode,type,
% 8.55/4.72 nat_prod_decode: nat > product_prod_nat_nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 8.55/4.72 nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 8.55/4.72 nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 8.55/4.72 nat_prod_encode: product_prod_nat_nat > nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat__Bijection_Oset__decode,type,
% 8.55/4.72 nat_set_decode: nat > set_nat ).
% 8.55/4.72
% 8.55/4.72 thf(sy_c_Nat__Bijection_Oset__encode,type,
% 8.55/4.72 nat_set_encode: set_nat > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Nat__Bijection_Osum__decode,type,
% 8.55/4.73 nat_sum_decode: nat > sum_sum_nat_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Nat__Bijection_Osum__encode,type,
% 8.55/4.73 nat_sum_encode: sum_sum_nat_nat > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Nat__Bijection_Otriangle,type,
% 8.55/4.73 nat_triangle: nat > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_NthRoot_Oroot,type,
% 8.55/4.73 root: nat > real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_NthRoot_Osqrt,type,
% 8.55/4.73 sqrt: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_OBitM,type,
% 8.55/4.73 bitM: num > num ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oinc,type,
% 8.55/4.73 inc: num > num ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Onat__of__num,type,
% 8.55/4.73 nat_of_num: num > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
% 8.55/4.73 neg_nu8804712462038260780nteger: code_integer > code_integer ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
% 8.55/4.73 neg_nu7009210354673126013omplex: complex > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
% 8.55/4.73 neg_numeral_dbl_int: int > int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
% 8.55/4.73 neg_numeral_dbl_rat: rat > rat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
% 8.55/4.73 neg_numeral_dbl_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
% 8.55/4.73 neg_nu7757733837767384882nteger: code_integer > code_integer ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
% 8.55/4.73 neg_nu6511756317524482435omplex: complex > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 8.55/4.73 neg_nu3811975205180677377ec_int: int > int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 8.55/4.73 neg_nu3179335615603231917ec_rat: rat > rat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
% 8.55/4.73 neg_nu6075765906172075777c_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
% 8.55/4.73 neg_nu5831290666863070958nteger: code_integer > code_integer ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
% 8.55/4.73 neg_nu8557863876264182079omplex: complex > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
% 8.55/4.73 neg_nu5851722552734809277nc_int: int > int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
% 8.55/4.73 neg_nu5219082963157363817nc_rat: rat > rat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
% 8.55/4.73 neg_nu8295874005876285629c_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Code____Numeral__Ointeger,type,
% 8.55/4.73 neg_nu5755505904847501662nteger: num > num > code_integer ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Complex__Ocomplex,type,
% 8.55/4.73 neg_nu8416839295433526191omplex: num > num > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
% 8.55/4.73 neg_numeral_sub_int: num > num > int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Rat__Orat,type,
% 8.55/4.73 neg_numeral_sub_rat: num > num > rat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Real__Oreal,type,
% 8.55/4.73 neg_numeral_sub_real: num > num > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Onum_OBit0,type,
% 8.55/4.73 bit0: num > num ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Onum_OBit1,type,
% 8.55/4.73 bit1: num > num ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Onum_OOne,type,
% 8.55/4.73 one: num ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
% 8.55/4.73 case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Onum_Osize__num,type,
% 8.55/4.73 size_num: num > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Onum__of__nat,type,
% 8.55/4.73 num_of_nat: nat > num ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
% 8.55/4.73 numera6620942414471956472nteger: num > code_integer ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Onatural,type,
% 8.55/4.73 numera5444537566228673987atural: num > code_natural ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
% 8.55/4.73 numera6690914467698888265omplex: num > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
% 8.55/4.73 numera1916890842035813515d_enat: num > extended_enat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
% 8.55/4.73 numeral_numeral_int: num > int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 8.55/4.73 numeral_numeral_nat: num > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
% 8.55/4.73 numeral_numeral_rat: num > rat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 8.55/4.73 numeral_numeral_real: num > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Opow,type,
% 8.55/4.73 pow: num > num > num ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Opred__numeral,type,
% 8.55/4.73 pred_numeral: num > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Num_Osqr,type,
% 8.55/4.73 sqr: num > num ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
% 8.55/4.73 none_num: option_num ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.55/4.73 none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
% 8.55/4.73 some_num: num > option_num ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.55/4.73 some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.55/4.73 case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
% 8.55/4.73 case_option_int_num: int > ( num > int ) > option_num > int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
% 8.55/4.73 case_option_num_num: num > ( num > num ) > option_num > num ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
% 8.55/4.73 case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
% 8.55/4.73
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% 8.55/4.73
% 8.55/4.73 thf(sy_c_Product__Type_Oscomp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
% 8.55/4.73 produc5538323210962509403atural: ( produc7822875418678951345atural > produc5835291356934675326atural ) > ( code_natural > produc7822875418678951345atural > produc5835291356934675326atural ) > produc7822875418678951345atural > produc5835291356934675326atural ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Quotient_OQuotient3_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
% 8.55/4.73 quotie8700032322157300518t_real: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > real ) > ( real > nat > rat ) > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Quotient_OQuotient3_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 8.55/4.73 quotie6776551016481293500at_int: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > int ) > ( int > product_prod_nat_nat ) > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Random_Oinc__shift,type,
% 8.55/4.73 inc_shift: code_natural > code_natural > code_natural ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Random_Oiterate_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
% 8.55/4.73 iterat8892046348760725948atural: code_natural > ( code_natural > produc7822875418678951345atural > produc5835291356934675326atural ) > code_natural > produc7822875418678951345atural > produc5835291356934675326atural ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Random_Olog,type,
% 8.55/4.73 log: code_natural > code_natural > code_natural ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Random_Olog__rel,type,
% 8.55/4.73 log_rel: produc7822875418678951345atural > produc7822875418678951345atural > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Random_Ominus__shift,type,
% 8.55/4.73 minus_shift: code_natural > code_natural > code_natural > code_natural ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Random_Onext,type,
% 8.55/4.73 next: produc7822875418678951345atural > produc5835291356934675326atural ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Random_Orange,type,
% 8.55/4.73 range: code_natural > produc7822875418678951345atural > produc5835291356934675326atural ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Random_Osplit__seed,type,
% 8.55/4.73 split_seed: produc7822875418678951345atural > produc6756470014200349799atural ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rat_OAbs__Rat,type,
% 8.55/4.73 abs_Rat: product_prod_int_int > rat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rat_OFract,type,
% 8.55/4.73 fract: int > int > rat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rat_OFrct,type,
% 8.55/4.73 frct: product_prod_int_int > rat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rat_ORep__Rat,type,
% 8.55/4.73 rep_Rat: rat > product_prod_int_int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 8.55/4.73 field_5140801741446780682s_real: set_real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
% 8.55/4.73 field_7254667332652039916t_real: rat > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rat_Onormalize,type,
% 8.55/4.73 normalize: product_prod_int_int > product_prod_int_int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rat_Oof__int,type,
% 8.55/4.73 of_int: int > rat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rat_Opcr__rat,type,
% 8.55/4.73 pcr_rat: product_prod_int_int > rat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rat_Opositive,type,
% 8.55/4.73 positive: rat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rat_Oquotient__of,type,
% 8.55/4.73 quotient_of: rat > product_prod_int_int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rat_Oratrel,type,
% 8.55/4.73 ratrel: product_prod_int_int > product_prod_int_int > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real_ORatreal,type,
% 8.55/4.73 ratreal: rat > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real_OReal,type,
% 8.55/4.73 real2: ( nat > rat ) > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real_Ocauchy,type,
% 8.55/4.73 cauchy: ( nat > rat ) > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real_Ocr__real,type,
% 8.55/4.73 cr_real: ( nat > rat ) > real > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real_Opcr__real,type,
% 8.55/4.73 pcr_real: ( nat > rat ) > real > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real_Opositive,type,
% 8.55/4.73 positive2: real > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real_Orealrel,type,
% 8.55/4.73 realrel: ( nat > rat ) > ( nat > rat ) > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real_Orep__real,type,
% 8.55/4.73 rep_real: real > nat > rat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real_Ovanishes,type,
% 8.55/4.73 vanishes: ( nat > rat ) > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
% 8.55/4.73 real_V2521375963428798218omplex: set_complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real__Vector__Spaces_OReals_001t__Real__Oreal,type,
% 8.55/4.73 real_V470468836141973256s_real: set_real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
% 8.55/4.73 real_V5970128139526366754l_real: ( real > real ) > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
% 8.55/4.73 real_V3694042436643373181omplex: complex > complex > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
% 8.55/4.73 real_V975177566351809787t_real: real > real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real__Vector__Spaces_Olinear_001t__Real__Oreal_001t__Real__Oreal,type,
% 8.55/4.73 real_V4572627801940501177l_real: ( real > real ) > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
% 8.55/4.73 real_V1022390504157884413omplex: complex > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
% 8.55/4.73 real_V7735802525324610683m_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
% 8.55/4.73 real_V4546457046886955230omplex: real > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
% 8.55/4.73 real_V1803761363581548252l_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
% 8.55/4.73 real_V2046097035970521341omplex: real > complex > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
% 8.55/4.73 real_V1485227260804924795R_real: real > real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Relation_Otransp_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
% 8.55/4.73 transp_nat_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
% 8.55/4.73 algebr932160517623751201me_int: int > int > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
% 8.55/4.73 algebr934650988132801477me_nat: nat > nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
% 8.55/4.73 divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Onatural,type,
% 8.55/4.73 divide5121882707175180666atural: code_natural > code_natural > code_natural ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
% 8.55/4.73 divide1717551699836669952omplex: complex > complex > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
% 8.55/4.73 divide_divide_int: int > int > int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 8.55/4.73 divide_divide_nat: nat > nat > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
% 8.55/4.73 divide_divide_rat: rat > rat > rat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
% 8.55/4.73 divide_divide_real: real > real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
% 8.55/4.73 dvd_dvd_Code_integer: code_integer > code_integer > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Onatural,type,
% 8.55/4.73 dvd_dvd_Code_natural: code_natural > code_natural > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
% 8.55/4.73 dvd_dvd_complex: complex > complex > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
% 8.55/4.73 dvd_dvd_int: int > int > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
% 8.55/4.73 dvd_dvd_nat: nat > nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
% 8.55/4.73 dvd_dvd_rat: rat > rat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
% 8.55/4.73 dvd_dvd_real: real > real > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
% 8.55/4.73 modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Onatural,type,
% 8.55/4.73 modulo8411746178871703098atural: code_natural > code_natural > code_natural ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
% 8.55/4.73 modulo_modulo_int: int > int > int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
% 8.55/4.73 modulo_modulo_nat: nat > nat > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Onormalization__semidom__class_Onormalize_001t__Int__Oint,type,
% 8.55/4.73 normal571682499473568032ze_int: int > int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Onormalization__semidom__class_Onormalize_001t__Nat__Onat,type,
% 8.55/4.73 normal574172969982618308ze_nat: nat > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Ounit__factor__class_Ounit__factor_001t__Int__Oint,type,
% 8.55/4.73 unit_f2746056213392204926or_int: int > int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Ounit__factor__class_Ounit__factor_001t__Nat__Onat,type,
% 8.55/4.73 unit_f2748546683901255202or_nat: nat > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
% 8.55/4.73 zero_n356916108424825756nteger: $o > code_integer ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Onatural,type,
% 8.55/4.73 zero_n8403883297036319079atural: $o > code_natural ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
% 8.55/4.73 zero_n1201886186963655149omplex: $o > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
% 8.55/4.73 zero_n2684676970156552555ol_int: $o > int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
% 8.55/4.73 zero_n2687167440665602831ol_nat: $o > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
% 8.55/4.73 zero_n2052037380579107095ol_rat: $o > rat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
% 8.55/4.73 zero_n3304061248610475627l_real: $o > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
% 8.55/4.73 suminf_real: ( nat > real ) > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
% 8.55/4.73 summable_real: ( nat > real ) > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Series_Osums_001t__Real__Oreal,type,
% 8.55/4.73 sums_real: ( nat > real ) > real > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
% 8.55/4.73 collect_complex: ( complex > $o ) > set_complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_OCollect_001t__Int__Oint,type,
% 8.55/4.73 collect_int: ( int > $o ) > set_int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
% 8.55/4.73 collect_list_nat: ( list_nat > $o ) > set_list_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 8.55/4.73 collect_nat: ( nat > $o ) > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 8.55/4.73 collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.55/4.73 collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 8.55/4.73 collect_real: ( real > $o ) > set_real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 8.55/4.73 collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_OPow_001t__Nat__Onat,type,
% 8.55/4.73 pow_nat: set_nat > set_set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
% 8.55/4.73 image_int_int: ( int > int ) > set_int > set_int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
% 8.55/4.73 image_int_nat: ( int > nat ) > set_int > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
% 8.55/4.73 image_list_nat_nat: ( list_nat > nat ) > set_list_nat > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
% 8.55/4.73 image_nat_int: ( nat > int ) > set_nat > set_int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
% 8.55/4.73 image_nat_list_nat: ( nat > list_nat ) > set_nat > set_list_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.55/4.73 image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.55/4.73 image_5846123807819985514at_nat: ( nat > product_prod_nat_nat ) > set_nat > set_Pr1261947904930325089at_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
% 8.55/4.73 image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
% 8.55/4.73 image_nat_char: ( nat > char ) > set_nat > set_char ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.55/4.73 image_678696785212003926at_nat: ( nat > sum_sum_nat_nat ) > set_nat > set_Sum_sum_nat_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 8.55/4.73 image_2486076414777270412at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
% 8.55/4.73 image_real_real: ( real > real ) > set_real > set_real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
% 8.55/4.73 image_char_nat: ( char > nat ) > set_char > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 8.55/4.73 image_1320371278474632150at_nat: ( sum_sum_nat_nat > nat ) > set_Sum_sum_nat_nat > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
% 8.55/4.73 insert_int: int > set_int > set_int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
% 8.55/4.73 insert_nat: nat > set_nat > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
% 8.55/4.73 insert_real: real > set_real > set_real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.55/4.73 vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
% 8.55/4.73 set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
% 8.55/4.73 set_or1266510415728281911st_int: int > int > set_int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
% 8.55/4.73 set_or1269000886237332187st_nat: nat > nat > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
% 8.55/4.73 set_or1222579329274155063t_real: real > real > set_real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
% 8.55/4.73 set_or4662586982721622107an_int: int > int > set_int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
% 8.55/4.73 set_or4665077453230672383an_nat: nat > nat > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
% 8.55/4.73 set_ord_atLeast_nat: nat > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
% 8.55/4.73 set_ord_atLeast_real: real > set_real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
% 8.55/4.73 set_ord_atMost_nat: nat > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
% 8.55/4.73 set_or6656581121297822940st_int: int > int > set_int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
% 8.55/4.73 set_or6659071591806873216st_nat: nat > nat > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
% 8.55/4.73 set_or5832277885323065728an_int: int > int > set_int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
% 8.55/4.73 set_or5834768355832116004an_nat: nat > nat > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
% 8.55/4.73 set_or1633881224788618240n_real: real > real > set_real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
% 8.55/4.73 set_or1210151606488870762an_nat: nat > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
% 8.55/4.73 set_or5849166863359141190n_real: real > set_real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 8.55/4.73 set_ord_lessThan_nat: nat > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
% 8.55/4.73 set_or5984915006950818249n_real: real > set_real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_String_OCode_Oabort_001t__Real__Oreal,type,
% 8.55/4.73 abort_real: literal > ( product_unit > real ) > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_String_OLiteral,type,
% 8.55/4.73 literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_String_Oascii__of,type,
% 8.55/4.73 ascii_of: char > char ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_String_Ochar_OChar,type,
% 8.55/4.73 char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_String_Ochar_Osize__char,type,
% 8.55/4.73 size_char: char > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
% 8.55/4.73 comm_s629917340098488124ar_nat: char > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_String_Ointeger__of__char,type,
% 8.55/4.73 integer_of_char: char > code_integer ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
% 8.55/4.73 unique3096191561947761185of_nat: nat > char ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Sum__Type_OInl_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.55/4.73 sum_Inl_nat_nat: nat > sum_sum_nat_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.55/4.73 sum_Inr_nat_nat: nat > sum_sum_nat_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
% 8.55/4.73 sum_ca7763040182479039464nt_nat: ( nat > int ) > ( nat > int ) > sum_sum_nat_nat > int ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.55/4.73 sum_ca6763686470577984908at_nat: ( nat > nat ) > ( nat > nat ) > sum_sum_nat_nat > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
% 8.55/4.73 topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 8.55/4.73 topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 8.55/4.73 topolo6980174941875973593q_real: ( nat > real ) > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 8.55/4.73 topolo2177554685111907308n_real: real > set_real > filter_real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Real__Oreal,type,
% 8.55/4.73 topolo7531315842566124627t_real: ( nat > real ) > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 8.55/4.73 topolo2815343760600316023s_real: real > filter_real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
% 8.55/4.73 topolo6517432010174082258omplex: ( nat > complex ) > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
% 8.55/4.73 topolo4055970368930404560y_real: ( nat > real ) > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Oarccos,type,
% 8.55/4.73 arccos: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 8.55/4.73 arcosh_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Oarcsin,type,
% 8.55/4.73 arcsin: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Oarctan,type,
% 8.55/4.73 arctan: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 8.55/4.73 arsinh_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 8.55/4.73 artanh_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
% 8.55/4.73 cos_complex: complex > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
% 8.55/4.73 cos_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Ocos__coeff,type,
% 8.55/4.73 cos_coeff: nat > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
% 8.55/4.73 cosh_complex: complex > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 8.55/4.73 cosh_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
% 8.55/4.73 cot_complex: complex > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
% 8.55/4.73 cot_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
% 8.55/4.73 exp_complex: complex > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
% 8.55/4.73 exp_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 8.55/4.73 ln_ln_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Olog,type,
% 8.55/4.73 log2: real > real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Opi,type,
% 8.55/4.73 pi: real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 8.55/4.73 powr_real: real > real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Opowr__real,type,
% 8.55/4.73 powr_real2: real > real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
% 8.55/4.73 sin_complex: complex > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 8.55/4.73 sin_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Osin__coeff,type,
% 8.55/4.73 sin_coeff: nat > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
% 8.55/4.73 sinh_complex: complex > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 8.55/4.73 sinh_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
% 8.55/4.73 tan_complex: complex > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 8.55/4.73 tan_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
% 8.55/4.73 tanh_complex: complex > complex ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 8.55/4.73 tanh_real: real > real ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 8.55/4.73 vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 8.55/4.73 vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 8.55/4.73 vEBT_size_VEBT: vEBT_VEBT > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 8.55/4.73 vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 8.55/4.73 vEBT_VEBT_high: nat > nat > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 8.55/4.73 vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 8.55/4.73 vEBT_VEBT_low: nat > nat > nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 8.55/4.73 vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 8.55/4.73 vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 8.55/4.73 vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 8.55/4.73 vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 8.55/4.73 vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 8.55/4.73 vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 8.55/4.73 vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 8.55/4.73 vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 8.55/4.73 vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 8.55/4.73 vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
% 8.55/4.73 accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
% 8.55/4.73 accp_nat: ( nat > nat > $o ) > nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
% 8.55/4.73 accp_P8126237942716283194atural: ( produc7822875418678951345atural > produc7822875418678951345atural > $o ) > produc7822875418678951345atural > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 8.55/4.73 accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.55/4.73 accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 8.55/4.73 accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 8.55/4.73 accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_member_001_Eo,type,
% 8.55/4.73 member_o: $o > set_o > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_member_001t__Code____Numeral__Ointeger,type,
% 8.55/4.73 member_Code_integer: code_integer > set_Code_integer > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_member_001t__Complex__Ocomplex,type,
% 8.55/4.73 member_complex: complex > set_complex > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_member_001t__Int__Oint,type,
% 8.55/4.73 member_int: int > set_int > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
% 8.55/4.73 member_list_nat: list_nat > set_list_nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_member_001t__Nat__Onat,type,
% 8.55/4.73 member_nat: nat > set_nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_member_001t__Rat__Orat,type,
% 8.55/4.73 member_rat: rat > set_rat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_member_001t__Real__Oreal,type,
% 8.55/4.73 member_real: real > set_real > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 8.55/4.73 member_set_nat: set_nat > set_set_nat > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 8.55/4.73 member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 8.55/4.73
% 8.55/4.73 thf(sy_v_va____,type,
% 8.55/4.73 va: nat ).
% 8.55/4.73
% 8.55/4.73 thf(sy_v_y____,type,
% 8.55/4.73 y: nat ).
% 8.55/4.73
% 8.55/4.73 thf(fact_0_True,axiom,
% 8.55/4.73 dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_1__C3_OIH_C_I1_J,axiom,
% 8.55/4.73 ! [X: nat,Xa: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
% 8.55/4.73 => ( ( X
% 8.55/4.73 = ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.73 => ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X ) @ Xa ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_2__C3_OIH_C_I4_J,axiom,
% 8.55/4.73 ! [X: nat,Xa: nat] :
% 8.55/4.73 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
% 8.55/4.73 => ( ( X
% 8.55/4.73 = ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.73 => ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ ( suc @ X ) ) @ Xa ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_3__C3_OIH_C_I3_J,axiom,
% 8.55/4.73 ! [X: nat,Xa: nat] :
% 8.55/4.73 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ va ) ) )
% 8.55/4.73 => ( ( X
% 8.55/4.73 = ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.73 => ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ X ) @ Xa ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_4_div2__Suc__Suc,axiom,
% 8.55/4.73 ! [M: nat] :
% 8.55/4.73 ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.73 = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_5_semiring__norm_I85_J,axiom,
% 8.55/4.73 ! [M: num] :
% 8.55/4.73 ( ( bit0 @ M )
% 8.55/4.73 != one ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_6_semiring__norm_I83_J,axiom,
% 8.55/4.73 ! [N: num] :
% 8.55/4.73 ( one
% 8.55/4.73 != ( bit0 @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_7_buildup__nothing__in__leaf,axiom,
% 8.55/4.73 ! [N: nat,X: nat] :
% 8.55/4.73 ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_8_numeral__Bit0__div__2,axiom,
% 8.55/4.73 ! [N: num] :
% 8.55/4.73 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.73 = ( numeral_numeral_nat @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_9_numeral__Bit0__div__2,axiom,
% 8.55/4.73 ! [N: num] :
% 8.55/4.73 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.55/4.73 = ( numeral_numeral_int @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_10_odd__Suc__div__two,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.73 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.73 = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_11_even__Suc__div__two,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.73 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.73 = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_12_even__Suc,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
% 8.55/4.73 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_13_even__Suc__Suc__iff,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
% 8.55/4.73 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_14_divide__numeral__1,axiom,
% 8.55/4.73 ! [A: complex] :
% 8.55/4.73 ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_15_divide__numeral__1,axiom,
% 8.55/4.73 ! [A: real] :
% 8.55/4.73 ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_16_divide__numeral__1,axiom,
% 8.55/4.73 ! [A: rat] :
% 8.55/4.73 ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_17_nat_Oinject,axiom,
% 8.55/4.73 ! [X2: nat,Y2: nat] :
% 8.55/4.73 ( ( ( suc @ X2 )
% 8.55/4.73 = ( suc @ Y2 ) )
% 8.55/4.73 = ( X2 = Y2 ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_18_old_Onat_Oinject,axiom,
% 8.55/4.73 ! [Nat: nat,Nat2: nat] :
% 8.55/4.73 ( ( ( suc @ Nat )
% 8.55/4.73 = ( suc @ Nat2 ) )
% 8.55/4.73 = ( Nat = Nat2 ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_19_numeral__eq__iff,axiom,
% 8.55/4.73 ! [M: num,N: num] :
% 8.55/4.73 ( ( ( numera6690914467698888265omplex @ M )
% 8.55/4.73 = ( numera6690914467698888265omplex @ N ) )
% 8.55/4.73 = ( M = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_20_numeral__eq__iff,axiom,
% 8.55/4.73 ! [M: num,N: num] :
% 8.55/4.73 ( ( ( numeral_numeral_real @ M )
% 8.55/4.73 = ( numeral_numeral_real @ N ) )
% 8.55/4.73 = ( M = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_21_numeral__eq__iff,axiom,
% 8.55/4.73 ! [M: num,N: num] :
% 8.55/4.73 ( ( ( numeral_numeral_rat @ M )
% 8.55/4.73 = ( numeral_numeral_rat @ N ) )
% 8.55/4.73 = ( M = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_22_numeral__eq__iff,axiom,
% 8.55/4.73 ! [M: num,N: num] :
% 8.55/4.73 ( ( ( numeral_numeral_nat @ M )
% 8.55/4.73 = ( numeral_numeral_nat @ N ) )
% 8.55/4.73 = ( M = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_23_numeral__eq__iff,axiom,
% 8.55/4.73 ! [M: num,N: num] :
% 8.55/4.73 ( ( ( numeral_numeral_int @ M )
% 8.55/4.73 = ( numeral_numeral_int @ N ) )
% 8.55/4.73 = ( M = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_24_semiring__norm_I87_J,axiom,
% 8.55/4.73 ! [M: num,N: num] :
% 8.55/4.73 ( ( ( bit0 @ M )
% 8.55/4.73 = ( bit0 @ N ) )
% 8.55/4.73 = ( M = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_25_dvd__antisym,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ M @ N )
% 8.55/4.73 => ( ( dvd_dvd_nat @ N @ M )
% 8.55/4.73 => ( M = N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_26_n__not__Suc__n,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( N
% 8.55/4.73 != ( suc @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_27_Suc__inject,axiom,
% 8.55/4.73 ! [X: nat,Y: nat] :
% 8.55/4.73 ( ( ( suc @ X )
% 8.55/4.73 = ( suc @ Y ) )
% 8.55/4.73 => ( X = Y ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_28_even__numeral,axiom,
% 8.55/4.73 ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_29_even__numeral,axiom,
% 8.55/4.73 ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_30_even__numeral,axiom,
% 8.55/4.73 ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_31_div2__even__ext__nat,axiom,
% 8.55/4.73 ! [X: nat,Y: nat] :
% 8.55/4.73 ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.73 = ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.73 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
% 8.55/4.73 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
% 8.55/4.73 => ( X = Y ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_32_div__dvd__div,axiom,
% 8.55/4.73 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.55/4.73 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.55/4.73 => ( ( dvd_dvd_Code_integer @ A @ C )
% 8.55/4.73 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 8.55/4.73 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_33_div__dvd__div,axiom,
% 8.55/4.73 ! [A: nat,B: nat,C: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.73 => ( ( dvd_dvd_nat @ A @ C )
% 8.55/4.73 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
% 8.55/4.73 = ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_34_div__dvd__div,axiom,
% 8.55/4.73 ! [A: int,B: int,C: int] :
% 8.55/4.73 ( ( dvd_dvd_int @ A @ B )
% 8.55/4.73 => ( ( dvd_dvd_int @ A @ C )
% 8.55/4.73 => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
% 8.55/4.73 = ( dvd_dvd_int @ B @ C ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_35_bit__eq__rec,axiom,
% 8.55/4.73 ( ( ^ [Y3: code_integer,Z: code_integer] : ( Y3 = Z ) )
% 8.55/4.73 = ( ^ [A2: code_integer,B2: code_integer] :
% 8.55/4.73 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
% 8.55/4.73 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
% 8.55/4.73 & ( ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.55/4.73 = ( divide6298287555418463151nteger @ B2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_36_bit__eq__rec,axiom,
% 8.55/4.73 ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 8.55/4.73 = ( ^ [A2: nat,B2: nat] :
% 8.55/4.73 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
% 8.55/4.73 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
% 8.55/4.73 & ( ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.73 = ( divide_divide_nat @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_37_bit__eq__rec,axiom,
% 8.55/4.73 ( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 8.55/4.73 = ( ^ [A2: int,B2: int] :
% 8.55/4.73 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
% 8.55/4.73 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
% 8.55/4.73 & ( ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.55/4.73 = ( divide_divide_int @ B2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_38_zdiv__numeral__Bit0,axiom,
% 8.55/4.73 ! [V: num,W: num] :
% 8.55/4.73 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 8.55/4.73 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_39_verit__eq__simplify_I8_J,axiom,
% 8.55/4.73 ! [X2: num,Y2: num] :
% 8.55/4.73 ( ( ( bit0 @ X2 )
% 8.55/4.73 = ( bit0 @ Y2 ) )
% 8.55/4.73 = ( X2 = Y2 ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_40_even__of__nat,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
% 8.55/4.73 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_41_even__of__nat,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
% 8.55/4.73 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_42_even__of__nat,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
% 8.55/4.73 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_43_div__div__div__same,axiom,
% 8.55/4.73 ! [D: code_integer,B: code_integer,A: code_integer] :
% 8.55/4.73 ( ( dvd_dvd_Code_integer @ D @ B )
% 8.55/4.73 => ( ( dvd_dvd_Code_integer @ B @ A )
% 8.55/4.73 => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
% 8.55/4.73 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_44_div__div__div__same,axiom,
% 8.55/4.73 ! [D: nat,B: nat,A: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ D @ B )
% 8.55/4.73 => ( ( dvd_dvd_nat @ B @ A )
% 8.55/4.73 => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
% 8.55/4.73 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_45_div__div__div__same,axiom,
% 8.55/4.73 ! [D: int,B: int,A: int] :
% 8.55/4.73 ( ( dvd_dvd_int @ D @ B )
% 8.55/4.73 => ( ( dvd_dvd_int @ B @ A )
% 8.55/4.73 => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
% 8.55/4.73 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_46_dvd__div__eq__cancel,axiom,
% 8.55/4.73 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.55/4.73 ( ( ( divide6298287555418463151nteger @ A @ C )
% 8.55/4.73 = ( divide6298287555418463151nteger @ B @ C ) )
% 8.55/4.73 => ( ( dvd_dvd_Code_integer @ C @ A )
% 8.55/4.73 => ( ( dvd_dvd_Code_integer @ C @ B )
% 8.55/4.73 => ( A = B ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_47_dvd__div__eq__cancel,axiom,
% 8.55/4.73 ! [A: complex,C: complex,B: complex] :
% 8.55/4.73 ( ( ( divide1717551699836669952omplex @ A @ C )
% 8.55/4.73 = ( divide1717551699836669952omplex @ B @ C ) )
% 8.55/4.73 => ( ( dvd_dvd_complex @ C @ A )
% 8.55/4.73 => ( ( dvd_dvd_complex @ C @ B )
% 8.55/4.73 => ( A = B ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_48_dvd__div__eq__cancel,axiom,
% 8.55/4.73 ! [A: real,C: real,B: real] :
% 8.55/4.73 ( ( ( divide_divide_real @ A @ C )
% 8.55/4.73 = ( divide_divide_real @ B @ C ) )
% 8.55/4.73 => ( ( dvd_dvd_real @ C @ A )
% 8.55/4.73 => ( ( dvd_dvd_real @ C @ B )
% 8.55/4.73 => ( A = B ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_49_dvd__div__eq__cancel,axiom,
% 8.55/4.73 ! [A: rat,C: rat,B: rat] :
% 8.55/4.73 ( ( ( divide_divide_rat @ A @ C )
% 8.55/4.73 = ( divide_divide_rat @ B @ C ) )
% 8.55/4.73 => ( ( dvd_dvd_rat @ C @ A )
% 8.55/4.73 => ( ( dvd_dvd_rat @ C @ B )
% 8.55/4.73 => ( A = B ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_50_dvd__div__eq__cancel,axiom,
% 8.55/4.73 ! [A: nat,C: nat,B: nat] :
% 8.55/4.73 ( ( ( divide_divide_nat @ A @ C )
% 8.55/4.73 = ( divide_divide_nat @ B @ C ) )
% 8.55/4.73 => ( ( dvd_dvd_nat @ C @ A )
% 8.55/4.73 => ( ( dvd_dvd_nat @ C @ B )
% 8.55/4.73 => ( A = B ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_51_dvd__div__eq__cancel,axiom,
% 8.55/4.73 ! [A: int,C: int,B: int] :
% 8.55/4.73 ( ( ( divide_divide_int @ A @ C )
% 8.55/4.73 = ( divide_divide_int @ B @ C ) )
% 8.55/4.73 => ( ( dvd_dvd_int @ C @ A )
% 8.55/4.73 => ( ( dvd_dvd_int @ C @ B )
% 8.55/4.73 => ( A = B ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_52_dvd__div__eq__iff,axiom,
% 8.55/4.73 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.55/4.73 ( ( dvd_dvd_Code_integer @ C @ A )
% 8.55/4.73 => ( ( dvd_dvd_Code_integer @ C @ B )
% 8.55/4.73 => ( ( ( divide6298287555418463151nteger @ A @ C )
% 8.55/4.73 = ( divide6298287555418463151nteger @ B @ C ) )
% 8.55/4.73 = ( A = B ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_53_dvd__div__eq__iff,axiom,
% 8.55/4.73 ! [C: complex,A: complex,B: complex] :
% 8.55/4.73 ( ( dvd_dvd_complex @ C @ A )
% 8.55/4.73 => ( ( dvd_dvd_complex @ C @ B )
% 8.55/4.73 => ( ( ( divide1717551699836669952omplex @ A @ C )
% 8.55/4.73 = ( divide1717551699836669952omplex @ B @ C ) )
% 8.55/4.73 = ( A = B ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_54_dvd__div__eq__iff,axiom,
% 8.55/4.73 ! [C: real,A: real,B: real] :
% 8.55/4.73 ( ( dvd_dvd_real @ C @ A )
% 8.55/4.73 => ( ( dvd_dvd_real @ C @ B )
% 8.55/4.73 => ( ( ( divide_divide_real @ A @ C )
% 8.55/4.73 = ( divide_divide_real @ B @ C ) )
% 8.55/4.73 = ( A = B ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_55_dvd__div__eq__iff,axiom,
% 8.55/4.73 ! [C: rat,A: rat,B: rat] :
% 8.55/4.73 ( ( dvd_dvd_rat @ C @ A )
% 8.55/4.73 => ( ( dvd_dvd_rat @ C @ B )
% 8.55/4.73 => ( ( ( divide_divide_rat @ A @ C )
% 8.55/4.73 = ( divide_divide_rat @ B @ C ) )
% 8.55/4.73 = ( A = B ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_56_dvd__div__eq__iff,axiom,
% 8.55/4.73 ! [C: nat,A: nat,B: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ C @ A )
% 8.55/4.73 => ( ( dvd_dvd_nat @ C @ B )
% 8.55/4.73 => ( ( ( divide_divide_nat @ A @ C )
% 8.55/4.73 = ( divide_divide_nat @ B @ C ) )
% 8.55/4.73 = ( A = B ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_57_dvd__div__eq__iff,axiom,
% 8.55/4.73 ! [C: int,A: int,B: int] :
% 8.55/4.73 ( ( dvd_dvd_int @ C @ A )
% 8.55/4.73 => ( ( dvd_dvd_int @ C @ B )
% 8.55/4.73 => ( ( ( divide_divide_int @ A @ C )
% 8.55/4.73 = ( divide_divide_int @ B @ C ) )
% 8.55/4.73 = ( A = B ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_58_verit__eq__simplify_I10_J,axiom,
% 8.55/4.73 ! [X2: num] :
% 8.55/4.73 ( one
% 8.55/4.73 != ( bit0 @ X2 ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_59_set__decode__Suc,axiom,
% 8.55/4.73 ! [N: nat,X: nat] :
% 8.55/4.73 ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
% 8.55/4.73 = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_60_of__nat__eq__iff,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( ( semiri8010041392384452111omplex @ M )
% 8.55/4.73 = ( semiri8010041392384452111omplex @ N ) )
% 8.55/4.73 = ( M = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_61_of__nat__eq__iff,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( ( semiri5074537144036343181t_real @ M )
% 8.55/4.73 = ( semiri5074537144036343181t_real @ N ) )
% 8.55/4.73 = ( M = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_62_of__nat__eq__iff,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( ( semiri681578069525770553at_rat @ M )
% 8.55/4.73 = ( semiri681578069525770553at_rat @ N ) )
% 8.55/4.73 = ( M = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_63_of__nat__eq__iff,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( ( semiri1316708129612266289at_nat @ M )
% 8.55/4.73 = ( semiri1316708129612266289at_nat @ N ) )
% 8.55/4.73 = ( M = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_64_of__nat__eq__iff,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( ( semiri1314217659103216013at_int @ M )
% 8.55/4.73 = ( semiri1314217659103216013at_int @ N ) )
% 8.55/4.73 = ( M = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_65_div__0,axiom,
% 8.55/4.73 ! [A: complex] :
% 8.55/4.73 ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 8.55/4.73 = zero_zero_complex ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_66_div__0,axiom,
% 8.55/4.73 ! [A: real] :
% 8.55/4.73 ( ( divide_divide_real @ zero_zero_real @ A )
% 8.55/4.73 = zero_zero_real ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_67_div__0,axiom,
% 8.55/4.73 ! [A: rat] :
% 8.55/4.73 ( ( divide_divide_rat @ zero_zero_rat @ A )
% 8.55/4.73 = zero_zero_rat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_68_div__0,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 8.55/4.73 = zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_69_div__0,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( divide_divide_int @ zero_zero_int @ A )
% 8.55/4.73 = zero_zero_int ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_70_div__by__0,axiom,
% 8.55/4.73 ! [A: complex] :
% 8.55/4.73 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 8.55/4.73 = zero_zero_complex ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_71_div__by__0,axiom,
% 8.55/4.73 ! [A: real] :
% 8.55/4.73 ( ( divide_divide_real @ A @ zero_zero_real )
% 8.55/4.73 = zero_zero_real ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_72_div__by__0,axiom,
% 8.55/4.73 ! [A: rat] :
% 8.55/4.73 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 8.55/4.73 = zero_zero_rat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_73_div__by__0,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 8.55/4.73 = zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_74_div__by__0,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( divide_divide_int @ A @ zero_zero_int )
% 8.55/4.73 = zero_zero_int ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_75_bits__div__0,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 8.55/4.73 = zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_76_bits__div__0,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( divide_divide_int @ zero_zero_int @ A )
% 8.55/4.73 = zero_zero_int ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_77_bits__div__by__0,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 8.55/4.73 = zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_78_bits__div__by__0,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( divide_divide_int @ A @ zero_zero_int )
% 8.55/4.73 = zero_zero_int ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_79_dvd__0__left__iff,axiom,
% 8.55/4.73 ! [A: code_integer] :
% 8.55/4.73 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 8.55/4.73 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_80_dvd__0__left__iff,axiom,
% 8.55/4.73 ! [A: real] :
% 8.55/4.73 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 8.55/4.73 = ( A = zero_zero_real ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_81_dvd__0__left__iff,axiom,
% 8.55/4.73 ! [A: rat] :
% 8.55/4.73 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 8.55/4.73 = ( A = zero_zero_rat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_82_dvd__0__left__iff,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 8.55/4.73 = ( A = zero_zero_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_83_dvd__0__left__iff,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 8.55/4.73 = ( A = zero_zero_int ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_84_dvd__0__right,axiom,
% 8.55/4.73 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_85_dvd__0__right,axiom,
% 8.55/4.73 ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_86_dvd__0__right,axiom,
% 8.55/4.73 ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_87_dvd__0__right,axiom,
% 8.55/4.73 ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_88_dvd__0__right,axiom,
% 8.55/4.73 ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_89_of__nat__0,axiom,
% 8.55/4.73 ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 8.55/4.73 = zero_zero_complex ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_90_of__nat__0,axiom,
% 8.55/4.73 ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 8.55/4.73 = zero_zero_real ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_91_of__nat__0,axiom,
% 8.55/4.73 ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 8.55/4.73 = zero_zero_rat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_92_of__nat__0,axiom,
% 8.55/4.73 ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 8.55/4.73 = zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_93_of__nat__0,axiom,
% 8.55/4.73 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 8.55/4.73 = zero_zero_int ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_94_mem__Collect__eq,axiom,
% 8.55/4.73 ! [A: complex,P: complex > $o] :
% 8.55/4.73 ( ( member_complex @ A @ ( collect_complex @ P ) )
% 8.55/4.73 = ( P @ A ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_95_mem__Collect__eq,axiom,
% 8.55/4.73 ! [A: real,P: real > $o] :
% 8.55/4.73 ( ( member_real @ A @ ( collect_real @ P ) )
% 8.55/4.73 = ( P @ A ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_96_mem__Collect__eq,axiom,
% 8.55/4.73 ! [A: list_nat,P: list_nat > $o] :
% 8.55/4.73 ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
% 8.55/4.73 = ( P @ A ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_97_mem__Collect__eq,axiom,
% 8.55/4.73 ! [A: set_nat,P: set_nat > $o] :
% 8.55/4.73 ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
% 8.55/4.73 = ( P @ A ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_98_mem__Collect__eq,axiom,
% 8.55/4.73 ! [A: nat,P: nat > $o] :
% 8.55/4.73 ( ( member_nat @ A @ ( collect_nat @ P ) )
% 8.55/4.73 = ( P @ A ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_99_mem__Collect__eq,axiom,
% 8.55/4.73 ! [A: int,P: int > $o] :
% 8.55/4.73 ( ( member_int @ A @ ( collect_int @ P ) )
% 8.55/4.73 = ( P @ A ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_100_Collect__mem__eq,axiom,
% 8.55/4.73 ! [A3: set_complex] :
% 8.55/4.73 ( ( collect_complex
% 8.55/4.73 @ ^ [X3: complex] : ( member_complex @ X3 @ A3 ) )
% 8.55/4.73 = A3 ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_101_Collect__mem__eq,axiom,
% 8.55/4.73 ! [A3: set_real] :
% 8.55/4.73 ( ( collect_real
% 8.55/4.73 @ ^ [X3: real] : ( member_real @ X3 @ A3 ) )
% 8.55/4.73 = A3 ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_102_Collect__mem__eq,axiom,
% 8.55/4.73 ! [A3: set_list_nat] :
% 8.55/4.73 ( ( collect_list_nat
% 8.55/4.73 @ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A3 ) )
% 8.55/4.73 = A3 ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_103_Collect__mem__eq,axiom,
% 8.55/4.73 ! [A3: set_set_nat] :
% 8.55/4.73 ( ( collect_set_nat
% 8.55/4.73 @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A3 ) )
% 8.55/4.73 = A3 ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_104_Collect__mem__eq,axiom,
% 8.55/4.73 ! [A3: set_nat] :
% 8.55/4.73 ( ( collect_nat
% 8.55/4.73 @ ^ [X3: nat] : ( member_nat @ X3 @ A3 ) )
% 8.55/4.73 = A3 ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_105_Collect__mem__eq,axiom,
% 8.55/4.73 ! [A3: set_int] :
% 8.55/4.73 ( ( collect_int
% 8.55/4.73 @ ^ [X3: int] : ( member_int @ X3 @ A3 ) )
% 8.55/4.73 = A3 ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_106_Collect__cong,axiom,
% 8.55/4.73 ! [P: real > $o,Q: real > $o] :
% 8.55/4.73 ( ! [X4: real] :
% 8.55/4.73 ( ( P @ X4 )
% 8.55/4.73 = ( Q @ X4 ) )
% 8.55/4.73 => ( ( collect_real @ P )
% 8.55/4.73 = ( collect_real @ Q ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_107_Collect__cong,axiom,
% 8.55/4.73 ! [P: list_nat > $o,Q: list_nat > $o] :
% 8.55/4.73 ( ! [X4: list_nat] :
% 8.55/4.73 ( ( P @ X4 )
% 8.55/4.73 = ( Q @ X4 ) )
% 8.55/4.73 => ( ( collect_list_nat @ P )
% 8.55/4.73 = ( collect_list_nat @ Q ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_108_Collect__cong,axiom,
% 8.55/4.73 ! [P: set_nat > $o,Q: set_nat > $o] :
% 8.55/4.73 ( ! [X4: set_nat] :
% 8.55/4.73 ( ( P @ X4 )
% 8.55/4.73 = ( Q @ X4 ) )
% 8.55/4.73 => ( ( collect_set_nat @ P )
% 8.55/4.73 = ( collect_set_nat @ Q ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_109_Collect__cong,axiom,
% 8.55/4.73 ! [P: nat > $o,Q: nat > $o] :
% 8.55/4.73 ( ! [X4: nat] :
% 8.55/4.73 ( ( P @ X4 )
% 8.55/4.73 = ( Q @ X4 ) )
% 8.55/4.73 => ( ( collect_nat @ P )
% 8.55/4.73 = ( collect_nat @ Q ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_110_Collect__cong,axiom,
% 8.55/4.73 ! [P: int > $o,Q: int > $o] :
% 8.55/4.73 ( ! [X4: int] :
% 8.55/4.73 ( ( P @ X4 )
% 8.55/4.73 = ( Q @ X4 ) )
% 8.55/4.73 => ( ( collect_int @ P )
% 8.55/4.73 = ( collect_int @ Q ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_111_of__nat__0__eq__iff,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( zero_zero_complex
% 8.55/4.73 = ( semiri8010041392384452111omplex @ N ) )
% 8.55/4.73 = ( zero_zero_nat = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_112_of__nat__0__eq__iff,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( zero_zero_real
% 8.55/4.73 = ( semiri5074537144036343181t_real @ N ) )
% 8.55/4.73 = ( zero_zero_nat = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_113_of__nat__0__eq__iff,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( zero_zero_rat
% 8.55/4.73 = ( semiri681578069525770553at_rat @ N ) )
% 8.55/4.73 = ( zero_zero_nat = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_114_of__nat__0__eq__iff,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( zero_zero_nat
% 8.55/4.73 = ( semiri1316708129612266289at_nat @ N ) )
% 8.55/4.73 = ( zero_zero_nat = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_115_of__nat__0__eq__iff,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( zero_zero_int
% 8.55/4.73 = ( semiri1314217659103216013at_int @ N ) )
% 8.55/4.73 = ( zero_zero_nat = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_116_of__nat__eq__0__iff,axiom,
% 8.55/4.73 ! [M: nat] :
% 8.55/4.73 ( ( ( semiri8010041392384452111omplex @ M )
% 8.55/4.73 = zero_zero_complex )
% 8.55/4.73 = ( M = zero_zero_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_117_of__nat__eq__0__iff,axiom,
% 8.55/4.73 ! [M: nat] :
% 8.55/4.73 ( ( ( semiri5074537144036343181t_real @ M )
% 8.55/4.73 = zero_zero_real )
% 8.55/4.73 = ( M = zero_zero_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_118_of__nat__eq__0__iff,axiom,
% 8.55/4.73 ! [M: nat] :
% 8.55/4.73 ( ( ( semiri681578069525770553at_rat @ M )
% 8.55/4.73 = zero_zero_rat )
% 8.55/4.73 = ( M = zero_zero_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_119_of__nat__eq__0__iff,axiom,
% 8.55/4.73 ! [M: nat] :
% 8.55/4.73 ( ( ( semiri1316708129612266289at_nat @ M )
% 8.55/4.73 = zero_zero_nat )
% 8.55/4.73 = ( M = zero_zero_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_120_of__nat__eq__0__iff,axiom,
% 8.55/4.73 ! [M: nat] :
% 8.55/4.73 ( ( ( semiri1314217659103216013at_int @ M )
% 8.55/4.73 = zero_zero_int )
% 8.55/4.73 = ( M = zero_zero_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_121_of__nat__numeral,axiom,
% 8.55/4.73 ! [N: num] :
% 8.55/4.73 ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
% 8.55/4.73 = ( numera6690914467698888265omplex @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_122_of__nat__numeral,axiom,
% 8.55/4.73 ! [N: num] :
% 8.55/4.73 ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
% 8.55/4.73 = ( numeral_numeral_real @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_123_of__nat__numeral,axiom,
% 8.55/4.73 ! [N: num] :
% 8.55/4.73 ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
% 8.55/4.73 = ( numeral_numeral_rat @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_124_of__nat__numeral,axiom,
% 8.55/4.73 ! [N: num] :
% 8.55/4.73 ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
% 8.55/4.73 = ( numeral_numeral_nat @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_125_of__nat__numeral,axiom,
% 8.55/4.73 ! [N: num] :
% 8.55/4.73 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 8.55/4.73 = ( numeral_numeral_int @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_126_dvd__1__left,axiom,
% 8.55/4.73 ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_127_dvd__1__iff__1,axiom,
% 8.55/4.73 ! [M: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 8.55/4.73 = ( M
% 8.55/4.73 = ( suc @ zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_128_div__by__Suc__0,axiom,
% 8.55/4.73 ! [M: nat] :
% 8.55/4.73 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 8.55/4.73 = M ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_129_set__decode__0,axiom,
% 8.55/4.73 ! [X: nat] :
% 8.55/4.73 ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
% 8.55/4.73 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_130_of__nat__neq__0,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
% 8.55/4.73 != zero_zero_complex ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_131_of__nat__neq__0,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 8.55/4.73 != zero_zero_real ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_132_of__nat__neq__0,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 8.55/4.73 != zero_zero_rat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_133_of__nat__neq__0,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 8.55/4.73 != zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_134_of__nat__neq__0,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 8.55/4.73 != zero_zero_int ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_135_int__ops_I3_J,axiom,
% 8.55/4.73 ! [N: num] :
% 8.55/4.73 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 8.55/4.73 = ( numeral_numeral_int @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_136_zdiv__int,axiom,
% 8.55/4.73 ! [A: nat,B: nat] :
% 8.55/4.73 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
% 8.55/4.73 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_137_list__decode_Ocases,axiom,
% 8.55/4.73 ! [X: nat] :
% 8.55/4.73 ( ( X != zero_zero_nat )
% 8.55/4.73 => ~ ! [N2: nat] :
% 8.55/4.73 ( X
% 8.55/4.73 != ( suc @ N2 ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_138_dvd__0__left,axiom,
% 8.55/4.73 ! [A: code_integer] :
% 8.55/4.73 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 8.55/4.73 => ( A = zero_z3403309356797280102nteger ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_139_dvd__0__left,axiom,
% 8.55/4.73 ! [A: real] :
% 8.55/4.73 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 8.55/4.73 => ( A = zero_zero_real ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_140_dvd__0__left,axiom,
% 8.55/4.73 ! [A: rat] :
% 8.55/4.73 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 8.55/4.73 => ( A = zero_zero_rat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_141_dvd__0__left,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 8.55/4.73 => ( A = zero_zero_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_142_dvd__0__left,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 8.55/4.73 => ( A = zero_zero_int ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_143_zero__neq__numeral,axiom,
% 8.55/4.73 ! [N: num] :
% 8.55/4.73 ( zero_zero_complex
% 8.55/4.73 != ( numera6690914467698888265omplex @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_144_zero__neq__numeral,axiom,
% 8.55/4.73 ! [N: num] :
% 8.55/4.73 ( zero_zero_real
% 8.55/4.73 != ( numeral_numeral_real @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_145_zero__neq__numeral,axiom,
% 8.55/4.73 ! [N: num] :
% 8.55/4.73 ( zero_zero_rat
% 8.55/4.73 != ( numeral_numeral_rat @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_146_zero__neq__numeral,axiom,
% 8.55/4.73 ! [N: num] :
% 8.55/4.73 ( zero_zero_nat
% 8.55/4.73 != ( numeral_numeral_nat @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_147_zero__neq__numeral,axiom,
% 8.55/4.73 ! [N: num] :
% 8.55/4.73 ( zero_zero_int
% 8.55/4.73 != ( numeral_numeral_int @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_148_vebt__buildup_Ocases,axiom,
% 8.55/4.73 ! [X: nat] :
% 8.55/4.73 ( ( X != zero_zero_nat )
% 8.55/4.73 => ( ( X
% 8.55/4.73 != ( suc @ zero_zero_nat ) )
% 8.55/4.73 => ~ ! [Va: nat] :
% 8.55/4.73 ( X
% 8.55/4.73 != ( suc @ ( suc @ Va ) ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_149_nat_Odistinct_I1_J,axiom,
% 8.55/4.73 ! [X2: nat] :
% 8.55/4.73 ( zero_zero_nat
% 8.55/4.73 != ( suc @ X2 ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_150_old_Onat_Odistinct_I2_J,axiom,
% 8.55/4.73 ! [Nat2: nat] :
% 8.55/4.73 ( ( suc @ Nat2 )
% 8.55/4.73 != zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_151_old_Onat_Odistinct_I1_J,axiom,
% 8.55/4.73 ! [Nat2: nat] :
% 8.55/4.73 ( zero_zero_nat
% 8.55/4.73 != ( suc @ Nat2 ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_152_nat_OdiscI,axiom,
% 8.55/4.73 ! [Nat: nat,X2: nat] :
% 8.55/4.73 ( ( Nat
% 8.55/4.73 = ( suc @ X2 ) )
% 8.55/4.73 => ( Nat != zero_zero_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_153_old_Onat_Oexhaust,axiom,
% 8.55/4.73 ! [Y: nat] :
% 8.55/4.73 ( ( Y != zero_zero_nat )
% 8.55/4.73 => ~ ! [Nat3: nat] :
% 8.55/4.73 ( Y
% 8.55/4.73 != ( suc @ Nat3 ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_154_nat__induct,axiom,
% 8.55/4.73 ! [P: nat > $o,N: nat] :
% 8.55/4.73 ( ( P @ zero_zero_nat )
% 8.55/4.73 => ( ! [N2: nat] :
% 8.55/4.73 ( ( P @ N2 )
% 8.55/4.73 => ( P @ ( suc @ N2 ) ) )
% 8.55/4.73 => ( P @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_155_diff__induct,axiom,
% 8.55/4.73 ! [P: nat > nat > $o,M: nat,N: nat] :
% 8.55/4.73 ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
% 8.55/4.73 => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
% 8.55/4.73 => ( ! [X4: nat,Y4: nat] :
% 8.55/4.73 ( ( P @ X4 @ Y4 )
% 8.55/4.73 => ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
% 8.55/4.73 => ( P @ M @ N ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_156_zero__induct,axiom,
% 8.55/4.73 ! [P: nat > $o,K: nat] :
% 8.55/4.73 ( ( P @ K )
% 8.55/4.73 => ( ! [N2: nat] :
% 8.55/4.73 ( ( P @ ( suc @ N2 ) )
% 8.55/4.73 => ( P @ N2 ) )
% 8.55/4.73 => ( P @ zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_157_Suc__neq__Zero,axiom,
% 8.55/4.73 ! [M: nat] :
% 8.55/4.73 ( ( suc @ M )
% 8.55/4.73 != zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_158_Zero__neq__Suc,axiom,
% 8.55/4.73 ! [M: nat] :
% 8.55/4.73 ( zero_zero_nat
% 8.55/4.73 != ( suc @ M ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_159_Zero__not__Suc,axiom,
% 8.55/4.73 ! [M: nat] :
% 8.55/4.73 ( zero_zero_nat
% 8.55/4.73 != ( suc @ M ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_160_not0__implies__Suc,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( N != zero_zero_nat )
% 8.55/4.73 => ? [M2: nat] :
% 8.55/4.73 ( N
% 8.55/4.73 = ( suc @ M2 ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_161_dvd__div__eq__0__iff,axiom,
% 8.55/4.73 ! [B: code_integer,A: code_integer] :
% 8.55/4.73 ( ( dvd_dvd_Code_integer @ B @ A )
% 8.55/4.73 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 8.55/4.73 = zero_z3403309356797280102nteger )
% 8.55/4.73 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_162_dvd__div__eq__0__iff,axiom,
% 8.55/4.73 ! [B: complex,A: complex] :
% 8.55/4.73 ( ( dvd_dvd_complex @ B @ A )
% 8.55/4.73 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 8.55/4.73 = zero_zero_complex )
% 8.55/4.73 = ( A = zero_zero_complex ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_163_dvd__div__eq__0__iff,axiom,
% 8.55/4.73 ! [B: real,A: real] :
% 8.55/4.73 ( ( dvd_dvd_real @ B @ A )
% 8.55/4.73 => ( ( ( divide_divide_real @ A @ B )
% 8.55/4.73 = zero_zero_real )
% 8.55/4.73 = ( A = zero_zero_real ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_164_dvd__div__eq__0__iff,axiom,
% 8.55/4.73 ! [B: rat,A: rat] :
% 8.55/4.73 ( ( dvd_dvd_rat @ B @ A )
% 8.55/4.73 => ( ( ( divide_divide_rat @ A @ B )
% 8.55/4.73 = zero_zero_rat )
% 8.55/4.73 = ( A = zero_zero_rat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_165_dvd__div__eq__0__iff,axiom,
% 8.55/4.73 ! [B: nat,A: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ B @ A )
% 8.55/4.73 => ( ( ( divide_divide_nat @ A @ B )
% 8.55/4.73 = zero_zero_nat )
% 8.55/4.73 = ( A = zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_166_dvd__div__eq__0__iff,axiom,
% 8.55/4.73 ! [B: int,A: int] :
% 8.55/4.73 ( ( dvd_dvd_int @ B @ A )
% 8.55/4.73 => ( ( ( divide_divide_int @ A @ B )
% 8.55/4.73 = zero_zero_int )
% 8.55/4.73 = ( A = zero_zero_int ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_167_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
% 8.55/4.73 = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_168_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
% 8.55/4.73 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_169_of__nat__dvd__iff,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
% 8.55/4.73 = ( dvd_dvd_nat @ M @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_170_of__nat__dvd__iff,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 8.55/4.73 = ( dvd_dvd_nat @ M @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_171_of__nat__dvd__iff,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 8.55/4.73 = ( dvd_dvd_nat @ M @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_172_dvd__trans,axiom,
% 8.55/4.73 ! [A: nat,B: nat,C: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.73 => ( ( dvd_dvd_nat @ B @ C )
% 8.55/4.73 => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_173_dvd__trans,axiom,
% 8.55/4.73 ! [A: int,B: int,C: int] :
% 8.55/4.73 ( ( dvd_dvd_int @ A @ B )
% 8.55/4.73 => ( ( dvd_dvd_int @ B @ C )
% 8.55/4.73 => ( dvd_dvd_int @ A @ C ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_174_dvd__trans,axiom,
% 8.55/4.73 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.55/4.73 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.55/4.73 => ( ( dvd_dvd_Code_integer @ B @ C )
% 8.55/4.73 => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_175_dvd__refl,axiom,
% 8.55/4.73 ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_176_dvd__refl,axiom,
% 8.55/4.73 ! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_177_dvd__refl,axiom,
% 8.55/4.73 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_178_numeral__1__eq__Suc__0,axiom,
% 8.55/4.73 ( ( numeral_numeral_nat @ one )
% 8.55/4.73 = ( suc @ zero_zero_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_179_even__zero,axiom,
% 8.55/4.73 dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 8.55/4.73
% 8.55/4.73 thf(fact_180_even__zero,axiom,
% 8.55/4.73 dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 8.55/4.73
% 8.55/4.73 thf(fact_181_even__zero,axiom,
% 8.55/4.73 dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 8.55/4.73
% 8.55/4.73 thf(fact_182_numeral__2__eq__2,axiom,
% 8.55/4.73 ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 8.55/4.73 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_183_int__eq__iff__numeral,axiom,
% 8.55/4.73 ! [M: nat,V: num] :
% 8.55/4.73 ( ( ( semiri1314217659103216013at_int @ M )
% 8.55/4.73 = ( numeral_numeral_int @ V ) )
% 8.55/4.73 = ( M
% 8.55/4.73 = ( numeral_numeral_nat @ V ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_184_divide__eq__0__iff,axiom,
% 8.55/4.73 ! [A: complex,B: complex] :
% 8.55/4.73 ( ( ( divide1717551699836669952omplex @ A @ B )
% 8.55/4.73 = zero_zero_complex )
% 8.55/4.73 = ( ( A = zero_zero_complex )
% 8.55/4.73 | ( B = zero_zero_complex ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_185_divide__eq__0__iff,axiom,
% 8.55/4.73 ! [A: real,B: real] :
% 8.55/4.73 ( ( ( divide_divide_real @ A @ B )
% 8.55/4.73 = zero_zero_real )
% 8.55/4.73 = ( ( A = zero_zero_real )
% 8.55/4.73 | ( B = zero_zero_real ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_186_divide__eq__0__iff,axiom,
% 8.55/4.73 ! [A: rat,B: rat] :
% 8.55/4.73 ( ( ( divide_divide_rat @ A @ B )
% 8.55/4.73 = zero_zero_rat )
% 8.55/4.73 = ( ( A = zero_zero_rat )
% 8.55/4.73 | ( B = zero_zero_rat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_187_divide__cancel__left,axiom,
% 8.55/4.73 ! [C: complex,A: complex,B: complex] :
% 8.55/4.73 ( ( ( divide1717551699836669952omplex @ C @ A )
% 8.55/4.73 = ( divide1717551699836669952omplex @ C @ B ) )
% 8.55/4.73 = ( ( C = zero_zero_complex )
% 8.55/4.73 | ( A = B ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_188_divide__cancel__left,axiom,
% 8.55/4.73 ! [C: real,A: real,B: real] :
% 8.55/4.73 ( ( ( divide_divide_real @ C @ A )
% 8.55/4.73 = ( divide_divide_real @ C @ B ) )
% 8.55/4.73 = ( ( C = zero_zero_real )
% 8.55/4.73 | ( A = B ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_189_divide__cancel__left,axiom,
% 8.55/4.73 ! [C: rat,A: rat,B: rat] :
% 8.55/4.73 ( ( ( divide_divide_rat @ C @ A )
% 8.55/4.73 = ( divide_divide_rat @ C @ B ) )
% 8.55/4.73 = ( ( C = zero_zero_rat )
% 8.55/4.73 | ( A = B ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_190_divide__cancel__right,axiom,
% 8.55/4.73 ! [A: complex,C: complex,B: complex] :
% 8.55/4.73 ( ( ( divide1717551699836669952omplex @ A @ C )
% 8.55/4.73 = ( divide1717551699836669952omplex @ B @ C ) )
% 8.55/4.73 = ( ( C = zero_zero_complex )
% 8.55/4.73 | ( A = B ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_191_divide__cancel__right,axiom,
% 8.55/4.73 ! [A: real,C: real,B: real] :
% 8.55/4.73 ( ( ( divide_divide_real @ A @ C )
% 8.55/4.73 = ( divide_divide_real @ B @ C ) )
% 8.55/4.73 = ( ( C = zero_zero_real )
% 8.55/4.73 | ( A = B ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_192_divide__cancel__right,axiom,
% 8.55/4.73 ! [A: rat,C: rat,B: rat] :
% 8.55/4.73 ( ( ( divide_divide_rat @ A @ C )
% 8.55/4.73 = ( divide_divide_rat @ B @ C ) )
% 8.55/4.73 = ( ( C = zero_zero_rat )
% 8.55/4.73 | ( A = B ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_193_division__ring__divide__zero,axiom,
% 8.55/4.73 ! [A: complex] :
% 8.55/4.73 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 8.55/4.73 = zero_zero_complex ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_194_division__ring__divide__zero,axiom,
% 8.55/4.73 ! [A: real] :
% 8.55/4.73 ( ( divide_divide_real @ A @ zero_zero_real )
% 8.55/4.73 = zero_zero_real ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_195_division__ring__divide__zero,axiom,
% 8.55/4.73 ! [A: rat] :
% 8.55/4.73 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 8.55/4.73 = zero_zero_rat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_196_even__set__bit__iff,axiom,
% 8.55/4.73 ! [M: nat,A: code_integer] :
% 8.55/4.73 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 8.55/4.73 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.55/4.73 & ( M != zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_197_even__set__bit__iff,axiom,
% 8.55/4.73 ! [M: nat,A: int] :
% 8.55/4.73 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 8.55/4.73 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.55/4.73 & ( M != zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_198_even__set__bit__iff,axiom,
% 8.55/4.73 ! [M: nat,A: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 8.55/4.73 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.55/4.73 & ( M != zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_199_even__flip__bit__iff,axiom,
% 8.55/4.73 ! [M: nat,A: code_integer] :
% 8.55/4.73 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 8.55/4.73 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.55/4.73 != ( M = zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_200_even__flip__bit__iff,axiom,
% 8.55/4.73 ! [M: nat,A: int] :
% 8.55/4.73 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 8.55/4.73 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.55/4.73 != ( M = zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_201_even__flip__bit__iff,axiom,
% 8.55/4.73 ! [M: nat,A: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 8.55/4.73 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.55/4.73 != ( M = zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_202_even__unset__bit__iff,axiom,
% 8.55/4.73 ! [M: nat,A: code_integer] :
% 8.55/4.73 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 8.55/4.73 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.55/4.73 | ( M = zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_203_even__unset__bit__iff,axiom,
% 8.55/4.73 ! [M: nat,A: int] :
% 8.55/4.73 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 8.55/4.73 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.55/4.73 | ( M = zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_204_even__unset__bit__iff,axiom,
% 8.55/4.73 ! [M: nat,A: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 8.55/4.73 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.55/4.73 | ( M = zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_205_int__dvd__int__iff,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 8.55/4.73 = ( dvd_dvd_nat @ M @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_206_odd__Suc__minus__one,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.73 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 8.55/4.73 = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_207_real__of__nat__div,axiom,
% 8.55/4.73 ! [D: nat,N: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ D @ N )
% 8.55/4.73 => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
% 8.55/4.73 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_208_diff__self,axiom,
% 8.55/4.73 ! [A: real] :
% 8.55/4.73 ( ( minus_minus_real @ A @ A )
% 8.55/4.73 = zero_zero_real ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_209_diff__self,axiom,
% 8.55/4.73 ! [A: rat] :
% 8.55/4.73 ( ( minus_minus_rat @ A @ A )
% 8.55/4.73 = zero_zero_rat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_210_diff__self,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( minus_minus_int @ A @ A )
% 8.55/4.73 = zero_zero_int ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_211_diff__0__right,axiom,
% 8.55/4.73 ! [A: real] :
% 8.55/4.73 ( ( minus_minus_real @ A @ zero_zero_real )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_212_diff__0__right,axiom,
% 8.55/4.73 ! [A: rat] :
% 8.55/4.73 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_213_diff__0__right,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( minus_minus_int @ A @ zero_zero_int )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_214_zero__diff,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 8.55/4.73 = zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_215_diff__zero,axiom,
% 8.55/4.73 ! [A: real] :
% 8.55/4.73 ( ( minus_minus_real @ A @ zero_zero_real )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_216_diff__zero,axiom,
% 8.55/4.73 ! [A: rat] :
% 8.55/4.73 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_217_diff__zero,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_218_diff__zero,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( minus_minus_int @ A @ zero_zero_int )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_219_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 8.55/4.73 ! [A: real] :
% 8.55/4.73 ( ( minus_minus_real @ A @ A )
% 8.55/4.73 = zero_zero_real ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_220_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 8.55/4.73 ! [A: rat] :
% 8.55/4.73 ( ( minus_minus_rat @ A @ A )
% 8.55/4.73 = zero_zero_rat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_221_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( minus_minus_nat @ A @ A )
% 8.55/4.73 = zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_222_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( minus_minus_int @ A @ A )
% 8.55/4.73 = zero_zero_int ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_223_Suc__diff__diff,axiom,
% 8.55/4.73 ! [M: nat,N: nat,K: nat] :
% 8.55/4.73 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 8.55/4.73 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_224_diff__Suc__Suc,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 8.55/4.73 = ( minus_minus_nat @ M @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_225_diff__self__eq__0,axiom,
% 8.55/4.73 ! [M: nat] :
% 8.55/4.73 ( ( minus_minus_nat @ M @ M )
% 8.55/4.73 = zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_226_diff__0__eq__0,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 8.55/4.73 = zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_227_div__diff,axiom,
% 8.55/4.73 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.55/4.73 ( ( dvd_dvd_Code_integer @ C @ A )
% 8.55/4.73 => ( ( dvd_dvd_Code_integer @ C @ B )
% 8.55/4.73 => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 8.55/4.73 = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_228_div__diff,axiom,
% 8.55/4.73 ! [C: int,A: int,B: int] :
% 8.55/4.73 ( ( dvd_dvd_int @ C @ A )
% 8.55/4.73 => ( ( dvd_dvd_int @ C @ B )
% 8.55/4.73 => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
% 8.55/4.73 = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_229_int__int__eq,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( ( semiri1314217659103216013at_int @ M )
% 8.55/4.73 = ( semiri1314217659103216013at_int @ N ) )
% 8.55/4.73 = ( M = N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_230_int__if,axiom,
% 8.55/4.73 ! [P: $o,A: nat,B: nat] :
% 8.55/4.73 ( ( P
% 8.55/4.73 => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
% 8.55/4.73 = ( semiri1314217659103216013at_int @ A ) ) )
% 8.55/4.73 & ( ~ P
% 8.55/4.73 => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
% 8.55/4.73 = ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_231_nat__int__comparison_I1_J,axiom,
% 8.55/4.73 ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 8.55/4.73 = ( ^ [A2: nat,B2: nat] :
% 8.55/4.73 ( ( semiri1314217659103216013at_int @ A2 )
% 8.55/4.73 = ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_232_diff__eq__diff__eq,axiom,
% 8.55/4.73 ! [A: real,B: real,C: real,D: real] :
% 8.55/4.73 ( ( ( minus_minus_real @ A @ B )
% 8.55/4.73 = ( minus_minus_real @ C @ D ) )
% 8.55/4.73 => ( ( A = B )
% 8.55/4.73 = ( C = D ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_233_diff__eq__diff__eq,axiom,
% 8.55/4.73 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.55/4.73 ( ( ( minus_minus_rat @ A @ B )
% 8.55/4.73 = ( minus_minus_rat @ C @ D ) )
% 8.55/4.73 => ( ( A = B )
% 8.55/4.73 = ( C = D ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_234_diff__eq__diff__eq,axiom,
% 8.55/4.73 ! [A: int,B: int,C: int,D: int] :
% 8.55/4.73 ( ( ( minus_minus_int @ A @ B )
% 8.55/4.73 = ( minus_minus_int @ C @ D ) )
% 8.55/4.73 => ( ( A = B )
% 8.55/4.73 = ( C = D ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_235_diff__right__commute,axiom,
% 8.55/4.73 ! [A: real,C: real,B: real] :
% 8.55/4.73 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 8.55/4.73 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_236_diff__right__commute,axiom,
% 8.55/4.73 ! [A: rat,C: rat,B: rat] :
% 8.55/4.73 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
% 8.55/4.73 = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_237_diff__right__commute,axiom,
% 8.55/4.73 ! [A: nat,C: nat,B: nat] :
% 8.55/4.73 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 8.55/4.73 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_238_diff__right__commute,axiom,
% 8.55/4.73 ! [A: int,C: int,B: int] :
% 8.55/4.73 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 8.55/4.73 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_239_diff__commute,axiom,
% 8.55/4.73 ! [I: nat,J: nat,K: nat] :
% 8.55/4.73 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 8.55/4.73 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_240_eq__iff__diff__eq__0,axiom,
% 8.55/4.73 ( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
% 8.55/4.73 = ( ^ [A2: real,B2: real] :
% 8.55/4.73 ( ( minus_minus_real @ A2 @ B2 )
% 8.55/4.73 = zero_zero_real ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_241_eq__iff__diff__eq__0,axiom,
% 8.55/4.73 ( ( ^ [Y3: rat,Z: rat] : ( Y3 = Z ) )
% 8.55/4.73 = ( ^ [A2: rat,B2: rat] :
% 8.55/4.73 ( ( minus_minus_rat @ A2 @ B2 )
% 8.55/4.73 = zero_zero_rat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_242_eq__iff__diff__eq__0,axiom,
% 8.55/4.73 ( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 8.55/4.73 = ( ^ [A2: int,B2: int] :
% 8.55/4.73 ( ( minus_minus_int @ A2 @ B2 )
% 8.55/4.73 = zero_zero_int ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_243_diff__divide__distrib,axiom,
% 8.55/4.73 ! [A: complex,B: complex,C: complex] :
% 8.55/4.73 ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
% 8.55/4.73 = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_244_diff__divide__distrib,axiom,
% 8.55/4.73 ! [A: real,B: real,C: real] :
% 8.55/4.73 ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
% 8.55/4.73 = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_245_diff__divide__distrib,axiom,
% 8.55/4.73 ! [A: rat,B: rat,C: rat] :
% 8.55/4.73 ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 8.55/4.73 = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_246_dvd__diff__commute,axiom,
% 8.55/4.73 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.55/4.73 ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
% 8.55/4.73 = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_247_dvd__diff__commute,axiom,
% 8.55/4.73 ! [A: int,C: int,B: int] :
% 8.55/4.73 ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
% 8.55/4.73 = ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_248_dvd__diff,axiom,
% 8.55/4.73 ! [X: code_integer,Y: code_integer,Z2: code_integer] :
% 8.55/4.73 ( ( dvd_dvd_Code_integer @ X @ Y )
% 8.55/4.73 => ( ( dvd_dvd_Code_integer @ X @ Z2 )
% 8.55/4.73 => ( dvd_dvd_Code_integer @ X @ ( minus_8373710615458151222nteger @ Y @ Z2 ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_249_dvd__diff,axiom,
% 8.55/4.73 ! [X: real,Y: real,Z2: real] :
% 8.55/4.73 ( ( dvd_dvd_real @ X @ Y )
% 8.55/4.73 => ( ( dvd_dvd_real @ X @ Z2 )
% 8.55/4.73 => ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z2 ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_250_dvd__diff,axiom,
% 8.55/4.73 ! [X: rat,Y: rat,Z2: rat] :
% 8.55/4.73 ( ( dvd_dvd_rat @ X @ Y )
% 8.55/4.73 => ( ( dvd_dvd_rat @ X @ Z2 )
% 8.55/4.73 => ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y @ Z2 ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_251_dvd__diff,axiom,
% 8.55/4.73 ! [X: int,Y: int,Z2: int] :
% 8.55/4.73 ( ( dvd_dvd_int @ X @ Y )
% 8.55/4.73 => ( ( dvd_dvd_int @ X @ Z2 )
% 8.55/4.73 => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z2 ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_252_zero__induct__lemma,axiom,
% 8.55/4.73 ! [P: nat > $o,K: nat,I: nat] :
% 8.55/4.73 ( ( P @ K )
% 8.55/4.73 => ( ! [N2: nat] :
% 8.55/4.73 ( ( P @ ( suc @ N2 ) )
% 8.55/4.73 => ( P @ N2 ) )
% 8.55/4.73 => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_253_diffs0__imp__equal,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( ( minus_minus_nat @ M @ N )
% 8.55/4.73 = zero_zero_nat )
% 8.55/4.73 => ( ( ( minus_minus_nat @ N @ M )
% 8.55/4.73 = zero_zero_nat )
% 8.55/4.73 => ( M = N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_254_minus__nat_Odiff__0,axiom,
% 8.55/4.73 ! [M: nat] :
% 8.55/4.73 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 8.55/4.73 = M ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_255_dvd__diff__nat,axiom,
% 8.55/4.73 ! [K: nat,M: nat,N: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ K @ M )
% 8.55/4.73 => ( ( dvd_dvd_nat @ K @ N )
% 8.55/4.73 => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_256_int__ops_I1_J,axiom,
% 8.55/4.73 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 8.55/4.73 = zero_zero_int ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_257_zero__reorient,axiom,
% 8.55/4.73 ! [X: literal] :
% 8.55/4.73 ( ( zero_zero_literal = X )
% 8.55/4.73 = ( X = zero_zero_literal ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_258_zero__reorient,axiom,
% 8.55/4.73 ! [X: real] :
% 8.55/4.73 ( ( zero_zero_real = X )
% 8.55/4.73 = ( X = zero_zero_real ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_259_zero__reorient,axiom,
% 8.55/4.73 ! [X: rat] :
% 8.55/4.73 ( ( zero_zero_rat = X )
% 8.55/4.73 = ( X = zero_zero_rat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_260_zero__reorient,axiom,
% 8.55/4.73 ! [X: nat] :
% 8.55/4.73 ( ( zero_zero_nat = X )
% 8.55/4.73 = ( X = zero_zero_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_261_zero__reorient,axiom,
% 8.55/4.73 ! [X: int] :
% 8.55/4.73 ( ( zero_zero_int = X )
% 8.55/4.73 = ( X = zero_zero_int ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_262_dvd__field__iff,axiom,
% 8.55/4.73 ( dvd_dvd_real
% 8.55/4.73 = ( ^ [A2: real,B2: real] :
% 8.55/4.73 ( ( A2 = zero_zero_real )
% 8.55/4.73 => ( B2 = zero_zero_real ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_263_dvd__field__iff,axiom,
% 8.55/4.73 ( dvd_dvd_rat
% 8.55/4.73 = ( ^ [A2: rat,B2: rat] :
% 8.55/4.73 ( ( A2 = zero_zero_rat )
% 8.55/4.73 => ( B2 = zero_zero_rat ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_264_unset__bit__0,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 8.55/4.73 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_265_unset__bit__0,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 8.55/4.73 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_266_gcd__nat_Oextremum,axiom,
% 8.55/4.73 ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_267_gcd__nat_Oextremum__strict,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 8.55/4.73 & ( zero_zero_nat != A ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_268_gcd__nat_Oextremum__unique,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 8.55/4.73 = ( A = zero_zero_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_269_gcd__nat_Onot__eq__extremum,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( A != zero_zero_nat )
% 8.55/4.73 = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
% 8.55/4.73 & ( A != zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_270_gcd__nat_Oextremum__uniqueI,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 8.55/4.73 => ( A = zero_zero_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_271_exists__least__lemma,axiom,
% 8.55/4.73 ! [P: nat > $o] :
% 8.55/4.73 ( ~ ( P @ zero_zero_nat )
% 8.55/4.73 => ( ? [X_1: nat] : ( P @ X_1 )
% 8.55/4.73 => ? [N2: nat] :
% 8.55/4.73 ( ~ ( P @ N2 )
% 8.55/4.73 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_272_one__div__two__eq__zero,axiom,
% 8.55/4.73 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.73 = zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_273_one__div__two__eq__zero,axiom,
% 8.55/4.73 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.55/4.73 = zero_zero_int ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_274_bits__1__div__2,axiom,
% 8.55/4.73 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.73 = zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_275_bits__1__div__2,axiom,
% 8.55/4.73 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.55/4.73 = zero_zero_int ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_276_even__push__bit__iff,axiom,
% 8.55/4.73 ! [N: nat,A: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 8.55/4.73 = ( ( N != zero_zero_nat )
% 8.55/4.73 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_277_even__push__bit__iff,axiom,
% 8.55/4.73 ! [N: nat,A: int] :
% 8.55/4.73 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se545348938243370406it_int @ N @ A ) )
% 8.55/4.73 = ( ( N != zero_zero_nat )
% 8.55/4.73 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_278_even__push__bit__iff,axiom,
% 8.55/4.73 ! [N: nat,A: code_integer] :
% 8.55/4.73 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se7788150548672797655nteger @ N @ A ) )
% 8.55/4.73 = ( ( N != zero_zero_nat )
% 8.55/4.73 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_279_even__push__bit__iff,axiom,
% 8.55/4.73 ! [N: nat,A: code_natural] :
% 8.55/4.73 ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se6611745700429515170atural @ N @ A ) )
% 8.55/4.73 = ( ( N != zero_zero_nat )
% 8.55/4.73 | ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_280_semiring__norm_I13_J,axiom,
% 8.55/4.73 ! [M: num,N: num] :
% 8.55/4.73 ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.55/4.73 = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_281_semiring__norm_I12_J,axiom,
% 8.55/4.73 ! [N: num] :
% 8.55/4.73 ( ( times_times_num @ one @ N )
% 8.55/4.73 = N ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_282_semiring__norm_I11_J,axiom,
% 8.55/4.73 ! [M: num] :
% 8.55/4.73 ( ( times_times_num @ M @ one )
% 8.55/4.73 = M ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_283_mult__is__0,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( ( times_times_nat @ M @ N )
% 8.55/4.73 = zero_zero_nat )
% 8.55/4.73 = ( ( M = zero_zero_nat )
% 8.55/4.73 | ( N = zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_284_mult__0__right,axiom,
% 8.55/4.73 ! [M: nat] :
% 8.55/4.73 ( ( times_times_nat @ M @ zero_zero_nat )
% 8.55/4.73 = zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_285_mult__cancel1,axiom,
% 8.55/4.73 ! [K: nat,M: nat,N: nat] :
% 8.55/4.73 ( ( ( times_times_nat @ K @ M )
% 8.55/4.73 = ( times_times_nat @ K @ N ) )
% 8.55/4.73 = ( ( M = N )
% 8.55/4.73 | ( K = zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_286_mult__cancel2,axiom,
% 8.55/4.73 ! [M: nat,K: nat,N: nat] :
% 8.55/4.73 ( ( ( times_times_nat @ M @ K )
% 8.55/4.73 = ( times_times_nat @ N @ K ) )
% 8.55/4.73 = ( ( M = N )
% 8.55/4.73 | ( K = zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_287_nat__dvd__1__iff__1,axiom,
% 8.55/4.73 ! [M: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ M @ one_one_nat )
% 8.55/4.73 = ( M = one_one_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_288_mult__cancel__right,axiom,
% 8.55/4.73 ! [A: real,C: real,B: real] :
% 8.55/4.73 ( ( ( times_times_real @ A @ C )
% 8.55/4.73 = ( times_times_real @ B @ C ) )
% 8.55/4.73 = ( ( C = zero_zero_real )
% 8.55/4.73 | ( A = B ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_289_mult__cancel__right,axiom,
% 8.55/4.73 ! [A: rat,C: rat,B: rat] :
% 8.55/4.73 ( ( ( times_times_rat @ A @ C )
% 8.55/4.73 = ( times_times_rat @ B @ C ) )
% 8.55/4.73 = ( ( C = zero_zero_rat )
% 8.55/4.73 | ( A = B ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_290_mult__cancel__right,axiom,
% 8.55/4.73 ! [A: nat,C: nat,B: nat] :
% 8.55/4.73 ( ( ( times_times_nat @ A @ C )
% 8.55/4.73 = ( times_times_nat @ B @ C ) )
% 8.55/4.73 = ( ( C = zero_zero_nat )
% 8.55/4.73 | ( A = B ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_291_mult__cancel__right,axiom,
% 8.55/4.73 ! [A: int,C: int,B: int] :
% 8.55/4.73 ( ( ( times_times_int @ A @ C )
% 8.55/4.73 = ( times_times_int @ B @ C ) )
% 8.55/4.73 = ( ( C = zero_zero_int )
% 8.55/4.73 | ( A = B ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_292_mult__cancel__left,axiom,
% 8.55/4.73 ! [C: real,A: real,B: real] :
% 8.55/4.73 ( ( ( times_times_real @ C @ A )
% 8.55/4.73 = ( times_times_real @ C @ B ) )
% 8.55/4.73 = ( ( C = zero_zero_real )
% 8.55/4.73 | ( A = B ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_293_mult__cancel__left,axiom,
% 8.55/4.73 ! [C: rat,A: rat,B: rat] :
% 8.55/4.73 ( ( ( times_times_rat @ C @ A )
% 8.55/4.73 = ( times_times_rat @ C @ B ) )
% 8.55/4.73 = ( ( C = zero_zero_rat )
% 8.55/4.73 | ( A = B ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_294_mult__cancel__left,axiom,
% 8.55/4.73 ! [C: nat,A: nat,B: nat] :
% 8.55/4.73 ( ( ( times_times_nat @ C @ A )
% 8.55/4.73 = ( times_times_nat @ C @ B ) )
% 8.55/4.73 = ( ( C = zero_zero_nat )
% 8.55/4.73 | ( A = B ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_295_mult__cancel__left,axiom,
% 8.55/4.73 ! [C: int,A: int,B: int] :
% 8.55/4.73 ( ( ( times_times_int @ C @ A )
% 8.55/4.73 = ( times_times_int @ C @ B ) )
% 8.55/4.73 = ( ( C = zero_zero_int )
% 8.55/4.73 | ( A = B ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_296_mult__eq__0__iff,axiom,
% 8.55/4.73 ! [A: real,B: real] :
% 8.55/4.73 ( ( ( times_times_real @ A @ B )
% 8.55/4.73 = zero_zero_real )
% 8.55/4.73 = ( ( A = zero_zero_real )
% 8.55/4.73 | ( B = zero_zero_real ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_297_mult__eq__0__iff,axiom,
% 8.55/4.73 ! [A: rat,B: rat] :
% 8.55/4.73 ( ( ( times_times_rat @ A @ B )
% 8.55/4.73 = zero_zero_rat )
% 8.55/4.73 = ( ( A = zero_zero_rat )
% 8.55/4.73 | ( B = zero_zero_rat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_298_mult__eq__0__iff,axiom,
% 8.55/4.73 ! [A: nat,B: nat] :
% 8.55/4.73 ( ( ( times_times_nat @ A @ B )
% 8.55/4.73 = zero_zero_nat )
% 8.55/4.73 = ( ( A = zero_zero_nat )
% 8.55/4.73 | ( B = zero_zero_nat ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_299_mult__eq__0__iff,axiom,
% 8.55/4.73 ! [A: int,B: int] :
% 8.55/4.73 ( ( ( times_times_int @ A @ B )
% 8.55/4.73 = zero_zero_int )
% 8.55/4.73 = ( ( A = zero_zero_int )
% 8.55/4.73 | ( B = zero_zero_int ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_300_mult__zero__right,axiom,
% 8.55/4.73 ! [A: real] :
% 8.55/4.73 ( ( times_times_real @ A @ zero_zero_real )
% 8.55/4.73 = zero_zero_real ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_301_mult__zero__right,axiom,
% 8.55/4.73 ! [A: rat] :
% 8.55/4.73 ( ( times_times_rat @ A @ zero_zero_rat )
% 8.55/4.73 = zero_zero_rat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_302_mult__zero__right,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( times_times_nat @ A @ zero_zero_nat )
% 8.55/4.73 = zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_303_mult__zero__right,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( times_times_int @ A @ zero_zero_int )
% 8.55/4.73 = zero_zero_int ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_304_mult__zero__left,axiom,
% 8.55/4.73 ! [A: real] :
% 8.55/4.73 ( ( times_times_real @ zero_zero_real @ A )
% 8.55/4.73 = zero_zero_real ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_305_mult__zero__left,axiom,
% 8.55/4.73 ! [A: rat] :
% 8.55/4.73 ( ( times_times_rat @ zero_zero_rat @ A )
% 8.55/4.73 = zero_zero_rat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_306_mult__zero__left,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( times_times_nat @ zero_zero_nat @ A )
% 8.55/4.73 = zero_zero_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_307_mult__zero__left,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( times_times_int @ zero_zero_int @ A )
% 8.55/4.73 = zero_zero_int ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_308_mult__numeral__left__semiring__numeral,axiom,
% 8.55/4.73 ! [V: num,W: num,Z2: complex] :
% 8.55/4.73 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z2 ) )
% 8.55/4.73 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_309_mult__numeral__left__semiring__numeral,axiom,
% 8.55/4.73 ! [V: num,W: num,Z2: real] :
% 8.55/4.73 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
% 8.55/4.73 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_310_mult__numeral__left__semiring__numeral,axiom,
% 8.55/4.73 ! [V: num,W: num,Z2: rat] :
% 8.55/4.73 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z2 ) )
% 8.55/4.73 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_311_mult__numeral__left__semiring__numeral,axiom,
% 8.55/4.73 ! [V: num,W: num,Z2: nat] :
% 8.55/4.73 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
% 8.55/4.73 = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_312_mult__numeral__left__semiring__numeral,axiom,
% 8.55/4.73 ! [V: num,W: num,Z2: int] :
% 8.55/4.73 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
% 8.55/4.73 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_313_numeral__times__numeral,axiom,
% 8.55/4.73 ! [M: num,N: num] :
% 8.55/4.73 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 8.55/4.73 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_314_numeral__times__numeral,axiom,
% 8.55/4.73 ! [M: num,N: num] :
% 8.55/4.73 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 8.55/4.73 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_315_numeral__times__numeral,axiom,
% 8.55/4.73 ! [M: num,N: num] :
% 8.55/4.73 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 8.55/4.73 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_316_numeral__times__numeral,axiom,
% 8.55/4.73 ! [M: num,N: num] :
% 8.55/4.73 ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.55/4.73 = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_317_numeral__times__numeral,axiom,
% 8.55/4.73 ! [M: num,N: num] :
% 8.55/4.73 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.55/4.73 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_318_mult_Oright__neutral,axiom,
% 8.55/4.73 ! [A: complex] :
% 8.55/4.73 ( ( times_times_complex @ A @ one_one_complex )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_319_mult_Oright__neutral,axiom,
% 8.55/4.73 ! [A: real] :
% 8.55/4.73 ( ( times_times_real @ A @ one_one_real )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_320_mult_Oright__neutral,axiom,
% 8.55/4.73 ! [A: rat] :
% 8.55/4.73 ( ( times_times_rat @ A @ one_one_rat )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_321_mult_Oright__neutral,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( times_times_nat @ A @ one_one_nat )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_322_mult_Oright__neutral,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( times_times_int @ A @ one_one_int )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_323_mult__1,axiom,
% 8.55/4.73 ! [A: complex] :
% 8.55/4.73 ( ( times_times_complex @ one_one_complex @ A )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_324_mult__1,axiom,
% 8.55/4.73 ! [A: real] :
% 8.55/4.73 ( ( times_times_real @ one_one_real @ A )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_325_mult__1,axiom,
% 8.55/4.73 ! [A: rat] :
% 8.55/4.73 ( ( times_times_rat @ one_one_rat @ A )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_326_mult__1,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( times_times_nat @ one_one_nat @ A )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_327_mult__1,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( times_times_int @ one_one_int @ A )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_328_num__double,axiom,
% 8.55/4.73 ! [N: num] :
% 8.55/4.73 ( ( times_times_num @ ( bit0 @ one ) @ N )
% 8.55/4.73 = ( bit0 @ N ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_329_times__divide__eq__left,axiom,
% 8.55/4.73 ! [B: complex,C: complex,A: complex] :
% 8.55/4.73 ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
% 8.55/4.73 = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_330_times__divide__eq__left,axiom,
% 8.55/4.73 ! [B: real,C: real,A: real] :
% 8.55/4.73 ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
% 8.55/4.73 = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_331_times__divide__eq__left,axiom,
% 8.55/4.73 ! [B: rat,C: rat,A: rat] :
% 8.55/4.73 ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 8.55/4.73 = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_332_divide__divide__eq__left,axiom,
% 8.55/4.73 ! [A: complex,B: complex,C: complex] :
% 8.55/4.73 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 8.55/4.73 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_333_divide__divide__eq__left,axiom,
% 8.55/4.73 ! [A: real,B: real,C: real] :
% 8.55/4.73 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 8.55/4.73 = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_334_divide__divide__eq__left,axiom,
% 8.55/4.73 ! [A: rat,B: rat,C: rat] :
% 8.55/4.73 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 8.55/4.73 = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_335_divide__divide__eq__right,axiom,
% 8.55/4.73 ! [A: complex,B: complex,C: complex] :
% 8.55/4.73 ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 8.55/4.73 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_336_divide__divide__eq__right,axiom,
% 8.55/4.73 ! [A: real,B: real,C: real] :
% 8.55/4.73 ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
% 8.55/4.73 = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_337_divide__divide__eq__right,axiom,
% 8.55/4.73 ! [A: rat,B: rat,C: rat] :
% 8.55/4.73 ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 8.55/4.73 = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_338_times__divide__eq__right,axiom,
% 8.55/4.73 ! [A: complex,B: complex,C: complex] :
% 8.55/4.73 ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 8.55/4.73 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_339_times__divide__eq__right,axiom,
% 8.55/4.73 ! [A: real,B: real,C: real] :
% 8.55/4.73 ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
% 8.55/4.73 = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_340_times__divide__eq__right,axiom,
% 8.55/4.73 ! [A: rat,B: rat,C: rat] :
% 8.55/4.73 ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 8.55/4.73 = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_341_div__by__1,axiom,
% 8.55/4.73 ! [A: complex] :
% 8.55/4.73 ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_342_div__by__1,axiom,
% 8.55/4.73 ! [A: real] :
% 8.55/4.73 ( ( divide_divide_real @ A @ one_one_real )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_343_div__by__1,axiom,
% 8.55/4.73 ! [A: rat] :
% 8.55/4.73 ( ( divide_divide_rat @ A @ one_one_rat )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_344_div__by__1,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( divide_divide_nat @ A @ one_one_nat )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_345_div__by__1,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( divide_divide_int @ A @ one_one_int )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_346_bits__div__by__1,axiom,
% 8.55/4.73 ! [A: nat] :
% 8.55/4.73 ( ( divide_divide_nat @ A @ one_one_nat )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_347_bits__div__by__1,axiom,
% 8.55/4.73 ! [A: int] :
% 8.55/4.73 ( ( divide_divide_int @ A @ one_one_int )
% 8.55/4.73 = A ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_348_of__nat__mult,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
% 8.55/4.73 = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_349_of__nat__mult,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 8.55/4.73 = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_350_of__nat__mult,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 8.55/4.73 = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_351_of__nat__mult,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 8.55/4.73 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_352_of__nat__mult,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 8.55/4.73 = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_353_of__nat__1,axiom,
% 8.55/4.73 ( ( semiri8010041392384452111omplex @ one_one_nat )
% 8.55/4.73 = one_one_complex ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_354_of__nat__1,axiom,
% 8.55/4.73 ( ( semiri5074537144036343181t_real @ one_one_nat )
% 8.55/4.73 = one_one_real ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_355_of__nat__1,axiom,
% 8.55/4.73 ( ( semiri681578069525770553at_rat @ one_one_nat )
% 8.55/4.73 = one_one_rat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_356_of__nat__1,axiom,
% 8.55/4.73 ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 8.55/4.73 = one_one_nat ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_357_of__nat__1,axiom,
% 8.55/4.73 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 8.55/4.73 = one_one_int ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_358_of__nat__1__eq__iff,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( one_one_complex
% 8.55/4.73 = ( semiri8010041392384452111omplex @ N ) )
% 8.55/4.73 = ( N = one_one_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_359_of__nat__1__eq__iff,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( one_one_real
% 8.55/4.73 = ( semiri5074537144036343181t_real @ N ) )
% 8.55/4.73 = ( N = one_one_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_360_of__nat__1__eq__iff,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( one_one_rat
% 8.55/4.73 = ( semiri681578069525770553at_rat @ N ) )
% 8.55/4.73 = ( N = one_one_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_361_of__nat__1__eq__iff,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( one_one_nat
% 8.55/4.73 = ( semiri1316708129612266289at_nat @ N ) )
% 8.55/4.73 = ( N = one_one_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_362_of__nat__1__eq__iff,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( one_one_int
% 8.55/4.73 = ( semiri1314217659103216013at_int @ N ) )
% 8.55/4.73 = ( N = one_one_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_363_of__nat__eq__1__iff,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( ( semiri8010041392384452111omplex @ N )
% 8.55/4.73 = one_one_complex )
% 8.55/4.73 = ( N = one_one_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_364_of__nat__eq__1__iff,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( ( semiri5074537144036343181t_real @ N )
% 8.55/4.73 = one_one_real )
% 8.55/4.73 = ( N = one_one_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_365_of__nat__eq__1__iff,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( ( semiri681578069525770553at_rat @ N )
% 8.55/4.73 = one_one_rat )
% 8.55/4.73 = ( N = one_one_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_366_of__nat__eq__1__iff,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( ( semiri1316708129612266289at_nat @ N )
% 8.55/4.73 = one_one_nat )
% 8.55/4.73 = ( N = one_one_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_367_of__nat__eq__1__iff,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( ( semiri1314217659103216013at_int @ N )
% 8.55/4.73 = one_one_int )
% 8.55/4.73 = ( N = one_one_nat ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_368_one__eq__mult__iff,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( ( suc @ zero_zero_nat )
% 8.55/4.73 = ( times_times_nat @ M @ N ) )
% 8.55/4.73 = ( ( M
% 8.55/4.73 = ( suc @ zero_zero_nat ) )
% 8.55/4.73 & ( N
% 8.55/4.73 = ( suc @ zero_zero_nat ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_369_mult__eq__1__iff,axiom,
% 8.55/4.73 ! [M: nat,N: nat] :
% 8.55/4.73 ( ( ( times_times_nat @ M @ N )
% 8.55/4.73 = ( suc @ zero_zero_nat ) )
% 8.55/4.73 = ( ( M
% 8.55/4.73 = ( suc @ zero_zero_nat ) )
% 8.55/4.73 & ( N
% 8.55/4.73 = ( suc @ zero_zero_nat ) ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_370_diff__Suc__1,axiom,
% 8.55/4.73 ! [N: nat] :
% 8.55/4.73 ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 8.55/4.73 = N ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_371_nat__mult__dvd__cancel__disj,axiom,
% 8.55/4.73 ! [K: nat,M: nat,N: nat] :
% 8.55/4.73 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.55/4.73 = ( ( K = zero_zero_nat )
% 8.55/4.73 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 8.55/4.73
% 8.55/4.73 thf(fact_372_nat__mult__div__cancel__disj,axiom,
% 8.55/4.73 ! [K: nat,M: nat,N: nat] :
% 8.55/4.73 ( ( ( K = zero_zero_nat )
% 8.55/4.73 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.55/4.74 = zero_zero_nat ) )
% 8.55/4.74 & ( ( K != zero_zero_nat )
% 8.55/4.74 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.55/4.74 = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_373_push__bit__eq__0__iff,axiom,
% 8.55/4.74 ! [N: nat,A: nat] :
% 8.55/4.74 ( ( ( bit_se547839408752420682it_nat @ N @ A )
% 8.55/4.74 = zero_zero_nat )
% 8.55/4.74 = ( A = zero_zero_nat ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_374_push__bit__eq__0__iff,axiom,
% 8.55/4.74 ! [N: nat,A: int] :
% 8.55/4.74 ( ( ( bit_se545348938243370406it_int @ N @ A )
% 8.55/4.74 = zero_zero_int )
% 8.55/4.74 = ( A = zero_zero_int ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_375_push__bit__eq__0__iff,axiom,
% 8.55/4.74 ! [N: nat,A: code_integer] :
% 8.55/4.74 ( ( ( bit_se7788150548672797655nteger @ N @ A )
% 8.55/4.74 = zero_z3403309356797280102nteger )
% 8.55/4.74 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_376_push__bit__eq__0__iff,axiom,
% 8.55/4.74 ! [N: nat,A: code_natural] :
% 8.55/4.74 ( ( ( bit_se6611745700429515170atural @ N @ A )
% 8.55/4.74 = zero_z2226904508553997617atural )
% 8.55/4.74 = ( A = zero_z2226904508553997617atural ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_377_push__bit__of__0,axiom,
% 8.55/4.74 ! [N: nat] :
% 8.55/4.74 ( ( bit_se547839408752420682it_nat @ N @ zero_zero_nat )
% 8.55/4.74 = zero_zero_nat ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_378_push__bit__of__0,axiom,
% 8.55/4.74 ! [N: nat] :
% 8.55/4.74 ( ( bit_se545348938243370406it_int @ N @ zero_zero_int )
% 8.55/4.74 = zero_zero_int ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_379_push__bit__of__0,axiom,
% 8.55/4.74 ! [N: nat] :
% 8.55/4.74 ( ( bit_se7788150548672797655nteger @ N @ zero_z3403309356797280102nteger )
% 8.55/4.74 = zero_z3403309356797280102nteger ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_380_push__bit__of__0,axiom,
% 8.55/4.74 ! [N: nat] :
% 8.55/4.74 ( ( bit_se6611745700429515170atural @ N @ zero_z2226904508553997617atural )
% 8.55/4.74 = zero_z2226904508553997617atural ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_381_mult__cancel__right2,axiom,
% 8.55/4.74 ! [A: complex,C: complex] :
% 8.55/4.74 ( ( ( times_times_complex @ A @ C )
% 8.55/4.74 = C )
% 8.55/4.74 = ( ( C = zero_zero_complex )
% 8.55/4.74 | ( A = one_one_complex ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_382_mult__cancel__right2,axiom,
% 8.55/4.74 ! [A: real,C: real] :
% 8.55/4.74 ( ( ( times_times_real @ A @ C )
% 8.55/4.74 = C )
% 8.55/4.74 = ( ( C = zero_zero_real )
% 8.55/4.74 | ( A = one_one_real ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_383_mult__cancel__right2,axiom,
% 8.55/4.74 ! [A: rat,C: rat] :
% 8.55/4.74 ( ( ( times_times_rat @ A @ C )
% 8.55/4.74 = C )
% 8.55/4.74 = ( ( C = zero_zero_rat )
% 8.55/4.74 | ( A = one_one_rat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_384_mult__cancel__right2,axiom,
% 8.55/4.74 ! [A: int,C: int] :
% 8.55/4.74 ( ( ( times_times_int @ A @ C )
% 8.55/4.74 = C )
% 8.55/4.74 = ( ( C = zero_zero_int )
% 8.55/4.74 | ( A = one_one_int ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_385_mult__cancel__right1,axiom,
% 8.55/4.74 ! [C: complex,B: complex] :
% 8.55/4.74 ( ( C
% 8.55/4.74 = ( times_times_complex @ B @ C ) )
% 8.55/4.74 = ( ( C = zero_zero_complex )
% 8.55/4.74 | ( B = one_one_complex ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_386_mult__cancel__right1,axiom,
% 8.55/4.74 ! [C: real,B: real] :
% 8.55/4.74 ( ( C
% 8.55/4.74 = ( times_times_real @ B @ C ) )
% 8.55/4.74 = ( ( C = zero_zero_real )
% 8.55/4.74 | ( B = one_one_real ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_387_mult__cancel__right1,axiom,
% 8.55/4.74 ! [C: rat,B: rat] :
% 8.55/4.74 ( ( C
% 8.55/4.74 = ( times_times_rat @ B @ C ) )
% 8.55/4.74 = ( ( C = zero_zero_rat )
% 8.55/4.74 | ( B = one_one_rat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_388_mult__cancel__right1,axiom,
% 8.55/4.74 ! [C: int,B: int] :
% 8.55/4.74 ( ( C
% 8.55/4.74 = ( times_times_int @ B @ C ) )
% 8.55/4.74 = ( ( C = zero_zero_int )
% 8.55/4.74 | ( B = one_one_int ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_389_mult__cancel__left2,axiom,
% 8.55/4.74 ! [C: complex,A: complex] :
% 8.55/4.74 ( ( ( times_times_complex @ C @ A )
% 8.55/4.74 = C )
% 8.55/4.74 = ( ( C = zero_zero_complex )
% 8.55/4.74 | ( A = one_one_complex ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_390_mult__cancel__left2,axiom,
% 8.55/4.74 ! [C: real,A: real] :
% 8.55/4.74 ( ( ( times_times_real @ C @ A )
% 8.55/4.74 = C )
% 8.55/4.74 = ( ( C = zero_zero_real )
% 8.55/4.74 | ( A = one_one_real ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_391_mult__cancel__left2,axiom,
% 8.55/4.74 ! [C: rat,A: rat] :
% 8.55/4.74 ( ( ( times_times_rat @ C @ A )
% 8.55/4.74 = C )
% 8.55/4.74 = ( ( C = zero_zero_rat )
% 8.55/4.74 | ( A = one_one_rat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_392_mult__cancel__left2,axiom,
% 8.55/4.74 ! [C: int,A: int] :
% 8.55/4.74 ( ( ( times_times_int @ C @ A )
% 8.55/4.74 = C )
% 8.55/4.74 = ( ( C = zero_zero_int )
% 8.55/4.74 | ( A = one_one_int ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_393_mult__cancel__left1,axiom,
% 8.55/4.74 ! [C: complex,B: complex] :
% 8.55/4.74 ( ( C
% 8.55/4.74 = ( times_times_complex @ C @ B ) )
% 8.55/4.74 = ( ( C = zero_zero_complex )
% 8.55/4.74 | ( B = one_one_complex ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_394_mult__cancel__left1,axiom,
% 8.55/4.74 ! [C: real,B: real] :
% 8.55/4.74 ( ( C
% 8.55/4.74 = ( times_times_real @ C @ B ) )
% 8.55/4.74 = ( ( C = zero_zero_real )
% 8.55/4.74 | ( B = one_one_real ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_395_mult__cancel__left1,axiom,
% 8.55/4.74 ! [C: rat,B: rat] :
% 8.55/4.74 ( ( C
% 8.55/4.74 = ( times_times_rat @ C @ B ) )
% 8.55/4.74 = ( ( C = zero_zero_rat )
% 8.55/4.74 | ( B = one_one_rat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_396_mult__cancel__left1,axiom,
% 8.55/4.74 ! [C: int,B: int] :
% 8.55/4.74 ( ( C
% 8.55/4.74 = ( times_times_int @ C @ B ) )
% 8.55/4.74 = ( ( C = zero_zero_int )
% 8.55/4.74 | ( B = one_one_int ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_397_diff__numeral__special_I9_J,axiom,
% 8.55/4.74 ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 8.55/4.74 = zero_zero_complex ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_398_diff__numeral__special_I9_J,axiom,
% 8.55/4.74 ( ( minus_minus_real @ one_one_real @ one_one_real )
% 8.55/4.74 = zero_zero_real ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_399_diff__numeral__special_I9_J,axiom,
% 8.55/4.74 ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 8.55/4.74 = zero_zero_rat ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_400_diff__numeral__special_I9_J,axiom,
% 8.55/4.74 ( ( minus_minus_int @ one_one_int @ one_one_int )
% 8.55/4.74 = zero_zero_int ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_401_nonzero__mult__div__cancel__left,axiom,
% 8.55/4.74 ! [A: complex,B: complex] :
% 8.55/4.74 ( ( A != zero_zero_complex )
% 8.55/4.74 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
% 8.55/4.74 = B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_402_nonzero__mult__div__cancel__left,axiom,
% 8.55/4.74 ! [A: real,B: real] :
% 8.55/4.74 ( ( A != zero_zero_real )
% 8.55/4.74 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
% 8.55/4.74 = B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_403_nonzero__mult__div__cancel__left,axiom,
% 8.55/4.74 ! [A: rat,B: rat] :
% 8.55/4.74 ( ( A != zero_zero_rat )
% 8.55/4.74 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
% 8.55/4.74 = B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_404_nonzero__mult__div__cancel__left,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( A != zero_zero_nat )
% 8.55/4.74 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 8.55/4.74 = B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_405_nonzero__mult__div__cancel__left,axiom,
% 8.55/4.74 ! [A: int,B: int] :
% 8.55/4.74 ( ( A != zero_zero_int )
% 8.55/4.74 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
% 8.55/4.74 = B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_406_nonzero__mult__div__cancel__right,axiom,
% 8.55/4.74 ! [B: complex,A: complex] :
% 8.55/4.74 ( ( B != zero_zero_complex )
% 8.55/4.74 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
% 8.55/4.74 = A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_407_nonzero__mult__div__cancel__right,axiom,
% 8.55/4.74 ! [B: real,A: real] :
% 8.55/4.74 ( ( B != zero_zero_real )
% 8.55/4.74 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
% 8.55/4.74 = A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_408_nonzero__mult__div__cancel__right,axiom,
% 8.55/4.74 ! [B: rat,A: rat] :
% 8.55/4.74 ( ( B != zero_zero_rat )
% 8.55/4.74 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
% 8.55/4.74 = A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_409_nonzero__mult__div__cancel__right,axiom,
% 8.55/4.74 ! [B: nat,A: nat] :
% 8.55/4.74 ( ( B != zero_zero_nat )
% 8.55/4.74 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 8.55/4.74 = A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_410_nonzero__mult__div__cancel__right,axiom,
% 8.55/4.74 ! [B: int,A: int] :
% 8.55/4.74 ( ( B != zero_zero_int )
% 8.55/4.74 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
% 8.55/4.74 = A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_411_div__mult__mult1__if,axiom,
% 8.55/4.74 ! [C: nat,A: nat,B: nat] :
% 8.55/4.74 ( ( ( C = zero_zero_nat )
% 8.55/4.74 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 8.55/4.74 = zero_zero_nat ) )
% 8.55/4.74 & ( ( C != zero_zero_nat )
% 8.55/4.74 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 8.55/4.74 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_412_div__mult__mult1__if,axiom,
% 8.55/4.74 ! [C: int,A: int,B: int] :
% 8.55/4.74 ( ( ( C = zero_zero_int )
% 8.55/4.74 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.55/4.74 = zero_zero_int ) )
% 8.55/4.74 & ( ( C != zero_zero_int )
% 8.55/4.74 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.55/4.74 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_413_div__mult__mult2,axiom,
% 8.55/4.74 ! [C: nat,A: nat,B: nat] :
% 8.55/4.74 ( ( C != zero_zero_nat )
% 8.55/4.74 => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 8.55/4.74 = ( divide_divide_nat @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_414_div__mult__mult2,axiom,
% 8.55/4.74 ! [C: int,A: int,B: int] :
% 8.55/4.74 ( ( C != zero_zero_int )
% 8.55/4.74 => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 8.55/4.74 = ( divide_divide_int @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_415_div__mult__mult1,axiom,
% 8.55/4.74 ! [C: nat,A: nat,B: nat] :
% 8.55/4.74 ( ( C != zero_zero_nat )
% 8.55/4.74 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 8.55/4.74 = ( divide_divide_nat @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_416_div__mult__mult1,axiom,
% 8.55/4.74 ! [C: int,A: int,B: int] :
% 8.55/4.74 ( ( C != zero_zero_int )
% 8.55/4.74 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.55/4.74 = ( divide_divide_int @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_417_mult__divide__mult__cancel__left__if,axiom,
% 8.55/4.74 ! [C: complex,A: complex,B: complex] :
% 8.55/4.74 ( ( ( C = zero_zero_complex )
% 8.55/4.74 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 8.55/4.74 = zero_zero_complex ) )
% 8.55/4.74 & ( ( C != zero_zero_complex )
% 8.55/4.74 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 8.55/4.74 = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_418_mult__divide__mult__cancel__left__if,axiom,
% 8.55/4.74 ! [C: real,A: real,B: real] :
% 8.55/4.74 ( ( ( C = zero_zero_real )
% 8.55/4.74 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.55/4.74 = zero_zero_real ) )
% 8.55/4.74 & ( ( C != zero_zero_real )
% 8.55/4.74 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.55/4.74 = ( divide_divide_real @ A @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_419_mult__divide__mult__cancel__left__if,axiom,
% 8.55/4.74 ! [C: rat,A: rat,B: rat] :
% 8.55/4.74 ( ( ( C = zero_zero_rat )
% 8.55/4.74 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.55/4.74 = zero_zero_rat ) )
% 8.55/4.74 & ( ( C != zero_zero_rat )
% 8.55/4.74 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.55/4.74 = ( divide_divide_rat @ A @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_420_nonzero__mult__divide__mult__cancel__left,axiom,
% 8.55/4.74 ! [C: complex,A: complex,B: complex] :
% 8.55/4.74 ( ( C != zero_zero_complex )
% 8.55/4.74 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 8.55/4.74 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_421_nonzero__mult__divide__mult__cancel__left,axiom,
% 8.55/4.74 ! [C: real,A: real,B: real] :
% 8.55/4.74 ( ( C != zero_zero_real )
% 8.55/4.74 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.55/4.74 = ( divide_divide_real @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_422_nonzero__mult__divide__mult__cancel__left,axiom,
% 8.55/4.74 ! [C: rat,A: rat,B: rat] :
% 8.55/4.74 ( ( C != zero_zero_rat )
% 8.55/4.74 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.55/4.74 = ( divide_divide_rat @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_423_nonzero__mult__divide__mult__cancel__left2,axiom,
% 8.55/4.74 ! [C: complex,A: complex,B: complex] :
% 8.55/4.74 ( ( C != zero_zero_complex )
% 8.55/4.74 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
% 8.55/4.74 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_424_nonzero__mult__divide__mult__cancel__left2,axiom,
% 8.55/4.74 ! [C: real,A: real,B: real] :
% 8.55/4.74 ( ( C != zero_zero_real )
% 8.55/4.74 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
% 8.55/4.74 = ( divide_divide_real @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_425_nonzero__mult__divide__mult__cancel__left2,axiom,
% 8.55/4.74 ! [C: rat,A: rat,B: rat] :
% 8.55/4.74 ( ( C != zero_zero_rat )
% 8.55/4.74 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
% 8.55/4.74 = ( divide_divide_rat @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_426_nonzero__mult__divide__mult__cancel__right,axiom,
% 8.55/4.74 ! [C: complex,A: complex,B: complex] :
% 8.55/4.74 ( ( C != zero_zero_complex )
% 8.55/4.74 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 8.55/4.74 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_427_nonzero__mult__divide__mult__cancel__right,axiom,
% 8.55/4.74 ! [C: real,A: real,B: real] :
% 8.55/4.74 ( ( C != zero_zero_real )
% 8.55/4.74 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 8.55/4.74 = ( divide_divide_real @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_428_nonzero__mult__divide__mult__cancel__right,axiom,
% 8.55/4.74 ! [C: rat,A: rat,B: rat] :
% 8.55/4.74 ( ( C != zero_zero_rat )
% 8.55/4.74 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 8.55/4.74 = ( divide_divide_rat @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_429_nonzero__mult__divide__mult__cancel__right2,axiom,
% 8.55/4.74 ! [C: complex,A: complex,B: complex] :
% 8.55/4.74 ( ( C != zero_zero_complex )
% 8.55/4.74 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
% 8.55/4.74 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_430_nonzero__mult__divide__mult__cancel__right2,axiom,
% 8.55/4.74 ! [C: real,A: real,B: real] :
% 8.55/4.74 ( ( C != zero_zero_real )
% 8.55/4.74 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
% 8.55/4.74 = ( divide_divide_real @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_431_nonzero__mult__divide__mult__cancel__right2,axiom,
% 8.55/4.74 ! [C: rat,A: rat,B: rat] :
% 8.55/4.74 ( ( C != zero_zero_rat )
% 8.55/4.74 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
% 8.55/4.74 = ( divide_divide_rat @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_432_div__self,axiom,
% 8.55/4.74 ! [A: complex] :
% 8.55/4.74 ( ( A != zero_zero_complex )
% 8.55/4.74 => ( ( divide1717551699836669952omplex @ A @ A )
% 8.55/4.74 = one_one_complex ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_433_div__self,axiom,
% 8.55/4.74 ! [A: real] :
% 8.55/4.74 ( ( A != zero_zero_real )
% 8.55/4.74 => ( ( divide_divide_real @ A @ A )
% 8.55/4.74 = one_one_real ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_434_div__self,axiom,
% 8.55/4.74 ! [A: rat] :
% 8.55/4.74 ( ( A != zero_zero_rat )
% 8.55/4.74 => ( ( divide_divide_rat @ A @ A )
% 8.55/4.74 = one_one_rat ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_435_div__self,axiom,
% 8.55/4.74 ! [A: nat] :
% 8.55/4.74 ( ( A != zero_zero_nat )
% 8.55/4.74 => ( ( divide_divide_nat @ A @ A )
% 8.55/4.74 = one_one_nat ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_436_div__self,axiom,
% 8.55/4.74 ! [A: int] :
% 8.55/4.74 ( ( A != zero_zero_int )
% 8.55/4.74 => ( ( divide_divide_int @ A @ A )
% 8.55/4.74 = one_one_int ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_437_divide__eq__1__iff,axiom,
% 8.55/4.74 ! [A: complex,B: complex] :
% 8.55/4.74 ( ( ( divide1717551699836669952omplex @ A @ B )
% 8.55/4.74 = one_one_complex )
% 8.55/4.74 = ( ( B != zero_zero_complex )
% 8.55/4.74 & ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_438_divide__eq__1__iff,axiom,
% 8.55/4.74 ! [A: real,B: real] :
% 8.55/4.74 ( ( ( divide_divide_real @ A @ B )
% 8.55/4.74 = one_one_real )
% 8.55/4.74 = ( ( B != zero_zero_real )
% 8.55/4.74 & ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_439_divide__eq__1__iff,axiom,
% 8.55/4.74 ! [A: rat,B: rat] :
% 8.55/4.74 ( ( ( divide_divide_rat @ A @ B )
% 8.55/4.74 = one_one_rat )
% 8.55/4.74 = ( ( B != zero_zero_rat )
% 8.55/4.74 & ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_440_one__eq__divide__iff,axiom,
% 8.55/4.74 ! [A: complex,B: complex] :
% 8.55/4.74 ( ( one_one_complex
% 8.55/4.74 = ( divide1717551699836669952omplex @ A @ B ) )
% 8.55/4.74 = ( ( B != zero_zero_complex )
% 8.55/4.74 & ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_441_one__eq__divide__iff,axiom,
% 8.55/4.74 ! [A: real,B: real] :
% 8.55/4.74 ( ( one_one_real
% 8.55/4.74 = ( divide_divide_real @ A @ B ) )
% 8.55/4.74 = ( ( B != zero_zero_real )
% 8.55/4.74 & ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_442_one__eq__divide__iff,axiom,
% 8.55/4.74 ! [A: rat,B: rat] :
% 8.55/4.74 ( ( one_one_rat
% 8.55/4.74 = ( divide_divide_rat @ A @ B ) )
% 8.55/4.74 = ( ( B != zero_zero_rat )
% 8.55/4.74 & ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_443_divide__self,axiom,
% 8.55/4.74 ! [A: complex] :
% 8.55/4.74 ( ( A != zero_zero_complex )
% 8.55/4.74 => ( ( divide1717551699836669952omplex @ A @ A )
% 8.55/4.74 = one_one_complex ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_444_divide__self,axiom,
% 8.55/4.74 ! [A: real] :
% 8.55/4.74 ( ( A != zero_zero_real )
% 8.55/4.74 => ( ( divide_divide_real @ A @ A )
% 8.55/4.74 = one_one_real ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_445_divide__self,axiom,
% 8.55/4.74 ! [A: rat] :
% 8.55/4.74 ( ( A != zero_zero_rat )
% 8.55/4.74 => ( ( divide_divide_rat @ A @ A )
% 8.55/4.74 = one_one_rat ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_446_divide__self__if,axiom,
% 8.55/4.74 ! [A: complex] :
% 8.55/4.74 ( ( ( A = zero_zero_complex )
% 8.55/4.74 => ( ( divide1717551699836669952omplex @ A @ A )
% 8.55/4.74 = zero_zero_complex ) )
% 8.55/4.74 & ( ( A != zero_zero_complex )
% 8.55/4.74 => ( ( divide1717551699836669952omplex @ A @ A )
% 8.55/4.74 = one_one_complex ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_447_divide__self__if,axiom,
% 8.55/4.74 ! [A: real] :
% 8.55/4.74 ( ( ( A = zero_zero_real )
% 8.55/4.74 => ( ( divide_divide_real @ A @ A )
% 8.55/4.74 = zero_zero_real ) )
% 8.55/4.74 & ( ( A != zero_zero_real )
% 8.55/4.74 => ( ( divide_divide_real @ A @ A )
% 8.55/4.74 = one_one_real ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_448_divide__self__if,axiom,
% 8.55/4.74 ! [A: rat] :
% 8.55/4.74 ( ( ( A = zero_zero_rat )
% 8.55/4.74 => ( ( divide_divide_rat @ A @ A )
% 8.55/4.74 = zero_zero_rat ) )
% 8.55/4.74 & ( ( A != zero_zero_rat )
% 8.55/4.74 => ( ( divide_divide_rat @ A @ A )
% 8.55/4.74 = one_one_rat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_449_divide__eq__eq__1,axiom,
% 8.55/4.74 ! [B: real,A: real] :
% 8.55/4.74 ( ( ( divide_divide_real @ B @ A )
% 8.55/4.74 = one_one_real )
% 8.55/4.74 = ( ( A != zero_zero_real )
% 8.55/4.74 & ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_450_divide__eq__eq__1,axiom,
% 8.55/4.74 ! [B: rat,A: rat] :
% 8.55/4.74 ( ( ( divide_divide_rat @ B @ A )
% 8.55/4.74 = one_one_rat )
% 8.55/4.74 = ( ( A != zero_zero_rat )
% 8.55/4.74 & ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_451_eq__divide__eq__1,axiom,
% 8.55/4.74 ! [B: real,A: real] :
% 8.55/4.74 ( ( one_one_real
% 8.55/4.74 = ( divide_divide_real @ B @ A ) )
% 8.55/4.74 = ( ( A != zero_zero_real )
% 8.55/4.74 & ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_452_eq__divide__eq__1,axiom,
% 8.55/4.74 ! [B: rat,A: rat] :
% 8.55/4.74 ( ( one_one_rat
% 8.55/4.74 = ( divide_divide_rat @ B @ A ) )
% 8.55/4.74 = ( ( A != zero_zero_rat )
% 8.55/4.74 & ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_453_one__divide__eq__0__iff,axiom,
% 8.55/4.74 ! [A: real] :
% 8.55/4.74 ( ( ( divide_divide_real @ one_one_real @ A )
% 8.55/4.74 = zero_zero_real )
% 8.55/4.74 = ( A = zero_zero_real ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_454_one__divide__eq__0__iff,axiom,
% 8.55/4.74 ! [A: rat] :
% 8.55/4.74 ( ( ( divide_divide_rat @ one_one_rat @ A )
% 8.55/4.74 = zero_zero_rat )
% 8.55/4.74 = ( A = zero_zero_rat ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_455_zero__eq__1__divide__iff,axiom,
% 8.55/4.74 ! [A: real] :
% 8.55/4.74 ( ( zero_zero_real
% 8.55/4.74 = ( divide_divide_real @ one_one_real @ A ) )
% 8.55/4.74 = ( A = zero_zero_real ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_456_zero__eq__1__divide__iff,axiom,
% 8.55/4.74 ! [A: rat] :
% 8.55/4.74 ( ( zero_zero_rat
% 8.55/4.74 = ( divide_divide_rat @ one_one_rat @ A ) )
% 8.55/4.74 = ( A = zero_zero_rat ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_457_right__diff__distrib__numeral,axiom,
% 8.55/4.74 ! [V: num,B: complex,C: complex] :
% 8.55/4.74 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
% 8.55/4.74 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_458_right__diff__distrib__numeral,axiom,
% 8.55/4.74 ! [V: num,B: real,C: real] :
% 8.55/4.74 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
% 8.55/4.74 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_459_right__diff__distrib__numeral,axiom,
% 8.55/4.74 ! [V: num,B: rat,C: rat] :
% 8.55/4.74 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
% 8.55/4.74 = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_460_right__diff__distrib__numeral,axiom,
% 8.55/4.74 ! [V: num,B: int,C: int] :
% 8.55/4.74 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
% 8.55/4.74 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_461_left__diff__distrib__numeral,axiom,
% 8.55/4.74 ! [A: complex,B: complex,V: num] :
% 8.55/4.74 ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 8.55/4.74 = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_462_left__diff__distrib__numeral,axiom,
% 8.55/4.74 ! [A: real,B: real,V: num] :
% 8.55/4.74 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 8.55/4.74 = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_463_left__diff__distrib__numeral,axiom,
% 8.55/4.74 ! [A: rat,B: rat,V: num] :
% 8.55/4.74 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 8.55/4.74 = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_464_left__diff__distrib__numeral,axiom,
% 8.55/4.74 ! [A: int,B: int,V: num] :
% 8.55/4.74 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 8.55/4.74 = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_465_numeral__eq__one__iff,axiom,
% 8.55/4.74 ! [N: num] :
% 8.55/4.74 ( ( ( numera6690914467698888265omplex @ N )
% 8.55/4.74 = one_one_complex )
% 8.55/4.74 = ( N = one ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_466_numeral__eq__one__iff,axiom,
% 8.55/4.74 ! [N: num] :
% 8.55/4.74 ( ( ( numeral_numeral_real @ N )
% 8.55/4.74 = one_one_real )
% 8.55/4.74 = ( N = one ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_467_numeral__eq__one__iff,axiom,
% 8.55/4.74 ! [N: num] :
% 8.55/4.74 ( ( ( numeral_numeral_rat @ N )
% 8.55/4.74 = one_one_rat )
% 8.55/4.74 = ( N = one ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_468_numeral__eq__one__iff,axiom,
% 8.55/4.74 ! [N: num] :
% 8.55/4.74 ( ( ( numeral_numeral_nat @ N )
% 8.55/4.74 = one_one_nat )
% 8.55/4.74 = ( N = one ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_469_numeral__eq__one__iff,axiom,
% 8.55/4.74 ! [N: num] :
% 8.55/4.74 ( ( ( numeral_numeral_int @ N )
% 8.55/4.74 = one_one_int )
% 8.55/4.74 = ( N = one ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_470_one__eq__numeral__iff,axiom,
% 8.55/4.74 ! [N: num] :
% 8.55/4.74 ( ( one_one_complex
% 8.55/4.74 = ( numera6690914467698888265omplex @ N ) )
% 8.55/4.74 = ( one = N ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_471_one__eq__numeral__iff,axiom,
% 8.55/4.74 ! [N: num] :
% 8.55/4.74 ( ( one_one_real
% 8.55/4.74 = ( numeral_numeral_real @ N ) )
% 8.55/4.74 = ( one = N ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_472_one__eq__numeral__iff,axiom,
% 8.55/4.74 ! [N: num] :
% 8.55/4.74 ( ( one_one_rat
% 8.55/4.74 = ( numeral_numeral_rat @ N ) )
% 8.55/4.74 = ( one = N ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_473_one__eq__numeral__iff,axiom,
% 8.55/4.74 ! [N: num] :
% 8.55/4.74 ( ( one_one_nat
% 8.55/4.74 = ( numeral_numeral_nat @ N ) )
% 8.55/4.74 = ( one = N ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_474_one__eq__numeral__iff,axiom,
% 8.55/4.74 ! [N: num] :
% 8.55/4.74 ( ( one_one_int
% 8.55/4.74 = ( numeral_numeral_int @ N ) )
% 8.55/4.74 = ( one = N ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_475_dvd__mult__cancel__left,axiom,
% 8.55/4.74 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 8.55/4.74 = ( ( C = zero_z3403309356797280102nteger )
% 8.55/4.74 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_476_dvd__mult__cancel__left,axiom,
% 8.55/4.74 ! [C: real,A: real,B: real] :
% 8.55/4.74 ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.55/4.74 = ( ( C = zero_zero_real )
% 8.55/4.74 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_477_dvd__mult__cancel__left,axiom,
% 8.55/4.74 ! [C: rat,A: rat,B: rat] :
% 8.55/4.74 ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.55/4.74 = ( ( C = zero_zero_rat )
% 8.55/4.74 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_478_dvd__mult__cancel__left,axiom,
% 8.55/4.74 ! [C: int,A: int,B: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.55/4.74 = ( ( C = zero_zero_int )
% 8.55/4.74 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_479_dvd__mult__cancel__right,axiom,
% 8.55/4.74 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 8.55/4.74 = ( ( C = zero_z3403309356797280102nteger )
% 8.55/4.74 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_480_dvd__mult__cancel__right,axiom,
% 8.55/4.74 ! [A: real,C: real,B: real] :
% 8.55/4.74 ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 8.55/4.74 = ( ( C = zero_zero_real )
% 8.55/4.74 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_481_dvd__mult__cancel__right,axiom,
% 8.55/4.74 ! [A: rat,C: rat,B: rat] :
% 8.55/4.74 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 8.55/4.74 = ( ( C = zero_zero_rat )
% 8.55/4.74 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_482_dvd__mult__cancel__right,axiom,
% 8.55/4.74 ! [A: int,C: int,B: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 8.55/4.74 = ( ( C = zero_zero_int )
% 8.55/4.74 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_483_dvd__times__left__cancel__iff,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.55/4.74 ( ( A != zero_z3403309356797280102nteger )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 8.55/4.74 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_484_dvd__times__left__cancel__iff,axiom,
% 8.55/4.74 ! [A: nat,B: nat,C: nat] :
% 8.55/4.74 ( ( A != zero_zero_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
% 8.55/4.74 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_485_dvd__times__left__cancel__iff,axiom,
% 8.55/4.74 ! [A: int,B: int,C: int] :
% 8.55/4.74 ( ( A != zero_zero_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
% 8.55/4.74 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_486_dvd__times__right__cancel__iff,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.55/4.74 ( ( A != zero_z3403309356797280102nteger )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 8.55/4.74 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_487_dvd__times__right__cancel__iff,axiom,
% 8.55/4.74 ! [A: nat,B: nat,C: nat] :
% 8.55/4.74 ( ( A != zero_zero_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
% 8.55/4.74 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_488_dvd__times__right__cancel__iff,axiom,
% 8.55/4.74 ! [A: int,B: int,C: int] :
% 8.55/4.74 ( ( A != zero_zero_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
% 8.55/4.74 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_489_unit__prod,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.55/4.74 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_490_unit__prod,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.55/4.74 => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_491_unit__prod,axiom,
% 8.55/4.74 ! [A: int,B: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ B @ one_one_int )
% 8.55/4.74 => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_492_dvd__div__mult__self,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.55/4.74 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 8.55/4.74 = B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_493_dvd__div__mult__self,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.74 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 8.55/4.74 = B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_494_dvd__div__mult__self,axiom,
% 8.55/4.74 ! [A: int,B: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ B )
% 8.55/4.74 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 8.55/4.74 = B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_495_dvd__mult__div__cancel,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.55/4.74 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
% 8.55/4.74 = B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_496_dvd__mult__div__cancel,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.74 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
% 8.55/4.74 = B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_497_dvd__mult__div__cancel,axiom,
% 8.55/4.74 ! [A: int,B: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ B )
% 8.55/4.74 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
% 8.55/4.74 = B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_498_unit__div,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.55/4.74 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_499_unit__div,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.55/4.74 => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_500_unit__div,axiom,
% 8.55/4.74 ! [A: int,B: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ B @ one_one_int )
% 8.55/4.74 => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_501_unit__div__1__unit,axiom,
% 8.55/4.74 ! [A: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.55/4.74 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_502_unit__div__1__unit,axiom,
% 8.55/4.74 ! [A: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.55/4.74 => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_503_unit__div__1__unit,axiom,
% 8.55/4.74 ! [A: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.55/4.74 => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_504_unit__div__1__div__1,axiom,
% 8.55/4.74 ! [A: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.55/4.74 => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 8.55/4.74 = A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_505_unit__div__1__div__1,axiom,
% 8.55/4.74 ! [A: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.55/4.74 => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 8.55/4.74 = A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_506_unit__div__1__div__1,axiom,
% 8.55/4.74 ! [A: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.55/4.74 => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 8.55/4.74 = A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_507_divide__eq__eq__numeral1_I1_J,axiom,
% 8.55/4.74 ! [B: complex,W: num,A: complex] :
% 8.55/4.74 ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
% 8.55/4.74 = A )
% 8.55/4.74 = ( ( ( ( numera6690914467698888265omplex @ W )
% 8.55/4.74 != zero_zero_complex )
% 8.55/4.74 => ( B
% 8.55/4.74 = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
% 8.55/4.74 & ( ( ( numera6690914467698888265omplex @ W )
% 8.55/4.74 = zero_zero_complex )
% 8.55/4.74 => ( A = zero_zero_complex ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_508_divide__eq__eq__numeral1_I1_J,axiom,
% 8.55/4.74 ! [B: real,W: num,A: real] :
% 8.55/4.74 ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
% 8.55/4.74 = A )
% 8.55/4.74 = ( ( ( ( numeral_numeral_real @ W )
% 8.55/4.74 != zero_zero_real )
% 8.55/4.74 => ( B
% 8.55/4.74 = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
% 8.55/4.74 & ( ( ( numeral_numeral_real @ W )
% 8.55/4.74 = zero_zero_real )
% 8.55/4.74 => ( A = zero_zero_real ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_509_divide__eq__eq__numeral1_I1_J,axiom,
% 8.55/4.74 ! [B: rat,W: num,A: rat] :
% 8.55/4.74 ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
% 8.55/4.74 = A )
% 8.55/4.74 = ( ( ( ( numeral_numeral_rat @ W )
% 8.55/4.74 != zero_zero_rat )
% 8.55/4.74 => ( B
% 8.55/4.74 = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
% 8.55/4.74 & ( ( ( numeral_numeral_rat @ W )
% 8.55/4.74 = zero_zero_rat )
% 8.55/4.74 => ( A = zero_zero_rat ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_510_eq__divide__eq__numeral1_I1_J,axiom,
% 8.55/4.74 ! [A: complex,B: complex,W: num] :
% 8.55/4.74 ( ( A
% 8.55/4.74 = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
% 8.55/4.74 = ( ( ( ( numera6690914467698888265omplex @ W )
% 8.55/4.74 != zero_zero_complex )
% 8.55/4.74 => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
% 8.55/4.74 = B ) )
% 8.55/4.74 & ( ( ( numera6690914467698888265omplex @ W )
% 8.55/4.74 = zero_zero_complex )
% 8.55/4.74 => ( A = zero_zero_complex ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_511_eq__divide__eq__numeral1_I1_J,axiom,
% 8.55/4.74 ! [A: real,B: real,W: num] :
% 8.55/4.74 ( ( A
% 8.55/4.74 = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 8.55/4.74 = ( ( ( ( numeral_numeral_real @ W )
% 8.55/4.74 != zero_zero_real )
% 8.55/4.74 => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
% 8.55/4.74 = B ) )
% 8.55/4.74 & ( ( ( numeral_numeral_real @ W )
% 8.55/4.74 = zero_zero_real )
% 8.55/4.74 => ( A = zero_zero_real ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_512_eq__divide__eq__numeral1_I1_J,axiom,
% 8.55/4.74 ! [A: rat,B: rat,W: num] :
% 8.55/4.74 ( ( A
% 8.55/4.74 = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 8.55/4.74 = ( ( ( ( numeral_numeral_rat @ W )
% 8.55/4.74 != zero_zero_rat )
% 8.55/4.74 => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
% 8.55/4.74 = B ) )
% 8.55/4.74 & ( ( ( numeral_numeral_rat @ W )
% 8.55/4.74 = zero_zero_rat )
% 8.55/4.74 => ( A = zero_zero_rat ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_513_nonzero__divide__mult__cancel__left,axiom,
% 8.55/4.74 ! [A: complex,B: complex] :
% 8.55/4.74 ( ( A != zero_zero_complex )
% 8.55/4.74 => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
% 8.55/4.74 = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_514_nonzero__divide__mult__cancel__left,axiom,
% 8.55/4.74 ! [A: real,B: real] :
% 8.55/4.74 ( ( A != zero_zero_real )
% 8.55/4.74 => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
% 8.55/4.74 = ( divide_divide_real @ one_one_real @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_515_nonzero__divide__mult__cancel__left,axiom,
% 8.55/4.74 ! [A: rat,B: rat] :
% 8.55/4.74 ( ( A != zero_zero_rat )
% 8.55/4.74 => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
% 8.55/4.74 = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_516_nonzero__divide__mult__cancel__right,axiom,
% 8.55/4.74 ! [B: complex,A: complex] :
% 8.55/4.74 ( ( B != zero_zero_complex )
% 8.55/4.74 => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
% 8.55/4.74 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_517_nonzero__divide__mult__cancel__right,axiom,
% 8.55/4.74 ! [B: real,A: real] :
% 8.55/4.74 ( ( B != zero_zero_real )
% 8.55/4.74 => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
% 8.55/4.74 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_518_nonzero__divide__mult__cancel__right,axiom,
% 8.55/4.74 ! [B: rat,A: rat] :
% 8.55/4.74 ( ( B != zero_zero_rat )
% 8.55/4.74 => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
% 8.55/4.74 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_519_unit__div__mult__self,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.55/4.74 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 8.55/4.74 = B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_520_unit__div__mult__self,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.55/4.74 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 8.55/4.74 = B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_521_unit__div__mult__self,axiom,
% 8.55/4.74 ! [A: int,B: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.55/4.74 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 8.55/4.74 = B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_522_unit__mult__div__div,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.55/4.74 => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 8.55/4.74 = ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_523_unit__mult__div__div,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.55/4.74 => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
% 8.55/4.74 = ( divide_divide_nat @ B @ A ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_524_unit__mult__div__div,axiom,
% 8.55/4.74 ! [A: int,B: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.55/4.74 => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
% 8.55/4.74 = ( divide_divide_int @ B @ A ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_525_Suc__1,axiom,
% 8.55/4.74 ( ( suc @ one_one_nat )
% 8.55/4.74 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_526_push__bit__Suc__numeral,axiom,
% 8.55/4.74 ! [N: nat,K: num] :
% 8.55/4.74 ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 8.55/4.74 = ( bit_se547839408752420682it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_527_push__bit__Suc__numeral,axiom,
% 8.55/4.74 ! [N: nat,K: num] :
% 8.55/4.74 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( numeral_numeral_int @ K ) )
% 8.55/4.74 = ( bit_se545348938243370406it_int @ N @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_528_push__bit__Suc__numeral,axiom,
% 8.55/4.74 ! [N: nat,K: num] :
% 8.55/4.74 ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ ( numera6620942414471956472nteger @ K ) )
% 8.55/4.74 = ( bit_se7788150548672797655nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_529_push__bit__Suc__numeral,axiom,
% 8.55/4.74 ! [N: nat,K: num] :
% 8.55/4.74 ( ( bit_se6611745700429515170atural @ ( suc @ N ) @ ( numera5444537566228673987atural @ K ) )
% 8.55/4.74 = ( bit_se6611745700429515170atural @ N @ ( numera5444537566228673987atural @ ( bit0 @ K ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_530_even__mult__iff,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
% 8.55/4.74 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.55/4.74 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_531_even__mult__iff,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
% 8.55/4.74 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.55/4.74 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_532_even__mult__iff,axiom,
% 8.55/4.74 ! [A: int,B: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
% 8.55/4.74 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.55/4.74 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_533_push__bit__Suc,axiom,
% 8.55/4.74 ! [N: nat,A: nat] :
% 8.55/4.74 ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ A )
% 8.55/4.74 = ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_534_push__bit__Suc,axiom,
% 8.55/4.74 ! [N: nat,A: int] :
% 8.55/4.74 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ A )
% 8.55/4.74 = ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_535_push__bit__Suc,axiom,
% 8.55/4.74 ! [N: nat,A: code_integer] :
% 8.55/4.74 ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ A )
% 8.55/4.74 = ( bit_se7788150548672797655nteger @ N @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_536_push__bit__Suc,axiom,
% 8.55/4.74 ! [N: nat,A: code_natural] :
% 8.55/4.74 ( ( bit_se6611745700429515170atural @ ( suc @ N ) @ A )
% 8.55/4.74 = ( bit_se6611745700429515170atural @ N @ ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_537_odd__two__times__div__two__nat,axiom,
% 8.55/4.74 ! [N: nat] :
% 8.55/4.74 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.74 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.74 = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_538_times__int__code_I1_J,axiom,
% 8.55/4.74 ! [K: int] :
% 8.55/4.74 ( ( times_times_int @ K @ zero_zero_int )
% 8.55/4.74 = zero_zero_int ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_539_times__int__code_I2_J,axiom,
% 8.55/4.74 ! [L: int] :
% 8.55/4.74 ( ( times_times_int @ zero_zero_int @ L )
% 8.55/4.74 = zero_zero_int ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_540_minus__int__code_I1_J,axiom,
% 8.55/4.74 ! [K: int] :
% 8.55/4.74 ( ( minus_minus_int @ K @ zero_zero_int )
% 8.55/4.74 = K ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_541_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 8.55/4.74 ! [A: real,B: real,C: real] :
% 8.55/4.74 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 8.55/4.74 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_542_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 8.55/4.74 ! [A: rat,B: rat,C: rat] :
% 8.55/4.74 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 8.55/4.74 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_543_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 8.55/4.74 ! [A: nat,B: nat,C: nat] :
% 8.55/4.74 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 8.55/4.74 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_544_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 8.55/4.74 ! [A: int,B: int,C: int] :
% 8.55/4.74 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 8.55/4.74 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_545_comm__monoid__mult__class_Omult__1,axiom,
% 8.55/4.74 ! [A: complex] :
% 8.55/4.74 ( ( times_times_complex @ one_one_complex @ A )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_546_comm__monoid__mult__class_Omult__1,axiom,
% 8.55/4.74 ! [A: real] :
% 8.55/4.74 ( ( times_times_real @ one_one_real @ A )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_547_comm__monoid__mult__class_Omult__1,axiom,
% 8.55/4.74 ! [A: rat] :
% 8.55/4.74 ( ( times_times_rat @ one_one_rat @ A )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_548_comm__monoid__mult__class_Omult__1,axiom,
% 8.55/4.74 ! [A: nat] :
% 8.55/4.74 ( ( times_times_nat @ one_one_nat @ A )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_549_comm__monoid__mult__class_Omult__1,axiom,
% 8.55/4.74 ! [A: int] :
% 8.55/4.74 ( ( times_times_int @ one_one_int @ A )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_550_mult_Oassoc,axiom,
% 8.55/4.74 ! [A: real,B: real,C: real] :
% 8.55/4.74 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 8.55/4.74 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_551_mult_Oassoc,axiom,
% 8.55/4.74 ! [A: rat,B: rat,C: rat] :
% 8.55/4.74 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 8.55/4.74 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_552_mult_Oassoc,axiom,
% 8.55/4.74 ! [A: nat,B: nat,C: nat] :
% 8.55/4.74 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 8.55/4.74 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_553_mult_Oassoc,axiom,
% 8.55/4.74 ! [A: int,B: int,C: int] :
% 8.55/4.74 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 8.55/4.74 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_554_mult_Ocommute,axiom,
% 8.55/4.74 ( times_times_real
% 8.55/4.74 = ( ^ [A2: real,B2: real] : ( times_times_real @ B2 @ A2 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_555_mult_Ocommute,axiom,
% 8.55/4.74 ( times_times_rat
% 8.55/4.74 = ( ^ [A2: rat,B2: rat] : ( times_times_rat @ B2 @ A2 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_556_mult_Ocommute,axiom,
% 8.55/4.74 ( times_times_nat
% 8.55/4.74 = ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_557_mult_Ocommute,axiom,
% 8.55/4.74 ( times_times_int
% 8.55/4.74 = ( ^ [A2: int,B2: int] : ( times_times_int @ B2 @ A2 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_558_mult_Ocomm__neutral,axiom,
% 8.55/4.74 ! [A: complex] :
% 8.55/4.74 ( ( times_times_complex @ A @ one_one_complex )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_559_mult_Ocomm__neutral,axiom,
% 8.55/4.74 ! [A: real] :
% 8.55/4.74 ( ( times_times_real @ A @ one_one_real )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_560_mult_Ocomm__neutral,axiom,
% 8.55/4.74 ! [A: rat] :
% 8.55/4.74 ( ( times_times_rat @ A @ one_one_rat )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_561_mult_Ocomm__neutral,axiom,
% 8.55/4.74 ! [A: nat] :
% 8.55/4.74 ( ( times_times_nat @ A @ one_one_nat )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_562_mult_Ocomm__neutral,axiom,
% 8.55/4.74 ! [A: int] :
% 8.55/4.74 ( ( times_times_int @ A @ one_one_int )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_563_mult_Oleft__commute,axiom,
% 8.55/4.74 ! [B: real,A: real,C: real] :
% 8.55/4.74 ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 8.55/4.74 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_564_mult_Oleft__commute,axiom,
% 8.55/4.74 ! [B: rat,A: rat,C: rat] :
% 8.55/4.74 ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
% 8.55/4.74 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_565_mult_Oleft__commute,axiom,
% 8.55/4.74 ! [B: nat,A: nat,C: nat] :
% 8.55/4.74 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 8.55/4.74 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_566_mult_Oleft__commute,axiom,
% 8.55/4.74 ! [B: int,A: int,C: int] :
% 8.55/4.74 ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
% 8.55/4.74 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_567_one__reorient,axiom,
% 8.55/4.74 ! [X: complex] :
% 8.55/4.74 ( ( one_one_complex = X )
% 8.55/4.74 = ( X = one_one_complex ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_568_one__reorient,axiom,
% 8.55/4.74 ! [X: real] :
% 8.55/4.74 ( ( one_one_real = X )
% 8.55/4.74 = ( X = one_one_real ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_569_one__reorient,axiom,
% 8.55/4.74 ! [X: rat] :
% 8.55/4.74 ( ( one_one_rat = X )
% 8.55/4.74 = ( X = one_one_rat ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_570_one__reorient,axiom,
% 8.55/4.74 ! [X: nat] :
% 8.55/4.74 ( ( one_one_nat = X )
% 8.55/4.74 = ( X = one_one_nat ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_571_one__reorient,axiom,
% 8.55/4.74 ! [X: int] :
% 8.55/4.74 ( ( one_one_int = X )
% 8.55/4.74 = ( X = one_one_int ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_572_push__bit__of__nat,axiom,
% 8.55/4.74 ! [N: nat,M: nat] :
% 8.55/4.74 ( ( bit_se547839408752420682it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 8.55/4.74 = ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_573_push__bit__of__nat,axiom,
% 8.55/4.74 ! [N: nat,M: nat] :
% 8.55/4.74 ( ( bit_se545348938243370406it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 8.55/4.74 = ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_574_push__bit__of__nat,axiom,
% 8.55/4.74 ! [N: nat,M: nat] :
% 8.55/4.74 ( ( bit_se7788150548672797655nteger @ N @ ( semiri4939895301339042750nteger @ M ) )
% 8.55/4.74 = ( semiri4939895301339042750nteger @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_575_push__bit__of__nat,axiom,
% 8.55/4.74 ! [N: nat,M: nat] :
% 8.55/4.74 ( ( bit_se6611745700429515170atural @ N @ ( semiri3763490453095760265atural @ M ) )
% 8.55/4.74 = ( semiri3763490453095760265atural @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_576_of__nat__push__bit,axiom,
% 8.55/4.74 ! [M: nat,N: nat] :
% 8.55/4.74 ( ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 8.55/4.74 = ( bit_se547839408752420682it_nat @ M @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_577_of__nat__push__bit,axiom,
% 8.55/4.74 ! [M: nat,N: nat] :
% 8.55/4.74 ( ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 8.55/4.74 = ( bit_se545348938243370406it_int @ M @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_578_of__nat__push__bit,axiom,
% 8.55/4.74 ! [M: nat,N: nat] :
% 8.55/4.74 ( ( semiri4939895301339042750nteger @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 8.55/4.74 = ( bit_se7788150548672797655nteger @ M @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_579_of__nat__push__bit,axiom,
% 8.55/4.74 ! [M: nat,N: nat] :
% 8.55/4.74 ( ( semiri3763490453095760265atural @ ( bit_se547839408752420682it_nat @ M @ N ) )
% 8.55/4.74 = ( bit_se6611745700429515170atural @ M @ ( semiri3763490453095760265atural @ N ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_580_is__unit__mult__iff,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
% 8.55/4.74 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.55/4.74 & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_581_is__unit__mult__iff,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
% 8.55/4.74 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.55/4.74 & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_582_is__unit__mult__iff,axiom,
% 8.55/4.74 ! [A: int,B: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
% 8.55/4.74 = ( ( dvd_dvd_int @ A @ one_one_int )
% 8.55/4.74 & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_583_dvd__mult__unit__iff,axiom,
% 8.55/4.74 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
% 8.55/4.74 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_584_dvd__mult__unit__iff,axiom,
% 8.55/4.74 ! [B: nat,A: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
% 8.55/4.74 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_585_dvd__mult__unit__iff,axiom,
% 8.55/4.74 ! [B: int,A: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
% 8.55/4.74 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_586_mult__unit__dvd__iff,axiom,
% 8.55/4.74 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 8.55/4.74 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_587_mult__unit__dvd__iff,axiom,
% 8.55/4.74 ! [B: nat,A: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 8.55/4.74 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_588_mult__unit__dvd__iff,axiom,
% 8.55/4.74 ! [B: int,A: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 8.55/4.74 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_589_dvd__mult__unit__iff_H,axiom,
% 8.55/4.74 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 8.55/4.74 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_590_dvd__mult__unit__iff_H,axiom,
% 8.55/4.74 ! [B: nat,A: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 8.55/4.74 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_591_dvd__mult__unit__iff_H,axiom,
% 8.55/4.74 ! [B: int,A: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 8.55/4.74 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_592_mult__unit__dvd__iff_H,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 8.55/4.74 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_593_mult__unit__dvd__iff_H,axiom,
% 8.55/4.74 ! [A: nat,B: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 8.55/4.74 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_594_mult__unit__dvd__iff_H,axiom,
% 8.55/4.74 ! [A: int,B: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 8.55/4.74 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_595_unit__mult__left__cancel,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.55/4.74 => ( ( ( times_3573771949741848930nteger @ A @ B )
% 8.55/4.74 = ( times_3573771949741848930nteger @ A @ C ) )
% 8.55/4.74 = ( B = C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_596_unit__mult__left__cancel,axiom,
% 8.55/4.74 ! [A: nat,B: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.55/4.74 => ( ( ( times_times_nat @ A @ B )
% 8.55/4.74 = ( times_times_nat @ A @ C ) )
% 8.55/4.74 = ( B = C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_597_unit__mult__left__cancel,axiom,
% 8.55/4.74 ! [A: int,B: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.55/4.74 => ( ( ( times_times_int @ A @ B )
% 8.55/4.74 = ( times_times_int @ A @ C ) )
% 8.55/4.74 = ( B = C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_598_unit__mult__right__cancel,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.55/4.74 => ( ( ( times_3573771949741848930nteger @ B @ A )
% 8.55/4.74 = ( times_3573771949741848930nteger @ C @ A ) )
% 8.55/4.74 = ( B = C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_599_unit__mult__right__cancel,axiom,
% 8.55/4.74 ! [A: nat,B: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.55/4.74 => ( ( ( times_times_nat @ B @ A )
% 8.55/4.74 = ( times_times_nat @ C @ A ) )
% 8.55/4.74 = ( B = C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_600_unit__mult__right__cancel,axiom,
% 8.55/4.74 ! [A: int,B: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.55/4.74 => ( ( ( times_times_int @ B @ A )
% 8.55/4.74 = ( times_times_int @ C @ A ) )
% 8.55/4.74 = ( B = C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_601_mult__eq__self__implies__10,axiom,
% 8.55/4.74 ! [M: nat,N: nat] :
% 8.55/4.74 ( ( M
% 8.55/4.74 = ( times_times_nat @ M @ N ) )
% 8.55/4.74 => ( ( N = one_one_nat )
% 8.55/4.74 | ( M = zero_zero_nat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_602_int__ops_I7_J,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
% 8.55/4.74 = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_603_int__ops_I2_J,axiom,
% 8.55/4.74 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 8.55/4.74 = one_one_int ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_604_division__decomp,axiom,
% 8.55/4.74 ! [A: nat,B: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 8.55/4.74 => ? [B3: nat,C2: nat] :
% 8.55/4.74 ( ( A
% 8.55/4.74 = ( times_times_nat @ B3 @ C2 ) )
% 8.55/4.74 & ( dvd_dvd_nat @ B3 @ B )
% 8.55/4.74 & ( dvd_dvd_nat @ C2 @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_605_division__decomp,axiom,
% 8.55/4.74 ! [A: int,B: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 8.55/4.74 => ? [B3: int,C2: int] :
% 8.55/4.74 ( ( A
% 8.55/4.74 = ( times_times_int @ B3 @ C2 ) )
% 8.55/4.74 & ( dvd_dvd_int @ B3 @ B )
% 8.55/4.74 & ( dvd_dvd_int @ C2 @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_606_dvd__productE,axiom,
% 8.55/4.74 ! [P2: nat,A: nat,B: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ P2 @ ( times_times_nat @ A @ B ) )
% 8.55/4.74 => ~ ! [X4: nat,Y4: nat] :
% 8.55/4.74 ( ( P2
% 8.55/4.74 = ( times_times_nat @ X4 @ Y4 ) )
% 8.55/4.74 => ( ( dvd_dvd_nat @ X4 @ A )
% 8.55/4.74 => ~ ( dvd_dvd_nat @ Y4 @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_607_dvd__productE,axiom,
% 8.55/4.74 ! [P2: int,A: int,B: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ P2 @ ( times_times_int @ A @ B ) )
% 8.55/4.74 => ~ ! [X4: int,Y4: int] :
% 8.55/4.74 ( ( P2
% 8.55/4.74 = ( times_times_int @ X4 @ Y4 ) )
% 8.55/4.74 => ( ( dvd_dvd_int @ X4 @ A )
% 8.55/4.74 => ~ ( dvd_dvd_int @ Y4 @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_608_unit__dvdE,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.55/4.74 => ~ ( ( A != zero_z3403309356797280102nteger )
% 8.55/4.74 => ! [C3: code_integer] :
% 8.55/4.74 ( B
% 8.55/4.74 != ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_609_unit__dvdE,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.55/4.74 => ~ ( ( A != zero_zero_nat )
% 8.55/4.74 => ! [C3: nat] :
% 8.55/4.74 ( B
% 8.55/4.74 != ( times_times_nat @ A @ C3 ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_610_unit__dvdE,axiom,
% 8.55/4.74 ! [A: int,B: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.55/4.74 => ~ ( ( A != zero_zero_int )
% 8.55/4.74 => ! [C3: int] :
% 8.55/4.74 ( B
% 8.55/4.74 != ( times_times_int @ A @ C3 ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_611_is__unit__div__mult2__eq,axiom,
% 8.55/4.74 ! [B: code_integer,C: code_integer,A: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 8.55/4.74 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 8.55/4.74 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_612_is__unit__div__mult2__eq,axiom,
% 8.55/4.74 ! [B: nat,C: nat,A: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 8.55/4.74 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 8.55/4.74 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_613_is__unit__div__mult2__eq,axiom,
% 8.55/4.74 ! [B: int,C: int,A: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ C @ one_one_int )
% 8.55/4.74 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 8.55/4.74 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_614_unit__div__mult__swap,axiom,
% 8.55/4.74 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 8.55/4.74 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 8.55/4.74 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_615_unit__div__mult__swap,axiom,
% 8.55/4.74 ! [C: nat,A: nat,B: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 8.55/4.74 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 8.55/4.74 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_616_unit__div__mult__swap,axiom,
% 8.55/4.74 ! [C: int,A: int,B: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ C @ one_one_int )
% 8.55/4.74 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 8.55/4.74 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_617_unit__div__commute,axiom,
% 8.55/4.74 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.55/4.74 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 8.55/4.74 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_618_unit__div__commute,axiom,
% 8.55/4.74 ! [B: nat,A: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.55/4.74 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 8.55/4.74 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_619_unit__div__commute,axiom,
% 8.55/4.74 ! [B: int,A: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.55/4.74 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
% 8.55/4.74 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_620_div__mult__unit2,axiom,
% 8.55/4.74 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ B @ A )
% 8.55/4.74 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 8.55/4.74 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_621_div__mult__unit2,axiom,
% 8.55/4.74 ! [C: nat,B: nat,A: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ B @ A )
% 8.55/4.74 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 8.55/4.74 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_622_div__mult__unit2,axiom,
% 8.55/4.74 ! [C: int,B: int,A: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ C @ one_one_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ B @ A )
% 8.55/4.74 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 8.55/4.74 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_623_unit__eq__div2,axiom,
% 8.55/4.74 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.55/4.74 => ( ( A
% 8.55/4.74 = ( divide6298287555418463151nteger @ C @ B ) )
% 8.55/4.74 = ( ( times_3573771949741848930nteger @ A @ B )
% 8.55/4.74 = C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_624_unit__eq__div2,axiom,
% 8.55/4.74 ! [B: nat,A: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.55/4.74 => ( ( A
% 8.55/4.74 = ( divide_divide_nat @ C @ B ) )
% 8.55/4.74 = ( ( times_times_nat @ A @ B )
% 8.55/4.74 = C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_625_unit__eq__div2,axiom,
% 8.55/4.74 ! [B: int,A: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.55/4.74 => ( ( A
% 8.55/4.74 = ( divide_divide_int @ C @ B ) )
% 8.55/4.74 = ( ( times_times_int @ A @ B )
% 8.55/4.74 = C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_626_unit__eq__div1,axiom,
% 8.55/4.74 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.55/4.74 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 8.55/4.74 = C )
% 8.55/4.74 = ( A
% 8.55/4.74 = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_627_unit__eq__div1,axiom,
% 8.55/4.74 ! [B: nat,A: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.55/4.74 => ( ( ( divide_divide_nat @ A @ B )
% 8.55/4.74 = C )
% 8.55/4.74 = ( A
% 8.55/4.74 = ( times_times_nat @ C @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_628_unit__eq__div1,axiom,
% 8.55/4.74 ! [B: int,A: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.55/4.74 => ( ( ( divide_divide_int @ A @ B )
% 8.55/4.74 = C )
% 8.55/4.74 = ( A
% 8.55/4.74 = ( times_times_int @ C @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_629_mult__right__cancel,axiom,
% 8.55/4.74 ! [C: real,A: real,B: real] :
% 8.55/4.74 ( ( C != zero_zero_real )
% 8.55/4.74 => ( ( ( times_times_real @ A @ C )
% 8.55/4.74 = ( times_times_real @ B @ C ) )
% 8.55/4.74 = ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_630_mult__right__cancel,axiom,
% 8.55/4.74 ! [C: rat,A: rat,B: rat] :
% 8.55/4.74 ( ( C != zero_zero_rat )
% 8.55/4.74 => ( ( ( times_times_rat @ A @ C )
% 8.55/4.74 = ( times_times_rat @ B @ C ) )
% 8.55/4.74 = ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_631_mult__right__cancel,axiom,
% 8.55/4.74 ! [C: nat,A: nat,B: nat] :
% 8.55/4.74 ( ( C != zero_zero_nat )
% 8.55/4.74 => ( ( ( times_times_nat @ A @ C )
% 8.55/4.74 = ( times_times_nat @ B @ C ) )
% 8.55/4.74 = ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_632_mult__right__cancel,axiom,
% 8.55/4.74 ! [C: int,A: int,B: int] :
% 8.55/4.74 ( ( C != zero_zero_int )
% 8.55/4.74 => ( ( ( times_times_int @ A @ C )
% 8.55/4.74 = ( times_times_int @ B @ C ) )
% 8.55/4.74 = ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_633_mult__left__cancel,axiom,
% 8.55/4.74 ! [C: real,A: real,B: real] :
% 8.55/4.74 ( ( C != zero_zero_real )
% 8.55/4.74 => ( ( ( times_times_real @ C @ A )
% 8.55/4.74 = ( times_times_real @ C @ B ) )
% 8.55/4.74 = ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_634_mult__left__cancel,axiom,
% 8.55/4.74 ! [C: rat,A: rat,B: rat] :
% 8.55/4.74 ( ( C != zero_zero_rat )
% 8.55/4.74 => ( ( ( times_times_rat @ C @ A )
% 8.55/4.74 = ( times_times_rat @ C @ B ) )
% 8.55/4.74 = ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_635_mult__left__cancel,axiom,
% 8.55/4.74 ! [C: nat,A: nat,B: nat] :
% 8.55/4.74 ( ( C != zero_zero_nat )
% 8.55/4.74 => ( ( ( times_times_nat @ C @ A )
% 8.55/4.74 = ( times_times_nat @ C @ B ) )
% 8.55/4.74 = ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_636_mult__left__cancel,axiom,
% 8.55/4.74 ! [C: int,A: int,B: int] :
% 8.55/4.74 ( ( C != zero_zero_int )
% 8.55/4.74 => ( ( ( times_times_int @ C @ A )
% 8.55/4.74 = ( times_times_int @ C @ B ) )
% 8.55/4.74 = ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_637_no__zero__divisors,axiom,
% 8.55/4.74 ! [A: real,B: real] :
% 8.55/4.74 ( ( A != zero_zero_real )
% 8.55/4.74 => ( ( B != zero_zero_real )
% 8.55/4.74 => ( ( times_times_real @ A @ B )
% 8.55/4.74 != zero_zero_real ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_638_no__zero__divisors,axiom,
% 8.55/4.74 ! [A: rat,B: rat] :
% 8.55/4.74 ( ( A != zero_zero_rat )
% 8.55/4.74 => ( ( B != zero_zero_rat )
% 8.55/4.74 => ( ( times_times_rat @ A @ B )
% 8.55/4.74 != zero_zero_rat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_639_no__zero__divisors,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( A != zero_zero_nat )
% 8.55/4.74 => ( ( B != zero_zero_nat )
% 8.55/4.74 => ( ( times_times_nat @ A @ B )
% 8.55/4.74 != zero_zero_nat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_640_no__zero__divisors,axiom,
% 8.55/4.74 ! [A: int,B: int] :
% 8.55/4.74 ( ( A != zero_zero_int )
% 8.55/4.74 => ( ( B != zero_zero_int )
% 8.55/4.74 => ( ( times_times_int @ A @ B )
% 8.55/4.74 != zero_zero_int ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_641_divisors__zero,axiom,
% 8.55/4.74 ! [A: real,B: real] :
% 8.55/4.74 ( ( ( times_times_real @ A @ B )
% 8.55/4.74 = zero_zero_real )
% 8.55/4.74 => ( ( A = zero_zero_real )
% 8.55/4.74 | ( B = zero_zero_real ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_642_divisors__zero,axiom,
% 8.55/4.74 ! [A: rat,B: rat] :
% 8.55/4.74 ( ( ( times_times_rat @ A @ B )
% 8.55/4.74 = zero_zero_rat )
% 8.55/4.74 => ( ( A = zero_zero_rat )
% 8.55/4.74 | ( B = zero_zero_rat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_643_divisors__zero,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( ( times_times_nat @ A @ B )
% 8.55/4.74 = zero_zero_nat )
% 8.55/4.74 => ( ( A = zero_zero_nat )
% 8.55/4.74 | ( B = zero_zero_nat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_644_divisors__zero,axiom,
% 8.55/4.74 ! [A: int,B: int] :
% 8.55/4.74 ( ( ( times_times_int @ A @ B )
% 8.55/4.74 = zero_zero_int )
% 8.55/4.74 => ( ( A = zero_zero_int )
% 8.55/4.74 | ( B = zero_zero_int ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_645_mult__not__zero,axiom,
% 8.55/4.74 ! [A: real,B: real] :
% 8.55/4.74 ( ( ( times_times_real @ A @ B )
% 8.55/4.74 != zero_zero_real )
% 8.55/4.74 => ( ( A != zero_zero_real )
% 8.55/4.74 & ( B != zero_zero_real ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_646_mult__not__zero,axiom,
% 8.55/4.74 ! [A: rat,B: rat] :
% 8.55/4.74 ( ( ( times_times_rat @ A @ B )
% 8.55/4.74 != zero_zero_rat )
% 8.55/4.74 => ( ( A != zero_zero_rat )
% 8.55/4.74 & ( B != zero_zero_rat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_647_mult__not__zero,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( ( times_times_nat @ A @ B )
% 8.55/4.74 != zero_zero_nat )
% 8.55/4.74 => ( ( A != zero_zero_nat )
% 8.55/4.74 & ( B != zero_zero_nat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_648_mult__not__zero,axiom,
% 8.55/4.74 ! [A: int,B: int] :
% 8.55/4.74 ( ( ( times_times_int @ A @ B )
% 8.55/4.74 != zero_zero_int )
% 8.55/4.74 => ( ( A != zero_zero_int )
% 8.55/4.74 & ( B != zero_zero_int ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_649_left__diff__distrib,axiom,
% 8.55/4.74 ! [A: real,B: real,C: real] :
% 8.55/4.74 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
% 8.55/4.74 = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_650_left__diff__distrib,axiom,
% 8.55/4.74 ! [A: rat,B: rat,C: rat] :
% 8.55/4.74 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 8.55/4.74 = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_651_left__diff__distrib,axiom,
% 8.55/4.74 ! [A: int,B: int,C: int] :
% 8.55/4.74 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
% 8.55/4.74 = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_652_right__diff__distrib,axiom,
% 8.55/4.74 ! [A: real,B: real,C: real] :
% 8.55/4.74 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 8.55/4.74 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_653_right__diff__distrib,axiom,
% 8.55/4.74 ! [A: rat,B: rat,C: rat] :
% 8.55/4.74 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 8.55/4.74 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_654_right__diff__distrib,axiom,
% 8.55/4.74 ! [A: int,B: int,C: int] :
% 8.55/4.74 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 8.55/4.74 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_655_left__diff__distrib_H,axiom,
% 8.55/4.74 ! [B: real,C: real,A: real] :
% 8.55/4.74 ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
% 8.55/4.74 = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_656_left__diff__distrib_H,axiom,
% 8.55/4.74 ! [B: rat,C: rat,A: rat] :
% 8.55/4.74 ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
% 8.55/4.74 = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_657_left__diff__distrib_H,axiom,
% 8.55/4.74 ! [B: nat,C: nat,A: nat] :
% 8.55/4.74 ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
% 8.55/4.74 = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_658_left__diff__distrib_H,axiom,
% 8.55/4.74 ! [B: int,C: int,A: int] :
% 8.55/4.74 ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
% 8.55/4.74 = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_659_right__diff__distrib_H,axiom,
% 8.55/4.74 ! [A: real,B: real,C: real] :
% 8.55/4.74 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 8.55/4.74 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_660_right__diff__distrib_H,axiom,
% 8.55/4.74 ! [A: rat,B: rat,C: rat] :
% 8.55/4.74 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 8.55/4.74 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_661_right__diff__distrib_H,axiom,
% 8.55/4.74 ! [A: nat,B: nat,C: nat] :
% 8.55/4.74 ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
% 8.55/4.74 = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_662_right__diff__distrib_H,axiom,
% 8.55/4.74 ! [A: int,B: int,C: int] :
% 8.55/4.74 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 8.55/4.74 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_663_times__divide__times__eq,axiom,
% 8.55/4.74 ! [X: complex,Y: complex,Z2: complex,W: complex] :
% 8.55/4.74 ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z2 @ W ) )
% 8.55/4.74 = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ Y @ W ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_664_times__divide__times__eq,axiom,
% 8.55/4.74 ! [X: real,Y: real,Z2: real,W: real] :
% 8.55/4.74 ( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z2 @ W ) )
% 8.55/4.74 = ( divide_divide_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ W ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_665_times__divide__times__eq,axiom,
% 8.55/4.74 ! [X: rat,Y: rat,Z2: rat,W: rat] :
% 8.55/4.74 ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z2 @ W ) )
% 8.55/4.74 = ( divide_divide_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ W ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_666_divide__divide__times__eq,axiom,
% 8.55/4.74 ! [X: complex,Y: complex,Z2: complex,W: complex] :
% 8.55/4.74 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z2 @ W ) )
% 8.55/4.74 = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y @ Z2 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_667_divide__divide__times__eq,axiom,
% 8.55/4.74 ! [X: real,Y: real,Z2: real,W: real] :
% 8.55/4.74 ( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z2 @ W ) )
% 8.55/4.74 = ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z2 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_668_divide__divide__times__eq,axiom,
% 8.55/4.74 ! [X: rat,Y: rat,Z2: rat,W: rat] :
% 8.55/4.74 ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z2 @ W ) )
% 8.55/4.74 = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y @ Z2 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_669_divide__divide__eq__left_H,axiom,
% 8.55/4.74 ! [A: complex,B: complex,C: complex] :
% 8.55/4.74 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 8.55/4.74 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_670_divide__divide__eq__left_H,axiom,
% 8.55/4.74 ! [A: real,B: real,C: real] :
% 8.55/4.74 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 8.55/4.74 = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_671_divide__divide__eq__left_H,axiom,
% 8.55/4.74 ! [A: rat,B: rat,C: rat] :
% 8.55/4.74 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 8.55/4.74 = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_672_Suc__mult__cancel1,axiom,
% 8.55/4.74 ! [K: nat,M: nat,N: nat] :
% 8.55/4.74 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 8.55/4.74 = ( times_times_nat @ ( suc @ K ) @ N ) )
% 8.55/4.74 = ( M = N ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_673_dvd__triv__right,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_674_dvd__triv__right,axiom,
% 8.55/4.74 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_675_dvd__triv__right,axiom,
% 8.55/4.74 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_676_dvd__triv__right,axiom,
% 8.55/4.74 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_677_dvd__triv__right,axiom,
% 8.55/4.74 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_678_dvd__mult__right,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 8.55/4.74 => ( dvd_dvd_Code_integer @ B @ C ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_679_dvd__mult__right,axiom,
% 8.55/4.74 ! [A: real,B: real,C: real] :
% 8.55/4.74 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 8.55/4.74 => ( dvd_dvd_real @ B @ C ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_680_dvd__mult__right,axiom,
% 8.55/4.74 ! [A: rat,B: rat,C: rat] :
% 8.55/4.74 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 8.55/4.74 => ( dvd_dvd_rat @ B @ C ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_681_dvd__mult__right,axiom,
% 8.55/4.74 ! [A: nat,B: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 8.55/4.74 => ( dvd_dvd_nat @ B @ C ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_682_dvd__mult__right,axiom,
% 8.55/4.74 ! [A: int,B: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 8.55/4.74 => ( dvd_dvd_int @ B @ C ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_683_mult__dvd__mono,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ C @ D )
% 8.55/4.74 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_684_mult__dvd__mono,axiom,
% 8.55/4.74 ! [A: real,B: real,C: real,D: real] :
% 8.55/4.74 ( ( dvd_dvd_real @ A @ B )
% 8.55/4.74 => ( ( dvd_dvd_real @ C @ D )
% 8.55/4.74 => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_685_mult__dvd__mono,axiom,
% 8.55/4.74 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.55/4.74 ( ( dvd_dvd_rat @ A @ B )
% 8.55/4.74 => ( ( dvd_dvd_rat @ C @ D )
% 8.55/4.74 => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_686_mult__dvd__mono,axiom,
% 8.55/4.74 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.74 => ( ( dvd_dvd_nat @ C @ D )
% 8.55/4.74 => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_687_mult__dvd__mono,axiom,
% 8.55/4.74 ! [A: int,B: int,C: int,D: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ B )
% 8.55/4.74 => ( ( dvd_dvd_int @ C @ D )
% 8.55/4.74 => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_688_dvd__triv__left,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_689_dvd__triv__left,axiom,
% 8.55/4.74 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_690_dvd__triv__left,axiom,
% 8.55/4.74 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_691_dvd__triv__left,axiom,
% 8.55/4.74 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_692_dvd__triv__left,axiom,
% 8.55/4.74 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_693_dvd__mult__left,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 8.55/4.74 => ( dvd_dvd_Code_integer @ A @ C ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_694_dvd__mult__left,axiom,
% 8.55/4.74 ! [A: real,B: real,C: real] :
% 8.55/4.74 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 8.55/4.74 => ( dvd_dvd_real @ A @ C ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_695_dvd__mult__left,axiom,
% 8.55/4.74 ! [A: rat,B: rat,C: rat] :
% 8.55/4.74 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 8.55/4.74 => ( dvd_dvd_rat @ A @ C ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_696_dvd__mult__left,axiom,
% 8.55/4.74 ! [A: nat,B: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 8.55/4.74 => ( dvd_dvd_nat @ A @ C ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_697_dvd__mult__left,axiom,
% 8.55/4.74 ! [A: int,B: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 8.55/4.74 => ( dvd_dvd_int @ A @ C ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_698_dvd__mult2,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.55/4.74 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_699_dvd__mult2,axiom,
% 8.55/4.74 ! [A: real,B: real,C: real] :
% 8.55/4.74 ( ( dvd_dvd_real @ A @ B )
% 8.55/4.74 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_700_dvd__mult2,axiom,
% 8.55/4.74 ! [A: rat,B: rat,C: rat] :
% 8.55/4.74 ( ( dvd_dvd_rat @ A @ B )
% 8.55/4.74 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_701_dvd__mult2,axiom,
% 8.55/4.74 ! [A: nat,B: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.74 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_702_dvd__mult2,axiom,
% 8.55/4.74 ! [A: int,B: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ B )
% 8.55/4.74 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_703_dvd__mult,axiom,
% 8.55/4.74 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ C )
% 8.55/4.74 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_704_dvd__mult,axiom,
% 8.55/4.74 ! [A: real,C: real,B: real] :
% 8.55/4.74 ( ( dvd_dvd_real @ A @ C )
% 8.55/4.74 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_705_dvd__mult,axiom,
% 8.55/4.74 ! [A: rat,C: rat,B: rat] :
% 8.55/4.74 ( ( dvd_dvd_rat @ A @ C )
% 8.55/4.74 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_706_dvd__mult,axiom,
% 8.55/4.74 ! [A: nat,C: nat,B: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ C )
% 8.55/4.74 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_707_dvd__mult,axiom,
% 8.55/4.74 ! [A: int,C: int,B: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ C )
% 8.55/4.74 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_708_dvd__def,axiom,
% 8.55/4.74 ( dvd_dvd_Code_integer
% 8.55/4.74 = ( ^ [B2: code_integer,A2: code_integer] :
% 8.55/4.74 ? [K2: code_integer] :
% 8.55/4.74 ( A2
% 8.55/4.74 = ( times_3573771949741848930nteger @ B2 @ K2 ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_709_dvd__def,axiom,
% 8.55/4.74 ( dvd_dvd_real
% 8.55/4.74 = ( ^ [B2: real,A2: real] :
% 8.55/4.74 ? [K2: real] :
% 8.55/4.74 ( A2
% 8.55/4.74 = ( times_times_real @ B2 @ K2 ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_710_dvd__def,axiom,
% 8.55/4.74 ( dvd_dvd_rat
% 8.55/4.74 = ( ^ [B2: rat,A2: rat] :
% 8.55/4.74 ? [K2: rat] :
% 8.55/4.74 ( A2
% 8.55/4.74 = ( times_times_rat @ B2 @ K2 ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_711_dvd__def,axiom,
% 8.55/4.74 ( dvd_dvd_nat
% 8.55/4.74 = ( ^ [B2: nat,A2: nat] :
% 8.55/4.74 ? [K2: nat] :
% 8.55/4.74 ( A2
% 8.55/4.74 = ( times_times_nat @ B2 @ K2 ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_712_dvd__def,axiom,
% 8.55/4.74 ( dvd_dvd_int
% 8.55/4.74 = ( ^ [B2: int,A2: int] :
% 8.55/4.74 ? [K2: int] :
% 8.55/4.74 ( A2
% 8.55/4.74 = ( times_times_int @ B2 @ K2 ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_713_dvdI,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer,K: code_integer] :
% 8.55/4.74 ( ( A
% 8.55/4.74 = ( times_3573771949741848930nteger @ B @ K ) )
% 8.55/4.74 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_714_dvdI,axiom,
% 8.55/4.74 ! [A: real,B: real,K: real] :
% 8.55/4.74 ( ( A
% 8.55/4.74 = ( times_times_real @ B @ K ) )
% 8.55/4.74 => ( dvd_dvd_real @ B @ A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_715_dvdI,axiom,
% 8.55/4.74 ! [A: rat,B: rat,K: rat] :
% 8.55/4.74 ( ( A
% 8.55/4.74 = ( times_times_rat @ B @ K ) )
% 8.55/4.74 => ( dvd_dvd_rat @ B @ A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_716_dvdI,axiom,
% 8.55/4.74 ! [A: nat,B: nat,K: nat] :
% 8.55/4.74 ( ( A
% 8.55/4.74 = ( times_times_nat @ B @ K ) )
% 8.55/4.74 => ( dvd_dvd_nat @ B @ A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_717_dvdI,axiom,
% 8.55/4.74 ! [A: int,B: int,K: int] :
% 8.55/4.74 ( ( A
% 8.55/4.74 = ( times_times_int @ B @ K ) )
% 8.55/4.74 => ( dvd_dvd_int @ B @ A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_718_dvdE,axiom,
% 8.55/4.74 ! [B: code_integer,A: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ B @ A )
% 8.55/4.74 => ~ ! [K3: code_integer] :
% 8.55/4.74 ( A
% 8.55/4.74 != ( times_3573771949741848930nteger @ B @ K3 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_719_dvdE,axiom,
% 8.55/4.74 ! [B: real,A: real] :
% 8.55/4.74 ( ( dvd_dvd_real @ B @ A )
% 8.55/4.74 => ~ ! [K3: real] :
% 8.55/4.74 ( A
% 8.55/4.74 != ( times_times_real @ B @ K3 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_720_dvdE,axiom,
% 8.55/4.74 ! [B: rat,A: rat] :
% 8.55/4.74 ( ( dvd_dvd_rat @ B @ A )
% 8.55/4.74 => ~ ! [K3: rat] :
% 8.55/4.74 ( A
% 8.55/4.74 != ( times_times_rat @ B @ K3 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_721_dvdE,axiom,
% 8.55/4.74 ! [B: nat,A: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ B @ A )
% 8.55/4.74 => ~ ! [K3: nat] :
% 8.55/4.74 ( A
% 8.55/4.74 != ( times_times_nat @ B @ K3 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_722_dvdE,axiom,
% 8.55/4.74 ! [B: int,A: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ B @ A )
% 8.55/4.74 => ~ ! [K3: int] :
% 8.55/4.74 ( A
% 8.55/4.74 != ( times_times_int @ B @ K3 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_723_mult__0,axiom,
% 8.55/4.74 ! [N: nat] :
% 8.55/4.74 ( ( times_times_nat @ zero_zero_nat @ N )
% 8.55/4.74 = zero_zero_nat ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_724_nat__mult__eq__cancel__disj,axiom,
% 8.55/4.74 ! [K: nat,M: nat,N: nat] :
% 8.55/4.74 ( ( ( times_times_nat @ K @ M )
% 8.55/4.74 = ( times_times_nat @ K @ N ) )
% 8.55/4.74 = ( ( K = zero_zero_nat )
% 8.55/4.74 | ( M = N ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_725_mult__of__nat__commute,axiom,
% 8.55/4.74 ! [X: nat,Y: complex] :
% 8.55/4.74 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
% 8.55/4.74 = ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_726_mult__of__nat__commute,axiom,
% 8.55/4.74 ! [X: nat,Y: real] :
% 8.55/4.74 ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
% 8.55/4.74 = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_727_mult__of__nat__commute,axiom,
% 8.55/4.74 ! [X: nat,Y: rat] :
% 8.55/4.74 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
% 8.55/4.74 = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_728_mult__of__nat__commute,axiom,
% 8.55/4.74 ! [X: nat,Y: nat] :
% 8.55/4.74 ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
% 8.55/4.74 = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_729_mult__of__nat__commute,axiom,
% 8.55/4.74 ! [X: nat,Y: int] :
% 8.55/4.74 ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
% 8.55/4.74 = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_730_zero__neq__one,axiom,
% 8.55/4.74 zero_zero_complex != one_one_complex ).
% 8.55/4.74
% 8.55/4.74 thf(fact_731_zero__neq__one,axiom,
% 8.55/4.74 zero_zero_real != one_one_real ).
% 8.55/4.74
% 8.55/4.74 thf(fact_732_zero__neq__one,axiom,
% 8.55/4.74 zero_zero_rat != one_one_rat ).
% 8.55/4.74
% 8.55/4.74 thf(fact_733_zero__neq__one,axiom,
% 8.55/4.74 zero_zero_nat != one_one_nat ).
% 8.55/4.74
% 8.55/4.74 thf(fact_734_zero__neq__one,axiom,
% 8.55/4.74 zero_zero_int != one_one_int ).
% 8.55/4.74
% 8.55/4.74 thf(fact_735_int__diff__cases,axiom,
% 8.55/4.74 ! [Z2: int] :
% 8.55/4.74 ~ ! [M2: nat,N2: nat] :
% 8.55/4.74 ( Z2
% 8.55/4.74 != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_736_diff__mult__distrib,axiom,
% 8.55/4.74 ! [M: nat,N: nat,K: nat] :
% 8.55/4.74 ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 8.55/4.74 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_737_diff__mult__distrib2,axiom,
% 8.55/4.74 ! [K: nat,M: nat,N: nat] :
% 8.55/4.74 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 8.55/4.74 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_738_dvd__unit__imp__unit,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.55/4.74 => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_739_dvd__unit__imp__unit,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.74 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.55/4.74 => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_740_dvd__unit__imp__unit,axiom,
% 8.55/4.74 ! [A: int,B: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ B )
% 8.55/4.74 => ( ( dvd_dvd_int @ B @ one_one_int )
% 8.55/4.74 => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_741_unit__imp__dvd,axiom,
% 8.55/4.74 ! [B: code_integer,A: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.55/4.74 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_742_unit__imp__dvd,axiom,
% 8.55/4.74 ! [B: nat,A: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.55/4.74 => ( dvd_dvd_nat @ B @ A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_743_unit__imp__dvd,axiom,
% 8.55/4.74 ! [B: int,A: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.55/4.74 => ( dvd_dvd_int @ B @ A ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_744_one__dvd,axiom,
% 8.55/4.74 ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_745_one__dvd,axiom,
% 8.55/4.74 ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_746_one__dvd,axiom,
% 8.55/4.74 ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_747_one__dvd,axiom,
% 8.55/4.74 ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_748_one__dvd,axiom,
% 8.55/4.74 ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_749_one__dvd,axiom,
% 8.55/4.74 ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_750_zdvd__zdiffD,axiom,
% 8.55/4.74 ! [K: int,M: int,N: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
% 8.55/4.74 => ( ( dvd_dvd_int @ K @ N )
% 8.55/4.74 => ( dvd_dvd_int @ K @ M ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_751_div__mult2__eq,axiom,
% 8.55/4.74 ! [M: nat,N: nat,Q2: nat] :
% 8.55/4.74 ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 8.55/4.74 = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_752_zdvd__mult__cancel,axiom,
% 8.55/4.74 ! [K: int,M: int,N: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
% 8.55/4.74 => ( ( K != zero_zero_int )
% 8.55/4.74 => ( dvd_dvd_int @ M @ N ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_753_bezout1__nat,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ? [D2: nat,X4: nat,Y4: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ D2 @ A )
% 8.55/4.74 & ( dvd_dvd_nat @ D2 @ B )
% 8.55/4.74 & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y4 ) )
% 8.55/4.74 = D2 )
% 8.55/4.74 | ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y4 ) )
% 8.55/4.74 = D2 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_754_push__bit__double,axiom,
% 8.55/4.74 ! [N: nat,A: nat] :
% 8.55/4.74 ( ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.74 = ( times_times_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_755_push__bit__double,axiom,
% 8.55/4.74 ! [N: nat,A: int] :
% 8.55/4.74 ( ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 8.55/4.74 = ( times_times_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_756_push__bit__double,axiom,
% 8.55/4.74 ! [N: nat,A: code_integer] :
% 8.55/4.74 ( ( bit_se7788150548672797655nteger @ N @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 8.55/4.74 = ( times_3573771949741848930nteger @ ( bit_se7788150548672797655nteger @ N @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_757_push__bit__double,axiom,
% 8.55/4.74 ! [N: nat,A: code_natural] :
% 8.55/4.74 ( ( bit_se6611745700429515170atural @ N @ ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
% 8.55/4.74 = ( times_2397367101498566445atural @ ( bit_se6611745700429515170atural @ N @ A ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_758_is__unit__div__mult__cancel__right,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer] :
% 8.55/4.74 ( ( A != zero_z3403309356797280102nteger )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.55/4.74 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
% 8.55/4.74 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_759_is__unit__div__mult__cancel__right,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( A != zero_zero_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.55/4.74 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
% 8.55/4.74 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_760_is__unit__div__mult__cancel__right,axiom,
% 8.55/4.74 ! [A: int,B: int] :
% 8.55/4.74 ( ( A != zero_zero_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ B @ one_one_int )
% 8.55/4.74 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
% 8.55/4.74 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_761_is__unit__div__mult__cancel__left,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer] :
% 8.55/4.74 ( ( A != zero_z3403309356797280102nteger )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.55/4.74 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
% 8.55/4.74 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_762_is__unit__div__mult__cancel__left,axiom,
% 8.55/4.74 ! [A: nat,B: nat] :
% 8.55/4.74 ( ( A != zero_zero_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.55/4.74 => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
% 8.55/4.74 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_763_is__unit__div__mult__cancel__left,axiom,
% 8.55/4.74 ! [A: int,B: int] :
% 8.55/4.74 ( ( A != zero_zero_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ B @ one_one_int )
% 8.55/4.74 => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
% 8.55/4.74 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_764_is__unitE,axiom,
% 8.55/4.74 ! [A: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.55/4.74 => ~ ( ( A != zero_z3403309356797280102nteger )
% 8.55/4.74 => ! [B4: code_integer] :
% 8.55/4.74 ( ( B4 != zero_z3403309356797280102nteger )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ B4 @ one_one_Code_integer )
% 8.55/4.74 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 8.55/4.74 = B4 )
% 8.55/4.74 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B4 )
% 8.55/4.74 = A )
% 8.55/4.74 => ( ( ( times_3573771949741848930nteger @ A @ B4 )
% 8.55/4.74 = one_one_Code_integer )
% 8.55/4.74 => ( ( divide6298287555418463151nteger @ C @ A )
% 8.55/4.74 != ( times_3573771949741848930nteger @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_765_is__unitE,axiom,
% 8.55/4.74 ! [A: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.55/4.74 => ~ ( ( A != zero_zero_nat )
% 8.55/4.74 => ! [B4: nat] :
% 8.55/4.74 ( ( B4 != zero_zero_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ B4 @ one_one_nat )
% 8.55/4.74 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 8.55/4.74 = B4 )
% 8.55/4.74 => ( ( ( divide_divide_nat @ one_one_nat @ B4 )
% 8.55/4.74 = A )
% 8.55/4.74 => ( ( ( times_times_nat @ A @ B4 )
% 8.55/4.74 = one_one_nat )
% 8.55/4.74 => ( ( divide_divide_nat @ C @ A )
% 8.55/4.74 != ( times_times_nat @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_766_is__unitE,axiom,
% 8.55/4.74 ! [A: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.55/4.74 => ~ ( ( A != zero_zero_int )
% 8.55/4.74 => ! [B4: int] :
% 8.55/4.74 ( ( B4 != zero_zero_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ B4 @ one_one_int )
% 8.55/4.74 => ( ( ( divide_divide_int @ one_one_int @ A )
% 8.55/4.74 = B4 )
% 8.55/4.74 => ( ( ( divide_divide_int @ one_one_int @ B4 )
% 8.55/4.74 = A )
% 8.55/4.74 => ( ( ( times_times_int @ A @ B4 )
% 8.55/4.74 = one_one_int )
% 8.55/4.74 => ( ( divide_divide_int @ C @ A )
% 8.55/4.74 != ( times_times_int @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_767_frac__eq__eq,axiom,
% 8.55/4.74 ! [Y: complex,Z2: complex,X: complex,W: complex] :
% 8.55/4.74 ( ( Y != zero_zero_complex )
% 8.55/4.74 => ( ( Z2 != zero_zero_complex )
% 8.55/4.74 => ( ( ( divide1717551699836669952omplex @ X @ Y )
% 8.55/4.74 = ( divide1717551699836669952omplex @ W @ Z2 ) )
% 8.55/4.74 = ( ( times_times_complex @ X @ Z2 )
% 8.55/4.74 = ( times_times_complex @ W @ Y ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_768_frac__eq__eq,axiom,
% 8.55/4.74 ! [Y: real,Z2: real,X: real,W: real] :
% 8.55/4.74 ( ( Y != zero_zero_real )
% 8.55/4.74 => ( ( Z2 != zero_zero_real )
% 8.55/4.74 => ( ( ( divide_divide_real @ X @ Y )
% 8.55/4.74 = ( divide_divide_real @ W @ Z2 ) )
% 8.55/4.74 = ( ( times_times_real @ X @ Z2 )
% 8.55/4.74 = ( times_times_real @ W @ Y ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_769_frac__eq__eq,axiom,
% 8.55/4.74 ! [Y: rat,Z2: rat,X: rat,W: rat] :
% 8.55/4.74 ( ( Y != zero_zero_rat )
% 8.55/4.74 => ( ( Z2 != zero_zero_rat )
% 8.55/4.74 => ( ( ( divide_divide_rat @ X @ Y )
% 8.55/4.74 = ( divide_divide_rat @ W @ Z2 ) )
% 8.55/4.74 = ( ( times_times_rat @ X @ Z2 )
% 8.55/4.74 = ( times_times_rat @ W @ Y ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_770_divide__eq__eq,axiom,
% 8.55/4.74 ! [B: complex,C: complex,A: complex] :
% 8.55/4.74 ( ( ( divide1717551699836669952omplex @ B @ C )
% 8.55/4.74 = A )
% 8.55/4.74 = ( ( ( C != zero_zero_complex )
% 8.55/4.74 => ( B
% 8.55/4.74 = ( times_times_complex @ A @ C ) ) )
% 8.55/4.74 & ( ( C = zero_zero_complex )
% 8.55/4.74 => ( A = zero_zero_complex ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_771_divide__eq__eq,axiom,
% 8.55/4.74 ! [B: real,C: real,A: real] :
% 8.55/4.74 ( ( ( divide_divide_real @ B @ C )
% 8.55/4.74 = A )
% 8.55/4.74 = ( ( ( C != zero_zero_real )
% 8.55/4.74 => ( B
% 8.55/4.74 = ( times_times_real @ A @ C ) ) )
% 8.55/4.74 & ( ( C = zero_zero_real )
% 8.55/4.74 => ( A = zero_zero_real ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_772_divide__eq__eq,axiom,
% 8.55/4.74 ! [B: rat,C: rat,A: rat] :
% 8.55/4.74 ( ( ( divide_divide_rat @ B @ C )
% 8.55/4.74 = A )
% 8.55/4.74 = ( ( ( C != zero_zero_rat )
% 8.55/4.74 => ( B
% 8.55/4.74 = ( times_times_rat @ A @ C ) ) )
% 8.55/4.74 & ( ( C = zero_zero_rat )
% 8.55/4.74 => ( A = zero_zero_rat ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_773_eq__divide__eq,axiom,
% 8.55/4.74 ! [A: complex,B: complex,C: complex] :
% 8.55/4.74 ( ( A
% 8.55/4.74 = ( divide1717551699836669952omplex @ B @ C ) )
% 8.55/4.74 = ( ( ( C != zero_zero_complex )
% 8.55/4.74 => ( ( times_times_complex @ A @ C )
% 8.55/4.74 = B ) )
% 8.55/4.74 & ( ( C = zero_zero_complex )
% 8.55/4.74 => ( A = zero_zero_complex ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_774_eq__divide__eq,axiom,
% 8.55/4.74 ! [A: real,B: real,C: real] :
% 8.55/4.74 ( ( A
% 8.55/4.74 = ( divide_divide_real @ B @ C ) )
% 8.55/4.74 = ( ( ( C != zero_zero_real )
% 8.55/4.74 => ( ( times_times_real @ A @ C )
% 8.55/4.74 = B ) )
% 8.55/4.74 & ( ( C = zero_zero_real )
% 8.55/4.74 => ( A = zero_zero_real ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_775_eq__divide__eq,axiom,
% 8.55/4.74 ! [A: rat,B: rat,C: rat] :
% 8.55/4.74 ( ( A
% 8.55/4.74 = ( divide_divide_rat @ B @ C ) )
% 8.55/4.74 = ( ( ( C != zero_zero_rat )
% 8.55/4.74 => ( ( times_times_rat @ A @ C )
% 8.55/4.74 = B ) )
% 8.55/4.74 & ( ( C = zero_zero_rat )
% 8.55/4.74 => ( A = zero_zero_rat ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_776_divide__eq__imp,axiom,
% 8.55/4.74 ! [C: complex,B: complex,A: complex] :
% 8.55/4.74 ( ( C != zero_zero_complex )
% 8.55/4.74 => ( ( B
% 8.55/4.74 = ( times_times_complex @ A @ C ) )
% 8.55/4.74 => ( ( divide1717551699836669952omplex @ B @ C )
% 8.55/4.74 = A ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_777_divide__eq__imp,axiom,
% 8.55/4.74 ! [C: real,B: real,A: real] :
% 8.55/4.74 ( ( C != zero_zero_real )
% 8.55/4.74 => ( ( B
% 8.55/4.74 = ( times_times_real @ A @ C ) )
% 8.55/4.74 => ( ( divide_divide_real @ B @ C )
% 8.55/4.74 = A ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_778_divide__eq__imp,axiom,
% 8.55/4.74 ! [C: rat,B: rat,A: rat] :
% 8.55/4.74 ( ( C != zero_zero_rat )
% 8.55/4.74 => ( ( B
% 8.55/4.74 = ( times_times_rat @ A @ C ) )
% 8.55/4.74 => ( ( divide_divide_rat @ B @ C )
% 8.55/4.74 = A ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_779_eq__divide__imp,axiom,
% 8.55/4.74 ! [C: complex,A: complex,B: complex] :
% 8.55/4.74 ( ( C != zero_zero_complex )
% 8.55/4.74 => ( ( ( times_times_complex @ A @ C )
% 8.55/4.74 = B )
% 8.55/4.74 => ( A
% 8.55/4.74 = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_780_eq__divide__imp,axiom,
% 8.55/4.74 ! [C: real,A: real,B: real] :
% 8.55/4.74 ( ( C != zero_zero_real )
% 8.55/4.74 => ( ( ( times_times_real @ A @ C )
% 8.55/4.74 = B )
% 8.55/4.74 => ( A
% 8.55/4.74 = ( divide_divide_real @ B @ C ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_781_eq__divide__imp,axiom,
% 8.55/4.74 ! [C: rat,A: rat,B: rat] :
% 8.55/4.74 ( ( C != zero_zero_rat )
% 8.55/4.74 => ( ( ( times_times_rat @ A @ C )
% 8.55/4.74 = B )
% 8.55/4.74 => ( A
% 8.55/4.74 = ( divide_divide_rat @ B @ C ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_782_nonzero__divide__eq__eq,axiom,
% 8.55/4.74 ! [C: complex,B: complex,A: complex] :
% 8.55/4.74 ( ( C != zero_zero_complex )
% 8.55/4.74 => ( ( ( divide1717551699836669952omplex @ B @ C )
% 8.55/4.74 = A )
% 8.55/4.74 = ( B
% 8.55/4.74 = ( times_times_complex @ A @ C ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_783_nonzero__divide__eq__eq,axiom,
% 8.55/4.74 ! [C: real,B: real,A: real] :
% 8.55/4.74 ( ( C != zero_zero_real )
% 8.55/4.74 => ( ( ( divide_divide_real @ B @ C )
% 8.55/4.74 = A )
% 8.55/4.74 = ( B
% 8.55/4.74 = ( times_times_real @ A @ C ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_784_nonzero__divide__eq__eq,axiom,
% 8.55/4.74 ! [C: rat,B: rat,A: rat] :
% 8.55/4.74 ( ( C != zero_zero_rat )
% 8.55/4.74 => ( ( ( divide_divide_rat @ B @ C )
% 8.55/4.74 = A )
% 8.55/4.74 = ( B
% 8.55/4.74 = ( times_times_rat @ A @ C ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_785_nonzero__eq__divide__eq,axiom,
% 8.55/4.74 ! [C: complex,A: complex,B: complex] :
% 8.55/4.74 ( ( C != zero_zero_complex )
% 8.55/4.74 => ( ( A
% 8.55/4.74 = ( divide1717551699836669952omplex @ B @ C ) )
% 8.55/4.74 = ( ( times_times_complex @ A @ C )
% 8.55/4.74 = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_786_nonzero__eq__divide__eq,axiom,
% 8.55/4.74 ! [C: real,A: real,B: real] :
% 8.55/4.74 ( ( C != zero_zero_real )
% 8.55/4.74 => ( ( A
% 8.55/4.74 = ( divide_divide_real @ B @ C ) )
% 8.55/4.74 = ( ( times_times_real @ A @ C )
% 8.55/4.74 = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_787_nonzero__eq__divide__eq,axiom,
% 8.55/4.74 ! [C: rat,A: rat,B: rat] :
% 8.55/4.74 ( ( C != zero_zero_rat )
% 8.55/4.74 => ( ( A
% 8.55/4.74 = ( divide_divide_rat @ B @ C ) )
% 8.55/4.74 = ( ( times_times_rat @ A @ C )
% 8.55/4.74 = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_788_mult__numeral__1,axiom,
% 8.55/4.74 ! [A: complex] :
% 8.55/4.74 ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_789_mult__numeral__1,axiom,
% 8.55/4.74 ! [A: real] :
% 8.55/4.74 ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_790_mult__numeral__1,axiom,
% 8.55/4.74 ! [A: rat] :
% 8.55/4.74 ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_791_mult__numeral__1,axiom,
% 8.55/4.74 ! [A: nat] :
% 8.55/4.74 ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_792_mult__numeral__1,axiom,
% 8.55/4.74 ! [A: int] :
% 8.55/4.74 ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_793_mult__numeral__1__right,axiom,
% 8.55/4.74 ! [A: complex] :
% 8.55/4.74 ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_794_mult__numeral__1__right,axiom,
% 8.55/4.74 ! [A: real] :
% 8.55/4.74 ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_795_mult__numeral__1__right,axiom,
% 8.55/4.74 ! [A: rat] :
% 8.55/4.74 ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_796_mult__numeral__1__right,axiom,
% 8.55/4.74 ! [A: nat] :
% 8.55/4.74 ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_797_mult__numeral__1__right,axiom,
% 8.55/4.74 ! [A: int] :
% 8.55/4.74 ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
% 8.55/4.74 = A ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_798_div__mult2__numeral__eq,axiom,
% 8.55/4.74 ! [A: nat,K: num,L: num] :
% 8.55/4.74 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
% 8.55/4.74 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_799_div__mult2__numeral__eq,axiom,
% 8.55/4.74 ! [A: int,K: num,L: num] :
% 8.55/4.74 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
% 8.55/4.74 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_800_dvd__div__mult,axiom,
% 8.55/4.74 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ C @ B )
% 8.55/4.74 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
% 8.55/4.74 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_801_dvd__div__mult,axiom,
% 8.55/4.74 ! [C: nat,B: nat,A: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ C @ B )
% 8.55/4.74 => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
% 8.55/4.74 = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_802_dvd__div__mult,axiom,
% 8.55/4.74 ! [C: int,B: int,A: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ C @ B )
% 8.55/4.74 => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
% 8.55/4.74 = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_803_div__mult__swap,axiom,
% 8.55/4.74 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ C @ B )
% 8.55/4.74 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 8.55/4.74 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_804_div__mult__swap,axiom,
% 8.55/4.74 ! [C: nat,B: nat,A: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ C @ B )
% 8.55/4.74 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 8.55/4.74 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_805_div__mult__swap,axiom,
% 8.55/4.74 ! [C: int,B: int,A: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ C @ B )
% 8.55/4.74 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 8.55/4.74 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_806_div__div__eq__right,axiom,
% 8.55/4.74 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ C @ B )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ B @ A )
% 8.55/4.74 => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 8.55/4.74 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_807_div__div__eq__right,axiom,
% 8.55/4.74 ! [C: nat,B: nat,A: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ C @ B )
% 8.55/4.74 => ( ( dvd_dvd_nat @ B @ A )
% 8.55/4.74 => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 8.55/4.74 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_808_div__div__eq__right,axiom,
% 8.55/4.74 ! [C: int,B: int,A: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ C @ B )
% 8.55/4.74 => ( ( dvd_dvd_int @ B @ A )
% 8.55/4.74 => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
% 8.55/4.74 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_809_dvd__div__mult2__eq,axiom,
% 8.55/4.74 ! [B: code_integer,C: code_integer,A: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
% 8.55/4.74 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 8.55/4.74 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_810_dvd__div__mult2__eq,axiom,
% 8.55/4.74 ! [B: nat,C: nat,A: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
% 8.55/4.74 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 8.55/4.74 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_811_dvd__div__mult2__eq,axiom,
% 8.55/4.74 ! [B: int,C: int,A: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
% 8.55/4.74 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 8.55/4.74 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_812_dvd__mult__imp__div,axiom,
% 8.55/4.74 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
% 8.55/4.74 => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_813_dvd__mult__imp__div,axiom,
% 8.55/4.74 ! [A: nat,C: nat,B: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
% 8.55/4.74 => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_814_dvd__mult__imp__div,axiom,
% 8.55/4.74 ! [A: int,C: int,B: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
% 8.55/4.74 => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_815_div__mult__div__if__dvd,axiom,
% 8.55/4.74 ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ B @ A )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ D @ C )
% 8.55/4.74 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 8.55/4.74 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_816_div__mult__div__if__dvd,axiom,
% 8.55/4.74 ! [B: nat,A: nat,D: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ B @ A )
% 8.55/4.74 => ( ( dvd_dvd_nat @ D @ C )
% 8.55/4.74 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
% 8.55/4.74 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_817_div__mult__div__if__dvd,axiom,
% 8.55/4.74 ! [B: int,A: int,D: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ B @ A )
% 8.55/4.74 => ( ( dvd_dvd_int @ D @ C )
% 8.55/4.74 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
% 8.55/4.74 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_818_div__mult2__eq_H,axiom,
% 8.55/4.74 ! [A: nat,M: nat,N: nat] :
% 8.55/4.74 ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 8.55/4.74 = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_819_div__mult2__eq_H,axiom,
% 8.55/4.74 ! [A: int,M: nat,N: nat] :
% 8.55/4.74 ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.55/4.74 = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_820_right__inverse__eq,axiom,
% 8.55/4.74 ! [B: complex,A: complex] :
% 8.55/4.74 ( ( B != zero_zero_complex )
% 8.55/4.74 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 8.55/4.74 = one_one_complex )
% 8.55/4.74 = ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_821_right__inverse__eq,axiom,
% 8.55/4.74 ! [B: real,A: real] :
% 8.55/4.74 ( ( B != zero_zero_real )
% 8.55/4.74 => ( ( ( divide_divide_real @ A @ B )
% 8.55/4.74 = one_one_real )
% 8.55/4.74 = ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_822_right__inverse__eq,axiom,
% 8.55/4.74 ! [B: rat,A: rat] :
% 8.55/4.74 ( ( B != zero_zero_rat )
% 8.55/4.74 => ( ( ( divide_divide_rat @ A @ B )
% 8.55/4.74 = one_one_rat )
% 8.55/4.74 = ( A = B ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_823_numeral__One,axiom,
% 8.55/4.74 ( ( numera6690914467698888265omplex @ one )
% 8.55/4.74 = one_one_complex ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_824_numeral__One,axiom,
% 8.55/4.74 ( ( numeral_numeral_real @ one )
% 8.55/4.74 = one_one_real ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_825_numeral__One,axiom,
% 8.55/4.74 ( ( numeral_numeral_rat @ one )
% 8.55/4.74 = one_one_rat ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_826_numeral__One,axiom,
% 8.55/4.74 ( ( numeral_numeral_nat @ one )
% 8.55/4.74 = one_one_nat ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_827_numeral__One,axiom,
% 8.55/4.74 ( ( numeral_numeral_int @ one )
% 8.55/4.74 = one_one_int ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_828_not__is__unit__0,axiom,
% 8.55/4.74 ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_829_not__is__unit__0,axiom,
% 8.55/4.74 ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_830_not__is__unit__0,axiom,
% 8.55/4.74 ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_831_unit__div__cancel,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.55/4.74 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 8.55/4.74 = ( divide6298287555418463151nteger @ C @ A ) )
% 8.55/4.74 = ( B = C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_832_unit__div__cancel,axiom,
% 8.55/4.74 ! [A: nat,B: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.55/4.74 => ( ( ( divide_divide_nat @ B @ A )
% 8.55/4.74 = ( divide_divide_nat @ C @ A ) )
% 8.55/4.74 = ( B = C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_833_unit__div__cancel,axiom,
% 8.55/4.74 ! [A: int,B: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.55/4.74 => ( ( ( divide_divide_int @ B @ A )
% 8.55/4.74 = ( divide_divide_int @ C @ A ) )
% 8.55/4.74 = ( B = C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_834_div__unit__dvd__iff,axiom,
% 8.55/4.74 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 8.55/4.74 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_835_div__unit__dvd__iff,axiom,
% 8.55/4.74 ! [B: nat,A: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 8.55/4.74 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_836_div__unit__dvd__iff,axiom,
% 8.55/4.74 ! [B: int,A: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 8.55/4.74 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_837_dvd__div__unit__iff,axiom,
% 8.55/4.74 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.55/4.74 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
% 8.55/4.74 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_838_dvd__div__unit__iff,axiom,
% 8.55/4.74 ! [B: nat,A: nat,C: nat] :
% 8.55/4.74 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
% 8.55/4.74 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_839_dvd__div__unit__iff,axiom,
% 8.55/4.74 ! [B: int,A: int,C: int] :
% 8.55/4.74 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
% 8.55/4.74 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_840_numerals_I1_J,axiom,
% 8.55/4.74 ( ( numeral_numeral_nat @ one )
% 8.55/4.74 = one_one_nat ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_841_One__nat__def,axiom,
% 8.55/4.74 ( one_one_nat
% 8.55/4.74 = ( suc @ zero_zero_nat ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_842_diff__Suc__eq__diff__pred,axiom,
% 8.55/4.74 ! [M: nat,N: nat] :
% 8.55/4.74 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 8.55/4.74 = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_843_divide__eq__eq__numeral_I1_J,axiom,
% 8.55/4.74 ! [B: complex,C: complex,W: num] :
% 8.55/4.74 ( ( ( divide1717551699836669952omplex @ B @ C )
% 8.55/4.74 = ( numera6690914467698888265omplex @ W ) )
% 8.55/4.74 = ( ( ( C != zero_zero_complex )
% 8.55/4.74 => ( B
% 8.55/4.74 = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
% 8.55/4.74 & ( ( C = zero_zero_complex )
% 8.55/4.74 => ( ( numera6690914467698888265omplex @ W )
% 8.55/4.74 = zero_zero_complex ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_844_divide__eq__eq__numeral_I1_J,axiom,
% 8.55/4.74 ! [B: real,C: real,W: num] :
% 8.55/4.74 ( ( ( divide_divide_real @ B @ C )
% 8.55/4.74 = ( numeral_numeral_real @ W ) )
% 8.55/4.74 = ( ( ( C != zero_zero_real )
% 8.55/4.74 => ( B
% 8.55/4.74 = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 8.55/4.74 & ( ( C = zero_zero_real )
% 8.55/4.74 => ( ( numeral_numeral_real @ W )
% 8.55/4.74 = zero_zero_real ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_845_divide__eq__eq__numeral_I1_J,axiom,
% 8.55/4.74 ! [B: rat,C: rat,W: num] :
% 8.55/4.74 ( ( ( divide_divide_rat @ B @ C )
% 8.55/4.74 = ( numeral_numeral_rat @ W ) )
% 8.55/4.74 = ( ( ( C != zero_zero_rat )
% 8.55/4.74 => ( B
% 8.55/4.74 = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 8.55/4.74 & ( ( C = zero_zero_rat )
% 8.55/4.74 => ( ( numeral_numeral_rat @ W )
% 8.55/4.74 = zero_zero_rat ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_846_eq__divide__eq__numeral_I1_J,axiom,
% 8.55/4.74 ! [W: num,B: complex,C: complex] :
% 8.55/4.74 ( ( ( numera6690914467698888265omplex @ W )
% 8.55/4.74 = ( divide1717551699836669952omplex @ B @ C ) )
% 8.55/4.74 = ( ( ( C != zero_zero_complex )
% 8.55/4.74 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
% 8.55/4.74 = B ) )
% 8.55/4.74 & ( ( C = zero_zero_complex )
% 8.55/4.74 => ( ( numera6690914467698888265omplex @ W )
% 8.55/4.74 = zero_zero_complex ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_847_eq__divide__eq__numeral_I1_J,axiom,
% 8.55/4.74 ! [W: num,B: real,C: real] :
% 8.55/4.74 ( ( ( numeral_numeral_real @ W )
% 8.55/4.74 = ( divide_divide_real @ B @ C ) )
% 8.55/4.74 = ( ( ( C != zero_zero_real )
% 8.55/4.74 => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
% 8.55/4.74 = B ) )
% 8.55/4.74 & ( ( C = zero_zero_real )
% 8.55/4.74 => ( ( numeral_numeral_real @ W )
% 8.55/4.74 = zero_zero_real ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_848_eq__divide__eq__numeral_I1_J,axiom,
% 8.55/4.74 ! [W: num,B: rat,C: rat] :
% 8.55/4.74 ( ( ( numeral_numeral_rat @ W )
% 8.55/4.74 = ( divide_divide_rat @ B @ C ) )
% 8.55/4.74 = ( ( ( C != zero_zero_rat )
% 8.55/4.74 => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
% 8.55/4.74 = B ) )
% 8.55/4.74 & ( ( C = zero_zero_rat )
% 8.55/4.74 => ( ( numeral_numeral_rat @ W )
% 8.55/4.74 = zero_zero_rat ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_849_add__divide__eq__if__simps_I4_J,axiom,
% 8.55/4.74 ! [Z2: complex,A: complex,B: complex] :
% 8.55/4.74 ( ( ( Z2 = zero_zero_complex )
% 8.55/4.74 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 8.55/4.74 = A ) )
% 8.55/4.74 & ( ( Z2 != zero_zero_complex )
% 8.55/4.74 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 8.55/4.74 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_850_add__divide__eq__if__simps_I4_J,axiom,
% 8.55/4.74 ! [Z2: real,A: real,B: real] :
% 8.55/4.74 ( ( ( Z2 = zero_zero_real )
% 8.55/4.74 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 8.55/4.74 = A ) )
% 8.55/4.74 & ( ( Z2 != zero_zero_real )
% 8.55/4.74 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 8.55/4.74 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_851_add__divide__eq__if__simps_I4_J,axiom,
% 8.55/4.74 ! [Z2: rat,A: rat,B: rat] :
% 8.55/4.74 ( ( ( Z2 = zero_zero_rat )
% 8.55/4.74 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 8.55/4.74 = A ) )
% 8.55/4.74 & ( ( Z2 != zero_zero_rat )
% 8.55/4.74 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 8.55/4.74 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_852_diff__frac__eq,axiom,
% 8.55/4.74 ! [Y: complex,Z2: complex,X: complex,W: complex] :
% 8.55/4.74 ( ( Y != zero_zero_complex )
% 8.55/4.74 => ( ( Z2 != zero_zero_complex )
% 8.55/4.74 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z2 ) )
% 8.55/4.74 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_853_diff__frac__eq,axiom,
% 8.55/4.74 ! [Y: real,Z2: real,X: real,W: real] :
% 8.55/4.74 ( ( Y != zero_zero_real )
% 8.55/4.74 => ( ( Z2 != zero_zero_real )
% 8.55/4.74 => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
% 8.55/4.74 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_854_diff__frac__eq,axiom,
% 8.55/4.74 ! [Y: rat,Z2: rat,X: rat,W: rat] :
% 8.55/4.74 ( ( Y != zero_zero_rat )
% 8.55/4.74 => ( ( Z2 != zero_zero_rat )
% 8.55/4.74 => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z2 ) )
% 8.55/4.74 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_855_diff__divide__eq__iff,axiom,
% 8.55/4.74 ! [Z2: complex,X: complex,Y: complex] :
% 8.55/4.74 ( ( Z2 != zero_zero_complex )
% 8.55/4.74 => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
% 8.55/4.74 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_856_diff__divide__eq__iff,axiom,
% 8.55/4.74 ! [Z2: real,X: real,Y: real] :
% 8.55/4.74 ( ( Z2 != zero_zero_real )
% 8.55/4.74 => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z2 ) )
% 8.55/4.74 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_857_diff__divide__eq__iff,axiom,
% 8.55/4.74 ! [Z2: rat,X: rat,Y: rat] :
% 8.55/4.74 ( ( Z2 != zero_zero_rat )
% 8.55/4.74 => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z2 ) )
% 8.55/4.74 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_858_divide__diff__eq__iff,axiom,
% 8.55/4.74 ! [Z2: complex,X: complex,Y: complex] :
% 8.55/4.74 ( ( Z2 != zero_zero_complex )
% 8.55/4.74 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z2 ) @ Y )
% 8.55/4.74 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_859_divide__diff__eq__iff,axiom,
% 8.55/4.74 ! [Z2: real,X: real,Y: real] :
% 8.55/4.74 ( ( Z2 != zero_zero_real )
% 8.55/4.74 => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z2 ) @ Y )
% 8.55/4.74 = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_860_divide__diff__eq__iff,axiom,
% 8.55/4.74 ! [Z2: rat,X: rat,Y: rat] :
% 8.55/4.74 ( ( Z2 != zero_zero_rat )
% 8.55/4.74 => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y )
% 8.55/4.74 = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_861_dvd__div__div__eq__mult,axiom,
% 8.55/4.74 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 8.55/4.74 ( ( A != zero_z3403309356797280102nteger )
% 8.55/4.74 => ( ( C != zero_z3403309356797280102nteger )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ A @ B )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ C @ D )
% 8.55/4.74 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 8.55/4.74 = ( divide6298287555418463151nteger @ D @ C ) )
% 8.55/4.74 = ( ( times_3573771949741848930nteger @ B @ C )
% 8.55/4.74 = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_862_dvd__div__div__eq__mult,axiom,
% 8.55/4.74 ! [A: nat,C: nat,B: nat,D: nat] :
% 8.55/4.74 ( ( A != zero_zero_nat )
% 8.55/4.74 => ( ( C != zero_zero_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.74 => ( ( dvd_dvd_nat @ C @ D )
% 8.55/4.74 => ( ( ( divide_divide_nat @ B @ A )
% 8.55/4.74 = ( divide_divide_nat @ D @ C ) )
% 8.55/4.74 = ( ( times_times_nat @ B @ C )
% 8.55/4.74 = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_863_dvd__div__div__eq__mult,axiom,
% 8.55/4.74 ! [A: int,C: int,B: int,D: int] :
% 8.55/4.74 ( ( A != zero_zero_int )
% 8.55/4.74 => ( ( C != zero_zero_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ A @ B )
% 8.55/4.74 => ( ( dvd_dvd_int @ C @ D )
% 8.55/4.74 => ( ( ( divide_divide_int @ B @ A )
% 8.55/4.74 = ( divide_divide_int @ D @ C ) )
% 8.55/4.74 = ( ( times_times_int @ B @ C )
% 8.55/4.74 = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_864_dvd__div__iff__mult,axiom,
% 8.55/4.74 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.55/4.74 ( ( C != zero_z3403309356797280102nteger )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ C @ B )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 8.55/4.74 = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_865_dvd__div__iff__mult,axiom,
% 8.55/4.74 ! [C: nat,B: nat,A: nat] :
% 8.55/4.74 ( ( C != zero_zero_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ C @ B )
% 8.55/4.74 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 8.55/4.74 = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_866_dvd__div__iff__mult,axiom,
% 8.55/4.74 ! [C: int,B: int,A: int] :
% 8.55/4.74 ( ( C != zero_zero_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ C @ B )
% 8.55/4.74 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
% 8.55/4.74 = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_867_div__dvd__iff__mult,axiom,
% 8.55/4.74 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.55/4.74 ( ( B != zero_z3403309356797280102nteger )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ B @ A )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 8.55/4.74 = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_868_div__dvd__iff__mult,axiom,
% 8.55/4.74 ! [B: nat,A: nat,C: nat] :
% 8.55/4.74 ( ( B != zero_zero_nat )
% 8.55/4.74 => ( ( dvd_dvd_nat @ B @ A )
% 8.55/4.74 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 8.55/4.74 = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_869_div__dvd__iff__mult,axiom,
% 8.55/4.74 ! [B: int,A: int,C: int] :
% 8.55/4.74 ( ( B != zero_zero_int )
% 8.55/4.74 => ( ( dvd_dvd_int @ B @ A )
% 8.55/4.74 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 8.55/4.74 = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% 8.55/4.74
% 8.55/4.74 thf(fact_870_dvd__div__eq__mult,axiom,
% 8.55/4.74 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.55/4.74 ( ( A != zero_z3403309356797280102nteger )
% 8.55/4.74 => ( ( dvd_dvd_Code_integer @ A @ B )
% 8.55/4.74 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 8.55/4.75 = C )
% 8.55/4.75 = ( B
% 8.55/4.75 = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_871_dvd__div__eq__mult,axiom,
% 8.55/4.75 ! [A: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( A != zero_zero_nat )
% 8.55/4.75 => ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 => ( ( ( divide_divide_nat @ B @ A )
% 8.55/4.75 = C )
% 8.55/4.75 = ( B
% 8.55/4.75 = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_872_dvd__div__eq__mult,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( A != zero_zero_int )
% 8.55/4.75 => ( ( dvd_dvd_int @ A @ B )
% 8.55/4.75 => ( ( ( divide_divide_int @ B @ A )
% 8.55/4.75 = C )
% 8.55/4.75 = ( B
% 8.55/4.75 = ( times_times_int @ C @ A ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_873_unit__div__eq__0__iff,axiom,
% 8.55/4.75 ! [B: code_integer,A: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.55/4.75 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 8.55/4.75 = zero_z3403309356797280102nteger )
% 8.55/4.75 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_874_unit__div__eq__0__iff,axiom,
% 8.55/4.75 ! [B: nat,A: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.55/4.75 => ( ( ( divide_divide_nat @ A @ B )
% 8.55/4.75 = zero_zero_nat )
% 8.55/4.75 = ( A = zero_zero_nat ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_875_unit__div__eq__0__iff,axiom,
% 8.55/4.75 ! [B: int,A: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.55/4.75 => ( ( ( divide_divide_int @ A @ B )
% 8.55/4.75 = zero_zero_int )
% 8.55/4.75 = ( A = zero_zero_int ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_876_gcd__nat_Onot__eq__order__implies__strict,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( A != B )
% 8.55/4.75 => ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 => ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 & ( A != B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_877_gcd__nat_Ostrict__implies__not__eq,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 & ( A != B ) )
% 8.55/4.75 => ( A != B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_878_gcd__nat_Ostrict__implies__order,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 & ( A != B ) )
% 8.55/4.75 => ( dvd_dvd_nat @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_879_gcd__nat_Ostrict__iff__order,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 & ( A != B ) )
% 8.55/4.75 = ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 & ( A != B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_880_gcd__nat_Oorder__iff__strict,axiom,
% 8.55/4.75 ( dvd_dvd_nat
% 8.55/4.75 = ( ^ [A2: nat,B2: nat] :
% 8.55/4.75 ( ( ( dvd_dvd_nat @ A2 @ B2 )
% 8.55/4.75 & ( A2 != B2 ) )
% 8.55/4.75 | ( A2 = B2 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_881_gcd__nat_Ostrict__iff__not,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 & ( A != B ) )
% 8.55/4.75 = ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_882_gcd__nat_Ostrict__trans2,axiom,
% 8.55/4.75 ! [A: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 & ( A != B ) )
% 8.55/4.75 => ( ( dvd_dvd_nat @ B @ C )
% 8.55/4.75 => ( ( dvd_dvd_nat @ A @ C )
% 8.55/4.75 & ( A != C ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_883_gcd__nat_Ostrict__trans1,axiom,
% 8.55/4.75 ! [A: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 => ( ( ( dvd_dvd_nat @ B @ C )
% 8.55/4.75 & ( B != C ) )
% 8.55/4.75 => ( ( dvd_dvd_nat @ A @ C )
% 8.55/4.75 & ( A != C ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_884_gcd__nat_Ostrict__trans,axiom,
% 8.55/4.75 ! [A: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 & ( A != B ) )
% 8.55/4.75 => ( ( ( dvd_dvd_nat @ B @ C )
% 8.55/4.75 & ( B != C ) )
% 8.55/4.75 => ( ( dvd_dvd_nat @ A @ C )
% 8.55/4.75 & ( A != C ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_885_gcd__nat_Oantisym,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 => ( ( dvd_dvd_nat @ B @ A )
% 8.55/4.75 => ( A = B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_886_gcd__nat_Oirrefl,axiom,
% 8.55/4.75 ! [A: nat] :
% 8.55/4.75 ~ ( ( dvd_dvd_nat @ A @ A )
% 8.55/4.75 & ( A != A ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_887_gcd__nat_Oeq__iff,axiom,
% 8.55/4.75 ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 8.55/4.75 = ( ^ [A2: nat,B2: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ A2 @ B2 )
% 8.55/4.75 & ( dvd_dvd_nat @ B2 @ A2 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_888_gcd__nat_Otrans,axiom,
% 8.55/4.75 ! [A: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 => ( ( dvd_dvd_nat @ B @ C )
% 8.55/4.75 => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_889_gcd__nat_Orefl,axiom,
% 8.55/4.75 ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_890_gcd__nat_Oasym,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 & ( A != B ) )
% 8.55/4.75 => ~ ( ( dvd_dvd_nat @ B @ A )
% 8.55/4.75 & ( B != A ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_891_evenE,axiom,
% 8.55/4.75 ! [A: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ~ ! [B4: code_integer] :
% 8.55/4.75 ( A
% 8.55/4.75 != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_892_evenE,axiom,
% 8.55/4.75 ! [A: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ~ ! [B4: nat] :
% 8.55/4.75 ( A
% 8.55/4.75 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_893_evenE,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ~ ! [B4: int] :
% 8.55/4.75 ( A
% 8.55/4.75 != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_894_Suc__double__not__eq__double,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.55/4.75 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_895_double__not__eq__Suc__double,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 8.55/4.75 != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_896_odd__one,axiom,
% 8.55/4.75 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_897_odd__one,axiom,
% 8.55/4.75 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_898_odd__one,axiom,
% 8.55/4.75 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_899_even__two__times__div__two,axiom,
% 8.55/4.75 ! [A: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 8.55/4.75 = A ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_900_even__two__times__div__two,axiom,
% 8.55/4.75 ! [A: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.75 = A ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_901_even__two__times__div__two,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 8.55/4.75 = A ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_902_dbl__simps_I3_J,axiom,
% 8.55/4.75 ( ( neg_nu7009210354673126013omplex @ one_one_complex )
% 8.55/4.75 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_903_dbl__simps_I3_J,axiom,
% 8.55/4.75 ( ( neg_numeral_dbl_real @ one_one_real )
% 8.55/4.75 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_904_dbl__simps_I3_J,axiom,
% 8.55/4.75 ( ( neg_numeral_dbl_rat @ one_one_rat )
% 8.55/4.75 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_905_dbl__simps_I3_J,axiom,
% 8.55/4.75 ( ( neg_numeral_dbl_int @ one_one_int )
% 8.55/4.75 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_906_triangle__def,axiom,
% 8.55/4.75 ( nat_triangle
% 8.55/4.75 = ( ^ [N3: nat] : ( divide_divide_nat @ ( times_times_nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_907_set__bit__0,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 8.55/4.75 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_908_set__bit__0,axiom,
% 8.55/4.75 ! [A: nat] :
% 8.55/4.75 ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 8.55/4.75 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_909_real__divide__square__eq,axiom,
% 8.55/4.75 ! [R: real,A: real] :
% 8.55/4.75 ( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
% 8.55/4.75 = ( divide_divide_real @ A @ R ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_910_odd__two__times__div__two__succ,axiom,
% 8.55/4.75 ! [A: code_integer] :
% 8.55/4.75 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 8.55/4.75 = A ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_911_odd__two__times__div__two__succ,axiom,
% 8.55/4.75 ! [A: nat] :
% 8.55/4.75 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 8.55/4.75 = A ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_912_odd__two__times__div__two__succ,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 8.55/4.75 = A ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_913_semiring__parity__class_Oeven__mask__iff,axiom,
% 8.55/4.75 ! [N: nat] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
% 8.55/4.75 = ( N = zero_zero_nat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_914_semiring__parity__class_Oeven__mask__iff,axiom,
% 8.55/4.75 ! [N: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
% 8.55/4.75 = ( N = zero_zero_nat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_915_semiring__parity__class_Oeven__mask__iff,axiom,
% 8.55/4.75 ! [N: nat] :
% 8.55/4.75 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 8.55/4.75 = ( N = zero_zero_nat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_916_zdvd__mono,axiom,
% 8.55/4.75 ! [K: int,M: int,T: int] :
% 8.55/4.75 ( ( K != zero_zero_int )
% 8.55/4.75 => ( ( dvd_dvd_int @ M @ T )
% 8.55/4.75 = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_917_even__succ__div__2,axiom,
% 8.55/4.75 ! [A: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.55/4.75 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_918_even__succ__div__2,axiom,
% 8.55/4.75 ! [A: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.75 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_919_even__succ__div__2,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.55/4.75 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_920_odd__succ__div__two,axiom,
% 8.55/4.75 ! [A: code_integer] :
% 8.55/4.75 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.55/4.75 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_921_odd__succ__div__two,axiom,
% 8.55/4.75 ! [A: nat] :
% 8.55/4.75 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.75 = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_922_odd__succ__div__two,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.55/4.75 = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_923_even__succ__div__two,axiom,
% 8.55/4.75 ! [A: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.55/4.75 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_924_even__succ__div__two,axiom,
% 8.55/4.75 ! [A: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.75 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_925_even__succ__div__two,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.55/4.75 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_926_inf__period_I1_J,axiom,
% 8.55/4.75 ! [P: real > $o,D3: real,Q: real > $o] :
% 8.55/4.75 ( ! [X4: real,K3: real] :
% 8.55/4.75 ( ( P @ X4 )
% 8.55/4.75 = ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D3 ) ) ) )
% 8.55/4.75 => ( ! [X4: real,K3: real] :
% 8.55/4.75 ( ( Q @ X4 )
% 8.55/4.75 = ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D3 ) ) ) )
% 8.55/4.75 => ! [X5: real,K4: real] :
% 8.55/4.75 ( ( ( P @ X5 )
% 8.55/4.75 & ( Q @ X5 ) )
% 8.55/4.75 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) )
% 8.55/4.75 & ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_927_inf__period_I1_J,axiom,
% 8.55/4.75 ! [P: rat > $o,D3: rat,Q: rat > $o] :
% 8.55/4.75 ( ! [X4: rat,K3: rat] :
% 8.55/4.75 ( ( P @ X4 )
% 8.55/4.75 = ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) ) )
% 8.55/4.75 => ( ! [X4: rat,K3: rat] :
% 8.55/4.75 ( ( Q @ X4 )
% 8.55/4.75 = ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) ) )
% 8.55/4.75 => ! [X5: rat,K4: rat] :
% 8.55/4.75 ( ( ( P @ X5 )
% 8.55/4.75 & ( Q @ X5 ) )
% 8.55/4.75 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) )
% 8.55/4.75 & ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_928_inf__period_I1_J,axiom,
% 8.55/4.75 ! [P: int > $o,D3: int,Q: int > $o] :
% 8.55/4.75 ( ! [X4: int,K3: int] :
% 8.55/4.75 ( ( P @ X4 )
% 8.55/4.75 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
% 8.55/4.75 => ( ! [X4: int,K3: int] :
% 8.55/4.75 ( ( Q @ X4 )
% 8.55/4.75 = ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
% 8.55/4.75 => ! [X5: int,K4: int] :
% 8.55/4.75 ( ( ( P @ X5 )
% 8.55/4.75 & ( Q @ X5 ) )
% 8.55/4.75 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
% 8.55/4.75 & ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_929_even__odd__cases,axiom,
% 8.55/4.75 ! [X: nat] :
% 8.55/4.75 ( ! [N2: nat] :
% 8.55/4.75 ( X
% 8.55/4.75 != ( plus_plus_nat @ N2 @ N2 ) )
% 8.55/4.75 => ~ ! [N2: nat] :
% 8.55/4.75 ( X
% 8.55/4.75 != ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_930_add__right__cancel,axiom,
% 8.55/4.75 ! [B: real,A: real,C: real] :
% 8.55/4.75 ( ( ( plus_plus_real @ B @ A )
% 8.55/4.75 = ( plus_plus_real @ C @ A ) )
% 8.55/4.75 = ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_931_add__right__cancel,axiom,
% 8.55/4.75 ! [B: rat,A: rat,C: rat] :
% 8.55/4.75 ( ( ( plus_plus_rat @ B @ A )
% 8.55/4.75 = ( plus_plus_rat @ C @ A ) )
% 8.55/4.75 = ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_932_add__right__cancel,axiom,
% 8.55/4.75 ! [B: nat,A: nat,C: nat] :
% 8.55/4.75 ( ( ( plus_plus_nat @ B @ A )
% 8.55/4.75 = ( plus_plus_nat @ C @ A ) )
% 8.55/4.75 = ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_933_add__right__cancel,axiom,
% 8.55/4.75 ! [B: int,A: int,C: int] :
% 8.55/4.75 ( ( ( plus_plus_int @ B @ A )
% 8.55/4.75 = ( plus_plus_int @ C @ A ) )
% 8.55/4.75 = ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_934_add__left__cancel,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( ( plus_plus_real @ A @ B )
% 8.55/4.75 = ( plus_plus_real @ A @ C ) )
% 8.55/4.75 = ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_935_add__left__cancel,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( ( plus_plus_rat @ A @ B )
% 8.55/4.75 = ( plus_plus_rat @ A @ C ) )
% 8.55/4.75 = ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_936_add__left__cancel,axiom,
% 8.55/4.75 ! [A: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( ( plus_plus_nat @ A @ B )
% 8.55/4.75 = ( plus_plus_nat @ A @ C ) )
% 8.55/4.75 = ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_937_add__left__cancel,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( ( plus_plus_int @ A @ B )
% 8.55/4.75 = ( plus_plus_int @ A @ C ) )
% 8.55/4.75 = ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_938_semiring__norm_I6_J,axiom,
% 8.55/4.75 ! [M: num,N: num] :
% 8.55/4.75 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.55/4.75 = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_939_add__Suc__right,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 8.55/4.75 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_940_Nat_Oadd__0__right,axiom,
% 8.55/4.75 ! [M: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 8.55/4.75 = M ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_941_add__is__0,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( ( plus_plus_nat @ M @ N )
% 8.55/4.75 = zero_zero_nat )
% 8.55/4.75 = ( ( M = zero_zero_nat )
% 8.55/4.75 & ( N = zero_zero_nat ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_942_diff__diff__left,axiom,
% 8.55/4.75 ! [I: nat,J: nat,K: nat] :
% 8.55/4.75 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 8.55/4.75 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_943_push__bit__push__bit,axiom,
% 8.55/4.75 ! [M: nat,N: nat,A: nat] :
% 8.55/4.75 ( ( bit_se547839408752420682it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 8.55/4.75 = ( bit_se547839408752420682it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_944_push__bit__push__bit,axiom,
% 8.55/4.75 ! [M: nat,N: nat,A: int] :
% 8.55/4.75 ( ( bit_se545348938243370406it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
% 8.55/4.75 = ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_945_push__bit__push__bit,axiom,
% 8.55/4.75 ! [M: nat,N: nat,A: code_integer] :
% 8.55/4.75 ( ( bit_se7788150548672797655nteger @ M @ ( bit_se7788150548672797655nteger @ N @ A ) )
% 8.55/4.75 = ( bit_se7788150548672797655nteger @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_946_push__bit__push__bit,axiom,
% 8.55/4.75 ! [M: nat,N: nat,A: code_natural] :
% 8.55/4.75 ( ( bit_se6611745700429515170atural @ M @ ( bit_se6611745700429515170atural @ N @ A ) )
% 8.55/4.75 = ( bit_se6611745700429515170atural @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_947_add_Oright__neutral,axiom,
% 8.55/4.75 ! [A: literal] :
% 8.55/4.75 ( ( plus_plus_literal @ A @ zero_zero_literal )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_948_add_Oright__neutral,axiom,
% 8.55/4.75 ! [A: real] :
% 8.55/4.75 ( ( plus_plus_real @ A @ zero_zero_real )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_949_add_Oright__neutral,axiom,
% 8.55/4.75 ! [A: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_950_add_Oright__neutral,axiom,
% 8.55/4.75 ! [A: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_951_add_Oright__neutral,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ( plus_plus_int @ A @ zero_zero_int )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_952_double__zero__sym,axiom,
% 8.55/4.75 ! [A: real] :
% 8.55/4.75 ( ( zero_zero_real
% 8.55/4.75 = ( plus_plus_real @ A @ A ) )
% 8.55/4.75 = ( A = zero_zero_real ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_953_double__zero__sym,axiom,
% 8.55/4.75 ! [A: rat] :
% 8.55/4.75 ( ( zero_zero_rat
% 8.55/4.75 = ( plus_plus_rat @ A @ A ) )
% 8.55/4.75 = ( A = zero_zero_rat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_954_double__zero__sym,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ( zero_zero_int
% 8.55/4.75 = ( plus_plus_int @ A @ A ) )
% 8.55/4.75 = ( A = zero_zero_int ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_955_add__cancel__left__left,axiom,
% 8.55/4.75 ! [B: real,A: real] :
% 8.55/4.75 ( ( ( plus_plus_real @ B @ A )
% 8.55/4.75 = A )
% 8.55/4.75 = ( B = zero_zero_real ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_956_add__cancel__left__left,axiom,
% 8.55/4.75 ! [B: rat,A: rat] :
% 8.55/4.75 ( ( ( plus_plus_rat @ B @ A )
% 8.55/4.75 = A )
% 8.55/4.75 = ( B = zero_zero_rat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_957_add__cancel__left__left,axiom,
% 8.55/4.75 ! [B: nat,A: nat] :
% 8.55/4.75 ( ( ( plus_plus_nat @ B @ A )
% 8.55/4.75 = A )
% 8.55/4.75 = ( B = zero_zero_nat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_958_add__cancel__left__left,axiom,
% 8.55/4.75 ! [B: int,A: int] :
% 8.55/4.75 ( ( ( plus_plus_int @ B @ A )
% 8.55/4.75 = A )
% 8.55/4.75 = ( B = zero_zero_int ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_959_add__cancel__left__right,axiom,
% 8.55/4.75 ! [A: real,B: real] :
% 8.55/4.75 ( ( ( plus_plus_real @ A @ B )
% 8.55/4.75 = A )
% 8.55/4.75 = ( B = zero_zero_real ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_960_add__cancel__left__right,axiom,
% 8.55/4.75 ! [A: rat,B: rat] :
% 8.55/4.75 ( ( ( plus_plus_rat @ A @ B )
% 8.55/4.75 = A )
% 8.55/4.75 = ( B = zero_zero_rat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_961_add__cancel__left__right,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( ( plus_plus_nat @ A @ B )
% 8.55/4.75 = A )
% 8.55/4.75 = ( B = zero_zero_nat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_962_add__cancel__left__right,axiom,
% 8.55/4.75 ! [A: int,B: int] :
% 8.55/4.75 ( ( ( plus_plus_int @ A @ B )
% 8.55/4.75 = A )
% 8.55/4.75 = ( B = zero_zero_int ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_963_add__cancel__right__left,axiom,
% 8.55/4.75 ! [A: real,B: real] :
% 8.55/4.75 ( ( A
% 8.55/4.75 = ( plus_plus_real @ B @ A ) )
% 8.55/4.75 = ( B = zero_zero_real ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_964_add__cancel__right__left,axiom,
% 8.55/4.75 ! [A: rat,B: rat] :
% 8.55/4.75 ( ( A
% 8.55/4.75 = ( plus_plus_rat @ B @ A ) )
% 8.55/4.75 = ( B = zero_zero_rat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_965_add__cancel__right__left,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( A
% 8.55/4.75 = ( plus_plus_nat @ B @ A ) )
% 8.55/4.75 = ( B = zero_zero_nat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_966_add__cancel__right__left,axiom,
% 8.55/4.75 ! [A: int,B: int] :
% 8.55/4.75 ( ( A
% 8.55/4.75 = ( plus_plus_int @ B @ A ) )
% 8.55/4.75 = ( B = zero_zero_int ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_967_add__cancel__right__right,axiom,
% 8.55/4.75 ! [A: real,B: real] :
% 8.55/4.75 ( ( A
% 8.55/4.75 = ( plus_plus_real @ A @ B ) )
% 8.55/4.75 = ( B = zero_zero_real ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_968_add__cancel__right__right,axiom,
% 8.55/4.75 ! [A: rat,B: rat] :
% 8.55/4.75 ( ( A
% 8.55/4.75 = ( plus_plus_rat @ A @ B ) )
% 8.55/4.75 = ( B = zero_zero_rat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_969_add__cancel__right__right,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( A
% 8.55/4.75 = ( plus_plus_nat @ A @ B ) )
% 8.55/4.75 = ( B = zero_zero_nat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_970_add__cancel__right__right,axiom,
% 8.55/4.75 ! [A: int,B: int] :
% 8.55/4.75 ( ( A
% 8.55/4.75 = ( plus_plus_int @ A @ B ) )
% 8.55/4.75 = ( B = zero_zero_int ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_971_add__eq__0__iff__both__eq__0,axiom,
% 8.55/4.75 ! [X: nat,Y: nat] :
% 8.55/4.75 ( ( ( plus_plus_nat @ X @ Y )
% 8.55/4.75 = zero_zero_nat )
% 8.55/4.75 = ( ( X = zero_zero_nat )
% 8.55/4.75 & ( Y = zero_zero_nat ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_972_zero__eq__add__iff__both__eq__0,axiom,
% 8.55/4.75 ! [X: nat,Y: nat] :
% 8.55/4.75 ( ( zero_zero_nat
% 8.55/4.75 = ( plus_plus_nat @ X @ Y ) )
% 8.55/4.75 = ( ( X = zero_zero_nat )
% 8.55/4.75 & ( Y = zero_zero_nat ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_973_add__0,axiom,
% 8.55/4.75 ! [A: literal] :
% 8.55/4.75 ( ( plus_plus_literal @ zero_zero_literal @ A )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_974_add__0,axiom,
% 8.55/4.75 ! [A: real] :
% 8.55/4.75 ( ( plus_plus_real @ zero_zero_real @ A )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_975_add__0,axiom,
% 8.55/4.75 ! [A: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_976_add__0,axiom,
% 8.55/4.75 ! [A: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_977_add__0,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ( plus_plus_int @ zero_zero_int @ A )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_978_double__eq__0__iff,axiom,
% 8.55/4.75 ! [A: real] :
% 8.55/4.75 ( ( ( plus_plus_real @ A @ A )
% 8.55/4.75 = zero_zero_real )
% 8.55/4.75 = ( A = zero_zero_real ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_979_double__eq__0__iff,axiom,
% 8.55/4.75 ! [A: rat] :
% 8.55/4.75 ( ( ( plus_plus_rat @ A @ A )
% 8.55/4.75 = zero_zero_rat )
% 8.55/4.75 = ( A = zero_zero_rat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_980_double__eq__0__iff,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ( ( plus_plus_int @ A @ A )
% 8.55/4.75 = zero_zero_int )
% 8.55/4.75 = ( A = zero_zero_int ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_981_numeral__plus__numeral,axiom,
% 8.55/4.75 ! [M: num,N: num] :
% 8.55/4.75 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 8.55/4.75 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_982_numeral__plus__numeral,axiom,
% 8.55/4.75 ! [M: num,N: num] :
% 8.55/4.75 ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 8.55/4.75 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_983_numeral__plus__numeral,axiom,
% 8.55/4.75 ! [M: num,N: num] :
% 8.55/4.75 ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 8.55/4.75 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_984_numeral__plus__numeral,axiom,
% 8.55/4.75 ! [M: num,N: num] :
% 8.55/4.75 ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.55/4.75 = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_985_numeral__plus__numeral,axiom,
% 8.55/4.75 ! [M: num,N: num] :
% 8.55/4.75 ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.55/4.75 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_986_add__numeral__left,axiom,
% 8.55/4.75 ! [V: num,W: num,Z2: complex] :
% 8.55/4.75 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z2 ) )
% 8.55/4.75 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_987_add__numeral__left,axiom,
% 8.55/4.75 ! [V: num,W: num,Z2: real] :
% 8.55/4.75 ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
% 8.55/4.75 = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_988_add__numeral__left,axiom,
% 8.55/4.75 ! [V: num,W: num,Z2: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z2 ) )
% 8.55/4.75 = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_989_add__numeral__left,axiom,
% 8.55/4.75 ! [V: num,W: num,Z2: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
% 8.55/4.75 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_990_add__numeral__left,axiom,
% 8.55/4.75 ! [V: num,W: num,Z2: int] :
% 8.55/4.75 ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
% 8.55/4.75 = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_991_semiring__norm_I2_J,axiom,
% 8.55/4.75 ( ( plus_plus_num @ one @ one )
% 8.55/4.75 = ( bit0 @ one ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_992_add__diff__cancel,axiom,
% 8.55/4.75 ! [A: real,B: real] :
% 8.55/4.75 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_993_add__diff__cancel,axiom,
% 8.55/4.75 ! [A: rat,B: rat] :
% 8.55/4.75 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_994_add__diff__cancel,axiom,
% 8.55/4.75 ! [A: int,B: int] :
% 8.55/4.75 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_995_diff__add__cancel,axiom,
% 8.55/4.75 ! [A: real,B: real] :
% 8.55/4.75 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_996_diff__add__cancel,axiom,
% 8.55/4.75 ! [A: rat,B: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_997_diff__add__cancel,axiom,
% 8.55/4.75 ! [A: int,B: int] :
% 8.55/4.75 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_998_add__diff__cancel__left,axiom,
% 8.55/4.75 ! [C: real,A: real,B: real] :
% 8.55/4.75 ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 8.55/4.75 = ( minus_minus_real @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_999_add__diff__cancel__left,axiom,
% 8.55/4.75 ! [C: rat,A: rat,B: rat] :
% 8.55/4.75 ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 8.55/4.75 = ( minus_minus_rat @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1000_add__diff__cancel__left,axiom,
% 8.55/4.75 ! [C: nat,A: nat,B: nat] :
% 8.55/4.75 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 8.55/4.75 = ( minus_minus_nat @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1001_add__diff__cancel__left,axiom,
% 8.55/4.75 ! [C: int,A: int,B: int] :
% 8.55/4.75 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 8.55/4.75 = ( minus_minus_int @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1002_add__diff__cancel__left_H,axiom,
% 8.55/4.75 ! [A: real,B: real] :
% 8.55/4.75 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 8.55/4.75 = B ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1003_add__diff__cancel__left_H,axiom,
% 8.55/4.75 ! [A: rat,B: rat] :
% 8.55/4.75 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
% 8.55/4.75 = B ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1004_add__diff__cancel__left_H,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 8.55/4.75 = B ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1005_add__diff__cancel__left_H,axiom,
% 8.55/4.75 ! [A: int,B: int] :
% 8.55/4.75 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 8.55/4.75 = B ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1006_add__diff__cancel__right,axiom,
% 8.55/4.75 ! [A: real,C: real,B: real] :
% 8.55/4.75 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 8.55/4.75 = ( minus_minus_real @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1007_add__diff__cancel__right,axiom,
% 8.55/4.75 ! [A: rat,C: rat,B: rat] :
% 8.55/4.75 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 8.55/4.75 = ( minus_minus_rat @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1008_add__diff__cancel__right,axiom,
% 8.55/4.75 ! [A: nat,C: nat,B: nat] :
% 8.55/4.75 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 8.55/4.75 = ( minus_minus_nat @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1009_add__diff__cancel__right,axiom,
% 8.55/4.75 ! [A: int,C: int,B: int] :
% 8.55/4.75 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 8.55/4.75 = ( minus_minus_int @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1010_add__diff__cancel__right_H,axiom,
% 8.55/4.75 ! [A: real,B: real] :
% 8.55/4.75 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1011_add__diff__cancel__right_H,axiom,
% 8.55/4.75 ! [A: rat,B: rat] :
% 8.55/4.75 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1012_add__diff__cancel__right_H,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1013_add__diff__cancel__right_H,axiom,
% 8.55/4.75 ! [A: int,B: int] :
% 8.55/4.75 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1014_dvd__add__triv__left__iff,axiom,
% 8.55/4.75 ! [A: code_integer,B: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 8.55/4.75 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1015_dvd__add__triv__left__iff,axiom,
% 8.55/4.75 ! [A: real,B: real] :
% 8.55/4.75 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
% 8.55/4.75 = ( dvd_dvd_real @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1016_dvd__add__triv__left__iff,axiom,
% 8.55/4.75 ! [A: rat,B: rat] :
% 8.55/4.75 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 8.55/4.75 = ( dvd_dvd_rat @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1017_dvd__add__triv__left__iff,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 8.55/4.75 = ( dvd_dvd_nat @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1018_dvd__add__triv__left__iff,axiom,
% 8.55/4.75 ! [A: int,B: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
% 8.55/4.75 = ( dvd_dvd_int @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1019_dvd__add__triv__right__iff,axiom,
% 8.55/4.75 ! [A: code_integer,B: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
% 8.55/4.75 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1020_dvd__add__triv__right__iff,axiom,
% 8.55/4.75 ! [A: real,B: real] :
% 8.55/4.75 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
% 8.55/4.75 = ( dvd_dvd_real @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1021_dvd__add__triv__right__iff,axiom,
% 8.55/4.75 ! [A: rat,B: rat] :
% 8.55/4.75 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 8.55/4.75 = ( dvd_dvd_rat @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1022_dvd__add__triv__right__iff,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 8.55/4.75 = ( dvd_dvd_nat @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1023_dvd__add__triv__right__iff,axiom,
% 8.55/4.75 ! [A: int,B: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
% 8.55/4.75 = ( dvd_dvd_int @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1024_of__nat__add,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N ) )
% 8.55/4.75 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1025_of__nat__add,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 8.55/4.75 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1026_of__nat__add,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 8.55/4.75 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1027_of__nat__add,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 8.55/4.75 = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1028_of__nat__add,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 8.55/4.75 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1029_mult__Suc__right,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( times_times_nat @ M @ ( suc @ N ) )
% 8.55/4.75 = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1030_nat__1__eq__mult__iff,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( one_one_nat
% 8.55/4.75 = ( times_times_nat @ M @ N ) )
% 8.55/4.75 = ( ( M = one_one_nat )
% 8.55/4.75 & ( N = one_one_nat ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1031_nat__mult__eq__1__iff,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( ( times_times_nat @ M @ N )
% 8.55/4.75 = one_one_nat )
% 8.55/4.75 = ( ( M = one_one_nat )
% 8.55/4.75 & ( N = one_one_nat ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1032_dbl__simps_I2_J,axiom,
% 8.55/4.75 ( ( neg_numeral_dbl_real @ zero_zero_real )
% 8.55/4.75 = zero_zero_real ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1033_dbl__simps_I2_J,axiom,
% 8.55/4.75 ( ( neg_numeral_dbl_rat @ zero_zero_rat )
% 8.55/4.75 = zero_zero_rat ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1034_dbl__simps_I2_J,axiom,
% 8.55/4.75 ( ( neg_numeral_dbl_int @ zero_zero_int )
% 8.55/4.75 = zero_zero_int ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1035_triangle__Suc,axiom,
% 8.55/4.75 ! [N: nat] :
% 8.55/4.75 ( ( nat_triangle @ ( suc @ N ) )
% 8.55/4.75 = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1036_triangle__0,axiom,
% 8.55/4.75 ( ( nat_triangle @ zero_zero_nat )
% 8.55/4.75 = zero_zero_nat ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1037_diff__add__zero,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 8.55/4.75 = zero_zero_nat ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1038_distrib__left__numeral,axiom,
% 8.55/4.75 ! [V: num,B: complex,C: complex] :
% 8.55/4.75 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
% 8.55/4.75 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1039_distrib__left__numeral,axiom,
% 8.55/4.75 ! [V: num,B: real,C: real] :
% 8.55/4.75 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
% 8.55/4.75 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1040_distrib__left__numeral,axiom,
% 8.55/4.75 ! [V: num,B: rat,C: rat] :
% 8.55/4.75 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
% 8.55/4.75 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1041_distrib__left__numeral,axiom,
% 8.55/4.75 ! [V: num,B: nat,C: nat] :
% 8.55/4.75 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1042_distrib__left__numeral,axiom,
% 8.55/4.75 ! [V: num,B: int,C: int] :
% 8.55/4.75 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
% 8.55/4.75 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1043_distrib__right__numeral,axiom,
% 8.55/4.75 ! [A: complex,B: complex,V: num] :
% 8.55/4.75 ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 8.55/4.75 = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1044_distrib__right__numeral,axiom,
% 8.55/4.75 ! [A: real,B: real,V: num] :
% 8.55/4.75 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 8.55/4.75 = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1045_distrib__right__numeral,axiom,
% 8.55/4.75 ! [A: rat,B: rat,V: num] :
% 8.55/4.75 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 8.55/4.75 = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1046_distrib__right__numeral,axiom,
% 8.55/4.75 ! [A: nat,B: nat,V: num] :
% 8.55/4.75 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1047_distrib__right__numeral,axiom,
% 8.55/4.75 ! [A: int,B: int,V: num] :
% 8.55/4.75 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 8.55/4.75 = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1048_dvd__add__times__triv__right__iff,axiom,
% 8.55/4.75 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 8.55/4.75 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1049_dvd__add__times__triv__right__iff,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
% 8.55/4.75 = ( dvd_dvd_real @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1050_dvd__add__times__triv__right__iff,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
% 8.55/4.75 = ( dvd_dvd_rat @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1051_dvd__add__times__triv__right__iff,axiom,
% 8.55/4.75 ! [A: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
% 8.55/4.75 = ( dvd_dvd_nat @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1052_dvd__add__times__triv__right__iff,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
% 8.55/4.75 = ( dvd_dvd_int @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1053_dvd__add__times__triv__left__iff,axiom,
% 8.55/4.75 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
% 8.55/4.75 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1054_dvd__add__times__triv__left__iff,axiom,
% 8.55/4.75 ! [A: real,C: real,B: real] :
% 8.55/4.75 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
% 8.55/4.75 = ( dvd_dvd_real @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1055_dvd__add__times__triv__left__iff,axiom,
% 8.55/4.75 ! [A: rat,C: rat,B: rat] :
% 8.55/4.75 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
% 8.55/4.75 = ( dvd_dvd_rat @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1056_dvd__add__times__triv__left__iff,axiom,
% 8.55/4.75 ! [A: nat,C: nat,B: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
% 8.55/4.75 = ( dvd_dvd_nat @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1057_dvd__add__times__triv__left__iff,axiom,
% 8.55/4.75 ! [A: int,C: int,B: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
% 8.55/4.75 = ( dvd_dvd_int @ A @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1058_div__add,axiom,
% 8.55/4.75 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ C @ A )
% 8.55/4.75 => ( ( dvd_dvd_Code_integer @ C @ B )
% 8.55/4.75 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 8.55/4.75 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1059_div__add,axiom,
% 8.55/4.75 ! [C: nat,A: nat,B: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ C @ A )
% 8.55/4.75 => ( ( dvd_dvd_nat @ C @ B )
% 8.55/4.75 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1060_div__add,axiom,
% 8.55/4.75 ! [C: int,A: int,B: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ C @ A )
% 8.55/4.75 => ( ( dvd_dvd_int @ C @ B )
% 8.55/4.75 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1061_Suc__numeral,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( suc @ ( numeral_numeral_nat @ N ) )
% 8.55/4.75 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1062_dbl__simps_I5_J,axiom,
% 8.55/4.75 ! [K: num] :
% 8.55/4.75 ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
% 8.55/4.75 = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1063_dbl__simps_I5_J,axiom,
% 8.55/4.75 ! [K: num] :
% 8.55/4.75 ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 8.55/4.75 = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1064_dbl__simps_I5_J,axiom,
% 8.55/4.75 ! [K: num] :
% 8.55/4.75 ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
% 8.55/4.75 = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1065_dbl__simps_I5_J,axiom,
% 8.55/4.75 ! [K: num] :
% 8.55/4.75 ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 8.55/4.75 = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1066_div__mult__self4,axiom,
% 8.55/4.75 ! [B: nat,C: nat,A: nat] :
% 8.55/4.75 ( ( B != zero_zero_nat )
% 8.55/4.75 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 8.55/4.75 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1067_div__mult__self4,axiom,
% 8.55/4.75 ! [B: int,C: int,A: int] :
% 8.55/4.75 ( ( B != zero_zero_int )
% 8.55/4.75 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 8.55/4.75 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1068_div__mult__self3,axiom,
% 8.55/4.75 ! [B: nat,C: nat,A: nat] :
% 8.55/4.75 ( ( B != zero_zero_nat )
% 8.55/4.75 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 8.55/4.75 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1069_div__mult__self3,axiom,
% 8.55/4.75 ! [B: int,C: int,A: int] :
% 8.55/4.75 ( ( B != zero_zero_int )
% 8.55/4.75 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 8.55/4.75 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1070_div__mult__self2,axiom,
% 8.55/4.75 ! [B: nat,A: nat,C: nat] :
% 8.55/4.75 ( ( B != zero_zero_nat )
% 8.55/4.75 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 8.55/4.75 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1071_div__mult__self2,axiom,
% 8.55/4.75 ! [B: int,A: int,C: int] :
% 8.55/4.75 ( ( B != zero_zero_int )
% 8.55/4.75 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 8.55/4.75 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1072_div__mult__self1,axiom,
% 8.55/4.75 ! [B: nat,A: nat,C: nat] :
% 8.55/4.75 ( ( B != zero_zero_nat )
% 8.55/4.75 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 8.55/4.75 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1073_div__mult__self1,axiom,
% 8.55/4.75 ! [B: int,A: int,C: int] :
% 8.55/4.75 ( ( B != zero_zero_int )
% 8.55/4.75 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 8.55/4.75 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1074_one__plus__numeral,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
% 8.55/4.75 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1075_one__plus__numeral,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 8.55/4.75 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1076_one__plus__numeral,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 8.55/4.75 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1077_one__plus__numeral,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 8.55/4.75 = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1078_one__plus__numeral,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 8.55/4.75 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1079_numeral__plus__one,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
% 8.55/4.75 = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1080_numeral__plus__one,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 8.55/4.75 = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1081_numeral__plus__one,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 8.55/4.75 = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1082_numeral__plus__one,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 8.55/4.75 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1083_numeral__plus__one,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 8.55/4.75 = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1084_add__2__eq__Suc_H,axiom,
% 8.55/4.75 ! [N: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.75 = ( suc @ ( suc @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1085_add__2__eq__Suc,axiom,
% 8.55/4.75 ! [N: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.75 = ( suc @ ( suc @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1086_of__nat__Suc,axiom,
% 8.55/4.75 ! [M: nat] :
% 8.55/4.75 ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 8.55/4.75 = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1087_of__nat__Suc,axiom,
% 8.55/4.75 ! [M: nat] :
% 8.55/4.75 ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 8.55/4.75 = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1088_of__nat__Suc,axiom,
% 8.55/4.75 ! [M: nat] :
% 8.55/4.75 ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 8.55/4.75 = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1089_of__nat__Suc,axiom,
% 8.55/4.75 ! [M: nat] :
% 8.55/4.75 ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 8.55/4.75 = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1090_of__nat__Suc,axiom,
% 8.55/4.75 ! [M: nat] :
% 8.55/4.75 ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 8.55/4.75 = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1091_add__self__div__2,axiom,
% 8.55/4.75 ! [M: nat] :
% 8.55/4.75 ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.75 = M ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1092_one__add__one,axiom,
% 8.55/4.75 ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 8.55/4.75 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1093_one__add__one,axiom,
% 8.55/4.75 ( ( plus_plus_real @ one_one_real @ one_one_real )
% 8.55/4.75 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1094_one__add__one,axiom,
% 8.55/4.75 ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 8.55/4.75 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1095_one__add__one,axiom,
% 8.55/4.75 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 8.55/4.75 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1096_one__add__one,axiom,
% 8.55/4.75 ( ( plus_plus_int @ one_one_int @ one_one_int )
% 8.55/4.75 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1097_even__add,axiom,
% 8.55/4.75 ! [A: code_integer,B: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 8.55/4.75 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1098_even__add,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
% 8.55/4.75 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1099_even__add,axiom,
% 8.55/4.75 ! [A: int,B: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
% 8.55/4.75 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1100_odd__add,axiom,
% 8.55/4.75 ! [A: code_integer,B: code_integer] :
% 8.55/4.75 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
% 8.55/4.75 = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 8.55/4.75 != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1101_odd__add,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
% 8.55/4.75 = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 8.55/4.75 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1102_odd__add,axiom,
% 8.55/4.75 ! [A: int,B: int] :
% 8.55/4.75 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
% 8.55/4.75 = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 8.55/4.75 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1103_even__plus__one__iff,axiom,
% 8.55/4.75 ! [A: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 8.55/4.75 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1104_even__plus__one__iff,axiom,
% 8.55/4.75 ! [A: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 8.55/4.75 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1105_even__plus__one__iff,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 8.55/4.75 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1106_even__diff,axiom,
% 8.55/4.75 ! [A: code_integer,B: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
% 8.55/4.75 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1107_even__diff,axiom,
% 8.55/4.75 ! [A: int,B: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
% 8.55/4.75 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1108_push__bit__of__1,axiom,
% 8.55/4.75 ! [N: nat] :
% 8.55/4.75 ( ( bit_se547839408752420682it_nat @ N @ one_one_nat )
% 8.55/4.75 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1109_push__bit__of__1,axiom,
% 8.55/4.75 ! [N: nat] :
% 8.55/4.75 ( ( bit_se545348938243370406it_int @ N @ one_one_int )
% 8.55/4.75 = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1110_push__bit__of__1,axiom,
% 8.55/4.75 ! [N: nat] :
% 8.55/4.75 ( ( bit_se7788150548672797655nteger @ N @ one_one_Code_integer )
% 8.55/4.75 = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1111_push__bit__of__1,axiom,
% 8.55/4.75 ! [N: nat] :
% 8.55/4.75 ( ( bit_se6611745700429515170atural @ N @ one_one_Code_natural )
% 8.55/4.75 = ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1112_push__bit__of__Suc__0,axiom,
% 8.55/4.75 ! [N: nat] :
% 8.55/4.75 ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 8.55/4.75 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1113_int__distrib_I4_J,axiom,
% 8.55/4.75 ! [W: int,Z1: int,Z22: int] :
% 8.55/4.75 ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 8.55/4.75 = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1114_int__distrib_I3_J,axiom,
% 8.55/4.75 ! [Z1: int,Z22: int,W: int] :
% 8.55/4.75 ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 8.55/4.75 = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1115_int__distrib_I2_J,axiom,
% 8.55/4.75 ! [W: int,Z1: int,Z22: int] :
% 8.55/4.75 ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
% 8.55/4.75 = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1116_int__distrib_I1_J,axiom,
% 8.55/4.75 ! [Z1: int,Z22: int,W: int] :
% 8.55/4.75 ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
% 8.55/4.75 = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1117_nat__mult__1,axiom,
% 8.55/4.75 ! [N: nat] :
% 8.55/4.75 ( ( times_times_nat @ one_one_nat @ N )
% 8.55/4.75 = N ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1118_add__mult__distrib,axiom,
% 8.55/4.75 ! [M: nat,N: nat,K: nat] :
% 8.55/4.75 ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1119_nat__mult__1__right,axiom,
% 8.55/4.75 ! [N: nat] :
% 8.55/4.75 ( ( times_times_nat @ N @ one_one_nat )
% 8.55/4.75 = N ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1120_add__mult__distrib2,axiom,
% 8.55/4.75 ! [K: nat,M: nat,N: nat] :
% 8.55/4.75 ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1121_left__add__mult__distrib,axiom,
% 8.55/4.75 ! [I: nat,U: nat,J: nat,K: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1122_dbl__def,axiom,
% 8.55/4.75 ( neg_numeral_dbl_real
% 8.55/4.75 = ( ^ [X3: real] : ( plus_plus_real @ X3 @ X3 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1123_dbl__def,axiom,
% 8.55/4.75 ( neg_numeral_dbl_rat
% 8.55/4.75 = ( ^ [X3: rat] : ( plus_plus_rat @ X3 @ X3 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1124_dbl__def,axiom,
% 8.55/4.75 ( neg_numeral_dbl_int
% 8.55/4.75 = ( ^ [X3: int] : ( plus_plus_int @ X3 @ X3 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1125_is__num__normalize_I1_J,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1126_is__num__normalize_I1_J,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1127_is__num__normalize_I1_J,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1128_add__right__imp__eq,axiom,
% 8.55/4.75 ! [B: real,A: real,C: real] :
% 8.55/4.75 ( ( ( plus_plus_real @ B @ A )
% 8.55/4.75 = ( plus_plus_real @ C @ A ) )
% 8.55/4.75 => ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1129_add__right__imp__eq,axiom,
% 8.55/4.75 ! [B: rat,A: rat,C: rat] :
% 8.55/4.75 ( ( ( plus_plus_rat @ B @ A )
% 8.55/4.75 = ( plus_plus_rat @ C @ A ) )
% 8.55/4.75 => ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1130_add__right__imp__eq,axiom,
% 8.55/4.75 ! [B: nat,A: nat,C: nat] :
% 8.55/4.75 ( ( ( plus_plus_nat @ B @ A )
% 8.55/4.75 = ( plus_plus_nat @ C @ A ) )
% 8.55/4.75 => ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1131_add__right__imp__eq,axiom,
% 8.55/4.75 ! [B: int,A: int,C: int] :
% 8.55/4.75 ( ( ( plus_plus_int @ B @ A )
% 8.55/4.75 = ( plus_plus_int @ C @ A ) )
% 8.55/4.75 => ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1132_add__left__imp__eq,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( ( plus_plus_real @ A @ B )
% 8.55/4.75 = ( plus_plus_real @ A @ C ) )
% 8.55/4.75 => ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1133_add__left__imp__eq,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( ( plus_plus_rat @ A @ B )
% 8.55/4.75 = ( plus_plus_rat @ A @ C ) )
% 8.55/4.75 => ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1134_add__left__imp__eq,axiom,
% 8.55/4.75 ! [A: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( ( plus_plus_nat @ A @ B )
% 8.55/4.75 = ( plus_plus_nat @ A @ C ) )
% 8.55/4.75 => ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1135_add__left__imp__eq,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( ( plus_plus_int @ A @ B )
% 8.55/4.75 = ( plus_plus_int @ A @ C ) )
% 8.55/4.75 => ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1136_add_Oleft__commute,axiom,
% 8.55/4.75 ! [B: real,A: real,C: real] :
% 8.55/4.75 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 8.55/4.75 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1137_add_Oleft__commute,axiom,
% 8.55/4.75 ! [B: rat,A: rat,C: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
% 8.55/4.75 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1138_add_Oleft__commute,axiom,
% 8.55/4.75 ! [B: nat,A: nat,C: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 8.55/4.75 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1139_add_Oleft__commute,axiom,
% 8.55/4.75 ! [B: int,A: int,C: int] :
% 8.55/4.75 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 8.55/4.75 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1140_add_Ocommute,axiom,
% 8.55/4.75 ( plus_plus_real
% 8.55/4.75 = ( ^ [A2: real,B2: real] : ( plus_plus_real @ B2 @ A2 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1141_add_Ocommute,axiom,
% 8.55/4.75 ( plus_plus_rat
% 8.55/4.75 = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ B2 @ A2 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1142_add_Ocommute,axiom,
% 8.55/4.75 ( plus_plus_nat
% 8.55/4.75 = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1143_add_Ocommute,axiom,
% 8.55/4.75 ( plus_plus_int
% 8.55/4.75 = ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1144_add_Oright__cancel,axiom,
% 8.55/4.75 ! [B: real,A: real,C: real] :
% 8.55/4.75 ( ( ( plus_plus_real @ B @ A )
% 8.55/4.75 = ( plus_plus_real @ C @ A ) )
% 8.55/4.75 = ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1145_add_Oright__cancel,axiom,
% 8.55/4.75 ! [B: rat,A: rat,C: rat] :
% 8.55/4.75 ( ( ( plus_plus_rat @ B @ A )
% 8.55/4.75 = ( plus_plus_rat @ C @ A ) )
% 8.55/4.75 = ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1146_add_Oright__cancel,axiom,
% 8.55/4.75 ! [B: int,A: int,C: int] :
% 8.55/4.75 ( ( ( plus_plus_int @ B @ A )
% 8.55/4.75 = ( plus_plus_int @ C @ A ) )
% 8.55/4.75 = ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1147_add_Oleft__cancel,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( ( plus_plus_real @ A @ B )
% 8.55/4.75 = ( plus_plus_real @ A @ C ) )
% 8.55/4.75 = ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1148_add_Oleft__cancel,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( ( plus_plus_rat @ A @ B )
% 8.55/4.75 = ( plus_plus_rat @ A @ C ) )
% 8.55/4.75 = ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1149_add_Oleft__cancel,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( ( plus_plus_int @ A @ B )
% 8.55/4.75 = ( plus_plus_int @ A @ C ) )
% 8.55/4.75 = ( B = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1150_add_Oassoc,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1151_add_Oassoc,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1152_add_Oassoc,axiom,
% 8.55/4.75 ! [A: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1153_add_Oassoc,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1154_group__cancel_Oadd2,axiom,
% 8.55/4.75 ! [B5: real,K: real,B: real,A: real] :
% 8.55/4.75 ( ( B5
% 8.55/4.75 = ( plus_plus_real @ K @ B ) )
% 8.55/4.75 => ( ( plus_plus_real @ A @ B5 )
% 8.55/4.75 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1155_group__cancel_Oadd2,axiom,
% 8.55/4.75 ! [B5: rat,K: rat,B: rat,A: rat] :
% 8.55/4.75 ( ( B5
% 8.55/4.75 = ( plus_plus_rat @ K @ B ) )
% 8.55/4.75 => ( ( plus_plus_rat @ A @ B5 )
% 8.55/4.75 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1156_group__cancel_Oadd2,axiom,
% 8.55/4.75 ! [B5: nat,K: nat,B: nat,A: nat] :
% 8.55/4.75 ( ( B5
% 8.55/4.75 = ( plus_plus_nat @ K @ B ) )
% 8.55/4.75 => ( ( plus_plus_nat @ A @ B5 )
% 8.55/4.75 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1157_group__cancel_Oadd2,axiom,
% 8.55/4.75 ! [B5: int,K: int,B: int,A: int] :
% 8.55/4.75 ( ( B5
% 8.55/4.75 = ( plus_plus_int @ K @ B ) )
% 8.55/4.75 => ( ( plus_plus_int @ A @ B5 )
% 8.55/4.75 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1158_group__cancel_Oadd1,axiom,
% 8.55/4.75 ! [A3: real,K: real,A: real,B: real] :
% 8.55/4.75 ( ( A3
% 8.55/4.75 = ( plus_plus_real @ K @ A ) )
% 8.55/4.75 => ( ( plus_plus_real @ A3 @ B )
% 8.55/4.75 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1159_group__cancel_Oadd1,axiom,
% 8.55/4.75 ! [A3: rat,K: rat,A: rat,B: rat] :
% 8.55/4.75 ( ( A3
% 8.55/4.75 = ( plus_plus_rat @ K @ A ) )
% 8.55/4.75 => ( ( plus_plus_rat @ A3 @ B )
% 8.55/4.75 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1160_group__cancel_Oadd1,axiom,
% 8.55/4.75 ! [A3: nat,K: nat,A: nat,B: nat] :
% 8.55/4.75 ( ( A3
% 8.55/4.75 = ( plus_plus_nat @ K @ A ) )
% 8.55/4.75 => ( ( plus_plus_nat @ A3 @ B )
% 8.55/4.75 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1161_group__cancel_Oadd1,axiom,
% 8.55/4.75 ! [A3: int,K: int,A: int,B: int] :
% 8.55/4.75 ( ( A3
% 8.55/4.75 = ( plus_plus_int @ K @ A ) )
% 8.55/4.75 => ( ( plus_plus_int @ A3 @ B )
% 8.55/4.75 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1162_add__mono__thms__linordered__semiring_I4_J,axiom,
% 8.55/4.75 ! [I: real,J: real,K: real,L: real] :
% 8.55/4.75 ( ( ( I = J )
% 8.55/4.75 & ( K = L ) )
% 8.55/4.75 => ( ( plus_plus_real @ I @ K )
% 8.55/4.75 = ( plus_plus_real @ J @ L ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1163_add__mono__thms__linordered__semiring_I4_J,axiom,
% 8.55/4.75 ! [I: rat,J: rat,K: rat,L: rat] :
% 8.55/4.75 ( ( ( I = J )
% 8.55/4.75 & ( K = L ) )
% 8.55/4.75 => ( ( plus_plus_rat @ I @ K )
% 8.55/4.75 = ( plus_plus_rat @ J @ L ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1164_add__mono__thms__linordered__semiring_I4_J,axiom,
% 8.55/4.75 ! [I: nat,J: nat,K: nat,L: nat] :
% 8.55/4.75 ( ( ( I = J )
% 8.55/4.75 & ( K = L ) )
% 8.55/4.75 => ( ( plus_plus_nat @ I @ K )
% 8.55/4.75 = ( plus_plus_nat @ J @ L ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1165_add__mono__thms__linordered__semiring_I4_J,axiom,
% 8.55/4.75 ! [I: int,J: int,K: int,L: int] :
% 8.55/4.75 ( ( ( I = J )
% 8.55/4.75 & ( K = L ) )
% 8.55/4.75 => ( ( plus_plus_int @ I @ K )
% 8.55/4.75 = ( plus_plus_int @ J @ L ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1166_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1167_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1168_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 8.55/4.75 ! [A: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1169_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1170_zadd__int__left,axiom,
% 8.55/4.75 ! [M: nat,N: nat,Z2: int] :
% 8.55/4.75 ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
% 8.55/4.75 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1171_int__plus,axiom,
% 8.55/4.75 ! [N: nat,M: nat] :
% 8.55/4.75 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
% 8.55/4.75 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1172_int__ops_I5_J,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
% 8.55/4.75 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1173_comm__monoid__add__class_Oadd__0,axiom,
% 8.55/4.75 ! [A: real] :
% 8.55/4.75 ( ( plus_plus_real @ zero_zero_real @ A )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1174_comm__monoid__add__class_Oadd__0,axiom,
% 8.55/4.75 ! [A: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1175_comm__monoid__add__class_Oadd__0,axiom,
% 8.55/4.75 ! [A: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1176_comm__monoid__add__class_Oadd__0,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ( plus_plus_int @ zero_zero_int @ A )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1177_add_Ocomm__neutral,axiom,
% 8.55/4.75 ! [A: real] :
% 8.55/4.75 ( ( plus_plus_real @ A @ zero_zero_real )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1178_add_Ocomm__neutral,axiom,
% 8.55/4.75 ! [A: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1179_add_Ocomm__neutral,axiom,
% 8.55/4.75 ! [A: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1180_add_Ocomm__neutral,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ( plus_plus_int @ A @ zero_zero_int )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1181_add_Ogroup__left__neutral,axiom,
% 8.55/4.75 ! [A: real] :
% 8.55/4.75 ( ( plus_plus_real @ zero_zero_real @ A )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1182_add_Ogroup__left__neutral,axiom,
% 8.55/4.75 ! [A: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1183_add_Ogroup__left__neutral,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ( plus_plus_int @ zero_zero_int @ A )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1184_verit__sum__simplify,axiom,
% 8.55/4.75 ! [A: real] :
% 8.55/4.75 ( ( plus_plus_real @ A @ zero_zero_real )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1185_verit__sum__simplify,axiom,
% 8.55/4.75 ! [A: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1186_verit__sum__simplify,axiom,
% 8.55/4.75 ! [A: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1187_verit__sum__simplify,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ( plus_plus_int @ A @ zero_zero_int )
% 8.55/4.75 = A ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1188_combine__common__factor,axiom,
% 8.55/4.75 ! [A: real,E: real,B: real,C: real] :
% 8.55/4.75 ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
% 8.55/4.75 = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1189_combine__common__factor,axiom,
% 8.55/4.75 ! [A: rat,E: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
% 8.55/4.75 = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1190_combine__common__factor,axiom,
% 8.55/4.75 ! [A: nat,E: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1191_combine__common__factor,axiom,
% 8.55/4.75 ! [A: int,E: int,B: int,C: int] :
% 8.55/4.75 ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
% 8.55/4.75 = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1192_distrib__right,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1193_distrib__right,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1194_distrib__right,axiom,
% 8.55/4.75 ! [A: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1195_distrib__right,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1196_distrib__left,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 8.55/4.75 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1197_distrib__left,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 8.55/4.75 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1198_distrib__left,axiom,
% 8.55/4.75 ! [A: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1199_distrib__left,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 8.55/4.75 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1200_comm__semiring__class_Odistrib,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1201_comm__semiring__class_Odistrib,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1202_comm__semiring__class_Odistrib,axiom,
% 8.55/4.75 ! [A: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1203_comm__semiring__class_Odistrib,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1204_ring__class_Oring__distribs_I1_J,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 8.55/4.75 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1205_ring__class_Oring__distribs_I1_J,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 8.55/4.75 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1206_ring__class_Oring__distribs_I1_J,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 8.55/4.75 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1207_ring__class_Oring__distribs_I2_J,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1208_ring__class_Oring__distribs_I2_J,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1209_ring__class_Oring__distribs_I2_J,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1210_group__cancel_Osub1,axiom,
% 8.55/4.75 ! [A3: real,K: real,A: real,B: real] :
% 8.55/4.75 ( ( A3
% 8.55/4.75 = ( plus_plus_real @ K @ A ) )
% 8.55/4.75 => ( ( minus_minus_real @ A3 @ B )
% 8.55/4.75 = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1211_group__cancel_Osub1,axiom,
% 8.55/4.75 ! [A3: rat,K: rat,A: rat,B: rat] :
% 8.55/4.75 ( ( A3
% 8.55/4.75 = ( plus_plus_rat @ K @ A ) )
% 8.55/4.75 => ( ( minus_minus_rat @ A3 @ B )
% 8.55/4.75 = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1212_group__cancel_Osub1,axiom,
% 8.55/4.75 ! [A3: int,K: int,A: int,B: int] :
% 8.55/4.75 ( ( A3
% 8.55/4.75 = ( plus_plus_int @ K @ A ) )
% 8.55/4.75 => ( ( minus_minus_int @ A3 @ B )
% 8.55/4.75 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1213_diff__eq__eq,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( ( minus_minus_real @ A @ B )
% 8.55/4.75 = C )
% 8.55/4.75 = ( A
% 8.55/4.75 = ( plus_plus_real @ C @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1214_diff__eq__eq,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( ( minus_minus_rat @ A @ B )
% 8.55/4.75 = C )
% 8.55/4.75 = ( A
% 8.55/4.75 = ( plus_plus_rat @ C @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1215_diff__eq__eq,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( ( minus_minus_int @ A @ B )
% 8.55/4.75 = C )
% 8.55/4.75 = ( A
% 8.55/4.75 = ( plus_plus_int @ C @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1216_eq__diff__eq,axiom,
% 8.55/4.75 ! [A: real,C: real,B: real] :
% 8.55/4.75 ( ( A
% 8.55/4.75 = ( minus_minus_real @ C @ B ) )
% 8.55/4.75 = ( ( plus_plus_real @ A @ B )
% 8.55/4.75 = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1217_eq__diff__eq,axiom,
% 8.55/4.75 ! [A: rat,C: rat,B: rat] :
% 8.55/4.75 ( ( A
% 8.55/4.75 = ( minus_minus_rat @ C @ B ) )
% 8.55/4.75 = ( ( plus_plus_rat @ A @ B )
% 8.55/4.75 = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1218_eq__diff__eq,axiom,
% 8.55/4.75 ! [A: int,C: int,B: int] :
% 8.55/4.75 ( ( A
% 8.55/4.75 = ( minus_minus_int @ C @ B ) )
% 8.55/4.75 = ( ( plus_plus_int @ A @ B )
% 8.55/4.75 = C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1219_add__diff__eq,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 8.55/4.75 = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1220_add__diff__eq,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 8.55/4.75 = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1221_add__diff__eq,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 8.55/4.75 = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1222_diff__diff__eq2,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 8.55/4.75 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1223_diff__diff__eq2,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 8.55/4.75 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1224_diff__diff__eq2,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 8.55/4.75 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1225_diff__add__eq,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 8.55/4.75 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1226_diff__add__eq,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 8.55/4.75 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1227_diff__add__eq,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 8.55/4.75 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1228_diff__add__eq__diff__diff__swap,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 8.55/4.75 = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1229_diff__add__eq__diff__diff__swap,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 8.55/4.75 = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1230_diff__add__eq__diff__diff__swap,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 8.55/4.75 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1231_add__implies__diff,axiom,
% 8.55/4.75 ! [C: real,B: real,A: real] :
% 8.55/4.75 ( ( ( plus_plus_real @ C @ B )
% 8.55/4.75 = A )
% 8.55/4.75 => ( C
% 8.55/4.75 = ( minus_minus_real @ A @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1232_add__implies__diff,axiom,
% 8.55/4.75 ! [C: rat,B: rat,A: rat] :
% 8.55/4.75 ( ( ( plus_plus_rat @ C @ B )
% 8.55/4.75 = A )
% 8.55/4.75 => ( C
% 8.55/4.75 = ( minus_minus_rat @ A @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1233_add__implies__diff,axiom,
% 8.55/4.75 ! [C: nat,B: nat,A: nat] :
% 8.55/4.75 ( ( ( plus_plus_nat @ C @ B )
% 8.55/4.75 = A )
% 8.55/4.75 => ( C
% 8.55/4.75 = ( minus_minus_nat @ A @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1234_add__implies__diff,axiom,
% 8.55/4.75 ! [C: int,B: int,A: int] :
% 8.55/4.75 ( ( ( plus_plus_int @ C @ B )
% 8.55/4.75 = A )
% 8.55/4.75 => ( C
% 8.55/4.75 = ( minus_minus_int @ A @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1235_add__diff__add,axiom,
% 8.55/4.75 ! [A: real,C: real,B: real,D: real] :
% 8.55/4.75 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
% 8.55/4.75 = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1236_add__diff__add,axiom,
% 8.55/4.75 ! [A: rat,C: rat,B: rat,D: rat] :
% 8.55/4.75 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
% 8.55/4.75 = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1237_add__diff__add,axiom,
% 8.55/4.75 ! [A: int,C: int,B: int,D: int] :
% 8.55/4.75 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
% 8.55/4.75 = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1238_diff__diff__eq,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 8.55/4.75 = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1239_diff__diff__eq,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 8.55/4.75 = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1240_diff__diff__eq,axiom,
% 8.55/4.75 ! [A: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 8.55/4.75 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1241_diff__diff__eq,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 8.55/4.75 = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1242_add__One__commute,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( plus_plus_num @ one @ N )
% 8.55/4.75 = ( plus_plus_num @ N @ one ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1243_add__divide__distrib,axiom,
% 8.55/4.75 ! [A: complex,B: complex,C: complex] :
% 8.55/4.75 ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1244_add__divide__distrib,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1245_add__divide__distrib,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1246_dvd__add,axiom,
% 8.55/4.75 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.55/4.75 => ( ( dvd_dvd_Code_integer @ A @ C )
% 8.55/4.75 => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1247_dvd__add,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( dvd_dvd_real @ A @ B )
% 8.55/4.75 => ( ( dvd_dvd_real @ A @ C )
% 8.55/4.75 => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1248_dvd__add,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( dvd_dvd_rat @ A @ B )
% 8.55/4.75 => ( ( dvd_dvd_rat @ A @ C )
% 8.55/4.75 => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1249_dvd__add,axiom,
% 8.55/4.75 ! [A: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 => ( ( dvd_dvd_nat @ A @ C )
% 8.55/4.75 => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1250_dvd__add,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ A @ B )
% 8.55/4.75 => ( ( dvd_dvd_int @ A @ C )
% 8.55/4.75 => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1251_dvd__add__left__iff,axiom,
% 8.55/4.75 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ A @ C )
% 8.55/4.75 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 8.55/4.75 = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1252_dvd__add__left__iff,axiom,
% 8.55/4.75 ! [A: real,C: real,B: real] :
% 8.55/4.75 ( ( dvd_dvd_real @ A @ C )
% 8.55/4.75 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 8.55/4.75 = ( dvd_dvd_real @ A @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1253_dvd__add__left__iff,axiom,
% 8.55/4.75 ! [A: rat,C: rat,B: rat] :
% 8.55/4.75 ( ( dvd_dvd_rat @ A @ C )
% 8.55/4.75 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 8.55/4.75 = ( dvd_dvd_rat @ A @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1254_dvd__add__left__iff,axiom,
% 8.55/4.75 ! [A: nat,C: nat,B: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ A @ C )
% 8.55/4.75 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 8.55/4.75 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1255_dvd__add__left__iff,axiom,
% 8.55/4.75 ! [A: int,C: int,B: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ A @ C )
% 8.55/4.75 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 8.55/4.75 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1256_dvd__add__right__iff,axiom,
% 8.55/4.75 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.55/4.75 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 8.55/4.75 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1257_dvd__add__right__iff,axiom,
% 8.55/4.75 ! [A: real,B: real,C: real] :
% 8.55/4.75 ( ( dvd_dvd_real @ A @ B )
% 8.55/4.75 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 8.55/4.75 = ( dvd_dvd_real @ A @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1258_dvd__add__right__iff,axiom,
% 8.55/4.75 ! [A: rat,B: rat,C: rat] :
% 8.55/4.75 ( ( dvd_dvd_rat @ A @ B )
% 8.55/4.75 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 8.55/4.75 = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1259_dvd__add__right__iff,axiom,
% 8.55/4.75 ! [A: nat,B: nat,C: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.75 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 8.55/4.75 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1260_dvd__add__right__iff,axiom,
% 8.55/4.75 ! [A: int,B: int,C: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ A @ B )
% 8.55/4.75 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 8.55/4.75 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1261_add__Suc__shift,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 8.55/4.75 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1262_add__Suc,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 8.55/4.75 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1263_nat__arith_Osuc1,axiom,
% 8.55/4.75 ! [A3: nat,K: nat,A: nat] :
% 8.55/4.75 ( ( A3
% 8.55/4.75 = ( plus_plus_nat @ K @ A ) )
% 8.55/4.75 => ( ( suc @ A3 )
% 8.55/4.75 = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1264_Suc__eq__plus1,axiom,
% 8.55/4.75 ( suc
% 8.55/4.75 = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1265_plus__1__eq__Suc,axiom,
% 8.55/4.75 ( ( plus_plus_nat @ one_one_nat )
% 8.55/4.75 = suc ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1266_Suc__eq__plus1__left,axiom,
% 8.55/4.75 ( suc
% 8.55/4.75 = ( plus_plus_nat @ one_one_nat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1267_mult__Suc,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( times_times_nat @ ( suc @ M ) @ N )
% 8.55/4.75 = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1268_add__eq__self__zero,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( ( plus_plus_nat @ M @ N )
% 8.55/4.75 = M )
% 8.55/4.75 => ( N = zero_zero_nat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1269_plus__nat_Oadd__0,axiom,
% 8.55/4.75 ! [N: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 8.55/4.75 = N ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1270_Euclid__induct,axiom,
% 8.55/4.75 ! [P: nat > nat > $o,A: nat,B: nat] :
% 8.55/4.75 ( ! [A4: nat,B4: nat] :
% 8.55/4.75 ( ( P @ A4 @ B4 )
% 8.55/4.75 = ( P @ B4 @ A4 ) )
% 8.55/4.75 => ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
% 8.55/4.75 => ( ! [A4: nat,B4: nat] :
% 8.55/4.75 ( ( P @ A4 @ B4 )
% 8.55/4.75 => ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
% 8.55/4.75 => ( P @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1271_plus__int__code_I2_J,axiom,
% 8.55/4.75 ! [L: int] :
% 8.55/4.75 ( ( plus_plus_int @ zero_zero_int @ L )
% 8.55/4.75 = L ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1272_plus__int__code_I1_J,axiom,
% 8.55/4.75 ! [K: int] :
% 8.55/4.75 ( ( plus_plus_int @ K @ zero_zero_int )
% 8.55/4.75 = K ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1273_odd__nonzero,axiom,
% 8.55/4.75 ! [Z2: int] :
% 8.55/4.75 ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
% 8.55/4.75 != zero_zero_int ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1274_diff__add__inverse2,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 8.55/4.75 = M ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1275_diff__add__inverse,axiom,
% 8.55/4.75 ! [N: nat,M: nat] :
% 8.55/4.75 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 8.55/4.75 = M ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1276_diff__cancel2,axiom,
% 8.55/4.75 ! [M: nat,K: nat,N: nat] :
% 8.55/4.75 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 8.55/4.75 = ( minus_minus_nat @ M @ N ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1277_Nat_Odiff__cancel,axiom,
% 8.55/4.75 ! [K: nat,M: nat,N: nat] :
% 8.55/4.75 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 8.55/4.75 = ( minus_minus_nat @ M @ N ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1278_bezout__lemma__nat,axiom,
% 8.55/4.75 ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ D @ A )
% 8.55/4.75 => ( ( dvd_dvd_nat @ D @ B )
% 8.55/4.75 => ( ( ( ( times_times_nat @ A @ X )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
% 8.55/4.75 | ( ( times_times_nat @ B @ X )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
% 8.55/4.75 => ? [X4: nat,Y4: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ D @ A )
% 8.55/4.75 & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
% 8.55/4.75 & ( ( ( times_times_nat @ A @ X4 )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y4 ) @ D ) )
% 8.55/4.75 | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X4 )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1279_bezout__add__nat,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ? [D2: nat,X4: nat,Y4: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ D2 @ A )
% 8.55/4.75 & ( dvd_dvd_nat @ D2 @ B )
% 8.55/4.75 & ( ( ( times_times_nat @ A @ X4 )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D2 ) )
% 8.55/4.75 | ( ( times_times_nat @ B @ X4 )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D2 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1280_zdvd__period,axiom,
% 8.55/4.75 ! [A: int,D: int,X: int,T: int,C: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ A @ D )
% 8.55/4.75 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
% 8.55/4.75 = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1281_zdvd__reduce,axiom,
% 8.55/4.75 ! [K: int,N: int,M: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
% 8.55/4.75 = ( dvd_dvd_int @ K @ N ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1282_push__bit__add,axiom,
% 8.55/4.75 ! [N: nat,A: nat,B: nat] :
% 8.55/4.75 ( ( bit_se547839408752420682it_nat @ N @ ( plus_plus_nat @ A @ B ) )
% 8.55/4.75 = ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1283_push__bit__add,axiom,
% 8.55/4.75 ! [N: nat,A: int,B: int] :
% 8.55/4.75 ( ( bit_se545348938243370406it_int @ N @ ( plus_plus_int @ A @ B ) )
% 8.55/4.75 = ( plus_plus_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1284_push__bit__add,axiom,
% 8.55/4.75 ! [N: nat,A: code_integer,B: code_integer] :
% 8.55/4.75 ( ( bit_se7788150548672797655nteger @ N @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 8.55/4.75 = ( plus_p5714425477246183910nteger @ ( bit_se7788150548672797655nteger @ N @ A ) @ ( bit_se7788150548672797655nteger @ N @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1285_push__bit__add,axiom,
% 8.55/4.75 ! [N: nat,A: code_natural,B: code_natural] :
% 8.55/4.75 ( ( bit_se6611745700429515170atural @ N @ ( plus_p4538020629002901425atural @ A @ B ) )
% 8.55/4.75 = ( plus_p4538020629002901425atural @ ( bit_se6611745700429515170atural @ N @ A ) @ ( bit_se6611745700429515170atural @ N @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1286_Suc__nat__number__of__add,axiom,
% 8.55/4.75 ! [V: num,N: nat] :
% 8.55/4.75 ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
% 8.55/4.75 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1287_exp__add__not__zero__imp__right,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 8.55/4.75 != zero_zero_nat )
% 8.55/4.75 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.75 != zero_zero_nat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1288_exp__add__not__zero__imp__right,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 8.55/4.75 != zero_zero_int )
% 8.55/4.75 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 8.55/4.75 != zero_zero_int ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1289_exp__add__not__zero__imp__left,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 8.55/4.75 != zero_zero_nat )
% 8.55/4.75 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 8.55/4.75 != zero_zero_nat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1290_exp__add__not__zero__imp__left,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 8.55/4.75 != zero_zero_int )
% 8.55/4.75 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 8.55/4.75 != zero_zero_int ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1291_div__exp__eq,axiom,
% 8.55/4.75 ! [A: nat,M: nat,N: nat] :
% 8.55/4.75 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.75 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1292_div__exp__eq,axiom,
% 8.55/4.75 ! [A: int,M: nat,N: nat] :
% 8.55/4.75 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.75 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1293_unity__coeff__ex,axiom,
% 8.55/4.75 ! [P: code_integer > $o,L: code_integer] :
% 8.55/4.75 ( ( ? [X3: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X3 ) ) )
% 8.55/4.75 = ( ? [X3: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X3 @ zero_z3403309356797280102nteger ) )
% 8.55/4.75 & ( P @ X3 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1294_unity__coeff__ex,axiom,
% 8.55/4.75 ! [P: real > $o,L: real] :
% 8.55/4.75 ( ( ? [X3: real] : ( P @ ( times_times_real @ L @ X3 ) ) )
% 8.55/4.75 = ( ? [X3: real] :
% 8.55/4.75 ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X3 @ zero_zero_real ) )
% 8.55/4.75 & ( P @ X3 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1295_unity__coeff__ex,axiom,
% 8.55/4.75 ! [P: rat > $o,L: rat] :
% 8.55/4.75 ( ( ? [X3: rat] : ( P @ ( times_times_rat @ L @ X3 ) ) )
% 8.55/4.75 = ( ? [X3: rat] :
% 8.55/4.75 ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X3 @ zero_zero_rat ) )
% 8.55/4.75 & ( P @ X3 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1296_unity__coeff__ex,axiom,
% 8.55/4.75 ! [P: nat > $o,L: nat] :
% 8.55/4.75 ( ( ? [X3: nat] : ( P @ ( times_times_nat @ L @ X3 ) ) )
% 8.55/4.75 = ( ? [X3: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X3 @ zero_zero_nat ) )
% 8.55/4.75 & ( P @ X3 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1297_unity__coeff__ex,axiom,
% 8.55/4.75 ! [P: int > $o,L: int] :
% 8.55/4.75 ( ( ? [X3: int] : ( P @ ( times_times_int @ L @ X3 ) ) )
% 8.55/4.75 = ( ? [X3: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X3 @ zero_zero_int ) )
% 8.55/4.75 & ( P @ X3 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1298_four__x__squared,axiom,
% 8.55/4.75 ! [X: real] :
% 8.55/4.75 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.75 = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1299_inf__period_I4_J,axiom,
% 8.55/4.75 ! [D: code_integer,D3: code_integer,T: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ D @ D3 )
% 8.55/4.75 => ! [X5: code_integer,K4: code_integer] :
% 8.55/4.75 ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) ) )
% 8.55/4.75 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1300_inf__period_I4_J,axiom,
% 8.55/4.75 ! [D: real,D3: real,T: real] :
% 8.55/4.75 ( ( dvd_dvd_real @ D @ D3 )
% 8.55/4.75 => ! [X5: real,K4: real] :
% 8.55/4.75 ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
% 8.55/4.75 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1301_inf__period_I4_J,axiom,
% 8.55/4.75 ! [D: rat,D3: rat,T: rat] :
% 8.55/4.75 ( ( dvd_dvd_rat @ D @ D3 )
% 8.55/4.75 => ! [X5: rat,K4: rat] :
% 8.55/4.75 ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
% 8.55/4.75 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1302_inf__period_I4_J,axiom,
% 8.55/4.75 ! [D: int,D3: int,T: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ D @ D3 )
% 8.55/4.75 => ! [X5: int,K4: int] :
% 8.55/4.75 ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
% 8.55/4.75 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1303_inf__period_I3_J,axiom,
% 8.55/4.75 ! [D: code_integer,D3: code_integer,T: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ D @ D3 )
% 8.55/4.75 => ! [X5: code_integer,K4: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) )
% 8.55/4.75 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1304_inf__period_I3_J,axiom,
% 8.55/4.75 ! [D: real,D3: real,T: real] :
% 8.55/4.75 ( ( dvd_dvd_real @ D @ D3 )
% 8.55/4.75 => ! [X5: real,K4: real] :
% 8.55/4.75 ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
% 8.55/4.75 = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1305_inf__period_I3_J,axiom,
% 8.55/4.75 ! [D: rat,D3: rat,T: rat] :
% 8.55/4.75 ( ( dvd_dvd_rat @ D @ D3 )
% 8.55/4.75 => ! [X5: rat,K4: rat] :
% 8.55/4.75 ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
% 8.55/4.75 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1306_inf__period_I3_J,axiom,
% 8.55/4.75 ! [D: int,D3: int,T: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ D @ D3 )
% 8.55/4.75 => ! [X5: int,K4: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 8.55/4.75 = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1307_one__plus__numeral__commute,axiom,
% 8.55/4.75 ! [X: num] :
% 8.55/4.75 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
% 8.55/4.75 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1308_one__plus__numeral__commute,axiom,
% 8.55/4.75 ! [X: num] :
% 8.55/4.75 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 8.55/4.75 = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1309_one__plus__numeral__commute,axiom,
% 8.55/4.75 ! [X: num] :
% 8.55/4.75 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 8.55/4.75 = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1310_one__plus__numeral__commute,axiom,
% 8.55/4.75 ! [X: num] :
% 8.55/4.75 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 8.55/4.75 = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1311_one__plus__numeral__commute,axiom,
% 8.55/4.75 ! [X: num] :
% 8.55/4.75 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 8.55/4.75 = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1312_numeral__Bit0,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 8.55/4.75 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1313_numeral__Bit0,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 8.55/4.75 = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1314_numeral__Bit0,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 8.55/4.75 = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1315_numeral__Bit0,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 8.55/4.75 = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1316_numeral__Bit0,axiom,
% 8.55/4.75 ! [N: num] :
% 8.55/4.75 ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 8.55/4.75 = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1317_square__diff__square__factored,axiom,
% 8.55/4.75 ! [X: real,Y: real] :
% 8.55/4.75 ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
% 8.55/4.75 = ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1318_square__diff__square__factored,axiom,
% 8.55/4.75 ! [X: rat,Y: rat] :
% 8.55/4.75 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 8.55/4.75 = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1319_square__diff__square__factored,axiom,
% 8.55/4.75 ! [X: int,Y: int] :
% 8.55/4.75 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 8.55/4.75 = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1320_eq__add__iff2,axiom,
% 8.55/4.75 ! [A: real,E: real,C: real,B: real,D: real] :
% 8.55/4.75 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 8.55/4.75 = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 8.55/4.75 = ( C
% 8.55/4.75 = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1321_eq__add__iff2,axiom,
% 8.55/4.75 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 8.55/4.75 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 8.55/4.75 = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 8.55/4.75 = ( C
% 8.55/4.75 = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1322_eq__add__iff2,axiom,
% 8.55/4.75 ! [A: int,E: int,C: int,B: int,D: int] :
% 8.55/4.75 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 8.55/4.75 = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 8.55/4.75 = ( C
% 8.55/4.75 = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1323_eq__add__iff1,axiom,
% 8.55/4.75 ! [A: real,E: real,C: real,B: real,D: real] :
% 8.55/4.75 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 8.55/4.75 = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 8.55/4.75 = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
% 8.55/4.75 = D ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1324_eq__add__iff1,axiom,
% 8.55/4.75 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 8.55/4.75 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 8.55/4.75 = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 8.55/4.75 = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
% 8.55/4.75 = D ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1325_eq__add__iff1,axiom,
% 8.55/4.75 ! [A: int,E: int,C: int,B: int,D: int] :
% 8.55/4.75 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 8.55/4.75 = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 8.55/4.75 = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
% 8.55/4.75 = D ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1326_mult__diff__mult,axiom,
% 8.55/4.75 ! [X: real,Y: real,A: real,B: real] :
% 8.55/4.75 ( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
% 8.55/4.75 = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1327_mult__diff__mult,axiom,
% 8.55/4.75 ! [X: rat,Y: rat,A: rat,B: rat] :
% 8.55/4.75 ( ( minus_minus_rat @ ( times_times_rat @ X @ Y ) @ ( times_times_rat @ A @ B ) )
% 8.55/4.75 = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1328_mult__diff__mult,axiom,
% 8.55/4.75 ! [X: int,Y: int,A: int,B: int] :
% 8.55/4.75 ( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
% 8.55/4.75 = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1329_div__plus__div__distrib__dvd__right,axiom,
% 8.55/4.75 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ C @ B )
% 8.55/4.75 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 8.55/4.75 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1330_div__plus__div__distrib__dvd__right,axiom,
% 8.55/4.75 ! [C: nat,B: nat,A: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ C @ B )
% 8.55/4.75 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1331_div__plus__div__distrib__dvd__right,axiom,
% 8.55/4.75 ! [C: int,B: int,A: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ C @ B )
% 8.55/4.75 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1332_div__plus__div__distrib__dvd__left,axiom,
% 8.55/4.75 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ C @ A )
% 8.55/4.75 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 8.55/4.75 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1333_div__plus__div__distrib__dvd__left,axiom,
% 8.55/4.75 ! [C: nat,A: nat,B: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ C @ A )
% 8.55/4.75 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1334_div__plus__div__distrib__dvd__left,axiom,
% 8.55/4.75 ! [C: int,A: int,B: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ C @ A )
% 8.55/4.75 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.55/4.75 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1335_add__is__1,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( ( plus_plus_nat @ M @ N )
% 8.55/4.75 = ( suc @ zero_zero_nat ) )
% 8.55/4.75 = ( ( ( M
% 8.55/4.75 = ( suc @ zero_zero_nat ) )
% 8.55/4.75 & ( N = zero_zero_nat ) )
% 8.55/4.75 | ( ( M = zero_zero_nat )
% 8.55/4.75 & ( N
% 8.55/4.75 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1336_one__is__add,axiom,
% 8.55/4.75 ! [M: nat,N: nat] :
% 8.55/4.75 ( ( ( suc @ zero_zero_nat )
% 8.55/4.75 = ( plus_plus_nat @ M @ N ) )
% 8.55/4.75 = ( ( ( M
% 8.55/4.75 = ( suc @ zero_zero_nat ) )
% 8.55/4.75 & ( N = zero_zero_nat ) )
% 8.55/4.75 | ( ( M = zero_zero_nat )
% 8.55/4.75 & ( N
% 8.55/4.75 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1337_diff__add__0,axiom,
% 8.55/4.75 ! [N: nat,M: nat] :
% 8.55/4.75 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 8.55/4.75 = zero_zero_nat ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1338_mult__eq__if,axiom,
% 8.55/4.75 ( times_times_nat
% 8.55/4.75 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1339_int__ops_I4_J,axiom,
% 8.55/4.75 ! [A: nat] :
% 8.55/4.75 ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 8.55/4.75 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1340_int__Suc,axiom,
% 8.55/4.75 ! [N: nat] :
% 8.55/4.75 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 8.55/4.75 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1341_bezout__add__strong__nat,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( A != zero_zero_nat )
% 8.55/4.75 => ? [D2: nat,X4: nat,Y4: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ D2 @ A )
% 8.55/4.75 & ( dvd_dvd_nat @ D2 @ B )
% 8.55/4.75 & ( ( times_times_nat @ A @ X4 )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D2 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1342_divide__add__eq__iff,axiom,
% 8.55/4.75 ! [Z2: complex,X: complex,Y: complex] :
% 8.55/4.75 ( ( Z2 != zero_zero_complex )
% 8.55/4.75 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z2 ) @ Y )
% 8.55/4.75 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1343_divide__add__eq__iff,axiom,
% 8.55/4.75 ! [Z2: real,X: real,Y: real] :
% 8.55/4.75 ( ( Z2 != zero_zero_real )
% 8.55/4.75 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z2 ) @ Y )
% 8.55/4.75 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1344_divide__add__eq__iff,axiom,
% 8.55/4.75 ! [Z2: rat,X: rat,Y: rat] :
% 8.55/4.75 ( ( Z2 != zero_zero_rat )
% 8.55/4.75 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y )
% 8.55/4.75 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1345_add__divide__eq__iff,axiom,
% 8.55/4.75 ! [Z2: complex,X: complex,Y: complex] :
% 8.55/4.75 ( ( Z2 != zero_zero_complex )
% 8.55/4.75 => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
% 8.55/4.75 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1346_add__divide__eq__iff,axiom,
% 8.55/4.75 ! [Z2: real,X: real,Y: real] :
% 8.55/4.75 ( ( Z2 != zero_zero_real )
% 8.55/4.75 => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z2 ) )
% 8.55/4.75 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1347_add__divide__eq__iff,axiom,
% 8.55/4.75 ! [Z2: rat,X: rat,Y: rat] :
% 8.55/4.75 ( ( Z2 != zero_zero_rat )
% 8.55/4.75 => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z2 ) )
% 8.55/4.75 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1348_add__num__frac,axiom,
% 8.55/4.75 ! [Y: complex,Z2: complex,X: complex] :
% 8.55/4.75 ( ( Y != zero_zero_complex )
% 8.55/4.75 => ( ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ X @ Y ) )
% 8.55/4.75 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1349_add__num__frac,axiom,
% 8.55/4.75 ! [Y: real,Z2: real,X: real] :
% 8.55/4.75 ( ( Y != zero_zero_real )
% 8.55/4.75 => ( ( plus_plus_real @ Z2 @ ( divide_divide_real @ X @ Y ) )
% 8.55/4.75 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1350_add__num__frac,axiom,
% 8.55/4.75 ! [Y: rat,Z2: rat,X: rat] :
% 8.55/4.75 ( ( Y != zero_zero_rat )
% 8.55/4.75 => ( ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) )
% 8.55/4.75 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1351_add__frac__num,axiom,
% 8.55/4.75 ! [Y: complex,X: complex,Z2: complex] :
% 8.55/4.75 ( ( Y != zero_zero_complex )
% 8.55/4.75 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z2 )
% 8.55/4.75 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1352_add__frac__num,axiom,
% 8.55/4.75 ! [Y: real,X: real,Z2: real] :
% 8.55/4.75 ( ( Y != zero_zero_real )
% 8.55/4.75 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z2 )
% 8.55/4.75 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1353_add__frac__num,axiom,
% 8.55/4.75 ! [Y: rat,X: rat,Z2: rat] :
% 8.55/4.75 ( ( Y != zero_zero_rat )
% 8.55/4.75 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 )
% 8.55/4.75 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1354_add__frac__eq,axiom,
% 8.55/4.75 ! [Y: complex,Z2: complex,X: complex,W: complex] :
% 8.55/4.75 ( ( Y != zero_zero_complex )
% 8.55/4.75 => ( ( Z2 != zero_zero_complex )
% 8.55/4.75 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z2 ) )
% 8.55/4.75 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1355_add__frac__eq,axiom,
% 8.55/4.75 ! [Y: real,Z2: real,X: real,W: real] :
% 8.55/4.75 ( ( Y != zero_zero_real )
% 8.55/4.75 => ( ( Z2 != zero_zero_real )
% 8.55/4.75 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
% 8.55/4.75 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1356_add__frac__eq,axiom,
% 8.55/4.75 ! [Y: rat,Z2: rat,X: rat,W: rat] :
% 8.55/4.75 ( ( Y != zero_zero_rat )
% 8.55/4.75 => ( ( Z2 != zero_zero_rat )
% 8.55/4.75 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z2 ) )
% 8.55/4.75 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1357_add__divide__eq__if__simps_I1_J,axiom,
% 8.55/4.75 ! [Z2: complex,A: complex,B: complex] :
% 8.55/4.75 ( ( ( Z2 = zero_zero_complex )
% 8.55/4.75 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 8.55/4.75 = A ) )
% 8.55/4.75 & ( ( Z2 != zero_zero_complex )
% 8.55/4.75 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 8.55/4.75 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1358_add__divide__eq__if__simps_I1_J,axiom,
% 8.55/4.75 ! [Z2: real,A: real,B: real] :
% 8.55/4.75 ( ( ( Z2 = zero_zero_real )
% 8.55/4.75 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 8.55/4.75 = A ) )
% 8.55/4.75 & ( ( Z2 != zero_zero_real )
% 8.55/4.75 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 8.55/4.75 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1359_add__divide__eq__if__simps_I1_J,axiom,
% 8.55/4.75 ! [Z2: rat,A: rat,B: rat] :
% 8.55/4.75 ( ( ( Z2 = zero_zero_rat )
% 8.55/4.75 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 8.55/4.75 = A ) )
% 8.55/4.75 & ( ( Z2 != zero_zero_rat )
% 8.55/4.75 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 8.55/4.75 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1360_add__divide__eq__if__simps_I2_J,axiom,
% 8.55/4.75 ! [Z2: complex,A: complex,B: complex] :
% 8.55/4.75 ( ( ( Z2 = zero_zero_complex )
% 8.55/4.75 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 8.55/4.75 = B ) )
% 8.55/4.75 & ( ( Z2 != zero_zero_complex )
% 8.55/4.75 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 8.55/4.75 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1361_add__divide__eq__if__simps_I2_J,axiom,
% 8.55/4.75 ! [Z2: real,A: real,B: real] :
% 8.55/4.75 ( ( ( Z2 = zero_zero_real )
% 8.55/4.75 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 8.55/4.75 = B ) )
% 8.55/4.75 & ( ( Z2 != zero_zero_real )
% 8.55/4.75 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 8.55/4.75 = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1362_add__divide__eq__if__simps_I2_J,axiom,
% 8.55/4.75 ! [Z2: rat,A: rat,B: rat] :
% 8.55/4.75 ( ( ( Z2 = zero_zero_rat )
% 8.55/4.75 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 8.55/4.75 = B ) )
% 8.55/4.75 & ( ( Z2 != zero_zero_rat )
% 8.55/4.75 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 8.55/4.75 = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1363_div__add__self2,axiom,
% 8.55/4.75 ! [B: nat,A: nat] :
% 8.55/4.75 ( ( B != zero_zero_nat )
% 8.55/4.75 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 8.55/4.75 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1364_div__add__self2,axiom,
% 8.55/4.75 ! [B: int,A: int] :
% 8.55/4.75 ( ( B != zero_zero_int )
% 8.55/4.75 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
% 8.55/4.75 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1365_div__add__self1,axiom,
% 8.55/4.75 ! [B: nat,A: nat] :
% 8.55/4.75 ( ( B != zero_zero_nat )
% 8.55/4.75 => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 8.55/4.75 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1366_div__add__self1,axiom,
% 8.55/4.75 ! [B: int,A: int] :
% 8.55/4.75 ( ( B != zero_zero_int )
% 8.55/4.75 => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
% 8.55/4.75 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1367_square__diff__one__factored,axiom,
% 8.55/4.75 ! [X: complex] :
% 8.55/4.75 ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
% 8.55/4.75 = ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1368_square__diff__one__factored,axiom,
% 8.55/4.75 ! [X: real] :
% 8.55/4.75 ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
% 8.55/4.75 = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1369_square__diff__one__factored,axiom,
% 8.55/4.75 ! [X: rat] :
% 8.55/4.75 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
% 8.55/4.75 = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1370_square__diff__one__factored,axiom,
% 8.55/4.75 ! [X: int] :
% 8.55/4.75 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
% 8.55/4.75 = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1371_push__bit__nat__def,axiom,
% 8.55/4.75 ( bit_se547839408752420682it_nat
% 8.55/4.75 = ( ^ [N3: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1372_push__bit__int__def,axiom,
% 8.55/4.75 ( bit_se545348938243370406it_int
% 8.55/4.75 = ( ^ [N3: nat,K2: int] : ( times_times_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1373_nat__1__add__1,axiom,
% 8.55/4.75 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 8.55/4.75 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1374_add__eq__if,axiom,
% 8.55/4.75 ( plus_plus_nat
% 8.55/4.75 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1375_left__add__twice,axiom,
% 8.55/4.75 ! [A: complex,B: complex] :
% 8.55/4.75 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
% 8.55/4.75 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1376_left__add__twice,axiom,
% 8.55/4.75 ! [A: real,B: real] :
% 8.55/4.75 ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
% 8.55/4.75 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1377_left__add__twice,axiom,
% 8.55/4.75 ! [A: rat,B: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 8.55/4.75 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1378_left__add__twice,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 8.55/4.75 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1379_left__add__twice,axiom,
% 8.55/4.75 ! [A: int,B: int] :
% 8.55/4.75 ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
% 8.55/4.75 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1380_mult__2__right,axiom,
% 8.55/4.75 ! [Z2: complex] :
% 8.55/4.75 ( ( times_times_complex @ Z2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 8.55/4.75 = ( plus_plus_complex @ Z2 @ Z2 ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1381_mult__2__right,axiom,
% 8.55/4.75 ! [Z2: real] :
% 8.55/4.75 ( ( times_times_real @ Z2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.55/4.75 = ( plus_plus_real @ Z2 @ Z2 ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1382_mult__2__right,axiom,
% 8.55/4.75 ! [Z2: rat] :
% 8.55/4.75 ( ( times_times_rat @ Z2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 8.55/4.75 = ( plus_plus_rat @ Z2 @ Z2 ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1383_mult__2__right,axiom,
% 8.55/4.75 ! [Z2: nat] :
% 8.55/4.75 ( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.75 = ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1384_mult__2__right,axiom,
% 8.55/4.75 ! [Z2: int] :
% 8.55/4.75 ( ( times_times_int @ Z2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.55/4.75 = ( plus_plus_int @ Z2 @ Z2 ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1385_mult__2,axiom,
% 8.55/4.75 ! [Z2: complex] :
% 8.55/4.75 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 )
% 8.55/4.75 = ( plus_plus_complex @ Z2 @ Z2 ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1386_mult__2,axiom,
% 8.55/4.75 ! [Z2: real] :
% 8.55/4.75 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 )
% 8.55/4.75 = ( plus_plus_real @ Z2 @ Z2 ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1387_mult__2,axiom,
% 8.55/4.75 ! [Z2: rat] :
% 8.55/4.75 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 )
% 8.55/4.75 = ( plus_plus_rat @ Z2 @ Z2 ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1388_mult__2,axiom,
% 8.55/4.75 ! [Z2: nat] :
% 8.55/4.75 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
% 8.55/4.75 = ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1389_mult__2,axiom,
% 8.55/4.75 ! [Z2: int] :
% 8.55/4.75 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z2 )
% 8.55/4.75 = ( plus_plus_int @ Z2 @ Z2 ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1390_field__sum__of__halves,axiom,
% 8.55/4.75 ! [X: real] :
% 8.55/4.75 ( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.55/4.75 = X ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1391_field__sum__of__halves,axiom,
% 8.55/4.75 ! [X: rat] :
% 8.55/4.75 ( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 8.55/4.75 = X ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1392_odd__even__add,axiom,
% 8.55/4.75 ! [A: code_integer,B: code_integer] :
% 8.55/4.75 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
% 8.55/4.75 => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1393_odd__even__add,axiom,
% 8.55/4.75 ! [A: nat,B: nat] :
% 8.55/4.75 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 8.55/4.75 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1394_odd__even__add,axiom,
% 8.55/4.75 ! [A: int,B: int] :
% 8.55/4.75 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
% 8.55/4.75 => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1395_nat__induct2,axiom,
% 8.55/4.75 ! [P: nat > $o,N: nat] :
% 8.55/4.75 ( ( P @ zero_zero_nat )
% 8.55/4.75 => ( ( P @ one_one_nat )
% 8.55/4.75 => ( ! [N2: nat] :
% 8.55/4.75 ( ( P @ N2 )
% 8.55/4.75 => ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.55/4.75 => ( P @ N ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1396_even__diff__iff,axiom,
% 8.55/4.75 ! [K: int,L: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
% 8.55/4.75 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1397_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 8.55/4.75 ! [N: nat,M: nat] :
% 8.55/4.75 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.75 != zero_zero_nat )
% 8.55/4.75 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 8.55/4.75 != zero_zero_nat ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1398_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 8.55/4.75 ! [N: nat,M: nat] :
% 8.55/4.75 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 8.55/4.75 != zero_zero_int )
% 8.55/4.75 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 8.55/4.75 != zero_zero_int ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1399_push__bit__eq__mult,axiom,
% 8.55/4.75 ( bit_se547839408752420682it_nat
% 8.55/4.75 = ( ^ [N3: nat,A2: nat] : ( times_times_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1400_push__bit__eq__mult,axiom,
% 8.55/4.75 ( bit_se545348938243370406it_int
% 8.55/4.75 = ( ^ [N3: nat,A2: int] : ( times_times_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1401_push__bit__eq__mult,axiom,
% 8.55/4.75 ( bit_se7788150548672797655nteger
% 8.55/4.75 = ( ^ [N3: nat,A2: code_integer] : ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1402_push__bit__eq__mult,axiom,
% 8.55/4.75 ( bit_se6611745700429515170atural
% 8.55/4.75 = ( ^ [N3: nat,A2: code_natural] : ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1403_exp__dvdE,axiom,
% 8.55/4.75 ! [N: nat,A: nat] :
% 8.55/4.75 ( ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A )
% 8.55/4.75 => ~ ! [B4: nat] :
% 8.55/4.75 ( A
% 8.55/4.75 != ( bit_se547839408752420682it_nat @ N @ B4 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1404_exp__dvdE,axiom,
% 8.55/4.75 ! [N: nat,A: int] :
% 8.55/4.75 ( ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A )
% 8.55/4.75 => ~ ! [B4: int] :
% 8.55/4.75 ( A
% 8.55/4.75 != ( bit_se545348938243370406it_int @ N @ B4 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1405_exp__dvdE,axiom,
% 8.55/4.75 ! [N: nat,A: code_integer] :
% 8.55/4.75 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A )
% 8.55/4.75 => ~ ! [B4: code_integer] :
% 8.55/4.75 ( A
% 8.55/4.75 != ( bit_se7788150548672797655nteger @ N @ B4 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1406_exp__dvdE,axiom,
% 8.55/4.75 ! [N: nat,A: code_natural] :
% 8.55/4.75 ( ( dvd_dvd_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) @ A )
% 8.55/4.75 => ~ ! [B4: code_natural] :
% 8.55/4.75 ( A
% 8.55/4.75 != ( bit_se6611745700429515170atural @ N @ B4 ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1407_oddE,axiom,
% 8.55/4.75 ! [A: code_integer] :
% 8.55/4.75 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ~ ! [B4: code_integer] :
% 8.55/4.75 ( A
% 8.55/4.75 != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) @ one_one_Code_integer ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1408_oddE,axiom,
% 8.55/4.75 ! [A: nat] :
% 8.55/4.75 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ~ ! [B4: nat] :
% 8.55/4.75 ( A
% 8.55/4.75 != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) @ one_one_nat ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1409_oddE,axiom,
% 8.55/4.75 ! [A: int] :
% 8.55/4.75 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.55/4.75 => ~ ! [B4: int] :
% 8.55/4.75 ( A
% 8.55/4.75 != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) @ one_one_int ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1410_inf__period_I2_J,axiom,
% 8.55/4.75 ! [P: real > $o,D3: real,Q: real > $o] :
% 8.55/4.75 ( ! [X4: real,K3: real] :
% 8.55/4.75 ( ( P @ X4 )
% 8.55/4.75 = ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D3 ) ) ) )
% 8.55/4.75 => ( ! [X4: real,K3: real] :
% 8.55/4.75 ( ( Q @ X4 )
% 8.55/4.75 = ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D3 ) ) ) )
% 8.55/4.75 => ! [X5: real,K4: real] :
% 8.55/4.75 ( ( ( P @ X5 )
% 8.55/4.75 | ( Q @ X5 ) )
% 8.55/4.75 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) )
% 8.55/4.75 | ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1411_inf__period_I2_J,axiom,
% 8.55/4.75 ! [P: rat > $o,D3: rat,Q: rat > $o] :
% 8.55/4.75 ( ! [X4: rat,K3: rat] :
% 8.55/4.75 ( ( P @ X4 )
% 8.55/4.75 = ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) ) )
% 8.55/4.75 => ( ! [X4: rat,K3: rat] :
% 8.55/4.75 ( ( Q @ X4 )
% 8.55/4.75 = ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) ) )
% 8.55/4.75 => ! [X5: rat,K4: rat] :
% 8.55/4.75 ( ( ( P @ X5 )
% 8.55/4.75 | ( Q @ X5 ) )
% 8.55/4.75 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) )
% 8.55/4.75 | ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1412_inf__period_I2_J,axiom,
% 8.55/4.75 ! [P: int > $o,D3: int,Q: int > $o] :
% 8.55/4.75 ( ! [X4: int,K3: int] :
% 8.55/4.75 ( ( P @ X4 )
% 8.55/4.75 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
% 8.55/4.75 => ( ! [X4: int,K3: int] :
% 8.55/4.75 ( ( Q @ X4 )
% 8.55/4.75 = ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
% 8.55/4.75 => ! [X5: int,K4: int] :
% 8.55/4.75 ( ( ( P @ X5 )
% 8.55/4.75 | ( Q @ X5 ) )
% 8.55/4.75 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
% 8.55/4.75 | ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1413_sum__power2__eq__zero__iff,axiom,
% 8.55/4.75 ! [X: rat,Y: rat] :
% 8.55/4.75 ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.75 = zero_zero_rat )
% 8.55/4.75 = ( ( X = zero_zero_rat )
% 8.55/4.75 & ( Y = zero_zero_rat ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1414_sum__power2__eq__zero__iff,axiom,
% 8.55/4.75 ! [X: real,Y: real] :
% 8.55/4.75 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.75 = zero_zero_real )
% 8.55/4.75 = ( ( X = zero_zero_real )
% 8.55/4.75 & ( Y = zero_zero_real ) ) ) ).
% 8.55/4.75
% 8.55/4.75 thf(fact_1415_sum__power2__eq__zero__iff,axiom,
% 8.55/4.75 ! [X: int,Y: int] :
% 8.55/4.75 ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.76 = zero_zero_int )
% 8.55/4.76 = ( ( X = zero_zero_int )
% 8.55/4.76 & ( Y = zero_zero_int ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1416_zero__eq__power2,axiom,
% 8.55/4.76 ! [A: rat] :
% 8.55/4.76 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = zero_zero_rat )
% 8.55/4.76 = ( A = zero_zero_rat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1417_zero__eq__power2,axiom,
% 8.55/4.76 ! [A: nat] :
% 8.55/4.76 ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = zero_zero_nat )
% 8.55/4.76 = ( A = zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1418_zero__eq__power2,axiom,
% 8.55/4.76 ! [A: real] :
% 8.55/4.76 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = zero_zero_real )
% 8.55/4.76 = ( A = zero_zero_real ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1419_zero__eq__power2,axiom,
% 8.55/4.76 ! [A: int] :
% 8.55/4.76 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = zero_zero_int )
% 8.55/4.76 = ( A = zero_zero_int ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1420_zero__eq__power2,axiom,
% 8.55/4.76 ! [A: complex] :
% 8.55/4.76 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = zero_zero_complex )
% 8.55/4.76 = ( A = zero_zero_complex ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1421_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 8.55/4.76 ! [Y: nat,X: num,N: nat] :
% 8.55/4.76 ( ( ( semiri8010041392384452111omplex @ Y )
% 8.55/4.76 = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
% 8.55/4.76 = ( Y
% 8.55/4.76 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1422_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 8.55/4.76 ! [Y: nat,X: num,N: nat] :
% 8.55/4.76 ( ( ( semiri5074537144036343181t_real @ Y )
% 8.55/4.76 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 8.55/4.76 = ( Y
% 8.55/4.76 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1423_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 8.55/4.76 ! [Y: nat,X: num,N: nat] :
% 8.55/4.76 ( ( ( semiri681578069525770553at_rat @ Y )
% 8.55/4.76 = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 8.55/4.76 = ( Y
% 8.55/4.76 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1424_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 8.55/4.76 ! [Y: nat,X: num,N: nat] :
% 8.55/4.76 ( ( ( semiri1316708129612266289at_nat @ Y )
% 8.55/4.76 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 8.55/4.76 = ( Y
% 8.55/4.76 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1425_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 8.55/4.76 ! [Y: nat,X: num,N: nat] :
% 8.55/4.76 ( ( ( semiri1314217659103216013at_int @ Y )
% 8.55/4.76 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 8.55/4.76 = ( Y
% 8.55/4.76 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1426_numeral__power__eq__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: num,N: nat,Y: nat] :
% 8.55/4.76 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
% 8.55/4.76 = ( semiri8010041392384452111omplex @ Y ) )
% 8.55/4.76 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 8.55/4.76 = Y ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1427_numeral__power__eq__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: num,N: nat,Y: nat] :
% 8.55/4.76 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 8.55/4.76 = ( semiri5074537144036343181t_real @ Y ) )
% 8.55/4.76 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 8.55/4.76 = Y ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1428_numeral__power__eq__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: num,N: nat,Y: nat] :
% 8.55/4.76 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
% 8.55/4.76 = ( semiri681578069525770553at_rat @ Y ) )
% 8.55/4.76 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 8.55/4.76 = Y ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1429_numeral__power__eq__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: num,N: nat,Y: nat] :
% 8.55/4.76 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 8.55/4.76 = ( semiri1316708129612266289at_nat @ Y ) )
% 8.55/4.76 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 8.55/4.76 = Y ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1430_numeral__power__eq__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: num,N: nat,Y: nat] :
% 8.55/4.76 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 8.55/4.76 = ( semiri1314217659103216013at_int @ Y ) )
% 8.55/4.76 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 8.55/4.76 = Y ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1431_power__mult__numeral,axiom,
% 8.55/4.76 ! [A: nat,M: num,N: num] :
% 8.55/4.76 ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 8.55/4.76 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1432_power__mult__numeral,axiom,
% 8.55/4.76 ! [A: real,M: num,N: num] :
% 8.55/4.76 ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 8.55/4.76 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1433_power__mult__numeral,axiom,
% 8.55/4.76 ! [A: int,M: num,N: num] :
% 8.55/4.76 ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 8.55/4.76 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1434_power__mult__numeral,axiom,
% 8.55/4.76 ! [A: complex,M: num,N: num] :
% 8.55/4.76 ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 8.55/4.76 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1435_power2__diff,axiom,
% 8.55/4.76 ! [X: complex,Y: complex] :
% 8.55/4.76 ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1436_power2__diff,axiom,
% 8.55/4.76 ! [X: real,Y: real] :
% 8.55/4.76 ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1437_power2__diff,axiom,
% 8.55/4.76 ! [X: rat,Y: rat] :
% 8.55/4.76 ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1438_power2__diff,axiom,
% 8.55/4.76 ! [X: int,Y: int] :
% 8.55/4.76 ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1439_real__average__minus__first,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 8.55/4.76 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1440_real__average__minus__second,axiom,
% 8.55/4.76 ! [B: real,A: real] :
% 8.55/4.76 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 8.55/4.76 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1441_power__Suc0__right,axiom,
% 8.55/4.76 ! [A: nat] :
% 8.55/4.76 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 8.55/4.76 = A ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1442_power__Suc0__right,axiom,
% 8.55/4.76 ! [A: real] :
% 8.55/4.76 ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 8.55/4.76 = A ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1443_power__Suc0__right,axiom,
% 8.55/4.76 ! [A: int] :
% 8.55/4.76 ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 8.55/4.76 = A ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1444_power__Suc0__right,axiom,
% 8.55/4.76 ! [A: complex] :
% 8.55/4.76 ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 8.55/4.76 = A ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1445_power__add__numeral,axiom,
% 8.55/4.76 ! [A: complex,M: num,N: num] :
% 8.55/4.76 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
% 8.55/4.76 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1446_power__add__numeral,axiom,
% 8.55/4.76 ! [A: real,M: num,N: num] :
% 8.55/4.76 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
% 8.55/4.76 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1447_power__add__numeral,axiom,
% 8.55/4.76 ! [A: rat,M: num,N: num] :
% 8.55/4.76 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
% 8.55/4.76 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1448_power__add__numeral,axiom,
% 8.55/4.76 ! [A: nat,M: num,N: num] :
% 8.55/4.76 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
% 8.55/4.76 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1449_power__add__numeral,axiom,
% 8.55/4.76 ! [A: int,M: num,N: num] :
% 8.55/4.76 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
% 8.55/4.76 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1450_power__add__numeral2,axiom,
% 8.55/4.76 ! [A: complex,M: num,N: num,B: complex] :
% 8.55/4.76 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 8.55/4.76 = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1451_power__add__numeral2,axiom,
% 8.55/4.76 ! [A: real,M: num,N: num,B: real] :
% 8.55/4.76 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 8.55/4.76 = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1452_power__add__numeral2,axiom,
% 8.55/4.76 ! [A: rat,M: num,N: num,B: rat] :
% 8.55/4.76 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 8.55/4.76 = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1453_power__add__numeral2,axiom,
% 8.55/4.76 ! [A: nat,M: num,N: num,B: nat] :
% 8.55/4.76 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 8.55/4.76 = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1454_power__add__numeral2,axiom,
% 8.55/4.76 ! [A: int,M: num,N: num,B: int] :
% 8.55/4.76 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 8.55/4.76 = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1455_power__one,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( power_power_rat @ one_one_rat @ N )
% 8.55/4.76 = one_one_rat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1456_power__one,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( power_power_nat @ one_one_nat @ N )
% 8.55/4.76 = one_one_nat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1457_power__one,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( power_power_real @ one_one_real @ N )
% 8.55/4.76 = one_one_real ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1458_power__one,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( power_power_int @ one_one_int @ N )
% 8.55/4.76 = one_one_int ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1459_power__one,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( power_power_complex @ one_one_complex @ N )
% 8.55/4.76 = one_one_complex ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1460_power__one__right,axiom,
% 8.55/4.76 ! [A: nat] :
% 8.55/4.76 ( ( power_power_nat @ A @ one_one_nat )
% 8.55/4.76 = A ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1461_power__one__right,axiom,
% 8.55/4.76 ! [A: real] :
% 8.55/4.76 ( ( power_power_real @ A @ one_one_nat )
% 8.55/4.76 = A ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1462_power__one__right,axiom,
% 8.55/4.76 ! [A: int] :
% 8.55/4.76 ( ( power_power_int @ A @ one_one_nat )
% 8.55/4.76 = A ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1463_power__one__right,axiom,
% 8.55/4.76 ! [A: complex] :
% 8.55/4.76 ( ( power_power_complex @ A @ one_one_nat )
% 8.55/4.76 = A ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1464_sum__squares__eq__zero__iff,axiom,
% 8.55/4.76 ! [X: real,Y: real] :
% 8.55/4.76 ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
% 8.55/4.76 = zero_zero_real )
% 8.55/4.76 = ( ( X = zero_zero_real )
% 8.55/4.76 & ( Y = zero_zero_real ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1465_sum__squares__eq__zero__iff,axiom,
% 8.55/4.76 ! [X: rat,Y: rat] :
% 8.55/4.76 ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 8.55/4.76 = zero_zero_rat )
% 8.55/4.76 = ( ( X = zero_zero_rat )
% 8.55/4.76 & ( Y = zero_zero_rat ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1466_sum__squares__eq__zero__iff,axiom,
% 8.55/4.76 ! [X: int,Y: int] :
% 8.55/4.76 ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 8.55/4.76 = zero_zero_int )
% 8.55/4.76 = ( ( X = zero_zero_int )
% 8.55/4.76 & ( Y = zero_zero_int ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1467_power__0__Suc,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 8.55/4.76 = zero_zero_rat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1468_power__0__Suc,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 8.55/4.76 = zero_zero_nat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1469_power__0__Suc,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 8.55/4.76 = zero_zero_real ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1470_power__0__Suc,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 8.55/4.76 = zero_zero_int ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1471_power__0__Suc,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 8.55/4.76 = zero_zero_complex ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1472_power__zero__numeral,axiom,
% 8.55/4.76 ! [K: num] :
% 8.55/4.76 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 8.55/4.76 = zero_zero_rat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1473_power__zero__numeral,axiom,
% 8.55/4.76 ! [K: num] :
% 8.55/4.76 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 8.55/4.76 = zero_zero_nat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1474_power__zero__numeral,axiom,
% 8.55/4.76 ! [K: num] :
% 8.55/4.76 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 8.55/4.76 = zero_zero_real ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1475_power__zero__numeral,axiom,
% 8.55/4.76 ! [K: num] :
% 8.55/4.76 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 8.55/4.76 = zero_zero_int ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1476_power__zero__numeral,axiom,
% 8.55/4.76 ! [K: num] :
% 8.55/4.76 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 8.55/4.76 = zero_zero_complex ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1477_nat__power__eq__Suc__0__iff,axiom,
% 8.55/4.76 ! [X: nat,M: nat] :
% 8.55/4.76 ( ( ( power_power_nat @ X @ M )
% 8.55/4.76 = ( suc @ zero_zero_nat ) )
% 8.55/4.76 = ( ( M = zero_zero_nat )
% 8.55/4.76 | ( X
% 8.55/4.76 = ( suc @ zero_zero_nat ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1478_power__Suc__0,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.55/4.76 = ( suc @ zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1479_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,B: nat,W: nat] :
% 8.55/4.76 ( ( ( semiri8010041392384452111omplex @ X )
% 8.55/4.76 = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
% 8.55/4.76 = ( X
% 8.55/4.76 = ( power_power_nat @ B @ W ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1480_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,B: nat,W: nat] :
% 8.55/4.76 ( ( ( semiri5074537144036343181t_real @ X )
% 8.55/4.76 = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 8.55/4.76 = ( X
% 8.55/4.76 = ( power_power_nat @ B @ W ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1481_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,B: nat,W: nat] :
% 8.55/4.76 ( ( ( semiri681578069525770553at_rat @ X )
% 8.55/4.76 = ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 8.55/4.76 = ( X
% 8.55/4.76 = ( power_power_nat @ B @ W ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1482_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,B: nat,W: nat] :
% 8.55/4.76 ( ( ( semiri1316708129612266289at_nat @ X )
% 8.55/4.76 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 8.55/4.76 = ( X
% 8.55/4.76 = ( power_power_nat @ B @ W ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1483_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,B: nat,W: nat] :
% 8.55/4.76 ( ( ( semiri1314217659103216013at_int @ X )
% 8.55/4.76 = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 8.55/4.76 = ( X
% 8.55/4.76 = ( power_power_nat @ B @ W ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1484_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 8.55/4.76 ! [B: nat,W: nat,X: nat] :
% 8.55/4.76 ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
% 8.55/4.76 = ( semiri8010041392384452111omplex @ X ) )
% 8.55/4.76 = ( ( power_power_nat @ B @ W )
% 8.55/4.76 = X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1485_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 8.55/4.76 ! [B: nat,W: nat,X: nat] :
% 8.55/4.76 ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
% 8.55/4.76 = ( semiri5074537144036343181t_real @ X ) )
% 8.55/4.76 = ( ( power_power_nat @ B @ W )
% 8.55/4.76 = X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1486_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 8.55/4.76 ! [B: nat,W: nat,X: nat] :
% 8.55/4.76 ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W )
% 8.55/4.76 = ( semiri681578069525770553at_rat @ X ) )
% 8.55/4.76 = ( ( power_power_nat @ B @ W )
% 8.55/4.76 = X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1487_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 8.55/4.76 ! [B: nat,W: nat,X: nat] :
% 8.55/4.76 ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
% 8.55/4.76 = ( semiri1316708129612266289at_nat @ X ) )
% 8.55/4.76 = ( ( power_power_nat @ B @ W )
% 8.55/4.76 = X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1488_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 8.55/4.76 ! [B: nat,W: nat,X: nat] :
% 8.55/4.76 ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
% 8.55/4.76 = ( semiri1314217659103216013at_int @ X ) )
% 8.55/4.76 = ( ( power_power_nat @ B @ W )
% 8.55/4.76 = X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1489_of__nat__power,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
% 8.55/4.76 = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1490_of__nat__power,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
% 8.55/4.76 = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1491_of__nat__power,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
% 8.55/4.76 = ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1492_of__nat__power,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
% 8.55/4.76 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1493_of__nat__power,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
% 8.55/4.76 = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1494_power__not__zero,axiom,
% 8.55/4.76 ! [A: rat,N: nat] :
% 8.55/4.76 ( ( A != zero_zero_rat )
% 8.55/4.76 => ( ( power_power_rat @ A @ N )
% 8.55/4.76 != zero_zero_rat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1495_power__not__zero,axiom,
% 8.55/4.76 ! [A: nat,N: nat] :
% 8.55/4.76 ( ( A != zero_zero_nat )
% 8.55/4.76 => ( ( power_power_nat @ A @ N )
% 8.55/4.76 != zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1496_power__not__zero,axiom,
% 8.55/4.76 ! [A: real,N: nat] :
% 8.55/4.76 ( ( A != zero_zero_real )
% 8.55/4.76 => ( ( power_power_real @ A @ N )
% 8.55/4.76 != zero_zero_real ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1497_power__not__zero,axiom,
% 8.55/4.76 ! [A: int,N: nat] :
% 8.55/4.76 ( ( A != zero_zero_int )
% 8.55/4.76 => ( ( power_power_int @ A @ N )
% 8.55/4.76 != zero_zero_int ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1498_power__not__zero,axiom,
% 8.55/4.76 ! [A: complex,N: nat] :
% 8.55/4.76 ( ( A != zero_zero_complex )
% 8.55/4.76 => ( ( power_power_complex @ A @ N )
% 8.55/4.76 != zero_zero_complex ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1499_power__commuting__commutes,axiom,
% 8.55/4.76 ! [X: complex,Y: complex,N: nat] :
% 8.55/4.76 ( ( ( times_times_complex @ X @ Y )
% 8.55/4.76 = ( times_times_complex @ Y @ X ) )
% 8.55/4.76 => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y )
% 8.55/4.76 = ( times_times_complex @ Y @ ( power_power_complex @ X @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1500_power__commuting__commutes,axiom,
% 8.55/4.76 ! [X: real,Y: real,N: nat] :
% 8.55/4.76 ( ( ( times_times_real @ X @ Y )
% 8.55/4.76 = ( times_times_real @ Y @ X ) )
% 8.55/4.76 => ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y )
% 8.55/4.76 = ( times_times_real @ Y @ ( power_power_real @ X @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1501_power__commuting__commutes,axiom,
% 8.55/4.76 ! [X: rat,Y: rat,N: nat] :
% 8.55/4.76 ( ( ( times_times_rat @ X @ Y )
% 8.55/4.76 = ( times_times_rat @ Y @ X ) )
% 8.55/4.76 => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y )
% 8.55/4.76 = ( times_times_rat @ Y @ ( power_power_rat @ X @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1502_power__commuting__commutes,axiom,
% 8.55/4.76 ! [X: nat,Y: nat,N: nat] :
% 8.55/4.76 ( ( ( times_times_nat @ X @ Y )
% 8.55/4.76 = ( times_times_nat @ Y @ X ) )
% 8.55/4.76 => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
% 8.55/4.76 = ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1503_power__commuting__commutes,axiom,
% 8.55/4.76 ! [X: int,Y: int,N: nat] :
% 8.55/4.76 ( ( ( times_times_int @ X @ Y )
% 8.55/4.76 = ( times_times_int @ Y @ X ) )
% 8.55/4.76 => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
% 8.55/4.76 = ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1504_power__mult__distrib,axiom,
% 8.55/4.76 ! [A: complex,B: complex,N: nat] :
% 8.55/4.76 ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
% 8.55/4.76 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1505_power__mult__distrib,axiom,
% 8.55/4.76 ! [A: real,B: real,N: nat] :
% 8.55/4.76 ( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
% 8.55/4.76 = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1506_power__mult__distrib,axiom,
% 8.55/4.76 ! [A: rat,B: rat,N: nat] :
% 8.55/4.76 ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
% 8.55/4.76 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1507_power__mult__distrib,axiom,
% 8.55/4.76 ! [A: nat,B: nat,N: nat] :
% 8.55/4.76 ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
% 8.55/4.76 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1508_power__mult__distrib,axiom,
% 8.55/4.76 ! [A: int,B: int,N: nat] :
% 8.55/4.76 ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
% 8.55/4.76 = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1509_power__commutes,axiom,
% 8.55/4.76 ! [A: complex,N: nat] :
% 8.55/4.76 ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
% 8.55/4.76 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1510_power__commutes,axiom,
% 8.55/4.76 ! [A: real,N: nat] :
% 8.55/4.76 ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
% 8.55/4.76 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1511_power__commutes,axiom,
% 8.55/4.76 ! [A: rat,N: nat] :
% 8.55/4.76 ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
% 8.55/4.76 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1512_power__commutes,axiom,
% 8.55/4.76 ! [A: nat,N: nat] :
% 8.55/4.76 ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
% 8.55/4.76 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1513_power__commutes,axiom,
% 8.55/4.76 ! [A: int,N: nat] :
% 8.55/4.76 ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
% 8.55/4.76 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1514_power__add,axiom,
% 8.55/4.76 ! [A: complex,M: nat,N: nat] :
% 8.55/4.76 ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
% 8.55/4.76 = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1515_power__add,axiom,
% 8.55/4.76 ! [A: real,M: nat,N: nat] :
% 8.55/4.76 ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
% 8.55/4.76 = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1516_power__add,axiom,
% 8.55/4.76 ! [A: rat,M: nat,N: nat] :
% 8.55/4.76 ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
% 8.55/4.76 = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1517_power__add,axiom,
% 8.55/4.76 ! [A: nat,M: nat,N: nat] :
% 8.55/4.76 ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
% 8.55/4.76 = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1518_power__add,axiom,
% 8.55/4.76 ! [A: int,M: nat,N: nat] :
% 8.55/4.76 ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
% 8.55/4.76 = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1519_power__divide,axiom,
% 8.55/4.76 ! [A: complex,B: complex,N: nat] :
% 8.55/4.76 ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
% 8.55/4.76 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1520_power__divide,axiom,
% 8.55/4.76 ! [A: real,B: real,N: nat] :
% 8.55/4.76 ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
% 8.55/4.76 = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1521_power__divide,axiom,
% 8.55/4.76 ! [A: rat,B: rat,N: nat] :
% 8.55/4.76 ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
% 8.55/4.76 = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1522_dvd__power__same,axiom,
% 8.55/4.76 ! [X: code_integer,Y: code_integer,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_Code_integer @ X @ Y )
% 8.55/4.76 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1523_dvd__power__same,axiom,
% 8.55/4.76 ! [X: nat,Y: nat,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_nat @ X @ Y )
% 8.55/4.76 => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1524_dvd__power__same,axiom,
% 8.55/4.76 ! [X: real,Y: real,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_real @ X @ Y )
% 8.55/4.76 => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1525_dvd__power__same,axiom,
% 8.55/4.76 ! [X: int,Y: int,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_int @ X @ Y )
% 8.55/4.76 => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1526_dvd__power__same,axiom,
% 8.55/4.76 ! [X: complex,Y: complex,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_complex @ X @ Y )
% 8.55/4.76 => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1527_power__mult,axiom,
% 8.55/4.76 ! [A: nat,M: nat,N: nat] :
% 8.55/4.76 ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
% 8.55/4.76 = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1528_power__mult,axiom,
% 8.55/4.76 ! [A: real,M: nat,N: nat] :
% 8.55/4.76 ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
% 8.55/4.76 = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1529_power__mult,axiom,
% 8.55/4.76 ! [A: int,M: nat,N: nat] :
% 8.55/4.76 ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
% 8.55/4.76 = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1530_power__mult,axiom,
% 8.55/4.76 ! [A: complex,M: nat,N: nat] :
% 8.55/4.76 ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
% 8.55/4.76 = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1531_left__right__inverse__power,axiom,
% 8.55/4.76 ! [X: complex,Y: complex,N: nat] :
% 8.55/4.76 ( ( ( times_times_complex @ X @ Y )
% 8.55/4.76 = one_one_complex )
% 8.55/4.76 => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
% 8.55/4.76 = one_one_complex ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1532_left__right__inverse__power,axiom,
% 8.55/4.76 ! [X: real,Y: real,N: nat] :
% 8.55/4.76 ( ( ( times_times_real @ X @ Y )
% 8.55/4.76 = one_one_real )
% 8.55/4.76 => ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
% 8.55/4.76 = one_one_real ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1533_left__right__inverse__power,axiom,
% 8.55/4.76 ! [X: rat,Y: rat,N: nat] :
% 8.55/4.76 ( ( ( times_times_rat @ X @ Y )
% 8.55/4.76 = one_one_rat )
% 8.55/4.76 => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
% 8.55/4.76 = one_one_rat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1534_left__right__inverse__power,axiom,
% 8.55/4.76 ! [X: nat,Y: nat,N: nat] :
% 8.55/4.76 ( ( ( times_times_nat @ X @ Y )
% 8.55/4.76 = one_one_nat )
% 8.55/4.76 => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
% 8.55/4.76 = one_one_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1535_left__right__inverse__power,axiom,
% 8.55/4.76 ! [X: int,Y: int,N: nat] :
% 8.55/4.76 ( ( ( times_times_int @ X @ Y )
% 8.55/4.76 = one_one_int )
% 8.55/4.76 => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
% 8.55/4.76 = one_one_int ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1536_power__one__over,axiom,
% 8.55/4.76 ! [A: complex,N: nat] :
% 8.55/4.76 ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
% 8.55/4.76 = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1537_power__one__over,axiom,
% 8.55/4.76 ! [A: real,N: nat] :
% 8.55/4.76 ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
% 8.55/4.76 = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1538_power__one__over,axiom,
% 8.55/4.76 ! [A: rat,N: nat] :
% 8.55/4.76 ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
% 8.55/4.76 = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1539_power__Suc2,axiom,
% 8.55/4.76 ! [A: complex,N: nat] :
% 8.55/4.76 ( ( power_power_complex @ A @ ( suc @ N ) )
% 8.55/4.76 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1540_power__Suc2,axiom,
% 8.55/4.76 ! [A: real,N: nat] :
% 8.55/4.76 ( ( power_power_real @ A @ ( suc @ N ) )
% 8.55/4.76 = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1541_power__Suc2,axiom,
% 8.55/4.76 ! [A: rat,N: nat] :
% 8.55/4.76 ( ( power_power_rat @ A @ ( suc @ N ) )
% 8.55/4.76 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1542_power__Suc2,axiom,
% 8.55/4.76 ! [A: nat,N: nat] :
% 8.55/4.76 ( ( power_power_nat @ A @ ( suc @ N ) )
% 8.55/4.76 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1543_power__Suc2,axiom,
% 8.55/4.76 ! [A: int,N: nat] :
% 8.55/4.76 ( ( power_power_int @ A @ ( suc @ N ) )
% 8.55/4.76 = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1544_power__Suc,axiom,
% 8.55/4.76 ! [A: complex,N: nat] :
% 8.55/4.76 ( ( power_power_complex @ A @ ( suc @ N ) )
% 8.55/4.76 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1545_power__Suc,axiom,
% 8.55/4.76 ! [A: real,N: nat] :
% 8.55/4.76 ( ( power_power_real @ A @ ( suc @ N ) )
% 8.55/4.76 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1546_power__Suc,axiom,
% 8.55/4.76 ! [A: rat,N: nat] :
% 8.55/4.76 ( ( power_power_rat @ A @ ( suc @ N ) )
% 8.55/4.76 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1547_power__Suc,axiom,
% 8.55/4.76 ! [A: nat,N: nat] :
% 8.55/4.76 ( ( power_power_nat @ A @ ( suc @ N ) )
% 8.55/4.76 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1548_power__Suc,axiom,
% 8.55/4.76 ! [A: int,N: nat] :
% 8.55/4.76 ( ( power_power_int @ A @ ( suc @ N ) )
% 8.55/4.76 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1549_power__0,axiom,
% 8.55/4.76 ! [A: rat] :
% 8.55/4.76 ( ( power_power_rat @ A @ zero_zero_nat )
% 8.55/4.76 = one_one_rat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1550_power__0,axiom,
% 8.55/4.76 ! [A: nat] :
% 8.55/4.76 ( ( power_power_nat @ A @ zero_zero_nat )
% 8.55/4.76 = one_one_nat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1551_power__0,axiom,
% 8.55/4.76 ! [A: real] :
% 8.55/4.76 ( ( power_power_real @ A @ zero_zero_nat )
% 8.55/4.76 = one_one_real ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1552_power__0,axiom,
% 8.55/4.76 ! [A: int] :
% 8.55/4.76 ( ( power_power_int @ A @ zero_zero_nat )
% 8.55/4.76 = one_one_int ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1553_power__0,axiom,
% 8.55/4.76 ! [A: complex] :
% 8.55/4.76 ( ( power_power_complex @ A @ zero_zero_nat )
% 8.55/4.76 = one_one_complex ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1554_div__power,axiom,
% 8.55/4.76 ! [B: code_integer,A: code_integer,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_Code_integer @ B @ A )
% 8.55/4.76 => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N )
% 8.55/4.76 = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1555_div__power,axiom,
% 8.55/4.76 ! [B: nat,A: nat,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_nat @ B @ A )
% 8.55/4.76 => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
% 8.55/4.76 = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1556_div__power,axiom,
% 8.55/4.76 ! [B: int,A: int,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_int @ B @ A )
% 8.55/4.76 => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
% 8.55/4.76 = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1557_power__0__left,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( ( N = zero_zero_nat )
% 8.55/4.76 => ( ( power_power_rat @ zero_zero_rat @ N )
% 8.55/4.76 = one_one_rat ) )
% 8.55/4.76 & ( ( N != zero_zero_nat )
% 8.55/4.76 => ( ( power_power_rat @ zero_zero_rat @ N )
% 8.55/4.76 = zero_zero_rat ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1558_power__0__left,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( ( N = zero_zero_nat )
% 8.55/4.76 => ( ( power_power_nat @ zero_zero_nat @ N )
% 8.55/4.76 = one_one_nat ) )
% 8.55/4.76 & ( ( N != zero_zero_nat )
% 8.55/4.76 => ( ( power_power_nat @ zero_zero_nat @ N )
% 8.55/4.76 = zero_zero_nat ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1559_power__0__left,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( ( N = zero_zero_nat )
% 8.55/4.76 => ( ( power_power_real @ zero_zero_real @ N )
% 8.55/4.76 = one_one_real ) )
% 8.55/4.76 & ( ( N != zero_zero_nat )
% 8.55/4.76 => ( ( power_power_real @ zero_zero_real @ N )
% 8.55/4.76 = zero_zero_real ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1560_power__0__left,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( ( N = zero_zero_nat )
% 8.55/4.76 => ( ( power_power_int @ zero_zero_int @ N )
% 8.55/4.76 = one_one_int ) )
% 8.55/4.76 & ( ( N != zero_zero_nat )
% 8.55/4.76 => ( ( power_power_int @ zero_zero_int @ N )
% 8.55/4.76 = zero_zero_int ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1561_power__0__left,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( ( N = zero_zero_nat )
% 8.55/4.76 => ( ( power_power_complex @ zero_zero_complex @ N )
% 8.55/4.76 = one_one_complex ) )
% 8.55/4.76 & ( ( N != zero_zero_nat )
% 8.55/4.76 => ( ( power_power_complex @ zero_zero_complex @ N )
% 8.55/4.76 = zero_zero_complex ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1562_is__unit__power__iff,axiom,
% 8.55/4.76 ! [A: code_integer,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
% 8.55/4.76 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.55/4.76 | ( N = zero_zero_nat ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1563_is__unit__power__iff,axiom,
% 8.55/4.76 ! [A: nat,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
% 8.55/4.76 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.55/4.76 | ( N = zero_zero_nat ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1564_is__unit__power__iff,axiom,
% 8.55/4.76 ! [A: int,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
% 8.55/4.76 = ( ( dvd_dvd_int @ A @ one_one_int )
% 8.55/4.76 | ( N = zero_zero_nat ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1565_zero__power2,axiom,
% 8.55/4.76 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = zero_zero_rat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1566_zero__power2,axiom,
% 8.55/4.76 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = zero_zero_nat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1567_zero__power2,axiom,
% 8.55/4.76 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = zero_zero_real ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1568_zero__power2,axiom,
% 8.55/4.76 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = zero_zero_int ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1569_zero__power2,axiom,
% 8.55/4.76 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = zero_zero_complex ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1570_power4__eq__xxxx,axiom,
% 8.55/4.76 ! [X: complex] :
% 8.55/4.76 ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.55/4.76 = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1571_power4__eq__xxxx,axiom,
% 8.55/4.76 ! [X: real] :
% 8.55/4.76 ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.55/4.76 = ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1572_power4__eq__xxxx,axiom,
% 8.55/4.76 ! [X: rat] :
% 8.55/4.76 ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.55/4.76 = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1573_power4__eq__xxxx,axiom,
% 8.55/4.76 ! [X: nat] :
% 8.55/4.76 ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.55/4.76 = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1574_power4__eq__xxxx,axiom,
% 8.55/4.76 ! [X: int] :
% 8.55/4.76 ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.55/4.76 = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1575_power2__eq__square,axiom,
% 8.55/4.76 ! [A: complex] :
% 8.55/4.76 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( times_times_complex @ A @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1576_power2__eq__square,axiom,
% 8.55/4.76 ! [A: real] :
% 8.55/4.76 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( times_times_real @ A @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1577_power2__eq__square,axiom,
% 8.55/4.76 ! [A: rat] :
% 8.55/4.76 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( times_times_rat @ A @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1578_power2__eq__square,axiom,
% 8.55/4.76 ! [A: nat] :
% 8.55/4.76 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( times_times_nat @ A @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1579_power2__eq__square,axiom,
% 8.55/4.76 ! [A: int] :
% 8.55/4.76 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( times_times_int @ A @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1580_one__power2,axiom,
% 8.55/4.76 ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = one_one_rat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1581_one__power2,axiom,
% 8.55/4.76 ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = one_one_nat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1582_one__power2,axiom,
% 8.55/4.76 ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = one_one_real ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1583_one__power2,axiom,
% 8.55/4.76 ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = one_one_int ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1584_one__power2,axiom,
% 8.55/4.76 ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = one_one_complex ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1585_power2__commute,axiom,
% 8.55/4.76 ! [X: complex,Y: complex] :
% 8.55/4.76 ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( power_power_complex @ ( minus_minus_complex @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1586_power2__commute,axiom,
% 8.55/4.76 ! [X: real,Y: real] :
% 8.55/4.76 ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( power_power_real @ ( minus_minus_real @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1587_power2__commute,axiom,
% 8.55/4.76 ! [X: rat,Y: rat] :
% 8.55/4.76 ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( power_power_rat @ ( minus_minus_rat @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1588_power2__commute,axiom,
% 8.55/4.76 ! [X: int,Y: int] :
% 8.55/4.76 ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( power_power_int @ ( minus_minus_int @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1589_power__even__eq,axiom,
% 8.55/4.76 ! [A: nat,N: nat] :
% 8.55/4.76 ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.76 = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1590_power__even__eq,axiom,
% 8.55/4.76 ! [A: real,N: nat] :
% 8.55/4.76 ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.76 = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1591_power__even__eq,axiom,
% 8.55/4.76 ! [A: int,N: nat] :
% 8.55/4.76 ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.76 = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1592_power__even__eq,axiom,
% 8.55/4.76 ! [A: complex,N: nat] :
% 8.55/4.76 ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.76 = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1593_power__eq__if,axiom,
% 8.55/4.76 ( power_power_complex
% 8.55/4.76 = ( ^ [P3: complex,M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P3 @ ( power_power_complex @ P3 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1594_power__eq__if,axiom,
% 8.55/4.76 ( power_power_real
% 8.55/4.76 = ( ^ [P3: real,M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P3 @ ( power_power_real @ P3 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1595_power__eq__if,axiom,
% 8.55/4.76 ( power_power_rat
% 8.55/4.76 = ( ^ [P3: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P3 @ ( power_power_rat @ P3 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1596_power__eq__if,axiom,
% 8.55/4.76 ( power_power_nat
% 8.55/4.76 = ( ^ [P3: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P3 @ ( power_power_nat @ P3 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1597_power__eq__if,axiom,
% 8.55/4.76 ( power_power_int
% 8.55/4.76 = ( ^ [P3: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P3 @ ( power_power_int @ P3 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1598_power2__sum,axiom,
% 8.55/4.76 ! [X: complex,Y: complex] :
% 8.55/4.76 ( ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1599_power2__sum,axiom,
% 8.55/4.76 ! [X: real,Y: real] :
% 8.55/4.76 ( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1600_power2__sum,axiom,
% 8.55/4.76 ! [X: rat,Y: rat] :
% 8.55/4.76 ( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1601_power2__sum,axiom,
% 8.55/4.76 ! [X: nat,Y: nat] :
% 8.55/4.76 ( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1602_power2__sum,axiom,
% 8.55/4.76 ! [X: int,Y: int] :
% 8.55/4.76 ( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1603_power__odd__eq,axiom,
% 8.55/4.76 ! [A: complex,N: nat] :
% 8.55/4.76 ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.55/4.76 = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1604_power__odd__eq,axiom,
% 8.55/4.76 ! [A: real,N: nat] :
% 8.55/4.76 ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.55/4.76 = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1605_power__odd__eq,axiom,
% 8.55/4.76 ! [A: rat,N: nat] :
% 8.55/4.76 ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.55/4.76 = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1606_power__odd__eq,axiom,
% 8.55/4.76 ! [A: nat,N: nat] :
% 8.55/4.76 ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.55/4.76 = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1607_power__odd__eq,axiom,
% 8.55/4.76 ! [A: int,N: nat] :
% 8.55/4.76 ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.55/4.76 = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1608_high__def,axiom,
% 8.55/4.76 ( vEBT_VEBT_high
% 8.55/4.76 = ( ^ [X3: nat,N3: nat] : ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1609_even__succ__div__exp,axiom,
% 8.55/4.76 ! [A: code_integer,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.55/4.76 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.76 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.76 = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1610_even__succ__div__exp,axiom,
% 8.55/4.76 ! [A: nat,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.55/4.76 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.76 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.76 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1611_even__succ__div__exp,axiom,
% 8.55/4.76 ! [A: int,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.55/4.76 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.76 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.76 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1612_flip__bit__0,axiom,
% 8.55/4.76 ! [A: code_integer] :
% 8.55/4.76 ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 8.55/4.76 = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1613_flip__bit__0,axiom,
% 8.55/4.76 ! [A: int] :
% 8.55/4.76 ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 8.55/4.76 = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1614_flip__bit__0,axiom,
% 8.55/4.76 ! [A: nat] :
% 8.55/4.76 ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 8.55/4.76 = ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1615_add__scale__eq__noteq,axiom,
% 8.55/4.76 ! [R: real,A: real,B: real,C: real,D: real] :
% 8.55/4.76 ( ( R != zero_zero_real )
% 8.55/4.76 => ( ( ( A = B )
% 8.55/4.76 & ( C != D ) )
% 8.55/4.76 => ( ( plus_plus_real @ A @ ( times_times_real @ R @ C ) )
% 8.55/4.76 != ( plus_plus_real @ B @ ( times_times_real @ R @ D ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1616_add__scale__eq__noteq,axiom,
% 8.55/4.76 ! [R: rat,A: rat,B: rat,C: rat,D: rat] :
% 8.55/4.76 ( ( R != zero_zero_rat )
% 8.55/4.76 => ( ( ( A = B )
% 8.55/4.76 & ( C != D ) )
% 8.55/4.76 => ( ( plus_plus_rat @ A @ ( times_times_rat @ R @ C ) )
% 8.55/4.76 != ( plus_plus_rat @ B @ ( times_times_rat @ R @ D ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1617_add__scale__eq__noteq,axiom,
% 8.55/4.76 ! [R: nat,A: nat,B: nat,C: nat,D: nat] :
% 8.55/4.76 ( ( R != zero_zero_nat )
% 8.55/4.76 => ( ( ( A = B )
% 8.55/4.76 & ( C != D ) )
% 8.55/4.76 => ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
% 8.55/4.76 != ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1618_add__scale__eq__noteq,axiom,
% 8.55/4.76 ! [R: int,A: int,B: int,C: int,D: int] :
% 8.55/4.76 ( ( R != zero_zero_int )
% 8.55/4.76 => ( ( ( A = B )
% 8.55/4.76 & ( C != D ) )
% 8.55/4.76 => ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
% 8.55/4.76 != ( plus_plus_int @ B @ ( times_times_int @ R @ D ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1619_num_Osize__gen_I2_J,axiom,
% 8.55/4.76 ! [X2: num] :
% 8.55/4.76 ( ( size_num @ ( bit0 @ X2 ) )
% 8.55/4.76 = ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1620_power__numeral,axiom,
% 8.55/4.76 ! [K: num,L: num] :
% 8.55/4.76 ( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L ) )
% 8.55/4.76 = ( numera6690914467698888265omplex @ ( pow @ K @ L ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1621_power__numeral,axiom,
% 8.55/4.76 ! [K: num,L: num] :
% 8.55/4.76 ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
% 8.55/4.76 = ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1622_power__numeral,axiom,
% 8.55/4.76 ! [K: num,L: num] :
% 8.55/4.76 ( ( power_power_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_nat @ L ) )
% 8.55/4.76 = ( numeral_numeral_rat @ ( pow @ K @ L ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1623_power__numeral,axiom,
% 8.55/4.76 ! [K: num,L: num] :
% 8.55/4.76 ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
% 8.55/4.76 = ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1624_power__numeral,axiom,
% 8.55/4.76 ! [K: num,L: num] :
% 8.55/4.76 ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
% 8.55/4.76 = ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1625_even__mask__div__iff,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 8.55/4.76 = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 8.55/4.76 = zero_z3403309356797280102nteger )
% 8.55/4.76 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1626_even__mask__div__iff,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.55/4.76 = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.76 = zero_zero_nat )
% 8.55/4.76 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1627_even__mask__div__iff,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 8.55/4.76 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 8.55/4.76 = zero_zero_int )
% 8.55/4.76 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1628_arith__geo__mean,axiom,
% 8.55/4.76 ! [U: real,X: real,Y: real] :
% 8.55/4.76 ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( times_times_real @ X @ Y ) )
% 8.55/4.76 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.55/4.76 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.55/4.76 => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1629_arith__geo__mean,axiom,
% 8.55/4.76 ! [U: rat,X: rat,Y: rat] :
% 8.55/4.76 ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( times_times_rat @ X @ Y ) )
% 8.55/4.76 => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 8.55/4.76 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 8.55/4.76 => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1630_zero__less__power__eq__numeral,axiom,
% 8.55/4.76 ! [A: real,W: num] :
% 8.55/4.76 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 8.55/4.76 = ( ( ( numeral_numeral_nat @ W )
% 8.55/4.76 = zero_zero_nat )
% 8.55/4.76 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.76 & ( A != zero_zero_real ) )
% 8.55/4.76 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.76 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1631_zero__less__power__eq__numeral,axiom,
% 8.55/4.76 ! [A: rat,W: num] :
% 8.55/4.76 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 8.55/4.76 = ( ( ( numeral_numeral_nat @ W )
% 8.55/4.76 = zero_zero_nat )
% 8.55/4.76 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.76 & ( A != zero_zero_rat ) )
% 8.55/4.76 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.76 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1632_zero__less__power__eq__numeral,axiom,
% 8.55/4.76 ! [A: int,W: num] :
% 8.55/4.76 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 8.55/4.76 = ( ( ( numeral_numeral_nat @ W )
% 8.55/4.76 = zero_zero_nat )
% 8.55/4.76 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.76 & ( A != zero_zero_int ) )
% 8.55/4.76 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.76 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1633_Bernoulli__inequality__even,axiom,
% 8.55/4.76 ! [N: nat,X: real] :
% 8.55/4.76 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.76 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1634_semiring__norm_I78_J,axiom,
% 8.55/4.76 ! [M: num,N: num] :
% 8.55/4.76 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.55/4.76 = ( ord_less_num @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1635_semiring__norm_I71_J,axiom,
% 8.55/4.76 ! [M: num,N: num] :
% 8.55/4.76 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.55/4.76 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1636_semiring__norm_I75_J,axiom,
% 8.55/4.76 ! [M: num] :
% 8.55/4.76 ~ ( ord_less_num @ M @ one ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1637_semiring__norm_I68_J,axiom,
% 8.55/4.76 ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1638_set__bit__negative__int__iff,axiom,
% 8.55/4.76 ! [N: nat,K: int] :
% 8.55/4.76 ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 8.55/4.76 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1639_flip__bit__negative__int__iff,axiom,
% 8.55/4.76 ! [N: nat,K: int] :
% 8.55/4.76 ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 8.55/4.76 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1640_unset__bit__negative__int__iff,axiom,
% 8.55/4.76 ! [N: nat,K: int] :
% 8.55/4.76 ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 8.55/4.76 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1641_set__bit__nonnegative__int__iff,axiom,
% 8.55/4.76 ! [N: nat,K: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
% 8.55/4.76 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1642_flip__bit__nonnegative__int__iff,axiom,
% 8.55/4.76 ! [N: nat,K: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
% 8.55/4.76 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1643_unset__bit__nonnegative__int__iff,axiom,
% 8.55/4.76 ! [N: nat,K: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
% 8.55/4.76 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1644_not__real__square__gt__zero,axiom,
% 8.55/4.76 ! [X: real] :
% 8.55/4.76 ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
% 8.55/4.76 = ( X = zero_zero_real ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1645_high__bound__aux,axiom,
% 8.55/4.76 ! [Ma: nat,N: nat,M: nat] :
% 8.55/4.76 ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 8.55/4.76 => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1646_le__zero__eq,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 8.55/4.76 = ( N = zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1647_not__gr__zero,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 8.55/4.76 = ( N = zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1648_numeral__le__iff,axiom,
% 8.55/4.76 ! [M: num,N: num] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 8.55/4.76 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1649_numeral__le__iff,axiom,
% 8.55/4.76 ! [M: num,N: num] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 8.55/4.76 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1650_numeral__le__iff,axiom,
% 8.55/4.76 ! [M: num,N: num] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.55/4.76 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1651_numeral__le__iff,axiom,
% 8.55/4.76 ! [M: num,N: num] :
% 8.55/4.76 ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.55/4.76 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1652_numeral__less__iff,axiom,
% 8.55/4.76 ! [M: num,N: num] :
% 8.55/4.76 ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 8.55/4.76 = ( ord_less_num @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1653_numeral__less__iff,axiom,
% 8.55/4.76 ! [M: num,N: num] :
% 8.55/4.76 ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 8.55/4.76 = ( ord_less_num @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1654_numeral__less__iff,axiom,
% 8.55/4.76 ! [M: num,N: num] :
% 8.55/4.76 ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.55/4.76 = ( ord_less_num @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1655_numeral__less__iff,axiom,
% 8.55/4.76 ! [M: num,N: num] :
% 8.55/4.76 ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.55/4.76 = ( ord_less_num @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1656_add__le__cancel__right,axiom,
% 8.55/4.76 ! [A: real,C: real,B: real] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 8.55/4.76 = ( ord_less_eq_real @ A @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1657_add__le__cancel__right,axiom,
% 8.55/4.76 ! [A: rat,C: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 8.55/4.76 = ( ord_less_eq_rat @ A @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1658_add__le__cancel__right,axiom,
% 8.55/4.76 ! [A: nat,C: nat,B: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 8.55/4.76 = ( ord_less_eq_nat @ A @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1659_add__le__cancel__right,axiom,
% 8.55/4.76 ! [A: int,C: int,B: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 8.55/4.76 = ( ord_less_eq_int @ A @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1660_add__le__cancel__left,axiom,
% 8.55/4.76 ! [C: real,A: real,B: real] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 8.55/4.76 = ( ord_less_eq_real @ A @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1661_add__le__cancel__left,axiom,
% 8.55/4.76 ! [C: rat,A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 8.55/4.76 = ( ord_less_eq_rat @ A @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1662_add__le__cancel__left,axiom,
% 8.55/4.76 ! [C: nat,A: nat,B: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 8.55/4.76 = ( ord_less_eq_nat @ A @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1663_add__le__cancel__left,axiom,
% 8.55/4.76 ! [C: int,A: int,B: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 8.55/4.76 = ( ord_less_eq_int @ A @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1664_add__less__cancel__right,axiom,
% 8.55/4.76 ! [A: real,C: real,B: real] :
% 8.55/4.76 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 8.55/4.76 = ( ord_less_real @ A @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1665_add__less__cancel__right,axiom,
% 8.55/4.76 ! [A: rat,C: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 8.55/4.76 = ( ord_less_rat @ A @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1666_add__less__cancel__right,axiom,
% 8.55/4.76 ! [A: nat,C: nat,B: nat] :
% 8.55/4.76 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 8.55/4.76 = ( ord_less_nat @ A @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1667_add__less__cancel__right,axiom,
% 8.55/4.76 ! [A: int,C: int,B: int] :
% 8.55/4.76 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 8.55/4.76 = ( ord_less_int @ A @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1668_add__less__cancel__left,axiom,
% 8.55/4.76 ! [C: real,A: real,B: real] :
% 8.55/4.76 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 8.55/4.76 = ( ord_less_real @ A @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1669_add__less__cancel__left,axiom,
% 8.55/4.76 ! [C: rat,A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 8.55/4.76 = ( ord_less_rat @ A @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1670_add__less__cancel__left,axiom,
% 8.55/4.76 ! [C: nat,A: nat,B: nat] :
% 8.55/4.76 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 8.55/4.76 = ( ord_less_nat @ A @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1671_add__less__cancel__left,axiom,
% 8.55/4.76 ! [C: int,A: int,B: int] :
% 8.55/4.76 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 8.55/4.76 = ( ord_less_int @ A @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1672_semiring__norm_I76_J,axiom,
% 8.55/4.76 ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1673_semiring__norm_I69_J,axiom,
% 8.55/4.76 ! [M: num] :
% 8.55/4.76 ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1674_Suc__less__eq,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 8.55/4.76 = ( ord_less_nat @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1675_Suc__mono,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ M @ N )
% 8.55/4.76 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1676_lessI,axiom,
% 8.55/4.76 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1677_less__nat__zero__code,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1678_neq0__conv,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( N != zero_zero_nat )
% 8.55/4.76 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1679_bot__nat__0_Onot__eq__extremum,axiom,
% 8.55/4.76 ! [A: nat] :
% 8.55/4.76 ( ( A != zero_zero_nat )
% 8.55/4.76 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1680_Suc__le__mono,axiom,
% 8.55/4.76 ! [N: nat,M: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 8.55/4.76 = ( ord_less_eq_nat @ N @ M ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1681_le0,axiom,
% 8.55/4.76 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1682_bot__nat__0_Oextremum,axiom,
% 8.55/4.76 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1683_nat__add__left__cancel__less,axiom,
% 8.55/4.76 ! [K: nat,M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 8.55/4.76 = ( ord_less_nat @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1684_nat__add__left__cancel__le,axiom,
% 8.55/4.76 ! [K: nat,M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 8.55/4.76 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1685_diff__diff__cancel,axiom,
% 8.55/4.76 ! [I: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ I @ N )
% 8.55/4.76 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 8.55/4.76 = I ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1686_of__bool__less__eq__iff,axiom,
% 8.55/4.76 ! [P: $o,Q: $o] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 8.55/4.76 = ( P
% 8.55/4.76 => Q ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1687_of__bool__less__eq__iff,axiom,
% 8.55/4.76 ! [P: $o,Q: $o] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 8.55/4.76 = ( P
% 8.55/4.76 => Q ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1688_of__bool__less__eq__iff,axiom,
% 8.55/4.76 ! [P: $o,Q: $o] :
% 8.55/4.76 ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 8.55/4.76 = ( P
% 8.55/4.76 => Q ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1689_of__bool__less__eq__iff,axiom,
% 8.55/4.76 ! [P: $o,Q: $o] :
% 8.55/4.76 ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 8.55/4.76 = ( P
% 8.55/4.76 => Q ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1690_of__bool__eq_I1_J,axiom,
% 8.55/4.76 ( ( zero_n3304061248610475627l_real @ $false )
% 8.55/4.76 = zero_zero_real ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1691_of__bool__eq_I1_J,axiom,
% 8.55/4.76 ( ( zero_n2052037380579107095ol_rat @ $false )
% 8.55/4.76 = zero_zero_rat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1692_of__bool__eq_I1_J,axiom,
% 8.55/4.76 ( ( zero_n2687167440665602831ol_nat @ $false )
% 8.55/4.76 = zero_zero_nat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1693_of__bool__eq_I1_J,axiom,
% 8.55/4.76 ( ( zero_n2684676970156552555ol_int @ $false )
% 8.55/4.76 = zero_zero_int ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1694_of__bool__eq_I1_J,axiom,
% 8.55/4.76 ( ( zero_n356916108424825756nteger @ $false )
% 8.55/4.76 = zero_z3403309356797280102nteger ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1695_of__bool__eq__0__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ( zero_n3304061248610475627l_real @ P )
% 8.55/4.76 = zero_zero_real )
% 8.55/4.76 = ~ P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1696_of__bool__eq__0__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 8.55/4.76 = zero_zero_rat )
% 8.55/4.76 = ~ P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1697_of__bool__eq__0__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 8.55/4.76 = zero_zero_nat )
% 8.55/4.76 = ~ P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1698_of__bool__eq__0__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ( zero_n2684676970156552555ol_int @ P )
% 8.55/4.76 = zero_zero_int )
% 8.55/4.76 = ~ P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1699_of__bool__eq__0__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ( zero_n356916108424825756nteger @ P )
% 8.55/4.76 = zero_z3403309356797280102nteger )
% 8.55/4.76 = ~ P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1700_of__bool__less__iff,axiom,
% 8.55/4.76 ! [P: $o,Q: $o] :
% 8.55/4.76 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 8.55/4.76 = ( ~ P
% 8.55/4.76 & Q ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1701_of__bool__less__iff,axiom,
% 8.55/4.76 ! [P: $o,Q: $o] :
% 8.55/4.76 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 8.55/4.76 = ( ~ P
% 8.55/4.76 & Q ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1702_of__bool__less__iff,axiom,
% 8.55/4.76 ! [P: $o,Q: $o] :
% 8.55/4.76 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 8.55/4.76 = ( ~ P
% 8.55/4.76 & Q ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1703_of__bool__less__iff,axiom,
% 8.55/4.76 ! [P: $o,Q: $o] :
% 8.55/4.76 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 8.55/4.76 = ( ~ P
% 8.55/4.76 & Q ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1704_of__bool__less__iff,axiom,
% 8.55/4.76 ! [P: $o,Q: $o] :
% 8.55/4.76 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 8.55/4.76 = ( ~ P
% 8.55/4.76 & Q ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1705_of__bool__eq__1__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ( zero_n1201886186963655149omplex @ P )
% 8.55/4.76 = one_one_complex )
% 8.55/4.76 = P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1706_of__bool__eq__1__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ( zero_n3304061248610475627l_real @ P )
% 8.55/4.76 = one_one_real )
% 8.55/4.76 = P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1707_of__bool__eq__1__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 8.55/4.76 = one_one_rat )
% 8.55/4.76 = P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1708_of__bool__eq__1__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 8.55/4.76 = one_one_nat )
% 8.55/4.76 = P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1709_of__bool__eq__1__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ( zero_n2684676970156552555ol_int @ P )
% 8.55/4.76 = one_one_int )
% 8.55/4.76 = P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1710_of__bool__eq__1__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ( zero_n356916108424825756nteger @ P )
% 8.55/4.76 = one_one_Code_integer )
% 8.55/4.76 = P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1711_of__bool__eq_I2_J,axiom,
% 8.55/4.76 ( ( zero_n1201886186963655149omplex @ $true )
% 8.55/4.76 = one_one_complex ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1712_of__bool__eq_I2_J,axiom,
% 8.55/4.76 ( ( zero_n3304061248610475627l_real @ $true )
% 8.55/4.76 = one_one_real ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1713_of__bool__eq_I2_J,axiom,
% 8.55/4.76 ( ( zero_n2052037380579107095ol_rat @ $true )
% 8.55/4.76 = one_one_rat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1714_of__bool__eq_I2_J,axiom,
% 8.55/4.76 ( ( zero_n2687167440665602831ol_nat @ $true )
% 8.55/4.76 = one_one_nat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1715_of__bool__eq_I2_J,axiom,
% 8.55/4.76 ( ( zero_n2684676970156552555ol_int @ $true )
% 8.55/4.76 = one_one_int ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1716_of__bool__eq_I2_J,axiom,
% 8.55/4.76 ( ( zero_n356916108424825756nteger @ $true )
% 8.55/4.76 = one_one_Code_integer ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1717_zle__add1__eq__le,axiom,
% 8.55/4.76 ! [W: int,Z2: int] :
% 8.55/4.76 ( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
% 8.55/4.76 = ( ord_less_eq_int @ W @ Z2 ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1718_div__neg__neg__trivial,axiom,
% 8.55/4.76 ! [K: int,L: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 8.55/4.76 => ( ( ord_less_int @ L @ K )
% 8.55/4.76 => ( ( divide_divide_int @ K @ L )
% 8.55/4.76 = zero_zero_int ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1719_div__pos__pos__trivial,axiom,
% 8.55/4.76 ! [K: int,L: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.55/4.76 => ( ( ord_less_int @ K @ L )
% 8.55/4.76 => ( ( divide_divide_int @ K @ L )
% 8.55/4.76 = zero_zero_int ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1720_of__nat__of__bool,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
% 8.55/4.76 = ( zero_n1201886186963655149omplex @ P ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1721_of__nat__of__bool,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 8.55/4.76 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1722_of__nat__of__bool,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 8.55/4.76 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1723_of__nat__of__bool,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 8.55/4.76 = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1724_of__nat__of__bool,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 8.55/4.76 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1725_of__nat__of__bool,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
% 8.55/4.76 = ( zero_n356916108424825756nteger @ P ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1726_zle__diff1__eq,axiom,
% 8.55/4.76 ! [W: int,Z2: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
% 8.55/4.76 = ( ord_less_int @ W @ Z2 ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1727_push__bit__negative__int__iff,axiom,
% 8.55/4.76 ! [N: nat,K: int] :
% 8.55/4.76 ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 8.55/4.76 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1728_push__bit__nonnegative__int__iff,axiom,
% 8.55/4.76 ! [N: nat,K: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
% 8.55/4.76 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1729_zero__le__double__add__iff__zero__le__single__add,axiom,
% 8.55/4.76 ! [A: real] :
% 8.55/4.76 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 8.55/4.76 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1730_zero__le__double__add__iff__zero__le__single__add,axiom,
% 8.55/4.76 ! [A: rat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 8.55/4.76 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1731_zero__le__double__add__iff__zero__le__single__add,axiom,
% 8.55/4.76 ! [A: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 8.55/4.76 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1732_double__add__le__zero__iff__single__add__le__zero,axiom,
% 8.55/4.76 ! [A: real] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 8.55/4.76 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1733_double__add__le__zero__iff__single__add__le__zero,axiom,
% 8.55/4.76 ! [A: rat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 8.55/4.76 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1734_double__add__le__zero__iff__single__add__le__zero,axiom,
% 8.55/4.76 ! [A: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 8.55/4.76 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1735_le__add__same__cancel2,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 8.55/4.76 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1736_le__add__same__cancel2,axiom,
% 8.55/4.76 ! [A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 8.55/4.76 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1737_le__add__same__cancel2,axiom,
% 8.55/4.76 ! [A: nat,B: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 8.55/4.76 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1738_le__add__same__cancel2,axiom,
% 8.55/4.76 ! [A: int,B: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 8.55/4.76 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1739_le__add__same__cancel1,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 8.55/4.76 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1740_le__add__same__cancel1,axiom,
% 8.55/4.76 ! [A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 8.55/4.76 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1741_le__add__same__cancel1,axiom,
% 8.55/4.76 ! [A: nat,B: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 8.55/4.76 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1742_le__add__same__cancel1,axiom,
% 8.55/4.76 ! [A: int,B: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 8.55/4.76 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1743_add__le__same__cancel2,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 8.55/4.76 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1744_add__le__same__cancel2,axiom,
% 8.55/4.76 ! [A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 8.55/4.76 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1745_add__le__same__cancel2,axiom,
% 8.55/4.76 ! [A: nat,B: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 8.55/4.76 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1746_add__le__same__cancel2,axiom,
% 8.55/4.76 ! [A: int,B: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 8.55/4.76 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1747_add__le__same__cancel1,axiom,
% 8.55/4.76 ! [B: real,A: real] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 8.55/4.76 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1748_add__le__same__cancel1,axiom,
% 8.55/4.76 ! [B: rat,A: rat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 8.55/4.76 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1749_add__le__same__cancel1,axiom,
% 8.55/4.76 ! [B: nat,A: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 8.55/4.76 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1750_add__le__same__cancel1,axiom,
% 8.55/4.76 ! [B: int,A: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 8.55/4.76 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1751_diff__ge__0__iff__ge,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 8.55/4.76 = ( ord_less_eq_real @ B @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1752_diff__ge__0__iff__ge,axiom,
% 8.55/4.76 ! [A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 8.55/4.76 = ( ord_less_eq_rat @ B @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1753_diff__ge__0__iff__ge,axiom,
% 8.55/4.76 ! [A: int,B: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 8.55/4.76 = ( ord_less_eq_int @ B @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1754_zero__less__double__add__iff__zero__less__single__add,axiom,
% 8.55/4.76 ! [A: real] :
% 8.55/4.76 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 8.55/4.76 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1755_zero__less__double__add__iff__zero__less__single__add,axiom,
% 8.55/4.76 ! [A: rat] :
% 8.55/4.76 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 8.55/4.76 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1756_zero__less__double__add__iff__zero__less__single__add,axiom,
% 8.55/4.76 ! [A: int] :
% 8.55/4.76 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 8.55/4.76 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1757_double__add__less__zero__iff__single__add__less__zero,axiom,
% 8.55/4.76 ! [A: real] :
% 8.55/4.76 ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 8.55/4.76 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1758_double__add__less__zero__iff__single__add__less__zero,axiom,
% 8.55/4.76 ! [A: rat] :
% 8.55/4.76 ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 8.55/4.76 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1759_double__add__less__zero__iff__single__add__less__zero,axiom,
% 8.55/4.76 ! [A: int] :
% 8.55/4.76 ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 8.55/4.76 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1760_less__add__same__cancel2,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 8.55/4.76 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1761_less__add__same__cancel2,axiom,
% 8.55/4.76 ! [A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 8.55/4.76 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1762_less__add__same__cancel2,axiom,
% 8.55/4.76 ! [A: nat,B: nat] :
% 8.55/4.76 ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 8.55/4.76 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1763_less__add__same__cancel2,axiom,
% 8.55/4.76 ! [A: int,B: int] :
% 8.55/4.76 ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 8.55/4.76 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1764_less__add__same__cancel1,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 8.55/4.76 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1765_less__add__same__cancel1,axiom,
% 8.55/4.76 ! [A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 8.55/4.76 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1766_less__add__same__cancel1,axiom,
% 8.55/4.76 ! [A: nat,B: nat] :
% 8.55/4.76 ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 8.55/4.76 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1767_less__add__same__cancel1,axiom,
% 8.55/4.76 ! [A: int,B: int] :
% 8.55/4.76 ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 8.55/4.76 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1768_add__less__same__cancel2,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 8.55/4.76 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1769_add__less__same__cancel2,axiom,
% 8.55/4.76 ! [A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 8.55/4.76 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1770_add__less__same__cancel2,axiom,
% 8.55/4.76 ! [A: nat,B: nat] :
% 8.55/4.76 ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 8.55/4.76 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1771_add__less__same__cancel2,axiom,
% 8.55/4.76 ! [A: int,B: int] :
% 8.55/4.76 ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 8.55/4.76 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1772_add__less__same__cancel1,axiom,
% 8.55/4.76 ! [B: real,A: real] :
% 8.55/4.76 ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 8.55/4.76 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1773_add__less__same__cancel1,axiom,
% 8.55/4.76 ! [B: rat,A: rat] :
% 8.55/4.76 ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 8.55/4.76 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1774_add__less__same__cancel1,axiom,
% 8.55/4.76 ! [B: nat,A: nat] :
% 8.55/4.76 ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 8.55/4.76 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1775_add__less__same__cancel1,axiom,
% 8.55/4.76 ! [B: int,A: int] :
% 8.55/4.76 ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 8.55/4.76 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1776_diff__gt__0__iff__gt,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 8.55/4.76 = ( ord_less_real @ B @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1777_diff__gt__0__iff__gt,axiom,
% 8.55/4.76 ! [A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 8.55/4.76 = ( ord_less_rat @ B @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1778_diff__gt__0__iff__gt,axiom,
% 8.55/4.76 ! [A: int,B: int] :
% 8.55/4.76 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 8.55/4.76 = ( ord_less_int @ B @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1779_le__add__diff__inverse2,axiom,
% 8.55/4.76 ! [B: real,A: real] :
% 8.55/4.76 ( ( ord_less_eq_real @ B @ A )
% 8.55/4.76 => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 8.55/4.76 = A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1780_le__add__diff__inverse2,axiom,
% 8.55/4.76 ! [B: rat,A: rat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ B @ A )
% 8.55/4.76 => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 8.55/4.76 = A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1781_le__add__diff__inverse2,axiom,
% 8.55/4.76 ! [B: nat,A: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ B @ A )
% 8.55/4.76 => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 8.55/4.76 = A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1782_le__add__diff__inverse2,axiom,
% 8.55/4.76 ! [B: int,A: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ B @ A )
% 8.55/4.76 => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 8.55/4.76 = A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1783_le__add__diff__inverse,axiom,
% 8.55/4.76 ! [B: real,A: real] :
% 8.55/4.76 ( ( ord_less_eq_real @ B @ A )
% 8.55/4.76 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 8.55/4.76 = A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1784_le__add__diff__inverse,axiom,
% 8.55/4.76 ! [B: rat,A: rat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ B @ A )
% 8.55/4.76 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 8.55/4.76 = A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1785_le__add__diff__inverse,axiom,
% 8.55/4.76 ! [B: nat,A: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ B @ A )
% 8.55/4.76 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 8.55/4.76 = A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1786_le__add__diff__inverse,axiom,
% 8.55/4.76 ! [B: int,A: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ B @ A )
% 8.55/4.76 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 8.55/4.76 = A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1787_power__inject__exp,axiom,
% 8.55/4.76 ! [A: real,M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_real @ one_one_real @ A )
% 8.55/4.76 => ( ( ( power_power_real @ A @ M )
% 8.55/4.76 = ( power_power_real @ A @ N ) )
% 8.55/4.76 = ( M = N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1788_power__inject__exp,axiom,
% 8.55/4.76 ! [A: rat,M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_rat @ one_one_rat @ A )
% 8.55/4.76 => ( ( ( power_power_rat @ A @ M )
% 8.55/4.76 = ( power_power_rat @ A @ N ) )
% 8.55/4.76 = ( M = N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1789_power__inject__exp,axiom,
% 8.55/4.76 ! [A: nat,M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ one_one_nat @ A )
% 8.55/4.76 => ( ( ( power_power_nat @ A @ M )
% 8.55/4.76 = ( power_power_nat @ A @ N ) )
% 8.55/4.76 = ( M = N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1790_power__inject__exp,axiom,
% 8.55/4.76 ! [A: int,M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_int @ one_one_int @ A )
% 8.55/4.76 => ( ( ( power_power_int @ A @ M )
% 8.55/4.76 = ( power_power_int @ A @ N ) )
% 8.55/4.76 = ( M = N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1791_of__nat__less__iff,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 8.55/4.76 = ( ord_less_nat @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1792_of__nat__less__iff,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 8.55/4.76 = ( ord_less_nat @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1793_of__nat__less__iff,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 8.55/4.76 = ( ord_less_nat @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1794_of__nat__less__iff,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 8.55/4.76 = ( ord_less_nat @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1795_of__nat__le__iff,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 8.55/4.76 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1796_of__nat__le__iff,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 8.55/4.76 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1797_of__nat__le__iff,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 8.55/4.76 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1798_of__nat__le__iff,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 8.55/4.76 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1799_less__Suc0,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 8.55/4.76 = ( N = zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1800_zero__less__Suc,axiom,
% 8.55/4.76 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1801_add__gr__0,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 8.55/4.76 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.55/4.76 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1802_zero__less__of__bool__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 8.55/4.76 = P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1803_zero__less__of__bool__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 8.55/4.76 = P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1804_zero__less__of__bool__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 8.55/4.76 = P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1805_zero__less__of__bool__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 8.55/4.76 = P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1806_zero__less__of__bool__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
% 8.55/4.76 = P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1807_zero__less__diff,axiom,
% 8.55/4.76 ! [N: nat,M: nat] :
% 8.55/4.76 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 8.55/4.76 = ( ord_less_nat @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1808_nat__mult__less__cancel__disj,axiom,
% 8.55/4.76 ! [K: nat,M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.55/4.76 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.55/4.76 & ( ord_less_nat @ M @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1809_nat__0__less__mult__iff,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 8.55/4.76 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.55/4.76 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1810_mult__less__cancel2,axiom,
% 8.55/4.76 ! [M: nat,K: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 8.55/4.76 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.55/4.76 & ( ord_less_nat @ M @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1811_diff__is__0__eq_H,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.76 => ( ( minus_minus_nat @ M @ N )
% 8.55/4.76 = zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1812_diff__is__0__eq,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( ( minus_minus_nat @ M @ N )
% 8.55/4.76 = zero_zero_nat )
% 8.55/4.76 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1813_of__bool__less__one__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 8.55/4.76 = ~ P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1814_of__bool__less__one__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 8.55/4.76 = ~ P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1815_of__bool__less__one__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 8.55/4.76 = ~ P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1816_of__bool__less__one__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 8.55/4.76 = ~ P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1817_of__bool__less__one__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
% 8.55/4.76 = ~ P ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1818_less__one,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ N @ one_one_nat )
% 8.55/4.76 = ( N = zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1819_of__bool__not__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( zero_n1201886186963655149omplex @ ~ P )
% 8.55/4.76 = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1820_of__bool__not__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( zero_n3304061248610475627l_real @ ~ P )
% 8.55/4.76 = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1821_of__bool__not__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 8.55/4.76 = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1822_of__bool__not__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( zero_n2684676970156552555ol_int @ ~ P )
% 8.55/4.76 = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1823_of__bool__not__iff,axiom,
% 8.55/4.76 ! [P: $o] :
% 8.55/4.76 ( ( zero_n356916108424825756nteger @ ~ P )
% 8.55/4.76 = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1824_Nat_Odiff__diff__right,axiom,
% 8.55/4.76 ! [K: nat,J: nat,I: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ K @ J )
% 8.55/4.76 => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 8.55/4.76 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1825_Nat_Oadd__diff__assoc2,axiom,
% 8.55/4.76 ! [K: nat,J: nat,I: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ K @ J )
% 8.55/4.76 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 8.55/4.76 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1826_Nat_Oadd__diff__assoc,axiom,
% 8.55/4.76 ! [K: nat,J: nat,I: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ K @ J )
% 8.55/4.76 => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 8.55/4.76 = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1827_div__less,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ M @ N )
% 8.55/4.76 => ( ( divide_divide_nat @ M @ N )
% 8.55/4.76 = zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1828_nat__zero__less__power__iff,axiom,
% 8.55/4.76 ! [X: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
% 8.55/4.76 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 8.55/4.76 | ( N = zero_zero_nat ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1829_zero__le__divide__1__iff,axiom,
% 8.55/4.76 ! [A: real] :
% 8.55/4.76 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 8.55/4.76 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1830_zero__le__divide__1__iff,axiom,
% 8.55/4.76 ! [A: rat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 8.55/4.76 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1831_divide__le__0__1__iff,axiom,
% 8.55/4.76 ! [A: real] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 8.55/4.76 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1832_divide__le__0__1__iff,axiom,
% 8.55/4.76 ! [A: rat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 8.55/4.76 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1833_numeral__le__one__iff,axiom,
% 8.55/4.76 ! [N: num] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 8.55/4.76 = ( ord_less_eq_num @ N @ one ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1834_numeral__le__one__iff,axiom,
% 8.55/4.76 ! [N: num] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 8.55/4.76 = ( ord_less_eq_num @ N @ one ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1835_numeral__le__one__iff,axiom,
% 8.55/4.76 ! [N: num] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 8.55/4.76 = ( ord_less_eq_num @ N @ one ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1836_numeral__le__one__iff,axiom,
% 8.55/4.76 ! [N: num] :
% 8.55/4.76 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 8.55/4.76 = ( ord_less_eq_num @ N @ one ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1837_zero__less__divide__1__iff,axiom,
% 8.55/4.76 ! [A: real] :
% 8.55/4.76 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 8.55/4.76 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1838_zero__less__divide__1__iff,axiom,
% 8.55/4.76 ! [A: rat] :
% 8.55/4.76 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 8.55/4.76 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1839_less__divide__eq__1__pos,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.76 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 8.55/4.76 = ( ord_less_real @ A @ B ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1840_less__divide__eq__1__pos,axiom,
% 8.55/4.76 ! [A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.76 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 8.55/4.76 = ( ord_less_rat @ A @ B ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1841_less__divide__eq__1__neg,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.76 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 8.55/4.76 = ( ord_less_real @ B @ A ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1842_less__divide__eq__1__neg,axiom,
% 8.55/4.76 ! [A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.76 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 8.55/4.76 = ( ord_less_rat @ B @ A ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1843_divide__less__eq__1__pos,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.76 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 8.55/4.76 = ( ord_less_real @ B @ A ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1844_divide__less__eq__1__pos,axiom,
% 8.55/4.76 ! [A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.76 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 8.55/4.76 = ( ord_less_rat @ B @ A ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1845_divide__less__eq__1__neg,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.76 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 8.55/4.76 = ( ord_less_real @ A @ B ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1846_divide__less__eq__1__neg,axiom,
% 8.55/4.76 ! [A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.76 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 8.55/4.76 = ( ord_less_rat @ A @ B ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1847_divide__less__0__1__iff,axiom,
% 8.55/4.76 ! [A: real] :
% 8.55/4.76 ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 8.55/4.76 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1848_divide__less__0__1__iff,axiom,
% 8.55/4.76 ! [A: rat] :
% 8.55/4.76 ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 8.55/4.76 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1849_le__divide__eq__numeral1_I1_J,axiom,
% 8.55/4.76 ! [A: real,B: real,W: num] :
% 8.55/4.76 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 8.55/4.76 = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1850_le__divide__eq__numeral1_I1_J,axiom,
% 8.55/4.76 ! [A: rat,B: rat,W: num] :
% 8.55/4.76 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 8.55/4.76 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1851_divide__le__eq__numeral1_I1_J,axiom,
% 8.55/4.76 ! [B: real,W: num,A: real] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 8.55/4.76 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1852_divide__le__eq__numeral1_I1_J,axiom,
% 8.55/4.76 ! [B: rat,W: num,A: rat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 8.55/4.76 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1853_one__less__numeral__iff,axiom,
% 8.55/4.76 ! [N: num] :
% 8.55/4.76 ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 8.55/4.76 = ( ord_less_num @ one @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1854_one__less__numeral__iff,axiom,
% 8.55/4.76 ! [N: num] :
% 8.55/4.76 ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 8.55/4.76 = ( ord_less_num @ one @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1855_one__less__numeral__iff,axiom,
% 8.55/4.76 ! [N: num] :
% 8.55/4.76 ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 8.55/4.76 = ( ord_less_num @ one @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1856_one__less__numeral__iff,axiom,
% 8.55/4.76 ! [N: num] :
% 8.55/4.76 ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 8.55/4.76 = ( ord_less_num @ one @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1857_less__divide__eq__numeral1_I1_J,axiom,
% 8.55/4.76 ! [A: real,B: real,W: num] :
% 8.55/4.76 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 8.55/4.76 = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1858_less__divide__eq__numeral1_I1_J,axiom,
% 8.55/4.76 ! [A: rat,B: rat,W: num] :
% 8.55/4.76 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 8.55/4.76 = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1859_divide__less__eq__numeral1_I1_J,axiom,
% 8.55/4.76 ! [B: real,W: num,A: real] :
% 8.55/4.76 ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 8.55/4.76 = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1860_divide__less__eq__numeral1_I1_J,axiom,
% 8.55/4.76 ! [B: rat,W: num,A: rat] :
% 8.55/4.76 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 8.55/4.76 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1861_of__nat__le__0__iff,axiom,
% 8.55/4.76 ! [M: nat] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 8.55/4.76 = ( M = zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1862_of__nat__le__0__iff,axiom,
% 8.55/4.76 ! [M: nat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 8.55/4.76 = ( M = zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1863_of__nat__le__0__iff,axiom,
% 8.55/4.76 ! [M: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 8.55/4.76 = ( M = zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1864_of__nat__le__0__iff,axiom,
% 8.55/4.76 ! [M: nat] :
% 8.55/4.76 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 8.55/4.76 = ( M = zero_zero_nat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1865_power__strict__increasing__iff,axiom,
% 8.55/4.76 ! [B: real,X: nat,Y: nat] :
% 8.55/4.76 ( ( ord_less_real @ one_one_real @ B )
% 8.55/4.76 => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
% 8.55/4.76 = ( ord_less_nat @ X @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1866_power__strict__increasing__iff,axiom,
% 8.55/4.76 ! [B: rat,X: nat,Y: nat] :
% 8.55/4.76 ( ( ord_less_rat @ one_one_rat @ B )
% 8.55/4.76 => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 8.55/4.76 = ( ord_less_nat @ X @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1867_power__strict__increasing__iff,axiom,
% 8.55/4.76 ! [B: nat,X: nat,Y: nat] :
% 8.55/4.76 ( ( ord_less_nat @ one_one_nat @ B )
% 8.55/4.76 => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 8.55/4.76 = ( ord_less_nat @ X @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1868_power__strict__increasing__iff,axiom,
% 8.55/4.76 ! [B: int,X: nat,Y: nat] :
% 8.55/4.76 ( ( ord_less_int @ one_one_int @ B )
% 8.55/4.76 => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 8.55/4.76 = ( ord_less_nat @ X @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1869_power__eq__0__iff,axiom,
% 8.55/4.76 ! [A: rat,N: nat] :
% 8.55/4.76 ( ( ( power_power_rat @ A @ N )
% 8.55/4.76 = zero_zero_rat )
% 8.55/4.76 = ( ( A = zero_zero_rat )
% 8.55/4.76 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1870_power__eq__0__iff,axiom,
% 8.55/4.76 ! [A: nat,N: nat] :
% 8.55/4.76 ( ( ( power_power_nat @ A @ N )
% 8.55/4.76 = zero_zero_nat )
% 8.55/4.76 = ( ( A = zero_zero_nat )
% 8.55/4.76 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1871_power__eq__0__iff,axiom,
% 8.55/4.76 ! [A: real,N: nat] :
% 8.55/4.76 ( ( ( power_power_real @ A @ N )
% 8.55/4.76 = zero_zero_real )
% 8.55/4.76 = ( ( A = zero_zero_real )
% 8.55/4.76 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1872_power__eq__0__iff,axiom,
% 8.55/4.76 ! [A: int,N: nat] :
% 8.55/4.76 ( ( ( power_power_int @ A @ N )
% 8.55/4.76 = zero_zero_int )
% 8.55/4.76 = ( ( A = zero_zero_int )
% 8.55/4.76 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1873_power__eq__0__iff,axiom,
% 8.55/4.76 ! [A: complex,N: nat] :
% 8.55/4.76 ( ( ( power_power_complex @ A @ N )
% 8.55/4.76 = zero_zero_complex )
% 8.55/4.76 = ( ( A = zero_zero_complex )
% 8.55/4.76 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1874_pow__divides__pow__iff,axiom,
% 8.55/4.76 ! [N: nat,A: nat,B: nat] :
% 8.55/4.76 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.76 => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 8.55/4.76 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1875_pow__divides__pow__iff,axiom,
% 8.55/4.76 ! [N: nat,A: int,B: int] :
% 8.55/4.76 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.76 => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 8.55/4.76 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1876_Suc__pred,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.76 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 8.55/4.76 = N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1877_one__le__mult__iff,axiom,
% 8.55/4.76 ! [M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 8.55/4.76 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 8.55/4.76 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1878_nat__mult__le__cancel__disj,axiom,
% 8.55/4.76 ! [K: nat,M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.55/4.76 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.55/4.76 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1879_mult__le__cancel2,axiom,
% 8.55/4.76 ! [M: nat,K: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 8.55/4.76 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.55/4.76 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1880_diff__Suc__diff__eq2,axiom,
% 8.55/4.76 ! [K: nat,J: nat,I: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ K @ J )
% 8.55/4.76 => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 8.55/4.76 = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1881_diff__Suc__diff__eq1,axiom,
% 8.55/4.76 ! [K: nat,J: nat,I: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ K @ J )
% 8.55/4.76 => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 8.55/4.76 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1882_div__mult__self1__is__m,axiom,
% 8.55/4.76 ! [N: nat,M: nat] :
% 8.55/4.76 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.76 => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 8.55/4.76 = M ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1883_div__mult__self__is__m,axiom,
% 8.55/4.76 ! [N: nat,M: nat] :
% 8.55/4.76 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.76 => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 8.55/4.76 = M ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1884_numeral__less__real__of__nat__iff,axiom,
% 8.55/4.76 ! [W: num,N: nat] :
% 8.55/4.76 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
% 8.55/4.76 = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1885_real__of__nat__less__numeral__iff,axiom,
% 8.55/4.76 ! [N: nat,W: num] :
% 8.55/4.76 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
% 8.55/4.76 = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1886_le__divide__eq__1__pos,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.76 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 8.55/4.76 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1887_le__divide__eq__1__pos,axiom,
% 8.55/4.76 ! [A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.76 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 8.55/4.76 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1888_le__divide__eq__1__neg,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.76 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 8.55/4.76 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1889_le__divide__eq__1__neg,axiom,
% 8.55/4.76 ! [A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.76 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 8.55/4.76 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1890_divide__le__eq__1__pos,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.76 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 8.55/4.76 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1891_divide__le__eq__1__pos,axiom,
% 8.55/4.76 ! [A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.76 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 8.55/4.76 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1892_divide__le__eq__1__neg,axiom,
% 8.55/4.76 ! [A: real,B: real] :
% 8.55/4.76 ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.76 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 8.55/4.76 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1893_divide__le__eq__1__neg,axiom,
% 8.55/4.76 ! [A: rat,B: rat] :
% 8.55/4.76 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.76 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 8.55/4.76 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1894_power__strict__decreasing__iff,axiom,
% 8.55/4.76 ! [B: real,M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_real @ zero_zero_real @ B )
% 8.55/4.76 => ( ( ord_less_real @ B @ one_one_real )
% 8.55/4.76 => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 8.55/4.76 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1895_power__strict__decreasing__iff,axiom,
% 8.55/4.76 ! [B: rat,M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_rat @ zero_zero_rat @ B )
% 8.55/4.76 => ( ( ord_less_rat @ B @ one_one_rat )
% 8.55/4.76 => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 8.55/4.76 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1896_power__strict__decreasing__iff,axiom,
% 8.55/4.76 ! [B: nat,M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.55/4.76 => ( ( ord_less_nat @ B @ one_one_nat )
% 8.55/4.76 => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 8.55/4.76 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1897_power__strict__decreasing__iff,axiom,
% 8.55/4.76 ! [B: int,M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_int @ zero_zero_int @ B )
% 8.55/4.76 => ( ( ord_less_int @ B @ one_one_int )
% 8.55/4.76 => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 8.55/4.76 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1898_power__increasing__iff,axiom,
% 8.55/4.76 ! [B: real,X: nat,Y: nat] :
% 8.55/4.76 ( ( ord_less_real @ one_one_real @ B )
% 8.55/4.76 => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
% 8.55/4.76 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1899_power__increasing__iff,axiom,
% 8.55/4.76 ! [B: rat,X: nat,Y: nat] :
% 8.55/4.76 ( ( ord_less_rat @ one_one_rat @ B )
% 8.55/4.76 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 8.55/4.76 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1900_power__increasing__iff,axiom,
% 8.55/4.76 ! [B: nat,X: nat,Y: nat] :
% 8.55/4.76 ( ( ord_less_nat @ one_one_nat @ B )
% 8.55/4.76 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 8.55/4.76 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1901_power__increasing__iff,axiom,
% 8.55/4.76 ! [B: int,X: nat,Y: nat] :
% 8.55/4.76 ( ( ord_less_int @ one_one_int @ B )
% 8.55/4.76 => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 8.55/4.76 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1902_power__mono__iff,axiom,
% 8.55/4.76 ! [A: real,B: real,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.76 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.55/4.76 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.76 => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 8.55/4.76 = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1903_power__mono__iff,axiom,
% 8.55/4.76 ! [A: rat,B: rat,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.76 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.55/4.76 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.76 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 8.55/4.76 = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1904_power__mono__iff,axiom,
% 8.55/4.76 ! [A: nat,B: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.76 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.55/4.76 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.76 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 8.55/4.76 = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1905_power__mono__iff,axiom,
% 8.55/4.76 ! [A: int,B: int,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.76 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.55/4.76 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.76 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 8.55/4.76 = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1906_of__nat__0__less__iff,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 8.55/4.76 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1907_of__nat__0__less__iff,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 8.55/4.76 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1908_of__nat__0__less__iff,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 8.55/4.76 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1909_of__nat__0__less__iff,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 8.55/4.76 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1910_odd__of__bool__self,axiom,
% 8.55/4.76 ! [P2: $o] :
% 8.55/4.76 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P2 ) ) )
% 8.55/4.76 = P2 ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1911_odd__of__bool__self,axiom,
% 8.55/4.76 ! [P2: $o] :
% 8.55/4.76 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P2 ) ) )
% 8.55/4.76 = P2 ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1912_odd__of__bool__self,axiom,
% 8.55/4.76 ! [P2: $o] :
% 8.55/4.76 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P2 ) ) )
% 8.55/4.76 = P2 ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1913_of__nat__less__of__nat__power__cancel__iff,axiom,
% 8.55/4.76 ! [B: nat,W: nat,X: nat] :
% 8.55/4.76 ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 8.55/4.76 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1914_of__nat__less__of__nat__power__cancel__iff,axiom,
% 8.55/4.76 ! [B: nat,W: nat,X: nat] :
% 8.55/4.76 ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 8.55/4.76 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1915_of__nat__less__of__nat__power__cancel__iff,axiom,
% 8.55/4.76 ! [B: nat,W: nat,X: nat] :
% 8.55/4.76 ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 8.55/4.76 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1916_of__nat__less__of__nat__power__cancel__iff,axiom,
% 8.55/4.76 ! [B: nat,W: nat,X: nat] :
% 8.55/4.76 ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 8.55/4.76 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1917_of__nat__power__less__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,B: nat,W: nat] :
% 8.55/4.76 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 8.55/4.76 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1918_of__nat__power__less__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,B: nat,W: nat] :
% 8.55/4.76 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 8.55/4.76 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1919_of__nat__power__less__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,B: nat,W: nat] :
% 8.55/4.76 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 8.55/4.76 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1920_of__nat__power__less__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,B: nat,W: nat] :
% 8.55/4.76 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 8.55/4.76 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1921_of__nat__le__of__nat__power__cancel__iff,axiom,
% 8.55/4.76 ! [B: nat,W: nat,X: nat] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 8.55/4.76 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1922_of__nat__le__of__nat__power__cancel__iff,axiom,
% 8.55/4.76 ! [B: nat,W: nat,X: nat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 8.55/4.76 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1923_of__nat__le__of__nat__power__cancel__iff,axiom,
% 8.55/4.76 ! [B: nat,W: nat,X: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 8.55/4.76 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1924_of__nat__le__of__nat__power__cancel__iff,axiom,
% 8.55/4.76 ! [B: nat,W: nat,X: nat] :
% 8.55/4.76 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 8.55/4.76 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1925_of__nat__power__le__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,B: nat,W: nat] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 8.55/4.76 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1926_of__nat__power__le__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,B: nat,W: nat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 8.55/4.76 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1927_of__nat__power__le__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,B: nat,W: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 8.55/4.76 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1928_of__nat__power__le__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,B: nat,W: nat] :
% 8.55/4.76 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 8.55/4.76 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1929_Suc__diff__1,axiom,
% 8.55/4.76 ! [N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.76 => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 8.55/4.76 = N ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1930_half__negative__int__iff,axiom,
% 8.55/4.76 ! [K: int] :
% 8.55/4.76 ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 8.55/4.76 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1931_half__nonnegative__int__iff,axiom,
% 8.55/4.76 ! [K: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 8.55/4.76 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1932_numeral__le__real__of__nat__iff,axiom,
% 8.55/4.76 ! [N: num,M: nat] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 8.55/4.76 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1933_power__decreasing__iff,axiom,
% 8.55/4.76 ! [B: real,M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_real @ zero_zero_real @ B )
% 8.55/4.76 => ( ( ord_less_real @ B @ one_one_real )
% 8.55/4.76 => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 8.55/4.76 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1934_power__decreasing__iff,axiom,
% 8.55/4.76 ! [B: rat,M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_rat @ zero_zero_rat @ B )
% 8.55/4.76 => ( ( ord_less_rat @ B @ one_one_rat )
% 8.55/4.76 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 8.55/4.76 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1935_power__decreasing__iff,axiom,
% 8.55/4.76 ! [B: nat,M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.55/4.76 => ( ( ord_less_nat @ B @ one_one_nat )
% 8.55/4.76 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 8.55/4.76 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1936_power__decreasing__iff,axiom,
% 8.55/4.76 ! [B: int,M: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_int @ zero_zero_int @ B )
% 8.55/4.76 => ( ( ord_less_int @ B @ one_one_int )
% 8.55/4.76 => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 8.55/4.76 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1937_power2__eq__iff__nonneg,axiom,
% 8.55/4.76 ! [X: real,Y: real] :
% 8.55/4.76 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.55/4.76 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.55/4.76 => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.76 = ( X = Y ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1938_power2__eq__iff__nonneg,axiom,
% 8.55/4.76 ! [X: rat,Y: rat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 8.55/4.76 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 8.55/4.76 => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.76 = ( X = Y ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1939_power2__eq__iff__nonneg,axiom,
% 8.55/4.76 ! [X: nat,Y: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 8.55/4.76 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 8.55/4.76 => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.76 = ( X = Y ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1940_power2__eq__iff__nonneg,axiom,
% 8.55/4.76 ! [X: int,Y: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.55/4.76 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.55/4.76 => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.76 = ( X = Y ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1941_power2__less__eq__zero__iff,axiom,
% 8.55/4.76 ! [A: real] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 8.55/4.76 = ( A = zero_zero_real ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1942_power2__less__eq__zero__iff,axiom,
% 8.55/4.76 ! [A: rat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 8.55/4.76 = ( A = zero_zero_rat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1943_power2__less__eq__zero__iff,axiom,
% 8.55/4.76 ! [A: int] :
% 8.55/4.76 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 8.55/4.76 = ( A = zero_zero_int ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1944_zero__less__power2,axiom,
% 8.55/4.76 ! [A: real] :
% 8.55/4.76 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.76 = ( A != zero_zero_real ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1945_zero__less__power2,axiom,
% 8.55/4.76 ! [A: rat] :
% 8.55/4.76 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.76 = ( A != zero_zero_rat ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1946_zero__less__power2,axiom,
% 8.55/4.76 ! [A: int] :
% 8.55/4.76 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.76 = ( A != zero_zero_int ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1947_of__nat__zero__less__power__iff,axiom,
% 8.55/4.76 ! [X: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
% 8.55/4.76 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 8.55/4.76 | ( N = zero_zero_nat ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1948_of__nat__zero__less__power__iff,axiom,
% 8.55/4.76 ! [X: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
% 8.55/4.76 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 8.55/4.76 | ( N = zero_zero_nat ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1949_of__nat__zero__less__power__iff,axiom,
% 8.55/4.76 ! [X: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
% 8.55/4.76 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 8.55/4.76 | ( N = zero_zero_nat ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1950_of__nat__zero__less__power__iff,axiom,
% 8.55/4.76 ! [X: nat,N: nat] :
% 8.55/4.76 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
% 8.55/4.76 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 8.55/4.76 | ( N = zero_zero_nat ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1951_of__bool__half__eq__0,axiom,
% 8.55/4.76 ! [B: $o] :
% 8.55/4.76 ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.76 = zero_zero_nat ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1952_of__bool__half__eq__0,axiom,
% 8.55/4.76 ! [B: $o] :
% 8.55/4.76 ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.55/4.76 = zero_zero_int ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1953_of__bool__half__eq__0,axiom,
% 8.55/4.76 ! [B: $o] :
% 8.55/4.76 ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.55/4.76 = zero_z3403309356797280102nteger ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1954_zero__le__power__eq__numeral,axiom,
% 8.55/4.76 ! [A: real,W: num] :
% 8.55/4.76 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 8.55/4.76 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.76 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.76 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1955_zero__le__power__eq__numeral,axiom,
% 8.55/4.76 ! [A: rat,W: num] :
% 8.55/4.76 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 8.55/4.76 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.76 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.76 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1956_zero__le__power__eq__numeral,axiom,
% 8.55/4.76 ! [A: int,W: num] :
% 8.55/4.76 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 8.55/4.76 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.76 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.76 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1957_even__power,axiom,
% 8.55/4.76 ! [A: code_integer,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 8.55/4.76 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.55/4.76 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1958_even__power,axiom,
% 8.55/4.76 ! [A: nat,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
% 8.55/4.76 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.55/4.76 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1959_even__power,axiom,
% 8.55/4.76 ! [A: int,N: nat] :
% 8.55/4.76 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
% 8.55/4.76 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.55/4.76 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1960_power__less__zero__eq__numeral,axiom,
% 8.55/4.76 ! [A: real,W: num] :
% 8.55/4.76 ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 8.55/4.76 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.76 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1961_power__less__zero__eq__numeral,axiom,
% 8.55/4.76 ! [A: rat,W: num] :
% 8.55/4.76 ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 8.55/4.76 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.76 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1962_power__less__zero__eq__numeral,axiom,
% 8.55/4.76 ! [A: int,W: num] :
% 8.55/4.76 ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 8.55/4.76 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.76 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1963_power__less__zero__eq,axiom,
% 8.55/4.76 ! [A: real,N: nat] :
% 8.55/4.76 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 8.55/4.76 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.76 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1964_power__less__zero__eq,axiom,
% 8.55/4.76 ! [A: rat,N: nat] :
% 8.55/4.76 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 8.55/4.76 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.76 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1965_power__less__zero__eq,axiom,
% 8.55/4.76 ! [A: int,N: nat] :
% 8.55/4.76 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 8.55/4.76 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.76 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1966_of__nat__less__numeral__power__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,I: num,N: nat] :
% 8.55/4.76 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 8.55/4.76 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1967_of__nat__less__numeral__power__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,I: num,N: nat] :
% 8.55/4.76 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 8.55/4.76 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1968_of__nat__less__numeral__power__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,I: num,N: nat] :
% 8.55/4.76 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 8.55/4.76 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1969_of__nat__less__numeral__power__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,I: num,N: nat] :
% 8.55/4.76 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 8.55/4.76 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1970_numeral__power__less__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [I: num,N: nat,X: nat] :
% 8.55/4.76 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 8.55/4.76 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1971_numeral__power__less__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [I: num,N: nat,X: nat] :
% 8.55/4.76 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 8.55/4.76 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1972_numeral__power__less__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [I: num,N: nat,X: nat] :
% 8.55/4.76 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 8.55/4.76 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1973_numeral__power__less__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [I: num,N: nat,X: nat] :
% 8.55/4.76 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 8.55/4.76 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1974_of__nat__le__numeral__power__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,I: num,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 8.55/4.76 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1975_of__nat__le__numeral__power__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,I: num,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 8.55/4.76 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1976_of__nat__le__numeral__power__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,I: num,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 8.55/4.76 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1977_of__nat__le__numeral__power__cancel__iff,axiom,
% 8.55/4.76 ! [X: nat,I: num,N: nat] :
% 8.55/4.76 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 8.55/4.76 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1978_numeral__power__le__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [I: num,N: nat,X: nat] :
% 8.55/4.76 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 8.55/4.76 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1979_numeral__power__le__of__nat__cancel__iff,axiom,
% 8.55/4.76 ! [I: num,N: nat,X: nat] :
% 8.55/4.76 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 8.55/4.76 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 8.55/4.76
% 8.55/4.76 thf(fact_1980_numeral__power__le__of__nat__cancel__iff,axiom,
% 8.55/4.77 ! [I: num,N: nat,X: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 8.55/4.77 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1981_numeral__power__le__of__nat__cancel__iff,axiom,
% 8.55/4.77 ! [I: num,N: nat,X: nat] :
% 8.55/4.77 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 8.55/4.77 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1982_even__diff__nat,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 8.55/4.77 = ( ( ord_less_nat @ M @ N )
% 8.55/4.77 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1983_one__div__2__pow__eq,axiom,
% 8.55/4.77 ! [N: nat] :
% 8.55/4.77 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.77 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1984_one__div__2__pow__eq,axiom,
% 8.55/4.77 ! [N: nat] :
% 8.55/4.77 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.77 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1985_one__div__2__pow__eq,axiom,
% 8.55/4.77 ! [N: nat] :
% 8.55/4.77 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.77 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1986_bits__1__div__exp,axiom,
% 8.55/4.77 ! [N: nat] :
% 8.55/4.77 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.77 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1987_bits__1__div__exp,axiom,
% 8.55/4.77 ! [N: nat] :
% 8.55/4.77 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.77 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1988_bits__1__div__exp,axiom,
% 8.55/4.77 ! [N: nat] :
% 8.55/4.77 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.77 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1989_power__le__zero__eq__numeral,axiom,
% 8.55/4.77 ! [A: real,W: num] :
% 8.55/4.77 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 8.55/4.77 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 8.55/4.77 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.77 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 8.55/4.77 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.77 & ( A = zero_zero_real ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1990_power__le__zero__eq__numeral,axiom,
% 8.55/4.77 ! [A: rat,W: num] :
% 8.55/4.77 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 8.55/4.77 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 8.55/4.77 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.77 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 8.55/4.77 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.77 & ( A = zero_zero_rat ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1991_power__le__zero__eq__numeral,axiom,
% 8.55/4.77 ! [A: int,W: num] :
% 8.55/4.77 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 8.55/4.77 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 8.55/4.77 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.77 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 8.55/4.77 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.55/4.77 & ( A = zero_zero_int ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1992_zero__less__eq__of__bool,axiom,
% 8.55/4.77 ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1993_zero__less__eq__of__bool,axiom,
% 8.55/4.77 ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1994_zero__less__eq__of__bool,axiom,
% 8.55/4.77 ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1995_zero__less__eq__of__bool,axiom,
% 8.55/4.77 ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1996_zero__less__eq__of__bool,axiom,
% 8.55/4.77 ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1997_of__bool__less__eq__one,axiom,
% 8.55/4.77 ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1998_of__bool__less__eq__one,axiom,
% 8.55/4.77 ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_1999_of__bool__less__eq__one,axiom,
% 8.55/4.77 ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2000_of__bool__less__eq__one,axiom,
% 8.55/4.77 ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2001_of__bool__less__eq__one,axiom,
% 8.55/4.77 ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2002_verit__la__disequality,axiom,
% 8.55/4.77 ! [A: rat,B: rat] :
% 8.55/4.77 ( ( A = B )
% 8.55/4.77 | ~ ( ord_less_eq_rat @ A @ B )
% 8.55/4.77 | ~ ( ord_less_eq_rat @ B @ A ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2003_verit__la__disequality,axiom,
% 8.55/4.77 ! [A: num,B: num] :
% 8.55/4.77 ( ( A = B )
% 8.55/4.77 | ~ ( ord_less_eq_num @ A @ B )
% 8.55/4.77 | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2004_verit__la__disequality,axiom,
% 8.55/4.77 ! [A: nat,B: nat] :
% 8.55/4.77 ( ( A = B )
% 8.55/4.77 | ~ ( ord_less_eq_nat @ A @ B )
% 8.55/4.77 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2005_verit__la__disequality,axiom,
% 8.55/4.77 ! [A: int,B: int] :
% 8.55/4.77 ( ( A = B )
% 8.55/4.77 | ~ ( ord_less_eq_int @ A @ B )
% 8.55/4.77 | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2006_linorder__neqE__linordered__idom,axiom,
% 8.55/4.77 ! [X: real,Y: real] :
% 8.55/4.77 ( ( X != Y )
% 8.55/4.77 => ( ~ ( ord_less_real @ X @ Y )
% 8.55/4.77 => ( ord_less_real @ Y @ X ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2007_linorder__neqE__linordered__idom,axiom,
% 8.55/4.77 ! [X: rat,Y: rat] :
% 8.55/4.77 ( ( X != Y )
% 8.55/4.77 => ( ~ ( ord_less_rat @ X @ Y )
% 8.55/4.77 => ( ord_less_rat @ Y @ X ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2008_linorder__neqE__linordered__idom,axiom,
% 8.55/4.77 ! [X: int,Y: int] :
% 8.55/4.77 ( ( X != Y )
% 8.55/4.77 => ( ~ ( ord_less_int @ X @ Y )
% 8.55/4.77 => ( ord_less_int @ Y @ X ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2009_of__bool__eq__iff,axiom,
% 8.55/4.77 ! [P2: $o,Q2: $o] :
% 8.55/4.77 ( ( ( zero_n2687167440665602831ol_nat @ P2 )
% 8.55/4.77 = ( zero_n2687167440665602831ol_nat @ Q2 ) )
% 8.55/4.77 = ( P2 = Q2 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2010_of__bool__eq__iff,axiom,
% 8.55/4.77 ! [P2: $o,Q2: $o] :
% 8.55/4.77 ( ( ( zero_n2684676970156552555ol_int @ P2 )
% 8.55/4.77 = ( zero_n2684676970156552555ol_int @ Q2 ) )
% 8.55/4.77 = ( P2 = Q2 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2011_of__bool__eq__iff,axiom,
% 8.55/4.77 ! [P2: $o,Q2: $o] :
% 8.55/4.77 ( ( ( zero_n356916108424825756nteger @ P2 )
% 8.55/4.77 = ( zero_n356916108424825756nteger @ Q2 ) )
% 8.55/4.77 = ( P2 = Q2 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2012_verit__comp__simplify1_I1_J,axiom,
% 8.55/4.77 ! [A: real] :
% 8.55/4.77 ~ ( ord_less_real @ A @ A ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2013_verit__comp__simplify1_I1_J,axiom,
% 8.55/4.77 ! [A: rat] :
% 8.55/4.77 ~ ( ord_less_rat @ A @ A ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2014_verit__comp__simplify1_I1_J,axiom,
% 8.55/4.77 ! [A: num] :
% 8.55/4.77 ~ ( ord_less_num @ A @ A ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2015_verit__comp__simplify1_I1_J,axiom,
% 8.55/4.77 ! [A: nat] :
% 8.55/4.77 ~ ( ord_less_nat @ A @ A ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2016_verit__comp__simplify1_I1_J,axiom,
% 8.55/4.77 ! [A: int] :
% 8.55/4.77 ~ ( ord_less_int @ A @ A ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2017_verit__comp__simplify1_I2_J,axiom,
% 8.55/4.77 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2018_verit__comp__simplify1_I2_J,axiom,
% 8.55/4.77 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2019_verit__comp__simplify1_I2_J,axiom,
% 8.55/4.77 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2020_verit__comp__simplify1_I2_J,axiom,
% 8.55/4.77 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2021_verit__comp__simplify1_I2_J,axiom,
% 8.55/4.77 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2022_verit__comp__simplify1_I3_J,axiom,
% 8.55/4.77 ! [B6: real,A5: real] :
% 8.55/4.77 ( ( ~ ( ord_less_eq_real @ B6 @ A5 ) )
% 8.55/4.77 = ( ord_less_real @ A5 @ B6 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2023_verit__comp__simplify1_I3_J,axiom,
% 8.55/4.77 ! [B6: rat,A5: rat] :
% 8.55/4.77 ( ( ~ ( ord_less_eq_rat @ B6 @ A5 ) )
% 8.55/4.77 = ( ord_less_rat @ A5 @ B6 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2024_verit__comp__simplify1_I3_J,axiom,
% 8.55/4.77 ! [B6: num,A5: num] :
% 8.55/4.77 ( ( ~ ( ord_less_eq_num @ B6 @ A5 ) )
% 8.55/4.77 = ( ord_less_num @ A5 @ B6 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2025_verit__comp__simplify1_I3_J,axiom,
% 8.55/4.77 ! [B6: nat,A5: nat] :
% 8.55/4.77 ( ( ~ ( ord_less_eq_nat @ B6 @ A5 ) )
% 8.55/4.77 = ( ord_less_nat @ A5 @ B6 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2026_verit__comp__simplify1_I3_J,axiom,
% 8.55/4.77 ! [B6: int,A5: int] :
% 8.55/4.77 ( ( ~ ( ord_less_eq_int @ B6 @ A5 ) )
% 8.55/4.77 = ( ord_less_int @ A5 @ B6 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2027_less__mono__imp__le__mono,axiom,
% 8.55/4.77 ! [F: nat > nat,I: nat,J: nat] :
% 8.55/4.77 ( ! [I2: nat,J2: nat] :
% 8.55/4.77 ( ( ord_less_nat @ I2 @ J2 )
% 8.55/4.77 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 8.55/4.77 => ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.77 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2028_le__neq__implies__less,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.77 => ( ( M != N )
% 8.55/4.77 => ( ord_less_nat @ M @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2029_Nat_Oex__has__greatest__nat,axiom,
% 8.55/4.77 ! [P: nat > $o,K: nat,B: nat] :
% 8.55/4.77 ( ( P @ K )
% 8.55/4.77 => ( ! [Y4: nat] :
% 8.55/4.77 ( ( P @ Y4 )
% 8.55/4.77 => ( ord_less_eq_nat @ Y4 @ B ) )
% 8.55/4.77 => ? [X4: nat] :
% 8.55/4.77 ( ( P @ X4 )
% 8.55/4.77 & ! [Y5: nat] :
% 8.55/4.77 ( ( P @ Y5 )
% 8.55/4.77 => ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2030_linorder__neqE__nat,axiom,
% 8.55/4.77 ! [X: nat,Y: nat] :
% 8.55/4.77 ( ( X != Y )
% 8.55/4.77 => ( ~ ( ord_less_nat @ X @ Y )
% 8.55/4.77 => ( ord_less_nat @ Y @ X ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2031_less__or__eq__imp__le,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ( ord_less_nat @ M @ N )
% 8.55/4.77 | ( M = N ) )
% 8.55/4.77 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2032_le__eq__less__or__eq,axiom,
% 8.55/4.77 ( ord_less_eq_nat
% 8.55/4.77 = ( ^ [M3: nat,N3: nat] :
% 8.55/4.77 ( ( ord_less_nat @ M3 @ N3 )
% 8.55/4.77 | ( M3 = N3 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2033_infinite__descent,axiom,
% 8.55/4.77 ! [P: nat > $o,N: nat] :
% 8.55/4.77 ( ! [N2: nat] :
% 8.55/4.77 ( ~ ( P @ N2 )
% 8.55/4.77 => ? [M4: nat] :
% 8.55/4.77 ( ( ord_less_nat @ M4 @ N2 )
% 8.55/4.77 & ~ ( P @ M4 ) ) )
% 8.55/4.77 => ( P @ N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2034_nat__less__induct,axiom,
% 8.55/4.77 ! [P: nat > $o,N: nat] :
% 8.55/4.77 ( ! [N2: nat] :
% 8.55/4.77 ( ! [M4: nat] :
% 8.55/4.77 ( ( ord_less_nat @ M4 @ N2 )
% 8.55/4.77 => ( P @ M4 ) )
% 8.55/4.77 => ( P @ N2 ) )
% 8.55/4.77 => ( P @ N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2035_less__irrefl__nat,axiom,
% 8.55/4.77 ! [N: nat] :
% 8.55/4.77 ~ ( ord_less_nat @ N @ N ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2036_less__imp__le__nat,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_nat @ M @ N )
% 8.55/4.77 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2037_less__not__refl3,axiom,
% 8.55/4.77 ! [S: nat,T: nat] :
% 8.55/4.77 ( ( ord_less_nat @ S @ T )
% 8.55/4.77 => ( S != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2038_less__not__refl2,axiom,
% 8.55/4.77 ! [N: nat,M: nat] :
% 8.55/4.77 ( ( ord_less_nat @ N @ M )
% 8.55/4.77 => ( M != N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2039_nat__le__linear,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.77 | ( ord_less_eq_nat @ N @ M ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2040_less__not__refl,axiom,
% 8.55/4.77 ! [N: nat] :
% 8.55/4.77 ~ ( ord_less_nat @ N @ N ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2041_nat__neq__iff,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( M != N )
% 8.55/4.77 = ( ( ord_less_nat @ M @ N )
% 8.55/4.77 | ( ord_less_nat @ N @ M ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2042_nat__less__le,axiom,
% 8.55/4.77 ( ord_less_nat
% 8.55/4.77 = ( ^ [M3: nat,N3: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ M3 @ N3 )
% 8.55/4.77 & ( M3 != N3 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2043_le__antisym,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.77 => ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.77 => ( M = N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2044_eq__imp__le,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( M = N )
% 8.55/4.77 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2045_le__trans,axiom,
% 8.55/4.77 ! [I: nat,J: nat,K: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.77 => ( ( ord_less_eq_nat @ J @ K )
% 8.55/4.77 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2046_le__refl,axiom,
% 8.55/4.77 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2047_less__eq__real__def,axiom,
% 8.55/4.77 ( ord_less_eq_real
% 8.55/4.77 = ( ^ [X3: real,Y6: real] :
% 8.55/4.77 ( ( ord_less_real @ X3 @ Y6 )
% 8.55/4.77 | ( X3 = Y6 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2048_complete__real,axiom,
% 8.55/4.77 ! [S2: set_real] :
% 8.55/4.77 ( ? [X5: real] : ( member_real @ X5 @ S2 )
% 8.55/4.77 => ( ? [Z3: real] :
% 8.55/4.77 ! [X4: real] :
% 8.55/4.77 ( ( member_real @ X4 @ S2 )
% 8.55/4.77 => ( ord_less_eq_real @ X4 @ Z3 ) )
% 8.55/4.77 => ? [Y4: real] :
% 8.55/4.77 ( ! [X5: real] :
% 8.55/4.77 ( ( member_real @ X5 @ S2 )
% 8.55/4.77 => ( ord_less_eq_real @ X5 @ Y4 ) )
% 8.55/4.77 & ! [Z3: real] :
% 8.55/4.77 ( ! [X4: real] :
% 8.55/4.77 ( ( member_real @ X4 @ S2 )
% 8.55/4.77 => ( ord_less_eq_real @ X4 @ Z3 ) )
% 8.55/4.77 => ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2049_linordered__field__no__ub,axiom,
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ? [X_12: real] : ( ord_less_real @ X5 @ X_12 ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2050_linordered__field__no__ub,axiom,
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ? [X_12: rat] : ( ord_less_rat @ X5 @ X_12 ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2051_linordered__field__no__lb,axiom,
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ? [Y4: real] : ( ord_less_real @ Y4 @ X5 ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2052_linordered__field__no__lb,axiom,
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X5 ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2053_pinf_I1_J,axiom,
% 8.55/4.77 ! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
% 8.55/4.77 ( ? [Z3: real] :
% 8.55/4.77 ! [X4: real] :
% 8.55/4.77 ( ( ord_less_real @ Z3 @ X4 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: real] :
% 8.55/4.77 ! [X4: real] :
% 8.55/4.77 ( ( ord_less_real @ Z3 @ X4 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: real] :
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ( ( ord_less_real @ Z4 @ X5 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 & ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2054_pinf_I1_J,axiom,
% 8.55/4.77 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
% 8.55/4.77 ( ? [Z3: rat] :
% 8.55/4.77 ! [X4: rat] :
% 8.55/4.77 ( ( ord_less_rat @ Z3 @ X4 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: rat] :
% 8.55/4.77 ! [X4: rat] :
% 8.55/4.77 ( ( ord_less_rat @ Z3 @ X4 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: rat] :
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ( ( ord_less_rat @ Z4 @ X5 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 & ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2055_pinf_I1_J,axiom,
% 8.55/4.77 ! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
% 8.55/4.77 ( ? [Z3: num] :
% 8.55/4.77 ! [X4: num] :
% 8.55/4.77 ( ( ord_less_num @ Z3 @ X4 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: num] :
% 8.55/4.77 ! [X4: num] :
% 8.55/4.77 ( ( ord_less_num @ Z3 @ X4 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: num] :
% 8.55/4.77 ! [X5: num] :
% 8.55/4.77 ( ( ord_less_num @ Z4 @ X5 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 & ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2056_pinf_I1_J,axiom,
% 8.55/4.77 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
% 8.55/4.77 ( ? [Z3: nat] :
% 8.55/4.77 ! [X4: nat] :
% 8.55/4.77 ( ( ord_less_nat @ Z3 @ X4 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: nat] :
% 8.55/4.77 ! [X4: nat] :
% 8.55/4.77 ( ( ord_less_nat @ Z3 @ X4 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: nat] :
% 8.55/4.77 ! [X5: nat] :
% 8.55/4.77 ( ( ord_less_nat @ Z4 @ X5 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 & ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2057_pinf_I1_J,axiom,
% 8.55/4.77 ! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
% 8.55/4.77 ( ? [Z3: int] :
% 8.55/4.77 ! [X4: int] :
% 8.55/4.77 ( ( ord_less_int @ Z3 @ X4 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: int] :
% 8.55/4.77 ! [X4: int] :
% 8.55/4.77 ( ( ord_less_int @ Z3 @ X4 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: int] :
% 8.55/4.77 ! [X5: int] :
% 8.55/4.77 ( ( ord_less_int @ Z4 @ X5 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 & ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2058_pinf_I2_J,axiom,
% 8.55/4.77 ! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
% 8.55/4.77 ( ? [Z3: real] :
% 8.55/4.77 ! [X4: real] :
% 8.55/4.77 ( ( ord_less_real @ Z3 @ X4 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: real] :
% 8.55/4.77 ! [X4: real] :
% 8.55/4.77 ( ( ord_less_real @ Z3 @ X4 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: real] :
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ( ( ord_less_real @ Z4 @ X5 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 | ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2059_pinf_I2_J,axiom,
% 8.55/4.77 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
% 8.55/4.77 ( ? [Z3: rat] :
% 8.55/4.77 ! [X4: rat] :
% 8.55/4.77 ( ( ord_less_rat @ Z3 @ X4 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: rat] :
% 8.55/4.77 ! [X4: rat] :
% 8.55/4.77 ( ( ord_less_rat @ Z3 @ X4 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: rat] :
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ( ( ord_less_rat @ Z4 @ X5 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 | ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2060_pinf_I2_J,axiom,
% 8.55/4.77 ! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
% 8.55/4.77 ( ? [Z3: num] :
% 8.55/4.77 ! [X4: num] :
% 8.55/4.77 ( ( ord_less_num @ Z3 @ X4 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: num] :
% 8.55/4.77 ! [X4: num] :
% 8.55/4.77 ( ( ord_less_num @ Z3 @ X4 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: num] :
% 8.55/4.77 ! [X5: num] :
% 8.55/4.77 ( ( ord_less_num @ Z4 @ X5 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 | ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2061_pinf_I2_J,axiom,
% 8.55/4.77 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
% 8.55/4.77 ( ? [Z3: nat] :
% 8.55/4.77 ! [X4: nat] :
% 8.55/4.77 ( ( ord_less_nat @ Z3 @ X4 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: nat] :
% 8.55/4.77 ! [X4: nat] :
% 8.55/4.77 ( ( ord_less_nat @ Z3 @ X4 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: nat] :
% 8.55/4.77 ! [X5: nat] :
% 8.55/4.77 ( ( ord_less_nat @ Z4 @ X5 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 | ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2062_pinf_I2_J,axiom,
% 8.55/4.77 ! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
% 8.55/4.77 ( ? [Z3: int] :
% 8.55/4.77 ! [X4: int] :
% 8.55/4.77 ( ( ord_less_int @ Z3 @ X4 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: int] :
% 8.55/4.77 ! [X4: int] :
% 8.55/4.77 ( ( ord_less_int @ Z3 @ X4 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: int] :
% 8.55/4.77 ! [X5: int] :
% 8.55/4.77 ( ( ord_less_int @ Z4 @ X5 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 | ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2063_pinf_I3_J,axiom,
% 8.55/4.77 ! [T: real] :
% 8.55/4.77 ? [Z4: real] :
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ( ( ord_less_real @ Z4 @ X5 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2064_pinf_I3_J,axiom,
% 8.55/4.77 ! [T: rat] :
% 8.55/4.77 ? [Z4: rat] :
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ( ( ord_less_rat @ Z4 @ X5 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2065_pinf_I3_J,axiom,
% 8.55/4.77 ! [T: num] :
% 8.55/4.77 ? [Z4: num] :
% 8.55/4.77 ! [X5: num] :
% 8.55/4.77 ( ( ord_less_num @ Z4 @ X5 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2066_pinf_I3_J,axiom,
% 8.55/4.77 ! [T: nat] :
% 8.55/4.77 ? [Z4: nat] :
% 8.55/4.77 ! [X5: nat] :
% 8.55/4.77 ( ( ord_less_nat @ Z4 @ X5 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2067_pinf_I3_J,axiom,
% 8.55/4.77 ! [T: int] :
% 8.55/4.77 ? [Z4: int] :
% 8.55/4.77 ! [X5: int] :
% 8.55/4.77 ( ( ord_less_int @ Z4 @ X5 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2068_pinf_I4_J,axiom,
% 8.55/4.77 ! [T: real] :
% 8.55/4.77 ? [Z4: real] :
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ( ( ord_less_real @ Z4 @ X5 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2069_pinf_I4_J,axiom,
% 8.55/4.77 ! [T: rat] :
% 8.55/4.77 ? [Z4: rat] :
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ( ( ord_less_rat @ Z4 @ X5 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2070_pinf_I4_J,axiom,
% 8.55/4.77 ! [T: num] :
% 8.55/4.77 ? [Z4: num] :
% 8.55/4.77 ! [X5: num] :
% 8.55/4.77 ( ( ord_less_num @ Z4 @ X5 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2071_pinf_I4_J,axiom,
% 8.55/4.77 ! [T: nat] :
% 8.55/4.77 ? [Z4: nat] :
% 8.55/4.77 ! [X5: nat] :
% 8.55/4.77 ( ( ord_less_nat @ Z4 @ X5 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2072_pinf_I4_J,axiom,
% 8.55/4.77 ! [T: int] :
% 8.55/4.77 ? [Z4: int] :
% 8.55/4.77 ! [X5: int] :
% 8.55/4.77 ( ( ord_less_int @ Z4 @ X5 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2073_pinf_I5_J,axiom,
% 8.55/4.77 ! [T: real] :
% 8.55/4.77 ? [Z4: real] :
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ( ( ord_less_real @ Z4 @ X5 )
% 8.55/4.77 => ~ ( ord_less_real @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2074_pinf_I5_J,axiom,
% 8.55/4.77 ! [T: rat] :
% 8.55/4.77 ? [Z4: rat] :
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ( ( ord_less_rat @ Z4 @ X5 )
% 8.55/4.77 => ~ ( ord_less_rat @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2075_pinf_I5_J,axiom,
% 8.55/4.77 ! [T: num] :
% 8.55/4.77 ? [Z4: num] :
% 8.55/4.77 ! [X5: num] :
% 8.55/4.77 ( ( ord_less_num @ Z4 @ X5 )
% 8.55/4.77 => ~ ( ord_less_num @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2076_pinf_I5_J,axiom,
% 8.55/4.77 ! [T: nat] :
% 8.55/4.77 ? [Z4: nat] :
% 8.55/4.77 ! [X5: nat] :
% 8.55/4.77 ( ( ord_less_nat @ Z4 @ X5 )
% 8.55/4.77 => ~ ( ord_less_nat @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2077_pinf_I5_J,axiom,
% 8.55/4.77 ! [T: int] :
% 8.55/4.77 ? [Z4: int] :
% 8.55/4.77 ! [X5: int] :
% 8.55/4.77 ( ( ord_less_int @ Z4 @ X5 )
% 8.55/4.77 => ~ ( ord_less_int @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2078_pinf_I6_J,axiom,
% 8.55/4.77 ! [T: real] :
% 8.55/4.77 ? [Z4: real] :
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ( ( ord_less_real @ Z4 @ X5 )
% 8.55/4.77 => ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2079_pinf_I6_J,axiom,
% 8.55/4.77 ! [T: rat] :
% 8.55/4.77 ? [Z4: rat] :
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ( ( ord_less_rat @ Z4 @ X5 )
% 8.55/4.77 => ~ ( ord_less_eq_rat @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2080_pinf_I6_J,axiom,
% 8.55/4.77 ! [T: num] :
% 8.55/4.77 ? [Z4: num] :
% 8.55/4.77 ! [X5: num] :
% 8.55/4.77 ( ( ord_less_num @ Z4 @ X5 )
% 8.55/4.77 => ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2081_pinf_I6_J,axiom,
% 8.55/4.77 ! [T: nat] :
% 8.55/4.77 ? [Z4: nat] :
% 8.55/4.77 ! [X5: nat] :
% 8.55/4.77 ( ( ord_less_nat @ Z4 @ X5 )
% 8.55/4.77 => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2082_pinf_I6_J,axiom,
% 8.55/4.77 ! [T: int] :
% 8.55/4.77 ? [Z4: int] :
% 8.55/4.77 ! [X5: int] :
% 8.55/4.77 ( ( ord_less_int @ Z4 @ X5 )
% 8.55/4.77 => ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2083_pinf_I7_J,axiom,
% 8.55/4.77 ! [T: real] :
% 8.55/4.77 ? [Z4: real] :
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ( ( ord_less_real @ Z4 @ X5 )
% 8.55/4.77 => ( ord_less_real @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2084_pinf_I7_J,axiom,
% 8.55/4.77 ! [T: rat] :
% 8.55/4.77 ? [Z4: rat] :
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ( ( ord_less_rat @ Z4 @ X5 )
% 8.55/4.77 => ( ord_less_rat @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2085_pinf_I7_J,axiom,
% 8.55/4.77 ! [T: num] :
% 8.55/4.77 ? [Z4: num] :
% 8.55/4.77 ! [X5: num] :
% 8.55/4.77 ( ( ord_less_num @ Z4 @ X5 )
% 8.55/4.77 => ( ord_less_num @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2086_pinf_I7_J,axiom,
% 8.55/4.77 ! [T: nat] :
% 8.55/4.77 ? [Z4: nat] :
% 8.55/4.77 ! [X5: nat] :
% 8.55/4.77 ( ( ord_less_nat @ Z4 @ X5 )
% 8.55/4.77 => ( ord_less_nat @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2087_pinf_I7_J,axiom,
% 8.55/4.77 ! [T: int] :
% 8.55/4.77 ? [Z4: int] :
% 8.55/4.77 ! [X5: int] :
% 8.55/4.77 ( ( ord_less_int @ Z4 @ X5 )
% 8.55/4.77 => ( ord_less_int @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2088_pinf_I8_J,axiom,
% 8.55/4.77 ! [T: real] :
% 8.55/4.77 ? [Z4: real] :
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ( ( ord_less_real @ Z4 @ X5 )
% 8.55/4.77 => ( ord_less_eq_real @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2089_pinf_I8_J,axiom,
% 8.55/4.77 ! [T: rat] :
% 8.55/4.77 ? [Z4: rat] :
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ( ( ord_less_rat @ Z4 @ X5 )
% 8.55/4.77 => ( ord_less_eq_rat @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2090_pinf_I8_J,axiom,
% 8.55/4.77 ! [T: num] :
% 8.55/4.77 ? [Z4: num] :
% 8.55/4.77 ! [X5: num] :
% 8.55/4.77 ( ( ord_less_num @ Z4 @ X5 )
% 8.55/4.77 => ( ord_less_eq_num @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2091_pinf_I8_J,axiom,
% 8.55/4.77 ! [T: nat] :
% 8.55/4.77 ? [Z4: nat] :
% 8.55/4.77 ! [X5: nat] :
% 8.55/4.77 ( ( ord_less_nat @ Z4 @ X5 )
% 8.55/4.77 => ( ord_less_eq_nat @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2092_pinf_I8_J,axiom,
% 8.55/4.77 ! [T: int] :
% 8.55/4.77 ? [Z4: int] :
% 8.55/4.77 ! [X5: int] :
% 8.55/4.77 ( ( ord_less_int @ Z4 @ X5 )
% 8.55/4.77 => ( ord_less_eq_int @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2093_minf_I1_J,axiom,
% 8.55/4.77 ! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
% 8.55/4.77 ( ? [Z3: real] :
% 8.55/4.77 ! [X4: real] :
% 8.55/4.77 ( ( ord_less_real @ X4 @ Z3 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: real] :
% 8.55/4.77 ! [X4: real] :
% 8.55/4.77 ( ( ord_less_real @ X4 @ Z3 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: real] :
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ( ( ord_less_real @ X5 @ Z4 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 & ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2094_minf_I1_J,axiom,
% 8.55/4.77 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
% 8.55/4.77 ( ? [Z3: rat] :
% 8.55/4.77 ! [X4: rat] :
% 8.55/4.77 ( ( ord_less_rat @ X4 @ Z3 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: rat] :
% 8.55/4.77 ! [X4: rat] :
% 8.55/4.77 ( ( ord_less_rat @ X4 @ Z3 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: rat] :
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ( ( ord_less_rat @ X5 @ Z4 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 & ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2095_minf_I1_J,axiom,
% 8.55/4.77 ! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
% 8.55/4.77 ( ? [Z3: num] :
% 8.55/4.77 ! [X4: num] :
% 8.55/4.77 ( ( ord_less_num @ X4 @ Z3 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: num] :
% 8.55/4.77 ! [X4: num] :
% 8.55/4.77 ( ( ord_less_num @ X4 @ Z3 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: num] :
% 8.55/4.77 ! [X5: num] :
% 8.55/4.77 ( ( ord_less_num @ X5 @ Z4 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 & ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2096_minf_I1_J,axiom,
% 8.55/4.77 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
% 8.55/4.77 ( ? [Z3: nat] :
% 8.55/4.77 ! [X4: nat] :
% 8.55/4.77 ( ( ord_less_nat @ X4 @ Z3 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: nat] :
% 8.55/4.77 ! [X4: nat] :
% 8.55/4.77 ( ( ord_less_nat @ X4 @ Z3 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: nat] :
% 8.55/4.77 ! [X5: nat] :
% 8.55/4.77 ( ( ord_less_nat @ X5 @ Z4 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 & ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2097_minf_I1_J,axiom,
% 8.55/4.77 ! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
% 8.55/4.77 ( ? [Z3: int] :
% 8.55/4.77 ! [X4: int] :
% 8.55/4.77 ( ( ord_less_int @ X4 @ Z3 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: int] :
% 8.55/4.77 ! [X4: int] :
% 8.55/4.77 ( ( ord_less_int @ X4 @ Z3 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: int] :
% 8.55/4.77 ! [X5: int] :
% 8.55/4.77 ( ( ord_less_int @ X5 @ Z4 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 & ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 & ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2098_minf_I2_J,axiom,
% 8.55/4.77 ! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
% 8.55/4.77 ( ? [Z3: real] :
% 8.55/4.77 ! [X4: real] :
% 8.55/4.77 ( ( ord_less_real @ X4 @ Z3 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: real] :
% 8.55/4.77 ! [X4: real] :
% 8.55/4.77 ( ( ord_less_real @ X4 @ Z3 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: real] :
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ( ( ord_less_real @ X5 @ Z4 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 | ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2099_minf_I2_J,axiom,
% 8.55/4.77 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
% 8.55/4.77 ( ? [Z3: rat] :
% 8.55/4.77 ! [X4: rat] :
% 8.55/4.77 ( ( ord_less_rat @ X4 @ Z3 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: rat] :
% 8.55/4.77 ! [X4: rat] :
% 8.55/4.77 ( ( ord_less_rat @ X4 @ Z3 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: rat] :
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ( ( ord_less_rat @ X5 @ Z4 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 | ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2100_minf_I2_J,axiom,
% 8.55/4.77 ! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
% 8.55/4.77 ( ? [Z3: num] :
% 8.55/4.77 ! [X4: num] :
% 8.55/4.77 ( ( ord_less_num @ X4 @ Z3 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: num] :
% 8.55/4.77 ! [X4: num] :
% 8.55/4.77 ( ( ord_less_num @ X4 @ Z3 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: num] :
% 8.55/4.77 ! [X5: num] :
% 8.55/4.77 ( ( ord_less_num @ X5 @ Z4 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 | ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2101_minf_I2_J,axiom,
% 8.55/4.77 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
% 8.55/4.77 ( ? [Z3: nat] :
% 8.55/4.77 ! [X4: nat] :
% 8.55/4.77 ( ( ord_less_nat @ X4 @ Z3 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: nat] :
% 8.55/4.77 ! [X4: nat] :
% 8.55/4.77 ( ( ord_less_nat @ X4 @ Z3 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: nat] :
% 8.55/4.77 ! [X5: nat] :
% 8.55/4.77 ( ( ord_less_nat @ X5 @ Z4 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 | ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2102_minf_I2_J,axiom,
% 8.55/4.77 ! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
% 8.55/4.77 ( ? [Z3: int] :
% 8.55/4.77 ! [X4: int] :
% 8.55/4.77 ( ( ord_less_int @ X4 @ Z3 )
% 8.55/4.77 => ( ( P @ X4 )
% 8.55/4.77 = ( P4 @ X4 ) ) )
% 8.55/4.77 => ( ? [Z3: int] :
% 8.55/4.77 ! [X4: int] :
% 8.55/4.77 ( ( ord_less_int @ X4 @ Z3 )
% 8.55/4.77 => ( ( Q @ X4 )
% 8.55/4.77 = ( Q3 @ X4 ) ) )
% 8.55/4.77 => ? [Z4: int] :
% 8.55/4.77 ! [X5: int] :
% 8.55/4.77 ( ( ord_less_int @ X5 @ Z4 )
% 8.55/4.77 => ( ( ( P @ X5 )
% 8.55/4.77 | ( Q @ X5 ) )
% 8.55/4.77 = ( ( P4 @ X5 )
% 8.55/4.77 | ( Q3 @ X5 ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2103_minf_I3_J,axiom,
% 8.55/4.77 ! [T: real] :
% 8.55/4.77 ? [Z4: real] :
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ( ( ord_less_real @ X5 @ Z4 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2104_minf_I3_J,axiom,
% 8.55/4.77 ! [T: rat] :
% 8.55/4.77 ? [Z4: rat] :
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ( ( ord_less_rat @ X5 @ Z4 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2105_minf_I3_J,axiom,
% 8.55/4.77 ! [T: num] :
% 8.55/4.77 ? [Z4: num] :
% 8.55/4.77 ! [X5: num] :
% 8.55/4.77 ( ( ord_less_num @ X5 @ Z4 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2106_minf_I3_J,axiom,
% 8.55/4.77 ! [T: nat] :
% 8.55/4.77 ? [Z4: nat] :
% 8.55/4.77 ! [X5: nat] :
% 8.55/4.77 ( ( ord_less_nat @ X5 @ Z4 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2107_minf_I3_J,axiom,
% 8.55/4.77 ! [T: int] :
% 8.55/4.77 ? [Z4: int] :
% 8.55/4.77 ! [X5: int] :
% 8.55/4.77 ( ( ord_less_int @ X5 @ Z4 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2108_minf_I4_J,axiom,
% 8.55/4.77 ! [T: real] :
% 8.55/4.77 ? [Z4: real] :
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ( ( ord_less_real @ X5 @ Z4 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2109_minf_I4_J,axiom,
% 8.55/4.77 ! [T: rat] :
% 8.55/4.77 ? [Z4: rat] :
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ( ( ord_less_rat @ X5 @ Z4 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2110_minf_I4_J,axiom,
% 8.55/4.77 ! [T: num] :
% 8.55/4.77 ? [Z4: num] :
% 8.55/4.77 ! [X5: num] :
% 8.55/4.77 ( ( ord_less_num @ X5 @ Z4 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2111_minf_I4_J,axiom,
% 8.55/4.77 ! [T: nat] :
% 8.55/4.77 ? [Z4: nat] :
% 8.55/4.77 ! [X5: nat] :
% 8.55/4.77 ( ( ord_less_nat @ X5 @ Z4 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2112_minf_I4_J,axiom,
% 8.55/4.77 ! [T: int] :
% 8.55/4.77 ? [Z4: int] :
% 8.55/4.77 ! [X5: int] :
% 8.55/4.77 ( ( ord_less_int @ X5 @ Z4 )
% 8.55/4.77 => ( X5 != T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2113_minf_I5_J,axiom,
% 8.55/4.77 ! [T: real] :
% 8.55/4.77 ? [Z4: real] :
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ( ( ord_less_real @ X5 @ Z4 )
% 8.55/4.77 => ( ord_less_real @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2114_minf_I5_J,axiom,
% 8.55/4.77 ! [T: rat] :
% 8.55/4.77 ? [Z4: rat] :
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ( ( ord_less_rat @ X5 @ Z4 )
% 8.55/4.77 => ( ord_less_rat @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2115_minf_I5_J,axiom,
% 8.55/4.77 ! [T: num] :
% 8.55/4.77 ? [Z4: num] :
% 8.55/4.77 ! [X5: num] :
% 8.55/4.77 ( ( ord_less_num @ X5 @ Z4 )
% 8.55/4.77 => ( ord_less_num @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2116_minf_I5_J,axiom,
% 8.55/4.77 ! [T: nat] :
% 8.55/4.77 ? [Z4: nat] :
% 8.55/4.77 ! [X5: nat] :
% 8.55/4.77 ( ( ord_less_nat @ X5 @ Z4 )
% 8.55/4.77 => ( ord_less_nat @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2117_minf_I5_J,axiom,
% 8.55/4.77 ! [T: int] :
% 8.55/4.77 ? [Z4: int] :
% 8.55/4.77 ! [X5: int] :
% 8.55/4.77 ( ( ord_less_int @ X5 @ Z4 )
% 8.55/4.77 => ( ord_less_int @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2118_minf_I6_J,axiom,
% 8.55/4.77 ! [T: real] :
% 8.55/4.77 ? [Z4: real] :
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ( ( ord_less_real @ X5 @ Z4 )
% 8.55/4.77 => ( ord_less_eq_real @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2119_minf_I6_J,axiom,
% 8.55/4.77 ! [T: rat] :
% 8.55/4.77 ? [Z4: rat] :
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ( ( ord_less_rat @ X5 @ Z4 )
% 8.55/4.77 => ( ord_less_eq_rat @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2120_minf_I6_J,axiom,
% 8.55/4.77 ! [T: num] :
% 8.55/4.77 ? [Z4: num] :
% 8.55/4.77 ! [X5: num] :
% 8.55/4.77 ( ( ord_less_num @ X5 @ Z4 )
% 8.55/4.77 => ( ord_less_eq_num @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2121_minf_I6_J,axiom,
% 8.55/4.77 ! [T: nat] :
% 8.55/4.77 ? [Z4: nat] :
% 8.55/4.77 ! [X5: nat] :
% 8.55/4.77 ( ( ord_less_nat @ X5 @ Z4 )
% 8.55/4.77 => ( ord_less_eq_nat @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2122_minf_I6_J,axiom,
% 8.55/4.77 ! [T: int] :
% 8.55/4.77 ? [Z4: int] :
% 8.55/4.77 ! [X5: int] :
% 8.55/4.77 ( ( ord_less_int @ X5 @ Z4 )
% 8.55/4.77 => ( ord_less_eq_int @ X5 @ T ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2123_minf_I7_J,axiom,
% 8.55/4.77 ! [T: real] :
% 8.55/4.77 ? [Z4: real] :
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ( ( ord_less_real @ X5 @ Z4 )
% 8.55/4.77 => ~ ( ord_less_real @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2124_minf_I7_J,axiom,
% 8.55/4.77 ! [T: rat] :
% 8.55/4.77 ? [Z4: rat] :
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ( ( ord_less_rat @ X5 @ Z4 )
% 8.55/4.77 => ~ ( ord_less_rat @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2125_minf_I7_J,axiom,
% 8.55/4.77 ! [T: num] :
% 8.55/4.77 ? [Z4: num] :
% 8.55/4.77 ! [X5: num] :
% 8.55/4.77 ( ( ord_less_num @ X5 @ Z4 )
% 8.55/4.77 => ~ ( ord_less_num @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2126_minf_I7_J,axiom,
% 8.55/4.77 ! [T: nat] :
% 8.55/4.77 ? [Z4: nat] :
% 8.55/4.77 ! [X5: nat] :
% 8.55/4.77 ( ( ord_less_nat @ X5 @ Z4 )
% 8.55/4.77 => ~ ( ord_less_nat @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2127_minf_I7_J,axiom,
% 8.55/4.77 ! [T: int] :
% 8.55/4.77 ? [Z4: int] :
% 8.55/4.77 ! [X5: int] :
% 8.55/4.77 ( ( ord_less_int @ X5 @ Z4 )
% 8.55/4.77 => ~ ( ord_less_int @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2128_minf_I8_J,axiom,
% 8.55/4.77 ! [T: real] :
% 8.55/4.77 ? [Z4: real] :
% 8.55/4.77 ! [X5: real] :
% 8.55/4.77 ( ( ord_less_real @ X5 @ Z4 )
% 8.55/4.77 => ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2129_minf_I8_J,axiom,
% 8.55/4.77 ! [T: rat] :
% 8.55/4.77 ? [Z4: rat] :
% 8.55/4.77 ! [X5: rat] :
% 8.55/4.77 ( ( ord_less_rat @ X5 @ Z4 )
% 8.55/4.77 => ~ ( ord_less_eq_rat @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2130_minf_I8_J,axiom,
% 8.55/4.77 ! [T: num] :
% 8.55/4.77 ? [Z4: num] :
% 8.55/4.77 ! [X5: num] :
% 8.55/4.77 ( ( ord_less_num @ X5 @ Z4 )
% 8.55/4.77 => ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2131_minf_I8_J,axiom,
% 8.55/4.77 ! [T: nat] :
% 8.55/4.77 ? [Z4: nat] :
% 8.55/4.77 ! [X5: nat] :
% 8.55/4.77 ( ( ord_less_nat @ X5 @ Z4 )
% 8.55/4.77 => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2132_minf_I8_J,axiom,
% 8.55/4.77 ! [T: int] :
% 8.55/4.77 ? [Z4: int] :
% 8.55/4.77 ! [X5: int] :
% 8.55/4.77 ( ( ord_less_int @ X5 @ Z4 )
% 8.55/4.77 => ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2133_add__less__le__mono,axiom,
% 8.55/4.77 ! [A: real,B: real,C: real,D: real] :
% 8.55/4.77 ( ( ord_less_real @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_real @ C @ D )
% 8.55/4.77 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2134_add__less__le__mono,axiom,
% 8.55/4.77 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.55/4.77 ( ( ord_less_rat @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_rat @ C @ D )
% 8.55/4.77 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2135_add__less__le__mono,axiom,
% 8.55/4.77 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.55/4.77 ( ( ord_less_nat @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_nat @ C @ D )
% 8.55/4.77 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2136_add__less__le__mono,axiom,
% 8.55/4.77 ! [A: int,B: int,C: int,D: int] :
% 8.55/4.77 ( ( ord_less_int @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_int @ C @ D )
% 8.55/4.77 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2137_add__le__less__mono,axiom,
% 8.55/4.77 ! [A: real,B: real,C: real,D: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ A @ B )
% 8.55/4.77 => ( ( ord_less_real @ C @ D )
% 8.55/4.77 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2138_add__le__less__mono,axiom,
% 8.55/4.77 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.77 => ( ( ord_less_rat @ C @ D )
% 8.55/4.77 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2139_add__le__less__mono,axiom,
% 8.55/4.77 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.77 => ( ( ord_less_nat @ C @ D )
% 8.55/4.77 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2140_add__le__less__mono,axiom,
% 8.55/4.77 ! [A: int,B: int,C: int,D: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ A @ B )
% 8.55/4.77 => ( ( ord_less_int @ C @ D )
% 8.55/4.77 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2141_add__mono__thms__linordered__field_I3_J,axiom,
% 8.55/4.77 ! [I: real,J: real,K: real,L: real] :
% 8.55/4.77 ( ( ( ord_less_real @ I @ J )
% 8.55/4.77 & ( ord_less_eq_real @ K @ L ) )
% 8.55/4.77 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2142_add__mono__thms__linordered__field_I3_J,axiom,
% 8.55/4.77 ! [I: rat,J: rat,K: rat,L: rat] :
% 8.55/4.77 ( ( ( ord_less_rat @ I @ J )
% 8.55/4.77 & ( ord_less_eq_rat @ K @ L ) )
% 8.55/4.77 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2143_add__mono__thms__linordered__field_I3_J,axiom,
% 8.55/4.77 ! [I: nat,J: nat,K: nat,L: nat] :
% 8.55/4.77 ( ( ( ord_less_nat @ I @ J )
% 8.55/4.77 & ( ord_less_eq_nat @ K @ L ) )
% 8.55/4.77 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2144_add__mono__thms__linordered__field_I3_J,axiom,
% 8.55/4.77 ! [I: int,J: int,K: int,L: int] :
% 8.55/4.77 ( ( ( ord_less_int @ I @ J )
% 8.55/4.77 & ( ord_less_eq_int @ K @ L ) )
% 8.55/4.77 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2145_add__mono__thms__linordered__field_I4_J,axiom,
% 8.55/4.77 ! [I: real,J: real,K: real,L: real] :
% 8.55/4.77 ( ( ( ord_less_eq_real @ I @ J )
% 8.55/4.77 & ( ord_less_real @ K @ L ) )
% 8.55/4.77 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2146_add__mono__thms__linordered__field_I4_J,axiom,
% 8.55/4.77 ! [I: rat,J: rat,K: rat,L: rat] :
% 8.55/4.77 ( ( ( ord_less_eq_rat @ I @ J )
% 8.55/4.77 & ( ord_less_rat @ K @ L ) )
% 8.55/4.77 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2147_add__mono__thms__linordered__field_I4_J,axiom,
% 8.55/4.77 ! [I: nat,J: nat,K: nat,L: nat] :
% 8.55/4.77 ( ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.77 & ( ord_less_nat @ K @ L ) )
% 8.55/4.77 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2148_add__mono__thms__linordered__field_I4_J,axiom,
% 8.55/4.77 ! [I: int,J: int,K: int,L: int] :
% 8.55/4.77 ( ( ( ord_less_eq_int @ I @ J )
% 8.55/4.77 & ( ord_less_int @ K @ L ) )
% 8.55/4.77 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2149_lift__Suc__mono__less,axiom,
% 8.55/4.77 ! [F: nat > real,N: nat,N4: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.55/4.77 => ( ( ord_less_nat @ N @ N4 )
% 8.55/4.77 => ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2150_lift__Suc__mono__less,axiom,
% 8.55/4.77 ! [F: nat > rat,N: nat,N4: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.55/4.77 => ( ( ord_less_nat @ N @ N4 )
% 8.55/4.77 => ( ord_less_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2151_lift__Suc__mono__less,axiom,
% 8.55/4.77 ! [F: nat > num,N: nat,N4: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.55/4.77 => ( ( ord_less_nat @ N @ N4 )
% 8.55/4.77 => ( ord_less_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2152_lift__Suc__mono__less,axiom,
% 8.55/4.77 ! [F: nat > nat,N: nat,N4: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.55/4.77 => ( ( ord_less_nat @ N @ N4 )
% 8.55/4.77 => ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2153_lift__Suc__mono__less,axiom,
% 8.55/4.77 ! [F: nat > int,N: nat,N4: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.55/4.77 => ( ( ord_less_nat @ N @ N4 )
% 8.55/4.77 => ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2154_lift__Suc__mono__less__iff,axiom,
% 8.55/4.77 ! [F: nat > real,N: nat,M: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.55/4.77 => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 8.55/4.77 = ( ord_less_nat @ N @ M ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2155_lift__Suc__mono__less__iff,axiom,
% 8.55/4.77 ! [F: nat > rat,N: nat,M: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.55/4.77 => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 8.55/4.77 = ( ord_less_nat @ N @ M ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2156_lift__Suc__mono__less__iff,axiom,
% 8.55/4.77 ! [F: nat > num,N: nat,M: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.55/4.77 => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 8.55/4.77 = ( ord_less_nat @ N @ M ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2157_lift__Suc__mono__less__iff,axiom,
% 8.55/4.77 ! [F: nat > nat,N: nat,M: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.55/4.77 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 8.55/4.77 = ( ord_less_nat @ N @ M ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2158_lift__Suc__mono__less__iff,axiom,
% 8.55/4.77 ! [F: nat > int,N: nat,M: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.55/4.77 => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 8.55/4.77 = ( ord_less_nat @ N @ M ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2159_lift__Suc__mono__le,axiom,
% 8.55/4.77 ! [F: nat > set_nat,N: nat,N4: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.55/4.77 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.55/4.77 => ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2160_lift__Suc__mono__le,axiom,
% 8.55/4.77 ! [F: nat > rat,N: nat,N4: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.55/4.77 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.55/4.77 => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2161_lift__Suc__mono__le,axiom,
% 8.55/4.77 ! [F: nat > num,N: nat,N4: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.55/4.77 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.55/4.77 => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2162_lift__Suc__mono__le,axiom,
% 8.55/4.77 ! [F: nat > nat,N: nat,N4: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.55/4.77 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.55/4.77 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2163_lift__Suc__mono__le,axiom,
% 8.55/4.77 ! [F: nat > int,N: nat,N4: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.55/4.77 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.55/4.77 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2164_lift__Suc__antimono__le,axiom,
% 8.55/4.77 ! [F: nat > set_nat,N: nat,N4: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 8.55/4.77 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.55/4.77 => ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2165_lift__Suc__antimono__le,axiom,
% 8.55/4.77 ! [F: nat > rat,N: nat,N4: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 8.55/4.77 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.55/4.77 => ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2166_lift__Suc__antimono__le,axiom,
% 8.55/4.77 ! [F: nat > num,N: nat,N4: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 8.55/4.77 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.55/4.77 => ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2167_lift__Suc__antimono__le,axiom,
% 8.55/4.77 ! [F: nat > nat,N: nat,N4: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 8.55/4.77 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.55/4.77 => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2168_lift__Suc__antimono__le,axiom,
% 8.55/4.77 ! [F: nat > int,N: nat,N4: nat] :
% 8.55/4.77 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 8.55/4.77 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.55/4.77 => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2169_Suc__leI,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_nat @ M @ N )
% 8.55/4.77 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2170_Suc__le__eq,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 8.55/4.77 = ( ord_less_nat @ M @ N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2171_dec__induct,axiom,
% 8.55/4.77 ! [I: nat,J: nat,P: nat > $o] :
% 8.55/4.77 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.77 => ( ( P @ I )
% 8.55/4.77 => ( ! [N2: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ I @ N2 )
% 8.55/4.77 => ( ( ord_less_nat @ N2 @ J )
% 8.55/4.77 => ( ( P @ N2 )
% 8.55/4.77 => ( P @ ( suc @ N2 ) ) ) ) )
% 8.55/4.77 => ( P @ J ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2172_inc__induct,axiom,
% 8.55/4.77 ! [I: nat,J: nat,P: nat > $o] :
% 8.55/4.77 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.77 => ( ( P @ J )
% 8.55/4.77 => ( ! [N2: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ I @ N2 )
% 8.55/4.77 => ( ( ord_less_nat @ N2 @ J )
% 8.55/4.77 => ( ( P @ ( suc @ N2 ) )
% 8.55/4.77 => ( P @ N2 ) ) ) )
% 8.55/4.77 => ( P @ I ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2173_Suc__le__lessD,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 8.55/4.77 => ( ord_less_nat @ M @ N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2174_le__less__Suc__eq,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.77 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 8.55/4.77 = ( N = M ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2175_less__Suc__eq__le,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 8.55/4.77 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2176_less__eq__Suc__le,axiom,
% 8.55/4.77 ( ord_less_nat
% 8.55/4.77 = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2177_le__imp__less__Suc,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.77 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2178_ex__least__nat__le,axiom,
% 8.55/4.77 ! [P: nat > $o,N: nat] :
% 8.55/4.77 ( ( P @ N )
% 8.55/4.77 => ( ~ ( P @ zero_zero_nat )
% 8.55/4.77 => ? [K3: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ K3 @ N )
% 8.55/4.77 & ! [I3: nat] :
% 8.55/4.77 ( ( ord_less_nat @ I3 @ K3 )
% 8.55/4.77 => ~ ( P @ I3 ) )
% 8.55/4.77 & ( P @ K3 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2179_of__nat__less__imp__less,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 8.55/4.77 => ( ord_less_nat @ M @ N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2180_of__nat__less__imp__less,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 8.55/4.77 => ( ord_less_nat @ M @ N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2181_of__nat__less__imp__less,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 8.55/4.77 => ( ord_less_nat @ M @ N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2182_of__nat__less__imp__less,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 8.55/4.77 => ( ord_less_nat @ M @ N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2183_less__imp__of__nat__less,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_nat @ M @ N )
% 8.55/4.77 => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2184_less__imp__of__nat__less,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_nat @ M @ N )
% 8.55/4.77 => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2185_less__imp__of__nat__less,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_nat @ M @ N )
% 8.55/4.77 => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2186_less__imp__of__nat__less,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_nat @ M @ N )
% 8.55/4.77 => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2187_of__nat__mono,axiom,
% 8.55/4.77 ! [I: nat,J: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.77 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2188_of__nat__mono,axiom,
% 8.55/4.77 ! [I: nat,J: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.77 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2189_of__nat__mono,axiom,
% 8.55/4.77 ! [I: nat,J: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.77 => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2190_of__nat__mono,axiom,
% 8.55/4.77 ! [I: nat,J: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.77 => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2191_mono__nat__linear__lb,axiom,
% 8.55/4.77 ! [F: nat > nat,M: nat,K: nat] :
% 8.55/4.77 ( ! [M2: nat,N2: nat] :
% 8.55/4.77 ( ( ord_less_nat @ M2 @ N2 )
% 8.55/4.77 => ( ord_less_nat @ ( F @ M2 ) @ ( F @ N2 ) ) )
% 8.55/4.77 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2192_diff__less__mono,axiom,
% 8.55/4.77 ! [A: nat,B: nat,C: nat] :
% 8.55/4.77 ( ( ord_less_nat @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_nat @ C @ A )
% 8.55/4.77 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2193_less__diff__iff,axiom,
% 8.55/4.77 ! [K: nat,M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ K @ M )
% 8.55/4.77 => ( ( ord_less_eq_nat @ K @ N )
% 8.55/4.77 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 8.55/4.77 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2194_nat__int__comparison_I2_J,axiom,
% 8.55/4.77 ( ord_less_nat
% 8.55/4.77 = ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2195_zle__int,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 8.55/4.77 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2196_nat__int__comparison_I3_J,axiom,
% 8.55/4.77 ( ord_less_eq_nat
% 8.55/4.77 = ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2197_power__le__imp__le__exp,axiom,
% 8.55/4.77 ! [A: real,M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_real @ one_one_real @ A )
% 8.55/4.77 => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 8.55/4.77 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2198_power__le__imp__le__exp,axiom,
% 8.55/4.77 ! [A: rat,M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_rat @ one_one_rat @ A )
% 8.55/4.77 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 8.55/4.77 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2199_power__le__imp__le__exp,axiom,
% 8.55/4.77 ! [A: nat,M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_nat @ one_one_nat @ A )
% 8.55/4.77 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 8.55/4.77 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2200_power__le__imp__le__exp,axiom,
% 8.55/4.77 ! [A: int,M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_int @ one_one_int @ A )
% 8.55/4.77 => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 8.55/4.77 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2201_subset__decode__imp__le,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 8.55/4.77 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2202_power__strict__mono,axiom,
% 8.55/4.77 ! [A: real,B: real,N: nat] :
% 8.55/4.77 ( ( ord_less_real @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.77 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.77 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2203_power__strict__mono,axiom,
% 8.55/4.77 ! [A: rat,B: rat,N: nat] :
% 8.55/4.77 ( ( ord_less_rat @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.77 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.77 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2204_power__strict__mono,axiom,
% 8.55/4.77 ! [A: nat,B: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_nat @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.77 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.77 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2205_power__strict__mono,axiom,
% 8.55/4.77 ! [A: int,B: int,N: nat] :
% 8.55/4.77 ( ( ord_less_int @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.77 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.77 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2206_mult__le__cancel__left,axiom,
% 8.55/4.77 ! [C: real,A: real,B: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.55/4.77 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.77 => ( ord_less_eq_real @ A @ B ) )
% 8.55/4.77 & ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.77 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2207_mult__le__cancel__left,axiom,
% 8.55/4.77 ! [C: rat,A: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.55/4.77 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.77 => ( ord_less_eq_rat @ A @ B ) )
% 8.55/4.77 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.77 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2208_mult__le__cancel__left,axiom,
% 8.55/4.77 ! [C: int,A: int,B: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.55/4.77 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 8.55/4.77 => ( ord_less_eq_int @ A @ B ) )
% 8.55/4.77 & ( ( ord_less_int @ C @ zero_zero_int )
% 8.55/4.77 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2209_mult__le__cancel__right,axiom,
% 8.55/4.77 ! [A: real,C: real,B: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 8.55/4.77 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.77 => ( ord_less_eq_real @ A @ B ) )
% 8.55/4.77 & ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.77 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2210_mult__le__cancel__right,axiom,
% 8.55/4.77 ! [A: rat,C: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 8.55/4.77 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.77 => ( ord_less_eq_rat @ A @ B ) )
% 8.55/4.77 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.77 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2211_mult__le__cancel__right,axiom,
% 8.55/4.77 ! [A: int,C: int,B: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 8.55/4.77 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 8.55/4.77 => ( ord_less_eq_int @ A @ B ) )
% 8.55/4.77 & ( ( ord_less_int @ C @ zero_zero_int )
% 8.55/4.77 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2212_mult__left__less__imp__less,axiom,
% 8.55/4.77 ! [C: real,A: real,B: real] :
% 8.55/4.77 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.55/4.77 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.77 => ( ord_less_real @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2213_mult__left__less__imp__less,axiom,
% 8.55/4.77 ! [C: rat,A: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.55/4.77 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.77 => ( ord_less_rat @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2214_mult__left__less__imp__less,axiom,
% 8.55/4.77 ! [C: nat,A: nat,B: nat] :
% 8.55/4.77 ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 8.55/4.77 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.55/4.77 => ( ord_less_nat @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2215_mult__left__less__imp__less,axiom,
% 8.55/4.77 ! [C: int,A: int,B: int] :
% 8.55/4.77 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.55/4.77 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.77 => ( ord_less_int @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2216_mult__strict__mono,axiom,
% 8.55/4.77 ! [A: real,B: real,C: real,D: real] :
% 8.55/4.77 ( ( ord_less_real @ A @ B )
% 8.55/4.77 => ( ( ord_less_real @ C @ D )
% 8.55/4.77 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.55/4.77 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.77 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2217_mult__strict__mono,axiom,
% 8.55/4.77 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.55/4.77 ( ( ord_less_rat @ A @ B )
% 8.55/4.77 => ( ( ord_less_rat @ C @ D )
% 8.55/4.77 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 8.55/4.77 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.77 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2218_mult__strict__mono,axiom,
% 8.55/4.77 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.55/4.77 ( ( ord_less_nat @ A @ B )
% 8.55/4.77 => ( ( ord_less_nat @ C @ D )
% 8.55/4.77 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.55/4.77 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.55/4.77 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2219_mult__strict__mono,axiom,
% 8.55/4.77 ! [A: int,B: int,C: int,D: int] :
% 8.55/4.77 ( ( ord_less_int @ A @ B )
% 8.55/4.77 => ( ( ord_less_int @ C @ D )
% 8.55/4.77 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.55/4.77 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.77 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2220_mult__less__cancel__left,axiom,
% 8.55/4.77 ! [C: real,A: real,B: real] :
% 8.55/4.77 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.55/4.77 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.77 => ( ord_less_real @ A @ B ) )
% 8.55/4.77 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.55/4.77 => ( ord_less_real @ B @ A ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2221_mult__less__cancel__left,axiom,
% 8.55/4.77 ! [C: rat,A: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.55/4.77 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.77 => ( ord_less_rat @ A @ B ) )
% 8.55/4.77 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.55/4.77 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2222_mult__less__cancel__left,axiom,
% 8.55/4.77 ! [C: int,A: int,B: int] :
% 8.55/4.77 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.55/4.77 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.77 => ( ord_less_int @ A @ B ) )
% 8.55/4.77 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.55/4.77 => ( ord_less_int @ B @ A ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2223_mult__right__less__imp__less,axiom,
% 8.55/4.77 ! [A: real,C: real,B: real] :
% 8.55/4.77 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 8.55/4.77 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.77 => ( ord_less_real @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2224_mult__right__less__imp__less,axiom,
% 8.55/4.77 ! [A: rat,C: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 8.55/4.77 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.77 => ( ord_less_rat @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2225_mult__right__less__imp__less,axiom,
% 8.55/4.77 ! [A: nat,C: nat,B: nat] :
% 8.55/4.77 ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 8.55/4.77 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.55/4.77 => ( ord_less_nat @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2226_mult__right__less__imp__less,axiom,
% 8.55/4.77 ! [A: int,C: int,B: int] :
% 8.55/4.77 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 8.55/4.77 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.77 => ( ord_less_int @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2227_mult__strict__mono_H,axiom,
% 8.55/4.77 ! [A: real,B: real,C: real,D: real] :
% 8.55/4.77 ( ( ord_less_real @ A @ B )
% 8.55/4.77 => ( ( ord_less_real @ C @ D )
% 8.55/4.77 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.77 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.77 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2228_mult__strict__mono_H,axiom,
% 8.55/4.77 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.55/4.77 ( ( ord_less_rat @ A @ B )
% 8.55/4.77 => ( ( ord_less_rat @ C @ D )
% 8.55/4.77 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.77 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.77 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2229_mult__strict__mono_H,axiom,
% 8.55/4.77 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.55/4.77 ( ( ord_less_nat @ A @ B )
% 8.55/4.77 => ( ( ord_less_nat @ C @ D )
% 8.55/4.77 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.77 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.55/4.77 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2230_mult__strict__mono_H,axiom,
% 8.55/4.77 ! [A: int,B: int,C: int,D: int] :
% 8.55/4.77 ( ( ord_less_int @ A @ B )
% 8.55/4.77 => ( ( ord_less_int @ C @ D )
% 8.55/4.77 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.77 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.77 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2231_mult__less__cancel__right,axiom,
% 8.55/4.77 ! [A: real,C: real,B: real] :
% 8.55/4.77 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 8.55/4.77 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.77 => ( ord_less_real @ A @ B ) )
% 8.55/4.77 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.55/4.77 => ( ord_less_real @ B @ A ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2232_mult__less__cancel__right,axiom,
% 8.55/4.77 ! [A: rat,C: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 8.55/4.77 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.77 => ( ord_less_rat @ A @ B ) )
% 8.55/4.77 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.55/4.77 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2233_mult__less__cancel__right,axiom,
% 8.55/4.77 ! [A: int,C: int,B: int] :
% 8.55/4.77 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 8.55/4.77 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.77 => ( ord_less_int @ A @ B ) )
% 8.55/4.77 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.55/4.77 => ( ord_less_int @ B @ A ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2234_mult__le__cancel__left__neg,axiom,
% 8.55/4.77 ! [C: real,A: real,B: real] :
% 8.55/4.77 ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.77 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.55/4.77 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2235_mult__le__cancel__left__neg,axiom,
% 8.55/4.77 ! [C: rat,A: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.77 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.55/4.77 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2236_mult__le__cancel__left__neg,axiom,
% 8.55/4.77 ! [C: int,A: int,B: int] :
% 8.55/4.77 ( ( ord_less_int @ C @ zero_zero_int )
% 8.55/4.77 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.55/4.77 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2237_mult__le__cancel__left__pos,axiom,
% 8.55/4.77 ! [C: real,A: real,B: real] :
% 8.55/4.77 ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.77 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.55/4.77 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2238_mult__le__cancel__left__pos,axiom,
% 8.55/4.77 ! [C: rat,A: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.77 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.55/4.77 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2239_mult__le__cancel__left__pos,axiom,
% 8.55/4.77 ! [C: int,A: int,B: int] :
% 8.55/4.77 ( ( ord_less_int @ zero_zero_int @ C )
% 8.55/4.77 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.55/4.77 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2240_mult__left__le__imp__le,axiom,
% 8.55/4.77 ! [C: real,A: real,B: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.55/4.77 => ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.77 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2241_mult__left__le__imp__le,axiom,
% 8.55/4.77 ! [C: rat,A: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.55/4.77 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.77 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2242_mult__left__le__imp__le,axiom,
% 8.55/4.77 ! [C: nat,A: nat,B: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 8.55/4.77 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 8.55/4.77 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2243_mult__left__le__imp__le,axiom,
% 8.55/4.77 ! [C: int,A: int,B: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.55/4.77 => ( ( ord_less_int @ zero_zero_int @ C )
% 8.55/4.77 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2244_mult__right__le__imp__le,axiom,
% 8.55/4.77 ! [A: real,C: real,B: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 8.55/4.77 => ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.77 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2245_mult__right__le__imp__le,axiom,
% 8.55/4.77 ! [A: rat,C: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 8.55/4.77 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.77 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2246_mult__right__le__imp__le,axiom,
% 8.55/4.77 ! [A: nat,C: nat,B: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 8.55/4.77 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 8.55/4.77 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2247_mult__right__le__imp__le,axiom,
% 8.55/4.77 ! [A: int,C: int,B: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 8.55/4.77 => ( ( ord_less_int @ zero_zero_int @ C )
% 8.55/4.77 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2248_mult__le__less__imp__less,axiom,
% 8.55/4.77 ! [A: real,B: real,C: real,D: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ A @ B )
% 8.55/4.77 => ( ( ord_less_real @ C @ D )
% 8.55/4.77 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.77 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.77 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2249_mult__le__less__imp__less,axiom,
% 8.55/4.77 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.77 => ( ( ord_less_rat @ C @ D )
% 8.55/4.77 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.77 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.77 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2250_mult__le__less__imp__less,axiom,
% 8.55/4.77 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.77 => ( ( ord_less_nat @ C @ D )
% 8.55/4.77 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.55/4.77 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.55/4.77 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2251_mult__le__less__imp__less,axiom,
% 8.55/4.77 ! [A: int,B: int,C: int,D: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ A @ B )
% 8.55/4.77 => ( ( ord_less_int @ C @ D )
% 8.55/4.77 => ( ( ord_less_int @ zero_zero_int @ A )
% 8.55/4.77 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.77 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2252_mult__less__le__imp__less,axiom,
% 8.55/4.77 ! [A: real,B: real,C: real,D: real] :
% 8.55/4.77 ( ( ord_less_real @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_real @ C @ D )
% 8.55/4.77 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.77 => ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.77 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2253_mult__less__le__imp__less,axiom,
% 8.55/4.77 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.55/4.77 ( ( ord_less_rat @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_rat @ C @ D )
% 8.55/4.77 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.77 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.77 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2254_mult__less__le__imp__less,axiom,
% 8.55/4.77 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.55/4.77 ( ( ord_less_nat @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_nat @ C @ D )
% 8.55/4.77 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.77 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 8.55/4.77 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2255_mult__less__le__imp__less,axiom,
% 8.55/4.77 ! [A: int,B: int,C: int,D: int] :
% 8.55/4.77 ( ( ord_less_int @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_int @ C @ D )
% 8.55/4.77 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.77 => ( ( ord_less_int @ zero_zero_int @ C )
% 8.55/4.77 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2256_add__strict__increasing2,axiom,
% 8.55/4.77 ! [A: real,B: real,C: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.77 => ( ( ord_less_real @ B @ C )
% 8.55/4.77 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2257_add__strict__increasing2,axiom,
% 8.55/4.77 ! [A: rat,B: rat,C: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.77 => ( ( ord_less_rat @ B @ C )
% 8.55/4.77 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2258_add__strict__increasing2,axiom,
% 8.55/4.77 ! [A: nat,B: nat,C: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.77 => ( ( ord_less_nat @ B @ C )
% 8.55/4.77 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2259_add__strict__increasing2,axiom,
% 8.55/4.77 ! [A: int,B: int,C: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.77 => ( ( ord_less_int @ B @ C )
% 8.55/4.77 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2260_add__strict__increasing,axiom,
% 8.55/4.77 ! [A: real,B: real,C: real] :
% 8.55/4.77 ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.77 => ( ( ord_less_eq_real @ B @ C )
% 8.55/4.77 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2261_add__strict__increasing,axiom,
% 8.55/4.77 ! [A: rat,B: rat,C: rat] :
% 8.55/4.77 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.77 => ( ( ord_less_eq_rat @ B @ C )
% 8.55/4.77 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2262_add__strict__increasing,axiom,
% 8.55/4.77 ! [A: nat,B: nat,C: nat] :
% 8.55/4.77 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.55/4.77 => ( ( ord_less_eq_nat @ B @ C )
% 8.55/4.77 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2263_add__strict__increasing,axiom,
% 8.55/4.77 ! [A: int,B: int,C: int] :
% 8.55/4.77 ( ( ord_less_int @ zero_zero_int @ A )
% 8.55/4.77 => ( ( ord_less_eq_int @ B @ C )
% 8.55/4.77 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2264_add__pos__nonneg,axiom,
% 8.55/4.77 ! [A: real,B: real] :
% 8.55/4.77 ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.77 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.55/4.77 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2265_add__pos__nonneg,axiom,
% 8.55/4.77 ! [A: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.77 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.55/4.77 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2266_add__pos__nonneg,axiom,
% 8.55/4.77 ! [A: nat,B: nat] :
% 8.55/4.77 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.55/4.77 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.55/4.77 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2267_add__pos__nonneg,axiom,
% 8.55/4.77 ! [A: int,B: int] :
% 8.55/4.77 ( ( ord_less_int @ zero_zero_int @ A )
% 8.55/4.77 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.55/4.77 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2268_add__nonpos__neg,axiom,
% 8.55/4.77 ! [A: real,B: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.55/4.77 => ( ( ord_less_real @ B @ zero_zero_real )
% 8.55/4.77 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2269_add__nonpos__neg,axiom,
% 8.55/4.77 ! [A: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.55/4.77 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 8.55/4.77 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2270_add__nonpos__neg,axiom,
% 8.55/4.77 ! [A: nat,B: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 8.55/4.77 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 8.55/4.77 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2271_add__nonpos__neg,axiom,
% 8.55/4.77 ! [A: int,B: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.55/4.77 => ( ( ord_less_int @ B @ zero_zero_int )
% 8.55/4.77 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2272_add__nonneg__pos,axiom,
% 8.55/4.77 ! [A: real,B: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.77 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.55/4.77 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2273_add__nonneg__pos,axiom,
% 8.55/4.77 ! [A: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.77 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 8.55/4.77 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2274_add__nonneg__pos,axiom,
% 8.55/4.77 ! [A: nat,B: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.77 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.55/4.77 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2275_add__nonneg__pos,axiom,
% 8.55/4.77 ! [A: int,B: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.77 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.55/4.77 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2276_add__neg__nonpos,axiom,
% 8.55/4.77 ! [A: real,B: real] :
% 8.55/4.77 ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.77 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 8.55/4.77 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2277_add__neg__nonpos,axiom,
% 8.55/4.77 ! [A: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.77 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 8.55/4.77 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2278_add__neg__nonpos,axiom,
% 8.55/4.77 ! [A: nat,B: nat] :
% 8.55/4.77 ( ( ord_less_nat @ A @ zero_zero_nat )
% 8.55/4.77 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 8.55/4.77 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2279_add__neg__nonpos,axiom,
% 8.55/4.77 ! [A: int,B: int] :
% 8.55/4.77 ( ( ord_less_int @ A @ zero_zero_int )
% 8.55/4.77 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 8.55/4.77 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2280_field__le__epsilon,axiom,
% 8.55/4.77 ! [X: real,Y: real] :
% 8.55/4.77 ( ! [E2: real] :
% 8.55/4.77 ( ( ord_less_real @ zero_zero_real @ E2 )
% 8.55/4.77 => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
% 8.55/4.77 => ( ord_less_eq_real @ X @ Y ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2281_field__le__epsilon,axiom,
% 8.55/4.77 ! [X: rat,Y: rat] :
% 8.55/4.77 ( ! [E2: rat] :
% 8.55/4.77 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 8.55/4.77 => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E2 ) ) )
% 8.55/4.77 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2282_divide__nonpos__pos,axiom,
% 8.55/4.77 ! [X: real,Y: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 8.55/4.77 => ( ( ord_less_real @ zero_zero_real @ Y )
% 8.55/4.77 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2283_divide__nonpos__pos,axiom,
% 8.55/4.77 ! [X: rat,Y: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 8.55/4.77 => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 8.55/4.77 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2284_divide__nonpos__neg,axiom,
% 8.55/4.77 ! [X: real,Y: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 8.55/4.77 => ( ( ord_less_real @ Y @ zero_zero_real )
% 8.55/4.77 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2285_divide__nonpos__neg,axiom,
% 8.55/4.77 ! [X: rat,Y: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 8.55/4.77 => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 8.55/4.77 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2286_divide__nonneg__pos,axiom,
% 8.55/4.77 ! [X: real,Y: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.55/4.77 => ( ( ord_less_real @ zero_zero_real @ Y )
% 8.55/4.77 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2287_divide__nonneg__pos,axiom,
% 8.55/4.77 ! [X: rat,Y: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 8.55/4.77 => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 8.55/4.77 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2288_divide__nonneg__neg,axiom,
% 8.55/4.77 ! [X: real,Y: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.55/4.77 => ( ( ord_less_real @ Y @ zero_zero_real )
% 8.55/4.77 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2289_divide__nonneg__neg,axiom,
% 8.55/4.77 ! [X: rat,Y: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 8.55/4.77 => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 8.55/4.77 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2290_divide__le__cancel,axiom,
% 8.55/4.77 ! [A: real,C: real,B: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 8.55/4.77 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.77 => ( ord_less_eq_real @ A @ B ) )
% 8.55/4.77 & ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.77 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2291_divide__le__cancel,axiom,
% 8.55/4.77 ! [A: rat,C: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 8.55/4.77 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.77 => ( ord_less_eq_rat @ A @ B ) )
% 8.55/4.77 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.77 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2292_frac__less2,axiom,
% 8.55/4.77 ! [X: real,Y: real,W: real,Z2: real] :
% 8.55/4.77 ( ( ord_less_real @ zero_zero_real @ X )
% 8.55/4.77 => ( ( ord_less_eq_real @ X @ Y )
% 8.55/4.77 => ( ( ord_less_real @ zero_zero_real @ W )
% 8.55/4.77 => ( ( ord_less_real @ W @ Z2 )
% 8.55/4.77 => ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2293_frac__less2,axiom,
% 8.55/4.77 ! [X: rat,Y: rat,W: rat,Z2: rat] :
% 8.55/4.77 ( ( ord_less_rat @ zero_zero_rat @ X )
% 8.55/4.77 => ( ( ord_less_eq_rat @ X @ Y )
% 8.55/4.77 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 8.55/4.77 => ( ( ord_less_rat @ W @ Z2 )
% 8.55/4.77 => ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2294_frac__less,axiom,
% 8.55/4.77 ! [X: real,Y: real,W: real,Z2: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.55/4.77 => ( ( ord_less_real @ X @ Y )
% 8.55/4.77 => ( ( ord_less_real @ zero_zero_real @ W )
% 8.55/4.77 => ( ( ord_less_eq_real @ W @ Z2 )
% 8.55/4.77 => ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2295_frac__less,axiom,
% 8.55/4.77 ! [X: rat,Y: rat,W: rat,Z2: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 8.55/4.77 => ( ( ord_less_rat @ X @ Y )
% 8.55/4.77 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 8.55/4.77 => ( ( ord_less_eq_rat @ W @ Z2 )
% 8.55/4.77 => ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2296_frac__le,axiom,
% 8.55/4.77 ! [Y: real,X: real,W: real,Z2: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.55/4.77 => ( ( ord_less_eq_real @ X @ Y )
% 8.55/4.77 => ( ( ord_less_real @ zero_zero_real @ W )
% 8.55/4.77 => ( ( ord_less_eq_real @ W @ Z2 )
% 8.55/4.77 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2297_frac__le,axiom,
% 8.55/4.77 ! [Y: rat,X: rat,W: rat,Z2: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 8.55/4.77 => ( ( ord_less_eq_rat @ X @ Y )
% 8.55/4.77 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 8.55/4.77 => ( ( ord_less_eq_rat @ W @ Z2 )
% 8.55/4.77 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2298_div__positive,axiom,
% 8.55/4.77 ! [B: nat,A: nat] :
% 8.55/4.77 ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.55/4.77 => ( ( ord_less_eq_nat @ B @ A )
% 8.55/4.77 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2299_div__positive,axiom,
% 8.55/4.77 ! [B: int,A: int] :
% 8.55/4.77 ( ( ord_less_int @ zero_zero_int @ B )
% 8.55/4.77 => ( ( ord_less_eq_int @ B @ A )
% 8.55/4.77 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2300_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 8.55/4.77 ! [A: nat,B: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.77 => ( ( ord_less_nat @ A @ B )
% 8.55/4.77 => ( ( divide_divide_nat @ A @ B )
% 8.55/4.77 = zero_zero_nat ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2301_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 8.55/4.77 ! [A: int,B: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.77 => ( ( ord_less_int @ A @ B )
% 8.55/4.77 => ( ( divide_divide_int @ A @ B )
% 8.55/4.77 = zero_zero_int ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2302_power__less__imp__less__base,axiom,
% 8.55/4.77 ! [A: real,N: nat,B: real] :
% 8.55/4.77 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 8.55/4.77 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.55/4.77 => ( ord_less_real @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2303_power__less__imp__less__base,axiom,
% 8.55/4.77 ! [A: rat,N: nat,B: rat] :
% 8.55/4.77 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 8.55/4.77 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.55/4.77 => ( ord_less_rat @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2304_power__less__imp__less__base,axiom,
% 8.55/4.77 ! [A: nat,N: nat,B: nat] :
% 8.55/4.77 ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 8.55/4.77 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.55/4.77 => ( ord_less_nat @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2305_power__less__imp__less__base,axiom,
% 8.55/4.77 ! [A: int,N: nat,B: int] :
% 8.55/4.77 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 8.55/4.77 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.55/4.77 => ( ord_less_int @ A @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2306_discrete,axiom,
% 8.55/4.77 ( ord_less_nat
% 8.55/4.77 = ( ^ [A2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2307_discrete,axiom,
% 8.55/4.77 ( ord_less_int
% 8.55/4.77 = ( ^ [A2: int] : ( ord_less_eq_int @ ( plus_plus_int @ A2 @ one_one_int ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2308_power__strict__increasing,axiom,
% 8.55/4.77 ! [N: nat,N5: nat,A: real] :
% 8.55/4.77 ( ( ord_less_nat @ N @ N5 )
% 8.55/4.77 => ( ( ord_less_real @ one_one_real @ A )
% 8.55/4.77 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2309_power__strict__increasing,axiom,
% 8.55/4.77 ! [N: nat,N5: nat,A: rat] :
% 8.55/4.77 ( ( ord_less_nat @ N @ N5 )
% 8.55/4.77 => ( ( ord_less_rat @ one_one_rat @ A )
% 8.55/4.77 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2310_power__strict__increasing,axiom,
% 8.55/4.77 ! [N: nat,N5: nat,A: nat] :
% 8.55/4.77 ( ( ord_less_nat @ N @ N5 )
% 8.55/4.77 => ( ( ord_less_nat @ one_one_nat @ A )
% 8.55/4.77 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2311_power__strict__increasing,axiom,
% 8.55/4.77 ! [N: nat,N5: nat,A: int] :
% 8.55/4.77 ( ( ord_less_nat @ N @ N5 )
% 8.55/4.77 => ( ( ord_less_int @ one_one_int @ A )
% 8.55/4.77 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2312_power__less__imp__less__exp,axiom,
% 8.55/4.77 ! [A: real,M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_real @ one_one_real @ A )
% 8.55/4.77 => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 8.55/4.77 => ( ord_less_nat @ M @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2313_power__less__imp__less__exp,axiom,
% 8.55/4.77 ! [A: rat,M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_rat @ one_one_rat @ A )
% 8.55/4.77 => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 8.55/4.77 => ( ord_less_nat @ M @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2314_power__less__imp__less__exp,axiom,
% 8.55/4.77 ! [A: nat,M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_nat @ one_one_nat @ A )
% 8.55/4.77 => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 8.55/4.77 => ( ord_less_nat @ M @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2315_power__less__imp__less__exp,axiom,
% 8.55/4.77 ! [A: int,M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_int @ one_one_int @ A )
% 8.55/4.77 => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 8.55/4.77 => ( ord_less_nat @ M @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2316_power__increasing,axiom,
% 8.55/4.77 ! [N: nat,N5: nat,A: real] :
% 8.55/4.77 ( ( ord_less_eq_nat @ N @ N5 )
% 8.55/4.77 => ( ( ord_less_eq_real @ one_one_real @ A )
% 8.55/4.77 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2317_power__increasing,axiom,
% 8.55/4.77 ! [N: nat,N5: nat,A: rat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ N @ N5 )
% 8.55/4.77 => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 8.55/4.77 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2318_power__increasing,axiom,
% 8.55/4.77 ! [N: nat,N5: nat,A: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ N @ N5 )
% 8.55/4.77 => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 8.55/4.77 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2319_power__increasing,axiom,
% 8.55/4.77 ! [N: nat,N5: nat,A: int] :
% 8.55/4.77 ( ( ord_less_eq_nat @ N @ N5 )
% 8.55/4.77 => ( ( ord_less_eq_int @ one_one_int @ A )
% 8.55/4.77 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2320_ex__least__nat__less,axiom,
% 8.55/4.77 ! [P: nat > $o,N: nat] :
% 8.55/4.77 ( ( P @ N )
% 8.55/4.77 => ( ~ ( P @ zero_zero_nat )
% 8.55/4.77 => ? [K3: nat] :
% 8.55/4.77 ( ( ord_less_nat @ K3 @ N )
% 8.55/4.77 & ! [I3: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ I3 @ K3 )
% 8.55/4.77 => ~ ( P @ I3 ) )
% 8.55/4.77 & ( P @ ( suc @ K3 ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2321_nat__mult__le__cancel1,axiom,
% 8.55/4.77 ! [K: nat,M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.55/4.77 => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.55/4.77 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2322_realpow__pos__nth__unique,axiom,
% 8.55/4.77 ! [N: nat,A: real] :
% 8.55/4.77 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.77 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.77 => ? [X4: real] :
% 8.55/4.77 ( ( ord_less_real @ zero_zero_real @ X4 )
% 8.55/4.77 & ( ( power_power_real @ X4 @ N )
% 8.55/4.77 = A )
% 8.55/4.77 & ! [Y5: real] :
% 8.55/4.77 ( ( ( ord_less_real @ zero_zero_real @ Y5 )
% 8.55/4.77 & ( ( power_power_real @ Y5 @ N )
% 8.55/4.77 = A ) )
% 8.55/4.77 => ( Y5 = X4 ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2323_realpow__pos__nth,axiom,
% 8.55/4.77 ! [N: nat,A: real] :
% 8.55/4.77 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.77 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.77 => ? [R2: real] :
% 8.55/4.77 ( ( ord_less_real @ zero_zero_real @ R2 )
% 8.55/4.77 & ( ( power_power_real @ R2 @ N )
% 8.55/4.77 = A ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2324_int__one__le__iff__zero__less,axiom,
% 8.55/4.77 ! [Z2: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ one_one_int @ Z2 )
% 8.55/4.77 = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2325_dvd__imp__le,axiom,
% 8.55/4.77 ! [K: nat,N: nat] :
% 8.55/4.77 ( ( dvd_dvd_nat @ K @ N )
% 8.55/4.77 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.77 => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2326_less__diff__conv2,axiom,
% 8.55/4.77 ! [K: nat,J: nat,I: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ K @ J )
% 8.55/4.77 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 8.55/4.77 = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2327_div__greater__zero__iff,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 8.55/4.77 = ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.77 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2328_div__le__mono2,axiom,
% 8.55/4.77 ! [M: nat,N: nat,K: nat] :
% 8.55/4.77 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.55/4.77 => ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.77 => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2329_add1__zle__eq,axiom,
% 8.55/4.77 ! [W: int,Z2: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
% 8.55/4.77 = ( ord_less_int @ W @ Z2 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2330_zless__imp__add1__zle,axiom,
% 8.55/4.77 ! [W: int,Z2: int] :
% 8.55/4.77 ( ( ord_less_int @ W @ Z2 )
% 8.55/4.77 => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2331_zdiv__mono1,axiom,
% 8.55/4.77 ! [A: int,A5: int,B: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ A @ A5 )
% 8.55/4.77 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.55/4.77 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2332_zdiv__mono2,axiom,
% 8.55/4.77 ! [A: int,B6: int,B: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.77 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 8.55/4.77 => ( ( ord_less_eq_int @ B6 @ B )
% 8.55/4.77 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B6 ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2333_zdiv__eq__0__iff,axiom,
% 8.55/4.77 ! [I: int,K: int] :
% 8.55/4.77 ( ( ( divide_divide_int @ I @ K )
% 8.55/4.77 = zero_zero_int )
% 8.55/4.77 = ( ( K = zero_zero_int )
% 8.55/4.77 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 8.55/4.77 & ( ord_less_int @ I @ K ) )
% 8.55/4.77 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 8.55/4.77 & ( ord_less_int @ K @ I ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2334_zdiv__mono1__neg,axiom,
% 8.55/4.77 ! [A: int,A5: int,B: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ A @ A5 )
% 8.55/4.77 => ( ( ord_less_int @ B @ zero_zero_int )
% 8.55/4.77 => ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2335_zdiv__mono2__neg,axiom,
% 8.55/4.77 ! [A: int,B6: int,B: int] :
% 8.55/4.77 ( ( ord_less_int @ A @ zero_zero_int )
% 8.55/4.77 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 8.55/4.77 => ( ( ord_less_eq_int @ B6 @ B )
% 8.55/4.77 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B6 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2336_div__int__pos__iff,axiom,
% 8.55/4.77 ! [K: int,L: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
% 8.55/4.77 = ( ( K = zero_zero_int )
% 8.55/4.77 | ( L = zero_zero_int )
% 8.55/4.77 | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.55/4.77 & ( ord_less_eq_int @ zero_zero_int @ L ) )
% 8.55/4.77 | ( ( ord_less_int @ K @ zero_zero_int )
% 8.55/4.77 & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2337_div__positive__int,axiom,
% 8.55/4.77 ! [L: int,K: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ L @ K )
% 8.55/4.77 => ( ( ord_less_int @ zero_zero_int @ L )
% 8.55/4.77 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2338_div__nonneg__neg__le0,axiom,
% 8.55/4.77 ! [A: int,B: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.77 => ( ( ord_less_int @ B @ zero_zero_int )
% 8.55/4.77 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2339_div__nonpos__pos__le0,axiom,
% 8.55/4.77 ! [A: int,B: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.55/4.77 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.55/4.77 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2340_pos__imp__zdiv__pos__iff,axiom,
% 8.55/4.77 ! [K: int,I: int] :
% 8.55/4.77 ( ( ord_less_int @ zero_zero_int @ K )
% 8.55/4.77 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 8.55/4.77 = ( ord_less_eq_int @ K @ I ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2341_neg__imp__zdiv__nonneg__iff,axiom,
% 8.55/4.77 ! [B: int,A: int] :
% 8.55/4.77 ( ( ord_less_int @ B @ zero_zero_int )
% 8.55/4.77 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 8.55/4.77 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2342_pos__imp__zdiv__nonneg__iff,axiom,
% 8.55/4.77 ! [B: int,A: int] :
% 8.55/4.77 ( ( ord_less_int @ zero_zero_int @ B )
% 8.55/4.77 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 8.55/4.77 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2343_nonneg1__imp__zdiv__pos__iff,axiom,
% 8.55/4.77 ! [A: int,B: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.77 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 8.55/4.77 = ( ( ord_less_eq_int @ B @ A )
% 8.55/4.77 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2344_zdvd__imp__le,axiom,
% 8.55/4.77 ! [Z2: int,N: int] :
% 8.55/4.77 ( ( dvd_dvd_int @ Z2 @ N )
% 8.55/4.77 => ( ( ord_less_int @ zero_zero_int @ N )
% 8.55/4.77 => ( ord_less_eq_int @ Z2 @ N ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2345_nat__le__real__less,axiom,
% 8.55/4.77 ( ord_less_eq_nat
% 8.55/4.77 = ( ^ [N3: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2346_nat__less__real__le,axiom,
% 8.55/4.77 ( ord_less_nat
% 8.55/4.77 = ( ^ [N3: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2347_of__bool__conj,axiom,
% 8.55/4.77 ! [P: $o,Q: $o] :
% 8.55/4.77 ( ( zero_n3304061248610475627l_real
% 8.55/4.77 @ ( P
% 8.55/4.77 & Q ) )
% 8.55/4.77 = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2348_of__bool__conj,axiom,
% 8.55/4.77 ! [P: $o,Q: $o] :
% 8.55/4.77 ( ( zero_n2052037380579107095ol_rat
% 8.55/4.77 @ ( P
% 8.55/4.77 & Q ) )
% 8.55/4.77 = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2349_of__bool__conj,axiom,
% 8.55/4.77 ! [P: $o,Q: $o] :
% 8.55/4.77 ( ( zero_n2687167440665602831ol_nat
% 8.55/4.77 @ ( P
% 8.55/4.77 & Q ) )
% 8.55/4.77 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2350_of__bool__conj,axiom,
% 8.55/4.77 ! [P: $o,Q: $o] :
% 8.55/4.77 ( ( zero_n2684676970156552555ol_int
% 8.55/4.77 @ ( P
% 8.55/4.77 & Q ) )
% 8.55/4.77 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2351_of__bool__conj,axiom,
% 8.55/4.77 ! [P: $o,Q: $o] :
% 8.55/4.77 ( ( zero_n356916108424825756nteger
% 8.55/4.77 @ ( P
% 8.55/4.77 & Q ) )
% 8.55/4.77 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2352_gr__zeroI,axiom,
% 8.55/4.77 ! [N: nat] :
% 8.55/4.77 ( ( N != zero_zero_nat )
% 8.55/4.77 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2353_not__less__zero,axiom,
% 8.55/4.77 ! [N: nat] :
% 8.55/4.77 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2354_gr__implies__not__zero,axiom,
% 8.55/4.77 ! [M: nat,N: nat] :
% 8.55/4.77 ( ( ord_less_nat @ M @ N )
% 8.55/4.77 => ( N != zero_zero_nat ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2355_zero__less__iff__neq__zero,axiom,
% 8.55/4.77 ! [N: nat] :
% 8.55/4.77 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.77 = ( N != zero_zero_nat ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2356_field__lbound__gt__zero,axiom,
% 8.55/4.77 ! [D1: real,D22: real] :
% 8.55/4.77 ( ( ord_less_real @ zero_zero_real @ D1 )
% 8.55/4.77 => ( ( ord_less_real @ zero_zero_real @ D22 )
% 8.55/4.77 => ? [E2: real] :
% 8.55/4.77 ( ( ord_less_real @ zero_zero_real @ E2 )
% 8.55/4.77 & ( ord_less_real @ E2 @ D1 )
% 8.55/4.77 & ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2357_field__lbound__gt__zero,axiom,
% 8.55/4.77 ! [D1: rat,D22: rat] :
% 8.55/4.77 ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 8.55/4.77 => ( ( ord_less_rat @ zero_zero_rat @ D22 )
% 8.55/4.77 => ? [E2: rat] :
% 8.55/4.77 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 8.55/4.77 & ( ord_less_rat @ E2 @ D1 )
% 8.55/4.77 & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2358_less__numeral__extra_I3_J,axiom,
% 8.55/4.77 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2359_less__numeral__extra_I3_J,axiom,
% 8.55/4.77 ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2360_less__numeral__extra_I3_J,axiom,
% 8.55/4.77 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2361_less__numeral__extra_I3_J,axiom,
% 8.55/4.77 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2362_zero__le,axiom,
% 8.55/4.77 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2363_le__numeral__extra_I3_J,axiom,
% 8.55/4.77 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2364_le__numeral__extra_I3_J,axiom,
% 8.55/4.77 ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2365_le__numeral__extra_I3_J,axiom,
% 8.55/4.77 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2366_le__numeral__extra_I3_J,axiom,
% 8.55/4.77 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2367_less__numeral__extra_I4_J,axiom,
% 8.55/4.77 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2368_less__numeral__extra_I4_J,axiom,
% 8.55/4.77 ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2369_less__numeral__extra_I4_J,axiom,
% 8.55/4.77 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2370_less__numeral__extra_I4_J,axiom,
% 8.55/4.77 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2371_add__less__imp__less__right,axiom,
% 8.55/4.77 ! [A: real,C: real,B: real] :
% 8.55/4.77 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 8.55/4.77 => ( ord_less_real @ A @ B ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2372_add__less__imp__less__right,axiom,
% 8.55/4.77 ! [A: rat,C: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 8.55/4.77 => ( ord_less_rat @ A @ B ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2373_add__less__imp__less__right,axiom,
% 8.55/4.77 ! [A: nat,C: nat,B: nat] :
% 8.55/4.77 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 8.55/4.77 => ( ord_less_nat @ A @ B ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2374_add__less__imp__less__right,axiom,
% 8.55/4.77 ! [A: int,C: int,B: int] :
% 8.55/4.77 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 8.55/4.77 => ( ord_less_int @ A @ B ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2375_add__less__imp__less__left,axiom,
% 8.55/4.77 ! [C: real,A: real,B: real] :
% 8.55/4.77 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 8.55/4.77 => ( ord_less_real @ A @ B ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2376_add__less__imp__less__left,axiom,
% 8.55/4.77 ! [C: rat,A: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 8.55/4.77 => ( ord_less_rat @ A @ B ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2377_add__less__imp__less__left,axiom,
% 8.55/4.77 ! [C: nat,A: nat,B: nat] :
% 8.55/4.77 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 8.55/4.77 => ( ord_less_nat @ A @ B ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2378_add__less__imp__less__left,axiom,
% 8.55/4.77 ! [C: int,A: int,B: int] :
% 8.55/4.77 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 8.55/4.77 => ( ord_less_int @ A @ B ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2379_add__strict__right__mono,axiom,
% 8.55/4.77 ! [A: real,B: real,C: real] :
% 8.55/4.77 ( ( ord_less_real @ A @ B )
% 8.55/4.77 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2380_add__strict__right__mono,axiom,
% 8.55/4.77 ! [A: rat,B: rat,C: rat] :
% 8.55/4.77 ( ( ord_less_rat @ A @ B )
% 8.55/4.77 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2381_add__strict__right__mono,axiom,
% 8.55/4.77 ! [A: nat,B: nat,C: nat] :
% 8.55/4.77 ( ( ord_less_nat @ A @ B )
% 8.55/4.77 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2382_add__strict__right__mono,axiom,
% 8.55/4.77 ! [A: int,B: int,C: int] :
% 8.55/4.77 ( ( ord_less_int @ A @ B )
% 8.55/4.77 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2383_add__strict__left__mono,axiom,
% 8.55/4.77 ! [A: real,B: real,C: real] :
% 8.55/4.77 ( ( ord_less_real @ A @ B )
% 8.55/4.77 => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2384_add__strict__left__mono,axiom,
% 8.55/4.77 ! [A: rat,B: rat,C: rat] :
% 8.55/4.77 ( ( ord_less_rat @ A @ B )
% 8.55/4.77 => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2385_add__strict__left__mono,axiom,
% 8.55/4.77 ! [A: nat,B: nat,C: nat] :
% 8.55/4.77 ( ( ord_less_nat @ A @ B )
% 8.55/4.77 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2386_add__strict__left__mono,axiom,
% 8.55/4.77 ! [A: int,B: int,C: int] :
% 8.55/4.77 ( ( ord_less_int @ A @ B )
% 8.55/4.77 => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2387_add__strict__mono,axiom,
% 8.55/4.77 ! [A: real,B: real,C: real,D: real] :
% 8.55/4.77 ( ( ord_less_real @ A @ B )
% 8.55/4.77 => ( ( ord_less_real @ C @ D )
% 8.55/4.77 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2388_add__strict__mono,axiom,
% 8.55/4.77 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.55/4.77 ( ( ord_less_rat @ A @ B )
% 8.55/4.77 => ( ( ord_less_rat @ C @ D )
% 8.55/4.77 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2389_add__strict__mono,axiom,
% 8.55/4.77 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.55/4.77 ( ( ord_less_nat @ A @ B )
% 8.55/4.77 => ( ( ord_less_nat @ C @ D )
% 8.55/4.77 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2390_add__strict__mono,axiom,
% 8.55/4.77 ! [A: int,B: int,C: int,D: int] :
% 8.55/4.77 ( ( ord_less_int @ A @ B )
% 8.55/4.77 => ( ( ord_less_int @ C @ D )
% 8.55/4.77 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2391_add__mono__thms__linordered__field_I1_J,axiom,
% 8.55/4.77 ! [I: real,J: real,K: real,L: real] :
% 8.55/4.77 ( ( ( ord_less_real @ I @ J )
% 8.55/4.77 & ( K = L ) )
% 8.55/4.77 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2392_add__mono__thms__linordered__field_I1_J,axiom,
% 8.55/4.77 ! [I: rat,J: rat,K: rat,L: rat] :
% 8.55/4.77 ( ( ( ord_less_rat @ I @ J )
% 8.55/4.77 & ( K = L ) )
% 8.55/4.77 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2393_add__mono__thms__linordered__field_I1_J,axiom,
% 8.55/4.77 ! [I: nat,J: nat,K: nat,L: nat] :
% 8.55/4.77 ( ( ( ord_less_nat @ I @ J )
% 8.55/4.77 & ( K = L ) )
% 8.55/4.77 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2394_add__mono__thms__linordered__field_I1_J,axiom,
% 8.55/4.77 ! [I: int,J: int,K: int,L: int] :
% 8.55/4.77 ( ( ( ord_less_int @ I @ J )
% 8.55/4.77 & ( K = L ) )
% 8.55/4.77 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2395_add__mono__thms__linordered__field_I2_J,axiom,
% 8.55/4.77 ! [I: real,J: real,K: real,L: real] :
% 8.55/4.77 ( ( ( I = J )
% 8.55/4.77 & ( ord_less_real @ K @ L ) )
% 8.55/4.77 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2396_add__mono__thms__linordered__field_I2_J,axiom,
% 8.55/4.77 ! [I: rat,J: rat,K: rat,L: rat] :
% 8.55/4.77 ( ( ( I = J )
% 8.55/4.77 & ( ord_less_rat @ K @ L ) )
% 8.55/4.77 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2397_add__mono__thms__linordered__field_I2_J,axiom,
% 8.55/4.77 ! [I: nat,J: nat,K: nat,L: nat] :
% 8.55/4.77 ( ( ( I = J )
% 8.55/4.77 & ( ord_less_nat @ K @ L ) )
% 8.55/4.77 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2398_add__mono__thms__linordered__field_I2_J,axiom,
% 8.55/4.77 ! [I: int,J: int,K: int,L: int] :
% 8.55/4.77 ( ( ( I = J )
% 8.55/4.77 & ( ord_less_int @ K @ L ) )
% 8.55/4.77 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2399_add__mono__thms__linordered__field_I5_J,axiom,
% 8.55/4.77 ! [I: real,J: real,K: real,L: real] :
% 8.55/4.77 ( ( ( ord_less_real @ I @ J )
% 8.55/4.77 & ( ord_less_real @ K @ L ) )
% 8.55/4.77 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2400_add__mono__thms__linordered__field_I5_J,axiom,
% 8.55/4.77 ! [I: rat,J: rat,K: rat,L: rat] :
% 8.55/4.77 ( ( ( ord_less_rat @ I @ J )
% 8.55/4.77 & ( ord_less_rat @ K @ L ) )
% 8.55/4.77 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2401_add__mono__thms__linordered__field_I5_J,axiom,
% 8.55/4.77 ! [I: nat,J: nat,K: nat,L: nat] :
% 8.55/4.77 ( ( ( ord_less_nat @ I @ J )
% 8.55/4.77 & ( ord_less_nat @ K @ L ) )
% 8.55/4.77 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2402_add__mono__thms__linordered__field_I5_J,axiom,
% 8.55/4.77 ! [I: int,J: int,K: int,L: int] :
% 8.55/4.77 ( ( ( ord_less_int @ I @ J )
% 8.55/4.77 & ( ord_less_int @ K @ L ) )
% 8.55/4.77 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2403_diff__strict__mono,axiom,
% 8.55/4.77 ! [A: real,B: real,D: real,C: real] :
% 8.55/4.77 ( ( ord_less_real @ A @ B )
% 8.55/4.77 => ( ( ord_less_real @ D @ C )
% 8.55/4.77 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2404_diff__strict__mono,axiom,
% 8.55/4.77 ! [A: rat,B: rat,D: rat,C: rat] :
% 8.55/4.77 ( ( ord_less_rat @ A @ B )
% 8.55/4.77 => ( ( ord_less_rat @ D @ C )
% 8.55/4.77 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2405_diff__strict__mono,axiom,
% 8.55/4.77 ! [A: int,B: int,D: int,C: int] :
% 8.55/4.77 ( ( ord_less_int @ A @ B )
% 8.55/4.77 => ( ( ord_less_int @ D @ C )
% 8.55/4.77 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2406_diff__eq__diff__less,axiom,
% 8.55/4.77 ! [A: real,B: real,C: real,D: real] :
% 8.55/4.77 ( ( ( minus_minus_real @ A @ B )
% 8.55/4.77 = ( minus_minus_real @ C @ D ) )
% 8.55/4.77 => ( ( ord_less_real @ A @ B )
% 8.55/4.77 = ( ord_less_real @ C @ D ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2407_diff__eq__diff__less,axiom,
% 8.55/4.77 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.55/4.77 ( ( ( minus_minus_rat @ A @ B )
% 8.55/4.77 = ( minus_minus_rat @ C @ D ) )
% 8.55/4.77 => ( ( ord_less_rat @ A @ B )
% 8.55/4.77 = ( ord_less_rat @ C @ D ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2408_diff__eq__diff__less,axiom,
% 8.55/4.77 ! [A: int,B: int,C: int,D: int] :
% 8.55/4.77 ( ( ( minus_minus_int @ A @ B )
% 8.55/4.77 = ( minus_minus_int @ C @ D ) )
% 8.55/4.77 => ( ( ord_less_int @ A @ B )
% 8.55/4.77 = ( ord_less_int @ C @ D ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2409_diff__strict__left__mono,axiom,
% 8.55/4.77 ! [B: real,A: real,C: real] :
% 8.55/4.77 ( ( ord_less_real @ B @ A )
% 8.55/4.77 => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2410_diff__strict__left__mono,axiom,
% 8.55/4.77 ! [B: rat,A: rat,C: rat] :
% 8.55/4.77 ( ( ord_less_rat @ B @ A )
% 8.55/4.77 => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2411_diff__strict__left__mono,axiom,
% 8.55/4.77 ! [B: int,A: int,C: int] :
% 8.55/4.77 ( ( ord_less_int @ B @ A )
% 8.55/4.77 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2412_diff__strict__right__mono,axiom,
% 8.55/4.77 ! [A: real,B: real,C: real] :
% 8.55/4.77 ( ( ord_less_real @ A @ B )
% 8.55/4.77 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2413_diff__strict__right__mono,axiom,
% 8.55/4.77 ! [A: rat,B: rat,C: rat] :
% 8.55/4.77 ( ( ord_less_rat @ A @ B )
% 8.55/4.77 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2414_diff__strict__right__mono,axiom,
% 8.55/4.77 ! [A: int,B: int,C: int] :
% 8.55/4.77 ( ( ord_less_int @ A @ B )
% 8.55/4.77 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2415_le__numeral__extra_I4_J,axiom,
% 8.55/4.77 ord_less_eq_real @ one_one_real @ one_one_real ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2416_le__numeral__extra_I4_J,axiom,
% 8.55/4.77 ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2417_le__numeral__extra_I4_J,axiom,
% 8.55/4.77 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2418_le__numeral__extra_I4_J,axiom,
% 8.55/4.77 ord_less_eq_int @ one_one_int @ one_one_int ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2419_add__le__imp__le__right,axiom,
% 8.55/4.77 ! [A: real,C: real,B: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 8.55/4.77 => ( ord_less_eq_real @ A @ B ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2420_add__le__imp__le__right,axiom,
% 8.55/4.77 ! [A: rat,C: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 8.55/4.77 => ( ord_less_eq_rat @ A @ B ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2421_add__le__imp__le__right,axiom,
% 8.55/4.77 ! [A: nat,C: nat,B: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 8.55/4.77 => ( ord_less_eq_nat @ A @ B ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2422_add__le__imp__le__right,axiom,
% 8.55/4.77 ! [A: int,C: int,B: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 8.55/4.77 => ( ord_less_eq_int @ A @ B ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2423_add__le__imp__le__left,axiom,
% 8.55/4.77 ! [C: real,A: real,B: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 8.55/4.77 => ( ord_less_eq_real @ A @ B ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2424_add__le__imp__le__left,axiom,
% 8.55/4.77 ! [C: rat,A: rat,B: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 8.55/4.77 => ( ord_less_eq_rat @ A @ B ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2425_add__le__imp__le__left,axiom,
% 8.55/4.77 ! [C: nat,A: nat,B: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 8.55/4.77 => ( ord_less_eq_nat @ A @ B ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2426_add__le__imp__le__left,axiom,
% 8.55/4.77 ! [C: int,A: int,B: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 8.55/4.77 => ( ord_less_eq_int @ A @ B ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2427_le__iff__add,axiom,
% 8.55/4.77 ( ord_less_eq_nat
% 8.55/4.77 = ( ^ [A2: nat,B2: nat] :
% 8.55/4.77 ? [C4: nat] :
% 8.55/4.77 ( B2
% 8.55/4.77 = ( plus_plus_nat @ A2 @ C4 ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2428_add__right__mono,axiom,
% 8.55/4.77 ! [A: real,B: real,C: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ A @ B )
% 8.55/4.77 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2429_add__right__mono,axiom,
% 8.55/4.77 ! [A: rat,B: rat,C: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.77 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2430_add__right__mono,axiom,
% 8.55/4.77 ! [A: nat,B: nat,C: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.77 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2431_add__right__mono,axiom,
% 8.55/4.77 ! [A: int,B: int,C: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ A @ B )
% 8.55/4.77 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2432_less__eqE,axiom,
% 8.55/4.77 ! [A: nat,B: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.77 => ~ ! [C3: nat] :
% 8.55/4.77 ( B
% 8.55/4.77 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2433_add__left__mono,axiom,
% 8.55/4.77 ! [A: real,B: real,C: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ A @ B )
% 8.55/4.77 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2434_add__left__mono,axiom,
% 8.55/4.77 ! [A: rat,B: rat,C: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.77 => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2435_add__left__mono,axiom,
% 8.55/4.77 ! [A: nat,B: nat,C: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.77 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2436_add__left__mono,axiom,
% 8.55/4.77 ! [A: int,B: int,C: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ A @ B )
% 8.55/4.77 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2437_add__mono,axiom,
% 8.55/4.77 ! [A: real,B: real,C: real,D: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_real @ C @ D )
% 8.55/4.77 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2438_add__mono,axiom,
% 8.55/4.77 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_rat @ C @ D )
% 8.55/4.77 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2439_add__mono,axiom,
% 8.55/4.77 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.55/4.77 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_nat @ C @ D )
% 8.55/4.77 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2440_add__mono,axiom,
% 8.55/4.77 ! [A: int,B: int,C: int,D: int] :
% 8.55/4.77 ( ( ord_less_eq_int @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_int @ C @ D )
% 8.55/4.77 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2441_add__mono__thms__linordered__semiring_I1_J,axiom,
% 8.55/4.77 ! [I: real,J: real,K: real,L: real] :
% 8.55/4.77 ( ( ( ord_less_eq_real @ I @ J )
% 8.55/4.77 & ( ord_less_eq_real @ K @ L ) )
% 8.55/4.77 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2442_add__mono__thms__linordered__semiring_I1_J,axiom,
% 8.55/4.77 ! [I: rat,J: rat,K: rat,L: rat] :
% 8.55/4.77 ( ( ( ord_less_eq_rat @ I @ J )
% 8.55/4.77 & ( ord_less_eq_rat @ K @ L ) )
% 8.55/4.77 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2443_add__mono__thms__linordered__semiring_I1_J,axiom,
% 8.55/4.77 ! [I: nat,J: nat,K: nat,L: nat] :
% 8.55/4.77 ( ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.77 & ( ord_less_eq_nat @ K @ L ) )
% 8.55/4.77 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2444_add__mono__thms__linordered__semiring_I1_J,axiom,
% 8.55/4.77 ! [I: int,J: int,K: int,L: int] :
% 8.55/4.77 ( ( ( ord_less_eq_int @ I @ J )
% 8.55/4.77 & ( ord_less_eq_int @ K @ L ) )
% 8.55/4.77 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2445_add__mono__thms__linordered__semiring_I2_J,axiom,
% 8.55/4.77 ! [I: real,J: real,K: real,L: real] :
% 8.55/4.77 ( ( ( I = J )
% 8.55/4.77 & ( ord_less_eq_real @ K @ L ) )
% 8.55/4.77 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2446_add__mono__thms__linordered__semiring_I2_J,axiom,
% 8.55/4.77 ! [I: rat,J: rat,K: rat,L: rat] :
% 8.55/4.77 ( ( ( I = J )
% 8.55/4.77 & ( ord_less_eq_rat @ K @ L ) )
% 8.55/4.77 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2447_add__mono__thms__linordered__semiring_I2_J,axiom,
% 8.55/4.77 ! [I: nat,J: nat,K: nat,L: nat] :
% 8.55/4.77 ( ( ( I = J )
% 8.55/4.77 & ( ord_less_eq_nat @ K @ L ) )
% 8.55/4.77 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2448_add__mono__thms__linordered__semiring_I2_J,axiom,
% 8.55/4.77 ! [I: int,J: int,K: int,L: int] :
% 8.55/4.77 ( ( ( I = J )
% 8.55/4.77 & ( ord_less_eq_int @ K @ L ) )
% 8.55/4.77 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2449_add__mono__thms__linordered__semiring_I3_J,axiom,
% 8.55/4.77 ! [I: real,J: real,K: real,L: real] :
% 8.55/4.77 ( ( ( ord_less_eq_real @ I @ J )
% 8.55/4.77 & ( K = L ) )
% 8.55/4.77 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2450_add__mono__thms__linordered__semiring_I3_J,axiom,
% 8.55/4.77 ! [I: rat,J: rat,K: rat,L: rat] :
% 8.55/4.77 ( ( ( ord_less_eq_rat @ I @ J )
% 8.55/4.77 & ( K = L ) )
% 8.55/4.77 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2451_add__mono__thms__linordered__semiring_I3_J,axiom,
% 8.55/4.77 ! [I: nat,J: nat,K: nat,L: nat] :
% 8.55/4.77 ( ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.77 & ( K = L ) )
% 8.55/4.77 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2452_add__mono__thms__linordered__semiring_I3_J,axiom,
% 8.55/4.77 ! [I: int,J: int,K: int,L: int] :
% 8.55/4.77 ( ( ( ord_less_eq_int @ I @ J )
% 8.55/4.77 & ( K = L ) )
% 8.55/4.77 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2453_le__num__One__iff,axiom,
% 8.55/4.77 ! [X: num] :
% 8.55/4.77 ( ( ord_less_eq_num @ X @ one )
% 8.55/4.77 = ( X = one ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2454_diff__mono,axiom,
% 8.55/4.77 ! [A: real,B: real,D: real,C: real] :
% 8.55/4.77 ( ( ord_less_eq_real @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_real @ D @ C )
% 8.55/4.77 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 8.55/4.77
% 8.55/4.77 thf(fact_2455_diff__mono,axiom,
% 8.55/4.77 ! [A: rat,B: rat,D: rat,C: rat] :
% 8.55/4.77 ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.77 => ( ( ord_less_eq_rat @ D @ C )
% 8.55/4.77 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2456_diff__mono,axiom,
% 8.55/4.78 ! [A: int,B: int,D: int,C: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_int @ D @ C )
% 8.55/4.78 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2457_diff__left__mono,axiom,
% 8.55/4.78 ! [B: real,A: real,C: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ B @ A )
% 8.55/4.78 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2458_diff__left__mono,axiom,
% 8.55/4.78 ! [B: rat,A: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ B @ A )
% 8.55/4.78 => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2459_diff__left__mono,axiom,
% 8.55/4.78 ! [B: int,A: int,C: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ B @ A )
% 8.55/4.78 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2460_diff__right__mono,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ A @ B )
% 8.55/4.78 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2461_diff__right__mono,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.78 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2462_diff__right__mono,axiom,
% 8.55/4.78 ! [A: int,B: int,C: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ A @ B )
% 8.55/4.78 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2463_diff__eq__diff__less__eq,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real,D: real] :
% 8.55/4.78 ( ( ( minus_minus_real @ A @ B )
% 8.55/4.78 = ( minus_minus_real @ C @ D ) )
% 8.55/4.78 => ( ( ord_less_eq_real @ A @ B )
% 8.55/4.78 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2464_diff__eq__diff__less__eq,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.55/4.78 ( ( ( minus_minus_rat @ A @ B )
% 8.55/4.78 = ( minus_minus_rat @ C @ D ) )
% 8.55/4.78 => ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.78 = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2465_diff__eq__diff__less__eq,axiom,
% 8.55/4.78 ! [A: int,B: int,C: int,D: int] :
% 8.55/4.78 ( ( ( minus_minus_int @ A @ B )
% 8.55/4.78 = ( minus_minus_int @ C @ D ) )
% 8.55/4.78 => ( ( ord_less_eq_int @ A @ B )
% 8.55/4.78 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2466_reals__Archimedean2,axiom,
% 8.55/4.78 ! [X: real] :
% 8.55/4.78 ? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2467_reals__Archimedean2,axiom,
% 8.55/4.78 ! [X: rat] :
% 8.55/4.78 ? [N2: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2468_not__less__less__Suc__eq,axiom,
% 8.55/4.78 ! [N: nat,M: nat] :
% 8.55/4.78 ( ~ ( ord_less_nat @ N @ M )
% 8.55/4.78 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 8.55/4.78 = ( N = M ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2469_strict__inc__induct,axiom,
% 8.55/4.78 ! [I: nat,J: nat,P: nat > $o] :
% 8.55/4.78 ( ( ord_less_nat @ I @ J )
% 8.55/4.78 => ( ! [I2: nat] :
% 8.55/4.78 ( ( J
% 8.55/4.78 = ( suc @ I2 ) )
% 8.55/4.78 => ( P @ I2 ) )
% 8.55/4.78 => ( ! [I2: nat] :
% 8.55/4.78 ( ( ord_less_nat @ I2 @ J )
% 8.55/4.78 => ( ( P @ ( suc @ I2 ) )
% 8.55/4.78 => ( P @ I2 ) ) )
% 8.55/4.78 => ( P @ I ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2470_less__Suc__induct,axiom,
% 8.55/4.78 ! [I: nat,J: nat,P: nat > nat > $o] :
% 8.55/4.78 ( ( ord_less_nat @ I @ J )
% 8.55/4.78 => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 8.55/4.78 => ( ! [I2: nat,J2: nat,K3: nat] :
% 8.55/4.78 ( ( ord_less_nat @ I2 @ J2 )
% 8.55/4.78 => ( ( ord_less_nat @ J2 @ K3 )
% 8.55/4.78 => ( ( P @ I2 @ J2 )
% 8.55/4.78 => ( ( P @ J2 @ K3 )
% 8.55/4.78 => ( P @ I2 @ K3 ) ) ) ) )
% 8.55/4.78 => ( P @ I @ J ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2471_less__trans__Suc,axiom,
% 8.55/4.78 ! [I: nat,J: nat,K: nat] :
% 8.55/4.78 ( ( ord_less_nat @ I @ J )
% 8.55/4.78 => ( ( ord_less_nat @ J @ K )
% 8.55/4.78 => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2472_Suc__less__SucD,axiom,
% 8.55/4.78 ! [M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 8.55/4.78 => ( ord_less_nat @ M @ N ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2473_less__antisym,axiom,
% 8.55/4.78 ! [N: nat,M: nat] :
% 8.55/4.78 ( ~ ( ord_less_nat @ N @ M )
% 8.55/4.78 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 8.55/4.78 => ( M = N ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2474_Suc__less__eq2,axiom,
% 8.55/4.78 ! [N: nat,M: nat] :
% 8.55/4.78 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 8.55/4.78 = ( ? [M5: nat] :
% 8.55/4.78 ( ( M
% 8.55/4.78 = ( suc @ M5 ) )
% 8.55/4.78 & ( ord_less_nat @ N @ M5 ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2475_All__less__Suc,axiom,
% 8.55/4.78 ! [N: nat,P: nat > $o] :
% 8.55/4.78 ( ( ! [I4: nat] :
% 8.55/4.78 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 8.55/4.78 => ( P @ I4 ) ) )
% 8.55/4.78 = ( ( P @ N )
% 8.55/4.78 & ! [I4: nat] :
% 8.55/4.78 ( ( ord_less_nat @ I4 @ N )
% 8.55/4.78 => ( P @ I4 ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2476_not__less__eq,axiom,
% 8.55/4.78 ! [M: nat,N: nat] :
% 8.55/4.78 ( ( ~ ( ord_less_nat @ M @ N ) )
% 8.55/4.78 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2477_less__Suc__eq,axiom,
% 8.55/4.78 ! [M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 8.55/4.78 = ( ( ord_less_nat @ M @ N )
% 8.55/4.78 | ( M = N ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2478_Ex__less__Suc,axiom,
% 8.55/4.78 ! [N: nat,P: nat > $o] :
% 8.55/4.78 ( ( ? [I4: nat] :
% 8.55/4.78 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 8.55/4.78 & ( P @ I4 ) ) )
% 8.55/4.78 = ( ( P @ N )
% 8.55/4.78 | ? [I4: nat] :
% 8.55/4.78 ( ( ord_less_nat @ I4 @ N )
% 8.55/4.78 & ( P @ I4 ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2479_less__SucI,axiom,
% 8.55/4.78 ! [M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_nat @ M @ N )
% 8.55/4.78 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2480_less__SucE,axiom,
% 8.55/4.78 ! [M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 8.55/4.78 => ( ~ ( ord_less_nat @ M @ N )
% 8.55/4.78 => ( M = N ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2481_Suc__lessI,axiom,
% 8.55/4.78 ! [M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_nat @ M @ N )
% 8.55/4.78 => ( ( ( suc @ M )
% 8.55/4.78 != N )
% 8.55/4.78 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2482_Suc__lessE,axiom,
% 8.55/4.78 ! [I: nat,K: nat] :
% 8.55/4.78 ( ( ord_less_nat @ ( suc @ I ) @ K )
% 8.55/4.78 => ~ ! [J2: nat] :
% 8.55/4.78 ( ( ord_less_nat @ I @ J2 )
% 8.55/4.78 => ( K
% 8.55/4.78 != ( suc @ J2 ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2483_Suc__lessD,axiom,
% 8.55/4.78 ! [M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 8.55/4.78 => ( ord_less_nat @ M @ N ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2484_Nat_OlessE,axiom,
% 8.55/4.78 ! [I: nat,K: nat] :
% 8.55/4.78 ( ( ord_less_nat @ I @ K )
% 8.55/4.78 => ( ( K
% 8.55/4.78 != ( suc @ I ) )
% 8.55/4.78 => ~ ! [J2: nat] :
% 8.55/4.78 ( ( ord_less_nat @ I @ J2 )
% 8.55/4.78 => ( K
% 8.55/4.78 != ( suc @ J2 ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2485_infinite__descent0,axiom,
% 8.55/4.78 ! [P: nat > $o,N: nat] :
% 8.55/4.78 ( ( P @ zero_zero_nat )
% 8.55/4.78 => ( ! [N2: nat] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 8.55/4.78 => ( ~ ( P @ N2 )
% 8.55/4.78 => ? [M4: nat] :
% 8.55/4.78 ( ( ord_less_nat @ M4 @ N2 )
% 8.55/4.78 & ~ ( P @ M4 ) ) ) )
% 8.55/4.78 => ( P @ N ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2486_gr__implies__not0,axiom,
% 8.55/4.78 ! [M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_nat @ M @ N )
% 8.55/4.78 => ( N != zero_zero_nat ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2487_less__zeroE,axiom,
% 8.55/4.78 ! [N: nat] :
% 8.55/4.78 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2488_not__less0,axiom,
% 8.55/4.78 ! [N: nat] :
% 8.55/4.78 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2489_not__gr0,axiom,
% 8.55/4.78 ! [N: nat] :
% 8.55/4.78 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 8.55/4.78 = ( N = zero_zero_nat ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2490_gr0I,axiom,
% 8.55/4.78 ! [N: nat] :
% 8.55/4.78 ( ( N != zero_zero_nat )
% 8.55/4.78 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2491_bot__nat__0_Oextremum__strict,axiom,
% 8.55/4.78 ! [A: nat] :
% 8.55/4.78 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2492_transitive__stepwise__le,axiom,
% 8.55/4.78 ! [M: nat,N: nat,R3: nat > nat > $o] :
% 8.55/4.78 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.78 => ( ! [X4: nat] : ( R3 @ X4 @ X4 )
% 8.55/4.78 => ( ! [X4: nat,Y4: nat,Z4: nat] :
% 8.55/4.78 ( ( R3 @ X4 @ Y4 )
% 8.55/4.78 => ( ( R3 @ Y4 @ Z4 )
% 8.55/4.78 => ( R3 @ X4 @ Z4 ) ) )
% 8.55/4.78 => ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
% 8.55/4.78 => ( R3 @ M @ N ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2493_nat__induct__at__least,axiom,
% 8.55/4.78 ! [M: nat,N: nat,P: nat > $o] :
% 8.55/4.78 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.78 => ( ( P @ M )
% 8.55/4.78 => ( ! [N2: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ M @ N2 )
% 8.55/4.78 => ( ( P @ N2 )
% 8.55/4.78 => ( P @ ( suc @ N2 ) ) ) )
% 8.55/4.78 => ( P @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2494_full__nat__induct,axiom,
% 8.55/4.78 ! [P: nat > $o,N: nat] :
% 8.55/4.78 ( ! [N2: nat] :
% 8.55/4.78 ( ! [M4: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
% 8.55/4.78 => ( P @ M4 ) )
% 8.55/4.78 => ( P @ N2 ) )
% 8.55/4.78 => ( P @ N ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2495_not__less__eq__eq,axiom,
% 8.55/4.78 ! [M: nat,N: nat] :
% 8.55/4.78 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 8.55/4.78 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2496_Suc__n__not__le__n,axiom,
% 8.55/4.78 ! [N: nat] :
% 8.55/4.78 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2497_le__Suc__eq,axiom,
% 8.55/4.78 ! [M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 8.55/4.78 = ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.78 | ( M
% 8.55/4.78 = ( suc @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2498_Suc__le__D,axiom,
% 8.55/4.78 ! [N: nat,M6: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
% 8.55/4.78 => ? [M2: nat] :
% 8.55/4.78 ( M6
% 8.55/4.78 = ( suc @ M2 ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2499_le__SucI,axiom,
% 8.55/4.78 ! [M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.78 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2500_le__SucE,axiom,
% 8.55/4.78 ! [M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 8.55/4.78 => ( ~ ( ord_less_eq_nat @ M @ N )
% 8.55/4.78 => ( M
% 8.55/4.78 = ( suc @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2501_Suc__leD,axiom,
% 8.55/4.78 ! [M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 8.55/4.78 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2502_le__0__eq,axiom,
% 8.55/4.78 ! [N: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 8.55/4.78 = ( N = zero_zero_nat ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2503_bot__nat__0_Oextremum__uniqueI,axiom,
% 8.55/4.78 ! [A: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 8.55/4.78 => ( A = zero_zero_nat ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2504_bot__nat__0_Oextremum__unique,axiom,
% 8.55/4.78 ! [A: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 8.55/4.78 = ( A = zero_zero_nat ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2505_less__eq__nat_Osimps_I1_J,axiom,
% 8.55/4.78 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2506_real__arch__simple,axiom,
% 8.55/4.78 ! [X: real] :
% 8.55/4.78 ? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2507_real__arch__simple,axiom,
% 8.55/4.78 ! [X: rat] :
% 8.55/4.78 ? [N2: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2508_less__int__code_I1_J,axiom,
% 8.55/4.78 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2509_imp__le__cong,axiom,
% 8.55/4.78 ! [X: int,X6: int,P: $o,P4: $o] :
% 8.55/4.78 ( ( X = X6 )
% 8.55/4.78 => ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
% 8.55/4.78 => ( P = P4 ) )
% 8.55/4.78 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.55/4.78 => P )
% 8.55/4.78 = ( ( ord_less_eq_int @ zero_zero_int @ X6 )
% 8.55/4.78 => P4 ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2510_conj__le__cong,axiom,
% 8.55/4.78 ! [X: int,X6: int,P: $o,P4: $o] :
% 8.55/4.78 ( ( X = X6 )
% 8.55/4.78 => ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
% 8.55/4.78 => ( P = P4 ) )
% 8.55/4.78 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.55/4.78 & P )
% 8.55/4.78 = ( ( ord_less_eq_int @ zero_zero_int @ X6 )
% 8.55/4.78 & P4 ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2511_less__eq__int__code_I1_J,axiom,
% 8.55/4.78 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2512_add__lessD1,axiom,
% 8.55/4.78 ! [I: nat,J: nat,K: nat] :
% 8.55/4.78 ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 8.55/4.78 => ( ord_less_nat @ I @ K ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2513_add__less__mono,axiom,
% 8.55/4.78 ! [I: nat,J: nat,K: nat,L: nat] :
% 8.55/4.78 ( ( ord_less_nat @ I @ J )
% 8.55/4.78 => ( ( ord_less_nat @ K @ L )
% 8.55/4.78 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2514_not__add__less1,axiom,
% 8.55/4.78 ! [I: nat,J: nat] :
% 8.55/4.78 ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2515_not__add__less2,axiom,
% 8.55/4.78 ! [J: nat,I: nat] :
% 8.55/4.78 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2516_add__less__mono1,axiom,
% 8.55/4.78 ! [I: nat,J: nat,K: nat] :
% 8.55/4.78 ( ( ord_less_nat @ I @ J )
% 8.55/4.78 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2517_trans__less__add1,axiom,
% 8.55/4.78 ! [I: nat,J: nat,M: nat] :
% 8.55/4.78 ( ( ord_less_nat @ I @ J )
% 8.55/4.78 => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2518_trans__less__add2,axiom,
% 8.55/4.78 ! [I: nat,J: nat,M: nat] :
% 8.55/4.78 ( ( ord_less_nat @ I @ J )
% 8.55/4.78 => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2519_less__add__eq__less,axiom,
% 8.55/4.78 ! [K: nat,L: nat,M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_nat @ K @ L )
% 8.55/4.78 => ( ( ( plus_plus_nat @ M @ L )
% 8.55/4.78 = ( plus_plus_nat @ K @ N ) )
% 8.55/4.78 => ( ord_less_nat @ M @ N ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2520_add__leE,axiom,
% 8.55/4.78 ! [M: nat,K: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 8.55/4.78 => ~ ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.78 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2521_le__add1,axiom,
% 8.55/4.78 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2522_le__add2,axiom,
% 8.55/4.78 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2523_add__leD1,axiom,
% 8.55/4.78 ! [M: nat,K: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 8.55/4.78 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2524_add__leD2,axiom,
% 8.55/4.78 ! [M: nat,K: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 8.55/4.78 => ( ord_less_eq_nat @ K @ N ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2525_le__Suc__ex,axiom,
% 8.55/4.78 ! [K: nat,L: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ K @ L )
% 8.55/4.78 => ? [N2: nat] :
% 8.55/4.78 ( L
% 8.55/4.78 = ( plus_plus_nat @ K @ N2 ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2526_add__le__mono,axiom,
% 8.55/4.78 ! [I: nat,J: nat,K: nat,L: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.78 => ( ( ord_less_eq_nat @ K @ L )
% 8.55/4.78 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2527_add__le__mono1,axiom,
% 8.55/4.78 ! [I: nat,J: nat,K: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.78 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2528_trans__le__add1,axiom,
% 8.55/4.78 ! [I: nat,J: nat,M: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.78 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2529_trans__le__add2,axiom,
% 8.55/4.78 ! [I: nat,J: nat,M: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.78 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2530_nat__le__iff__add,axiom,
% 8.55/4.78 ( ord_less_eq_nat
% 8.55/4.78 = ( ^ [M3: nat,N3: nat] :
% 8.55/4.78 ? [K2: nat] :
% 8.55/4.78 ( N3
% 8.55/4.78 = ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2531_less__imp__diff__less,axiom,
% 8.55/4.78 ! [J: nat,K: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_nat @ J @ K )
% 8.55/4.78 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2532_diff__less__mono2,axiom,
% 8.55/4.78 ! [M: nat,N: nat,L: nat] :
% 8.55/4.78 ( ( ord_less_nat @ M @ N )
% 8.55/4.78 => ( ( ord_less_nat @ M @ L )
% 8.55/4.78 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2533_diff__le__mono2,axiom,
% 8.55/4.78 ! [M: nat,N: nat,L: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.78 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2534_le__diff__iff_H,axiom,
% 8.55/4.78 ! [A: nat,C: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ A @ C )
% 8.55/4.78 => ( ( ord_less_eq_nat @ B @ C )
% 8.55/4.78 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 8.55/4.78 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2535_diff__le__self,axiom,
% 8.55/4.78 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2536_diff__le__mono,axiom,
% 8.55/4.78 ! [M: nat,N: nat,L: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.78 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2537_Nat_Odiff__diff__eq,axiom,
% 8.55/4.78 ! [K: nat,M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ K @ M )
% 8.55/4.78 => ( ( ord_less_eq_nat @ K @ N )
% 8.55/4.78 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 8.55/4.78 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2538_le__diff__iff,axiom,
% 8.55/4.78 ! [K: nat,M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ K @ M )
% 8.55/4.78 => ( ( ord_less_eq_nat @ K @ N )
% 8.55/4.78 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 8.55/4.78 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2539_eq__diff__iff,axiom,
% 8.55/4.78 ! [K: nat,M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ K @ M )
% 8.55/4.78 => ( ( ord_less_eq_nat @ K @ N )
% 8.55/4.78 => ( ( ( minus_minus_nat @ M @ K )
% 8.55/4.78 = ( minus_minus_nat @ N @ K ) )
% 8.55/4.78 = ( M = N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2540_le__cube,axiom,
% 8.55/4.78 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2541_le__square,axiom,
% 8.55/4.78 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2542_mult__le__mono,axiom,
% 8.55/4.78 ! [I: nat,J: nat,K: nat,L: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.78 => ( ( ord_less_eq_nat @ K @ L )
% 8.55/4.78 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2543_mult__le__mono1,axiom,
% 8.55/4.78 ! [I: nat,J: nat,K: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.78 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2544_mult__le__mono2,axiom,
% 8.55/4.78 ! [I: nat,J: nat,K: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.78 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2545_real__arch__pow,axiom,
% 8.55/4.78 ! [X: real,Y: real] :
% 8.55/4.78 ( ( ord_less_real @ one_one_real @ X )
% 8.55/4.78 => ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2546_real__arch__pow__inv,axiom,
% 8.55/4.78 ! [Y: real,X: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ Y )
% 8.55/4.78 => ( ( ord_less_real @ X @ one_one_real )
% 8.55/4.78 => ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2547_div__le__dividend,axiom,
% 8.55/4.78 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2548_div__le__mono,axiom,
% 8.55/4.78 ! [M: nat,N: nat,K: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.78 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2549_mult__le__cancel__left1,axiom,
% 8.55/4.78 ! [C: real,B: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_eq_real @ one_one_real @ B ) )
% 8.55/4.78 & ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2550_mult__le__cancel__left1,axiom,
% 8.55/4.78 ! [C: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 8.55/4.78 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2551_mult__le__cancel__left1,axiom,
% 8.55/4.78 ! [C: int,B: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 8.55/4.78 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ord_less_eq_int @ one_one_int @ B ) )
% 8.55/4.78 & ( ( ord_less_int @ C @ zero_zero_int )
% 8.55/4.78 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2552_mult__le__cancel__left2,axiom,
% 8.55/4.78 ! [C: real,A: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_eq_real @ A @ one_one_real ) )
% 8.55/4.78 & ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2553_mult__le__cancel__left2,axiom,
% 8.55/4.78 ! [C: rat,A: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 8.55/4.78 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2554_mult__le__cancel__left2,axiom,
% 8.55/4.78 ! [C: int,A: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 8.55/4.78 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ord_less_eq_int @ A @ one_one_int ) )
% 8.55/4.78 & ( ( ord_less_int @ C @ zero_zero_int )
% 8.55/4.78 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2555_mult__le__cancel__right1,axiom,
% 8.55/4.78 ! [C: real,B: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_eq_real @ one_one_real @ B ) )
% 8.55/4.78 & ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2556_mult__le__cancel__right1,axiom,
% 8.55/4.78 ! [C: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 8.55/4.78 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2557_mult__le__cancel__right1,axiom,
% 8.55/4.78 ! [C: int,B: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 8.55/4.78 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ord_less_eq_int @ one_one_int @ B ) )
% 8.55/4.78 & ( ( ord_less_int @ C @ zero_zero_int )
% 8.55/4.78 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2558_mult__le__cancel__right2,axiom,
% 8.55/4.78 ! [A: real,C: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_eq_real @ A @ one_one_real ) )
% 8.55/4.78 & ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2559_mult__le__cancel__right2,axiom,
% 8.55/4.78 ! [A: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 8.55/4.78 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2560_mult__le__cancel__right2,axiom,
% 8.55/4.78 ! [A: int,C: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 8.55/4.78 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ord_less_eq_int @ A @ one_one_int ) )
% 8.55/4.78 & ( ( ord_less_int @ C @ zero_zero_int )
% 8.55/4.78 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2561_mult__less__cancel__left1,axiom,
% 8.55/4.78 ! [C: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
% 8.55/4.78 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_real @ one_one_real @ B ) )
% 8.55/4.78 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2562_mult__less__cancel__left1,axiom,
% 8.55/4.78 ! [C: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 8.55/4.78 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_rat @ one_one_rat @ B ) )
% 8.55/4.78 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2563_mult__less__cancel__left1,axiom,
% 8.55/4.78 ! [C: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 8.55/4.78 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ord_less_int @ one_one_int @ B ) )
% 8.55/4.78 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.55/4.78 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2564_mult__less__cancel__left2,axiom,
% 8.55/4.78 ! [C: real,A: real] :
% 8.55/4.78 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 8.55/4.78 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_real @ A @ one_one_real ) )
% 8.55/4.78 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2565_mult__less__cancel__left2,axiom,
% 8.55/4.78 ! [C: rat,A: rat] :
% 8.55/4.78 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 8.55/4.78 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_rat @ A @ one_one_rat ) )
% 8.55/4.78 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2566_mult__less__cancel__left2,axiom,
% 8.55/4.78 ! [C: int,A: int] :
% 8.55/4.78 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 8.55/4.78 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ord_less_int @ A @ one_one_int ) )
% 8.55/4.78 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.55/4.78 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2567_mult__less__cancel__right1,axiom,
% 8.55/4.78 ! [C: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
% 8.55/4.78 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_real @ one_one_real @ B ) )
% 8.55/4.78 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2568_mult__less__cancel__right1,axiom,
% 8.55/4.78 ! [C: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 8.55/4.78 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_rat @ one_one_rat @ B ) )
% 8.55/4.78 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2569_mult__less__cancel__right1,axiom,
% 8.55/4.78 ! [C: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 8.55/4.78 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ord_less_int @ one_one_int @ B ) )
% 8.55/4.78 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.55/4.78 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2570_mult__less__cancel__right2,axiom,
% 8.55/4.78 ! [A: real,C: real] :
% 8.55/4.78 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 8.55/4.78 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_real @ A @ one_one_real ) )
% 8.55/4.78 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2571_mult__less__cancel__right2,axiom,
% 8.55/4.78 ! [A: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 8.55/4.78 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_rat @ A @ one_one_rat ) )
% 8.55/4.78 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2572_mult__less__cancel__right2,axiom,
% 8.55/4.78 ! [A: int,C: int] :
% 8.55/4.78 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 8.55/4.78 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ord_less_int @ A @ one_one_int ) )
% 8.55/4.78 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.55/4.78 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2573_field__le__mult__one__interval,axiom,
% 8.55/4.78 ! [X: real,Y: real] :
% 8.55/4.78 ( ! [Z4: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ Z4 )
% 8.55/4.78 => ( ( ord_less_real @ Z4 @ one_one_real )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ Z4 @ X ) @ Y ) ) )
% 8.55/4.78 => ( ord_less_eq_real @ X @ Y ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2574_field__le__mult__one__interval,axiom,
% 8.55/4.78 ! [X: rat,Y: rat] :
% 8.55/4.78 ( ! [Z4: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ Z4 )
% 8.55/4.78 => ( ( ord_less_rat @ Z4 @ one_one_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ ( times_times_rat @ Z4 @ X ) @ Y ) ) )
% 8.55/4.78 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2575_divide__left__mono__neg,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 8.55/4.78 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2576_divide__left__mono__neg,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 8.55/4.78 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2577_mult__imp__le__div__pos,axiom,
% 8.55/4.78 ! [Y: real,Z2: real,X: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ Y )
% 8.55/4.78 => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ Y ) @ X )
% 8.55/4.78 => ( ord_less_eq_real @ Z2 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2578_mult__imp__le__div__pos,axiom,
% 8.55/4.78 ! [Y: rat,Z2: rat,X: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 8.55/4.78 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ Y ) @ X )
% 8.55/4.78 => ( ord_less_eq_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2579_mult__imp__div__pos__le,axiom,
% 8.55/4.78 ! [Y: real,X: real,Z2: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ Y )
% 8.55/4.78 => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z2 @ Y ) )
% 8.55/4.78 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z2 ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2580_mult__imp__div__pos__le,axiom,
% 8.55/4.78 ! [Y: rat,X: rat,Z2: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 8.55/4.78 => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z2 @ Y ) )
% 8.55/4.78 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2581_pos__le__divide__eq,axiom,
% 8.55/4.78 ! [C: real,A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 8.55/4.78 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2582_pos__le__divide__eq,axiom,
% 8.55/4.78 ! [C: rat,A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 8.55/4.78 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2583_pos__divide__le__eq,axiom,
% 8.55/4.78 ! [C: real,B: real,A: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 8.55/4.78 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2584_pos__divide__le__eq,axiom,
% 8.55/4.78 ! [C: rat,B: rat,A: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 8.55/4.78 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2585_neg__le__divide__eq,axiom,
% 8.55/4.78 ! [C: real,A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 8.55/4.78 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2586_neg__le__divide__eq,axiom,
% 8.55/4.78 ! [C: rat,A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 8.55/4.78 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2587_neg__divide__le__eq,axiom,
% 8.55/4.78 ! [C: real,B: real,A: real] :
% 8.55/4.78 ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 8.55/4.78 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2588_neg__divide__le__eq,axiom,
% 8.55/4.78 ! [C: rat,B: rat,A: rat] :
% 8.55/4.78 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 8.55/4.78 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2589_divide__left__mono,axiom,
% 8.55/4.78 ! [B: real,A: real,C: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ B @ A )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 8.55/4.78 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2590_divide__left__mono,axiom,
% 8.55/4.78 ! [B: rat,A: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ B @ A )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 8.55/4.78 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2591_le__divide__eq,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 8.55/4.78 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 8.55/4.78 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2592_le__divide__eq,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 8.55/4.78 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 8.55/4.78 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2593_divide__le__eq,axiom,
% 8.55/4.78 ! [B: real,C: real,A: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 8.55/4.78 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 8.55/4.78 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2594_divide__le__eq,axiom,
% 8.55/4.78 ! [B: rat,C: rat,A: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 8.55/4.78 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 8.55/4.78 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2595_le__divide__eq__1,axiom,
% 8.55/4.78 ! [B: real,A: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.78 & ( ord_less_eq_real @ A @ B ) )
% 8.55/4.78 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.78 & ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2596_le__divide__eq__1,axiom,
% 8.55/4.78 ! [B: rat,A: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.78 & ( ord_less_eq_rat @ A @ B ) )
% 8.55/4.78 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.78 & ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2597_divide__le__eq__1,axiom,
% 8.55/4.78 ! [B: real,A: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.78 & ( ord_less_eq_real @ B @ A ) )
% 8.55/4.78 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.78 & ( ord_less_eq_real @ A @ B ) )
% 8.55/4.78 | ( A = zero_zero_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2598_divide__le__eq__1,axiom,
% 8.55/4.78 ! [B: rat,A: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.78 & ( ord_less_eq_rat @ B @ A ) )
% 8.55/4.78 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.78 & ( ord_less_eq_rat @ A @ B ) )
% 8.55/4.78 | ( A = zero_zero_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2599_power__strict__decreasing,axiom,
% 8.55/4.78 ! [N: nat,N5: nat,A: real] :
% 8.55/4.78 ( ( ord_less_nat @ N @ N5 )
% 8.55/4.78 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ( ord_less_real @ A @ one_one_real )
% 8.55/4.78 => ( ord_less_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2600_power__strict__decreasing,axiom,
% 8.55/4.78 ! [N: nat,N5: nat,A: rat] :
% 8.55/4.78 ( ( ord_less_nat @ N @ N5 )
% 8.55/4.78 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ( ord_less_rat @ A @ one_one_rat )
% 8.55/4.78 => ( ord_less_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2601_power__strict__decreasing,axiom,
% 8.55/4.78 ! [N: nat,N5: nat,A: nat] :
% 8.55/4.78 ( ( ord_less_nat @ N @ N5 )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ( ord_less_nat @ A @ one_one_nat )
% 8.55/4.78 => ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2602_power__strict__decreasing,axiom,
% 8.55/4.78 ! [N: nat,N5: nat,A: int] :
% 8.55/4.78 ( ( ord_less_nat @ N @ N5 )
% 8.55/4.78 => ( ( ord_less_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ( ord_less_int @ A @ one_one_int )
% 8.55/4.78 => ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2603_power__decreasing,axiom,
% 8.55/4.78 ! [N: nat,N5: nat,A: real] :
% 8.55/4.78 ( ( ord_less_eq_nat @ N @ N5 )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ( ord_less_eq_real @ A @ one_one_real )
% 8.55/4.78 => ( ord_less_eq_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2604_power__decreasing,axiom,
% 8.55/4.78 ! [N: nat,N5: nat,A: rat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ N @ N5 )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2605_power__decreasing,axiom,
% 8.55/4.78 ! [N: nat,N5: nat,A: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ N @ N5 )
% 8.55/4.78 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 8.55/4.78 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2606_power__decreasing,axiom,
% 8.55/4.78 ! [N: nat,N5: nat,A: int] :
% 8.55/4.78 ( ( ord_less_eq_nat @ N @ N5 )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ( ord_less_eq_int @ A @ one_one_int )
% 8.55/4.78 => ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2607_power__eq__imp__eq__base,axiom,
% 8.55/4.78 ! [A: real,N: nat,B: real] :
% 8.55/4.78 ( ( ( power_power_real @ A @ N )
% 8.55/4.78 = ( power_power_real @ B @ N ) )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( A = B ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2608_power__eq__imp__eq__base,axiom,
% 8.55/4.78 ! [A: rat,N: nat,B: rat] :
% 8.55/4.78 ( ( ( power_power_rat @ A @ N )
% 8.55/4.78 = ( power_power_rat @ B @ N ) )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( A = B ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2609_power__eq__imp__eq__base,axiom,
% 8.55/4.78 ! [A: nat,N: nat,B: nat] :
% 8.55/4.78 ( ( ( power_power_nat @ A @ N )
% 8.55/4.78 = ( power_power_nat @ B @ N ) )
% 8.55/4.78 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( A = B ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2610_power__eq__imp__eq__base,axiom,
% 8.55/4.78 ! [A: int,N: nat,B: int] :
% 8.55/4.78 ( ( ( power_power_int @ A @ N )
% 8.55/4.78 = ( power_power_int @ B @ N ) )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( A = B ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2611_power__eq__iff__eq__base,axiom,
% 8.55/4.78 ! [N: nat,A: real,B: real] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.55/4.78 => ( ( ( power_power_real @ A @ N )
% 8.55/4.78 = ( power_power_real @ B @ N ) )
% 8.55/4.78 = ( A = B ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2612_power__eq__iff__eq__base,axiom,
% 8.55/4.78 ! [N: nat,A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.55/4.78 => ( ( ( power_power_rat @ A @ N )
% 8.55/4.78 = ( power_power_rat @ B @ N ) )
% 8.55/4.78 = ( A = B ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2613_power__eq__iff__eq__base,axiom,
% 8.55/4.78 ! [N: nat,A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.55/4.78 => ( ( ( power_power_nat @ A @ N )
% 8.55/4.78 = ( power_power_nat @ B @ N ) )
% 8.55/4.78 = ( A = B ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2614_power__eq__iff__eq__base,axiom,
% 8.55/4.78 ! [N: nat,A: int,B: int] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.55/4.78 => ( ( ( power_power_int @ A @ N )
% 8.55/4.78 = ( power_power_int @ B @ N ) )
% 8.55/4.78 = ( A = B ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2615_one__less__power,axiom,
% 8.55/4.78 ! [A: real,N: nat] :
% 8.55/4.78 ( ( ord_less_real @ one_one_real @ A )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2616_one__less__power,axiom,
% 8.55/4.78 ! [A: rat,N: nat] :
% 8.55/4.78 ( ( ord_less_rat @ one_one_rat @ A )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2617_one__less__power,axiom,
% 8.55/4.78 ! [A: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_nat @ one_one_nat @ A )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2618_one__less__power,axiom,
% 8.55/4.78 ! [A: int,N: nat] :
% 8.55/4.78 ( ( ord_less_int @ one_one_int @ A )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2619_self__le__power,axiom,
% 8.55/4.78 ! [A: real,N: nat] :
% 8.55/4.78 ( ( ord_less_eq_real @ one_one_real @ A )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2620_self__le__power,axiom,
% 8.55/4.78 ! [A: rat,N: nat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2621_self__le__power,axiom,
% 8.55/4.78 ! [A: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2622_self__le__power,axiom,
% 8.55/4.78 ! [A: int,N: nat] :
% 8.55/4.78 ( ( ord_less_eq_int @ one_one_int @ A )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2623_zero__less__imp__eq__int,axiom,
% 8.55/4.78 ! [K: int] :
% 8.55/4.78 ( ( ord_less_int @ zero_zero_int @ K )
% 8.55/4.78 => ? [N2: nat] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 8.55/4.78 & ( K
% 8.55/4.78 = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2624_pos__int__cases,axiom,
% 8.55/4.78 ! [K: int] :
% 8.55/4.78 ( ( ord_less_int @ zero_zero_int @ K )
% 8.55/4.78 => ~ ! [N2: nat] :
% 8.55/4.78 ( ( K
% 8.55/4.78 = ( semiri1314217659103216013at_int @ N2 ) )
% 8.55/4.78 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2625_nat__less__add__iff2,axiom,
% 8.55/4.78 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.78 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 8.55/4.78 = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2626_nat__less__add__iff1,axiom,
% 8.55/4.78 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ J @ I )
% 8.55/4.78 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 8.55/4.78 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2627_zmult__zless__mono2__lemma,axiom,
% 8.55/4.78 ! [I: int,J: int,K: nat] :
% 8.55/4.78 ( ( ord_less_int @ I @ J )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.55/4.78 => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2628_le__imp__0__less,axiom,
% 8.55/4.78 ! [Z2: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.55/4.78 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2629_div__nat__eqI,axiom,
% 8.55/4.78 ! [N: nat,Q2: nat,M: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
% 8.55/4.78 => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
% 8.55/4.78 => ( ( divide_divide_nat @ M @ N )
% 8.55/4.78 = Q2 ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2630_less__eq__div__iff__mult__less__eq,axiom,
% 8.55/4.78 ! [Q2: nat,M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 8.55/4.78 => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
% 8.55/4.78 = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2631_unique__quotient__lemma__neg,axiom,
% 8.55/4.78 ! [B: int,Q4: int,R4: int,Q2: int,R: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 8.55/4.78 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 8.55/4.78 => ( ( ord_less_int @ B @ R )
% 8.55/4.78 => ( ( ord_less_int @ B @ R4 )
% 8.55/4.78 => ( ord_less_eq_int @ Q2 @ Q4 ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2632_unique__quotient__lemma,axiom,
% 8.55/4.78 ! [B: int,Q4: int,R4: int,Q2: int,R: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 8.55/4.78 => ( ( ord_less_int @ R4 @ B )
% 8.55/4.78 => ( ( ord_less_int @ R @ B )
% 8.55/4.78 => ( ord_less_eq_int @ Q4 @ Q2 ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2633_zdiv__mono2__neg__lemma,axiom,
% 8.55/4.78 ! [B: int,Q2: int,R: int,B6: int,Q4: int,R4: int] :
% 8.55/4.78 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R )
% 8.55/4.78 = ( plus_plus_int @ ( times_times_int @ B6 @ Q4 ) @ R4 ) )
% 8.55/4.78 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q4 ) @ R4 ) @ zero_zero_int )
% 8.55/4.78 => ( ( ord_less_int @ R @ B )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 8.55/4.78 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 8.55/4.78 => ( ( ord_less_eq_int @ B6 @ B )
% 8.55/4.78 => ( ord_less_eq_int @ Q4 @ Q2 ) ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2634_incr__mult__lemma,axiom,
% 8.55/4.78 ! [D: int,P: int > $o,K: int] :
% 8.55/4.78 ( ( ord_less_int @ zero_zero_int @ D )
% 8.55/4.78 => ( ! [X4: int] :
% 8.55/4.78 ( ( P @ X4 )
% 8.55/4.78 => ( P @ ( plus_plus_int @ X4 @ D ) ) )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.55/4.78 => ! [X5: int] :
% 8.55/4.78 ( ( P @ X5 )
% 8.55/4.78 => ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2635_zdiv__mono2__lemma,axiom,
% 8.55/4.78 ! [B: int,Q2: int,R: int,B6: int,Q4: int,R4: int] :
% 8.55/4.78 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R )
% 8.55/4.78 = ( plus_plus_int @ ( times_times_int @ B6 @ Q4 ) @ R4 ) )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q4 ) @ R4 ) )
% 8.55/4.78 => ( ( ord_less_int @ R4 @ B6 )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 8.55/4.78 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 8.55/4.78 => ( ( ord_less_eq_int @ B6 @ B )
% 8.55/4.78 => ( ord_less_eq_int @ Q2 @ Q4 ) ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2636_q__pos__lemma,axiom,
% 8.55/4.78 ! [B6: int,Q4: int,R4: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q4 ) @ R4 ) )
% 8.55/4.78 => ( ( ord_less_int @ R4 @ B6 )
% 8.55/4.78 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 8.55/4.78 => ( ord_less_eq_int @ zero_zero_int @ Q4 ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2637_real__archimedian__rdiv__eq__0,axiom,
% 8.55/4.78 ! [X: real,C: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ! [M2: nat] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ M2 )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ X ) @ C ) )
% 8.55/4.78 => ( X = zero_zero_real ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2638_decr__mult__lemma,axiom,
% 8.55/4.78 ! [D: int,P: int > $o,K: int] :
% 8.55/4.78 ( ( ord_less_int @ zero_zero_int @ D )
% 8.55/4.78 => ( ! [X4: int] :
% 8.55/4.78 ( ( P @ X4 )
% 8.55/4.78 => ( P @ ( minus_minus_int @ X4 @ D ) ) )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.55/4.78 => ! [X5: int] :
% 8.55/4.78 ( ( P @ X5 )
% 8.55/4.78 => ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2639_power__dvd__imp__le,axiom,
% 8.55/4.78 ! [I: nat,M: nat,N: nat] :
% 8.55/4.78 ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 8.55/4.78 => ( ( ord_less_nat @ one_one_nat @ I )
% 8.55/4.78 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2640_convex__bound__lt,axiom,
% 8.55/4.78 ! [X: real,A: real,Y: real,U: real,V: real] :
% 8.55/4.78 ( ( ord_less_real @ X @ A )
% 8.55/4.78 => ( ( ord_less_real @ Y @ A )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 8.55/4.78 => ( ( ( plus_plus_real @ U @ V )
% 8.55/4.78 = one_one_real )
% 8.55/4.78 => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2641_convex__bound__lt,axiom,
% 8.55/4.78 ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
% 8.55/4.78 ( ( ord_less_rat @ X @ A )
% 8.55/4.78 => ( ( ord_less_rat @ Y @ A )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 8.55/4.78 => ( ( ( plus_plus_rat @ U @ V )
% 8.55/4.78 = one_one_rat )
% 8.55/4.78 => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2642_convex__bound__lt,axiom,
% 8.55/4.78 ! [X: int,A: int,Y: int,U: int,V: int] :
% 8.55/4.78 ( ( ord_less_int @ X @ A )
% 8.55/4.78 => ( ( ord_less_int @ Y @ A )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 8.55/4.78 => ( ( ( plus_plus_int @ U @ V )
% 8.55/4.78 = one_one_int )
% 8.55/4.78 => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2643_divide__le__eq__numeral_I1_J,axiom,
% 8.55/4.78 ! [B: real,C: real,W: num] :
% 8.55/4.78 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 8.55/4.78 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 8.55/4.78 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2644_divide__le__eq__numeral_I1_J,axiom,
% 8.55/4.78 ! [B: rat,C: rat,W: num] :
% 8.55/4.78 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 8.55/4.78 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 8.55/4.78 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2645_le__divide__eq__numeral_I1_J,axiom,
% 8.55/4.78 ! [W: num,B: real,C: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 8.55/4.78 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 8.55/4.78 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2646_le__divide__eq__numeral_I1_J,axiom,
% 8.55/4.78 ! [W: num,B: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 8.55/4.78 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 8.55/4.78 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2647_of__bool__def,axiom,
% 8.55/4.78 ( zero_n1201886186963655149omplex
% 8.55/4.78 = ( ^ [P3: $o] : ( if_complex @ P3 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2648_of__bool__def,axiom,
% 8.55/4.78 ( zero_n3304061248610475627l_real
% 8.55/4.78 = ( ^ [P3: $o] : ( if_real @ P3 @ one_one_real @ zero_zero_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2649_of__bool__def,axiom,
% 8.55/4.78 ( zero_n2052037380579107095ol_rat
% 8.55/4.78 = ( ^ [P3: $o] : ( if_rat @ P3 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2650_of__bool__def,axiom,
% 8.55/4.78 ( zero_n2687167440665602831ol_nat
% 8.55/4.78 = ( ^ [P3: $o] : ( if_nat @ P3 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2651_of__bool__def,axiom,
% 8.55/4.78 ( zero_n2684676970156552555ol_int
% 8.55/4.78 = ( ^ [P3: $o] : ( if_int @ P3 @ one_one_int @ zero_zero_int ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2652_of__bool__def,axiom,
% 8.55/4.78 ( zero_n356916108424825756nteger
% 8.55/4.78 = ( ^ [P3: $o] : ( if_Code_integer @ P3 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2653_split__of__bool,axiom,
% 8.55/4.78 ! [P: complex > $o,P2: $o] :
% 8.55/4.78 ( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
% 8.55/4.78 = ( ( P2
% 8.55/4.78 => ( P @ one_one_complex ) )
% 8.55/4.78 & ( ~ P2
% 8.55/4.78 => ( P @ zero_zero_complex ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2654_split__of__bool,axiom,
% 8.55/4.78 ! [P: real > $o,P2: $o] :
% 8.55/4.78 ( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
% 8.55/4.78 = ( ( P2
% 8.55/4.78 => ( P @ one_one_real ) )
% 8.55/4.78 & ( ~ P2
% 8.55/4.78 => ( P @ zero_zero_real ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2655_split__of__bool,axiom,
% 8.55/4.78 ! [P: rat > $o,P2: $o] :
% 8.55/4.78 ( ( P @ ( zero_n2052037380579107095ol_rat @ P2 ) )
% 8.55/4.78 = ( ( P2
% 8.55/4.78 => ( P @ one_one_rat ) )
% 8.55/4.78 & ( ~ P2
% 8.55/4.78 => ( P @ zero_zero_rat ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2656_split__of__bool,axiom,
% 8.55/4.78 ! [P: nat > $o,P2: $o] :
% 8.55/4.78 ( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
% 8.55/4.78 = ( ( P2
% 8.55/4.78 => ( P @ one_one_nat ) )
% 8.55/4.78 & ( ~ P2
% 8.55/4.78 => ( P @ zero_zero_nat ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2657_split__of__bool,axiom,
% 8.55/4.78 ! [P: int > $o,P2: $o] :
% 8.55/4.78 ( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
% 8.55/4.78 = ( ( P2
% 8.55/4.78 => ( P @ one_one_int ) )
% 8.55/4.78 & ( ~ P2
% 8.55/4.78 => ( P @ zero_zero_int ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2658_split__of__bool,axiom,
% 8.55/4.78 ! [P: code_integer > $o,P2: $o] :
% 8.55/4.78 ( ( P @ ( zero_n356916108424825756nteger @ P2 ) )
% 8.55/4.78 = ( ( P2
% 8.55/4.78 => ( P @ one_one_Code_integer ) )
% 8.55/4.78 & ( ~ P2
% 8.55/4.78 => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2659_split__of__bool__asm,axiom,
% 8.55/4.78 ! [P: complex > $o,P2: $o] :
% 8.55/4.78 ( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
% 8.55/4.78 = ( ~ ( ( P2
% 8.55/4.78 & ~ ( P @ one_one_complex ) )
% 8.55/4.78 | ( ~ P2
% 8.55/4.78 & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2660_split__of__bool__asm,axiom,
% 8.55/4.78 ! [P: real > $o,P2: $o] :
% 8.55/4.78 ( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
% 8.55/4.78 = ( ~ ( ( P2
% 8.55/4.78 & ~ ( P @ one_one_real ) )
% 8.55/4.78 | ( ~ P2
% 8.55/4.78 & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2661_split__of__bool__asm,axiom,
% 8.55/4.78 ! [P: rat > $o,P2: $o] :
% 8.55/4.78 ( ( P @ ( zero_n2052037380579107095ol_rat @ P2 ) )
% 8.55/4.78 = ( ~ ( ( P2
% 8.55/4.78 & ~ ( P @ one_one_rat ) )
% 8.55/4.78 | ( ~ P2
% 8.55/4.78 & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2662_split__of__bool__asm,axiom,
% 8.55/4.78 ! [P: nat > $o,P2: $o] :
% 8.55/4.78 ( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
% 8.55/4.78 = ( ~ ( ( P2
% 8.55/4.78 & ~ ( P @ one_one_nat ) )
% 8.55/4.78 | ( ~ P2
% 8.55/4.78 & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2663_split__of__bool__asm,axiom,
% 8.55/4.78 ! [P: int > $o,P2: $o] :
% 8.55/4.78 ( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
% 8.55/4.78 = ( ~ ( ( P2
% 8.55/4.78 & ~ ( P @ one_one_int ) )
% 8.55/4.78 | ( ~ P2
% 8.55/4.78 & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2664_split__of__bool__asm,axiom,
% 8.55/4.78 ! [P: code_integer > $o,P2: $o] :
% 8.55/4.78 ( ( P @ ( zero_n356916108424825756nteger @ P2 ) )
% 8.55/4.78 = ( ~ ( ( P2
% 8.55/4.78 & ~ ( P @ one_one_Code_integer ) )
% 8.55/4.78 | ( ~ P2
% 8.55/4.78 & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2665_inverse__of__nat__le,axiom,
% 8.55/4.78 ! [N: nat,M: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.78 => ( ( N != zero_zero_nat )
% 8.55/4.78 => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2666_inverse__of__nat__le,axiom,
% 8.55/4.78 ! [N: nat,M: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.78 => ( ( N != zero_zero_nat )
% 8.55/4.78 => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2667_le__div__geq,axiom,
% 8.55/4.78 ! [N: nat,M: nat] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.78 => ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.78 => ( ( divide_divide_nat @ M @ N )
% 8.55/4.78 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2668_split__div_H,axiom,
% 8.55/4.78 ! [P: nat > $o,M: nat,N: nat] :
% 8.55/4.78 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 8.55/4.78 = ( ( ( N = zero_zero_nat )
% 8.55/4.78 & ( P @ zero_zero_nat ) )
% 8.55/4.78 | ? [Q5: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q5 ) @ M )
% 8.55/4.78 & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q5 ) ) )
% 8.55/4.78 & ( P @ Q5 ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2669_zero__less__numeral,axiom,
% 8.55/4.78 ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2670_zero__less__numeral,axiom,
% 8.55/4.78 ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2671_zero__less__numeral,axiom,
% 8.55/4.78 ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2672_zero__less__numeral,axiom,
% 8.55/4.78 ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2673_not__numeral__less__zero,axiom,
% 8.55/4.78 ! [N: num] :
% 8.55/4.78 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2674_not__numeral__less__zero,axiom,
% 8.55/4.78 ! [N: num] :
% 8.55/4.78 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2675_not__numeral__less__zero,axiom,
% 8.55/4.78 ! [N: num] :
% 8.55/4.78 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2676_not__numeral__less__zero,axiom,
% 8.55/4.78 ! [N: num] :
% 8.55/4.78 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2677_mult__neg__neg,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.78 => ( ( ord_less_real @ B @ zero_zero_real )
% 8.55/4.78 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2678_mult__neg__neg,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.78 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2679_mult__neg__neg,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ A @ zero_zero_int )
% 8.55/4.78 => ( ( ord_less_int @ B @ zero_zero_int )
% 8.55/4.78 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2680_not__square__less__zero,axiom,
% 8.55/4.78 ! [A: real] :
% 8.55/4.78 ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2681_not__square__less__zero,axiom,
% 8.55/4.78 ! [A: rat] :
% 8.55/4.78 ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2682_not__square__less__zero,axiom,
% 8.55/4.78 ! [A: int] :
% 8.55/4.78 ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2683_mult__less__0__iff,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.78 & ( ord_less_real @ B @ zero_zero_real ) )
% 8.55/4.78 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.78 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2684_mult__less__0__iff,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.78 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 8.55/4.78 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.78 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2685_mult__less__0__iff,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 8.55/4.78 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 8.55/4.78 & ( ord_less_int @ B @ zero_zero_int ) )
% 8.55/4.78 | ( ( ord_less_int @ A @ zero_zero_int )
% 8.55/4.78 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2686_mult__neg__pos,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.78 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.55/4.78 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2687_mult__neg__pos,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.78 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 8.55/4.78 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2688_mult__neg__pos,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_nat @ A @ zero_zero_nat )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.55/4.78 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2689_mult__neg__pos,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ A @ zero_zero_int )
% 8.55/4.78 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.55/4.78 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2690_mult__pos__neg,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ( ord_less_real @ B @ zero_zero_real )
% 8.55/4.78 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2691_mult__pos__neg,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2692_mult__pos__neg,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 8.55/4.78 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2693_mult__pos__neg,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ( ord_less_int @ B @ zero_zero_int )
% 8.55/4.78 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2694_mult__pos__pos,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.55/4.78 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2695_mult__pos__pos,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 8.55/4.78 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2696_mult__pos__pos,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.55/4.78 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2697_mult__pos__pos,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.55/4.78 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2698_mult__pos__neg2,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ( ord_less_real @ B @ zero_zero_real )
% 8.55/4.78 => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2699_mult__pos__neg2,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2700_mult__pos__neg2,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 8.55/4.78 => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2701_mult__pos__neg2,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ( ord_less_int @ B @ zero_zero_int )
% 8.55/4.78 => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2702_zero__less__mult__iff,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.78 & ( ord_less_real @ zero_zero_real @ B ) )
% 8.55/4.78 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.78 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2703_zero__less__mult__iff,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.78 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 8.55/4.78 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.78 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2704_zero__less__mult__iff,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 8.55/4.78 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 8.55/4.78 & ( ord_less_int @ zero_zero_int @ B ) )
% 8.55/4.78 | ( ( ord_less_int @ A @ zero_zero_int )
% 8.55/4.78 & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2705_zero__less__mult__pos,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 8.55/4.78 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2706_zero__less__mult__pos,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 8.55/4.78 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2707_zero__less__mult__pos,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2708_zero__less__mult__pos,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 8.55/4.78 => ( ( ord_less_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2709_zero__less__mult__pos2,axiom,
% 8.55/4.78 ! [B: real,A: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
% 8.55/4.78 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2710_zero__less__mult__pos2,axiom,
% 8.55/4.78 ! [B: rat,A: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
% 8.55/4.78 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2711_zero__less__mult__pos2,axiom,
% 8.55/4.78 ! [B: nat,A: nat] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2712_zero__less__mult__pos2,axiom,
% 8.55/4.78 ! [B: int,A: int] :
% 8.55/4.78 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
% 8.55/4.78 => ( ( ord_less_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2713_mult__less__cancel__left__neg,axiom,
% 8.55/4.78 ! [C: real,A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.55/4.78 = ( ord_less_real @ B @ A ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2714_mult__less__cancel__left__neg,axiom,
% 8.55/4.78 ! [C: rat,A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.55/4.78 = ( ord_less_rat @ B @ A ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2715_mult__less__cancel__left__neg,axiom,
% 8.55/4.78 ! [C: int,A: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ C @ zero_zero_int )
% 8.55/4.78 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.55/4.78 = ( ord_less_int @ B @ A ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2716_mult__less__cancel__left__pos,axiom,
% 8.55/4.78 ! [C: real,A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.55/4.78 = ( ord_less_real @ A @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2717_mult__less__cancel__left__pos,axiom,
% 8.55/4.78 ! [C: rat,A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.55/4.78 = ( ord_less_rat @ A @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2718_mult__less__cancel__left__pos,axiom,
% 8.55/4.78 ! [C: int,A: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.55/4.78 = ( ord_less_int @ A @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2719_mult__strict__left__mono__neg,axiom,
% 8.55/4.78 ! [B: real,A: real,C: real] :
% 8.55/4.78 ( ( ord_less_real @ B @ A )
% 8.55/4.78 => ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2720_mult__strict__left__mono__neg,axiom,
% 8.55/4.78 ! [B: rat,A: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_rat @ B @ A )
% 8.55/4.78 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2721_mult__strict__left__mono__neg,axiom,
% 8.55/4.78 ! [B: int,A: int,C: int] :
% 8.55/4.78 ( ( ord_less_int @ B @ A )
% 8.55/4.78 => ( ( ord_less_int @ C @ zero_zero_int )
% 8.55/4.78 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2722_mult__strict__left__mono,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real] :
% 8.55/4.78 ( ( ord_less_real @ A @ B )
% 8.55/4.78 => ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2723_mult__strict__left__mono,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_rat @ A @ B )
% 8.55/4.78 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2724_mult__strict__left__mono,axiom,
% 8.55/4.78 ! [A: nat,B: nat,C: nat] :
% 8.55/4.78 ( ( ord_less_nat @ A @ B )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 8.55/4.78 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2725_mult__strict__left__mono,axiom,
% 8.55/4.78 ! [A: int,B: int,C: int] :
% 8.55/4.78 ( ( ord_less_int @ A @ B )
% 8.55/4.78 => ( ( ord_less_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2726_mult__less__cancel__left__disj,axiom,
% 8.55/4.78 ! [C: real,A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 & ( ord_less_real @ A @ B ) )
% 8.55/4.78 | ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 & ( ord_less_real @ B @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2727_mult__less__cancel__left__disj,axiom,
% 8.55/4.78 ! [C: rat,A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 & ( ord_less_rat @ A @ B ) )
% 8.55/4.78 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2728_mult__less__cancel__left__disj,axiom,
% 8.55/4.78 ! [C: int,A: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.55/4.78 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 8.55/4.78 & ( ord_less_int @ A @ B ) )
% 8.55/4.78 | ( ( ord_less_int @ C @ zero_zero_int )
% 8.55/4.78 & ( ord_less_int @ B @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2729_mult__strict__right__mono__neg,axiom,
% 8.55/4.78 ! [B: real,A: real,C: real] :
% 8.55/4.78 ( ( ord_less_real @ B @ A )
% 8.55/4.78 => ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2730_mult__strict__right__mono__neg,axiom,
% 8.55/4.78 ! [B: rat,A: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_rat @ B @ A )
% 8.55/4.78 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2731_mult__strict__right__mono__neg,axiom,
% 8.55/4.78 ! [B: int,A: int,C: int] :
% 8.55/4.78 ( ( ord_less_int @ B @ A )
% 8.55/4.78 => ( ( ord_less_int @ C @ zero_zero_int )
% 8.55/4.78 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2732_mult__strict__right__mono,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real] :
% 8.55/4.78 ( ( ord_less_real @ A @ B )
% 8.55/4.78 => ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2733_mult__strict__right__mono,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_rat @ A @ B )
% 8.55/4.78 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2734_mult__strict__right__mono,axiom,
% 8.55/4.78 ! [A: nat,B: nat,C: nat] :
% 8.55/4.78 ( ( ord_less_nat @ A @ B )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 8.55/4.78 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2735_mult__strict__right__mono,axiom,
% 8.55/4.78 ! [A: int,B: int,C: int] :
% 8.55/4.78 ( ( ord_less_int @ A @ B )
% 8.55/4.78 => ( ( ord_less_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2736_mult__less__cancel__right__disj,axiom,
% 8.55/4.78 ! [A: real,C: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 & ( ord_less_real @ A @ B ) )
% 8.55/4.78 | ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 & ( ord_less_real @ B @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2737_mult__less__cancel__right__disj,axiom,
% 8.55/4.78 ! [A: rat,C: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 & ( ord_less_rat @ A @ B ) )
% 8.55/4.78 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2738_mult__less__cancel__right__disj,axiom,
% 8.55/4.78 ! [A: int,C: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 8.55/4.78 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 8.55/4.78 & ( ord_less_int @ A @ B ) )
% 8.55/4.78 | ( ( ord_less_int @ C @ zero_zero_int )
% 8.55/4.78 & ( ord_less_int @ B @ A ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2739_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real] :
% 8.55/4.78 ( ( ord_less_real @ A @ B )
% 8.55/4.78 => ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2740_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_rat @ A @ B )
% 8.55/4.78 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2741_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 8.55/4.78 ! [A: nat,B: nat,C: nat] :
% 8.55/4.78 ( ( ord_less_nat @ A @ B )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 8.55/4.78 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2742_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 8.55/4.78 ! [A: int,B: int,C: int] :
% 8.55/4.78 ( ( ord_less_int @ A @ B )
% 8.55/4.78 => ( ( ord_less_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2743_zdiff__int__split,axiom,
% 8.55/4.78 ! [P: int > $o,X: nat,Y: nat] :
% 8.55/4.78 ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
% 8.55/4.78 = ( ( ( ord_less_eq_nat @ Y @ X )
% 8.55/4.78 => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
% 8.55/4.78 & ( ( ord_less_nat @ X @ Y )
% 8.55/4.78 => ( P @ zero_zero_int ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2744_int__div__pos__eq,axiom,
% 8.55/4.78 ! [A: int,B: int,Q2: int,R: int] :
% 8.55/4.78 ( ( A
% 8.55/4.78 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 8.55/4.78 => ( ( ord_less_int @ R @ B )
% 8.55/4.78 => ( ( divide_divide_int @ A @ B )
% 8.55/4.78 = Q2 ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2745_int__div__neg__eq,axiom,
% 8.55/4.78 ! [A: int,B: int,Q2: int,R: int] :
% 8.55/4.78 ( ( A
% 8.55/4.78 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 8.55/4.78 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 8.55/4.78 => ( ( ord_less_int @ B @ R )
% 8.55/4.78 => ( ( divide_divide_int @ A @ B )
% 8.55/4.78 = Q2 ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2746_split__zdiv,axiom,
% 8.55/4.78 ! [P: int > $o,N: int,K: int] :
% 8.55/4.78 ( ( P @ ( divide_divide_int @ N @ K ) )
% 8.55/4.78 = ( ( ( K = zero_zero_int )
% 8.55/4.78 => ( P @ zero_zero_int ) )
% 8.55/4.78 & ( ( ord_less_int @ zero_zero_int @ K )
% 8.55/4.78 => ! [I4: int,J3: int] :
% 8.55/4.78 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 8.55/4.78 & ( ord_less_int @ J3 @ K )
% 8.55/4.78 & ( N
% 8.55/4.78 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 8.55/4.78 => ( P @ I4 ) ) )
% 8.55/4.78 & ( ( ord_less_int @ K @ zero_zero_int )
% 8.55/4.78 => ! [I4: int,J3: int] :
% 8.55/4.78 ( ( ( ord_less_int @ K @ J3 )
% 8.55/4.78 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 8.55/4.78 & ( N
% 8.55/4.78 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 8.55/4.78 => ( P @ I4 ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2747_zero__less__one,axiom,
% 8.55/4.78 ord_less_real @ zero_zero_real @ one_one_real ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2748_zero__less__one,axiom,
% 8.55/4.78 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2749_zero__less__one,axiom,
% 8.55/4.78 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2750_zero__less__one,axiom,
% 8.55/4.78 ord_less_int @ zero_zero_int @ one_one_int ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2751_not__one__less__zero,axiom,
% 8.55/4.78 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2752_not__one__less__zero,axiom,
% 8.55/4.78 ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2753_not__one__less__zero,axiom,
% 8.55/4.78 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2754_not__one__less__zero,axiom,
% 8.55/4.78 ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2755_less__numeral__extra_I1_J,axiom,
% 8.55/4.78 ord_less_real @ zero_zero_real @ one_one_real ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2756_less__numeral__extra_I1_J,axiom,
% 8.55/4.78 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2757_less__numeral__extra_I1_J,axiom,
% 8.55/4.78 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2758_less__numeral__extra_I1_J,axiom,
% 8.55/4.78 ord_less_int @ zero_zero_int @ one_one_int ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2759_pos__add__strict,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ( ord_less_real @ B @ C )
% 8.55/4.78 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2760_pos__add__strict,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ( ord_less_rat @ B @ C )
% 8.55/4.78 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2761_pos__add__strict,axiom,
% 8.55/4.78 ! [A: nat,B: nat,C: nat] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ( ord_less_nat @ B @ C )
% 8.55/4.78 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2762_pos__add__strict,axiom,
% 8.55/4.78 ! [A: int,B: int,C: int] :
% 8.55/4.78 ( ( ord_less_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ( ord_less_int @ B @ C )
% 8.55/4.78 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2763_canonically__ordered__monoid__add__class_OlessE,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_nat @ A @ B )
% 8.55/4.78 => ~ ! [C3: nat] :
% 8.55/4.78 ( ( B
% 8.55/4.78 = ( plus_plus_nat @ A @ C3 ) )
% 8.55/4.78 => ( C3 = zero_zero_nat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2764_add__pos__pos,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.55/4.78 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2765_add__pos__pos,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 8.55/4.78 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2766_add__pos__pos,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.55/4.78 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2767_add__pos__pos,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.55/4.78 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2768_add__neg__neg,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.78 => ( ( ord_less_real @ B @ zero_zero_real )
% 8.55/4.78 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2769_add__neg__neg,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.78 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2770_add__neg__neg,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_nat @ A @ zero_zero_nat )
% 8.55/4.78 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 8.55/4.78 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2771_add__neg__neg,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ A @ zero_zero_int )
% 8.55/4.78 => ( ( ord_less_int @ B @ zero_zero_int )
% 8.55/4.78 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2772_add__less__zeroD,axiom,
% 8.55/4.78 ! [X: real,Y: real] :
% 8.55/4.78 ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 8.55/4.78 => ( ( ord_less_real @ X @ zero_zero_real )
% 8.55/4.78 | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2773_add__less__zeroD,axiom,
% 8.55/4.78 ! [X: rat,Y: rat] :
% 8.55/4.78 ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
% 8.55/4.78 => ( ( ord_less_rat @ X @ zero_zero_rat )
% 8.55/4.78 | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2774_add__less__zeroD,axiom,
% 8.55/4.78 ! [X: int,Y: int] :
% 8.55/4.78 ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
% 8.55/4.78 => ( ( ord_less_int @ X @ zero_zero_int )
% 8.55/4.78 | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2775_less__iff__diff__less__0,axiom,
% 8.55/4.78 ( ord_less_real
% 8.55/4.78 = ( ^ [A2: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2776_less__iff__diff__less__0,axiom,
% 8.55/4.78 ( ord_less_rat
% 8.55/4.78 = ( ^ [A2: rat,B2: rat] : ( ord_less_rat @ ( minus_minus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2777_less__iff__diff__less__0,axiom,
% 8.55/4.78 ( ord_less_int
% 8.55/4.78 = ( ^ [A2: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2778_divide__strict__right__mono__neg,axiom,
% 8.55/4.78 ! [B: real,A: real,C: real] :
% 8.55/4.78 ( ( ord_less_real @ B @ A )
% 8.55/4.78 => ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2779_divide__strict__right__mono__neg,axiom,
% 8.55/4.78 ! [B: rat,A: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_rat @ B @ A )
% 8.55/4.78 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2780_divide__strict__right__mono,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real] :
% 8.55/4.78 ( ( ord_less_real @ A @ B )
% 8.55/4.78 => ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2781_divide__strict__right__mono,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_rat @ A @ B )
% 8.55/4.78 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2782_zero__less__divide__iff,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.78 & ( ord_less_real @ zero_zero_real @ B ) )
% 8.55/4.78 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.78 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2783_zero__less__divide__iff,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.78 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 8.55/4.78 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.78 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2784_divide__less__cancel,axiom,
% 8.55/4.78 ! [A: real,C: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_real @ A @ B ) )
% 8.55/4.78 & ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_real @ B @ A ) )
% 8.55/4.78 & ( C != zero_zero_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2785_divide__less__cancel,axiom,
% 8.55/4.78 ! [A: rat,C: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_rat @ A @ B ) )
% 8.55/4.78 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_rat @ B @ A ) )
% 8.55/4.78 & ( C != zero_zero_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2786_divide__less__0__iff,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 8.55/4.78 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.78 & ( ord_less_real @ B @ zero_zero_real ) )
% 8.55/4.78 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.78 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2787_divide__less__0__iff,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 8.55/4.78 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.78 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 8.55/4.78 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.78 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2788_divide__pos__pos,axiom,
% 8.55/4.78 ! [X: real,Y: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ X )
% 8.55/4.78 => ( ( ord_less_real @ zero_zero_real @ Y )
% 8.55/4.78 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2789_divide__pos__pos,axiom,
% 8.55/4.78 ! [X: rat,Y: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ X )
% 8.55/4.78 => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 8.55/4.78 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2790_divide__pos__neg,axiom,
% 8.55/4.78 ! [X: real,Y: real] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ X )
% 8.55/4.78 => ( ( ord_less_real @ Y @ zero_zero_real )
% 8.55/4.78 => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2791_divide__pos__neg,axiom,
% 8.55/4.78 ! [X: rat,Y: rat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ X )
% 8.55/4.78 => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2792_divide__neg__pos,axiom,
% 8.55/4.78 ! [X: real,Y: real] :
% 8.55/4.78 ( ( ord_less_real @ X @ zero_zero_real )
% 8.55/4.78 => ( ( ord_less_real @ zero_zero_real @ Y )
% 8.55/4.78 => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2793_divide__neg__pos,axiom,
% 8.55/4.78 ! [X: rat,Y: rat] :
% 8.55/4.78 ( ( ord_less_rat @ X @ zero_zero_rat )
% 8.55/4.78 => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 8.55/4.78 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2794_divide__neg__neg,axiom,
% 8.55/4.78 ! [X: real,Y: real] :
% 8.55/4.78 ( ( ord_less_real @ X @ zero_zero_real )
% 8.55/4.78 => ( ( ord_less_real @ Y @ zero_zero_real )
% 8.55/4.78 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2795_divide__neg__neg,axiom,
% 8.55/4.78 ! [X: rat,Y: rat] :
% 8.55/4.78 ( ( ord_less_rat @ X @ zero_zero_rat )
% 8.55/4.78 => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2796_not__numeral__less__one,axiom,
% 8.55/4.78 ! [N: num] :
% 8.55/4.78 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2797_not__numeral__less__one,axiom,
% 8.55/4.78 ! [N: num] :
% 8.55/4.78 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2798_not__numeral__less__one,axiom,
% 8.55/4.78 ! [N: num] :
% 8.55/4.78 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2799_not__numeral__less__one,axiom,
% 8.55/4.78 ! [N: num] :
% 8.55/4.78 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2800_less__1__mult,axiom,
% 8.55/4.78 ! [M: real,N: real] :
% 8.55/4.78 ( ( ord_less_real @ one_one_real @ M )
% 8.55/4.78 => ( ( ord_less_real @ one_one_real @ N )
% 8.55/4.78 => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2801_less__1__mult,axiom,
% 8.55/4.78 ! [M: rat,N: rat] :
% 8.55/4.78 ( ( ord_less_rat @ one_one_rat @ M )
% 8.55/4.78 => ( ( ord_less_rat @ one_one_rat @ N )
% 8.55/4.78 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2802_less__1__mult,axiom,
% 8.55/4.78 ! [M: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_nat @ one_one_nat @ M )
% 8.55/4.78 => ( ( ord_less_nat @ one_one_nat @ N )
% 8.55/4.78 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2803_less__1__mult,axiom,
% 8.55/4.78 ! [M: int,N: int] :
% 8.55/4.78 ( ( ord_less_int @ one_one_int @ M )
% 8.55/4.78 => ( ( ord_less_int @ one_one_int @ N )
% 8.55/4.78 => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2804_zero__le__numeral,axiom,
% 8.55/4.78 ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2805_zero__le__numeral,axiom,
% 8.55/4.78 ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2806_zero__le__numeral,axiom,
% 8.55/4.78 ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2807_zero__le__numeral,axiom,
% 8.55/4.78 ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2808_not__numeral__le__zero,axiom,
% 8.55/4.78 ! [N: num] :
% 8.55/4.78 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2809_not__numeral__le__zero,axiom,
% 8.55/4.78 ! [N: num] :
% 8.55/4.78 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2810_not__numeral__le__zero,axiom,
% 8.55/4.78 ! [N: num] :
% 8.55/4.78 ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2811_not__numeral__le__zero,axiom,
% 8.55/4.78 ! [N: num] :
% 8.55/4.78 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2812_zero__less__power,axiom,
% 8.55/4.78 ! [A: real,N: nat] :
% 8.55/4.78 ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2813_zero__less__power,axiom,
% 8.55/4.78 ! [A: rat,N: nat] :
% 8.55/4.78 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2814_zero__less__power,axiom,
% 8.55/4.78 ! [A: nat,N: nat] :
% 8.55/4.78 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2815_zero__less__power,axiom,
% 8.55/4.78 ! [A: int,N: nat] :
% 8.55/4.78 ( ( ord_less_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2816_mult__mono,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real,D: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_real @ C @ D )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2817_mult__mono,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_rat @ C @ D )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2818_mult__mono,axiom,
% 8.55/4.78 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_nat @ C @ D )
% 8.55/4.78 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.55/4.78 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.55/4.78 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2819_mult__mono,axiom,
% 8.55/4.78 ! [A: int,B: int,C: int,D: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_int @ C @ D )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2820_mult__mono_H,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real,D: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_real @ C @ D )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2821_mult__mono_H,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_rat @ C @ D )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2822_mult__mono_H,axiom,
% 8.55/4.78 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_nat @ C @ D )
% 8.55/4.78 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.55/4.78 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2823_mult__mono_H,axiom,
% 8.55/4.78 ! [A: int,B: int,C: int,D: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_int @ C @ D )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2824_zero__le__square,axiom,
% 8.55/4.78 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2825_zero__le__square,axiom,
% 8.55/4.78 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2826_zero__le__square,axiom,
% 8.55/4.78 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2827_split__mult__pos__le,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.78 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 8.55/4.78 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.55/4.78 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 8.55/4.78 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2828_split__mult__pos__le,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.78 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 8.55/4.78 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.55/4.78 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 8.55/4.78 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2829_split__mult__pos__le,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.78 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 8.55/4.78 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.55/4.78 & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 8.55/4.78 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2830_mult__left__mono__neg,axiom,
% 8.55/4.78 ! [B: real,A: real,C: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ B @ A )
% 8.55/4.78 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2831_mult__left__mono__neg,axiom,
% 8.55/4.78 ! [B: rat,A: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ B @ A )
% 8.55/4.78 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2832_mult__left__mono__neg,axiom,
% 8.55/4.78 ! [B: int,A: int,C: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ B @ A )
% 8.55/4.78 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.55/4.78 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2833_mult__nonpos__nonpos,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.55/4.78 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2834_mult__nonpos__nonpos,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.55/4.78 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2835_mult__nonpos__nonpos,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.55/4.78 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 8.55/4.78 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2836_mult__left__mono,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2837_mult__left__mono,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2838_mult__left__mono,axiom,
% 8.55/4.78 ! [A: nat,B: nat,C: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.55/4.78 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2839_mult__left__mono,axiom,
% 8.55/4.78 ! [A: int,B: int,C: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2840_mult__right__mono__neg,axiom,
% 8.55/4.78 ! [B: real,A: real,C: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ B @ A )
% 8.55/4.78 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2841_mult__right__mono__neg,axiom,
% 8.55/4.78 ! [B: rat,A: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ B @ A )
% 8.55/4.78 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2842_mult__right__mono__neg,axiom,
% 8.55/4.78 ! [B: int,A: int,C: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ B @ A )
% 8.55/4.78 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.55/4.78 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2843_mult__right__mono,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2844_mult__right__mono,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2845_mult__right__mono,axiom,
% 8.55/4.78 ! [A: nat,B: nat,C: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.55/4.78 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2846_mult__right__mono,axiom,
% 8.55/4.78 ! [A: int,B: int,C: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2847_mult__le__0__iff,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 8.55/4.78 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.78 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 8.55/4.78 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.55/4.78 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2848_mult__le__0__iff,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 8.55/4.78 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.78 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 8.55/4.78 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.55/4.78 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2849_mult__le__0__iff,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 8.55/4.78 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.78 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 8.55/4.78 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.55/4.78 & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2850_split__mult__neg__le,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.78 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 8.55/4.78 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.55/4.78 & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2851_split__mult__neg__le,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.78 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 8.55/4.78 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.55/4.78 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 8.55/4.78 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2852_split__mult__neg__le,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.78 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 8.55/4.78 | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 8.55/4.78 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 8.55/4.78 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2853_split__mult__neg__le,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.78 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 8.55/4.78 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.55/4.78 & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 8.55/4.78 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2854_mult__nonneg__nonneg,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.55/4.78 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2855_mult__nonneg__nonneg,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.55/4.78 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2856_mult__nonneg__nonneg,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.55/4.78 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2857_mult__nonneg__nonneg,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.55/4.78 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2858_mult__nonneg__nonpos,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2859_mult__nonneg__nonpos,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2860_mult__nonneg__nonpos,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 8.55/4.78 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2861_mult__nonneg__nonpos,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 8.55/4.78 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2862_mult__nonpos__nonneg,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2863_mult__nonpos__nonneg,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.55/4.78 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2864_mult__nonpos__nonneg,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 8.55/4.78 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.55/4.78 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2865_mult__nonpos__nonneg,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.55/4.78 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2866_mult__nonneg__nonpos2,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2867_mult__nonneg__nonpos2,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2868_mult__nonneg__nonpos2,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 8.55/4.78 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2869_mult__nonneg__nonpos2,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 8.55/4.78 => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2870_zero__le__mult__iff,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 8.55/4.78 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.78 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 8.55/4.78 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.55/4.78 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2871_zero__le__mult__iff,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 8.55/4.78 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.78 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 8.55/4.78 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.55/4.78 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2872_zero__le__mult__iff,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 8.55/4.78 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.78 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 8.55/4.78 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.55/4.78 & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2873_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2874_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2875_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 8.55/4.78 ! [A: nat,B: nat,C: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.55/4.78 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2876_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 8.55/4.78 ! [A: int,B: int,C: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2877_add__mono1,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ A @ B )
% 8.55/4.78 => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2878_add__mono1,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ A @ B )
% 8.55/4.78 => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2879_add__mono1,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_nat @ A @ B )
% 8.55/4.78 => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2880_add__mono1,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ A @ B )
% 8.55/4.78 => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2881_less__add__one,axiom,
% 8.55/4.78 ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2882_less__add__one,axiom,
% 8.55/4.78 ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2883_less__add__one,axiom,
% 8.55/4.78 ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2884_less__add__one,axiom,
% 8.55/4.78 ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2885_zero__less__one__class_Ozero__le__one,axiom,
% 8.55/4.78 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2886_zero__less__one__class_Ozero__le__one,axiom,
% 8.55/4.78 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2887_zero__less__one__class_Ozero__le__one,axiom,
% 8.55/4.78 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2888_zero__less__one__class_Ozero__le__one,axiom,
% 8.55/4.78 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2889_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 8.55/4.78 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2890_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 8.55/4.78 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2891_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 8.55/4.78 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2892_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 8.55/4.78 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2893_not__one__le__zero,axiom,
% 8.55/4.78 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2894_not__one__le__zero,axiom,
% 8.55/4.78 ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2895_not__one__le__zero,axiom,
% 8.55/4.78 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2896_not__one__le__zero,axiom,
% 8.55/4.78 ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2897_add__nonpos__eq__0__iff,axiom,
% 8.55/4.78 ! [X: real,Y: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 8.55/4.78 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 8.55/4.78 => ( ( ( plus_plus_real @ X @ Y )
% 8.55/4.78 = zero_zero_real )
% 8.55/4.78 = ( ( X = zero_zero_real )
% 8.55/4.78 & ( Y = zero_zero_real ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2898_add__nonpos__eq__0__iff,axiom,
% 8.55/4.78 ! [X: rat,Y: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 8.55/4.78 => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 8.55/4.78 => ( ( ( plus_plus_rat @ X @ Y )
% 8.55/4.78 = zero_zero_rat )
% 8.55/4.78 = ( ( X = zero_zero_rat )
% 8.55/4.78 & ( Y = zero_zero_rat ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2899_add__nonpos__eq__0__iff,axiom,
% 8.55/4.78 ! [X: nat,Y: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 8.55/4.78 => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
% 8.55/4.78 => ( ( ( plus_plus_nat @ X @ Y )
% 8.55/4.78 = zero_zero_nat )
% 8.55/4.78 = ( ( X = zero_zero_nat )
% 8.55/4.78 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2900_add__nonpos__eq__0__iff,axiom,
% 8.55/4.78 ! [X: int,Y: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ X @ zero_zero_int )
% 8.55/4.78 => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 8.55/4.78 => ( ( ( plus_plus_int @ X @ Y )
% 8.55/4.78 = zero_zero_int )
% 8.55/4.78 = ( ( X = zero_zero_int )
% 8.55/4.78 & ( Y = zero_zero_int ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2901_add__nonneg__eq__0__iff,axiom,
% 8.55/4.78 ! [X: real,Y: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.55/4.78 => ( ( ( plus_plus_real @ X @ Y )
% 8.55/4.78 = zero_zero_real )
% 8.55/4.78 = ( ( X = zero_zero_real )
% 8.55/4.78 & ( Y = zero_zero_real ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2902_add__nonneg__eq__0__iff,axiom,
% 8.55/4.78 ! [X: rat,Y: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 8.55/4.78 => ( ( ( plus_plus_rat @ X @ Y )
% 8.55/4.78 = zero_zero_rat )
% 8.55/4.78 = ( ( X = zero_zero_rat )
% 8.55/4.78 & ( Y = zero_zero_rat ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2903_add__nonneg__eq__0__iff,axiom,
% 8.55/4.78 ! [X: nat,Y: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 8.55/4.78 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 8.55/4.78 => ( ( ( plus_plus_nat @ X @ Y )
% 8.55/4.78 = zero_zero_nat )
% 8.55/4.78 = ( ( X = zero_zero_nat )
% 8.55/4.78 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2904_add__nonneg__eq__0__iff,axiom,
% 8.55/4.78 ! [X: int,Y: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.55/4.78 => ( ( ( plus_plus_int @ X @ Y )
% 8.55/4.78 = zero_zero_int )
% 8.55/4.78 = ( ( X = zero_zero_int )
% 8.55/4.78 & ( Y = zero_zero_int ) ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2905_add__nonpos__nonpos,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.55/4.78 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2906_add__nonpos__nonpos,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.55/4.78 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2907_add__nonpos__nonpos,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 8.55/4.78 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 8.55/4.78 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2908_add__nonpos__nonpos,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.55/4.78 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 8.55/4.78 => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2909_add__nonneg__nonneg,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.55/4.78 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2910_add__nonneg__nonneg,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.55/4.78 => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2911_add__nonneg__nonneg,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.55/4.78 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2912_add__nonneg__nonneg,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.55/4.78 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2913_add__increasing2,axiom,
% 8.55/4.78 ! [C: real,B: real,A: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.78 => ( ( ord_less_eq_real @ B @ A )
% 8.55/4.78 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2914_add__increasing2,axiom,
% 8.55/4.78 ! [C: rat,B: rat,A: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.78 => ( ( ord_less_eq_rat @ B @ A )
% 8.55/4.78 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2915_add__increasing2,axiom,
% 8.55/4.78 ! [C: nat,B: nat,A: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.55/4.78 => ( ( ord_less_eq_nat @ B @ A )
% 8.55/4.78 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2916_add__increasing2,axiom,
% 8.55/4.78 ! [C: int,B: int,A: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.78 => ( ( ord_less_eq_int @ B @ A )
% 8.55/4.78 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2917_add__decreasing2,axiom,
% 8.55/4.78 ! [C: real,A: real,B: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ( ord_less_eq_real @ A @ B )
% 8.55/4.78 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2918_add__decreasing2,axiom,
% 8.55/4.78 ! [C: rat,A: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.78 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2919_add__decreasing2,axiom,
% 8.55/4.78 ! [C: nat,A: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 8.55/4.78 => ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.78 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2920_add__decreasing2,axiom,
% 8.55/4.78 ! [C: int,A: int,B: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.55/4.78 => ( ( ord_less_eq_int @ A @ B )
% 8.55/4.78 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2921_add__increasing,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.78 => ( ( ord_less_eq_real @ B @ C )
% 8.55/4.78 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2922_add__increasing,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.78 => ( ( ord_less_eq_rat @ B @ C )
% 8.55/4.78 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2923_add__increasing,axiom,
% 8.55/4.78 ! [A: nat,B: nat,C: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.78 => ( ( ord_less_eq_nat @ B @ C )
% 8.55/4.78 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2924_add__increasing,axiom,
% 8.55/4.78 ! [A: int,B: int,C: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.78 => ( ( ord_less_eq_int @ B @ C )
% 8.55/4.78 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2925_add__decreasing,axiom,
% 8.55/4.78 ! [A: real,C: real,B: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.55/4.78 => ( ( ord_less_eq_real @ C @ B )
% 8.55/4.78 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2926_add__decreasing,axiom,
% 8.55/4.78 ! [A: rat,C: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.55/4.78 => ( ( ord_less_eq_rat @ C @ B )
% 8.55/4.78 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2927_add__decreasing,axiom,
% 8.55/4.78 ! [A: nat,C: nat,B: nat] :
% 8.55/4.78 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 8.55/4.78 => ( ( ord_less_eq_nat @ C @ B )
% 8.55/4.78 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2928_add__decreasing,axiom,
% 8.55/4.78 ! [A: int,C: int,B: int] :
% 8.55/4.78 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.55/4.78 => ( ( ord_less_eq_int @ C @ B )
% 8.55/4.78 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2929_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 8.55/4.78 ! [A: real,B: real] :
% 8.55/4.78 ( ~ ( ord_less_real @ A @ B )
% 8.55/4.78 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 8.55/4.78 = A ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2930_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 8.55/4.78 ! [A: rat,B: rat] :
% 8.55/4.78 ( ~ ( ord_less_rat @ A @ B )
% 8.55/4.78 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 8.55/4.78 = A ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2931_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 8.55/4.78 ! [A: nat,B: nat] :
% 8.55/4.78 ( ~ ( ord_less_nat @ A @ B )
% 8.55/4.78 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 8.55/4.78 = A ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2932_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 8.55/4.78 ! [A: int,B: int] :
% 8.55/4.78 ( ~ ( ord_less_int @ A @ B )
% 8.55/4.78 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 8.55/4.78 = A ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2933_diff__less__eq,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real] :
% 8.55/4.78 ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
% 8.55/4.78 = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2934_diff__less__eq,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 8.55/4.78 = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2935_diff__less__eq,axiom,
% 8.55/4.78 ! [A: int,B: int,C: int] :
% 8.55/4.78 ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
% 8.55/4.78 = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2936_less__diff__eq,axiom,
% 8.55/4.78 ! [A: real,C: real,B: real] :
% 8.55/4.78 ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
% 8.55/4.78 = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2937_less__diff__eq,axiom,
% 8.55/4.78 ! [A: rat,C: rat,B: rat] :
% 8.55/4.78 ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 8.55/4.78 = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2938_less__diff__eq,axiom,
% 8.55/4.78 ! [A: int,C: int,B: int] :
% 8.55/4.78 ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
% 8.55/4.78 = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2939_le__iff__diff__le__0,axiom,
% 8.55/4.78 ( ord_less_eq_real
% 8.55/4.78 = ( ^ [A2: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2940_le__iff__diff__le__0,axiom,
% 8.55/4.78 ( ord_less_eq_rat
% 8.55/4.78 = ( ^ [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2941_le__iff__diff__le__0,axiom,
% 8.55/4.78 ( ord_less_eq_int
% 8.55/4.78 = ( ^ [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2942_of__nat__less__0__iff,axiom,
% 8.55/4.78 ! [M: nat] :
% 8.55/4.78 ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2943_of__nat__less__0__iff,axiom,
% 8.55/4.78 ! [M: nat] :
% 8.55/4.78 ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2944_of__nat__less__0__iff,axiom,
% 8.55/4.78 ! [M: nat] :
% 8.55/4.78 ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2945_of__nat__less__0__iff,axiom,
% 8.55/4.78 ! [M: nat] :
% 8.55/4.78 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2946_divide__right__mono__neg,axiom,
% 8.55/4.78 ! [A: real,B: real,C: real] :
% 8.55/4.78 ( ( ord_less_eq_real @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.55/4.78 => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 8.55/4.78
% 8.55/4.78 thf(fact_2947_divide__right__mono__neg,axiom,
% 8.55/4.78 ! [A: rat,B: rat,C: rat] :
% 8.55/4.78 ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.78 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.55/4.78 => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2948_divide__nonpos__nonpos,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 8.55/4.79 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 8.55/4.79 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2949_divide__nonpos__nonpos,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 8.55/4.79 => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 8.55/4.79 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2950_divide__nonpos__nonneg,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.55/4.79 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2951_divide__nonpos__nonneg,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 8.55/4.79 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2952_divide__nonneg__nonpos,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.55/4.79 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 8.55/4.79 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2953_divide__nonneg__nonpos,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 8.55/4.79 => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 8.55/4.79 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2954_divide__nonneg__nonneg,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.55/4.79 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2955_divide__nonneg__nonneg,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 8.55/4.79 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2956_zero__le__divide__iff,axiom,
% 8.55/4.79 ! [A: real,B: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 8.55/4.79 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.79 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 8.55/4.79 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.55/4.79 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2957_zero__le__divide__iff,axiom,
% 8.55/4.79 ! [A: rat,B: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 8.55/4.79 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.79 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 8.55/4.79 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.55/4.79 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2958_divide__right__mono,axiom,
% 8.55/4.79 ! [A: real,B: real,C: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ A @ B )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.55/4.79 => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2959_divide__right__mono,axiom,
% 8.55/4.79 ! [A: rat,B: rat,C: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.55/4.79 => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2960_divide__le__0__iff,axiom,
% 8.55/4.79 ! [A: real,B: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 8.55/4.79 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.79 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 8.55/4.79 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.55/4.79 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2961_divide__le__0__iff,axiom,
% 8.55/4.79 ! [A: rat,B: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 8.55/4.79 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.79 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 8.55/4.79 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.55/4.79 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2962_one__le__numeral,axiom,
% 8.55/4.79 ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2963_one__le__numeral,axiom,
% 8.55/4.79 ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2964_one__le__numeral,axiom,
% 8.55/4.79 ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2965_one__le__numeral,axiom,
% 8.55/4.79 ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2966_zero__le__power,axiom,
% 8.55/4.79 ! [A: real,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.79 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2967_zero__le__power,axiom,
% 8.55/4.79 ! [A: rat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.79 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2968_zero__le__power,axiom,
% 8.55/4.79 ! [A: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.79 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2969_zero__le__power,axiom,
% 8.55/4.79 ! [A: int,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.79 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2970_power__mono,axiom,
% 8.55/4.79 ! [A: real,B: real,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_real @ A @ B )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.79 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2971_power__mono,axiom,
% 8.55/4.79 ! [A: rat,B: rat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.79 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2972_power__mono,axiom,
% 8.55/4.79 ! [A: nat,B: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.79 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.79 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2973_power__mono,axiom,
% 8.55/4.79 ! [A: int,B: int,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_int @ A @ B )
% 8.55/4.79 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.79 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2974_pinf_I9_J,axiom,
% 8.55/4.79 ! [D: code_integer,S: code_integer] :
% 8.55/4.79 ? [Z4: code_integer] :
% 8.55/4.79 ! [X5: code_integer] :
% 8.55/4.79 ( ( ord_le6747313008572928689nteger @ Z4 @ X5 )
% 8.55/4.79 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) )
% 8.55/4.79 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2975_pinf_I9_J,axiom,
% 8.55/4.79 ! [D: real,S: real] :
% 8.55/4.79 ? [Z4: real] :
% 8.55/4.79 ! [X5: real] :
% 8.55/4.79 ( ( ord_less_real @ Z4 @ X5 )
% 8.55/4.79 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
% 8.55/4.79 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2976_pinf_I9_J,axiom,
% 8.55/4.79 ! [D: rat,S: rat] :
% 8.55/4.79 ? [Z4: rat] :
% 8.55/4.79 ! [X5: rat] :
% 8.55/4.79 ( ( ord_less_rat @ Z4 @ X5 )
% 8.55/4.79 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
% 8.55/4.79 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2977_pinf_I9_J,axiom,
% 8.55/4.79 ! [D: nat,S: nat] :
% 8.55/4.79 ? [Z4: nat] :
% 8.55/4.79 ! [X5: nat] :
% 8.55/4.79 ( ( ord_less_nat @ Z4 @ X5 )
% 8.55/4.79 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
% 8.55/4.79 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2978_pinf_I9_J,axiom,
% 8.55/4.79 ! [D: int,S: int] :
% 8.55/4.79 ? [Z4: int] :
% 8.55/4.79 ! [X5: int] :
% 8.55/4.79 ( ( ord_less_int @ Z4 @ X5 )
% 8.55/4.79 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
% 8.55/4.79 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2979_pinf_I10_J,axiom,
% 8.55/4.79 ! [D: code_integer,S: code_integer] :
% 8.55/4.79 ? [Z4: code_integer] :
% 8.55/4.79 ! [X5: code_integer] :
% 8.55/4.79 ( ( ord_le6747313008572928689nteger @ Z4 @ X5 )
% 8.55/4.79 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) )
% 8.55/4.79 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2980_pinf_I10_J,axiom,
% 8.55/4.79 ! [D: real,S: real] :
% 8.55/4.79 ? [Z4: real] :
% 8.55/4.79 ! [X5: real] :
% 8.55/4.79 ( ( ord_less_real @ Z4 @ X5 )
% 8.55/4.79 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
% 8.55/4.79 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2981_pinf_I10_J,axiom,
% 8.55/4.79 ! [D: rat,S: rat] :
% 8.55/4.79 ? [Z4: rat] :
% 8.55/4.79 ! [X5: rat] :
% 8.55/4.79 ( ( ord_less_rat @ Z4 @ X5 )
% 8.55/4.79 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
% 8.55/4.79 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2982_pinf_I10_J,axiom,
% 8.55/4.79 ! [D: nat,S: nat] :
% 8.55/4.79 ? [Z4: nat] :
% 8.55/4.79 ! [X5: nat] :
% 8.55/4.79 ( ( ord_less_nat @ Z4 @ X5 )
% 8.55/4.79 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
% 8.55/4.79 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2983_pinf_I10_J,axiom,
% 8.55/4.79 ! [D: int,S: int] :
% 8.55/4.79 ? [Z4: int] :
% 8.55/4.79 ! [X5: int] :
% 8.55/4.79 ( ( ord_less_int @ Z4 @ X5 )
% 8.55/4.79 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
% 8.55/4.79 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2984_minf_I9_J,axiom,
% 8.55/4.79 ! [D: code_integer,S: code_integer] :
% 8.55/4.79 ? [Z4: code_integer] :
% 8.55/4.79 ! [X5: code_integer] :
% 8.55/4.79 ( ( ord_le6747313008572928689nteger @ X5 @ Z4 )
% 8.55/4.79 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) )
% 8.55/4.79 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2985_minf_I9_J,axiom,
% 8.55/4.79 ! [D: real,S: real] :
% 8.55/4.79 ? [Z4: real] :
% 8.55/4.79 ! [X5: real] :
% 8.55/4.79 ( ( ord_less_real @ X5 @ Z4 )
% 8.55/4.79 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
% 8.55/4.79 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2986_minf_I9_J,axiom,
% 8.55/4.79 ! [D: rat,S: rat] :
% 8.55/4.79 ? [Z4: rat] :
% 8.55/4.79 ! [X5: rat] :
% 8.55/4.79 ( ( ord_less_rat @ X5 @ Z4 )
% 8.55/4.79 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
% 8.55/4.79 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2987_minf_I9_J,axiom,
% 8.55/4.79 ! [D: nat,S: nat] :
% 8.55/4.79 ? [Z4: nat] :
% 8.55/4.79 ! [X5: nat] :
% 8.55/4.79 ( ( ord_less_nat @ X5 @ Z4 )
% 8.55/4.79 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
% 8.55/4.79 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2988_minf_I9_J,axiom,
% 8.55/4.79 ! [D: int,S: int] :
% 8.55/4.79 ? [Z4: int] :
% 8.55/4.79 ! [X5: int] :
% 8.55/4.79 ( ( ord_less_int @ X5 @ Z4 )
% 8.55/4.79 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
% 8.55/4.79 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2989_minf_I10_J,axiom,
% 8.55/4.79 ! [D: code_integer,S: code_integer] :
% 8.55/4.79 ? [Z4: code_integer] :
% 8.55/4.79 ! [X5: code_integer] :
% 8.55/4.79 ( ( ord_le6747313008572928689nteger @ X5 @ Z4 )
% 8.55/4.79 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) )
% 8.55/4.79 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2990_minf_I10_J,axiom,
% 8.55/4.79 ! [D: real,S: real] :
% 8.55/4.79 ? [Z4: real] :
% 8.55/4.79 ! [X5: real] :
% 8.55/4.79 ( ( ord_less_real @ X5 @ Z4 )
% 8.55/4.79 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
% 8.55/4.79 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2991_minf_I10_J,axiom,
% 8.55/4.79 ! [D: rat,S: rat] :
% 8.55/4.79 ? [Z4: rat] :
% 8.55/4.79 ! [X5: rat] :
% 8.55/4.79 ( ( ord_less_rat @ X5 @ Z4 )
% 8.55/4.79 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
% 8.55/4.79 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2992_minf_I10_J,axiom,
% 8.55/4.79 ! [D: nat,S: nat] :
% 8.55/4.79 ? [Z4: nat] :
% 8.55/4.79 ! [X5: nat] :
% 8.55/4.79 ( ( ord_less_nat @ X5 @ Z4 )
% 8.55/4.79 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
% 8.55/4.79 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2993_minf_I10_J,axiom,
% 8.55/4.79 ! [D: int,S: int] :
% 8.55/4.79 ? [Z4: int] :
% 8.55/4.79 ! [X5: int] :
% 8.55/4.79 ( ( ord_less_int @ X5 @ Z4 )
% 8.55/4.79 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
% 8.55/4.79 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2994_add__le__add__imp__diff__le,axiom,
% 8.55/4.79 ! [I: real,K: real,N: real,J: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 8.55/4.79 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 8.55/4.79 => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 8.55/4.79 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 8.55/4.79 => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2995_add__le__add__imp__diff__le,axiom,
% 8.55/4.79 ! [I: rat,K: rat,N: rat,J: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 8.55/4.79 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 8.55/4.79 => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 8.55/4.79 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 8.55/4.79 => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2996_add__le__add__imp__diff__le,axiom,
% 8.55/4.79 ! [I: nat,K: nat,N: nat,J: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 8.55/4.79 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 8.55/4.79 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 8.55/4.79 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 8.55/4.79 => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2997_add__le__add__imp__diff__le,axiom,
% 8.55/4.79 ! [I: int,K: int,N: int,J: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 8.55/4.79 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 8.55/4.79 => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 8.55/4.79 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 8.55/4.79 => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2998_add__le__imp__le__diff,axiom,
% 8.55/4.79 ! [I: real,K: real,N: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 8.55/4.79 => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_2999_add__le__imp__le__diff,axiom,
% 8.55/4.79 ! [I: rat,K: rat,N: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 8.55/4.79 => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3000_add__le__imp__le__diff,axiom,
% 8.55/4.79 ! [I: nat,K: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 8.55/4.79 => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3001_add__le__imp__le__diff,axiom,
% 8.55/4.79 ! [I: int,K: int,N: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 8.55/4.79 => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3002_diff__le__eq,axiom,
% 8.55/4.79 ! [A: real,B: real,C: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 8.55/4.79 = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3003_diff__le__eq,axiom,
% 8.55/4.79 ! [A: rat,B: rat,C: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 8.55/4.79 = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3004_diff__le__eq,axiom,
% 8.55/4.79 ! [A: int,B: int,C: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 8.55/4.79 = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3005_le__diff__eq,axiom,
% 8.55/4.79 ! [A: real,C: real,B: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 8.55/4.79 = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3006_le__diff__eq,axiom,
% 8.55/4.79 ! [A: rat,C: rat,B: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 8.55/4.79 = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3007_le__diff__eq,axiom,
% 8.55/4.79 ! [A: int,C: int,B: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 8.55/4.79 = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3008_diff__add,axiom,
% 8.55/4.79 ! [A: nat,B: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.79 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 8.55/4.79 = B ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3009_le__add__diff,axiom,
% 8.55/4.79 ! [A: nat,B: nat,C: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.79 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3010_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 8.55/4.79 ! [A: nat,B: nat,C: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.79 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 8.55/4.79 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3011_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 8.55/4.79 ! [A: nat,B: nat,C: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.79 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 8.55/4.79 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3012_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 8.55/4.79 ! [A: nat,B: nat,C: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.79 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 8.55/4.79 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3013_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 8.55/4.79 ! [A: nat,B: nat,C: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.79 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 8.55/4.79 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3014_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 8.55/4.79 ! [A: nat,B: nat,C: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.79 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 8.55/4.79 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3015_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 8.55/4.79 ! [A: nat,B: nat,C: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.79 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 8.55/4.79 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3016_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 8.55/4.79 ! [A: nat,B: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.79 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 8.55/4.79 = B ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3017_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 8.55/4.79 ! [A: nat,B: nat,C: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.79 => ( ( ord_less_eq_nat @ A @ B )
% 8.55/4.79 => ( ( ( minus_minus_nat @ B @ A )
% 8.55/4.79 = C )
% 8.55/4.79 = ( B
% 8.55/4.79 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3018_of__nat__0__le__iff,axiom,
% 8.55/4.79 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3019_of__nat__0__le__iff,axiom,
% 8.55/4.79 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3020_of__nat__0__le__iff,axiom,
% 8.55/4.79 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3021_of__nat__0__le__iff,axiom,
% 8.55/4.79 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3022_one__le__power,axiom,
% 8.55/4.79 ! [A: real,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_real @ one_one_real @ A )
% 8.55/4.79 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3023_one__le__power,axiom,
% 8.55/4.79 ! [A: rat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 8.55/4.79 => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3024_one__le__power,axiom,
% 8.55/4.79 ! [A: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 8.55/4.79 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3025_one__le__power,axiom,
% 8.55/4.79 ! [A: int,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_int @ one_one_int @ A )
% 8.55/4.79 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3026_Ex__less__Suc2,axiom,
% 8.55/4.79 ! [N: nat,P: nat > $o] :
% 8.55/4.79 ( ( ? [I4: nat] :
% 8.55/4.79 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 8.55/4.79 & ( P @ I4 ) ) )
% 8.55/4.79 = ( ( P @ zero_zero_nat )
% 8.55/4.79 | ? [I4: nat] :
% 8.55/4.79 ( ( ord_less_nat @ I4 @ N )
% 8.55/4.79 & ( P @ ( suc @ I4 ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3027_gr0__conv__Suc,axiom,
% 8.55/4.79 ! [N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 = ( ? [M3: nat] :
% 8.55/4.79 ( N
% 8.55/4.79 = ( suc @ M3 ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3028_All__less__Suc2,axiom,
% 8.55/4.79 ! [N: nat,P: nat > $o] :
% 8.55/4.79 ( ( ! [I4: nat] :
% 8.55/4.79 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 8.55/4.79 => ( P @ I4 ) ) )
% 8.55/4.79 = ( ( P @ zero_zero_nat )
% 8.55/4.79 & ! [I4: nat] :
% 8.55/4.79 ( ( ord_less_nat @ I4 @ N )
% 8.55/4.79 => ( P @ ( suc @ I4 ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3029_gr0__implies__Suc,axiom,
% 8.55/4.79 ! [N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ? [M2: nat] :
% 8.55/4.79 ( N
% 8.55/4.79 = ( suc @ M2 ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3030_less__Suc__eq__0__disj,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 8.55/4.79 = ( ( M = zero_zero_nat )
% 8.55/4.79 | ? [J3: nat] :
% 8.55/4.79 ( ( M
% 8.55/4.79 = ( suc @ J3 ) )
% 8.55/4.79 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3031_le__imp__power__dvd,axiom,
% 8.55/4.79 ! [M: nat,N: nat,A: code_integer] :
% 8.55/4.79 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.79 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3032_le__imp__power__dvd,axiom,
% 8.55/4.79 ! [M: nat,N: nat,A: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.79 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3033_le__imp__power__dvd,axiom,
% 8.55/4.79 ! [M: nat,N: nat,A: real] :
% 8.55/4.79 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.79 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3034_le__imp__power__dvd,axiom,
% 8.55/4.79 ! [M: nat,N: nat,A: int] :
% 8.55/4.79 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.79 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3035_le__imp__power__dvd,axiom,
% 8.55/4.79 ! [M: nat,N: nat,A: complex] :
% 8.55/4.79 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.79 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3036_power__le__dvd,axiom,
% 8.55/4.79 ! [A: code_integer,N: nat,B: code_integer,M: nat] :
% 8.55/4.79 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
% 8.55/4.79 => ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.79 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3037_power__le__dvd,axiom,
% 8.55/4.79 ! [A: nat,N: nat,B: nat,M: nat] :
% 8.55/4.79 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
% 8.55/4.79 => ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.79 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3038_power__le__dvd,axiom,
% 8.55/4.79 ! [A: real,N: nat,B: real,M: nat] :
% 8.55/4.79 ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
% 8.55/4.79 => ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.79 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3039_power__le__dvd,axiom,
% 8.55/4.79 ! [A: int,N: nat,B: int,M: nat] :
% 8.55/4.79 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
% 8.55/4.79 => ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.79 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3040_power__le__dvd,axiom,
% 8.55/4.79 ! [A: complex,N: nat,B: complex,M: nat] :
% 8.55/4.79 ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
% 8.55/4.79 => ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.79 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3041_dvd__power__le,axiom,
% 8.55/4.79 ! [X: code_integer,Y: code_integer,N: nat,M: nat] :
% 8.55/4.79 ( ( dvd_dvd_Code_integer @ X @ Y )
% 8.55/4.79 => ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3042_dvd__power__le,axiom,
% 8.55/4.79 ! [X: nat,Y: nat,N: nat,M: nat] :
% 8.55/4.79 ( ( dvd_dvd_nat @ X @ Y )
% 8.55/4.79 => ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3043_dvd__power__le,axiom,
% 8.55/4.79 ! [X: real,Y: real,N: nat,M: nat] :
% 8.55/4.79 ( ( dvd_dvd_real @ X @ Y )
% 8.55/4.79 => ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ M ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3044_dvd__power__le,axiom,
% 8.55/4.79 ! [X: int,Y: int,N: nat,M: nat] :
% 8.55/4.79 ( ( dvd_dvd_int @ X @ Y )
% 8.55/4.79 => ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3045_dvd__power__le,axiom,
% 8.55/4.79 ! [X: complex,Y: complex,N: nat,M: nat] :
% 8.55/4.79 ( ( dvd_dvd_complex @ X @ Y )
% 8.55/4.79 => ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ M ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3046_less__natE,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ M @ N )
% 8.55/4.79 => ~ ! [Q6: nat] :
% 8.55/4.79 ( N
% 8.55/4.79 != ( suc @ ( plus_plus_nat @ M @ Q6 ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3047_less__add__Suc1,axiom,
% 8.55/4.79 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3048_less__add__Suc2,axiom,
% 8.55/4.79 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3049_less__iff__Suc__add,axiom,
% 8.55/4.79 ( ord_less_nat
% 8.55/4.79 = ( ^ [M3: nat,N3: nat] :
% 8.55/4.79 ? [K2: nat] :
% 8.55/4.79 ( N3
% 8.55/4.79 = ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3050_less__imp__Suc__add,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ M @ N )
% 8.55/4.79 => ? [K3: nat] :
% 8.55/4.79 ( N
% 8.55/4.79 = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3051_less__imp__add__positive,axiom,
% 8.55/4.79 ! [I: nat,J: nat] :
% 8.55/4.79 ( ( ord_less_nat @ I @ J )
% 8.55/4.79 => ? [K3: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 8.55/4.79 & ( ( plus_plus_nat @ I @ K3 )
% 8.55/4.79 = J ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3052_Suc__diff__Suc,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_nat @ N @ M )
% 8.55/4.79 => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 8.55/4.79 = ( minus_minus_nat @ M @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3053_diff__less__Suc,axiom,
% 8.55/4.79 ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3054_diff__less,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.55/4.79 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3055_Suc__mult__less__cancel1,axiom,
% 8.55/4.79 ! [K: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 8.55/4.79 = ( ord_less_nat @ M @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3056_nat__mult__less__cancel1,axiom,
% 8.55/4.79 ! [K: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.55/4.79 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.55/4.79 = ( ord_less_nat @ M @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3057_nat__mult__eq__cancel1,axiom,
% 8.55/4.79 ! [K: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.55/4.79 => ( ( ( times_times_nat @ K @ M )
% 8.55/4.79 = ( times_times_nat @ K @ N ) )
% 8.55/4.79 = ( M = N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3058_mult__less__mono2,axiom,
% 8.55/4.79 ! [I: nat,J: nat,K: nat] :
% 8.55/4.79 ( ( ord_less_nat @ I @ J )
% 8.55/4.79 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.55/4.79 => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3059_mult__less__mono1,axiom,
% 8.55/4.79 ! [I: nat,J: nat,K: nat] :
% 8.55/4.79 ( ( ord_less_nat @ I @ J )
% 8.55/4.79 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.55/4.79 => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3060_Suc__diff__le,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 8.55/4.79 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3061_Suc__mult__le__cancel1,axiom,
% 8.55/4.79 ! [K: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 8.55/4.79 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3062_realpow__pos__nth2,axiom,
% 8.55/4.79 ! [A: real,N: nat] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.79 => ? [R2: real] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ R2 )
% 8.55/4.79 & ( ( power_power_real @ R2 @ ( suc @ N ) )
% 8.55/4.79 = A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3063_nat__dvd__not__less,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.55/4.79 => ( ( ord_less_nat @ M @ N )
% 8.55/4.79 => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3064_dvd__pos__nat,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( dvd_dvd_nat @ M @ N )
% 8.55/4.79 => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3065_add__diff__inverse__nat,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ~ ( ord_less_nat @ M @ N )
% 8.55/4.79 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 8.55/4.79 = M ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3066_less__diff__conv,axiom,
% 8.55/4.79 ! [I: nat,J: nat,K: nat] :
% 8.55/4.79 ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 8.55/4.79 = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3067_zmult__zless__mono2,axiom,
% 8.55/4.79 ! [I: int,J: int,K: int] :
% 8.55/4.79 ( ( ord_less_int @ I @ J )
% 8.55/4.79 => ( ( ord_less_int @ zero_zero_int @ K )
% 8.55/4.79 => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3068_Nat_Ole__imp__diff__is__add,axiom,
% 8.55/4.79 ! [I: nat,J: nat,K: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.79 => ( ( ( minus_minus_nat @ J @ I )
% 8.55/4.79 = K )
% 8.55/4.79 = ( J
% 8.55/4.79 = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3069_Nat_Odiff__add__assoc2,axiom,
% 8.55/4.79 ! [K: nat,J: nat,I: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ K @ J )
% 8.55/4.79 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 8.55/4.79 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3070_Nat_Odiff__add__assoc,axiom,
% 8.55/4.79 ! [K: nat,J: nat,I: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ K @ J )
% 8.55/4.79 => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 8.55/4.79 = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3071_Nat_Ole__diff__conv2,axiom,
% 8.55/4.79 ! [K: nat,J: nat,I: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ K @ J )
% 8.55/4.79 => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 8.55/4.79 = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3072_le__diff__conv,axiom,
% 8.55/4.79 ! [J: nat,K: nat,I: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 8.55/4.79 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3073_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( ( divide_divide_nat @ M @ N )
% 8.55/4.79 = zero_zero_nat )
% 8.55/4.79 = ( ( ord_less_nat @ M @ N )
% 8.55/4.79 | ( N = zero_zero_nat ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3074_Suc__div__le__mono,axiom,
% 8.55/4.79 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3075_nat__power__less__imp__less,axiom,
% 8.55/4.79 ! [I: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ I )
% 8.55/4.79 => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 8.55/4.79 => ( ord_less_nat @ M @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3076_int__gr__induct,axiom,
% 8.55/4.79 ! [K: int,I: int,P: int > $o] :
% 8.55/4.79 ( ( ord_less_int @ K @ I )
% 8.55/4.79 => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 8.55/4.79 => ( ! [I2: int] :
% 8.55/4.79 ( ( ord_less_int @ K @ I2 )
% 8.55/4.79 => ( ( P @ I2 )
% 8.55/4.79 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 8.55/4.79 => ( P @ I ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3077_zless__add1__eq,axiom,
% 8.55/4.79 ! [W: int,Z2: int] :
% 8.55/4.79 ( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
% 8.55/4.79 = ( ( ord_less_int @ W @ Z2 )
% 8.55/4.79 | ( W = Z2 ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3078_zero__le__imp__eq__int,axiom,
% 8.55/4.79 ! [K: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.55/4.79 => ? [N2: nat] :
% 8.55/4.79 ( K
% 8.55/4.79 = ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3079_nonneg__int__cases,axiom,
% 8.55/4.79 ! [K: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.55/4.79 => ~ ! [N2: nat] :
% 8.55/4.79 ( K
% 8.55/4.79 != ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3080_dvd__minus__self,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 8.55/4.79 = ( ( ord_less_nat @ N @ M )
% 8.55/4.79 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3081_pos__imp__zdiv__neg__iff,axiom,
% 8.55/4.79 ! [B: int,A: int] :
% 8.55/4.79 ( ( ord_less_int @ zero_zero_int @ B )
% 8.55/4.79 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 8.55/4.79 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3082_neg__imp__zdiv__neg__iff,axiom,
% 8.55/4.79 ! [B: int,A: int] :
% 8.55/4.79 ( ( ord_less_int @ B @ zero_zero_int )
% 8.55/4.79 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 8.55/4.79 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3083_div__neg__pos__less0,axiom,
% 8.55/4.79 ! [A: int,B: int] :
% 8.55/4.79 ( ( ord_less_int @ A @ zero_zero_int )
% 8.55/4.79 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.55/4.79 => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3084_dvd__diffD,axiom,
% 8.55/4.79 ! [K: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 8.55/4.79 => ( ( dvd_dvd_nat @ K @ N )
% 8.55/4.79 => ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3085_dvd__diffD1,axiom,
% 8.55/4.79 ! [K: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 8.55/4.79 => ( ( dvd_dvd_nat @ K @ M )
% 8.55/4.79 => ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3086_less__eq__dvd__minus,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.79 => ( ( dvd_dvd_nat @ M @ N )
% 8.55/4.79 = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3087_zdvd__not__zless,axiom,
% 8.55/4.79 ! [M: int,N: int] :
% 8.55/4.79 ( ( ord_less_int @ zero_zero_int @ M )
% 8.55/4.79 => ( ( ord_less_int @ M @ N )
% 8.55/4.79 => ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3088_int__ge__induct,axiom,
% 8.55/4.79 ! [K: int,I: int,P: int > $o] :
% 8.55/4.79 ( ( ord_less_eq_int @ K @ I )
% 8.55/4.79 => ( ( P @ K )
% 8.55/4.79 => ( ! [I2: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ K @ I2 )
% 8.55/4.79 => ( ( P @ I2 )
% 8.55/4.79 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 8.55/4.79 => ( P @ I ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3089_int__less__induct,axiom,
% 8.55/4.79 ! [I: int,K: int,P: int > $o] :
% 8.55/4.79 ( ( ord_less_int @ I @ K )
% 8.55/4.79 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 8.55/4.79 => ( ! [I2: int] :
% 8.55/4.79 ( ( ord_less_int @ I2 @ K )
% 8.55/4.79 => ( ( P @ I2 )
% 8.55/4.79 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 8.55/4.79 => ( P @ I ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3090_less__mult__imp__div__less,axiom,
% 8.55/4.79 ! [M: nat,I: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 8.55/4.79 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3091_div__times__less__eq__dividend,axiom,
% 8.55/4.79 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3092_times__div__less__eq__dividend,axiom,
% 8.55/4.79 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3093_zdvd__antisym__nonneg,axiom,
% 8.55/4.79 ! [M: int,N: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 8.55/4.79 => ( ( ord_less_eq_int @ zero_zero_int @ N )
% 8.55/4.79 => ( ( dvd_dvd_int @ M @ N )
% 8.55/4.79 => ( ( dvd_dvd_int @ N @ M )
% 8.55/4.79 => ( M = N ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3094_reals__Archimedean3,axiom,
% 8.55/4.79 ! [X: real] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ X )
% 8.55/4.79 => ! [Y5: real] :
% 8.55/4.79 ? [N2: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3095_zle__iff__zadd,axiom,
% 8.55/4.79 ( ord_less_eq_int
% 8.55/4.79 = ( ^ [W2: int,Z5: int] :
% 8.55/4.79 ? [N3: nat] :
% 8.55/4.79 ( Z5
% 8.55/4.79 = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3096_int__le__induct,axiom,
% 8.55/4.79 ! [I: int,K: int,P: int > $o] :
% 8.55/4.79 ( ( ord_less_eq_int @ I @ K )
% 8.55/4.79 => ( ( P @ K )
% 8.55/4.79 => ( ! [I2: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ I2 @ K )
% 8.55/4.79 => ( ( P @ I2 )
% 8.55/4.79 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 8.55/4.79 => ( P @ I ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3097_power2__less__imp__less,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.55/4.79 => ( ord_less_real @ X @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3098_power2__less__imp__less,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 8.55/4.79 => ( ord_less_rat @ X @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3099_power2__less__imp__less,axiom,
% 8.55/4.79 ! [X: nat,Y: nat] :
% 8.55/4.79 ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.79 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 8.55/4.79 => ( ord_less_nat @ X @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3100_power2__less__imp__less,axiom,
% 8.55/4.79 ! [X: int,Y: int] :
% 8.55/4.79 ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.79 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.55/4.79 => ( ord_less_int @ X @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3101_pow_Osimps_I1_J,axiom,
% 8.55/4.79 ! [X: num] :
% 8.55/4.79 ( ( pow @ X @ one )
% 8.55/4.79 = X ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3102_div__pos__geq,axiom,
% 8.55/4.79 ! [L: int,K: int] :
% 8.55/4.79 ( ( ord_less_int @ zero_zero_int @ L )
% 8.55/4.79 => ( ( ord_less_eq_int @ L @ K )
% 8.55/4.79 => ( ( divide_divide_int @ K @ L )
% 8.55/4.79 = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3103_not__sum__squares__lt__zero,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3104_not__sum__squares__lt__zero,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3105_not__sum__squares__lt__zero,axiom,
% 8.55/4.79 ! [X: int,Y: int] :
% 8.55/4.79 ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3106_sum__squares__gt__zero__iff,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
% 8.55/4.79 = ( ( X != zero_zero_real )
% 8.55/4.79 | ( Y != zero_zero_real ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3107_sum__squares__gt__zero__iff,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
% 8.55/4.79 = ( ( X != zero_zero_rat )
% 8.55/4.79 | ( Y != zero_zero_rat ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3108_sum__squares__gt__zero__iff,axiom,
% 8.55/4.79 ! [X: int,Y: int] :
% 8.55/4.79 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
% 8.55/4.79 = ( ( X != zero_zero_int )
% 8.55/4.79 | ( Y != zero_zero_int ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3109_zero__less__two,axiom,
% 8.55/4.79 ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3110_zero__less__two,axiom,
% 8.55/4.79 ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3111_zero__less__two,axiom,
% 8.55/4.79 ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3112_zero__less__two,axiom,
% 8.55/4.79 ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3113_ex__power__ivl2,axiom,
% 8.55/4.79 ! [B: nat,K: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 8.55/4.79 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 8.55/4.79 => ? [N2: nat] :
% 8.55/4.79 ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 8.55/4.79 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3114_ex__power__ivl1,axiom,
% 8.55/4.79 ! [B: nat,K: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 8.55/4.79 => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 8.55/4.79 => ? [N2: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 8.55/4.79 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3115_divide__strict__left__mono__neg,axiom,
% 8.55/4.79 ! [A: real,B: real,C: real] :
% 8.55/4.79 ( ( ord_less_real @ A @ B )
% 8.55/4.79 => ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.79 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 8.55/4.79 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3116_divide__strict__left__mono__neg,axiom,
% 8.55/4.79 ! [A: rat,B: rat,C: rat] :
% 8.55/4.79 ( ( ord_less_rat @ A @ B )
% 8.55/4.79 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.79 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 8.55/4.79 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3117_divide__strict__left__mono,axiom,
% 8.55/4.79 ! [B: real,A: real,C: real] :
% 8.55/4.79 ( ( ord_less_real @ B @ A )
% 8.55/4.79 => ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.79 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 8.55/4.79 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3118_divide__strict__left__mono,axiom,
% 8.55/4.79 ! [B: rat,A: rat,C: rat] :
% 8.55/4.79 ( ( ord_less_rat @ B @ A )
% 8.55/4.79 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.79 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 8.55/4.79 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3119_mult__imp__less__div__pos,axiom,
% 8.55/4.79 ! [Y: real,Z2: real,X: real] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ Y )
% 8.55/4.79 => ( ( ord_less_real @ ( times_times_real @ Z2 @ Y ) @ X )
% 8.55/4.79 => ( ord_less_real @ Z2 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3120_mult__imp__less__div__pos,axiom,
% 8.55/4.79 ! [Y: rat,Z2: rat,X: rat] :
% 8.55/4.79 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 8.55/4.79 => ( ( ord_less_rat @ ( times_times_rat @ Z2 @ Y ) @ X )
% 8.55/4.79 => ( ord_less_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3121_mult__imp__div__pos__less,axiom,
% 8.55/4.79 ! [Y: real,X: real,Z2: real] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ Y )
% 8.55/4.79 => ( ( ord_less_real @ X @ ( times_times_real @ Z2 @ Y ) )
% 8.55/4.79 => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z2 ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3122_mult__imp__div__pos__less,axiom,
% 8.55/4.79 ! [Y: rat,X: rat,Z2: rat] :
% 8.55/4.79 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 8.55/4.79 => ( ( ord_less_rat @ X @ ( times_times_rat @ Z2 @ Y ) )
% 8.55/4.79 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3123_pos__less__divide__eq,axiom,
% 8.55/4.79 ! [C: real,A: real,B: real] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.79 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 8.55/4.79 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3124_pos__less__divide__eq,axiom,
% 8.55/4.79 ! [C: rat,A: rat,B: rat] :
% 8.55/4.79 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.79 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 8.55/4.79 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3125_pos__divide__less__eq,axiom,
% 8.55/4.79 ! [C: real,B: real,A: real] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.79 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 8.55/4.79 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3126_pos__divide__less__eq,axiom,
% 8.55/4.79 ! [C: rat,B: rat,A: rat] :
% 8.55/4.79 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.79 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 8.55/4.79 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3127_neg__less__divide__eq,axiom,
% 8.55/4.79 ! [C: real,A: real,B: real] :
% 8.55/4.79 ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.79 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 8.55/4.79 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3128_neg__less__divide__eq,axiom,
% 8.55/4.79 ! [C: rat,A: rat,B: rat] :
% 8.55/4.79 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.79 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 8.55/4.79 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3129_neg__divide__less__eq,axiom,
% 8.55/4.79 ! [C: real,B: real,A: real] :
% 8.55/4.79 ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.79 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 8.55/4.79 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3130_neg__divide__less__eq,axiom,
% 8.55/4.79 ! [C: rat,B: rat,A: rat] :
% 8.55/4.79 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.79 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 8.55/4.79 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3131_less__divide__eq,axiom,
% 8.55/4.79 ! [A: real,B: real,C: real] :
% 8.55/4.79 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 8.55/4.79 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.79 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 8.55/4.79 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.79 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.79 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 8.55/4.79 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.79 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3132_less__divide__eq,axiom,
% 8.55/4.79 ! [A: rat,B: rat,C: rat] :
% 8.55/4.79 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 8.55/4.79 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.79 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 8.55/4.79 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.79 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.79 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 8.55/4.79 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.79 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3133_divide__less__eq,axiom,
% 8.55/4.79 ! [B: real,C: real,A: real] :
% 8.55/4.79 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 8.55/4.79 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.79 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 8.55/4.79 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.79 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.79 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 8.55/4.79 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.79 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3134_divide__less__eq,axiom,
% 8.55/4.79 ! [B: rat,C: rat,A: rat] :
% 8.55/4.79 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 8.55/4.79 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.79 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 8.55/4.79 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.79 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.79 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 8.55/4.79 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.79 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3135_less__divide__eq__1,axiom,
% 8.55/4.79 ! [B: real,A: real] :
% 8.55/4.79 ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 8.55/4.79 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.79 & ( ord_less_real @ A @ B ) )
% 8.55/4.79 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.79 & ( ord_less_real @ B @ A ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3136_less__divide__eq__1,axiom,
% 8.55/4.79 ! [B: rat,A: rat] :
% 8.55/4.79 ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 8.55/4.79 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.79 & ( ord_less_rat @ A @ B ) )
% 8.55/4.79 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.79 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3137_divide__less__eq__1,axiom,
% 8.55/4.79 ! [B: real,A: real] :
% 8.55/4.79 ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 8.55/4.79 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.79 & ( ord_less_real @ B @ A ) )
% 8.55/4.79 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.79 & ( ord_less_real @ A @ B ) )
% 8.55/4.79 | ( A = zero_zero_real ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3138_divide__less__eq__1,axiom,
% 8.55/4.79 ! [B: rat,A: rat] :
% 8.55/4.79 ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 8.55/4.79 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.79 & ( ord_less_rat @ B @ A ) )
% 8.55/4.79 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.79 & ( ord_less_rat @ A @ B ) )
% 8.55/4.79 | ( A = zero_zero_rat ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3139_less__add__iff1,axiom,
% 8.55/4.79 ! [A: real,E: real,C: real,B: real,D: real] :
% 8.55/4.79 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 8.55/4.79 = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3140_less__add__iff1,axiom,
% 8.55/4.79 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 8.55/4.79 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 8.55/4.79 = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3141_less__add__iff1,axiom,
% 8.55/4.79 ! [A: int,E: int,C: int,B: int,D: int] :
% 8.55/4.79 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 8.55/4.79 = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3142_less__add__iff2,axiom,
% 8.55/4.79 ! [A: real,E: real,C: real,B: real,D: real] :
% 8.55/4.79 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 8.55/4.79 = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3143_less__add__iff2,axiom,
% 8.55/4.79 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 8.55/4.79 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 8.55/4.79 = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3144_less__add__iff2,axiom,
% 8.55/4.79 ! [A: int,E: int,C: int,B: int,D: int] :
% 8.55/4.79 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 8.55/4.79 = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3145_mult__left__le__one__le,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.55/4.79 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.55/4.79 => ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3146_mult__left__le__one__le,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 8.55/4.79 => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 8.55/4.79 => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3147_mult__left__le__one__le,axiom,
% 8.55/4.79 ! [X: int,Y: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.55/4.79 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.55/4.79 => ( ( ord_less_eq_int @ Y @ one_one_int )
% 8.55/4.79 => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3148_mult__right__le__one__le,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.55/4.79 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.55/4.79 => ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3149_mult__right__le__one__le,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 8.55/4.79 => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 8.55/4.79 => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3150_mult__right__le__one__le,axiom,
% 8.55/4.79 ! [X: int,Y: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.55/4.79 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.55/4.79 => ( ( ord_less_eq_int @ Y @ one_one_int )
% 8.55/4.79 => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3151_mult__le__one,axiom,
% 8.55/4.79 ! [A: real,B: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ A @ one_one_real )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.55/4.79 => ( ( ord_less_eq_real @ B @ one_one_real )
% 8.55/4.79 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3152_mult__le__one,axiom,
% 8.55/4.79 ! [A: rat,B: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ A @ one_one_rat )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.55/4.79 => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 8.55/4.79 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3153_mult__le__one,axiom,
% 8.55/4.79 ! [A: nat,B: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ A @ one_one_nat )
% 8.55/4.79 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.55/4.79 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 8.55/4.79 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3154_mult__le__one,axiom,
% 8.55/4.79 ! [A: int,B: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ A @ one_one_int )
% 8.55/4.79 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.55/4.79 => ( ( ord_less_eq_int @ B @ one_one_int )
% 8.55/4.79 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3155_mult__left__le,axiom,
% 8.55/4.79 ! [C: real,A: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ C @ one_one_real )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.79 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3156_mult__left__le,axiom,
% 8.55/4.79 ! [C: rat,A: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ C @ one_one_rat )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.79 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3157_mult__left__le,axiom,
% 8.55/4.79 ! [C: nat,A: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 8.55/4.79 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.79 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3158_mult__left__le,axiom,
% 8.55/4.79 ! [C: int,A: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ C @ one_one_int )
% 8.55/4.79 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.79 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3159_sum__squares__ge__zero,axiom,
% 8.55/4.79 ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3160_sum__squares__ge__zero,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3161_sum__squares__ge__zero,axiom,
% 8.55/4.79 ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3162_sum__squares__le__zero__iff,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
% 8.55/4.79 = ( ( X = zero_zero_real )
% 8.55/4.79 & ( Y = zero_zero_real ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3163_sum__squares__le__zero__iff,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
% 8.55/4.79 = ( ( X = zero_zero_rat )
% 8.55/4.79 & ( Y = zero_zero_rat ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3164_sum__squares__le__zero__iff,axiom,
% 8.55/4.79 ! [X: int,Y: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
% 8.55/4.79 = ( ( X = zero_zero_int )
% 8.55/4.79 & ( Y = zero_zero_int ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3165_ex__less__of__nat__mult,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ X )
% 8.55/4.79 => ? [N2: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3166_ex__less__of__nat__mult,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ( ( ord_less_rat @ zero_zero_rat @ X )
% 8.55/4.79 => ? [N2: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ X ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3167_less__half__sum,axiom,
% 8.55/4.79 ! [A: real,B: real] :
% 8.55/4.79 ( ( ord_less_real @ A @ B )
% 8.55/4.79 => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3168_less__half__sum,axiom,
% 8.55/4.79 ! [A: rat,B: rat] :
% 8.55/4.79 ( ( ord_less_rat @ A @ B )
% 8.55/4.79 => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3169_gt__half__sum,axiom,
% 8.55/4.79 ! [A: real,B: real] :
% 8.55/4.79 ( ( ord_less_real @ A @ B )
% 8.55/4.79 => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3170_gt__half__sum,axiom,
% 8.55/4.79 ! [A: rat,B: rat] :
% 8.55/4.79 ( ( ord_less_rat @ A @ B )
% 8.55/4.79 => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3171_power__gt1__lemma,axiom,
% 8.55/4.79 ! [A: real,N: nat] :
% 8.55/4.79 ( ( ord_less_real @ one_one_real @ A )
% 8.55/4.79 => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3172_power__gt1__lemma,axiom,
% 8.55/4.79 ! [A: rat,N: nat] :
% 8.55/4.79 ( ( ord_less_rat @ one_one_rat @ A )
% 8.55/4.79 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3173_power__gt1__lemma,axiom,
% 8.55/4.79 ! [A: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ one_one_nat @ A )
% 8.55/4.79 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3174_power__gt1__lemma,axiom,
% 8.55/4.79 ! [A: int,N: nat] :
% 8.55/4.79 ( ( ord_less_int @ one_one_int @ A )
% 8.55/4.79 => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3175_power__less__power__Suc,axiom,
% 8.55/4.79 ! [A: real,N: nat] :
% 8.55/4.79 ( ( ord_less_real @ one_one_real @ A )
% 8.55/4.79 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3176_power__less__power__Suc,axiom,
% 8.55/4.79 ! [A: rat,N: nat] :
% 8.55/4.79 ( ( ord_less_rat @ one_one_rat @ A )
% 8.55/4.79 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3177_power__less__power__Suc,axiom,
% 8.55/4.79 ! [A: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ one_one_nat @ A )
% 8.55/4.79 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3178_power__less__power__Suc,axiom,
% 8.55/4.79 ! [A: int,N: nat] :
% 8.55/4.79 ( ( ord_less_int @ one_one_int @ A )
% 8.55/4.79 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3179_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 8.55/4.79 ! [C: nat,A: nat,B: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.55/4.79 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 8.55/4.79 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3180_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 8.55/4.79 ! [C: int,A: int,B: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.79 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 8.55/4.79 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3181_power__le__one,axiom,
% 8.55/4.79 ! [A: real,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.79 => ( ( ord_less_eq_real @ A @ one_one_real )
% 8.55/4.79 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3182_power__le__one,axiom,
% 8.55/4.79 ! [A: rat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.79 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 8.55/4.79 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3183_power__le__one,axiom,
% 8.55/4.79 ! [A: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.79 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 8.55/4.79 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3184_power__le__one,axiom,
% 8.55/4.79 ! [A: int,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.79 => ( ( ord_less_eq_int @ A @ one_one_int )
% 8.55/4.79 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3185_power__gt1,axiom,
% 8.55/4.79 ! [A: real,N: nat] :
% 8.55/4.79 ( ( ord_less_real @ one_one_real @ A )
% 8.55/4.79 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3186_power__gt1,axiom,
% 8.55/4.79 ! [A: rat,N: nat] :
% 8.55/4.79 ( ( ord_less_rat @ one_one_rat @ A )
% 8.55/4.79 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3187_power__gt1,axiom,
% 8.55/4.79 ! [A: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ one_one_nat @ A )
% 8.55/4.79 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3188_power__gt1,axiom,
% 8.55/4.79 ! [A: int,N: nat] :
% 8.55/4.79 ( ( ord_less_int @ one_one_int @ A )
% 8.55/4.79 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3189_ordered__ring__class_Ole__add__iff1,axiom,
% 8.55/4.79 ! [A: real,E: real,C: real,B: real,D: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 8.55/4.79 = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3190_ordered__ring__class_Ole__add__iff1,axiom,
% 8.55/4.79 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 8.55/4.79 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3191_ordered__ring__class_Ole__add__iff1,axiom,
% 8.55/4.79 ! [A: int,E: int,C: int,B: int,D: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 8.55/4.79 = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3192_ordered__ring__class_Ole__add__iff2,axiom,
% 8.55/4.79 ! [A: real,E: real,C: real,B: real,D: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 8.55/4.79 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3193_ordered__ring__class_Ole__add__iff2,axiom,
% 8.55/4.79 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 8.55/4.79 = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3194_ordered__ring__class_Ole__add__iff2,axiom,
% 8.55/4.79 ! [A: int,E: int,C: int,B: int,D: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 8.55/4.79 = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3195_int__power__div__base,axiom,
% 8.55/4.79 ! [M: nat,K: int] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.55/4.79 => ( ( ord_less_int @ zero_zero_int @ K )
% 8.55/4.79 => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 8.55/4.79 = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3196_power__inject__base,axiom,
% 8.55/4.79 ! [A: real,N: nat,B: real] :
% 8.55/4.79 ( ( ( power_power_real @ A @ ( suc @ N ) )
% 8.55/4.79 = ( power_power_real @ B @ ( suc @ N ) ) )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.55/4.79 => ( A = B ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3197_power__inject__base,axiom,
% 8.55/4.79 ! [A: rat,N: nat,B: rat] :
% 8.55/4.79 ( ( ( power_power_rat @ A @ ( suc @ N ) )
% 8.55/4.79 = ( power_power_rat @ B @ ( suc @ N ) ) )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.55/4.79 => ( A = B ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3198_power__inject__base,axiom,
% 8.55/4.79 ! [A: nat,N: nat,B: nat] :
% 8.55/4.79 ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 8.55/4.79 = ( power_power_nat @ B @ ( suc @ N ) ) )
% 8.55/4.79 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.79 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.55/4.79 => ( A = B ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3199_power__inject__base,axiom,
% 8.55/4.79 ! [A: int,N: nat,B: int] :
% 8.55/4.79 ( ( ( power_power_int @ A @ ( suc @ N ) )
% 8.55/4.79 = ( power_power_int @ B @ ( suc @ N ) ) )
% 8.55/4.79 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.79 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.55/4.79 => ( A = B ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3200_power__le__imp__le__base,axiom,
% 8.55/4.79 ! [A: real,N: nat,B: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.55/4.79 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3201_power__le__imp__le__base,axiom,
% 8.55/4.79 ! [A: rat,N: nat,B: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.55/4.79 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3202_power__le__imp__le__base,axiom,
% 8.55/4.79 ! [A: nat,N: nat,B: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
% 8.55/4.79 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.55/4.79 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3203_power__le__imp__le__base,axiom,
% 8.55/4.79 ! [A: int,N: nat,B: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
% 8.55/4.79 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.55/4.79 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3204_zero__power,axiom,
% 8.55/4.79 ! [N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( power_power_rat @ zero_zero_rat @ N )
% 8.55/4.79 = zero_zero_rat ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3205_zero__power,axiom,
% 8.55/4.79 ! [N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( power_power_nat @ zero_zero_nat @ N )
% 8.55/4.79 = zero_zero_nat ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3206_zero__power,axiom,
% 8.55/4.79 ! [N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( power_power_real @ zero_zero_real @ N )
% 8.55/4.79 = zero_zero_real ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3207_zero__power,axiom,
% 8.55/4.79 ! [N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( power_power_int @ zero_zero_int @ N )
% 8.55/4.79 = zero_zero_int ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3208_zero__power,axiom,
% 8.55/4.79 ! [N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( power_power_complex @ zero_zero_complex @ N )
% 8.55/4.79 = zero_zero_complex ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3209_diff__Suc__less,axiom,
% 8.55/4.79 ! [N: nat,I: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3210_n__less__n__mult__m,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 8.55/4.79 => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3211_n__less__m__mult__n,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 8.55/4.79 => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3212_one__less__mult,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.55/4.79 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 8.55/4.79 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3213_of__nat__diff,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N ) )
% 8.55/4.79 = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3214_of__nat__diff,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 8.55/4.79 = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3215_of__nat__diff,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 8.55/4.79 = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3216_of__nat__diff,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 8.55/4.79 = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3217_of__nat__diff,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 8.55/4.79 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3218_nat__induct__non__zero,axiom,
% 8.55/4.79 ! [N: nat,P: nat > $o] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( P @ one_one_nat )
% 8.55/4.79 => ( ! [N2: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 8.55/4.79 => ( ( P @ N2 )
% 8.55/4.79 => ( P @ ( suc @ N2 ) ) ) )
% 8.55/4.79 => ( P @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3219_nat__diff__split,axiom,
% 8.55/4.79 ! [P: nat > $o,A: nat,B: nat] :
% 8.55/4.79 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 8.55/4.79 = ( ( ( ord_less_nat @ A @ B )
% 8.55/4.79 => ( P @ zero_zero_nat ) )
% 8.55/4.79 & ! [D4: nat] :
% 8.55/4.79 ( ( A
% 8.55/4.79 = ( plus_plus_nat @ B @ D4 ) )
% 8.55/4.79 => ( P @ D4 ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3220_nat__diff__split__asm,axiom,
% 8.55/4.79 ! [P: nat > $o,A: nat,B: nat] :
% 8.55/4.79 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 8.55/4.79 = ( ~ ( ( ( ord_less_nat @ A @ B )
% 8.55/4.79 & ~ ( P @ zero_zero_nat ) )
% 8.55/4.79 | ? [D4: nat] :
% 8.55/4.79 ( ( A
% 8.55/4.79 = ( plus_plus_nat @ B @ D4 ) )
% 8.55/4.79 & ~ ( P @ D4 ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3221_power__gt__expt,axiom,
% 8.55/4.79 ! [N: nat,K: nat] :
% 8.55/4.79 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.55/4.79 => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3222_zless__iff__Suc__zadd,axiom,
% 8.55/4.79 ( ord_less_int
% 8.55/4.79 = ( ^ [W2: int,Z5: int] :
% 8.55/4.79 ? [N3: nat] :
% 8.55/4.79 ( Z5
% 8.55/4.79 = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3223_nat__one__le__power,axiom,
% 8.55/4.79 ! [I: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 8.55/4.79 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3224_pos__zmult__eq__1__iff,axiom,
% 8.55/4.79 ! [M: int,N: int] :
% 8.55/4.79 ( ( ord_less_int @ zero_zero_int @ M )
% 8.55/4.79 => ( ( ( times_times_int @ M @ N )
% 8.55/4.79 = one_one_int )
% 8.55/4.79 = ( ( M = one_one_int )
% 8.55/4.79 & ( N = one_one_int ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3225_nat__mult__dvd__cancel1,axiom,
% 8.55/4.79 ! [K: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.55/4.79 => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.55/4.79 = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3226_dvd__mult__cancel,axiom,
% 8.55/4.79 ! [K: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.55/4.79 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.55/4.79 => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3227_odd__less__0__iff,axiom,
% 8.55/4.79 ! [Z2: int] :
% 8.55/4.79 ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
% 8.55/4.79 = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3228_div__less__iff__less__mult,axiom,
% 8.55/4.79 ! [Q2: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 8.55/4.79 => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
% 8.55/4.79 = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3229_nat__mult__div__cancel1,axiom,
% 8.55/4.79 ! [K: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.55/4.79 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.55/4.79 = ( divide_divide_nat @ M @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3230_nat__eq__add__iff1,axiom,
% 8.55/4.79 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ J @ I )
% 8.55/4.79 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 8.55/4.79 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 8.55/4.79 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 8.55/4.79 = N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3231_nat__eq__add__iff2,axiom,
% 8.55/4.79 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.79 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 8.55/4.79 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 8.55/4.79 = ( M
% 8.55/4.79 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3232_nat__le__add__iff1,axiom,
% 8.55/4.79 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ J @ I )
% 8.55/4.79 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 8.55/4.79 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3233_nat__le__add__iff2,axiom,
% 8.55/4.79 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.79 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 8.55/4.79 = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3234_nat__diff__add__eq1,axiom,
% 8.55/4.79 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ J @ I )
% 8.55/4.79 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 8.55/4.79 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3235_nat__diff__add__eq2,axiom,
% 8.55/4.79 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ I @ J )
% 8.55/4.79 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 8.55/4.79 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3236_div__eq__dividend__iff,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.55/4.79 => ( ( ( divide_divide_nat @ M @ N )
% 8.55/4.79 = M )
% 8.55/4.79 = ( N = one_one_nat ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3237_div__less__dividend,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_nat @ one_one_nat @ N )
% 8.55/4.79 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.55/4.79 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3238_int__div__less__self,axiom,
% 8.55/4.79 ! [X: int,K: int] :
% 8.55/4.79 ( ( ord_less_int @ zero_zero_int @ X )
% 8.55/4.79 => ( ( ord_less_int @ one_one_int @ K )
% 8.55/4.79 => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3239_plusinfinity,axiom,
% 8.55/4.79 ! [D: int,P4: int > $o,P: int > $o] :
% 8.55/4.79 ( ( ord_less_int @ zero_zero_int @ D )
% 8.55/4.79 => ( ! [X4: int,K3: int] :
% 8.55/4.79 ( ( P4 @ X4 )
% 8.55/4.79 = ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) )
% 8.55/4.79 => ( ? [Z3: int] :
% 8.55/4.79 ! [X4: int] :
% 8.55/4.79 ( ( ord_less_int @ Z3 @ X4 )
% 8.55/4.79 => ( ( P @ X4 )
% 8.55/4.79 = ( P4 @ X4 ) ) )
% 8.55/4.79 => ( ? [X_1: int] : ( P4 @ X_1 )
% 8.55/4.79 => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3240_minusinfinity,axiom,
% 8.55/4.79 ! [D: int,P1: int > $o,P: int > $o] :
% 8.55/4.79 ( ( ord_less_int @ zero_zero_int @ D )
% 8.55/4.79 => ( ! [X4: int,K3: int] :
% 8.55/4.79 ( ( P1 @ X4 )
% 8.55/4.79 = ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) )
% 8.55/4.79 => ( ? [Z3: int] :
% 8.55/4.79 ! [X4: int] :
% 8.55/4.79 ( ( ord_less_int @ X4 @ Z3 )
% 8.55/4.79 => ( ( P @ X4 )
% 8.55/4.79 = ( P1 @ X4 ) ) )
% 8.55/4.79 => ( ? [X_1: int] : ( P1 @ X_1 )
% 8.55/4.79 => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3241_zdiv__zmult2__eq,axiom,
% 8.55/4.79 ! [C: int,A: int,B: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.55/4.79 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 8.55/4.79 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3242_int__induct,axiom,
% 8.55/4.79 ! [P: int > $o,K: int,I: int] :
% 8.55/4.79 ( ( P @ K )
% 8.55/4.79 => ( ! [I2: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ K @ I2 )
% 8.55/4.79 => ( ( P @ I2 )
% 8.55/4.79 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 8.55/4.79 => ( ! [I2: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ I2 @ K )
% 8.55/4.79 => ( ( P @ I2 )
% 8.55/4.79 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 8.55/4.79 => ( P @ I ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3243_real__of__nat__div4,axiom,
% 8.55/4.79 ! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3244_power__le__zero__eq,axiom,
% 8.55/4.79 ! [A: real,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 8.55/4.79 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 8.55/4.79 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 & ( A = zero_zero_real ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3245_power__le__zero__eq,axiom,
% 8.55/4.79 ! [A: rat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 8.55/4.79 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 8.55/4.79 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 & ( A = zero_zero_rat ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3246_power__le__zero__eq,axiom,
% 8.55/4.79 ! [A: int,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 8.55/4.79 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 8.55/4.79 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 & ( A = zero_zero_int ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3247_exp__div__exp__eq,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.79 = ( times_times_nat
% 8.55/4.79 @ ( zero_n2687167440665602831ol_nat
% 8.55/4.79 @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 8.55/4.79 != zero_zero_nat )
% 8.55/4.79 & ( ord_less_eq_nat @ N @ M ) ) )
% 8.55/4.79 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3248_exp__div__exp__eq,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.79 = ( times_times_int
% 8.55/4.79 @ ( zero_n2684676970156552555ol_int
% 8.55/4.79 @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 8.55/4.79 != zero_zero_int )
% 8.55/4.79 & ( ord_less_eq_nat @ N @ M ) ) )
% 8.55/4.79 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3249_exp__div__exp__eq,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.79 = ( times_3573771949741848930nteger
% 8.55/4.79 @ ( zero_n356916108424825756nteger
% 8.55/4.79 @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 8.55/4.79 != zero_z3403309356797280102nteger )
% 8.55/4.79 & ( ord_less_eq_nat @ N @ M ) ) )
% 8.55/4.79 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3250_less__divide__eq__numeral_I1_J,axiom,
% 8.55/4.79 ! [W: num,B: real,C: real] :
% 8.55/4.79 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 8.55/4.79 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.79 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 8.55/4.79 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.79 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.79 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 8.55/4.79 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.79 => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3251_less__divide__eq__numeral_I1_J,axiom,
% 8.55/4.79 ! [W: num,B: rat,C: rat] :
% 8.55/4.79 ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 8.55/4.79 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.79 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 8.55/4.79 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.79 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.79 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 8.55/4.79 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.79 => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3252_divide__less__eq__numeral_I1_J,axiom,
% 8.55/4.79 ! [B: real,C: real,W: num] :
% 8.55/4.79 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 8.55/4.79 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.79 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 8.55/4.79 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.55/4.79 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.79 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 8.55/4.79 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.55/4.79 => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3253_divide__less__eq__numeral_I1_J,axiom,
% 8.55/4.79 ! [B: rat,C: rat,W: num] :
% 8.55/4.79 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 8.55/4.79 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.79 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 8.55/4.79 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.55/4.79 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.79 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 8.55/4.79 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.55/4.79 => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3254_frac__less__eq,axiom,
% 8.55/4.79 ! [Y: real,Z2: real,X: real,W: real] :
% 8.55/4.79 ( ( Y != zero_zero_real )
% 8.55/4.79 => ( ( Z2 != zero_zero_real )
% 8.55/4.79 => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
% 8.55/4.79 = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3255_frac__less__eq,axiom,
% 8.55/4.79 ! [Y: rat,Z2: rat,X: rat,W: rat] :
% 8.55/4.79 ( ( Y != zero_zero_rat )
% 8.55/4.79 => ( ( Z2 != zero_zero_rat )
% 8.55/4.79 => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z2 ) )
% 8.55/4.79 = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3256_power__Suc__less,axiom,
% 8.55/4.79 ! [A: real,N: nat] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.79 => ( ( ord_less_real @ A @ one_one_real )
% 8.55/4.79 => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3257_power__Suc__less,axiom,
% 8.55/4.79 ! [A: rat,N: nat] :
% 8.55/4.79 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.79 => ( ( ord_less_rat @ A @ one_one_rat )
% 8.55/4.79 => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3258_power__Suc__less,axiom,
% 8.55/4.79 ! [A: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.55/4.79 => ( ( ord_less_nat @ A @ one_one_nat )
% 8.55/4.79 => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3259_power__Suc__less,axiom,
% 8.55/4.79 ! [A: int,N: nat] :
% 8.55/4.79 ( ( ord_less_int @ zero_zero_int @ A )
% 8.55/4.79 => ( ( ord_less_int @ A @ one_one_int )
% 8.55/4.79 => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3260_power__Suc__less__one,axiom,
% 8.55/4.79 ! [A: real,N: nat] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.79 => ( ( ord_less_real @ A @ one_one_real )
% 8.55/4.79 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3261_power__Suc__less__one,axiom,
% 8.55/4.79 ! [A: rat,N: nat] :
% 8.55/4.79 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.79 => ( ( ord_less_rat @ A @ one_one_rat )
% 8.55/4.79 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3262_power__Suc__less__one,axiom,
% 8.55/4.79 ! [A: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.55/4.79 => ( ( ord_less_nat @ A @ one_one_nat )
% 8.55/4.79 => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3263_power__Suc__less__one,axiom,
% 8.55/4.79 ! [A: int,N: nat] :
% 8.55/4.79 ( ( ord_less_int @ zero_zero_int @ A )
% 8.55/4.79 => ( ( ord_less_int @ A @ one_one_int )
% 8.55/4.79 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3264_convex__bound__le,axiom,
% 8.55/4.79 ! [X: real,A: real,Y: real,U: real,V: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ X @ A )
% 8.55/4.79 => ( ( ord_less_eq_real @ Y @ A )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 8.55/4.79 => ( ( ( plus_plus_real @ U @ V )
% 8.55/4.79 = one_one_real )
% 8.55/4.79 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3265_convex__bound__le,axiom,
% 8.55/4.79 ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ X @ A )
% 8.55/4.79 => ( ( ord_less_eq_rat @ Y @ A )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 8.55/4.79 => ( ( ( plus_plus_rat @ U @ V )
% 8.55/4.79 = one_one_rat )
% 8.55/4.79 => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3266_convex__bound__le,axiom,
% 8.55/4.79 ! [X: int,A: int,Y: int,U: int,V: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ X @ A )
% 8.55/4.79 => ( ( ord_less_eq_int @ Y @ A )
% 8.55/4.79 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 8.55/4.79 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 8.55/4.79 => ( ( ( plus_plus_int @ U @ V )
% 8.55/4.79 = one_one_int )
% 8.55/4.79 => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3267_frac__le__eq,axiom,
% 8.55/4.79 ! [Y: real,Z2: real,X: real,W: real] :
% 8.55/4.79 ( ( Y != zero_zero_real )
% 8.55/4.79 => ( ( Z2 != zero_zero_real )
% 8.55/4.79 => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
% 8.55/4.79 = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3268_frac__le__eq,axiom,
% 8.55/4.79 ! [Y: rat,Z2: rat,X: rat,W: rat] :
% 8.55/4.79 ( ( Y != zero_zero_rat )
% 8.55/4.79 => ( ( Z2 != zero_zero_rat )
% 8.55/4.79 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z2 ) )
% 8.55/4.79 = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3269_power__Suc__le__self,axiom,
% 8.55/4.79 ! [A: real,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.55/4.79 => ( ( ord_less_eq_real @ A @ one_one_real )
% 8.55/4.79 => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3270_power__Suc__le__self,axiom,
% 8.55/4.79 ! [A: rat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.55/4.79 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 8.55/4.79 => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3271_power__Suc__le__self,axiom,
% 8.55/4.79 ! [A: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.55/4.79 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 8.55/4.79 => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3272_power__Suc__le__self,axiom,
% 8.55/4.79 ! [A: int,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.79 => ( ( ord_less_eq_int @ A @ one_one_int )
% 8.55/4.79 => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3273_dvd__power__iff,axiom,
% 8.55/4.79 ! [X: code_integer,M: nat,N: nat] :
% 8.55/4.79 ( ( X != zero_z3403309356797280102nteger )
% 8.55/4.79 => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
% 8.55/4.79 = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
% 8.55/4.79 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3274_dvd__power__iff,axiom,
% 8.55/4.79 ! [X: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( X != zero_zero_nat )
% 8.55/4.79 => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
% 8.55/4.79 = ( ( dvd_dvd_nat @ X @ one_one_nat )
% 8.55/4.79 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3275_dvd__power__iff,axiom,
% 8.55/4.79 ! [X: int,M: nat,N: nat] :
% 8.55/4.79 ( ( X != zero_zero_int )
% 8.55/4.79 => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
% 8.55/4.79 = ( ( dvd_dvd_int @ X @ one_one_int )
% 8.55/4.79 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3276_pos2,axiom,
% 8.55/4.79 ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3277_dvd__power,axiom,
% 8.55/4.79 ! [N: nat,X: code_integer] :
% 8.55/4.79 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 | ( X = one_one_Code_integer ) )
% 8.55/4.79 => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3278_dvd__power,axiom,
% 8.55/4.79 ! [N: nat,X: rat] :
% 8.55/4.79 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 | ( X = one_one_rat ) )
% 8.55/4.79 => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3279_dvd__power,axiom,
% 8.55/4.79 ! [N: nat,X: nat] :
% 8.55/4.79 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 | ( X = one_one_nat ) )
% 8.55/4.79 => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3280_dvd__power,axiom,
% 8.55/4.79 ! [N: nat,X: real] :
% 8.55/4.79 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 | ( X = one_one_real ) )
% 8.55/4.79 => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3281_dvd__power,axiom,
% 8.55/4.79 ! [N: nat,X: int] :
% 8.55/4.79 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 | ( X = one_one_int ) )
% 8.55/4.79 => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3282_dvd__power,axiom,
% 8.55/4.79 ! [N: nat,X: complex] :
% 8.55/4.79 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 | ( X = one_one_complex ) )
% 8.55/4.79 => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3283_power__diff,axiom,
% 8.55/4.79 ! [A: complex,N: nat,M: nat] :
% 8.55/4.79 ( ( A != zero_zero_complex )
% 8.55/4.79 => ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
% 8.55/4.79 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3284_power__diff,axiom,
% 8.55/4.79 ! [A: real,N: nat,M: nat] :
% 8.55/4.79 ( ( A != zero_zero_real )
% 8.55/4.79 => ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
% 8.55/4.79 = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3285_power__diff,axiom,
% 8.55/4.79 ! [A: rat,N: nat,M: nat] :
% 8.55/4.79 ( ( A != zero_zero_rat )
% 8.55/4.79 => ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
% 8.55/4.79 = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3286_power__diff,axiom,
% 8.55/4.79 ! [A: nat,N: nat,M: nat] :
% 8.55/4.79 ( ( A != zero_zero_nat )
% 8.55/4.79 => ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
% 8.55/4.79 = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3287_power__diff,axiom,
% 8.55/4.79 ! [A: int,N: nat,M: nat] :
% 8.55/4.79 ( ( A != zero_zero_int )
% 8.55/4.79 => ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
% 8.55/4.79 = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3288_less__exp,axiom,
% 8.55/4.79 ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3289_power2__nat__le__imp__le,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 8.55/4.79 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3290_power2__nat__le__eq__le,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.79 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3291_self__le__ge2__pow,axiom,
% 8.55/4.79 ! [K: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 8.55/4.79 => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3292_two__realpow__ge__one,axiom,
% 8.55/4.79 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3293_Suc__diff__eq__diff__pred,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 8.55/4.79 = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3294_Suc__pred_H,axiom,
% 8.55/4.79 ! [N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( N
% 8.55/4.79 = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3295_div__if,axiom,
% 8.55/4.79 ( divide_divide_nat
% 8.55/4.79 = ( ^ [M3: nat,N3: nat] :
% 8.55/4.79 ( if_nat
% 8.55/4.79 @ ( ( ord_less_nat @ M3 @ N3 )
% 8.55/4.79 | ( N3 = zero_zero_nat ) )
% 8.55/4.79 @ zero_zero_nat
% 8.55/4.79 @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3296_div__geq,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ~ ( ord_less_nat @ M @ N )
% 8.55/4.79 => ( ( divide_divide_nat @ M @ N )
% 8.55/4.79 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3297_dividend__less__times__div,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3298_dividend__less__div__times,axiom,
% 8.55/4.79 ! [N: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3299_split__div,axiom,
% 8.55/4.79 ! [P: nat > $o,M: nat,N: nat] :
% 8.55/4.79 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 8.55/4.79 = ( ( ( N = zero_zero_nat )
% 8.55/4.79 => ( P @ zero_zero_nat ) )
% 8.55/4.79 & ( ( N != zero_zero_nat )
% 8.55/4.79 => ! [I4: nat,J3: nat] :
% 8.55/4.79 ( ( ord_less_nat @ J3 @ N )
% 8.55/4.79 => ( ( M
% 8.55/4.79 = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 8.55/4.79 => ( P @ I4 ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3300_dvd__mult__cancel2,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.55/4.79 => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 8.55/4.79 = ( N = one_one_nat ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3301_dvd__mult__cancel1,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.55/4.79 => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 8.55/4.79 = ( N = one_one_nat ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3302_int__ops_I6_J,axiom,
% 8.55/4.79 ! [A: nat,B: nat] :
% 8.55/4.79 ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 8.55/4.79 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 8.55/4.79 = zero_zero_int ) )
% 8.55/4.79 & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 8.55/4.79 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 8.55/4.79 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3303_dvd__minus__add,axiom,
% 8.55/4.79 ! [Q2: nat,N: nat,R: nat,M: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ Q2 @ N )
% 8.55/4.79 => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R @ M ) )
% 8.55/4.79 => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
% 8.55/4.79 = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q2 ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3304_even__mult__exp__div__exp__iff,axiom,
% 8.55/4.79 ! [A: code_integer,M: nat,N: nat] :
% 8.55/4.79 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 8.55/4.79 = ( ( ord_less_nat @ N @ M )
% 8.55/4.79 | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 = zero_z3403309356797280102nteger )
% 8.55/4.79 | ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.79 & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3305_even__mult__exp__div__exp__iff,axiom,
% 8.55/4.79 ! [A: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.55/4.79 = ( ( ord_less_nat @ N @ M )
% 8.55/4.79 | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 = zero_zero_nat )
% 8.55/4.79 | ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.79 & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3306_even__mult__exp__div__exp__iff,axiom,
% 8.55/4.79 ! [A: int,M: nat,N: nat] :
% 8.55/4.79 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 8.55/4.79 = ( ( ord_less_nat @ N @ M )
% 8.55/4.79 | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 = zero_zero_int )
% 8.55/4.79 | ( ( ord_less_eq_nat @ M @ N )
% 8.55/4.79 & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3307_real__of__nat__div2,axiom,
% 8.55/4.79 ! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3308_real__of__nat__div3,axiom,
% 8.55/4.79 ! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3309_half__gt__zero__iff,axiom,
% 8.55/4.79 ! [A: real] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.55/4.79 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3310_half__gt__zero__iff,axiom,
% 8.55/4.79 ! [A: rat] :
% 8.55/4.79 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 8.55/4.79 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3311_half__gt__zero,axiom,
% 8.55/4.79 ! [A: real] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ A )
% 8.55/4.79 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3312_half__gt__zero,axiom,
% 8.55/4.79 ! [A: rat] :
% 8.55/4.79 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.55/4.79 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3313_field__less__half__sum,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ( ( ord_less_real @ X @ Y )
% 8.55/4.79 => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3314_field__less__half__sum,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ( ( ord_less_rat @ X @ Y )
% 8.55/4.79 => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3315_power2__less__0,axiom,
% 8.55/4.79 ! [A: real] :
% 8.55/4.79 ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3316_power2__less__0,axiom,
% 8.55/4.79 ! [A: rat] :
% 8.55/4.79 ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3317_power2__less__0,axiom,
% 8.55/4.79 ! [A: int] :
% 8.55/4.79 ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3318_nat__approx__posE,axiom,
% 8.55/4.79 ! [E: real] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ E )
% 8.55/4.79 => ~ ! [N2: nat] :
% 8.55/4.79 ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ E ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3319_nat__approx__posE,axiom,
% 8.55/4.79 ! [E: rat] :
% 8.55/4.79 ( ( ord_less_rat @ zero_zero_rat @ E )
% 8.55/4.79 => ~ ! [N2: nat] :
% 8.55/4.79 ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ E ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3320_scaling__mono,axiom,
% 8.55/4.79 ! [U: real,V: real,R: real,S: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ U @ V )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ R )
% 8.55/4.79 => ( ( ord_less_eq_real @ R @ S )
% 8.55/4.79 => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3321_scaling__mono,axiom,
% 8.55/4.79 ! [U: rat,V: rat,R: rat,S: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ U @ V )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ R )
% 8.55/4.79 => ( ( ord_less_eq_rat @ R @ S )
% 8.55/4.79 => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3322_of__nat__less__two__power,axiom,
% 8.55/4.79 ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3323_of__nat__less__two__power,axiom,
% 8.55/4.79 ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3324_of__nat__less__two__power,axiom,
% 8.55/4.79 ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3325_power2__le__imp__le,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.55/4.79 => ( ord_less_eq_real @ X @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3326_power2__le__imp__le,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 8.55/4.79 => ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3327_power2__le__imp__le,axiom,
% 8.55/4.79 ! [X: nat,Y: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.79 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 8.55/4.79 => ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3328_power2__le__imp__le,axiom,
% 8.55/4.79 ! [X: int,Y: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.79 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.55/4.79 => ( ord_less_eq_int @ X @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3329_power2__eq__imp__eq,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.79 = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.55/4.79 => ( X = Y ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3330_power2__eq__imp__eq,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.79 = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 8.55/4.79 => ( X = Y ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3331_power2__eq__imp__eq,axiom,
% 8.55/4.79 ! [X: nat,Y: nat] :
% 8.55/4.79 ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.79 = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.79 => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 8.55/4.79 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 8.55/4.79 => ( X = Y ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3332_power2__eq__imp__eq,axiom,
% 8.55/4.79 ! [X: int,Y: int] :
% 8.55/4.79 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.79 = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.55/4.79 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.55/4.79 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.55/4.79 => ( X = Y ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3333_zero__le__power2,axiom,
% 8.55/4.79 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3334_zero__le__power2,axiom,
% 8.55/4.79 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3335_zero__le__power2,axiom,
% 8.55/4.79 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3336_num_Osize__gen_I1_J,axiom,
% 8.55/4.79 ( ( size_num @ one )
% 8.55/4.79 = zero_zero_nat ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3337_less__2__cases__iff,axiom,
% 8.55/4.79 ! [N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.79 = ( ( N = zero_zero_nat )
% 8.55/4.79 | ( N
% 8.55/4.79 = ( suc @ zero_zero_nat ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3338_less__2__cases,axiom,
% 8.55/4.79 ! [N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.79 => ( ( N = zero_zero_nat )
% 8.55/4.79 | ( N
% 8.55/4.79 = ( suc @ zero_zero_nat ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3339_power__diff__power__eq,axiom,
% 8.55/4.79 ! [A: nat,N: nat,M: nat] :
% 8.55/4.79 ( ( A != zero_zero_nat )
% 8.55/4.79 => ( ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 8.55/4.79 = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 8.55/4.79 & ( ~ ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 8.55/4.79 = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3340_power__diff__power__eq,axiom,
% 8.55/4.79 ! [A: int,N: nat,M: nat] :
% 8.55/4.79 ( ( A != zero_zero_int )
% 8.55/4.79 => ( ( ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 8.55/4.79 = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 8.55/4.79 & ( ~ ( ord_less_eq_nat @ N @ M )
% 8.55/4.79 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 8.55/4.79 = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3341_power__mono__odd,axiom,
% 8.55/4.79 ! [N: nat,A: real,B: real] :
% 8.55/4.79 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 => ( ( ord_less_eq_real @ A @ B )
% 8.55/4.79 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3342_power__mono__odd,axiom,
% 8.55/4.79 ! [N: nat,A: rat,B: rat] :
% 8.55/4.79 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 => ( ( ord_less_eq_rat @ A @ B )
% 8.55/4.79 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3343_power__mono__odd,axiom,
% 8.55/4.79 ! [N: nat,A: int,B: int] :
% 8.55/4.79 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 => ( ( ord_less_eq_int @ A @ B )
% 8.55/4.79 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3344_odd__pos,axiom,
% 8.55/4.79 ! [N: nat] :
% 8.55/4.79 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3345_power__minus__mult,axiom,
% 8.55/4.79 ! [N: nat,A: complex] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 8.55/4.79 = ( power_power_complex @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3346_power__minus__mult,axiom,
% 8.55/4.79 ! [N: nat,A: real] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 8.55/4.79 = ( power_power_real @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3347_power__minus__mult,axiom,
% 8.55/4.79 ! [N: nat,A: rat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 8.55/4.79 = ( power_power_rat @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3348_power__minus__mult,axiom,
% 8.55/4.79 ! [N: nat,A: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 8.55/4.79 = ( power_power_nat @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3349_power__minus__mult,axiom,
% 8.55/4.79 ! [N: nat,A: int] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 8.55/4.79 = ( power_power_int @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3350_diff__le__diff__pow,axiom,
% 8.55/4.79 ! [K: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 8.55/4.79 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3351_dvd__power__iff__le,axiom,
% 8.55/4.79 ! [K: nat,M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 8.55/4.79 => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 8.55/4.79 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3352_not__exp__less__eq__0__int,axiom,
% 8.55/4.79 ! [N: nat] :
% 8.55/4.79 ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3353_bits__induct,axiom,
% 8.55/4.79 ! [P: nat > $o,A: nat] :
% 8.55/4.79 ( ! [A4: nat] :
% 8.55/4.79 ( ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.79 = A4 )
% 8.55/4.79 => ( P @ A4 ) )
% 8.55/4.79 => ( ! [A4: nat,B4: $o] :
% 8.55/4.79 ( ( P @ A4 )
% 8.55/4.79 => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.79 = A4 )
% 8.55/4.79 => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 8.55/4.79 => ( P @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3354_bits__induct,axiom,
% 8.55/4.79 ! [P: int > $o,A: int] :
% 8.55/4.79 ( ! [A4: int] :
% 8.55/4.79 ( ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.55/4.79 = A4 )
% 8.55/4.79 => ( P @ A4 ) )
% 8.55/4.79 => ( ! [A4: int,B4: $o] :
% 8.55/4.79 ( ( P @ A4 )
% 8.55/4.79 => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.55/4.79 = A4 )
% 8.55/4.79 => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 8.55/4.79 => ( P @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3355_bits__induct,axiom,
% 8.55/4.79 ! [P: code_integer > $o,A: code_integer] :
% 8.55/4.79 ( ! [A4: code_integer] :
% 8.55/4.79 ( ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.55/4.79 = A4 )
% 8.55/4.79 => ( P @ A4 ) )
% 8.55/4.79 => ( ! [A4: code_integer,B4: $o] :
% 8.55/4.79 ( ( P @ A4 )
% 8.55/4.79 => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.55/4.79 = A4 )
% 8.55/4.79 => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 8.55/4.79 => ( P @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3356_sum__power2__gt__zero__iff,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.55/4.79 = ( ( X != zero_zero_real )
% 8.55/4.79 | ( Y != zero_zero_real ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3357_sum__power2__gt__zero__iff,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.55/4.79 = ( ( X != zero_zero_rat )
% 8.55/4.79 | ( Y != zero_zero_rat ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3358_sum__power2__gt__zero__iff,axiom,
% 8.55/4.79 ! [X: int,Y: int] :
% 8.55/4.79 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.55/4.79 = ( ( X != zero_zero_int )
% 8.55/4.79 | ( Y != zero_zero_int ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3359_not__sum__power2__lt__zero,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3360_not__sum__power2__lt__zero,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3361_not__sum__power2__lt__zero,axiom,
% 8.55/4.79 ! [X: int,Y: int] :
% 8.55/4.79 ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3362_sum__power2__le__zero__iff,axiom,
% 8.55/4.79 ! [X: real,Y: real] :
% 8.55/4.79 ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 8.55/4.79 = ( ( X = zero_zero_real )
% 8.55/4.79 & ( Y = zero_zero_real ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3363_sum__power2__le__zero__iff,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 8.55/4.79 = ( ( X = zero_zero_rat )
% 8.55/4.79 & ( Y = zero_zero_rat ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3364_sum__power2__le__zero__iff,axiom,
% 8.55/4.79 ! [X: int,Y: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 8.55/4.79 = ( ( X = zero_zero_int )
% 8.55/4.79 & ( Y = zero_zero_int ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3365_sum__power2__ge__zero,axiom,
% 8.55/4.79 ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3366_sum__power2__ge__zero,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3367_sum__power2__ge__zero,axiom,
% 8.55/4.79 ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3368_zero__le__even__power_H,axiom,
% 8.55/4.79 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3369_zero__le__even__power_H,axiom,
% 8.55/4.79 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3370_zero__le__even__power_H,axiom,
% 8.55/4.79 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3371_zero__le__even__power,axiom,
% 8.55/4.79 ! [N: nat,A: real] :
% 8.55/4.79 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3372_zero__le__even__power,axiom,
% 8.55/4.79 ! [N: nat,A: rat] :
% 8.55/4.79 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3373_zero__le__even__power,axiom,
% 8.55/4.79 ! [N: nat,A: int] :
% 8.55/4.79 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3374_zero__le__odd__power,axiom,
% 8.55/4.79 ! [N: nat,A: real] :
% 8.55/4.79 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 8.55/4.79 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3375_zero__le__odd__power,axiom,
% 8.55/4.79 ! [N: nat,A: rat] :
% 8.55/4.79 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 8.55/4.79 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3376_zero__le__odd__power,axiom,
% 8.55/4.79 ! [N: nat,A: int] :
% 8.55/4.79 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 8.55/4.79 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3377_zero__le__power__eq,axiom,
% 8.55/4.79 ! [A: real,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 8.55/4.79 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3378_zero__le__power__eq,axiom,
% 8.55/4.79 ! [A: rat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 8.55/4.79 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3379_zero__le__power__eq,axiom,
% 8.55/4.79 ! [A: int,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 8.55/4.79 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3380_nat__bit__induct,axiom,
% 8.55/4.79 ! [P: nat > $o,N: nat] :
% 8.55/4.79 ( ( P @ zero_zero_nat )
% 8.55/4.79 => ( ! [N2: nat] :
% 8.55/4.79 ( ( P @ N2 )
% 8.55/4.79 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 8.55/4.79 => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 8.55/4.79 => ( ! [N2: nat] :
% 8.55/4.79 ( ( P @ N2 )
% 8.55/4.79 => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 8.55/4.79 => ( P @ N ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3381_Suc__n__div__2__gt__zero,axiom,
% 8.55/4.79 ! [N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3382_div__2__gt__zero,axiom,
% 8.55/4.79 ! [N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.55/4.79 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3383_add__0__iff,axiom,
% 8.55/4.79 ! [B: real,A: real] :
% 8.55/4.79 ( ( B
% 8.55/4.79 = ( plus_plus_real @ B @ A ) )
% 8.55/4.79 = ( A = zero_zero_real ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3384_add__0__iff,axiom,
% 8.55/4.79 ! [B: rat,A: rat] :
% 8.55/4.79 ( ( B
% 8.55/4.79 = ( plus_plus_rat @ B @ A ) )
% 8.55/4.79 = ( A = zero_zero_rat ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3385_add__0__iff,axiom,
% 8.55/4.79 ! [B: nat,A: nat] :
% 8.55/4.79 ( ( B
% 8.55/4.79 = ( plus_plus_nat @ B @ A ) )
% 8.55/4.79 = ( A = zero_zero_nat ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3386_add__0__iff,axiom,
% 8.55/4.79 ! [B: int,A: int] :
% 8.55/4.79 ( ( B
% 8.55/4.79 = ( plus_plus_int @ B @ A ) )
% 8.55/4.79 = ( A = zero_zero_int ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3387_crossproduct__eq,axiom,
% 8.55/4.79 ! [W: real,Y: real,X: real,Z2: real] :
% 8.55/4.79 ( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z2 ) )
% 8.55/4.79 = ( plus_plus_real @ ( times_times_real @ W @ Z2 ) @ ( times_times_real @ X @ Y ) ) )
% 8.55/4.79 = ( ( W = X )
% 8.55/4.79 | ( Y = Z2 ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3388_crossproduct__eq,axiom,
% 8.55/4.79 ! [W: rat,Y: rat,X: rat,Z2: rat] :
% 8.55/4.79 ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z2 ) )
% 8.55/4.79 = ( plus_plus_rat @ ( times_times_rat @ W @ Z2 ) @ ( times_times_rat @ X @ Y ) ) )
% 8.55/4.79 = ( ( W = X )
% 8.55/4.79 | ( Y = Z2 ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3389_crossproduct__eq,axiom,
% 8.55/4.79 ! [W: nat,Y: nat,X: nat,Z2: nat] :
% 8.55/4.79 ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z2 ) )
% 8.55/4.79 = ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X @ Y ) ) )
% 8.55/4.79 = ( ( W = X )
% 8.55/4.79 | ( Y = Z2 ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3390_crossproduct__eq,axiom,
% 8.55/4.79 ! [W: int,Y: int,X: int,Z2: int] :
% 8.55/4.79 ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z2 ) )
% 8.55/4.79 = ( plus_plus_int @ ( times_times_int @ W @ Z2 ) @ ( times_times_int @ X @ Y ) ) )
% 8.55/4.79 = ( ( W = X )
% 8.55/4.79 | ( Y = Z2 ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3391_crossproduct__noteq,axiom,
% 8.55/4.79 ! [A: real,B: real,C: real,D: real] :
% 8.55/4.79 ( ( ( A != B )
% 8.55/4.79 & ( C != D ) )
% 8.55/4.79 = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
% 8.55/4.79 != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3392_crossproduct__noteq,axiom,
% 8.55/4.79 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.55/4.79 ( ( ( A != B )
% 8.55/4.79 & ( C != D ) )
% 8.55/4.79 = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
% 8.55/4.79 != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3393_crossproduct__noteq,axiom,
% 8.55/4.79 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.55/4.79 ( ( ( A != B )
% 8.55/4.79 & ( C != D ) )
% 8.55/4.79 = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
% 8.55/4.79 != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3394_crossproduct__noteq,axiom,
% 8.55/4.79 ! [A: int,B: int,C: int,D: int] :
% 8.55/4.79 ( ( ( A != B )
% 8.55/4.79 & ( C != D ) )
% 8.55/4.79 = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
% 8.55/4.79 != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3395_odd__power__less__zero,axiom,
% 8.55/4.79 ! [A: real,N: nat] :
% 8.55/4.79 ( ( ord_less_real @ A @ zero_zero_real )
% 8.55/4.79 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3396_odd__power__less__zero,axiom,
% 8.55/4.79 ! [A: rat,N: nat] :
% 8.55/4.79 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.55/4.79 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3397_odd__power__less__zero,axiom,
% 8.55/4.79 ! [A: int,N: nat] :
% 8.55/4.79 ( ( ord_less_int @ A @ zero_zero_int )
% 8.55/4.79 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3398_sum__squares__bound,axiom,
% 8.55/4.79 ! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3399_sum__squares__bound,axiom,
% 8.55/4.79 ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3400_zero__less__power__eq,axiom,
% 8.55/4.79 ! [A: real,N: nat] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 8.55/4.79 = ( ( N = zero_zero_nat )
% 8.55/4.79 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 & ( A != zero_zero_real ) )
% 8.55/4.79 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3401_zero__less__power__eq,axiom,
% 8.55/4.79 ! [A: rat,N: nat] :
% 8.55/4.79 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 8.55/4.79 = ( ( N = zero_zero_nat )
% 8.55/4.79 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 & ( A != zero_zero_rat ) )
% 8.55/4.79 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3402_zero__less__power__eq,axiom,
% 8.55/4.79 ! [A: int,N: nat] :
% 8.55/4.79 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 8.55/4.79 = ( ( N = zero_zero_nat )
% 8.55/4.79 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 & ( A != zero_zero_int ) )
% 8.55/4.79 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.55/4.79 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3403_odd__0__le__power__imp__0__le,axiom,
% 8.55/4.79 ! [A: real,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 8.55/4.79 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3404_odd__0__le__power__imp__0__le,axiom,
% 8.55/4.79 ! [A: rat,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 8.55/4.79 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3405_odd__0__le__power__imp__0__le,axiom,
% 8.55/4.79 ! [A: int,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 8.55/4.79 => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3406_L2__set__mult__ineq__lemma,axiom,
% 8.55/4.79 ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3407_even__mask__div__iff_H,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 8.55/4.79 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3408_even__mask__div__iff_H,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.55/4.79 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3409_even__mask__div__iff_H,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 8.55/4.79 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3410_pos__zdiv__mult__2,axiom,
% 8.55/4.79 ! [A: int,B: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.55/4.79 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 8.55/4.79 = ( divide_divide_int @ B @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3411_neg__zdiv__mult__2,axiom,
% 8.55/4.79 ! [A: int,B: int] :
% 8.55/4.79 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.55/4.79 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 8.55/4.79 = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3412_linear__plus__1__le__power,axiom,
% 8.55/4.79 ! [X: real,N: nat] :
% 8.55/4.79 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.55/4.79 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3413_enat__ord__number_I1_J,axiom,
% 8.55/4.79 ! [M: num,N: num] :
% 8.55/4.79 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 8.55/4.79 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3414_enat__ord__number_I2_J,axiom,
% 8.55/4.79 ! [M: num,N: num] :
% 8.55/4.79 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 8.55/4.79 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3415_mult__le__cancel__iff1,axiom,
% 8.55/4.79 ! [Z2: real,X: real,Y: real] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ Z2 )
% 8.55/4.79 => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
% 8.55/4.79 = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3416_mult__le__cancel__iff1,axiom,
% 8.55/4.79 ! [Z2: rat,X: rat,Y: rat] :
% 8.55/4.79 ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 8.55/4.79 => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
% 8.55/4.79 = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3417_mult__le__cancel__iff1,axiom,
% 8.55/4.79 ! [Z2: int,X: int,Y: int] :
% 8.55/4.79 ( ( ord_less_int @ zero_zero_int @ Z2 )
% 8.55/4.79 => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
% 8.55/4.79 = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3418_mult__le__cancel__iff2,axiom,
% 8.55/4.79 ! [Z2: real,X: real,Y: real] :
% 8.55/4.79 ( ( ord_less_real @ zero_zero_real @ Z2 )
% 8.55/4.79 => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ ( times_times_real @ Z2 @ Y ) )
% 8.55/4.79 = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3419_mult__le__cancel__iff2,axiom,
% 8.55/4.79 ! [Z2: rat,X: rat,Y: rat] :
% 8.55/4.79 ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 8.55/4.79 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ ( times_times_rat @ Z2 @ Y ) )
% 8.55/4.79 = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3420_mult__le__cancel__iff2,axiom,
% 8.55/4.79 ! [Z2: int,X: int,Y: int] :
% 8.55/4.79 ( ( ord_less_int @ zero_zero_int @ Z2 )
% 8.55/4.79 => ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X ) @ ( times_times_int @ Z2 @ Y ) )
% 8.55/4.79 = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3421_lemma__termdiff3,axiom,
% 8.55/4.79 ! [H: real,Z2: real,K5: real,N: nat] :
% 8.55/4.79 ( ( H != zero_zero_real )
% 8.55/4.79 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ K5 )
% 8.55/4.79 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z2 @ H ) ) @ K5 )
% 8.55/4.79 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H ) @ N ) @ ( power_power_real @ Z2 @ N ) ) @ H ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3422_lemma__termdiff3,axiom,
% 8.55/4.79 ! [H: complex,Z2: complex,K5: real,N: nat] :
% 8.55/4.79 ( ( H != zero_zero_complex )
% 8.55/4.79 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ K5 )
% 8.55/4.79 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z2 @ H ) ) @ K5 )
% 8.55/4.79 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H ) @ N ) @ ( power_power_complex @ Z2 @ N ) ) @ H ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3423_even__succ__mod__exp,axiom,
% 8.55/4.79 ! [A: nat,N: nat] :
% 8.55/4.79 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.55/4.79 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.79 = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3424_even__succ__mod__exp,axiom,
% 8.55/4.79 ! [A: int,N: nat] :
% 8.55/4.79 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.55/4.79 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.79 = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3425_even__succ__mod__exp,axiom,
% 8.55/4.79 ! [A: code_integer,N: nat] :
% 8.55/4.79 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.55/4.79 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.79 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3426_even__succ__mod__exp,axiom,
% 8.55/4.79 ! [A: code_natural,N: nat] :
% 8.55/4.79 ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
% 8.55/4.79 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.55/4.79 => ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
% 8.55/4.79 = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3427_mod__less,axiom,
% 8.55/4.79 ! [M: nat,N: nat] :
% 8.55/4.79 ( ( ord_less_nat @ M @ N )
% 8.55/4.79 => ( ( modulo_modulo_nat @ M @ N )
% 8.55/4.79 = M ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3428_mod__mod__trivial,axiom,
% 8.55/4.79 ! [A: nat,B: nat] :
% 8.55/4.79 ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 8.55/4.79 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3429_mod__mod__trivial,axiom,
% 8.55/4.79 ! [A: int,B: int] :
% 8.55/4.79 ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 8.55/4.79 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3430_mod__mod__trivial,axiom,
% 8.55/4.79 ! [A: code_integer,B: code_integer] :
% 8.55/4.79 ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 8.55/4.79 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3431_mod__mod__trivial,axiom,
% 8.55/4.79 ! [A: code_natural,B: code_natural] :
% 8.55/4.79 ( ( modulo8411746178871703098atural @ ( modulo8411746178871703098atural @ A @ B ) @ B )
% 8.55/4.79 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3432_bits__mod__0,axiom,
% 8.55/4.79 ! [A: nat] :
% 8.55/4.79 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 8.55/4.79 = zero_zero_nat ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3433_bits__mod__0,axiom,
% 8.55/4.79 ! [A: int] :
% 8.55/4.79 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 8.55/4.79 = zero_zero_int ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3434_bits__mod__0,axiom,
% 8.55/4.79 ! [A: code_integer] :
% 8.55/4.79 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 8.55/4.79 = zero_z3403309356797280102nteger ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3435_bits__mod__0,axiom,
% 8.55/4.79 ! [A: code_natural] :
% 8.55/4.79 ( ( modulo8411746178871703098atural @ zero_z2226904508553997617atural @ A )
% 8.55/4.79 = zero_z2226904508553997617atural ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3436_mod__self,axiom,
% 8.55/4.79 ! [A: nat] :
% 8.55/4.79 ( ( modulo_modulo_nat @ A @ A )
% 8.55/4.79 = zero_zero_nat ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3437_mod__self,axiom,
% 8.55/4.79 ! [A: int] :
% 8.55/4.79 ( ( modulo_modulo_int @ A @ A )
% 8.55/4.79 = zero_zero_int ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3438_mod__self,axiom,
% 8.55/4.79 ! [A: code_integer] :
% 8.55/4.79 ( ( modulo364778990260209775nteger @ A @ A )
% 8.55/4.79 = zero_z3403309356797280102nteger ) ).
% 8.55/4.79
% 8.55/4.79 thf(fact_3439_mod__self,axiom,
% 8.55/4.80 ! [A: code_natural] :
% 8.55/4.80 ( ( modulo8411746178871703098atural @ A @ A )
% 8.55/4.80 = zero_z2226904508553997617atural ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3440_mod__by__0,axiom,
% 8.55/4.80 ! [A: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 8.55/4.80 = A ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3441_mod__by__0,axiom,
% 8.55/4.80 ! [A: int] :
% 8.55/4.80 ( ( modulo_modulo_int @ A @ zero_zero_int )
% 8.55/4.80 = A ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3442_mod__by__0,axiom,
% 8.55/4.80 ! [A: code_integer] :
% 8.55/4.80 ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 8.55/4.80 = A ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3443_mod__by__0,axiom,
% 8.55/4.80 ! [A: code_natural] :
% 8.55/4.80 ( ( modulo8411746178871703098atural @ A @ zero_z2226904508553997617atural )
% 8.55/4.80 = A ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3444_mod__0,axiom,
% 8.55/4.80 ! [A: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 8.55/4.80 = zero_zero_nat ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3445_mod__0,axiom,
% 8.55/4.80 ! [A: int] :
% 8.55/4.80 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 8.55/4.80 = zero_zero_int ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3446_mod__0,axiom,
% 8.55/4.80 ! [A: code_integer] :
% 8.55/4.80 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 8.55/4.80 = zero_z3403309356797280102nteger ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3447_mod__0,axiom,
% 8.55/4.80 ! [A: code_natural] :
% 8.55/4.80 ( ( modulo8411746178871703098atural @ zero_z2226904508553997617atural @ A )
% 8.55/4.80 = zero_z2226904508553997617atural ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3448_mod__by__Suc__0,axiom,
% 8.55/4.80 ! [M: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 8.55/4.80 = zero_zero_nat ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3449_mod__add__self1,axiom,
% 8.55/4.80 ! [B: nat,A: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 8.55/4.80 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3450_mod__add__self1,axiom,
% 8.55/4.80 ! [B: int,A: int] :
% 8.55/4.80 ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
% 8.55/4.80 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3451_mod__add__self1,axiom,
% 8.55/4.80 ! [B: code_integer,A: code_integer] :
% 8.55/4.80 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 8.55/4.80 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3452_mod__add__self1,axiom,
% 8.55/4.80 ! [B: code_natural,A: code_natural] :
% 8.55/4.80 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ B @ A ) @ B )
% 8.55/4.80 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3453_mod__add__self2,axiom,
% 8.55/4.80 ! [A: nat,B: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 8.55/4.80 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3454_mod__add__self2,axiom,
% 8.55/4.80 ! [A: int,B: int] :
% 8.55/4.80 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
% 8.55/4.80 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3455_mod__add__self2,axiom,
% 8.55/4.80 ! [A: code_integer,B: code_integer] :
% 8.55/4.80 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 8.55/4.80 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3456_mod__add__self2,axiom,
% 8.55/4.80 ! [A: code_natural,B: code_natural] :
% 8.55/4.80 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ B )
% 8.55/4.80 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3457_minus__mod__self2,axiom,
% 8.55/4.80 ! [A: int,B: int] :
% 8.55/4.80 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
% 8.55/4.80 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3458_minus__mod__self2,axiom,
% 8.55/4.80 ! [A: code_integer,B: code_integer] :
% 8.55/4.80 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
% 8.55/4.80 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3459_mod__mult__self1__is__0,axiom,
% 8.55/4.80 ! [B: nat,A: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
% 8.55/4.80 = zero_zero_nat ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3460_mod__mult__self1__is__0,axiom,
% 8.55/4.80 ! [B: int,A: int] :
% 8.55/4.80 ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
% 8.55/4.80 = zero_zero_int ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3461_mod__mult__self1__is__0,axiom,
% 8.55/4.80 ! [B: code_integer,A: code_integer] :
% 8.55/4.80 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
% 8.55/4.80 = zero_z3403309356797280102nteger ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3462_mod__mult__self1__is__0,axiom,
% 8.55/4.80 ! [B: code_natural,A: code_natural] :
% 8.55/4.80 ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ B @ A ) @ B )
% 8.55/4.80 = zero_z2226904508553997617atural ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3463_mod__mult__self2__is__0,axiom,
% 8.55/4.80 ! [A: nat,B: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
% 8.55/4.80 = zero_zero_nat ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3464_mod__mult__self2__is__0,axiom,
% 8.55/4.80 ! [A: int,B: int] :
% 8.55/4.80 ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
% 8.55/4.80 = zero_zero_int ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3465_mod__mult__self2__is__0,axiom,
% 8.55/4.80 ! [A: code_integer,B: code_integer] :
% 8.55/4.80 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 8.55/4.80 = zero_z3403309356797280102nteger ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3466_mod__mult__self2__is__0,axiom,
% 8.55/4.80 ! [A: code_natural,B: code_natural] :
% 8.55/4.80 ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ B )
% 8.55/4.80 = zero_z2226904508553997617atural ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3467_bits__mod__by__1,axiom,
% 8.55/4.80 ! [A: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 8.55/4.80 = zero_zero_nat ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3468_bits__mod__by__1,axiom,
% 8.55/4.80 ! [A: int] :
% 8.55/4.80 ( ( modulo_modulo_int @ A @ one_one_int )
% 8.55/4.80 = zero_zero_int ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3469_bits__mod__by__1,axiom,
% 8.55/4.80 ! [A: code_integer] :
% 8.55/4.80 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 8.55/4.80 = zero_z3403309356797280102nteger ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3470_bits__mod__by__1,axiom,
% 8.55/4.80 ! [A: code_natural] :
% 8.55/4.80 ( ( modulo8411746178871703098atural @ A @ one_one_Code_natural )
% 8.55/4.80 = zero_z2226904508553997617atural ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3471_mod__by__1,axiom,
% 8.55/4.80 ! [A: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 8.55/4.80 = zero_zero_nat ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3472_mod__by__1,axiom,
% 8.55/4.80 ! [A: int] :
% 8.55/4.80 ( ( modulo_modulo_int @ A @ one_one_int )
% 8.55/4.80 = zero_zero_int ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3473_mod__by__1,axiom,
% 8.55/4.80 ! [A: code_integer] :
% 8.55/4.80 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 8.55/4.80 = zero_z3403309356797280102nteger ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3474_mod__by__1,axiom,
% 8.55/4.80 ! [A: code_natural] :
% 8.55/4.80 ( ( modulo8411746178871703098atural @ A @ one_one_Code_natural )
% 8.55/4.80 = zero_z2226904508553997617atural ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3475_bits__mod__div__trivial,axiom,
% 8.55/4.80 ! [A: nat,B: nat] :
% 8.55/4.80 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 8.55/4.80 = zero_zero_nat ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3476_bits__mod__div__trivial,axiom,
% 8.55/4.80 ! [A: int,B: int] :
% 8.55/4.80 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 8.55/4.80 = zero_zero_int ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3477_bits__mod__div__trivial,axiom,
% 8.55/4.80 ! [A: code_integer,B: code_integer] :
% 8.55/4.80 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 8.55/4.80 = zero_z3403309356797280102nteger ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3478_bits__mod__div__trivial,axiom,
% 8.55/4.80 ! [A: code_natural,B: code_natural] :
% 8.55/4.80 ( ( divide5121882707175180666atural @ ( modulo8411746178871703098atural @ A @ B ) @ B )
% 8.55/4.80 = zero_z2226904508553997617atural ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3479_mod__div__trivial,axiom,
% 8.55/4.80 ! [A: nat,B: nat] :
% 8.55/4.80 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 8.55/4.80 = zero_zero_nat ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3480_mod__div__trivial,axiom,
% 8.55/4.80 ! [A: int,B: int] :
% 8.55/4.80 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 8.55/4.80 = zero_zero_int ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3481_mod__div__trivial,axiom,
% 8.55/4.80 ! [A: code_integer,B: code_integer] :
% 8.55/4.80 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 8.55/4.80 = zero_z3403309356797280102nteger ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3482_mod__div__trivial,axiom,
% 8.55/4.80 ! [A: code_natural,B: code_natural] :
% 8.55/4.80 ( ( divide5121882707175180666atural @ ( modulo8411746178871703098atural @ A @ B ) @ B )
% 8.55/4.80 = zero_z2226904508553997617atural ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3483_mod__mult__self1,axiom,
% 8.55/4.80 ! [A: nat,C: nat,B: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 8.55/4.80 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3484_mod__mult__self1,axiom,
% 8.55/4.80 ! [A: int,C: int,B: int] :
% 8.55/4.80 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 8.55/4.80 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3485_mod__mult__self1,axiom,
% 8.55/4.80 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.55/4.80 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 8.55/4.80 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3486_mod__mult__self1,axiom,
% 8.55/4.80 ! [A: code_natural,C: code_natural,B: code_natural] :
% 8.55/4.80 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ C @ B ) ) @ B )
% 8.55/4.80 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3487_mod__mult__self2,axiom,
% 8.55/4.80 ! [A: nat,B: nat,C: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 8.55/4.80 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3488_mod__mult__self2,axiom,
% 8.55/4.80 ! [A: int,B: int,C: int] :
% 8.55/4.80 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 8.55/4.80 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3489_mod__mult__self2,axiom,
% 8.55/4.80 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.55/4.80 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 8.55/4.80 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3490_mod__mult__self2,axiom,
% 8.55/4.80 ! [A: code_natural,B: code_natural,C: code_natural] :
% 8.55/4.80 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ B @ C ) ) @ B )
% 8.55/4.80 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3491_mod__mult__self3,axiom,
% 8.55/4.80 ! [C: nat,B: nat,A: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 8.55/4.80 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3492_mod__mult__self3,axiom,
% 8.55/4.80 ! [C: int,B: int,A: int] :
% 8.55/4.80 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 8.55/4.80 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3493_mod__mult__self3,axiom,
% 8.55/4.80 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.55/4.80 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 8.55/4.80 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3494_mod__mult__self3,axiom,
% 8.55/4.80 ! [C: code_natural,B: code_natural,A: code_natural] :
% 8.55/4.80 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C @ B ) @ A ) @ B )
% 8.55/4.80 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3495_mod__mult__self4,axiom,
% 8.55/4.80 ! [B: nat,C: nat,A: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 8.55/4.80 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3496_mod__mult__self4,axiom,
% 8.55/4.80 ! [B: int,C: int,A: int] :
% 8.55/4.80 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 8.55/4.80 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3497_mod__mult__self4,axiom,
% 8.55/4.80 ! [B: code_integer,C: code_integer,A: code_integer] :
% 8.55/4.80 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 8.55/4.80 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3498_mod__mult__self4,axiom,
% 8.55/4.80 ! [B: code_natural,C: code_natural,A: code_natural] :
% 8.55/4.80 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ C ) @ A ) @ B )
% 8.55/4.80 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3499_dvd__imp__mod__0,axiom,
% 8.55/4.80 ! [A: nat,B: nat] :
% 8.55/4.80 ( ( dvd_dvd_nat @ A @ B )
% 8.55/4.80 => ( ( modulo_modulo_nat @ B @ A )
% 8.55/4.80 = zero_zero_nat ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3500_dvd__imp__mod__0,axiom,
% 8.55/4.80 ! [A: int,B: int] :
% 8.55/4.80 ( ( dvd_dvd_int @ A @ B )
% 8.55/4.80 => ( ( modulo_modulo_int @ B @ A )
% 8.55/4.80 = zero_zero_int ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3501_dvd__imp__mod__0,axiom,
% 8.55/4.80 ! [A: code_integer,B: code_integer] :
% 8.55/4.80 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.55/4.80 => ( ( modulo364778990260209775nteger @ B @ A )
% 8.55/4.80 = zero_z3403309356797280102nteger ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3502_dvd__imp__mod__0,axiom,
% 8.55/4.80 ! [A: code_natural,B: code_natural] :
% 8.55/4.80 ( ( dvd_dvd_Code_natural @ A @ B )
% 8.55/4.80 => ( ( modulo8411746178871703098atural @ B @ A )
% 8.55/4.80 = zero_z2226904508553997617atural ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3503_Suc__mod__mult__self1,axiom,
% 8.55/4.80 ! [M: nat,K: nat,N: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
% 8.55/4.80 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3504_Suc__mod__mult__self2,axiom,
% 8.55/4.80 ! [M: nat,N: nat,K: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
% 8.55/4.80 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3505_Suc__mod__mult__self3,axiom,
% 8.55/4.80 ! [K: nat,N: nat,M: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
% 8.55/4.80 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3506_Suc__mod__mult__self4,axiom,
% 8.55/4.80 ! [N: nat,K: nat,M: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
% 8.55/4.80 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3507_mod__pos__pos__trivial,axiom,
% 8.55/4.80 ! [K: int,L: int] :
% 8.55/4.80 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.55/4.80 => ( ( ord_less_int @ K @ L )
% 8.55/4.80 => ( ( modulo_modulo_int @ K @ L )
% 8.55/4.80 = K ) ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3508_mod__neg__neg__trivial,axiom,
% 8.55/4.80 ! [K: int,L: int] :
% 8.55/4.80 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 8.55/4.80 => ( ( ord_less_int @ L @ K )
% 8.55/4.80 => ( ( modulo_modulo_int @ K @ L )
% 8.55/4.80 = K ) ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3509_Suc__0__mod__eq,axiom,
% 8.55/4.80 ! [N: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.55/4.80 = ( zero_n2687167440665602831ol_nat
% 8.55/4.80 @ ( N
% 8.55/4.80 != ( suc @ zero_zero_nat ) ) ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3510_mod2__Suc__Suc,axiom,
% 8.55/4.80 ! [M: nat] :
% 8.55/4.80 ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.80 = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3511_zmod__numeral__Bit0,axiom,
% 8.55/4.80 ! [V: num,W: num] :
% 8.55/4.80 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 8.55/4.80 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3512_Suc__times__numeral__mod__eq,axiom,
% 8.55/4.80 ! [K: num,N: nat] :
% 8.55/4.80 ( ( ( numeral_numeral_nat @ K )
% 8.55/4.80 != one_one_nat )
% 8.55/4.80 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
% 8.55/4.80 = one_one_nat ) ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3513_bits__one__mod__two__eq__one,axiom,
% 8.55/4.80 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.55/4.80 = one_one_nat ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3514_bits__one__mod__two__eq__one,axiom,
% 8.55/4.80 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.55/4.80 = one_one_int ) ).
% 8.55/4.80
% 8.55/4.80 thf(fact_3515_bits__one__mod__two__eq__one,axiom,
% 8.55/4.80 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.55/4.80 = one_one_Code_integer ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3516_bits__one__mod__two__eq__one,axiom,
% 8.58/4.80 ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_Code_natural ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3517_one__mod__two__eq__one,axiom,
% 8.58/4.80 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_nat ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3518_one__mod__two__eq__one,axiom,
% 8.58/4.80 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_int ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3519_one__mod__two__eq__one,axiom,
% 8.58/4.80 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_Code_integer ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3520_one__mod__two__eq__one,axiom,
% 8.58/4.80 ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_Code_natural ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3521_not__mod2__eq__Suc__0__eq__0,axiom,
% 8.58/4.80 ! [N: nat] :
% 8.58/4.80 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 != ( suc @ zero_zero_nat ) )
% 8.58/4.80 = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 = zero_zero_nat ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3522_even__mod__2__iff,axiom,
% 8.58/4.80 ! [A: nat] :
% 8.58/4.80 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.80 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3523_even__mod__2__iff,axiom,
% 8.58/4.80 ! [A: int] :
% 8.58/4.80 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 8.58/4.80 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3524_even__mod__2__iff,axiom,
% 8.58/4.80 ! [A: code_integer] :
% 8.58/4.80 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 8.58/4.80 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3525_even__mod__2__iff,axiom,
% 8.58/4.80 ! [A: code_natural] :
% 8.58/4.80 ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
% 8.58/4.80 = ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3526_add__self__mod__2,axiom,
% 8.58/4.80 ! [M: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 = zero_zero_nat ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3527_not__mod__2__eq__0__eq__1,axiom,
% 8.58/4.80 ! [A: nat] :
% 8.58/4.80 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 != zero_zero_nat )
% 8.58/4.80 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_nat ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3528_not__mod__2__eq__0__eq__1,axiom,
% 8.58/4.80 ! [A: int] :
% 8.58/4.80 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.80 != zero_zero_int )
% 8.58/4.80 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_int ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3529_not__mod__2__eq__0__eq__1,axiom,
% 8.58/4.80 ! [A: code_integer] :
% 8.58/4.80 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.80 != zero_z3403309356797280102nteger )
% 8.58/4.80 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_Code_integer ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3530_not__mod__2__eq__0__eq__1,axiom,
% 8.58/4.80 ! [A: code_natural] :
% 8.58/4.80 ( ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 8.58/4.80 != zero_z2226904508553997617atural )
% 8.58/4.80 = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_Code_natural ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3531_not__mod__2__eq__1__eq__0,axiom,
% 8.58/4.80 ! [A: nat] :
% 8.58/4.80 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 != one_one_nat )
% 8.58/4.80 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 = zero_zero_nat ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3532_not__mod__2__eq__1__eq__0,axiom,
% 8.58/4.80 ! [A: int] :
% 8.58/4.80 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.80 != one_one_int )
% 8.58/4.80 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.80 = zero_zero_int ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3533_not__mod__2__eq__1__eq__0,axiom,
% 8.58/4.80 ! [A: code_integer] :
% 8.58/4.80 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.80 != one_one_Code_integer )
% 8.58/4.80 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.80 = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3534_not__mod__2__eq__1__eq__0,axiom,
% 8.58/4.80 ! [A: code_natural] :
% 8.58/4.80 ( ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 8.58/4.80 != one_one_Code_natural )
% 8.58/4.80 = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 8.58/4.80 = zero_z2226904508553997617atural ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3535_mod2__gr__0,axiom,
% 8.58/4.80 ! [M: nat] :
% 8.58/4.80 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.80 = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_nat ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3536_one__mod__2__pow__eq,axiom,
% 8.58/4.80 ! [N: nat] :
% 8.58/4.80 ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.80 = ( zero_n8403883297036319079atural @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3537_one__mod__2__pow__eq,axiom,
% 8.58/4.80 ! [N: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.80 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3538_one__mod__2__pow__eq,axiom,
% 8.58/4.80 ! [N: nat] :
% 8.58/4.80 ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.80 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3539_one__mod__2__pow__eq,axiom,
% 8.58/4.80 ! [N: nat] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.80 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3540_add__diff__assoc__enat,axiom,
% 8.58/4.80 ! [Z2: extended_enat,Y: extended_enat,X: extended_enat] :
% 8.58/4.80 ( ( ord_le2932123472753598470d_enat @ Z2 @ Y )
% 8.58/4.80 => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z2 ) )
% 8.58/4.80 = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z2 ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3541_verit__la__generic,axiom,
% 8.58/4.80 ! [A: int,X: int] :
% 8.58/4.80 ( ( ord_less_eq_int @ A @ X )
% 8.58/4.80 | ( A = X )
% 8.58/4.80 | ( ord_less_eq_int @ X @ A ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3542_unset__bit__less__eq,axiom,
% 8.58/4.80 ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3543_set__bit__greater__eq,axiom,
% 8.58/4.80 ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3544_enat__0__less__mult__iff,axiom,
% 8.58/4.80 ! [M: extended_enat,N: extended_enat] :
% 8.58/4.80 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 8.58/4.80 = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 8.58/4.80 & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3545_of__nat__mod,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3546_of__nat__mod,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ( semiri3763490453095760265atural @ ( modulo_modulo_nat @ M @ N ) )
% 8.58/4.80 = ( modulo8411746178871703098atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3547_of__nat__mod,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
% 8.58/4.80 = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3548_of__nat__mod,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
% 8.58/4.80 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3549_zmod__int,axiom,
% 8.58/4.80 ! [A: nat,B: nat] :
% 8.58/4.80 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
% 8.58/4.80 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3550_mod__mult__eq,axiom,
% 8.58/4.80 ! [A: nat,C: nat,B: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3551_mod__mult__eq,axiom,
% 8.58/4.80 ! [A: int,C: int,B: int] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3552_mod__mult__eq,axiom,
% 8.58/4.80 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3553_mod__mult__eq,axiom,
% 8.58/4.80 ! [A: code_natural,C: code_natural,B: code_natural] :
% 8.58/4.80 ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ C ) @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3554_mod__mult__cong,axiom,
% 8.58/4.80 ! [A: nat,C: nat,A5: nat,B: nat,B6: nat] :
% 8.58/4.80 ( ( ( modulo_modulo_nat @ A @ C )
% 8.58/4.80 = ( modulo_modulo_nat @ A5 @ C ) )
% 8.58/4.80 => ( ( ( modulo_modulo_nat @ B @ C )
% 8.58/4.80 = ( modulo_modulo_nat @ B6 @ C ) )
% 8.58/4.80 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
% 8.58/4.80 = ( modulo_modulo_nat @ ( times_times_nat @ A5 @ B6 ) @ C ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3555_mod__mult__cong,axiom,
% 8.58/4.80 ! [A: int,C: int,A5: int,B: int,B6: int] :
% 8.58/4.80 ( ( ( modulo_modulo_int @ A @ C )
% 8.58/4.80 = ( modulo_modulo_int @ A5 @ C ) )
% 8.58/4.80 => ( ( ( modulo_modulo_int @ B @ C )
% 8.58/4.80 = ( modulo_modulo_int @ B6 @ C ) )
% 8.58/4.80 => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
% 8.58/4.80 = ( modulo_modulo_int @ ( times_times_int @ A5 @ B6 ) @ C ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3556_mod__mult__cong,axiom,
% 8.58/4.80 ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B6: code_integer] :
% 8.58/4.80 ( ( ( modulo364778990260209775nteger @ A @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ A5 @ C ) )
% 8.58/4.80 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 8.58/4.80 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A5 @ B6 ) @ C ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3557_mod__mult__cong,axiom,
% 8.58/4.80 ! [A: code_natural,C: code_natural,A5: code_natural,B: code_natural,B6: code_natural] :
% 8.58/4.80 ( ( ( modulo8411746178871703098atural @ A @ C )
% 8.58/4.80 = ( modulo8411746178871703098atural @ A5 @ C ) )
% 8.58/4.80 => ( ( ( modulo8411746178871703098atural @ B @ C )
% 8.58/4.80 = ( modulo8411746178871703098atural @ B6 @ C ) )
% 8.58/4.80 => ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C )
% 8.58/4.80 = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A5 @ B6 ) @ C ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3558_mod__mult__mult2,axiom,
% 8.58/4.80 ! [A: nat,C: nat,B: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 8.58/4.80 = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3559_mod__mult__mult2,axiom,
% 8.58/4.80 ! [A: int,C: int,B: int] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 8.58/4.80 = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3560_mod__mult__mult2,axiom,
% 8.58/4.80 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 8.58/4.80 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3561_mod__mult__mult2,axiom,
% 8.58/4.80 ! [A: code_natural,C: code_natural,B: code_natural] :
% 8.58/4.80 ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) )
% 8.58/4.80 = ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3562_mult__mod__right,axiom,
% 8.58/4.80 ! [C: nat,A: nat,B: nat] :
% 8.58/4.80 ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 8.58/4.80 = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3563_mult__mod__right,axiom,
% 8.58/4.80 ! [C: int,A: int,B: int] :
% 8.58/4.80 ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 8.58/4.80 = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3564_mult__mod__right,axiom,
% 8.58/4.80 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3565_mult__mod__right,axiom,
% 8.58/4.80 ! [C: code_natural,A: code_natural,B: code_natural] :
% 8.58/4.80 ( ( times_2397367101498566445atural @ C @ ( modulo8411746178871703098atural @ A @ B ) )
% 8.58/4.80 = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3566_mod__mult__left__eq,axiom,
% 8.58/4.80 ! [A: nat,C: nat,B: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 8.58/4.80 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3567_mod__mult__left__eq,axiom,
% 8.58/4.80 ! [A: int,C: int,B: int] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 8.58/4.80 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3568_mod__mult__left__eq,axiom,
% 8.58/4.80 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3569_mod__mult__left__eq,axiom,
% 8.58/4.80 ! [A: code_natural,C: code_natural,B: code_natural] :
% 8.58/4.80 ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ C ) @ B ) @ C )
% 8.58/4.80 = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3570_mod__mult__right__eq,axiom,
% 8.58/4.80 ! [A: nat,B: nat,C: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3571_mod__mult__right__eq,axiom,
% 8.58/4.80 ! [A: int,B: int,C: int] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3572_mod__mult__right__eq,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3573_mod__mult__right__eq,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural,C: code_natural] :
% 8.58/4.80 ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3574_mod__add__eq,axiom,
% 8.58/4.80 ! [A: nat,C: nat,B: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3575_mod__add__eq,axiom,
% 8.58/4.80 ! [A: int,C: int,B: int] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3576_mod__add__eq,axiom,
% 8.58/4.80 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3577_mod__add__eq,axiom,
% 8.58/4.80 ! [A: code_natural,C: code_natural,B: code_natural] :
% 8.58/4.80 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C ) @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3578_mod__add__cong,axiom,
% 8.58/4.80 ! [A: nat,C: nat,A5: nat,B: nat,B6: nat] :
% 8.58/4.80 ( ( ( modulo_modulo_nat @ A @ C )
% 8.58/4.80 = ( modulo_modulo_nat @ A5 @ C ) )
% 8.58/4.80 => ( ( ( modulo_modulo_nat @ B @ C )
% 8.58/4.80 = ( modulo_modulo_nat @ B6 @ C ) )
% 8.58/4.80 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.58/4.80 = ( modulo_modulo_nat @ ( plus_plus_nat @ A5 @ B6 ) @ C ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3579_mod__add__cong,axiom,
% 8.58/4.80 ! [A: int,C: int,A5: int,B: int,B6: int] :
% 8.58/4.80 ( ( ( modulo_modulo_int @ A @ C )
% 8.58/4.80 = ( modulo_modulo_int @ A5 @ C ) )
% 8.58/4.80 => ( ( ( modulo_modulo_int @ B @ C )
% 8.58/4.80 = ( modulo_modulo_int @ B6 @ C ) )
% 8.58/4.80 => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.58/4.80 = ( modulo_modulo_int @ ( plus_plus_int @ A5 @ B6 ) @ C ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3580_mod__add__cong,axiom,
% 8.58/4.80 ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B6: code_integer] :
% 8.58/4.80 ( ( ( modulo364778990260209775nteger @ A @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ A5 @ C ) )
% 8.58/4.80 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 8.58/4.80 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A5 @ B6 ) @ C ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3581_mod__add__cong,axiom,
% 8.58/4.80 ! [A: code_natural,C: code_natural,A5: code_natural,B: code_natural,B6: code_natural] :
% 8.58/4.80 ( ( ( modulo8411746178871703098atural @ A @ C )
% 8.58/4.80 = ( modulo8411746178871703098atural @ A5 @ C ) )
% 8.58/4.80 => ( ( ( modulo8411746178871703098atural @ B @ C )
% 8.58/4.80 = ( modulo8411746178871703098atural @ B6 @ C ) )
% 8.58/4.80 => ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
% 8.58/4.80 = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A5 @ B6 ) @ C ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3582_mod__add__left__eq,axiom,
% 8.58/4.80 ! [A: nat,C: nat,B: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 8.58/4.80 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3583_mod__add__left__eq,axiom,
% 8.58/4.80 ! [A: int,C: int,B: int] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 8.58/4.80 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3584_mod__add__left__eq,axiom,
% 8.58/4.80 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3585_mod__add__left__eq,axiom,
% 8.58/4.80 ! [A: code_natural,C: code_natural,B: code_natural] :
% 8.58/4.80 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C ) @ B ) @ C )
% 8.58/4.80 = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3586_mod__add__right__eq,axiom,
% 8.58/4.80 ! [A: nat,B: nat,C: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3587_mod__add__right__eq,axiom,
% 8.58/4.80 ! [A: int,B: int,C: int] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3588_mod__add__right__eq,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3589_mod__add__right__eq,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural,C: code_natural] :
% 8.58/4.80 ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3590_mod__diff__eq,axiom,
% 8.58/4.80 ! [A: int,C: int,B: int] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3591_mod__diff__eq,axiom,
% 8.58/4.80 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3592_mod__diff__cong,axiom,
% 8.58/4.80 ! [A: int,C: int,A5: int,B: int,B6: int] :
% 8.58/4.80 ( ( ( modulo_modulo_int @ A @ C )
% 8.58/4.80 = ( modulo_modulo_int @ A5 @ C ) )
% 8.58/4.80 => ( ( ( modulo_modulo_int @ B @ C )
% 8.58/4.80 = ( modulo_modulo_int @ B6 @ C ) )
% 8.58/4.80 => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
% 8.58/4.80 = ( modulo_modulo_int @ ( minus_minus_int @ A5 @ B6 ) @ C ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3593_mod__diff__cong,axiom,
% 8.58/4.80 ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B6: code_integer] :
% 8.58/4.80 ( ( ( modulo364778990260209775nteger @ A @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ A5 @ C ) )
% 8.58/4.80 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 8.58/4.80 => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A5 @ B6 ) @ C ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3594_mod__diff__left__eq,axiom,
% 8.58/4.80 ! [A: int,C: int,B: int] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 8.58/4.80 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3595_mod__diff__left__eq,axiom,
% 8.58/4.80 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3596_mod__diff__right__eq,axiom,
% 8.58/4.80 ! [A: int,B: int,C: int] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3597_mod__diff__right__eq,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3598_power__mod,axiom,
% 8.58/4.80 ! [A: nat,B: nat,N: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
% 8.58/4.80 = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3599_power__mod,axiom,
% 8.58/4.80 ! [A: int,B: int,N: nat] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
% 8.58/4.80 = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3600_power__mod,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer,N: nat] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3601_power__mod,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural,N: nat] :
% 8.58/4.80 ( ( modulo8411746178871703098atural @ ( power_7079662738309270450atural @ ( modulo8411746178871703098atural @ A @ B ) @ N ) @ B )
% 8.58/4.80 = ( modulo8411746178871703098atural @ ( power_7079662738309270450atural @ A @ N ) @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3602_dvd__mod__imp__dvd,axiom,
% 8.58/4.80 ! [C: nat,A: nat,B: nat] :
% 8.58/4.80 ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 8.58/4.80 => ( ( dvd_dvd_nat @ C @ B )
% 8.58/4.80 => ( dvd_dvd_nat @ C @ A ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3603_dvd__mod__imp__dvd,axiom,
% 8.58/4.80 ! [C: int,A: int,B: int] :
% 8.58/4.80 ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 8.58/4.80 => ( ( dvd_dvd_int @ C @ B )
% 8.58/4.80 => ( dvd_dvd_int @ C @ A ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3604_dvd__mod__imp__dvd,axiom,
% 8.58/4.80 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 8.58/4.80 => ( ( dvd_dvd_Code_integer @ C @ B )
% 8.58/4.80 => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3605_dvd__mod__imp__dvd,axiom,
% 8.58/4.80 ! [C: code_natural,A: code_natural,B: code_natural] :
% 8.58/4.80 ( ( dvd_dvd_Code_natural @ C @ ( modulo8411746178871703098atural @ A @ B ) )
% 8.58/4.80 => ( ( dvd_dvd_Code_natural @ C @ B )
% 8.58/4.80 => ( dvd_dvd_Code_natural @ C @ A ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3606_dvd__mod__iff,axiom,
% 8.58/4.80 ! [C: nat,B: nat,A: nat] :
% 8.58/4.80 ( ( dvd_dvd_nat @ C @ B )
% 8.58/4.80 => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 8.58/4.80 = ( dvd_dvd_nat @ C @ A ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3607_dvd__mod__iff,axiom,
% 8.58/4.80 ! [C: int,B: int,A: int] :
% 8.58/4.80 ( ( dvd_dvd_int @ C @ B )
% 8.58/4.80 => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 8.58/4.80 = ( dvd_dvd_int @ C @ A ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3608_dvd__mod__iff,axiom,
% 8.58/4.80 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.58/4.80 ( ( dvd_dvd_Code_integer @ C @ B )
% 8.58/4.80 => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 8.58/4.80 = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3609_dvd__mod__iff,axiom,
% 8.58/4.80 ! [C: code_natural,B: code_natural,A: code_natural] :
% 8.58/4.80 ( ( dvd_dvd_Code_natural @ C @ B )
% 8.58/4.80 => ( ( dvd_dvd_Code_natural @ C @ ( modulo8411746178871703098atural @ A @ B ) )
% 8.58/4.80 = ( dvd_dvd_Code_natural @ C @ A ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3610_mod__mod__cancel,axiom,
% 8.58/4.80 ! [C: nat,B: nat,A: nat] :
% 8.58/4.80 ( ( dvd_dvd_nat @ C @ B )
% 8.58/4.80 => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
% 8.58/4.80 = ( modulo_modulo_nat @ A @ C ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3611_mod__mod__cancel,axiom,
% 8.58/4.80 ! [C: int,B: int,A: int] :
% 8.58/4.80 ( ( dvd_dvd_int @ C @ B )
% 8.58/4.80 => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
% 8.58/4.80 = ( modulo_modulo_int @ A @ C ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3612_mod__mod__cancel,axiom,
% 8.58/4.80 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.58/4.80 ( ( dvd_dvd_Code_integer @ C @ B )
% 8.58/4.80 => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ A @ C ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3613_mod__mod__cancel,axiom,
% 8.58/4.80 ! [C: code_natural,B: code_natural,A: code_natural] :
% 8.58/4.80 ( ( dvd_dvd_Code_natural @ C @ B )
% 8.58/4.80 => ( ( modulo8411746178871703098atural @ ( modulo8411746178871703098atural @ A @ B ) @ C )
% 8.58/4.80 = ( modulo8411746178871703098atural @ A @ C ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3614_dvd__mod,axiom,
% 8.58/4.80 ! [K: nat,M: nat,N: nat] :
% 8.58/4.80 ( ( dvd_dvd_nat @ K @ M )
% 8.58/4.80 => ( ( dvd_dvd_nat @ K @ N )
% 8.58/4.80 => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3615_dvd__mod,axiom,
% 8.58/4.80 ! [K: int,M: int,N: int] :
% 8.58/4.80 ( ( dvd_dvd_int @ K @ M )
% 8.58/4.80 => ( ( dvd_dvd_int @ K @ N )
% 8.58/4.80 => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3616_dvd__mod,axiom,
% 8.58/4.80 ! [K: code_integer,M: code_integer,N: code_integer] :
% 8.58/4.80 ( ( dvd_dvd_Code_integer @ K @ M )
% 8.58/4.80 => ( ( dvd_dvd_Code_integer @ K @ N )
% 8.58/4.80 => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3617_dvd__mod,axiom,
% 8.58/4.80 ! [K: code_natural,M: code_natural,N: code_natural] :
% 8.58/4.80 ( ( dvd_dvd_Code_natural @ K @ M )
% 8.58/4.80 => ( ( dvd_dvd_Code_natural @ K @ N )
% 8.58/4.80 => ( dvd_dvd_Code_natural @ K @ ( modulo8411746178871703098atural @ M @ N ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3618_mod__Suc__eq,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
% 8.58/4.80 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3619_mod__Suc__Suc__eq,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
% 8.58/4.80 = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3620_zmod__le__nonneg__dividend,axiom,
% 8.58/4.80 ! [M: int,K: int] :
% 8.58/4.80 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 8.58/4.80 => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3621_neg__mod__bound,axiom,
% 8.58/4.80 ! [L: int,K: int] :
% 8.58/4.80 ( ( ord_less_int @ L @ zero_zero_int )
% 8.58/4.80 => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3622_Euclidean__Division_Opos__mod__bound,axiom,
% 8.58/4.80 ! [L: int,K: int] :
% 8.58/4.80 ( ( ord_less_int @ zero_zero_int @ L )
% 8.58/4.80 => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3623_neg__mod__conj,axiom,
% 8.58/4.80 ! [B: int,A: int] :
% 8.58/4.80 ( ( ord_less_int @ B @ zero_zero_int )
% 8.58/4.80 => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
% 8.58/4.80 & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3624_pos__mod__conj,axiom,
% 8.58/4.80 ! [B: int,A: int] :
% 8.58/4.80 ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.80 => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
% 8.58/4.80 & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3625_zmod__trivial__iff,axiom,
% 8.58/4.80 ! [I: int,K: int] :
% 8.58/4.80 ( ( ( modulo_modulo_int @ I @ K )
% 8.58/4.80 = I )
% 8.58/4.80 = ( ( K = zero_zero_int )
% 8.58/4.80 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 8.58/4.80 & ( ord_less_int @ I @ K ) )
% 8.58/4.80 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 8.58/4.80 & ( ord_less_int @ K @ I ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3626_Euclidean__Division_Opos__mod__sign,axiom,
% 8.58/4.80 ! [L: int,K: int] :
% 8.58/4.80 ( ( ord_less_int @ zero_zero_int @ L )
% 8.58/4.80 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3627_neg__mod__sign,axiom,
% 8.58/4.80 ! [L: int,K: int] :
% 8.58/4.80 ( ( ord_less_int @ L @ zero_zero_int )
% 8.58/4.80 => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3628_mod__less__eq__dividend,axiom,
% 8.58/4.80 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3629_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 8.58/4.80 => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3630_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 8.58/4.80 ! [A: nat,B: nat] :
% 8.58/4.80 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.58/4.80 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3631_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.58/4.80 => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3632_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 8.58/4.80 ! [B: nat,A: nat] :
% 8.58/4.80 ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.58/4.80 => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3633_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 8.58/4.80 ! [B: int,A: int] :
% 8.58/4.80 ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.80 => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3634_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 8.58/4.80 ! [B: code_integer,A: code_integer] :
% 8.58/4.80 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 8.58/4.80 => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3635_mod__eq__self__iff__div__eq__0,axiom,
% 8.58/4.80 ! [A: nat,B: nat] :
% 8.58/4.80 ( ( ( modulo_modulo_nat @ A @ B )
% 8.58/4.80 = A )
% 8.58/4.80 = ( ( divide_divide_nat @ A @ B )
% 8.58/4.80 = zero_zero_nat ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3636_mod__eq__self__iff__div__eq__0,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( ( ( modulo_modulo_int @ A @ B )
% 8.58/4.80 = A )
% 8.58/4.80 = ( ( divide_divide_int @ A @ B )
% 8.58/4.80 = zero_zero_int ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3637_mod__eq__self__iff__div__eq__0,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( ( modulo364778990260209775nteger @ A @ B )
% 8.58/4.80 = A )
% 8.58/4.80 = ( ( divide6298287555418463151nteger @ A @ B )
% 8.58/4.80 = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3638_mod__eq__self__iff__div__eq__0,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural] :
% 8.58/4.80 ( ( ( modulo8411746178871703098atural @ A @ B )
% 8.58/4.80 = A )
% 8.58/4.80 = ( ( divide5121882707175180666atural @ A @ B )
% 8.58/4.80 = zero_z2226904508553997617atural ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3639_cong__exp__iff__simps_I9_J,axiom,
% 8.58/4.80 ! [M: num,Q2: num,N: num] :
% 8.58/4.80 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 8.58/4.80 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 8.58/4.80 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 8.58/4.80 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3640_cong__exp__iff__simps_I9_J,axiom,
% 8.58/4.80 ! [M: num,Q2: num,N: num] :
% 8.58/4.80 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 8.58/4.80 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 8.58/4.80 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 8.58/4.80 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3641_cong__exp__iff__simps_I9_J,axiom,
% 8.58/4.80 ! [M: num,Q2: num,N: num] :
% 8.58/4.80 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 8.58/4.80 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3642_mod__eqE,axiom,
% 8.58/4.80 ! [A: int,C: int,B: int] :
% 8.58/4.80 ( ( ( modulo_modulo_int @ A @ C )
% 8.58/4.80 = ( modulo_modulo_int @ B @ C ) )
% 8.58/4.80 => ~ ! [D2: int] :
% 8.58/4.80 ( B
% 8.58/4.80 != ( plus_plus_int @ A @ ( times_times_int @ C @ D2 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3643_mod__eqE,axiom,
% 8.58/4.80 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.58/4.80 ( ( ( modulo364778990260209775nteger @ A @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ B @ C ) )
% 8.58/4.80 => ~ ! [D2: code_integer] :
% 8.58/4.80 ( B
% 8.58/4.80 != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D2 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3644_cong__exp__iff__simps_I4_J,axiom,
% 8.58/4.80 ! [M: num,N: num] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
% 8.58/4.80 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3645_cong__exp__iff__simps_I4_J,axiom,
% 8.58/4.80 ! [M: num,N: num] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
% 8.58/4.80 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3646_cong__exp__iff__simps_I4_J,axiom,
% 8.58/4.80 ! [M: num,N: num] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
% 8.58/4.80 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3647_mod__0__imp__dvd,axiom,
% 8.58/4.80 ! [A: nat,B: nat] :
% 8.58/4.80 ( ( ( modulo_modulo_nat @ A @ B )
% 8.58/4.80 = zero_zero_nat )
% 8.58/4.80 => ( dvd_dvd_nat @ B @ A ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3648_mod__0__imp__dvd,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( ( ( modulo_modulo_int @ A @ B )
% 8.58/4.80 = zero_zero_int )
% 8.58/4.80 => ( dvd_dvd_int @ B @ A ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3649_mod__0__imp__dvd,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( ( modulo364778990260209775nteger @ A @ B )
% 8.58/4.80 = zero_z3403309356797280102nteger )
% 8.58/4.80 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3650_mod__0__imp__dvd,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural] :
% 8.58/4.80 ( ( ( modulo8411746178871703098atural @ A @ B )
% 8.58/4.80 = zero_z2226904508553997617atural )
% 8.58/4.80 => ( dvd_dvd_Code_natural @ B @ A ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3651_dvd__eq__mod__eq__0,axiom,
% 8.58/4.80 ( dvd_dvd_nat
% 8.58/4.80 = ( ^ [A2: nat,B2: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ B2 @ A2 )
% 8.58/4.80 = zero_zero_nat ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3652_dvd__eq__mod__eq__0,axiom,
% 8.58/4.80 ( dvd_dvd_int
% 8.58/4.80 = ( ^ [A2: int,B2: int] :
% 8.58/4.80 ( ( modulo_modulo_int @ B2 @ A2 )
% 8.58/4.80 = zero_zero_int ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3653_dvd__eq__mod__eq__0,axiom,
% 8.58/4.80 ( dvd_dvd_Code_integer
% 8.58/4.80 = ( ^ [A2: code_integer,B2: code_integer] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ B2 @ A2 )
% 8.58/4.80 = zero_z3403309356797280102nteger ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3654_dvd__eq__mod__eq__0,axiom,
% 8.58/4.80 ( dvd_dvd_Code_natural
% 8.58/4.80 = ( ^ [A2: code_natural,B2: code_natural] :
% 8.58/4.80 ( ( modulo8411746178871703098atural @ B2 @ A2 )
% 8.58/4.80 = zero_z2226904508553997617atural ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3655_mod__eq__0__iff__dvd,axiom,
% 8.58/4.80 ! [A: nat,B: nat] :
% 8.58/4.80 ( ( ( modulo_modulo_nat @ A @ B )
% 8.58/4.80 = zero_zero_nat )
% 8.58/4.80 = ( dvd_dvd_nat @ B @ A ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3656_mod__eq__0__iff__dvd,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( ( ( modulo_modulo_int @ A @ B )
% 8.58/4.80 = zero_zero_int )
% 8.58/4.80 = ( dvd_dvd_int @ B @ A ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3657_mod__eq__0__iff__dvd,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( ( modulo364778990260209775nteger @ A @ B )
% 8.58/4.80 = zero_z3403309356797280102nteger )
% 8.58/4.80 = ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3658_mod__eq__0__iff__dvd,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural] :
% 8.58/4.80 ( ( ( modulo8411746178871703098atural @ A @ B )
% 8.58/4.80 = zero_z2226904508553997617atural )
% 8.58/4.80 = ( dvd_dvd_Code_natural @ B @ A ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3659_div__add1__eq,axiom,
% 8.58/4.80 ! [A: nat,B: nat,C: nat] :
% 8.58/4.80 ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.58/4.80 = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3660_div__add1__eq,axiom,
% 8.58/4.80 ! [A: int,B: int,C: int] :
% 8.58/4.80 ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.58/4.80 = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3661_div__add1__eq,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.58/4.80 ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 8.58/4.80 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3662_div__add1__eq,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural,C: code_natural] :
% 8.58/4.80 ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
% 8.58/4.80 = ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ C ) @ ( divide5121882707175180666atural @ B @ C ) ) @ ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C ) @ ( modulo8411746178871703098atural @ B @ C ) ) @ C ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3663_dvd__minus__mod,axiom,
% 8.58/4.80 ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3664_dvd__minus__mod,axiom,
% 8.58/4.80 ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3665_dvd__minus__mod,axiom,
% 8.58/4.80 ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3666_dvd__minus__mod,axiom,
% 8.58/4.80 ! [B: code_natural,A: code_natural] : ( dvd_dvd_Code_natural @ B @ ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3667_mod__eq__dvd__iff,axiom,
% 8.58/4.80 ! [A: int,C: int,B: int] :
% 8.58/4.80 ( ( ( modulo_modulo_int @ A @ C )
% 8.58/4.80 = ( modulo_modulo_int @ B @ C ) )
% 8.58/4.80 = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3668_mod__eq__dvd__iff,axiom,
% 8.58/4.80 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.58/4.80 ( ( ( modulo364778990260209775nteger @ A @ C )
% 8.58/4.80 = ( modulo364778990260209775nteger @ B @ C ) )
% 8.58/4.80 = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3669_mod__Suc,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 8.58/4.80 = N )
% 8.58/4.80 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 8.58/4.80 = zero_zero_nat ) )
% 8.58/4.80 & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 8.58/4.80 != N )
% 8.58/4.80 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 8.58/4.80 = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3670_mod__induct,axiom,
% 8.58/4.80 ! [P: nat > $o,N: nat,P2: nat,M: nat] :
% 8.58/4.80 ( ( P @ N )
% 8.58/4.80 => ( ( ord_less_nat @ N @ P2 )
% 8.58/4.80 => ( ( ord_less_nat @ M @ P2 )
% 8.58/4.80 => ( ! [N2: nat] :
% 8.58/4.80 ( ( ord_less_nat @ N2 @ P2 )
% 8.58/4.80 => ( ( P @ N2 )
% 8.58/4.80 => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P2 ) ) ) )
% 8.58/4.80 => ( P @ M ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3671_gcd__nat__induct,axiom,
% 8.58/4.80 ! [P: nat > nat > $o,M: nat,N: nat] :
% 8.58/4.80 ( ! [M2: nat] : ( P @ M2 @ zero_zero_nat )
% 8.58/4.80 => ( ! [M2: nat,N2: nat] :
% 8.58/4.80 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 8.58/4.80 => ( ( P @ N2 @ ( modulo_modulo_nat @ M2 @ N2 ) )
% 8.58/4.80 => ( P @ M2 @ N2 ) ) )
% 8.58/4.80 => ( P @ M @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3672_mod__less__divisor,axiom,
% 8.58/4.80 ! [N: nat,M: nat] :
% 8.58/4.80 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.80 => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3673_mod__Suc__le__divisor,axiom,
% 8.58/4.80 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3674_mod__eq__0D,axiom,
% 8.58/4.80 ! [M: nat,D: nat] :
% 8.58/4.80 ( ( ( modulo_modulo_nat @ M @ D )
% 8.58/4.80 = zero_zero_nat )
% 8.58/4.80 => ? [Q6: nat] :
% 8.58/4.80 ( M
% 8.58/4.80 = ( times_times_nat @ D @ Q6 ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3675_mod__if,axiom,
% 8.58/4.80 ( modulo_modulo_nat
% 8.58/4.80 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N3 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3676_mod__geq,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ~ ( ord_less_nat @ M @ N )
% 8.58/4.80 => ( ( modulo_modulo_nat @ M @ N )
% 8.58/4.80 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3677_le__mod__geq,axiom,
% 8.58/4.80 ! [N: nat,M: nat] :
% 8.58/4.80 ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.80 => ( ( modulo_modulo_nat @ M @ N )
% 8.58/4.80 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3678_zmod__eq__0__iff,axiom,
% 8.58/4.80 ! [M: int,D: int] :
% 8.58/4.80 ( ( ( modulo_modulo_int @ M @ D )
% 8.58/4.80 = zero_zero_int )
% 8.58/4.80 = ( ? [Q5: int] :
% 8.58/4.80 ( M
% 8.58/4.80 = ( times_times_int @ D @ Q5 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3679_zmod__eq__0D,axiom,
% 8.58/4.80 ! [M: int,D: int] :
% 8.58/4.80 ( ( ( modulo_modulo_int @ M @ D )
% 8.58/4.80 = zero_zero_int )
% 8.58/4.80 => ? [Q6: int] :
% 8.58/4.80 ( M
% 8.58/4.80 = ( times_times_int @ D @ Q6 ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3680_nat__mod__eq__iff,axiom,
% 8.58/4.80 ! [X: nat,N: nat,Y: nat] :
% 8.58/4.80 ( ( ( modulo_modulo_nat @ X @ N )
% 8.58/4.80 = ( modulo_modulo_nat @ Y @ N ) )
% 8.58/4.80 = ( ? [Q1: nat,Q22: nat] :
% 8.58/4.80 ( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
% 8.58/4.80 = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3681_mod__pos__neg__trivial,axiom,
% 8.58/4.80 ! [K: int,L: int] :
% 8.58/4.80 ( ( ord_less_int @ zero_zero_int @ K )
% 8.58/4.80 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 8.58/4.80 => ( ( modulo_modulo_int @ K @ L )
% 8.58/4.80 = ( plus_plus_int @ K @ L ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3682_mod__pos__geq,axiom,
% 8.58/4.80 ! [L: int,K: int] :
% 8.58/4.80 ( ( ord_less_int @ zero_zero_int @ L )
% 8.58/4.80 => ( ( ord_less_eq_int @ L @ K )
% 8.58/4.80 => ( ( modulo_modulo_int @ K @ L )
% 8.58/4.80 = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3683_zdiv__mono__strict,axiom,
% 8.58/4.80 ! [A3: int,B5: int,N: int] :
% 8.58/4.80 ( ( ord_less_int @ A3 @ B5 )
% 8.58/4.80 => ( ( ord_less_int @ zero_zero_int @ N )
% 8.58/4.80 => ( ( ( modulo_modulo_int @ A3 @ N )
% 8.58/4.80 = zero_zero_int )
% 8.58/4.80 => ( ( ( modulo_modulo_int @ B5 @ N )
% 8.58/4.80 = zero_zero_int )
% 8.58/4.80 => ( ord_less_int @ ( divide_divide_int @ A3 @ N ) @ ( divide_divide_int @ B5 @ N ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3684_mod__int__pos__iff,axiom,
% 8.58/4.80 ! [K: int,L: int] :
% 8.58/4.80 ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
% 8.58/4.80 = ( ( dvd_dvd_int @ L @ K )
% 8.58/4.80 | ( ( L = zero_zero_int )
% 8.58/4.80 & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 8.58/4.80 | ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3685_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 8.58/4.80 => ( ( ord_le6747313008572928689nteger @ A @ B )
% 8.58/4.80 => ( ( modulo364778990260209775nteger @ A @ B )
% 8.58/4.80 = A ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3686_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 8.58/4.80 ! [A: nat,B: nat] :
% 8.58/4.80 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.58/4.80 => ( ( ord_less_nat @ A @ B )
% 8.58/4.80 => ( ( modulo_modulo_nat @ A @ B )
% 8.58/4.80 = A ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3687_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.58/4.80 => ( ( ord_less_int @ A @ B )
% 8.58/4.80 => ( ( modulo_modulo_int @ A @ B )
% 8.58/4.80 = A ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3688_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 8.58/4.80 ! [B: code_integer,A: code_integer] :
% 8.58/4.80 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 8.58/4.80 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3689_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 8.58/4.80 ! [B: nat,A: nat] :
% 8.58/4.80 ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.58/4.80 => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3690_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 8.58/4.80 ! [B: int,A: int] :
% 8.58/4.80 ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.80 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3691_cong__exp__iff__simps_I2_J,axiom,
% 8.58/4.80 ! [N: num,Q2: num] :
% 8.58/4.80 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 8.58/4.80 = zero_zero_nat )
% 8.58/4.80 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 8.58/4.80 = zero_zero_nat ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3692_cong__exp__iff__simps_I2_J,axiom,
% 8.58/4.80 ! [N: num,Q2: num] :
% 8.58/4.80 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 8.58/4.80 = zero_zero_int )
% 8.58/4.80 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 8.58/4.80 = zero_zero_int ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3693_cong__exp__iff__simps_I2_J,axiom,
% 8.58/4.80 ! [N: num,Q2: num] :
% 8.58/4.80 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 8.58/4.80 = zero_z3403309356797280102nteger )
% 8.58/4.80 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 8.58/4.80 = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3694_cong__exp__iff__simps_I1_J,axiom,
% 8.58/4.80 ! [N: num] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 8.58/4.80 = zero_zero_nat ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3695_cong__exp__iff__simps_I1_J,axiom,
% 8.58/4.80 ! [N: num] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 8.58/4.80 = zero_zero_int ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3696_cong__exp__iff__simps_I1_J,axiom,
% 8.58/4.80 ! [N: num] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 8.58/4.80 = zero_z3403309356797280102nteger ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3697_cong__exp__iff__simps_I8_J,axiom,
% 8.58/4.80 ! [M: num,Q2: num] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 8.58/4.80 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3698_cong__exp__iff__simps_I8_J,axiom,
% 8.58/4.80 ! [M: num,Q2: num] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 8.58/4.80 != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3699_cong__exp__iff__simps_I8_J,axiom,
% 8.58/4.80 ! [M: num,Q2: num] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 8.58/4.80 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3700_cong__exp__iff__simps_I6_J,axiom,
% 8.58/4.80 ! [Q2: num,N: num] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 8.58/4.80 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3701_cong__exp__iff__simps_I6_J,axiom,
% 8.58/4.80 ! [Q2: num,N: num] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 8.58/4.80 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3702_cong__exp__iff__simps_I6_J,axiom,
% 8.58/4.80 ! [Q2: num,N: num] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 8.58/4.80 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3703_cancel__div__mod__rules_I2_J,axiom,
% 8.58/4.80 ! [B: nat,A: nat,C: nat] :
% 8.58/4.80 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 8.58/4.80 = ( plus_plus_nat @ A @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3704_cancel__div__mod__rules_I2_J,axiom,
% 8.58/4.80 ! [B: int,A: int,C: int] :
% 8.58/4.80 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 8.58/4.80 = ( plus_plus_int @ A @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3705_cancel__div__mod__rules_I2_J,axiom,
% 8.58/4.80 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.58/4.80 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 8.58/4.80 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3706_cancel__div__mod__rules_I2_J,axiom,
% 8.58/4.80 ! [B: code_natural,A: code_natural,C: code_natural] :
% 8.58/4.80 ( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C )
% 8.58/4.80 = ( plus_p4538020629002901425atural @ A @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3707_cancel__div__mod__rules_I1_J,axiom,
% 8.58/4.80 ! [A: nat,B: nat,C: nat] :
% 8.58/4.80 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 8.58/4.80 = ( plus_plus_nat @ A @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3708_cancel__div__mod__rules_I1_J,axiom,
% 8.58/4.80 ! [A: int,B: int,C: int] :
% 8.58/4.80 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 8.58/4.80 = ( plus_plus_int @ A @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3709_cancel__div__mod__rules_I1_J,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.58/4.80 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 8.58/4.80 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3710_cancel__div__mod__rules_I1_J,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural,C: code_natural] :
% 8.58/4.80 ( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C )
% 8.58/4.80 = ( plus_p4538020629002901425atural @ A @ C ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3711_mod__div__decomp,axiom,
% 8.58/4.80 ! [A: nat,B: nat] :
% 8.58/4.80 ( A
% 8.58/4.80 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3712_mod__div__decomp,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( A
% 8.58/4.80 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3713_mod__div__decomp,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer] :
% 8.58/4.80 ( A
% 8.58/4.80 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3714_mod__div__decomp,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural] :
% 8.58/4.80 ( A
% 8.58/4.80 = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3715_div__mult__mod__eq,axiom,
% 8.58/4.80 ! [A: nat,B: nat] :
% 8.58/4.80 ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
% 8.58/4.80 = A ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3716_div__mult__mod__eq,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
% 8.58/4.80 = A ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3717_div__mult__mod__eq,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 8.58/4.80 = A ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3718_div__mult__mod__eq,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural] :
% 8.58/4.80 ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) )
% 8.58/4.80 = A ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3719_mod__div__mult__eq,axiom,
% 8.58/4.80 ! [A: nat,B: nat] :
% 8.58/4.80 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 8.58/4.80 = A ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3720_mod__div__mult__eq,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 8.58/4.80 = A ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3721_mod__div__mult__eq,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 8.58/4.80 = A ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3722_mod__div__mult__eq,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural] :
% 8.58/4.80 ( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ B ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) )
% 8.58/4.80 = A ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3723_mod__mult__div__eq,axiom,
% 8.58/4.80 ! [A: nat,B: nat] :
% 8.58/4.80 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 8.58/4.80 = A ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3724_mod__mult__div__eq,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 8.58/4.80 = A ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3725_mod__mult__div__eq,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 8.58/4.80 = A ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3726_mod__mult__div__eq,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural] :
% 8.58/4.80 ( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ B ) @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) )
% 8.58/4.80 = A ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3727_mult__div__mod__eq,axiom,
% 8.58/4.80 ! [B: nat,A: nat] :
% 8.58/4.80 ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
% 8.58/4.80 = A ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3728_mult__div__mod__eq,axiom,
% 8.58/4.80 ! [B: int,A: int] :
% 8.58/4.80 ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
% 8.58/4.80 = A ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3729_mult__div__mod__eq,axiom,
% 8.58/4.80 ! [B: code_integer,A: code_integer] :
% 8.58/4.80 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 8.58/4.80 = A ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3730_mult__div__mod__eq,axiom,
% 8.58/4.80 ! [B: code_natural,A: code_natural] :
% 8.58/4.80 ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) )
% 8.58/4.80 = A ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3731_div__mult1__eq,axiom,
% 8.58/4.80 ! [A: nat,B: nat,C: nat] :
% 8.58/4.80 ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
% 8.58/4.80 = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3732_div__mult1__eq,axiom,
% 8.58/4.80 ! [A: int,B: int,C: int] :
% 8.58/4.80 ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
% 8.58/4.80 = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3733_div__mult1__eq,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.58/4.80 ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 8.58/4.80 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3734_div__mult1__eq,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural,C: code_natural] :
% 8.58/4.80 ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C )
% 8.58/4.80 = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C ) ) @ ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3735_unit__imp__mod__eq__0,axiom,
% 8.58/4.80 ! [B: nat,A: nat] :
% 8.58/4.80 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.58/4.80 => ( ( modulo_modulo_nat @ A @ B )
% 8.58/4.80 = zero_zero_nat ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3736_unit__imp__mod__eq__0,axiom,
% 8.58/4.80 ! [B: int,A: int] :
% 8.58/4.80 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.58/4.80 => ( ( modulo_modulo_int @ A @ B )
% 8.58/4.80 = zero_zero_int ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3737_unit__imp__mod__eq__0,axiom,
% 8.58/4.80 ! [B: code_integer,A: code_integer] :
% 8.58/4.80 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.58/4.80 => ( ( modulo364778990260209775nteger @ A @ B )
% 8.58/4.80 = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3738_unit__imp__mod__eq__0,axiom,
% 8.58/4.80 ! [B: code_natural,A: code_natural] :
% 8.58/4.80 ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
% 8.58/4.80 => ( ( modulo8411746178871703098atural @ A @ B )
% 8.58/4.80 = zero_z2226904508553997617atural ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3739_minus__mult__div__eq__mod,axiom,
% 8.58/4.80 ! [A: nat,B: nat] :
% 8.58/4.80 ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 8.58/4.80 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3740_minus__mult__div__eq__mod,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 8.58/4.80 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3741_minus__mult__div__eq__mod,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 8.58/4.80 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3742_minus__mult__div__eq__mod,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural] :
% 8.58/4.80 ( ( minus_7197305767214868737atural @ A @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) )
% 8.58/4.80 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3743_minus__mod__eq__mult__div,axiom,
% 8.58/4.80 ! [A: nat,B: nat] :
% 8.58/4.80 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 8.58/4.80 = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3744_minus__mod__eq__mult__div,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 8.58/4.80 = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3745_minus__mod__eq__mult__div,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 8.58/4.80 = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3746_minus__mod__eq__mult__div,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural] :
% 8.58/4.80 ( ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) )
% 8.58/4.80 = ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3747_minus__mod__eq__div__mult,axiom,
% 8.58/4.80 ! [A: nat,B: nat] :
% 8.58/4.80 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 8.58/4.80 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3748_minus__mod__eq__div__mult,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 8.58/4.80 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3749_minus__mod__eq__div__mult,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 8.58/4.80 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3750_minus__mod__eq__div__mult,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural] :
% 8.58/4.80 ( ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) )
% 8.58/4.80 = ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3751_minus__div__mult__eq__mod,axiom,
% 8.58/4.80 ! [A: nat,B: nat] :
% 8.58/4.80 ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 8.58/4.80 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3752_minus__div__mult__eq__mod,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 8.58/4.80 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3753_minus__div__mult__eq__mod,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 8.58/4.80 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3754_minus__div__mult__eq__mod,axiom,
% 8.58/4.80 ! [A: code_natural,B: code_natural] :
% 8.58/4.80 ( ( minus_7197305767214868737atural @ A @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) )
% 8.58/4.80 = ( modulo8411746178871703098atural @ A @ B ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3755_mod__le__divisor,axiom,
% 8.58/4.80 ! [N: nat,M: nat] :
% 8.58/4.80 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.80 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3756_mod__greater__zero__iff__not__dvd,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 8.58/4.80 = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3757_div__less__mono,axiom,
% 8.58/4.80 ! [A3: nat,B5: nat,N: nat] :
% 8.58/4.80 ( ( ord_less_nat @ A3 @ B5 )
% 8.58/4.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.80 => ( ( ( modulo_modulo_nat @ A3 @ N )
% 8.58/4.80 = zero_zero_nat )
% 8.58/4.80 => ( ( ( modulo_modulo_nat @ B5 @ N )
% 8.58/4.80 = zero_zero_nat )
% 8.58/4.80 => ( ord_less_nat @ ( divide_divide_nat @ A3 @ N ) @ ( divide_divide_nat @ B5 @ N ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3758_nat__mod__eq__lemma,axiom,
% 8.58/4.80 ! [X: nat,N: nat,Y: nat] :
% 8.58/4.80 ( ( ( modulo_modulo_nat @ X @ N )
% 8.58/4.80 = ( modulo_modulo_nat @ Y @ N ) )
% 8.58/4.80 => ( ( ord_less_eq_nat @ Y @ X )
% 8.58/4.80 => ? [Q6: nat] :
% 8.58/4.80 ( X
% 8.58/4.80 = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q6 ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3759_mod__eq__nat2E,axiom,
% 8.58/4.80 ! [M: nat,Q2: nat,N: nat] :
% 8.58/4.80 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 8.58/4.80 = ( modulo_modulo_nat @ N @ Q2 ) )
% 8.58/4.80 => ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.80 => ~ ! [S3: nat] :
% 8.58/4.80 ( N
% 8.58/4.80 != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3760_mod__eq__nat1E,axiom,
% 8.58/4.80 ! [M: nat,Q2: nat,N: nat] :
% 8.58/4.80 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 8.58/4.80 = ( modulo_modulo_nat @ N @ Q2 ) )
% 8.58/4.80 => ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.80 => ~ ! [S3: nat] :
% 8.58/4.80 ( M
% 8.58/4.80 != ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3761_mod__eq__dvd__iff__nat,axiom,
% 8.58/4.80 ! [N: nat,M: nat,Q2: nat] :
% 8.58/4.80 ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.80 => ( ( ( modulo_modulo_nat @ M @ Q2 )
% 8.58/4.80 = ( modulo_modulo_nat @ N @ Q2 ) )
% 8.58/4.80 = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3762_split__zmod,axiom,
% 8.58/4.80 ! [P: int > $o,N: int,K: int] :
% 8.58/4.80 ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 8.58/4.80 = ( ( ( K = zero_zero_int )
% 8.58/4.80 => ( P @ N ) )
% 8.58/4.80 & ( ( ord_less_int @ zero_zero_int @ K )
% 8.58/4.80 => ! [I4: int,J3: int] :
% 8.58/4.80 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 8.58/4.80 & ( ord_less_int @ J3 @ K )
% 8.58/4.80 & ( N
% 8.58/4.80 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 8.58/4.80 => ( P @ J3 ) ) )
% 8.58/4.80 & ( ( ord_less_int @ K @ zero_zero_int )
% 8.58/4.80 => ! [I4: int,J3: int] :
% 8.58/4.80 ( ( ( ord_less_int @ K @ J3 )
% 8.58/4.80 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 8.58/4.80 & ( N
% 8.58/4.80 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 8.58/4.80 => ( P @ J3 ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3763_int__mod__neg__eq,axiom,
% 8.58/4.80 ! [A: int,B: int,Q2: int,R: int] :
% 8.58/4.80 ( ( A
% 8.58/4.80 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 8.58/4.80 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 8.58/4.80 => ( ( ord_less_int @ B @ R )
% 8.58/4.80 => ( ( modulo_modulo_int @ A @ B )
% 8.58/4.80 = R ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3764_int__mod__pos__eq,axiom,
% 8.58/4.80 ! [A: int,B: int,Q2: int,R: int] :
% 8.58/4.80 ( ( A
% 8.58/4.80 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 8.58/4.80 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 8.58/4.80 => ( ( ord_less_int @ R @ B )
% 8.58/4.80 => ( ( modulo_modulo_int @ A @ B )
% 8.58/4.80 = R ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3765_div__mod__decomp,axiom,
% 8.58/4.80 ! [A3: nat,N: nat] :
% 8.58/4.80 ( A3
% 8.58/4.80 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A3 @ N ) @ N ) @ ( modulo_modulo_nat @ A3 @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3766_mod__mult2__eq,axiom,
% 8.58/4.80 ! [M: nat,N: nat,Q2: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 8.58/4.80 = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3767_modulo__nat__def,axiom,
% 8.58/4.80 ( modulo_modulo_nat
% 8.58/4.80 = ( ^ [M3: nat,N3: nat] : ( minus_minus_nat @ M3 @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N3 ) @ N3 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3768_div__mod__decomp__int,axiom,
% 8.58/4.80 ! [A3: int,N: int] :
% 8.58/4.80 ( A3
% 8.58/4.80 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A3 @ N ) @ N ) @ ( modulo_modulo_int @ A3 @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3769_mod__mult2__eq_H,axiom,
% 8.58/4.80 ! [A: code_integer,M: nat,N: nat] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
% 8.58/4.80 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3770_mod__mult2__eq_H,axiom,
% 8.58/4.80 ! [A: code_natural,M: nat,N: nat] :
% 8.58/4.80 ( ( modulo8411746178871703098atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) )
% 8.58/4.80 = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( modulo8411746178871703098atural @ ( divide5121882707175180666atural @ A @ ( semiri3763490453095760265atural @ M ) ) @ ( semiri3763490453095760265atural @ N ) ) ) @ ( modulo8411746178871703098atural @ A @ ( semiri3763490453095760265atural @ M ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3771_mod__mult2__eq_H,axiom,
% 8.58/4.80 ! [A: nat,M: nat,N: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 8.58/4.80 = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3772_mod__mult2__eq_H,axiom,
% 8.58/4.80 ! [A: int,M: nat,N: nat] :
% 8.58/4.80 ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.58/4.80 = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3773_even__even__mod__4__iff,axiom,
% 8.58/4.80 ! [N: nat] :
% 8.58/4.80 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.80 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3774_field__char__0__class_Oof__nat__div,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
% 8.58/4.80 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3775_field__char__0__class_Oof__nat__div,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
% 8.58/4.80 = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3776_field__char__0__class_Oof__nat__div,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
% 8.58/4.80 = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3777_split__mod,axiom,
% 8.58/4.80 ! [P: nat > $o,M: nat,N: nat] :
% 8.58/4.80 ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 8.58/4.80 = ( ( ( N = zero_zero_nat )
% 8.58/4.80 => ( P @ M ) )
% 8.58/4.80 & ( ( N != zero_zero_nat )
% 8.58/4.80 => ! [I4: nat,J3: nat] :
% 8.58/4.80 ( ( ord_less_nat @ J3 @ N )
% 8.58/4.80 => ( ( M
% 8.58/4.80 = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 8.58/4.80 => ( P @ J3 ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3778_mod__nat__eqI,axiom,
% 8.58/4.80 ! [R: nat,N: nat,M: nat] :
% 8.58/4.80 ( ( ord_less_nat @ R @ N )
% 8.58/4.80 => ( ( ord_less_eq_nat @ R @ M )
% 8.58/4.80 => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R ) )
% 8.58/4.80 => ( ( modulo_modulo_nat @ M @ N )
% 8.58/4.80 = R ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3779_verit__le__mono__div__int,axiom,
% 8.58/4.80 ! [A3: int,B5: int,N: int] :
% 8.58/4.80 ( ( ord_less_int @ A3 @ B5 )
% 8.58/4.80 => ( ( ord_less_int @ zero_zero_int @ N )
% 8.58/4.80 => ( ord_less_eq_int
% 8.58/4.80 @ ( plus_plus_int @ ( divide_divide_int @ A3 @ N )
% 8.58/4.80 @ ( if_int
% 8.58/4.80 @ ( ( modulo_modulo_int @ B5 @ N )
% 8.58/4.80 = zero_zero_int )
% 8.58/4.80 @ one_one_int
% 8.58/4.80 @ zero_zero_int ) )
% 8.58/4.80 @ ( divide_divide_int @ B5 @ N ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3780_zmod__zmult2__eq,axiom,
% 8.58/4.80 ! [C: int,A: int,B: int] :
% 8.58/4.80 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.58/4.80 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 8.58/4.80 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3781_split__neg__lemma,axiom,
% 8.58/4.80 ! [K: int,P: int > int > $o,N: int] :
% 8.58/4.80 ( ( ord_less_int @ K @ zero_zero_int )
% 8.58/4.80 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 8.58/4.80 = ( ! [I4: int,J3: int] :
% 8.58/4.80 ( ( ( ord_less_int @ K @ J3 )
% 8.58/4.80 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 8.58/4.80 & ( N
% 8.58/4.80 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 8.58/4.80 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3782_split__pos__lemma,axiom,
% 8.58/4.80 ! [K: int,P: int > int > $o,N: int] :
% 8.58/4.80 ( ( ord_less_int @ zero_zero_int @ K )
% 8.58/4.80 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 8.58/4.80 = ( ! [I4: int,J3: int] :
% 8.58/4.80 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 8.58/4.80 & ( ord_less_int @ J3 @ K )
% 8.58/4.80 & ( N
% 8.58/4.80 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 8.58/4.80 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3783_real__of__nat__div__aux,axiom,
% 8.58/4.80 ! [X: nat,D: nat] :
% 8.58/4.80 ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
% 8.58/4.80 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3784_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 8.58/4.80 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 8.58/4.80 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 8.58/4.80 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3785_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 8.58/4.80 ! [C: nat,A: nat,B: nat] :
% 8.58/4.80 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.58/4.80 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
% 8.58/4.80 = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3786_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 8.58/4.80 ! [C: int,A: int,B: int] :
% 8.58/4.80 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.58/4.80 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 8.58/4.80 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3787_even__iff__mod__2__eq__zero,axiom,
% 8.58/4.80 ! [A: nat] :
% 8.58/4.80 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 = zero_zero_nat ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3788_even__iff__mod__2__eq__zero,axiom,
% 8.58/4.80 ! [A: int] :
% 8.58/4.80 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.80 = zero_zero_int ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3789_even__iff__mod__2__eq__zero,axiom,
% 8.58/4.80 ! [A: code_integer] :
% 8.58/4.80 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.80 = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3790_even__iff__mod__2__eq__zero,axiom,
% 8.58/4.80 ! [A: code_natural] :
% 8.58/4.80 ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 8.58/4.80 = zero_z2226904508553997617atural ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3791_odd__iff__mod__2__eq__one,axiom,
% 8.58/4.80 ! [A: nat] :
% 8.58/4.80 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 8.58/4.80 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_nat ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3792_odd__iff__mod__2__eq__one,axiom,
% 8.58/4.80 ! [A: int] :
% 8.58/4.80 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 8.58/4.80 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_int ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3793_odd__iff__mod__2__eq__one,axiom,
% 8.58/4.80 ! [A: code_integer] :
% 8.58/4.80 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 8.58/4.80 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_Code_integer ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3794_odd__iff__mod__2__eq__one,axiom,
% 8.58/4.80 ! [A: code_natural] :
% 8.58/4.80 ( ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) )
% 8.58/4.80 = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_Code_natural ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3795_of__bool__odd__eq__mod__2,axiom,
% 8.58/4.80 ! [A: code_natural] :
% 8.58/4.80 ( ( zero_n8403883297036319079atural
% 8.58/4.80 @ ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) )
% 8.58/4.80 = ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3796_of__bool__odd__eq__mod__2,axiom,
% 8.58/4.80 ! [A: nat] :
% 8.58/4.80 ( ( zero_n2687167440665602831ol_nat
% 8.58/4.80 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 8.58/4.80 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3797_of__bool__odd__eq__mod__2,axiom,
% 8.58/4.80 ! [A: int] :
% 8.58/4.80 ( ( zero_n2684676970156552555ol_int
% 8.58/4.80 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 8.58/4.80 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3798_of__bool__odd__eq__mod__2,axiom,
% 8.58/4.80 ! [A: code_integer] :
% 8.58/4.80 ( ( zero_n356916108424825756nteger
% 8.58/4.80 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 8.58/4.80 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3799_Suc__times__mod__eq,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 8.58/4.80 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 8.58/4.80 = one_one_nat ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3800_divmod__digit__0_I2_J,axiom,
% 8.58/4.80 ! [B: nat,A: nat] :
% 8.58/4.80 ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.58/4.80 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.58/4.80 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
% 8.58/4.80 = ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3801_divmod__digit__0_I2_J,axiom,
% 8.58/4.80 ! [B: int,A: int] :
% 8.58/4.80 ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.80 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.58/4.80 => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
% 8.58/4.80 = ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3802_divmod__digit__0_I2_J,axiom,
% 8.58/4.80 ! [B: code_integer,A: code_integer] :
% 8.58/4.80 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 8.58/4.80 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.58/4.80 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
% 8.58/4.80 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3803_bits__stable__imp__add__self,axiom,
% 8.58/4.80 ! [A: nat] :
% 8.58/4.80 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 = A )
% 8.58/4.80 => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.80 = zero_zero_nat ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3804_bits__stable__imp__add__self,axiom,
% 8.58/4.80 ! [A: int] :
% 8.58/4.80 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.80 = A )
% 8.58/4.80 => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 8.58/4.80 = zero_zero_int ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3805_bits__stable__imp__add__self,axiom,
% 8.58/4.80 ! [A: code_integer] :
% 8.58/4.80 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.80 = A )
% 8.58/4.80 => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 8.58/4.80 = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3806_bits__stable__imp__add__self,axiom,
% 8.58/4.80 ! [A: code_natural] :
% 8.58/4.80 ( ( ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 8.58/4.80 = A )
% 8.58/4.80 => ( ( plus_p4538020629002901425atural @ A @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
% 8.58/4.80 = zero_z2226904508553997617atural ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3807_mod2__eq__if,axiom,
% 8.58/4.80 ! [A: nat] :
% 8.58/4.80 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 = zero_zero_nat ) )
% 8.58/4.80 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_nat ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3808_mod2__eq__if,axiom,
% 8.58/4.80 ! [A: int] :
% 8.58/4.80 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.80 = zero_zero_int ) )
% 8.58/4.80 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_int ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3809_mod2__eq__if,axiom,
% 8.58/4.80 ! [A: code_integer] :
% 8.58/4.80 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.80 = zero_z3403309356797280102nteger ) )
% 8.58/4.80 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_Code_integer ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3810_mod2__eq__if,axiom,
% 8.58/4.80 ! [A: code_natural] :
% 8.58/4.80 ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 8.58/4.80 = zero_z2226904508553997617atural ) )
% 8.58/4.80 & ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_Code_natural ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3811_parity__cases,axiom,
% 8.58/4.80 ! [A: nat] :
% 8.58/4.80 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 != zero_zero_nat ) )
% 8.58/4.80 => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 != one_one_nat ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3812_parity__cases,axiom,
% 8.58/4.80 ! [A: int] :
% 8.58/4.80 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.80 != zero_zero_int ) )
% 8.58/4.80 => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.80 != one_one_int ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3813_parity__cases,axiom,
% 8.58/4.80 ! [A: code_integer] :
% 8.58/4.80 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.80 != zero_z3403309356797280102nteger ) )
% 8.58/4.80 => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.80 != one_one_Code_integer ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3814_parity__cases,axiom,
% 8.58/4.80 ! [A: code_natural] :
% 8.58/4.80 ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 8.58/4.80 != zero_z2226904508553997617atural ) )
% 8.58/4.80 => ~ ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
% 8.58/4.80 => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
% 8.58/4.80 != one_one_Code_natural ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3815_div__exp__mod__exp__eq,axiom,
% 8.58/4.80 ! [A: nat,N: nat,M: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.80 = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3816_div__exp__mod__exp__eq,axiom,
% 8.58/4.80 ! [A: int,N: nat,M: nat] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.80 = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3817_div__exp__mod__exp__eq,axiom,
% 8.58/4.80 ! [A: code_integer,N: nat,M: nat] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.80 = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3818_div__exp__mod__exp__eq,axiom,
% 8.58/4.80 ! [A: code_natural,N: nat,M: nat] :
% 8.58/4.80 ( ( modulo8411746178871703098atural @ ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.80 = ( divide5121882707175180666atural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3819_verit__le__mono__div,axiom,
% 8.58/4.80 ! [A3: nat,B5: nat,N: nat] :
% 8.58/4.80 ( ( ord_less_nat @ A3 @ B5 )
% 8.58/4.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.80 => ( ord_less_eq_nat
% 8.58/4.80 @ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ N )
% 8.58/4.80 @ ( if_nat
% 8.58/4.80 @ ( ( modulo_modulo_nat @ B5 @ N )
% 8.58/4.80 = zero_zero_nat )
% 8.58/4.80 @ one_one_nat
% 8.58/4.80 @ zero_zero_nat ) )
% 8.58/4.80 @ ( divide_divide_nat @ B5 @ N ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3820_divmod__digit__0_I1_J,axiom,
% 8.58/4.80 ! [B: nat,A: nat] :
% 8.58/4.80 ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.58/4.80 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.58/4.80 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 8.58/4.80 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3821_divmod__digit__0_I1_J,axiom,
% 8.58/4.80 ! [B: int,A: int] :
% 8.58/4.80 ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.80 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.58/4.80 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 8.58/4.80 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3822_divmod__digit__0_I1_J,axiom,
% 8.58/4.80 ! [B: code_integer,A: code_integer] :
% 8.58/4.80 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 8.58/4.80 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.58/4.80 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 8.58/4.80 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3823_mult__exp__mod__exp__eq,axiom,
% 8.58/4.80 ! [M: nat,N: nat,A: nat] :
% 8.58/4.80 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.80 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.80 = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3824_mult__exp__mod__exp__eq,axiom,
% 8.58/4.80 ! [M: nat,N: nat,A: int] :
% 8.58/4.80 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.80 => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.80 = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3825_mult__exp__mod__exp__eq,axiom,
% 8.58/4.80 ! [M: nat,N: nat,A: code_integer] :
% 8.58/4.80 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.80 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.80 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3826_mult__exp__mod__exp__eq,axiom,
% 8.58/4.80 ! [M: nat,N: nat,A: code_natural] :
% 8.58/4.80 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.80 => ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.80 = ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3827_exp__mod__exp,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ( modulo8411746178871703098atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.80 = ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_nat @ M @ N ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3828_exp__mod__exp,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.80 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3829_exp__mod__exp,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.80 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3830_exp__mod__exp,axiom,
% 8.58/4.80 ! [M: nat,N: nat] :
% 8.58/4.80 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.80 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3831_pos__zmod__mult__2,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.58/4.80 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 8.58/4.80 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3832_Bolzano,axiom,
% 8.58/4.80 ! [A: real,B: real,P: real > real > $o] :
% 8.58/4.80 ( ( ord_less_eq_real @ A @ B )
% 8.58/4.80 => ( ! [A4: real,B4: real,C3: real] :
% 8.58/4.80 ( ( P @ A4 @ B4 )
% 8.58/4.80 => ( ( P @ B4 @ C3 )
% 8.58/4.80 => ( ( ord_less_eq_real @ A4 @ B4 )
% 8.58/4.80 => ( ( ord_less_eq_real @ B4 @ C3 )
% 8.58/4.80 => ( P @ A4 @ C3 ) ) ) ) )
% 8.58/4.80 => ( ! [X4: real] :
% 8.58/4.80 ( ( ord_less_eq_real @ A @ X4 )
% 8.58/4.80 => ( ( ord_less_eq_real @ X4 @ B )
% 8.58/4.80 => ? [D5: real] :
% 8.58/4.80 ( ( ord_less_real @ zero_zero_real @ D5 )
% 8.58/4.80 & ! [A4: real,B4: real] :
% 8.58/4.80 ( ( ( ord_less_eq_real @ A4 @ X4 )
% 8.58/4.80 & ( ord_less_eq_real @ X4 @ B4 )
% 8.58/4.80 & ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D5 ) )
% 8.58/4.80 => ( P @ A4 @ B4 ) ) ) ) )
% 8.58/4.80 => ( P @ A @ B ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3833_mod__double__modulus,axiom,
% 8.58/4.80 ! [M: code_integer,X: code_integer] :
% 8.58/4.80 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 8.58/4.80 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 8.58/4.80 => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.80 = ( modulo364778990260209775nteger @ X @ M ) )
% 8.58/4.80 | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.80 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3834_mod__double__modulus,axiom,
% 8.58/4.80 ! [M: nat,X: nat] :
% 8.58/4.80 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.58/4.80 => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 8.58/4.80 => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.80 = ( modulo_modulo_nat @ X @ M ) )
% 8.58/4.80 | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.80 = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3835_mod__double__modulus,axiom,
% 8.58/4.80 ! [M: int,X: int] :
% 8.58/4.80 ( ( ord_less_int @ zero_zero_int @ M )
% 8.58/4.80 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.58/4.80 => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.80 = ( modulo_modulo_int @ X @ M ) )
% 8.58/4.80 | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.80 = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3836_divmod__digit__1_I2_J,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 8.58/4.80 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 8.58/4.80 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 8.58/4.80 => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.58/4.80 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3837_divmod__digit__1_I2_J,axiom,
% 8.58/4.80 ! [A: nat,B: nat] :
% 8.58/4.80 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.58/4.80 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.58/4.80 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 8.58/4.80 => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.58/4.80 = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3838_divmod__digit__1_I2_J,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.58/4.80 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.80 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 8.58/4.80 => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.58/4.80 = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3839_unset__bit__Suc,axiom,
% 8.58/4.80 ! [N: nat,A: code_integer] :
% 8.58/4.80 ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
% 8.58/4.80 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3840_unset__bit__Suc,axiom,
% 8.58/4.80 ! [N: nat,A: code_natural] :
% 8.58/4.80 ( ( bit_se7083795435491715335atural @ ( suc @ N ) @ A )
% 8.58/4.80 = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se7083795435491715335atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3841_unset__bit__Suc,axiom,
% 8.58/4.80 ! [N: nat,A: int] :
% 8.58/4.80 ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
% 8.58/4.80 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3842_unset__bit__Suc,axiom,
% 8.58/4.80 ! [N: nat,A: nat] :
% 8.58/4.80 ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
% 8.58/4.80 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3843_set__bit__Suc,axiom,
% 8.58/4.80 ! [N: nat,A: code_integer] :
% 8.58/4.80 ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
% 8.58/4.80 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3844_set__bit__Suc,axiom,
% 8.58/4.80 ! [N: nat,A: code_natural] :
% 8.58/4.80 ( ( bit_se1617098188084679374atural @ ( suc @ N ) @ A )
% 8.58/4.80 = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se1617098188084679374atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3845_set__bit__Suc,axiom,
% 8.58/4.80 ! [N: nat,A: int] :
% 8.58/4.80 ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
% 8.58/4.80 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3846_set__bit__Suc,axiom,
% 8.58/4.80 ! [N: nat,A: nat] :
% 8.58/4.80 ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
% 8.58/4.80 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3847_flip__bit__Suc,axiom,
% 8.58/4.80 ! [N: nat,A: code_integer] :
% 8.58/4.80 ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
% 8.58/4.80 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3848_flip__bit__Suc,axiom,
% 8.58/4.80 ! [N: nat,A: code_natural] :
% 8.58/4.80 ( ( bit_se168947363167071951atural @ ( suc @ N ) @ A )
% 8.58/4.80 = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se168947363167071951atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3849_flip__bit__Suc,axiom,
% 8.58/4.80 ! [N: nat,A: int] :
% 8.58/4.80 ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
% 8.58/4.80 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3850_flip__bit__Suc,axiom,
% 8.58/4.80 ! [N: nat,A: nat] :
% 8.58/4.80 ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
% 8.58/4.80 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3851_even__mod__4__div__2,axiom,
% 8.58/4.80 ! [N: nat] :
% 8.58/4.80 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.58/4.80 = ( suc @ zero_zero_nat ) )
% 8.58/4.80 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3852_neg__zmod__mult__2,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.58/4.80 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 8.58/4.80 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3853_divmod__digit__1_I1_J,axiom,
% 8.58/4.80 ! [A: code_integer,B: code_integer] :
% 8.58/4.80 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 8.58/4.80 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 8.58/4.80 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 8.58/4.80 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
% 8.58/4.80 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3854_divmod__digit__1_I1_J,axiom,
% 8.58/4.80 ! [A: nat,B: nat] :
% 8.58/4.80 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.58/4.80 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.58/4.80 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 8.58/4.80 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
% 8.58/4.80 = ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3855_divmod__digit__1_I1_J,axiom,
% 8.58/4.80 ! [A: int,B: int] :
% 8.58/4.80 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.58/4.80 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.80 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 8.58/4.80 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
% 8.58/4.80 = ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3856_mult__less__iff1,axiom,
% 8.58/4.80 ! [Z2: real,X: real,Y: real] :
% 8.58/4.80 ( ( ord_less_real @ zero_zero_real @ Z2 )
% 8.58/4.80 => ( ( ord_less_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
% 8.58/4.80 = ( ord_less_real @ X @ Y ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3857_mult__less__iff1,axiom,
% 8.58/4.80 ! [Z2: rat,X: rat,Y: rat] :
% 8.58/4.80 ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 8.58/4.80 => ( ( ord_less_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
% 8.58/4.80 = ( ord_less_rat @ X @ Y ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3858_mult__less__iff1,axiom,
% 8.58/4.80 ! [Z2: int,X: int,Y: int] :
% 8.58/4.80 ( ( ord_less_int @ zero_zero_int @ Z2 )
% 8.58/4.80 => ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
% 8.58/4.80 = ( ord_less_int @ X @ Y ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3859_norm__divide__numeral,axiom,
% 8.58/4.80 ! [A: real,W: num] :
% 8.58/4.80 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
% 8.58/4.80 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3860_norm__divide__numeral,axiom,
% 8.58/4.80 ! [A: complex,W: num] :
% 8.58/4.80 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 8.58/4.80 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3861_norm__mult__numeral2,axiom,
% 8.58/4.80 ! [A: real,W: num] :
% 8.58/4.80 ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
% 8.58/4.80 = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3862_norm__mult__numeral2,axiom,
% 8.58/4.80 ! [A: complex,W: num] :
% 8.58/4.80 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 8.58/4.80 = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3863_norm__mult__numeral1,axiom,
% 8.58/4.80 ! [W: num,A: real] :
% 8.58/4.80 ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 8.58/4.80 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3864_norm__mult__numeral1,axiom,
% 8.58/4.80 ! [W: num,A: complex] :
% 8.58/4.80 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 8.58/4.80 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3865_norm__le__zero__iff,axiom,
% 8.58/4.80 ! [X: real] :
% 8.58/4.80 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
% 8.58/4.80 = ( X = zero_zero_real ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3866_norm__le__zero__iff,axiom,
% 8.58/4.80 ! [X: complex] :
% 8.58/4.80 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
% 8.58/4.80 = ( X = zero_zero_complex ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3867_zero__less__norm__iff,axiom,
% 8.58/4.80 ! [X: real] :
% 8.58/4.80 ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
% 8.58/4.80 = ( X != zero_zero_real ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3868_zero__less__norm__iff,axiom,
% 8.58/4.80 ! [X: complex] :
% 8.58/4.80 ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
% 8.58/4.80 = ( X != zero_zero_complex ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3869_norm__of__nat,axiom,
% 8.58/4.80 ! [N: nat] :
% 8.58/4.80 ( ( real_V7735802525324610683m_real @ ( semiri5074537144036343181t_real @ N ) )
% 8.58/4.80 = ( semiri5074537144036343181t_real @ N ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3870_norm__of__nat,axiom,
% 8.58/4.80 ! [N: nat] :
% 8.58/4.80 ( ( real_V1022390504157884413omplex @ ( semiri8010041392384452111omplex @ N ) )
% 8.58/4.80 = ( semiri5074537144036343181t_real @ N ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3871_norm__numeral,axiom,
% 8.58/4.80 ! [W: num] :
% 8.58/4.80 ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
% 8.58/4.80 = ( numeral_numeral_real @ W ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3872_norm__numeral,axiom,
% 8.58/4.80 ! [W: num] :
% 8.58/4.80 ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.58/4.80 = ( numeral_numeral_real @ W ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3873_norm__one,axiom,
% 8.58/4.80 ( ( real_V7735802525324610683m_real @ one_one_real )
% 8.58/4.80 = one_one_real ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3874_norm__one,axiom,
% 8.58/4.80 ( ( real_V1022390504157884413omplex @ one_one_complex )
% 8.58/4.80 = one_one_real ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3875_idiff__0,axiom,
% 8.58/4.80 ! [N: extended_enat] :
% 8.58/4.80 ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
% 8.58/4.80 = zero_z5237406670263579293d_enat ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3876_idiff__0__right,axiom,
% 8.58/4.80 ! [N: extended_enat] :
% 8.58/4.80 ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
% 8.58/4.80 = N ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3877_norm__zero,axiom,
% 8.58/4.80 ( ( real_V7735802525324610683m_real @ zero_zero_real )
% 8.58/4.80 = zero_zero_real ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3878_norm__zero,axiom,
% 8.58/4.80 ( ( real_V1022390504157884413omplex @ zero_zero_complex )
% 8.58/4.80 = zero_zero_real ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3879_norm__eq__zero,axiom,
% 8.58/4.80 ! [X: real] :
% 8.58/4.80 ( ( ( real_V7735802525324610683m_real @ X )
% 8.58/4.80 = zero_zero_real )
% 8.58/4.80 = ( X = zero_zero_real ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3880_norm__eq__zero,axiom,
% 8.58/4.80 ! [X: complex] :
% 8.58/4.80 ( ( ( real_V1022390504157884413omplex @ X )
% 8.58/4.80 = zero_zero_real )
% 8.58/4.80 = ( X = zero_zero_complex ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3881_zero__one__enat__neq_I1_J,axiom,
% 8.58/4.80 zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3882_imult__is__0,axiom,
% 8.58/4.80 ! [M: extended_enat,N: extended_enat] :
% 8.58/4.80 ( ( ( times_7803423173614009249d_enat @ M @ N )
% 8.58/4.80 = zero_z5237406670263579293d_enat )
% 8.58/4.80 = ( ( M = zero_z5237406670263579293d_enat )
% 8.58/4.80 | ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3883_iadd__is__0,axiom,
% 8.58/4.80 ! [M: extended_enat,N: extended_enat] :
% 8.58/4.80 ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 8.58/4.80 = zero_z5237406670263579293d_enat )
% 8.58/4.80 = ( ( M = zero_z5237406670263579293d_enat )
% 8.58/4.80 & ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3884_norm__minus__commute,axiom,
% 8.58/4.80 ! [A: real,B: real] :
% 8.58/4.80 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
% 8.58/4.80 = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3885_norm__minus__commute,axiom,
% 8.58/4.80 ! [A: complex,B: complex] :
% 8.58/4.80 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
% 8.58/4.80 = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3886_norm__mult,axiom,
% 8.58/4.80 ! [X: real,Y: real] :
% 8.58/4.80 ( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) )
% 8.58/4.80 = ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3887_norm__mult,axiom,
% 8.58/4.80 ! [X: complex,Y: complex] :
% 8.58/4.80 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) )
% 8.58/4.80 = ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3888_norm__divide,axiom,
% 8.58/4.80 ! [A: real,B: real] :
% 8.58/4.80 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 8.58/4.80 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3889_norm__divide,axiom,
% 8.58/4.80 ! [A: complex,B: complex] :
% 8.58/4.80 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.58/4.80 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3890_norm__power,axiom,
% 8.58/4.80 ! [X: real,N: nat] :
% 8.58/4.80 ( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) )
% 8.58/4.80 = ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3891_norm__power,axiom,
% 8.58/4.80 ! [X: complex,N: nat] :
% 8.58/4.80 ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) )
% 8.58/4.80 = ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3892_nonzero__norm__divide,axiom,
% 8.58/4.80 ! [B: real,A: real] :
% 8.58/4.80 ( ( B != zero_zero_real )
% 8.58/4.80 => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 8.58/4.80 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3893_nonzero__norm__divide,axiom,
% 8.58/4.80 ! [B: complex,A: complex] :
% 8.58/4.80 ( ( B != zero_zero_complex )
% 8.58/4.80 => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.58/4.80 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3894_power__eq__imp__eq__norm,axiom,
% 8.58/4.80 ! [W: real,N: nat,Z2: real] :
% 8.58/4.80 ( ( ( power_power_real @ W @ N )
% 8.58/4.80 = ( power_power_real @ Z2 @ N ) )
% 8.58/4.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.80 => ( ( real_V7735802525324610683m_real @ W )
% 8.58/4.80 = ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3895_power__eq__imp__eq__norm,axiom,
% 8.58/4.80 ! [W: complex,N: nat,Z2: complex] :
% 8.58/4.80 ( ( ( power_power_complex @ W @ N )
% 8.58/4.80 = ( power_power_complex @ Z2 @ N ) )
% 8.58/4.80 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.80 => ( ( real_V1022390504157884413omplex @ W )
% 8.58/4.80 = ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3896_norm__mult__less,axiom,
% 8.58/4.80 ! [X: real,R: real,Y: real,S: real] :
% 8.58/4.80 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R )
% 8.58/4.80 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
% 8.58/4.80 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R @ S ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3897_norm__mult__less,axiom,
% 8.58/4.80 ! [X: complex,R: real,Y: complex,S: real] :
% 8.58/4.80 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R )
% 8.58/4.80 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
% 8.58/4.80 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R @ S ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3898_norm__mult__ineq,axiom,
% 8.58/4.80 ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3899_norm__mult__ineq,axiom,
% 8.58/4.80 ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3900_norm__triangle__lt,axiom,
% 8.58/4.80 ! [X: real,Y: real,E: real] :
% 8.58/4.80 ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
% 8.58/4.80 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3901_norm__triangle__lt,axiom,
% 8.58/4.80 ! [X: complex,Y: complex,E: real] :
% 8.58/4.80 ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
% 8.58/4.80 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3902_norm__add__less,axiom,
% 8.58/4.80 ! [X: real,R: real,Y: real,S: real] :
% 8.58/4.80 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R )
% 8.58/4.80 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
% 8.58/4.80 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3903_norm__add__less,axiom,
% 8.58/4.80 ! [X: complex,R: real,Y: complex,S: real] :
% 8.58/4.80 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R )
% 8.58/4.80 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
% 8.58/4.80 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3904_norm__triangle__mono,axiom,
% 8.58/4.80 ! [A: real,R: real,B: real,S: real] :
% 8.58/4.80 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R )
% 8.58/4.80 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
% 8.58/4.80 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3905_norm__triangle__mono,axiom,
% 8.58/4.80 ! [A: complex,R: real,B: complex,S: real] :
% 8.58/4.80 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R )
% 8.58/4.80 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
% 8.58/4.80 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3906_norm__triangle__ineq,axiom,
% 8.58/4.80 ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3907_norm__triangle__ineq,axiom,
% 8.58/4.80 ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3908_norm__triangle__le,axiom,
% 8.58/4.80 ! [X: real,Y: real,E: real] :
% 8.58/4.80 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
% 8.58/4.80 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3909_norm__triangle__le,axiom,
% 8.58/4.80 ! [X: complex,Y: complex,E: real] :
% 8.58/4.80 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
% 8.58/4.80 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3910_norm__add__leD,axiom,
% 8.58/4.80 ! [A: real,B: real,C: real] :
% 8.58/4.80 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
% 8.58/4.80 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3911_norm__add__leD,axiom,
% 8.58/4.80 ! [A: complex,B: complex,C: real] :
% 8.58/4.80 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
% 8.58/4.80 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3912_norm__diff__triangle__less,axiom,
% 8.58/4.80 ! [X: real,Y: real,E1: real,Z2: real,E22: real] :
% 8.58/4.80 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
% 8.58/4.80 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z2 ) ) @ E22 )
% 8.58/4.80 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3913_norm__diff__triangle__less,axiom,
% 8.58/4.80 ! [X: complex,Y: complex,E1: real,Z2: complex,E22: real] :
% 8.58/4.80 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
% 8.58/4.80 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z2 ) ) @ E22 )
% 8.58/4.80 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3914_norm__power__ineq,axiom,
% 8.58/4.80 ! [X: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3915_norm__power__ineq,axiom,
% 8.58/4.80 ! [X: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3916_norm__triangle__le__diff,axiom,
% 8.58/4.80 ! [X: real,Y: real,E: real] :
% 8.58/4.80 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
% 8.58/4.80 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3917_norm__triangle__le__diff,axiom,
% 8.58/4.80 ! [X: complex,Y: complex,E: real] :
% 8.58/4.80 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
% 8.58/4.80 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3918_norm__diff__triangle__le,axiom,
% 8.58/4.80 ! [X: real,Y: real,E1: real,Z2: real,E22: real] :
% 8.58/4.80 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
% 8.58/4.80 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z2 ) ) @ E22 )
% 8.58/4.80 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3919_norm__diff__triangle__le,axiom,
% 8.58/4.80 ! [X: complex,Y: complex,E1: real,Z2: complex,E22: real] :
% 8.58/4.80 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
% 8.58/4.80 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z2 ) ) @ E22 )
% 8.58/4.80 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3920_norm__triangle__ineq4,axiom,
% 8.58/4.80 ! [A: real,B: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3921_norm__triangle__ineq4,axiom,
% 8.58/4.80 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3922_norm__triangle__sub,axiom,
% 8.58/4.80 ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3923_norm__triangle__sub,axiom,
% 8.58/4.80 ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3924_norm__diff__ineq,axiom,
% 8.58/4.80 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3925_norm__diff__ineq,axiom,
% 8.58/4.80 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3926_norm__triangle__ineq2,axiom,
% 8.58/4.80 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3927_norm__triangle__ineq2,axiom,
% 8.58/4.80 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3928_power__eq__1__iff,axiom,
% 8.58/4.80 ! [W: real,N: nat] :
% 8.58/4.80 ( ( ( power_power_real @ W @ N )
% 8.58/4.80 = one_one_real )
% 8.58/4.80 => ( ( ( real_V7735802525324610683m_real @ W )
% 8.58/4.80 = one_one_real )
% 8.58/4.80 | ( N = zero_zero_nat ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3929_power__eq__1__iff,axiom,
% 8.58/4.80 ! [W: complex,N: nat] :
% 8.58/4.80 ( ( ( power_power_complex @ W @ N )
% 8.58/4.80 = one_one_complex )
% 8.58/4.80 => ( ( ( real_V1022390504157884413omplex @ W )
% 8.58/4.80 = one_one_real )
% 8.58/4.80 | ( N = zero_zero_nat ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3930_norm__diff__triangle__ineq,axiom,
% 8.58/4.80 ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3931_norm__diff__triangle__ineq,axiom,
% 8.58/4.80 ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3932_square__norm__one,axiom,
% 8.58/4.80 ! [X: real] :
% 8.58/4.80 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_real )
% 8.58/4.80 => ( ( real_V7735802525324610683m_real @ X )
% 8.58/4.80 = one_one_real ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3933_square__norm__one,axiom,
% 8.58/4.80 ! [X: complex] :
% 8.58/4.80 ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.80 = one_one_complex )
% 8.58/4.80 => ( ( real_V1022390504157884413omplex @ X )
% 8.58/4.80 = one_one_real ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3934_norm__power__diff,axiom,
% 8.58/4.80 ! [Z2: real,W: real,M: nat] :
% 8.58/4.80 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ one_one_real )
% 8.58/4.80 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
% 8.58/4.80 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z2 @ M ) @ ( power_power_real @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z2 @ W ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3935_norm__power__diff,axiom,
% 8.58/4.80 ! [Z2: complex,W: complex,M: nat] :
% 8.58/4.80 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
% 8.58/4.80 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
% 8.58/4.80 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z2 @ M ) @ ( power_power_complex @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z2 @ W ) ) ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3936_arcosh__1,axiom,
% 8.58/4.80 ( ( arcosh_real @ one_one_real )
% 8.58/4.80 = zero_zero_real ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3937_low__def,axiom,
% 8.58/4.80 ( vEBT_VEBT_low
% 8.58/4.80 = ( ^ [X3: nat,N3: nat] : ( modulo_modulo_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3938_arsinh__0,axiom,
% 8.58/4.80 ( ( arsinh_real @ zero_zero_real )
% 8.58/4.80 = zero_zero_real ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3939_artanh__0,axiom,
% 8.58/4.80 ( ( artanh_real @ zero_zero_real )
% 8.58/4.80 = zero_zero_real ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3940_pochhammer__double,axiom,
% 8.58/4.80 ! [Z2: complex,N: nat] :
% 8.58/4.80 ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.80 = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s2602460028002588243omplex @ Z2 @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3941_pochhammer__double,axiom,
% 8.58/4.80 ! [Z2: real,N: nat] :
% 8.58/4.80 ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.80 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3942_pochhammer__double,axiom,
% 8.58/4.80 ! [Z2: rat,N: nat] :
% 8.58/4.80 ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.80 = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3943_central__binomial__lower__bound,axiom,
% 8.58/4.80 ! [N: nat] :
% 8.58/4.80 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.80 => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3944_pochhammer__0,axiom,
% 8.58/4.80 ! [A: complex] :
% 8.58/4.80 ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 8.58/4.80 = one_one_complex ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3945_pochhammer__0,axiom,
% 8.58/4.80 ! [A: real] :
% 8.58/4.80 ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 8.58/4.80 = one_one_real ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3946_pochhammer__0,axiom,
% 8.58/4.80 ! [A: rat] :
% 8.58/4.80 ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 8.58/4.80 = one_one_rat ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3947_pochhammer__0,axiom,
% 8.58/4.80 ! [A: nat] :
% 8.58/4.80 ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 8.58/4.80 = one_one_nat ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3948_pochhammer__0,axiom,
% 8.58/4.80 ! [A: int] :
% 8.58/4.80 ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 8.58/4.80 = one_one_int ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3949_pochhammer__of__nat,axiom,
% 8.58/4.80 ! [X: nat,N: nat] :
% 8.58/4.80 ( ( comm_s2602460028002588243omplex @ ( semiri8010041392384452111omplex @ X ) @ N )
% 8.58/4.80 = ( semiri8010041392384452111omplex @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3950_pochhammer__of__nat,axiom,
% 8.58/4.80 ! [X: nat,N: nat] :
% 8.58/4.80 ( ( comm_s7457072308508201937r_real @ ( semiri5074537144036343181t_real @ X ) @ N )
% 8.58/4.80 = ( semiri5074537144036343181t_real @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3951_pochhammer__of__nat,axiom,
% 8.58/4.80 ! [X: nat,N: nat] :
% 8.58/4.80 ( ( comm_s4028243227959126397er_rat @ ( semiri681578069525770553at_rat @ X ) @ N )
% 8.58/4.80 = ( semiri681578069525770553at_rat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3952_pochhammer__of__nat,axiom,
% 8.58/4.80 ! [X: nat,N: nat] :
% 8.58/4.80 ( ( comm_s4663373288045622133er_nat @ ( semiri1316708129612266289at_nat @ X ) @ N )
% 8.58/4.80 = ( semiri1316708129612266289at_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3953_pochhammer__of__nat,axiom,
% 8.58/4.80 ! [X: nat,N: nat] :
% 8.58/4.80 ( ( comm_s4660882817536571857er_int @ ( semiri1314217659103216013at_int @ X ) @ N )
% 8.58/4.80 = ( semiri1314217659103216013at_int @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3954_pochhammer__pos,axiom,
% 8.58/4.80 ! [X: real,N: nat] :
% 8.58/4.80 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.80 => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3955_pochhammer__pos,axiom,
% 8.58/4.80 ! [X: rat,N: nat] :
% 8.58/4.80 ( ( ord_less_rat @ zero_zero_rat @ X )
% 8.58/4.80 => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3956_pochhammer__pos,axiom,
% 8.58/4.80 ! [X: nat,N: nat] :
% 8.58/4.80 ( ( ord_less_nat @ zero_zero_nat @ X )
% 8.58/4.80 => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3957_pochhammer__pos,axiom,
% 8.58/4.80 ! [X: int,N: nat] :
% 8.58/4.80 ( ( ord_less_int @ zero_zero_int @ X )
% 8.58/4.80 => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3958_pochhammer__eq__0__mono,axiom,
% 8.58/4.80 ! [A: real,N: nat,M: nat] :
% 8.58/4.80 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 8.58/4.80 = zero_zero_real )
% 8.58/4.80 => ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.80 => ( ( comm_s7457072308508201937r_real @ A @ M )
% 8.58/4.80 = zero_zero_real ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3959_pochhammer__eq__0__mono,axiom,
% 8.58/4.80 ! [A: rat,N: nat,M: nat] :
% 8.58/4.80 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 8.58/4.80 = zero_zero_rat )
% 8.58/4.80 => ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.80 => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 8.58/4.80 = zero_zero_rat ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3960_pochhammer__neq__0__mono,axiom,
% 8.58/4.80 ! [A: real,M: nat,N: nat] :
% 8.58/4.80 ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 8.58/4.80 != zero_zero_real )
% 8.58/4.80 => ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.80 => ( ( comm_s7457072308508201937r_real @ A @ N )
% 8.58/4.80 != zero_zero_real ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3961_pochhammer__neq__0__mono,axiom,
% 8.58/4.80 ! [A: rat,M: nat,N: nat] :
% 8.58/4.80 ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 8.58/4.80 != zero_zero_rat )
% 8.58/4.80 => ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.80 => ( ( comm_s4028243227959126397er_rat @ A @ N )
% 8.58/4.80 != zero_zero_rat ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3962_pochhammer__nonneg,axiom,
% 8.58/4.80 ! [X: real,N: nat] :
% 8.58/4.80 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.80 => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3963_pochhammer__nonneg,axiom,
% 8.58/4.80 ! [X: rat,N: nat] :
% 8.58/4.80 ( ( ord_less_rat @ zero_zero_rat @ X )
% 8.58/4.80 => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3964_pochhammer__nonneg,axiom,
% 8.58/4.80 ! [X: nat,N: nat] :
% 8.58/4.80 ( ( ord_less_nat @ zero_zero_nat @ X )
% 8.58/4.80 => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3965_pochhammer__nonneg,axiom,
% 8.58/4.80 ! [X: int,N: nat] :
% 8.58/4.80 ( ( ord_less_int @ zero_zero_int @ X )
% 8.58/4.80 => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3966_pochhammer__0__left,axiom,
% 8.58/4.80 ! [N: nat] :
% 8.58/4.80 ( ( ( N = zero_zero_nat )
% 8.58/4.80 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 8.58/4.80 = one_one_complex ) )
% 8.58/4.80 & ( ( N != zero_zero_nat )
% 8.58/4.80 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 8.58/4.80 = zero_zero_complex ) ) ) ).
% 8.58/4.80
% 8.58/4.80 thf(fact_3967_pochhammer__0__left,axiom,
% 8.58/4.80 ! [N: nat] :
% 8.58/4.80 ( ( ( N = zero_zero_nat )
% 8.58/4.80 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 8.58/4.80 = one_one_real ) )
% 8.58/4.80 & ( ( N != zero_zero_nat )
% 8.58/4.80 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 8.58/4.80 = zero_zero_real ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3968_pochhammer__0__left,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( ( N = zero_zero_nat )
% 8.58/4.81 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 8.58/4.81 = one_one_rat ) )
% 8.58/4.81 & ( ( N != zero_zero_nat )
% 8.58/4.81 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 8.58/4.81 = zero_zero_rat ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3969_pochhammer__0__left,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( ( N = zero_zero_nat )
% 8.58/4.81 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 8.58/4.81 = one_one_nat ) )
% 8.58/4.81 & ( ( N != zero_zero_nat )
% 8.58/4.81 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 8.58/4.81 = zero_zero_nat ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3970_pochhammer__0__left,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( ( N = zero_zero_nat )
% 8.58/4.81 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 8.58/4.81 = one_one_int ) )
% 8.58/4.81 & ( ( N != zero_zero_nat )
% 8.58/4.81 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 8.58/4.81 = zero_zero_int ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3971_pochhammer__rec,axiom,
% 8.58/4.81 ! [A: complex,N: nat] :
% 8.58/4.81 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3972_pochhammer__rec,axiom,
% 8.58/4.81 ! [A: real,N: nat] :
% 8.58/4.81 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3973_pochhammer__rec,axiom,
% 8.58/4.81 ! [A: rat,N: nat] :
% 8.58/4.81 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3974_pochhammer__rec,axiom,
% 8.58/4.81 ! [A: nat,N: nat] :
% 8.58/4.81 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3975_pochhammer__rec,axiom,
% 8.58/4.81 ! [A: int,N: nat] :
% 8.58/4.81 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3976_pochhammer__rec_H,axiom,
% 8.58/4.81 ! [Z2: complex,N: nat] :
% 8.58/4.81 ( ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_complex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ N ) ) @ ( comm_s2602460028002588243omplex @ Z2 @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3977_pochhammer__rec_H,axiom,
% 8.58/4.81 ! [Z2: real,N: nat] :
% 8.58/4.81 ( ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3978_pochhammer__rec_H,axiom,
% 8.58/4.81 ! [Z2: rat,N: nat] :
% 8.58/4.81 ( ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3979_pochhammer__rec_H,axiom,
% 8.58/4.81 ! [Z2: nat,N: nat] :
% 8.58/4.81 ( ( comm_s4663373288045622133er_nat @ Z2 @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z2 @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3980_pochhammer__rec_H,axiom,
% 8.58/4.81 ! [Z2: int,N: nat] :
% 8.58/4.81 ( ( comm_s4660882817536571857er_int @ Z2 @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z2 @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3981_pochhammer__Suc,axiom,
% 8.58/4.81 ! [A: complex,N: nat] :
% 8.58/4.81 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3982_pochhammer__Suc,axiom,
% 8.58/4.81 ! [A: real,N: nat] :
% 8.58/4.81 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3983_pochhammer__Suc,axiom,
% 8.58/4.81 ! [A: rat,N: nat] :
% 8.58/4.81 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3984_pochhammer__Suc,axiom,
% 8.58/4.81 ! [A: nat,N: nat] :
% 8.58/4.81 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3985_pochhammer__Suc,axiom,
% 8.58/4.81 ! [A: int,N: nat] :
% 8.58/4.81 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3986_pochhammer__product_H,axiom,
% 8.58/4.81 ! [Z2: complex,N: nat,M: nat] :
% 8.58/4.81 ( ( comm_s2602460028002588243omplex @ Z2 @ ( plus_plus_nat @ N @ M ) )
% 8.58/4.81 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ N ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ N ) ) @ M ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3987_pochhammer__product_H,axiom,
% 8.58/4.81 ! [Z2: real,N: nat,M: nat] :
% 8.58/4.81 ( ( comm_s7457072308508201937r_real @ Z2 @ ( plus_plus_nat @ N @ M ) )
% 8.58/4.81 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3988_pochhammer__product_H,axiom,
% 8.58/4.81 ! [Z2: rat,N: nat,M: nat] :
% 8.58/4.81 ( ( comm_s4028243227959126397er_rat @ Z2 @ ( plus_plus_nat @ N @ M ) )
% 8.58/4.81 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3989_pochhammer__product_H,axiom,
% 8.58/4.81 ! [Z2: nat,N: nat,M: nat] :
% 8.58/4.81 ( ( comm_s4663373288045622133er_nat @ Z2 @ ( plus_plus_nat @ N @ M ) )
% 8.58/4.81 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3990_pochhammer__product_H,axiom,
% 8.58/4.81 ! [Z2: int,N: nat,M: nat] :
% 8.58/4.81 ( ( comm_s4660882817536571857er_int @ Z2 @ ( plus_plus_nat @ N @ M ) )
% 8.58/4.81 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3991_binomial__mono,axiom,
% 8.58/4.81 ! [K: nat,K6: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ K6 )
% 8.58/4.81 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 8.58/4.81 => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3992_binomial__maximum_H,axiom,
% 8.58/4.81 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3993_binomial__maximum,axiom,
% 8.58/4.81 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3994_binomial__antimono,axiom,
% 8.58/4.81 ! [K: nat,K6: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ K6 )
% 8.58/4.81 => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 8.58/4.81 => ( ( ord_less_eq_nat @ K6 @ N )
% 8.58/4.81 => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3995_pochhammer__product,axiom,
% 8.58/4.81 ! [M: nat,N: nat,Z2: complex] :
% 8.58/4.81 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.81 => ( ( comm_s2602460028002588243omplex @ Z2 @ N )
% 8.58/4.81 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3996_pochhammer__product,axiom,
% 8.58/4.81 ! [M: nat,N: nat,Z2: real] :
% 8.58/4.81 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.81 => ( ( comm_s7457072308508201937r_real @ Z2 @ N )
% 8.58/4.81 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3997_pochhammer__product,axiom,
% 8.58/4.81 ! [M: nat,N: nat,Z2: rat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.81 => ( ( comm_s4028243227959126397er_rat @ Z2 @ N )
% 8.58/4.81 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3998_pochhammer__product,axiom,
% 8.58/4.81 ! [M: nat,N: nat,Z2: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.81 => ( ( comm_s4663373288045622133er_nat @ Z2 @ N )
% 8.58/4.81 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_3999_pochhammer__product,axiom,
% 8.58/4.81 ! [M: nat,N: nat,Z2: int] :
% 8.58/4.81 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.81 => ( ( comm_s4660882817536571857er_int @ Z2 @ N )
% 8.58/4.81 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4000_binomial__strict__mono,axiom,
% 8.58/4.81 ! [K: nat,K6: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_nat @ K @ K6 )
% 8.58/4.81 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 8.58/4.81 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4001_binomial__strict__antimono,axiom,
% 8.58/4.81 ! [K: nat,K6: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_nat @ K @ K6 )
% 8.58/4.81 => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 8.58/4.81 => ( ( ord_less_eq_nat @ K6 @ N )
% 8.58/4.81 => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4002_binomial__less__binomial__Suc,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.81 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4003_central__binomial__odd,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.81 => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.81 = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4004_zero__less__binomial__iff,axiom,
% 8.58/4.81 ! [N: nat,K: nat] :
% 8.58/4.81 ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 8.58/4.81 = ( ord_less_eq_nat @ K @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4005_choose__two,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.81 = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4006_binomial__n__0,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( binomial @ N @ zero_zero_nat )
% 8.58/4.81 = one_one_nat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4007_binomial__Suc__Suc,axiom,
% 8.58/4.81 ! [N: nat,K: nat] :
% 8.58/4.81 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 8.58/4.81 = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4008_binomial__eq__0__iff,axiom,
% 8.58/4.81 ! [N: nat,K: nat] :
% 8.58/4.81 ( ( ( binomial @ N @ K )
% 8.58/4.81 = zero_zero_nat )
% 8.58/4.81 = ( ord_less_nat @ N @ K ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4009_binomial__0__Suc,axiom,
% 8.58/4.81 ! [K: nat] :
% 8.58/4.81 ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 8.58/4.81 = zero_zero_nat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4010_binomial__1,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
% 8.58/4.81 = N ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4011_binomial__addition__formula,axiom,
% 8.58/4.81 ! [N: nat,K: nat] :
% 8.58/4.81 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.81 => ( ( binomial @ N @ ( suc @ K ) )
% 8.58/4.81 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4012_binomial__Suc__n,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( binomial @ ( suc @ N ) @ N )
% 8.58/4.81 = ( suc @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4013_binomial__n__n,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( binomial @ N @ N )
% 8.58/4.81 = one_one_nat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4014_choose__one,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( binomial @ N @ one_one_nat )
% 8.58/4.81 = N ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4015_binomial__eq__0,axiom,
% 8.58/4.81 ! [N: nat,K: nat] :
% 8.58/4.81 ( ( ord_less_nat @ N @ K )
% 8.58/4.81 => ( ( binomial @ N @ K )
% 8.58/4.81 = zero_zero_nat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4016_Suc__times__binomial,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
% 8.58/4.81 = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4017_Suc__times__binomial__eq,axiom,
% 8.58/4.81 ! [N: nat,K: nat] :
% 8.58/4.81 ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
% 8.58/4.81 = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4018_binomial__symmetric,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( ( binomial @ N @ K )
% 8.58/4.81 = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4019_choose__mult__lemma,axiom,
% 8.58/4.81 ! [M: nat,R: nat,K: nat] :
% 8.58/4.81 ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
% 8.58/4.81 = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R ) @ M ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4020_binomial__le__pow,axiom,
% 8.58/4.81 ! [R: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ R @ N )
% 8.58/4.81 => ( ord_less_eq_nat @ ( binomial @ N @ R ) @ ( power_power_nat @ N @ R ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4021_zero__less__binomial,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4022_Suc__times__binomial__add,axiom,
% 8.58/4.81 ! [A: nat,B: nat] :
% 8.58/4.81 ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
% 8.58/4.81 = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4023_choose__mult,axiom,
% 8.58/4.81 ! [K: nat,M: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ M )
% 8.58/4.81 => ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.81 => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 8.58/4.81 = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4024_binomial__Suc__Suc__eq__times,axiom,
% 8.58/4.81 ! [N: nat,K: nat] :
% 8.58/4.81 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 8.58/4.81 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4025_binomial__absorb__comp,axiom,
% 8.58/4.81 ! [N: nat,K: nat] :
% 8.58/4.81 ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
% 8.58/4.81 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4026_binomial__absorption,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 8.58/4.81 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4027_binomial__ge__n__over__k__pow__k,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4028_binomial__ge__n__over__k__pow__k,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4029_binomial__le__pow2,axiom,
% 8.58/4.81 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4030_choose__reduce__nat,axiom,
% 8.58/4.81 ! [N: nat,K: nat] :
% 8.58/4.81 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.81 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.58/4.81 => ( ( binomial @ N @ K )
% 8.58/4.81 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4031_times__binomial__minus1__eq,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.58/4.81 => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 8.58/4.81 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4032_artanh__def,axiom,
% 8.58/4.81 ( artanh_real
% 8.58/4.81 = ( ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4033_concat__bit__Suc,axiom,
% 8.58/4.81 ! [N: nat,K: int,L: int] :
% 8.58/4.81 ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
% 8.58/4.81 = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4034_signed__take__bit__Suc,axiom,
% 8.58/4.81 ! [N: nat,A: code_integer] :
% 8.58/4.81 ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
% 8.58/4.81 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4035_signed__take__bit__Suc,axiom,
% 8.58/4.81 ! [N: nat,A: int] :
% 8.58/4.81 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
% 8.58/4.81 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4036_fact__double,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.81 = ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4037_fact__double,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.81 = ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4038_fact__double,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.81 = ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4039_modulo__int__unfold,axiom,
% 8.58/4.81 ! [L: int,K: int,N: nat,M: nat] :
% 8.58/4.81 ( ( ( ( ( sgn_sgn_int @ L )
% 8.58/4.81 = zero_zero_int )
% 8.58/4.81 | ( ( sgn_sgn_int @ K )
% 8.58/4.81 = zero_zero_int )
% 8.58/4.81 | ( N = zero_zero_nat ) )
% 8.58/4.81 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.58/4.81 = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 8.58/4.81 & ( ~ ( ( ( sgn_sgn_int @ L )
% 8.58/4.81 = zero_zero_int )
% 8.58/4.81 | ( ( sgn_sgn_int @ K )
% 8.58/4.81 = zero_zero_int )
% 8.58/4.81 | ( N = zero_zero_nat ) )
% 8.58/4.81 => ( ( ( ( sgn_sgn_int @ K )
% 8.58/4.81 = ( sgn_sgn_int @ L ) )
% 8.58/4.81 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.58/4.81 = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 8.58/4.81 & ( ( ( sgn_sgn_int @ K )
% 8.58/4.81 != ( sgn_sgn_int @ L ) )
% 8.58/4.81 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.58/4.81 = ( times_times_int @ ( sgn_sgn_int @ L )
% 8.58/4.81 @ ( minus_minus_int
% 8.58/4.81 @ ( semiri1314217659103216013at_int
% 8.58/4.81 @ ( times_times_nat @ N
% 8.58/4.81 @ ( zero_n2687167440665602831ol_nat
% 8.58/4.81 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 8.58/4.81 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4040_take__bit__rec,axiom,
% 8.58/4.81 ( bit_se1745604003318907178nteger
% 8.58/4.81 = ( ^ [N3: nat,A2: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4041_take__bit__rec,axiom,
% 8.58/4.81 ( bit_se569199155075624693atural
% 8.58/4.81 = ( ^ [N3: nat,A2: code_natural] : ( if_Code_natural @ ( N3 = zero_zero_nat ) @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4042_take__bit__rec,axiom,
% 8.58/4.81 ( bit_se2923211474154528505it_int
% 8.58/4.81 = ( ^ [N3: nat,A2: int] : ( if_int @ ( N3 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4043_take__bit__rec,axiom,
% 8.58/4.81 ( bit_se2925701944663578781it_nat
% 8.58/4.81 = ( ^ [N3: nat,A2: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4044_log2__of__power__le,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.81 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.58/4.81 => ( ord_less_eq_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4045_sgn__sgn,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( sgn_sgn_int @ ( sgn_sgn_int @ A ) )
% 8.58/4.81 = ( sgn_sgn_int @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4046_sgn__sgn,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
% 8.58/4.81 = ( sgn_sgn_real @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4047_sgn__sgn,axiom,
% 8.58/4.81 ! [A: complex] :
% 8.58/4.81 ( ( sgn_sgn_complex @ ( sgn_sgn_complex @ A ) )
% 8.58/4.81 = ( sgn_sgn_complex @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4048_sgn__sgn,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( sgn_sgn_rat @ ( sgn_sgn_rat @ A ) )
% 8.58/4.81 = ( sgn_sgn_rat @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4049_sgn__sgn,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( sgn_sgn_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 8.58/4.81 = ( sgn_sgn_Code_integer @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4050_take__bit__of__0,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
% 8.58/4.81 = zero_zero_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4051_take__bit__of__0,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
% 8.58/4.81 = zero_zero_nat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4052_sgn__0,axiom,
% 8.58/4.81 ( ( sgn_sgn_complex @ zero_zero_complex )
% 8.58/4.81 = zero_zero_complex ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4053_sgn__0,axiom,
% 8.58/4.81 ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
% 8.58/4.81 = zero_z3403309356797280102nteger ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4054_sgn__0,axiom,
% 8.58/4.81 ( ( sgn_sgn_real @ zero_zero_real )
% 8.58/4.81 = zero_zero_real ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4055_sgn__0,axiom,
% 8.58/4.81 ( ( sgn_sgn_rat @ zero_zero_rat )
% 8.58/4.81 = zero_zero_rat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4056_sgn__0,axiom,
% 8.58/4.81 ( ( sgn_sgn_int @ zero_zero_int )
% 8.58/4.81 = zero_zero_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4057_sgn__zero,axiom,
% 8.58/4.81 ( ( sgn_sgn_complex @ zero_zero_complex )
% 8.58/4.81 = zero_zero_complex ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4058_sgn__zero,axiom,
% 8.58/4.81 ( ( sgn_sgn_real @ zero_zero_real )
% 8.58/4.81 = zero_zero_real ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4059_sgn__1,axiom,
% 8.58/4.81 ( ( sgn_sgn_int @ one_one_int )
% 8.58/4.81 = one_one_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4060_sgn__1,axiom,
% 8.58/4.81 ( ( sgn_sgn_real @ one_one_real )
% 8.58/4.81 = one_one_real ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4061_sgn__1,axiom,
% 8.58/4.81 ( ( sgn_sgn_complex @ one_one_complex )
% 8.58/4.81 = one_one_complex ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4062_sgn__1,axiom,
% 8.58/4.81 ( ( sgn_sgn_rat @ one_one_rat )
% 8.58/4.81 = one_one_rat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4063_sgn__1,axiom,
% 8.58/4.81 ( ( sgn_sgn_Code_integer @ one_one_Code_integer )
% 8.58/4.81 = one_one_Code_integer ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4064_sgn__one,axiom,
% 8.58/4.81 ( ( sgn_sgn_real @ one_one_real )
% 8.58/4.81 = one_one_real ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4065_sgn__one,axiom,
% 8.58/4.81 ( ( sgn_sgn_complex @ one_one_complex )
% 8.58/4.81 = one_one_complex ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4066_sgn__divide,axiom,
% 8.58/4.81 ! [A: complex,B: complex] :
% 8.58/4.81 ( ( sgn_sgn_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.58/4.81 = ( divide1717551699836669952omplex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4067_sgn__divide,axiom,
% 8.58/4.81 ! [A: real,B: real] :
% 8.58/4.81 ( ( sgn_sgn_real @ ( divide_divide_real @ A @ B ) )
% 8.58/4.81 = ( divide_divide_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4068_sgn__divide,axiom,
% 8.58/4.81 ! [A: rat,B: rat] :
% 8.58/4.81 ( ( sgn_sgn_rat @ ( divide_divide_rat @ A @ B ) )
% 8.58/4.81 = ( divide_divide_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4069_power__sgn,axiom,
% 8.58/4.81 ! [A: rat,N: nat] :
% 8.58/4.81 ( ( sgn_sgn_rat @ ( power_power_rat @ A @ N ) )
% 8.58/4.81 = ( power_power_rat @ ( sgn_sgn_rat @ A ) @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4070_power__sgn,axiom,
% 8.58/4.81 ! [A: code_integer,N: nat] :
% 8.58/4.81 ( ( sgn_sgn_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 8.58/4.81 = ( power_8256067586552552935nteger @ ( sgn_sgn_Code_integer @ A ) @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4071_power__sgn,axiom,
% 8.58/4.81 ! [A: real,N: nat] :
% 8.58/4.81 ( ( sgn_sgn_real @ ( power_power_real @ A @ N ) )
% 8.58/4.81 = ( power_power_real @ ( sgn_sgn_real @ A ) @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4072_power__sgn,axiom,
% 8.58/4.81 ! [A: int,N: nat] :
% 8.58/4.81 ( ( sgn_sgn_int @ ( power_power_int @ A @ N ) )
% 8.58/4.81 = ( power_power_int @ ( sgn_sgn_int @ A ) @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4073_of__nat__fact,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri8010041392384452111omplex @ ( semiri1408675320244567234ct_nat @ N ) )
% 8.58/4.81 = ( semiri5044797733671781792omplex @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4074_of__nat__fact,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri681578069525770553at_rat @ ( semiri1408675320244567234ct_nat @ N ) )
% 8.58/4.81 = ( semiri773545260158071498ct_rat @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4075_of__nat__fact,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri1314217659103216013at_int @ ( semiri1408675320244567234ct_nat @ N ) )
% 8.58/4.81 = ( semiri1406184849735516958ct_int @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4076_of__nat__fact,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri1316708129612266289at_nat @ ( semiri1408675320244567234ct_nat @ N ) )
% 8.58/4.81 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4077_of__nat__fact,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri5074537144036343181t_real @ ( semiri1408675320244567234ct_nat @ N ) )
% 8.58/4.81 = ( semiri2265585572941072030t_real @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4078_signed__take__bit__of__0,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 8.58/4.81 = zero_zero_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4079_concat__bit__0,axiom,
% 8.58/4.81 ! [K: int,L: int] :
% 8.58/4.81 ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
% 8.58/4.81 = L ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4080_concat__bit__of__zero__2,axiom,
% 8.58/4.81 ! [N: nat,K: int] :
% 8.58/4.81 ( ( bit_concat_bit @ N @ K @ zero_zero_int )
% 8.58/4.81 = ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4081_sgn__greater,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
% 8.58/4.81 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4082_sgn__greater,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
% 8.58/4.81 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4083_sgn__greater,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A ) )
% 8.58/4.81 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4084_sgn__greater,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
% 8.58/4.81 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4085_sgn__less,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 8.58/4.81 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4086_sgn__less,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
% 8.58/4.81 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4087_sgn__less,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( ord_less_rat @ ( sgn_sgn_rat @ A ) @ zero_zero_rat )
% 8.58/4.81 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4088_sgn__less,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
% 8.58/4.81 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4089_take__bit__0,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
% 8.58/4.81 = zero_zero_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4090_take__bit__0,axiom,
% 8.58/4.81 ! [A: nat] :
% 8.58/4.81 ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
% 8.58/4.81 = zero_zero_nat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4091_ln__less__zero__iff,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 8.58/4.81 = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4092_ln__gt__zero__iff,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 8.58/4.81 = ( ord_less_real @ one_one_real @ X ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4093_ln__eq__zero__iff,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ( ln_ln_real @ X )
% 8.58/4.81 = zero_zero_real )
% 8.58/4.81 = ( X = one_one_real ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4094_divide__sgn,axiom,
% 8.58/4.81 ! [A: real,B: real] :
% 8.58/4.81 ( ( divide_divide_real @ A @ ( sgn_sgn_real @ B ) )
% 8.58/4.81 = ( times_times_real @ A @ ( sgn_sgn_real @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4095_divide__sgn,axiom,
% 8.58/4.81 ! [A: rat,B: rat] :
% 8.58/4.81 ( ( divide_divide_rat @ A @ ( sgn_sgn_rat @ B ) )
% 8.58/4.81 = ( times_times_rat @ A @ ( sgn_sgn_rat @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4096_take__bit__Suc__1,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
% 8.58/4.81 = one_one_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4097_take__bit__Suc__1,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
% 8.58/4.81 = one_one_nat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4098_take__bit__numeral__1,axiom,
% 8.58/4.81 ! [L: num] :
% 8.58/4.81 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
% 8.58/4.81 = one_one_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4099_take__bit__numeral__1,axiom,
% 8.58/4.81 ! [L: num] :
% 8.58/4.81 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
% 8.58/4.81 = one_one_nat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4100_ln__one,axiom,
% 8.58/4.81 ( ( ln_ln_real @ one_one_real )
% 8.58/4.81 = zero_zero_real ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4101_fact__0,axiom,
% 8.58/4.81 ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 8.58/4.81 = one_one_complex ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4102_fact__0,axiom,
% 8.58/4.81 ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 8.58/4.81 = one_one_rat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4103_fact__0,axiom,
% 8.58/4.81 ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 8.58/4.81 = one_one_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4104_fact__0,axiom,
% 8.58/4.81 ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 8.58/4.81 = one_one_nat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4105_fact__0,axiom,
% 8.58/4.81 ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 8.58/4.81 = one_one_real ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4106_fact__1,axiom,
% 8.58/4.81 ( ( semiri5044797733671781792omplex @ one_one_nat )
% 8.58/4.81 = one_one_complex ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4107_fact__1,axiom,
% 8.58/4.81 ( ( semiri773545260158071498ct_rat @ one_one_nat )
% 8.58/4.81 = one_one_rat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4108_fact__1,axiom,
% 8.58/4.81 ( ( semiri1406184849735516958ct_int @ one_one_nat )
% 8.58/4.81 = one_one_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4109_fact__1,axiom,
% 8.58/4.81 ( ( semiri1408675320244567234ct_nat @ one_one_nat )
% 8.58/4.81 = one_one_nat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4110_fact__1,axiom,
% 8.58/4.81 ( ( semiri2265585572941072030t_real @ one_one_nat )
% 8.58/4.81 = one_one_real ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4111_signed__take__bit__Suc__1,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
% 8.58/4.81 = one_one_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4112_signed__take__bit__numeral__of__1,axiom,
% 8.58/4.81 ! [K: num] :
% 8.58/4.81 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
% 8.58/4.81 = one_one_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4113_log__one,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( log2 @ A @ one_one_real )
% 8.58/4.81 = zero_zero_real ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4114_concat__bit__nonnegative__iff,axiom,
% 8.58/4.81 ! [N: nat,K: int,L: int] :
% 8.58/4.81 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
% 8.58/4.81 = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4115_concat__bit__negative__iff,axiom,
% 8.58/4.81 ! [N: nat,K: int,L: int] :
% 8.58/4.81 ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
% 8.58/4.81 = ( ord_less_int @ L @ zero_zero_int ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4116_concat__bit__of__zero__1,axiom,
% 8.58/4.81 ! [N: nat,L: int] :
% 8.58/4.81 ( ( bit_concat_bit @ N @ zero_zero_int @ L )
% 8.58/4.81 = ( bit_se545348938243370406it_int @ N @ L ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4117_sgn__pos,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 8.58/4.81 => ( ( sgn_sgn_Code_integer @ A )
% 8.58/4.81 = one_one_Code_integer ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4118_sgn__pos,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.81 => ( ( sgn_sgn_real @ A )
% 8.58/4.81 = one_one_real ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4119_sgn__pos,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.58/4.81 => ( ( sgn_sgn_rat @ A )
% 8.58/4.81 = one_one_rat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4120_sgn__pos,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( ord_less_int @ zero_zero_int @ A )
% 8.58/4.81 => ( ( sgn_sgn_int @ A )
% 8.58/4.81 = one_one_int ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4121_take__bit__of__1__eq__0__iff,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 8.58/4.81 = zero_zero_int )
% 8.58/4.81 = ( N = zero_zero_nat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4122_take__bit__of__1__eq__0__iff,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 8.58/4.81 = zero_zero_nat )
% 8.58/4.81 = ( N = zero_zero_nat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4123_ln__le__zero__iff,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 8.58/4.81 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4124_ln__ge__zero__iff,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 8.58/4.81 = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4125_fact__Suc__0,axiom,
% 8.58/4.81 ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 8.58/4.81 = one_one_complex ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4126_fact__Suc__0,axiom,
% 8.58/4.81 ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 8.58/4.81 = one_one_rat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4127_fact__Suc__0,axiom,
% 8.58/4.81 ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 8.58/4.81 = one_one_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4128_fact__Suc__0,axiom,
% 8.58/4.81 ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 8.58/4.81 = one_one_nat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4129_fact__Suc__0,axiom,
% 8.58/4.81 ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 8.58/4.81 = one_one_real ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4130_fact__Suc,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri5044797733671781792omplex @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4131_fact__Suc,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4132_fact__Suc,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4133_fact__Suc,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4134_fact__Suc,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
% 8.58/4.81 = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4135_sgn__mult__self__eq,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ A ) )
% 8.58/4.81 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4136_sgn__mult__self__eq,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ A ) )
% 8.58/4.81 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4137_sgn__mult__self__eq,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
% 8.58/4.81 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4138_sgn__mult__self__eq,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
% 8.58/4.81 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4139_zero__less__log__cancel__iff,axiom,
% 8.58/4.81 ! [A: real,X: real] :
% 8.58/4.81 ( ( ord_less_real @ one_one_real @ A )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ ( log2 @ A @ X ) )
% 8.58/4.81 = ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4140_log__less__zero__cancel__iff,axiom,
% 8.58/4.81 ! [A: real,X: real] :
% 8.58/4.81 ( ( ord_less_real @ one_one_real @ A )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_real @ ( log2 @ A @ X ) @ zero_zero_real )
% 8.58/4.81 = ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4141_one__less__log__cancel__iff,axiom,
% 8.58/4.81 ! [A: real,X: real] :
% 8.58/4.81 ( ( ord_less_real @ one_one_real @ A )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_real @ one_one_real @ ( log2 @ A @ X ) )
% 8.58/4.81 = ( ord_less_real @ A @ X ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4142_log__less__one__cancel__iff,axiom,
% 8.58/4.81 ! [A: real,X: real] :
% 8.58/4.81 ( ( ord_less_real @ one_one_real @ A )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_real @ ( log2 @ A @ X ) @ one_one_real )
% 8.58/4.81 = ( ord_less_real @ X @ A ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4143_log__less__cancel__iff,axiom,
% 8.58/4.81 ! [A: real,X: real,Y: real] :
% 8.58/4.81 ( ( ord_less_real @ one_one_real @ A )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ Y )
% 8.58/4.81 => ( ( ord_less_real @ ( log2 @ A @ X ) @ ( log2 @ A @ Y ) )
% 8.58/4.81 = ( ord_less_real @ X @ Y ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4144_log__eq__one,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.81 => ( ( A != one_one_real )
% 8.58/4.81 => ( ( log2 @ A @ A )
% 8.58/4.81 = one_one_real ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4145_take__bit__of__Suc__0,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 8.58/4.81 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4146_sgn__mult__dvd__iff,axiom,
% 8.58/4.81 ! [R: int,L: int,K: int] :
% 8.58/4.81 ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R ) @ L ) @ K )
% 8.58/4.81 = ( ( dvd_dvd_int @ L @ K )
% 8.58/4.81 & ( ( R = zero_zero_int )
% 8.58/4.81 => ( K = zero_zero_int ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4147_mult__sgn__dvd__iff,axiom,
% 8.58/4.81 ! [L: int,R: int,K: int] :
% 8.58/4.81 ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R ) ) @ K )
% 8.58/4.81 = ( ( dvd_dvd_int @ L @ K )
% 8.58/4.81 & ( ( R = zero_zero_int )
% 8.58/4.81 => ( K = zero_zero_int ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4148_dvd__sgn__mult__iff,axiom,
% 8.58/4.81 ! [L: int,R: int,K: int] :
% 8.58/4.81 ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R ) @ K ) )
% 8.58/4.81 = ( ( dvd_dvd_int @ L @ K )
% 8.58/4.81 | ( R = zero_zero_int ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4149_dvd__mult__sgn__iff,axiom,
% 8.58/4.81 ! [L: int,K: int,R: int] :
% 8.58/4.81 ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R ) ) )
% 8.58/4.81 = ( ( dvd_dvd_int @ L @ K )
% 8.58/4.81 | ( R = zero_zero_int ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4150_fact__2,axiom,
% 8.58/4.81 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.81 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4151_fact__2,axiom,
% 8.58/4.81 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.81 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4152_fact__2,axiom,
% 8.58/4.81 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.81 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4153_fact__2,axiom,
% 8.58/4.81 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.81 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4154_fact__2,axiom,
% 8.58/4.81 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.81 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4155_signed__take__bit__Suc__bit0,axiom,
% 8.58/4.81 ! [N: nat,K: num] :
% 8.58/4.81 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 8.58/4.81 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4156_take__bit__of__1,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
% 8.58/4.81 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4157_take__bit__of__1,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 8.58/4.81 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4158_take__bit__of__1,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 8.58/4.81 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4159_zero__le__log__cancel__iff,axiom,
% 8.58/4.81 ! [A: real,X: real] :
% 8.58/4.81 ( ( ord_less_real @ one_one_real @ A )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_eq_real @ zero_zero_real @ ( log2 @ A @ X ) )
% 8.58/4.81 = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4160_log__le__zero__cancel__iff,axiom,
% 8.58/4.81 ! [A: real,X: real] :
% 8.58/4.81 ( ( ord_less_real @ one_one_real @ A )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_eq_real @ ( log2 @ A @ X ) @ zero_zero_real )
% 8.58/4.81 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4161_one__le__log__cancel__iff,axiom,
% 8.58/4.81 ! [A: real,X: real] :
% 8.58/4.81 ( ( ord_less_real @ one_one_real @ A )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_eq_real @ one_one_real @ ( log2 @ A @ X ) )
% 8.58/4.81 = ( ord_less_eq_real @ A @ X ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4162_log__le__one__cancel__iff,axiom,
% 8.58/4.81 ! [A: real,X: real] :
% 8.58/4.81 ( ( ord_less_real @ one_one_real @ A )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_eq_real @ ( log2 @ A @ X ) @ one_one_real )
% 8.58/4.81 = ( ord_less_eq_real @ X @ A ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4163_log__le__cancel__iff,axiom,
% 8.58/4.81 ! [A: real,X: real,Y: real] :
% 8.58/4.81 ( ( ord_less_real @ one_one_real @ A )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ Y )
% 8.58/4.81 => ( ( ord_less_eq_real @ ( log2 @ A @ X ) @ ( log2 @ A @ Y ) )
% 8.58/4.81 = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4164_sgn__of__nat,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( sgn_sgn_real @ ( semiri5074537144036343181t_real @ N ) )
% 8.58/4.81 = ( zero_n3304061248610475627l_real @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4165_sgn__of__nat,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( sgn_sgn_rat @ ( semiri681578069525770553at_rat @ N ) )
% 8.58/4.81 = ( zero_n2052037380579107095ol_rat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4166_sgn__of__nat,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N ) )
% 8.58/4.81 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4167_sgn__of__nat,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( sgn_sgn_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 8.58/4.81 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4168_even__take__bit__eq,axiom,
% 8.58/4.81 ! [N: nat,A: code_integer] :
% 8.58/4.81 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
% 8.58/4.81 = ( ( N = zero_zero_nat )
% 8.58/4.81 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4169_even__take__bit__eq,axiom,
% 8.58/4.81 ! [N: nat,A: int] :
% 8.58/4.81 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 8.58/4.81 = ( ( N = zero_zero_nat )
% 8.58/4.81 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4170_even__take__bit__eq,axiom,
% 8.58/4.81 ! [N: nat,A: nat] :
% 8.58/4.81 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 8.58/4.81 = ( ( N = zero_zero_nat )
% 8.58/4.81 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4171_log__pow__cancel,axiom,
% 8.58/4.81 ! [A: real,B: nat] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.81 => ( ( A != one_one_real )
% 8.58/4.81 => ( ( log2 @ A @ ( power_power_real @ A @ B ) )
% 8.58/4.81 = ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4172_take__bit__Suc__0,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
% 8.58/4.81 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4173_take__bit__Suc__0,axiom,
% 8.58/4.81 ! [A: code_natural] :
% 8.58/4.81 ( ( bit_se569199155075624693atural @ ( suc @ zero_zero_nat ) @ A )
% 8.58/4.81 = ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4174_take__bit__Suc__0,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
% 8.58/4.81 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4175_take__bit__Suc__0,axiom,
% 8.58/4.81 ! [A: nat] :
% 8.58/4.81 ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
% 8.58/4.81 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4176_take__bit__of__exp,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.81 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4177_take__bit__of__exp,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.81 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4178_take__bit__of__exp,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.81 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4179_take__bit__of__2,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.81 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4180_take__bit__of__2,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.81 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4181_take__bit__of__2,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.81 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4182_take__bit__of__nat,axiom,
% 8.58/4.81 ! [N: nat,M: nat] :
% 8.58/4.81 ( ( bit_se2923211474154528505it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 8.58/4.81 = ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4183_take__bit__of__nat,axiom,
% 8.58/4.81 ! [N: nat,M: nat] :
% 8.58/4.81 ( ( bit_se2925701944663578781it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 8.58/4.81 = ( semiri1316708129612266289at_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4184_concat__bit__eq__iff,axiom,
% 8.58/4.81 ! [N: nat,K: int,L: int,R: int,S: int] :
% 8.58/4.81 ( ( ( bit_concat_bit @ N @ K @ L )
% 8.58/4.81 = ( bit_concat_bit @ N @ R @ S ) )
% 8.58/4.81 = ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 8.58/4.81 = ( bit_se2923211474154528505it_int @ N @ R ) )
% 8.58/4.81 & ( L = S ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4185_concat__bit__take__bit__eq,axiom,
% 8.58/4.81 ! [N: nat,B: int] :
% 8.58/4.81 ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B ) )
% 8.58/4.81 = ( bit_concat_bit @ N @ B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4186_signed__take__bit__take__bit,axiom,
% 8.58/4.81 ! [M: nat,N: nat,A: int] :
% 8.58/4.81 ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 8.58/4.81 = ( if_int_int @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4187_signed__take__bit__eq__iff__take__bit__eq,axiom,
% 8.58/4.81 ! [N: nat,A: int,B: int] :
% 8.58/4.81 ( ( ( bit_ri631733984087533419it_int @ N @ A )
% 8.58/4.81 = ( bit_ri631733984087533419it_int @ N @ B ) )
% 8.58/4.81 = ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 8.58/4.81 = ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4188_log__def,axiom,
% 8.58/4.81 ( log2
% 8.58/4.81 = ( ^ [A2: real,X3: real] : ( divide_divide_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ A2 ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4189_take__bit__add,axiom,
% 8.58/4.81 ! [N: nat,A: int,B: int] :
% 8.58/4.81 ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) )
% 8.58/4.81 = ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4190_take__bit__add,axiom,
% 8.58/4.81 ! [N: nat,A: nat,B: nat] :
% 8.58/4.81 ( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) )
% 8.58/4.81 = ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4191_take__bit__tightened,axiom,
% 8.58/4.81 ! [N: nat,A: int,B: int,M: nat] :
% 8.58/4.81 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 8.58/4.81 = ( bit_se2923211474154528505it_int @ N @ B ) )
% 8.58/4.81 => ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.81 => ( ( bit_se2923211474154528505it_int @ M @ A )
% 8.58/4.81 = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4192_take__bit__tightened,axiom,
% 8.58/4.81 ! [N: nat,A: nat,B: nat,M: nat] :
% 8.58/4.81 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 8.58/4.81 = ( bit_se2925701944663578781it_nat @ N @ B ) )
% 8.58/4.81 => ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.81 => ( ( bit_se2925701944663578781it_nat @ M @ A )
% 8.58/4.81 = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4193_take__bit__signed__take__bit,axiom,
% 8.58/4.81 ! [M: nat,N: nat,A: int] :
% 8.58/4.81 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 8.58/4.81 => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A ) )
% 8.58/4.81 = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4194_take__bit__nat__less__eq__self,axiom,
% 8.58/4.81 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4195_take__bit__tightened__less__eq__nat,axiom,
% 8.58/4.81 ! [M: nat,N: nat,Q2: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.81 => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4196_sgn__eq__0__iff,axiom,
% 8.58/4.81 ! [A: complex] :
% 8.58/4.81 ( ( ( sgn_sgn_complex @ A )
% 8.58/4.81 = zero_zero_complex )
% 8.58/4.81 = ( A = zero_zero_complex ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4197_sgn__eq__0__iff,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( ( sgn_sgn_Code_integer @ A )
% 8.58/4.81 = zero_z3403309356797280102nteger )
% 8.58/4.81 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4198_sgn__eq__0__iff,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ( sgn_sgn_real @ A )
% 8.58/4.81 = zero_zero_real )
% 8.58/4.81 = ( A = zero_zero_real ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4199_sgn__eq__0__iff,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( ( sgn_sgn_rat @ A )
% 8.58/4.81 = zero_zero_rat )
% 8.58/4.81 = ( A = zero_zero_rat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4200_sgn__eq__0__iff,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( ( sgn_sgn_int @ A )
% 8.58/4.81 = zero_zero_int )
% 8.58/4.81 = ( A = zero_zero_int ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4201_sgn__0__0,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( ( sgn_sgn_Code_integer @ A )
% 8.58/4.81 = zero_z3403309356797280102nteger )
% 8.58/4.81 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4202_sgn__0__0,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ( sgn_sgn_real @ A )
% 8.58/4.81 = zero_zero_real )
% 8.58/4.81 = ( A = zero_zero_real ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4203_sgn__0__0,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( ( sgn_sgn_rat @ A )
% 8.58/4.81 = zero_zero_rat )
% 8.58/4.81 = ( A = zero_zero_rat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4204_sgn__0__0,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( ( sgn_sgn_int @ A )
% 8.58/4.81 = zero_zero_int )
% 8.58/4.81 = ( A = zero_zero_int ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4205_sgn__zero__iff,axiom,
% 8.58/4.81 ! [X: complex] :
% 8.58/4.81 ( ( ( sgn_sgn_complex @ X )
% 8.58/4.81 = zero_zero_complex )
% 8.58/4.81 = ( X = zero_zero_complex ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4206_sgn__zero__iff,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ( sgn_sgn_real @ X )
% 8.58/4.81 = zero_zero_real )
% 8.58/4.81 = ( X = zero_zero_real ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4207_sgn__mult,axiom,
% 8.58/4.81 ! [A: complex,B: complex] :
% 8.58/4.81 ( ( sgn_sgn_complex @ ( times_times_complex @ A @ B ) )
% 8.58/4.81 = ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4208_sgn__mult,axiom,
% 8.58/4.81 ! [A: code_integer,B: code_integer] :
% 8.58/4.81 ( ( sgn_sgn_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 8.58/4.81 = ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4209_sgn__mult,axiom,
% 8.58/4.81 ! [A: real,B: real] :
% 8.58/4.81 ( ( sgn_sgn_real @ ( times_times_real @ A @ B ) )
% 8.58/4.81 = ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4210_sgn__mult,axiom,
% 8.58/4.81 ! [A: rat,B: rat] :
% 8.58/4.81 ( ( sgn_sgn_rat @ ( times_times_rat @ A @ B ) )
% 8.58/4.81 = ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4211_sgn__mult,axiom,
% 8.58/4.81 ! [A: int,B: int] :
% 8.58/4.81 ( ( sgn_sgn_int @ ( times_times_int @ A @ B ) )
% 8.58/4.81 = ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4212_Real__Vector__Spaces_Osgn__mult,axiom,
% 8.58/4.81 ! [X: complex,Y: complex] :
% 8.58/4.81 ( ( sgn_sgn_complex @ ( times_times_complex @ X @ Y ) )
% 8.58/4.81 = ( times_times_complex @ ( sgn_sgn_complex @ X ) @ ( sgn_sgn_complex @ Y ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4213_Real__Vector__Spaces_Osgn__mult,axiom,
% 8.58/4.81 ! [X: real,Y: real] :
% 8.58/4.81 ( ( sgn_sgn_real @ ( times_times_real @ X @ Y ) )
% 8.58/4.81 = ( times_times_real @ ( sgn_sgn_real @ X ) @ ( sgn_sgn_real @ Y ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4214_same__sgn__sgn__add,axiom,
% 8.58/4.81 ! [B: code_integer,A: code_integer] :
% 8.58/4.81 ( ( ( sgn_sgn_Code_integer @ B )
% 8.58/4.81 = ( sgn_sgn_Code_integer @ A ) )
% 8.58/4.81 => ( ( sgn_sgn_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 8.58/4.81 = ( sgn_sgn_Code_integer @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4215_same__sgn__sgn__add,axiom,
% 8.58/4.81 ! [B: real,A: real] :
% 8.58/4.81 ( ( ( sgn_sgn_real @ B )
% 8.58/4.81 = ( sgn_sgn_real @ A ) )
% 8.58/4.81 => ( ( sgn_sgn_real @ ( plus_plus_real @ A @ B ) )
% 8.58/4.81 = ( sgn_sgn_real @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4216_same__sgn__sgn__add,axiom,
% 8.58/4.81 ! [B: rat,A: rat] :
% 8.58/4.81 ( ( ( sgn_sgn_rat @ B )
% 8.58/4.81 = ( sgn_sgn_rat @ A ) )
% 8.58/4.81 => ( ( sgn_sgn_rat @ ( plus_plus_rat @ A @ B ) )
% 8.58/4.81 = ( sgn_sgn_rat @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4217_same__sgn__sgn__add,axiom,
% 8.58/4.81 ! [B: int,A: int] :
% 8.58/4.81 ( ( ( sgn_sgn_int @ B )
% 8.58/4.81 = ( sgn_sgn_int @ A ) )
% 8.58/4.81 => ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B ) )
% 8.58/4.81 = ( sgn_sgn_int @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4218_fact__nonzero,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri773545260158071498ct_rat @ N )
% 8.58/4.81 != zero_zero_rat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4219_fact__nonzero,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri1406184849735516958ct_int @ N )
% 8.58/4.81 != zero_zero_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4220_fact__nonzero,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri1408675320244567234ct_nat @ N )
% 8.58/4.81 != zero_zero_nat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4221_fact__nonzero,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( semiri2265585572941072030t_real @ N )
% 8.58/4.81 != zero_zero_real ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4222_take__bit__mult,axiom,
% 8.58/4.81 ! [N: nat,K: int,L: int] :
% 8.58/4.81 ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
% 8.58/4.81 = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4223_take__bit__diff,axiom,
% 8.58/4.81 ! [N: nat,K: int,L: int] :
% 8.58/4.81 ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
% 8.58/4.81 = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4224_signed__take__bit__mult,axiom,
% 8.58/4.81 ! [N: nat,K: int,L: int] :
% 8.58/4.81 ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 8.58/4.81 = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4225_signed__take__bit__add,axiom,
% 8.58/4.81 ! [N: nat,K: int,L: int] :
% 8.58/4.81 ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 8.58/4.81 = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4226_signed__take__bit__diff,axiom,
% 8.58/4.81 ! [N: nat,K: int,L: int] :
% 8.58/4.81 ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 8.58/4.81 = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4227_concat__bit__eq,axiom,
% 8.58/4.81 ( bit_concat_bit
% 8.58/4.81 = ( ^ [N3: nat,K2: int,L2: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N3 @ K2 ) @ ( bit_se545348938243370406it_int @ N3 @ L2 ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4228_concat__bit__assoc,axiom,
% 8.58/4.81 ! [N: nat,K: int,M: nat,L: int,R: int] :
% 8.58/4.81 ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R ) )
% 8.58/4.81 = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4229_fact__less__mono__nat,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.58/4.81 => ( ( ord_less_nat @ M @ N )
% 8.58/4.81 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4230_take__bit__tightened__less__eq__int,axiom,
% 8.58/4.81 ! [M: nat,N: nat,K: int] :
% 8.58/4.81 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.81 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4231_take__bit__nonnegative,axiom,
% 8.58/4.81 ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4232_take__bit__int__less__eq__self__iff,axiom,
% 8.58/4.81 ! [N: nat,K: int] :
% 8.58/4.81 ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 8.58/4.81 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4233_not__take__bit__negative,axiom,
% 8.58/4.81 ! [N: nat,K: int] :
% 8.58/4.81 ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4234_take__bit__int__greater__self__iff,axiom,
% 8.58/4.81 ! [K: int,N: nat] :
% 8.58/4.81 ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 8.58/4.81 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4235_push__bit__take__bit,axiom,
% 8.58/4.81 ! [M: nat,N: nat,A: code_integer] :
% 8.58/4.81 ( ( bit_se7788150548672797655nteger @ M @ ( bit_se1745604003318907178nteger @ N @ A ) )
% 8.58/4.81 = ( bit_se1745604003318907178nteger @ ( plus_plus_nat @ M @ N ) @ ( bit_se7788150548672797655nteger @ M @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4236_push__bit__take__bit,axiom,
% 8.58/4.81 ! [M: nat,N: nat,A: code_natural] :
% 8.58/4.81 ( ( bit_se6611745700429515170atural @ M @ ( bit_se569199155075624693atural @ N @ A ) )
% 8.58/4.81 = ( bit_se569199155075624693atural @ ( plus_plus_nat @ M @ N ) @ ( bit_se6611745700429515170atural @ M @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4237_push__bit__take__bit,axiom,
% 8.58/4.81 ! [M: nat,N: nat,A: int] :
% 8.58/4.81 ( ( bit_se545348938243370406it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 8.58/4.81 = ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ M @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4238_push__bit__take__bit,axiom,
% 8.58/4.81 ! [M: nat,N: nat,A: nat] :
% 8.58/4.81 ( ( bit_se547839408752420682it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 8.58/4.81 = ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N ) @ ( bit_se547839408752420682it_nat @ M @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4239_fact__ge__zero,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4240_fact__ge__zero,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4241_fact__ge__zero,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4242_fact__ge__zero,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4243_take__bit__push__bit,axiom,
% 8.58/4.81 ! [M: nat,N: nat,A: code_integer] :
% 8.58/4.81 ( ( bit_se1745604003318907178nteger @ M @ ( bit_se7788150548672797655nteger @ N @ A ) )
% 8.58/4.81 = ( bit_se7788150548672797655nteger @ N @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4244_take__bit__push__bit,axiom,
% 8.58/4.81 ! [M: nat,N: nat,A: code_natural] :
% 8.58/4.81 ( ( bit_se569199155075624693atural @ M @ ( bit_se6611745700429515170atural @ N @ A ) )
% 8.58/4.81 = ( bit_se6611745700429515170atural @ N @ ( bit_se569199155075624693atural @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4245_take__bit__push__bit,axiom,
% 8.58/4.81 ! [M: nat,N: nat,A: int] :
% 8.58/4.81 ( ( bit_se2923211474154528505it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
% 8.58/4.81 = ( bit_se545348938243370406it_int @ N @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4246_take__bit__push__bit,axiom,
% 8.58/4.81 ! [M: nat,N: nat,A: nat] :
% 8.58/4.81 ( ( bit_se2925701944663578781it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
% 8.58/4.81 = ( bit_se547839408752420682it_nat @ N @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4247_fact__gt__zero,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4248_fact__gt__zero,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4249_fact__gt__zero,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4250_fact__gt__zero,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4251_fact__not__neg,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4252_fact__not__neg,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4253_fact__not__neg,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4254_fact__not__neg,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4255_int__sgnE,axiom,
% 8.58/4.81 ! [K: int] :
% 8.58/4.81 ~ ! [N2: nat,L3: int] :
% 8.58/4.81 ( K
% 8.58/4.81 != ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4256_fact__ge__1,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4257_fact__ge__1,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4258_fact__ge__1,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4259_fact__ge__1,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4260_take__bit__unset__bit__eq,axiom,
% 8.58/4.81 ! [N: nat,M: nat,A: int] :
% 8.58/4.81 ( ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.81 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 8.58/4.81 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 8.58/4.81 & ( ~ ( ord_less_eq_nat @ N @ M )
% 8.58/4.81 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 8.58/4.81 = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4261_take__bit__unset__bit__eq,axiom,
% 8.58/4.81 ! [N: nat,M: nat,A: nat] :
% 8.58/4.81 ( ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.81 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 8.58/4.81 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 8.58/4.81 & ( ~ ( ord_less_eq_nat @ N @ M )
% 8.58/4.81 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 8.58/4.81 = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4262_take__bit__set__bit__eq,axiom,
% 8.58/4.81 ! [N: nat,M: nat,A: int] :
% 8.58/4.81 ( ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.81 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 8.58/4.81 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 8.58/4.81 & ( ~ ( ord_less_eq_nat @ N @ M )
% 8.58/4.81 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 8.58/4.81 = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4263_take__bit__set__bit__eq,axiom,
% 8.58/4.81 ! [N: nat,M: nat,A: nat] :
% 8.58/4.81 ( ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.81 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 8.58/4.81 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 8.58/4.81 & ( ~ ( ord_less_eq_nat @ N @ M )
% 8.58/4.81 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 8.58/4.81 = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4264_take__bit__flip__bit__eq,axiom,
% 8.58/4.81 ! [N: nat,M: nat,A: int] :
% 8.58/4.81 ( ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.81 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 8.58/4.81 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 8.58/4.81 & ( ~ ( ord_less_eq_nat @ N @ M )
% 8.58/4.81 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 8.58/4.81 = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4265_take__bit__flip__bit__eq,axiom,
% 8.58/4.81 ! [N: nat,M: nat,A: nat] :
% 8.58/4.81 ( ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.81 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 8.58/4.81 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 8.58/4.81 & ( ~ ( ord_less_eq_nat @ N @ M )
% 8.58/4.81 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 8.58/4.81 = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4266_fact__dvd,axiom,
% 8.58/4.81 ! [N: nat,M: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.81 => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4267_fact__dvd,axiom,
% 8.58/4.81 ! [N: nat,M: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.81 => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4268_fact__dvd,axiom,
% 8.58/4.81 ! [N: nat,M: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.81 => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4269_fact__dvd,axiom,
% 8.58/4.81 ! [N: nat,M: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.81 => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4270_pochhammer__fact,axiom,
% 8.58/4.81 ( semiri5044797733671781792omplex
% 8.58/4.81 = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4271_pochhammer__fact,axiom,
% 8.58/4.81 ( semiri773545260158071498ct_rat
% 8.58/4.81 = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4272_pochhammer__fact,axiom,
% 8.58/4.81 ( semiri1406184849735516958ct_int
% 8.58/4.81 = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4273_pochhammer__fact,axiom,
% 8.58/4.81 ( semiri1408675320244567234ct_nat
% 8.58/4.81 = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4274_pochhammer__fact,axiom,
% 8.58/4.81 ( semiri2265585572941072030t_real
% 8.58/4.81 = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4275_log__eq__div__ln__mult__log,axiom,
% 8.58/4.81 ! [A: real,B: real,X: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.81 => ( ( A != one_one_real )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.58/4.81 => ( ( B != one_one_real )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( log2 @ A @ X )
% 8.58/4.81 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log2 @ B @ X ) ) ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4276_ln__gt__zero__imp__gt__one,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ord_less_real @ one_one_real @ X ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4277_ln__less__zero,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_real @ X @ one_one_real )
% 8.58/4.81 => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4278_ln__gt__zero,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ord_less_real @ one_one_real @ X )
% 8.58/4.81 => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4279_ln__ge__zero,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ord_less_eq_real @ one_one_real @ X )
% 8.58/4.81 => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4280_fact__ge__Suc__0__nat,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4281_sgn__1__pos,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( ( sgn_sgn_Code_integer @ A )
% 8.58/4.81 = one_one_Code_integer )
% 8.58/4.81 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4282_sgn__1__pos,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ( sgn_sgn_real @ A )
% 8.58/4.81 = one_one_real )
% 8.58/4.81 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4283_sgn__1__pos,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( ( sgn_sgn_rat @ A )
% 8.58/4.81 = one_one_rat )
% 8.58/4.81 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4284_sgn__1__pos,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( ( sgn_sgn_int @ A )
% 8.58/4.81 = one_one_int )
% 8.58/4.81 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4285_dvd__fact,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 8.58/4.81 => ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.81 => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4286_take__bit__decr__eq,axiom,
% 8.58/4.81 ! [N: nat,K: int] :
% 8.58/4.81 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 8.58/4.81 != zero_zero_int )
% 8.58/4.81 => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
% 8.58/4.81 = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4287_fact__less__mono,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.58/4.81 => ( ( ord_less_nat @ M @ N )
% 8.58/4.81 => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4288_fact__less__mono,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.58/4.81 => ( ( ord_less_nat @ M @ N )
% 8.58/4.81 => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4289_fact__less__mono,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.58/4.81 => ( ( ord_less_nat @ M @ N )
% 8.58/4.81 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4290_fact__less__mono,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.58/4.81 => ( ( ord_less_nat @ M @ N )
% 8.58/4.81 => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4291_sgn__mod,axiom,
% 8.58/4.81 ! [L: int,K: int] :
% 8.58/4.81 ( ( L != zero_zero_int )
% 8.58/4.81 => ( ~ ( dvd_dvd_int @ L @ K )
% 8.58/4.81 => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
% 8.58/4.81 = ( sgn_sgn_int @ L ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4292_fact__fact__dvd__fact,axiom,
% 8.58/4.81 ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4293_fact__fact__dvd__fact,axiom,
% 8.58/4.81 ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4294_fact__fact__dvd__fact,axiom,
% 8.58/4.81 ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4295_fact__fact__dvd__fact,axiom,
% 8.58/4.81 ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4296_fact__fact__dvd__fact,axiom,
% 8.58/4.81 ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4297_fact__mod,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.81 => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
% 8.58/4.81 = zero_zero_int ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4298_fact__mod,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.81 => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
% 8.58/4.81 = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4299_fact__mod,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.81 => ( ( modulo8411746178871703098atural @ ( semiri2447717529341329178atural @ N ) @ ( semiri2447717529341329178atural @ M ) )
% 8.58/4.81 = zero_z2226904508553997617atural ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4300_fact__mod,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.81 => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 8.58/4.81 = zero_zero_nat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4301_fact__le__power,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4302_fact__le__power,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4303_fact__le__power,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4304_fact__le__power,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4305_signed__take__bit__eq__take__bit__shift,axiom,
% 8.58/4.81 ( bit_ri631733984087533419it_int
% 8.58/4.81 = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4306_ln__ge__zero__imp__ge__one,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4307_ln__add__one__self__le__self,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4308_ln__mult,axiom,
% 8.58/4.81 ! [X: real,Y: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ Y )
% 8.58/4.81 => ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
% 8.58/4.81 = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4309_ln__eq__minus__one,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ( ln_ln_real @ X )
% 8.58/4.81 = ( minus_minus_real @ X @ one_one_real ) )
% 8.58/4.81 => ( X = one_one_real ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4310_ln__div,axiom,
% 8.58/4.81 ! [X: real,Y: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ Y )
% 8.58/4.81 => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
% 8.58/4.81 = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4311_log__base__change,axiom,
% 8.58/4.81 ! [A: real,B: real,X: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.81 => ( ( A != one_one_real )
% 8.58/4.81 => ( ( log2 @ B @ X )
% 8.58/4.81 = ( divide_divide_real @ ( log2 @ A @ X ) @ ( log2 @ A @ B ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4312_less__log__of__power,axiom,
% 8.58/4.81 ! [B: real,N: nat,M: real] :
% 8.58/4.81 ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
% 8.58/4.81 => ( ( ord_less_real @ one_one_real @ B )
% 8.58/4.81 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ B @ M ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4313_log__of__power__eq,axiom,
% 8.58/4.81 ! [M: nat,B: real,N: nat] :
% 8.58/4.81 ( ( ( semiri5074537144036343181t_real @ M )
% 8.58/4.81 = ( power_power_real @ B @ N ) )
% 8.58/4.81 => ( ( ord_less_real @ one_one_real @ B )
% 8.58/4.81 => ( ( semiri5074537144036343181t_real @ N )
% 8.58/4.81 = ( log2 @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4314_fact__div__fact__le__pow,axiom,
% 8.58/4.81 ! [R: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ R @ N )
% 8.58/4.81 => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R ) ) ) @ ( power_power_nat @ N @ R ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4315_binomial__fact__lemma,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 8.58/4.81 = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4316_norm__sgn,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ( X = zero_zero_real )
% 8.58/4.81 => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
% 8.58/4.81 = zero_zero_real ) )
% 8.58/4.81 & ( ( X != zero_zero_real )
% 8.58/4.81 => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
% 8.58/4.81 = one_one_real ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4317_norm__sgn,axiom,
% 8.58/4.81 ! [X: complex] :
% 8.58/4.81 ( ( ( X = zero_zero_complex )
% 8.58/4.81 => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
% 8.58/4.81 = zero_zero_real ) )
% 8.58/4.81 & ( ( X != zero_zero_complex )
% 8.58/4.81 => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
% 8.58/4.81 = one_one_real ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4318_choose__dvd,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4319_choose__dvd,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4320_choose__dvd,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4321_choose__dvd,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4322_choose__dvd,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4323_take__bit__Suc__bit0,axiom,
% 8.58/4.81 ! [N: nat,K: num] :
% 8.58/4.81 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 8.58/4.81 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4324_take__bit__Suc__bit0,axiom,
% 8.58/4.81 ! [N: nat,K: num] :
% 8.58/4.81 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 8.58/4.81 = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4325_take__bit__eq__mod,axiom,
% 8.58/4.81 ( bit_se1745604003318907178nteger
% 8.58/4.81 = ( ^ [N3: nat,A2: code_integer] : ( modulo364778990260209775nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4326_take__bit__eq__mod,axiom,
% 8.58/4.81 ( bit_se569199155075624693atural
% 8.58/4.81 = ( ^ [N3: nat,A2: code_natural] : ( modulo8411746178871703098atural @ A2 @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4327_take__bit__eq__mod,axiom,
% 8.58/4.81 ( bit_se2923211474154528505it_int
% 8.58/4.81 = ( ^ [N3: nat,A2: int] : ( modulo_modulo_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4328_take__bit__eq__mod,axiom,
% 8.58/4.81 ( bit_se2925701944663578781it_nat
% 8.58/4.81 = ( ^ [N3: nat,A2: nat] : ( modulo_modulo_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4329_take__bit__nat__eq__self,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.81 => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 8.58/4.81 = M ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4330_take__bit__nat__less__exp,axiom,
% 8.58/4.81 ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4331_take__bit__nat__eq__self__iff,axiom,
% 8.58/4.81 ! [N: nat,M: nat] :
% 8.58/4.81 ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 8.58/4.81 = M )
% 8.58/4.81 = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4332_ln__le__minus__one,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4333_ln__diff__le,axiom,
% 8.58/4.81 ! [X: real,Y: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ Y )
% 8.58/4.81 => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4334_ln__realpow,axiom,
% 8.58/4.81 ! [X: real,N: nat] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
% 8.58/4.81 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4335_take__bit__nat__def,axiom,
% 8.58/4.81 ( bit_se2925701944663578781it_nat
% 8.58/4.81 = ( ^ [N3: nat,M3: nat] : ( modulo_modulo_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4336_log__mult,axiom,
% 8.58/4.81 ! [A: real,X: real,Y: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.81 => ( ( A != one_one_real )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ Y )
% 8.58/4.81 => ( ( log2 @ A @ ( times_times_real @ X @ Y ) )
% 8.58/4.81 = ( plus_plus_real @ ( log2 @ A @ X ) @ ( log2 @ A @ Y ) ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4337_log__divide,axiom,
% 8.58/4.81 ! [A: real,X: real,Y: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.81 => ( ( A != one_one_real )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_real @ zero_zero_real @ Y )
% 8.58/4.81 => ( ( log2 @ A @ ( divide_divide_real @ X @ Y ) )
% 8.58/4.81 = ( minus_minus_real @ ( log2 @ A @ X ) @ ( log2 @ A @ Y ) ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4338_le__log__of__power,axiom,
% 8.58/4.81 ! [B: real,N: nat,M: real] :
% 8.58/4.81 ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
% 8.58/4.81 => ( ( ord_less_real @ one_one_real @ B )
% 8.58/4.81 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ B @ M ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4339_log__base__pow,axiom,
% 8.58/4.81 ! [A: real,N: nat,X: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.81 => ( ( log2 @ ( power_power_real @ A @ N ) @ X )
% 8.58/4.81 = ( divide_divide_real @ ( log2 @ A @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4340_log__nat__power,axiom,
% 8.58/4.81 ! [X: real,B: real,N: nat] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( log2 @ B @ ( power_power_real @ X @ N ) )
% 8.58/4.81 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ B @ X ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4341_take__bit__int__less__exp,axiom,
% 8.58/4.81 ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4342_binomial__altdef__nat,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( ( binomial @ N @ K )
% 8.58/4.81 = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4343_take__bit__int__def,axiom,
% 8.58/4.81 ( bit_se2923211474154528505it_int
% 8.58/4.81 = ( ^ [N3: nat,K2: int] : ( modulo_modulo_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4344_signed__take__bit__int__less__exp,axiom,
% 8.58/4.81 ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4345_even__signed__take__bit__iff,axiom,
% 8.58/4.81 ! [M: nat,A: code_integer] :
% 8.58/4.81 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
% 8.58/4.81 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4346_even__signed__take__bit__iff,axiom,
% 8.58/4.81 ! [M: nat,A: int] :
% 8.58/4.81 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
% 8.58/4.81 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4347_take__bit__eq__0__iff,axiom,
% 8.58/4.81 ! [N: nat,A: code_integer] :
% 8.58/4.81 ( ( ( bit_se1745604003318907178nteger @ N @ A )
% 8.58/4.81 = zero_z3403309356797280102nteger )
% 8.58/4.81 = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4348_take__bit__eq__0__iff,axiom,
% 8.58/4.81 ! [N: nat,A: int] :
% 8.58/4.81 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 8.58/4.81 = zero_zero_int )
% 8.58/4.81 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4349_take__bit__eq__0__iff,axiom,
% 8.58/4.81 ! [N: nat,A: nat] :
% 8.58/4.81 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 8.58/4.81 = zero_zero_nat )
% 8.58/4.81 = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4350_take__bit__nat__less__self__iff,axiom,
% 8.58/4.81 ! [N: nat,M: nat] :
% 8.58/4.81 ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 8.58/4.81 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4351_square__fact__le__2__fact,axiom,
% 8.58/4.81 ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4352_log__of__power__less,axiom,
% 8.58/4.81 ! [M: nat,B: real,N: nat] :
% 8.58/4.81 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 8.58/4.81 => ( ( ord_less_real @ one_one_real @ B )
% 8.58/4.81 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.58/4.81 => ( ord_less_real @ ( log2 @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4353_log2__of__power__eq,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( M
% 8.58/4.81 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.81 => ( ( semiri5074537144036343181t_real @ N )
% 8.58/4.81 = ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4354_take__bit__int__less__self__iff,axiom,
% 8.58/4.81 ! [N: nat,K: int] :
% 8.58/4.81 ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 8.58/4.81 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4355_take__bit__int__greater__eq__self__iff,axiom,
% 8.58/4.81 ! [K: int,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 8.58/4.81 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4356_fact__num__eq__if,axiom,
% 8.58/4.81 ( semiri5044797733671781792omplex
% 8.58/4.81 = ( ^ [M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M3 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4357_fact__num__eq__if,axiom,
% 8.58/4.81 ( semiri773545260158071498ct_rat
% 8.58/4.81 = ( ^ [M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M3 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4358_fact__num__eq__if,axiom,
% 8.58/4.81 ( semiri1406184849735516958ct_int
% 8.58/4.81 = ( ^ [M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4359_fact__num__eq__if,axiom,
% 8.58/4.81 ( semiri1408675320244567234ct_nat
% 8.58/4.81 = ( ^ [M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4360_fact__num__eq__if,axiom,
% 8.58/4.81 ( semiri2265585572941072030t_real
% 8.58/4.81 = ( ^ [M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4361_fact__reduce,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.81 => ( ( semiri5044797733671781792omplex @ N )
% 8.58/4.81 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4362_fact__reduce,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.81 => ( ( semiri773545260158071498ct_rat @ N )
% 8.58/4.81 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4363_fact__reduce,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.81 => ( ( semiri1406184849735516958ct_int @ N )
% 8.58/4.81 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4364_fact__reduce,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.81 => ( ( semiri1408675320244567234ct_nat @ N )
% 8.58/4.81 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4365_fact__reduce,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.81 => ( ( semiri2265585572941072030t_real @ N )
% 8.58/4.81 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4366_signed__take__bit__int__less__self__iff,axiom,
% 8.58/4.81 ! [N: nat,K: int] :
% 8.58/4.81 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 8.58/4.81 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4367_signed__take__bit__int__greater__eq__self__iff,axiom,
% 8.58/4.81 ! [K: int,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 8.58/4.81 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4368_fact__binomial,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
% 8.58/4.81 = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4369_fact__binomial,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
% 8.58/4.81 = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4370_fact__binomial,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
% 8.58/4.81 = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4371_binomial__fact,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 8.58/4.81 = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4372_binomial__fact,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 8.58/4.81 = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4373_binomial__fact,axiom,
% 8.58/4.81 ! [K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.81 => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 8.58/4.81 = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4374_log__of__power__le,axiom,
% 8.58/4.81 ! [M: nat,B: real,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 8.58/4.81 => ( ( ord_less_real @ one_one_real @ B )
% 8.58/4.81 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.58/4.81 => ( ord_less_eq_real @ ( log2 @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4375_take__bit__int__eq__self,axiom,
% 8.58/4.81 ! [K: int,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.58/4.81 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.81 => ( ( bit_se2923211474154528505it_int @ N @ K )
% 8.58/4.81 = K ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4376_take__bit__int__eq__self__iff,axiom,
% 8.58/4.81 ! [N: nat,K: int] :
% 8.58/4.81 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 8.58/4.81 = K )
% 8.58/4.81 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.58/4.81 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4377_take__bit__incr__eq,axiom,
% 8.58/4.81 ! [N: nat,K: int] :
% 8.58/4.81 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 8.58/4.81 != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 8.58/4.81 => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 8.58/4.81 = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4378_signed__take__bit__int__less__eq,axiom,
% 8.58/4.81 ! [N: nat,K: int] :
% 8.58/4.81 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 8.58/4.81 => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4379_take__bit__Suc,axiom,
% 8.58/4.81 ! [N: nat,A: code_integer] :
% 8.58/4.81 ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
% 8.58/4.81 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4380_take__bit__Suc,axiom,
% 8.58/4.81 ! [N: nat,A: code_natural] :
% 8.58/4.81 ( ( bit_se569199155075624693atural @ ( suc @ N ) @ A )
% 8.58/4.81 = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4381_take__bit__Suc,axiom,
% 8.58/4.81 ! [N: nat,A: int] :
% 8.58/4.81 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 8.58/4.81 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4382_take__bit__Suc,axiom,
% 8.58/4.81 ! [N: nat,A: nat] :
% 8.58/4.81 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
% 8.58/4.81 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4383_less__log2__of__power,axiom,
% 8.58/4.81 ! [N: nat,M: nat] :
% 8.58/4.81 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 8.58/4.81 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4384_le__log2__of__power,axiom,
% 8.58/4.81 ! [N: nat,M: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 8.58/4.81 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4385_take__bit__int__less__eq,axiom,
% 8.58/4.81 ! [N: nat,K: int] :
% 8.58/4.81 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 8.58/4.81 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.81 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4386_take__bit__int__greater__eq,axiom,
% 8.58/4.81 ! [K: int,N: nat] :
% 8.58/4.81 ( ( ord_less_int @ K @ zero_zero_int )
% 8.58/4.81 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4387_stable__imp__take__bit__eq,axiom,
% 8.58/4.81 ! [A: code_integer,N: nat] :
% 8.58/4.81 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.81 = A )
% 8.58/4.81 => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.58/4.81 => ( ( bit_se1745604003318907178nteger @ N @ A )
% 8.58/4.81 = zero_z3403309356797280102nteger ) )
% 8.58/4.81 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.58/4.81 => ( ( bit_se1745604003318907178nteger @ N @ A )
% 8.58/4.81 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4388_stable__imp__take__bit__eq,axiom,
% 8.58/4.81 ! [A: int,N: nat] :
% 8.58/4.81 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.81 = A )
% 8.58/4.81 => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.58/4.81 => ( ( bit_se2923211474154528505it_int @ N @ A )
% 8.58/4.81 = zero_zero_int ) )
% 8.58/4.81 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.58/4.81 => ( ( bit_se2923211474154528505it_int @ N @ A )
% 8.58/4.81 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4389_stable__imp__take__bit__eq,axiom,
% 8.58/4.81 ! [A: nat,N: nat] :
% 8.58/4.81 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.81 = A )
% 8.58/4.81 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.58/4.81 => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 8.58/4.81 = zero_zero_nat ) )
% 8.58/4.81 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.58/4.81 => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 8.58/4.81 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4390_ln__one__plus__pos__lower__bound,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_eq_real @ X @ one_one_real )
% 8.58/4.81 => ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4391_log2__of__power__less,axiom,
% 8.58/4.81 ! [M: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.81 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.58/4.81 => ( ord_less_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4392_ln__2__less__1,axiom,
% 8.58/4.81 ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4393_binomial__code,axiom,
% 8.58/4.81 ( binomial
% 8.58/4.81 = ( ^ [N3: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N3 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N3 @ ( minus_minus_nat @ N3 @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N3 @ K2 ) @ one_one_nat ) @ N3 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4394_fact__diff__Suc,axiom,
% 8.58/4.81 ! [N: nat,M: nat] :
% 8.58/4.81 ( ( ord_less_nat @ N @ ( suc @ M ) )
% 8.58/4.81 => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 8.58/4.81 = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4395_ceiling__log__nat__eq__powr__iff,axiom,
% 8.58/4.81 ! [B: nat,K: nat,N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 8.58/4.81 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.58/4.81 => ( ( ( archim7802044766580827645g_real @ ( log2 @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 8.58/4.81 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 8.58/4.81 = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 8.58/4.81 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4396_ceiling__log__nat__eq__if,axiom,
% 8.58/4.81 ! [B: nat,N: nat,K: nat] :
% 8.58/4.81 ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 8.58/4.81 => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 8.58/4.81 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 8.58/4.81 => ( ( archim7802044766580827645g_real @ ( log2 @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 8.58/4.81 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4397_ceiling__log2__div2,axiom,
% 8.58/4.81 ! [N: nat] :
% 8.58/4.81 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.81 => ( ( archim7802044766580827645g_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 8.58/4.81 = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4398_tanh__ln__real,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( tanh_real @ ( ln_ln_real @ X ) )
% 8.58/4.81 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4399_ln__one__minus__pos__lower__bound,axiom,
% 8.58/4.81 ! [X: real] :
% 8.58/4.81 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.81 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.81 => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4400_verit__minus__simplify_I4_J,axiom,
% 8.58/4.81 ! [B: real] :
% 8.58/4.81 ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
% 8.58/4.81 = B ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4401_verit__minus__simplify_I4_J,axiom,
% 8.58/4.81 ! [B: int] :
% 8.58/4.81 ( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
% 8.58/4.81 = B ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4402_verit__minus__simplify_I4_J,axiom,
% 8.58/4.81 ! [B: complex] :
% 8.58/4.81 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B ) )
% 8.58/4.81 = B ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4403_verit__minus__simplify_I4_J,axiom,
% 8.58/4.81 ! [B: rat] :
% 8.58/4.81 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B ) )
% 8.58/4.81 = B ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4404_verit__minus__simplify_I4_J,axiom,
% 8.58/4.81 ! [B: code_integer] :
% 8.58/4.81 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B ) )
% 8.58/4.81 = B ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4405_neg__equal__iff__equal,axiom,
% 8.58/4.81 ! [A: real,B: real] :
% 8.58/4.81 ( ( ( uminus_uminus_real @ A )
% 8.58/4.81 = ( uminus_uminus_real @ B ) )
% 8.58/4.81 = ( A = B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4406_neg__equal__iff__equal,axiom,
% 8.58/4.81 ! [A: int,B: int] :
% 8.58/4.81 ( ( ( uminus_uminus_int @ A )
% 8.58/4.81 = ( uminus_uminus_int @ B ) )
% 8.58/4.81 = ( A = B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4407_neg__equal__iff__equal,axiom,
% 8.58/4.81 ! [A: complex,B: complex] :
% 8.58/4.81 ( ( ( uminus1482373934393186551omplex @ A )
% 8.58/4.81 = ( uminus1482373934393186551omplex @ B ) )
% 8.58/4.81 = ( A = B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4408_neg__equal__iff__equal,axiom,
% 8.58/4.81 ! [A: rat,B: rat] :
% 8.58/4.81 ( ( ( uminus_uminus_rat @ A )
% 8.58/4.81 = ( uminus_uminus_rat @ B ) )
% 8.58/4.81 = ( A = B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4409_neg__equal__iff__equal,axiom,
% 8.58/4.81 ! [A: code_integer,B: code_integer] :
% 8.58/4.81 ( ( ( uminus1351360451143612070nteger @ A )
% 8.58/4.81 = ( uminus1351360451143612070nteger @ B ) )
% 8.58/4.81 = ( A = B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4410_add_Oinverse__inverse,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 8.58/4.81 = A ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4411_add_Oinverse__inverse,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 8.58/4.81 = A ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4412_add_Oinverse__inverse,axiom,
% 8.58/4.81 ! [A: complex] :
% 8.58/4.81 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
% 8.58/4.81 = A ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4413_add_Oinverse__inverse,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 8.58/4.81 = A ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4414_add_Oinverse__inverse,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 8.58/4.81 = A ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4415_neg__le__iff__le,axiom,
% 8.58/4.81 ! [B: real,A: real] :
% 8.58/4.81 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 8.58/4.81 = ( ord_less_eq_real @ A @ B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4416_neg__le__iff__le,axiom,
% 8.58/4.81 ! [B: code_integer,A: code_integer] :
% 8.58/4.81 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 8.58/4.81 = ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4417_neg__le__iff__le,axiom,
% 8.58/4.81 ! [B: rat,A: rat] :
% 8.58/4.81 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 8.58/4.81 = ( ord_less_eq_rat @ A @ B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4418_neg__le__iff__le,axiom,
% 8.58/4.81 ! [B: int,A: int] :
% 8.58/4.81 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 8.58/4.81 = ( ord_less_eq_int @ A @ B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4419_add_Oinverse__neutral,axiom,
% 8.58/4.81 ( ( uminus_uminus_real @ zero_zero_real )
% 8.58/4.81 = zero_zero_real ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4420_add_Oinverse__neutral,axiom,
% 8.58/4.81 ( ( uminus_uminus_int @ zero_zero_int )
% 8.58/4.81 = zero_zero_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4421_add_Oinverse__neutral,axiom,
% 8.58/4.81 ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 8.58/4.81 = zero_zero_complex ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4422_add_Oinverse__neutral,axiom,
% 8.58/4.81 ( ( uminus_uminus_rat @ zero_zero_rat )
% 8.58/4.81 = zero_zero_rat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4423_add_Oinverse__neutral,axiom,
% 8.58/4.81 ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 8.58/4.81 = zero_z3403309356797280102nteger ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4424_neg__0__equal__iff__equal,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( zero_zero_real
% 8.58/4.81 = ( uminus_uminus_real @ A ) )
% 8.58/4.81 = ( zero_zero_real = A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4425_neg__0__equal__iff__equal,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( zero_zero_int
% 8.58/4.81 = ( uminus_uminus_int @ A ) )
% 8.58/4.81 = ( zero_zero_int = A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4426_neg__0__equal__iff__equal,axiom,
% 8.58/4.81 ! [A: complex] :
% 8.58/4.81 ( ( zero_zero_complex
% 8.58/4.81 = ( uminus1482373934393186551omplex @ A ) )
% 8.58/4.81 = ( zero_zero_complex = A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4427_neg__0__equal__iff__equal,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( zero_zero_rat
% 8.58/4.81 = ( uminus_uminus_rat @ A ) )
% 8.58/4.81 = ( zero_zero_rat = A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4428_neg__0__equal__iff__equal,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( zero_z3403309356797280102nteger
% 8.58/4.81 = ( uminus1351360451143612070nteger @ A ) )
% 8.58/4.81 = ( zero_z3403309356797280102nteger = A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4429_neg__equal__0__iff__equal,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ( uminus_uminus_real @ A )
% 8.58/4.81 = zero_zero_real )
% 8.58/4.81 = ( A = zero_zero_real ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4430_neg__equal__0__iff__equal,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( ( uminus_uminus_int @ A )
% 8.58/4.81 = zero_zero_int )
% 8.58/4.81 = ( A = zero_zero_int ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4431_neg__equal__0__iff__equal,axiom,
% 8.58/4.81 ! [A: complex] :
% 8.58/4.81 ( ( ( uminus1482373934393186551omplex @ A )
% 8.58/4.81 = zero_zero_complex )
% 8.58/4.81 = ( A = zero_zero_complex ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4432_neg__equal__0__iff__equal,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( ( uminus_uminus_rat @ A )
% 8.58/4.81 = zero_zero_rat )
% 8.58/4.81 = ( A = zero_zero_rat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4433_neg__equal__0__iff__equal,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( ( uminus1351360451143612070nteger @ A )
% 8.58/4.81 = zero_z3403309356797280102nteger )
% 8.58/4.81 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4434_equal__neg__zero,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( A
% 8.58/4.81 = ( uminus_uminus_real @ A ) )
% 8.58/4.81 = ( A = zero_zero_real ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4435_equal__neg__zero,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( A
% 8.58/4.81 = ( uminus_uminus_int @ A ) )
% 8.58/4.81 = ( A = zero_zero_int ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4436_equal__neg__zero,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( A
% 8.58/4.81 = ( uminus_uminus_rat @ A ) )
% 8.58/4.81 = ( A = zero_zero_rat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4437_equal__neg__zero,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( A
% 8.58/4.81 = ( uminus1351360451143612070nteger @ A ) )
% 8.58/4.81 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4438_neg__equal__zero,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ( uminus_uminus_real @ A )
% 8.58/4.81 = A )
% 8.58/4.81 = ( A = zero_zero_real ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4439_neg__equal__zero,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( ( uminus_uminus_int @ A )
% 8.58/4.81 = A )
% 8.58/4.81 = ( A = zero_zero_int ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4440_neg__equal__zero,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( ( uminus_uminus_rat @ A )
% 8.58/4.81 = A )
% 8.58/4.81 = ( A = zero_zero_rat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4441_neg__equal__zero,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( ( uminus1351360451143612070nteger @ A )
% 8.58/4.81 = A )
% 8.58/4.81 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4442_neg__less__iff__less,axiom,
% 8.58/4.81 ! [B: real,A: real] :
% 8.58/4.81 ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 8.58/4.81 = ( ord_less_real @ A @ B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4443_neg__less__iff__less,axiom,
% 8.58/4.81 ! [B: int,A: int] :
% 8.58/4.81 ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 8.58/4.81 = ( ord_less_int @ A @ B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4444_neg__less__iff__less,axiom,
% 8.58/4.81 ! [B: rat,A: rat] :
% 8.58/4.81 ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 8.58/4.81 = ( ord_less_rat @ A @ B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4445_neg__less__iff__less,axiom,
% 8.58/4.81 ! [B: code_integer,A: code_integer] :
% 8.58/4.81 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 8.58/4.81 = ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4446_neg__numeral__eq__iff,axiom,
% 8.58/4.81 ! [M: num,N: num] :
% 8.58/4.81 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 8.58/4.81 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.58/4.81 = ( M = N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4447_neg__numeral__eq__iff,axiom,
% 8.58/4.81 ! [M: num,N: num] :
% 8.58/4.81 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 8.58/4.81 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.81 = ( M = N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4448_neg__numeral__eq__iff,axiom,
% 8.58/4.81 ! [M: num,N: num] :
% 8.58/4.81 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 8.58/4.81 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.58/4.81 = ( M = N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4449_neg__numeral__eq__iff,axiom,
% 8.58/4.81 ! [M: num,N: num] :
% 8.58/4.81 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 8.58/4.81 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.58/4.81 = ( M = N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4450_neg__numeral__eq__iff,axiom,
% 8.58/4.81 ! [M: num,N: num] :
% 8.58/4.81 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 8.58/4.81 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.58/4.81 = ( M = N ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4451_mult__minus__right,axiom,
% 8.58/4.81 ! [A: real,B: real] :
% 8.58/4.81 ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
% 8.58/4.81 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4452_mult__minus__right,axiom,
% 8.58/4.81 ! [A: int,B: int] :
% 8.58/4.81 ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
% 8.58/4.81 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4453_mult__minus__right,axiom,
% 8.58/4.81 ! [A: complex,B: complex] :
% 8.58/4.81 ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 8.58/4.81 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4454_mult__minus__right,axiom,
% 8.58/4.81 ! [A: rat,B: rat] :
% 8.58/4.81 ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
% 8.58/4.81 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4455_mult__minus__right,axiom,
% 8.58/4.81 ! [A: code_integer,B: code_integer] :
% 8.58/4.81 ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 8.58/4.81 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4456_minus__mult__minus,axiom,
% 8.58/4.81 ! [A: real,B: real] :
% 8.58/4.81 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 8.58/4.81 = ( times_times_real @ A @ B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4457_minus__mult__minus,axiom,
% 8.58/4.81 ! [A: int,B: int] :
% 8.58/4.81 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 8.58/4.81 = ( times_times_int @ A @ B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4458_minus__mult__minus,axiom,
% 8.58/4.81 ! [A: complex,B: complex] :
% 8.58/4.81 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 8.58/4.81 = ( times_times_complex @ A @ B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4459_minus__mult__minus,axiom,
% 8.58/4.81 ! [A: rat,B: rat] :
% 8.58/4.81 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 8.58/4.81 = ( times_times_rat @ A @ B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4460_minus__mult__minus,axiom,
% 8.58/4.81 ! [A: code_integer,B: code_integer] :
% 8.58/4.81 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 8.58/4.81 = ( times_3573771949741848930nteger @ A @ B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4461_mult__minus__left,axiom,
% 8.58/4.81 ! [A: real,B: real] :
% 8.58/4.81 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 8.58/4.81 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4462_mult__minus__left,axiom,
% 8.58/4.81 ! [A: int,B: int] :
% 8.58/4.81 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 8.58/4.81 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4463_mult__minus__left,axiom,
% 8.58/4.81 ! [A: complex,B: complex] :
% 8.58/4.81 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 8.58/4.81 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4464_mult__minus__left,axiom,
% 8.58/4.81 ! [A: rat,B: rat] :
% 8.58/4.81 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 8.58/4.81 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4465_mult__minus__left,axiom,
% 8.58/4.81 ! [A: code_integer,B: code_integer] :
% 8.58/4.81 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 8.58/4.81 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4466_minus__add__distrib,axiom,
% 8.58/4.81 ! [A: real,B: real] :
% 8.58/4.81 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 8.58/4.81 = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4467_minus__add__distrib,axiom,
% 8.58/4.81 ! [A: int,B: int] :
% 8.58/4.81 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 8.58/4.81 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4468_minus__add__distrib,axiom,
% 8.58/4.81 ! [A: complex,B: complex] :
% 8.58/4.81 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 8.58/4.81 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4469_minus__add__distrib,axiom,
% 8.58/4.81 ! [A: rat,B: rat] :
% 8.58/4.81 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 8.58/4.81 = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4470_minus__add__distrib,axiom,
% 8.58/4.81 ! [A: code_integer,B: code_integer] :
% 8.58/4.81 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 8.58/4.81 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4471_minus__add__cancel,axiom,
% 8.58/4.81 ! [A: real,B: real] :
% 8.58/4.81 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 8.58/4.81 = B ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4472_minus__add__cancel,axiom,
% 8.58/4.81 ! [A: int,B: int] :
% 8.58/4.81 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 8.58/4.81 = B ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4473_minus__add__cancel,axiom,
% 8.58/4.81 ! [A: complex,B: complex] :
% 8.58/4.81 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
% 8.58/4.81 = B ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4474_minus__add__cancel,axiom,
% 8.58/4.81 ! [A: rat,B: rat] :
% 8.58/4.81 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
% 8.58/4.81 = B ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4475_minus__add__cancel,axiom,
% 8.58/4.81 ! [A: code_integer,B: code_integer] :
% 8.58/4.81 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 8.58/4.81 = B ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4476_add__minus__cancel,axiom,
% 8.58/4.81 ! [A: real,B: real] :
% 8.58/4.81 ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 8.58/4.81 = B ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4477_add__minus__cancel,axiom,
% 8.58/4.81 ! [A: int,B: int] :
% 8.58/4.81 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 8.58/4.81 = B ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4478_add__minus__cancel,axiom,
% 8.58/4.81 ! [A: complex,B: complex] :
% 8.58/4.81 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
% 8.58/4.81 = B ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4479_add__minus__cancel,axiom,
% 8.58/4.81 ! [A: rat,B: rat] :
% 8.58/4.81 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
% 8.58/4.81 = B ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4480_add__minus__cancel,axiom,
% 8.58/4.81 ! [A: code_integer,B: code_integer] :
% 8.58/4.81 ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
% 8.58/4.81 = B ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4481_minus__diff__eq,axiom,
% 8.58/4.81 ! [A: real,B: real] :
% 8.58/4.81 ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 8.58/4.81 = ( minus_minus_real @ B @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4482_minus__diff__eq,axiom,
% 8.58/4.81 ! [A: int,B: int] :
% 8.58/4.81 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 8.58/4.81 = ( minus_minus_int @ B @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4483_minus__diff__eq,axiom,
% 8.58/4.81 ! [A: complex,B: complex] :
% 8.58/4.81 ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
% 8.58/4.81 = ( minus_minus_complex @ B @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4484_minus__diff__eq,axiom,
% 8.58/4.81 ! [A: rat,B: rat] :
% 8.58/4.81 ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
% 8.58/4.81 = ( minus_minus_rat @ B @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4485_minus__diff__eq,axiom,
% 8.58/4.81 ! [A: code_integer,B: code_integer] :
% 8.58/4.81 ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
% 8.58/4.81 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4486_div__minus__minus,axiom,
% 8.58/4.81 ! [A: int,B: int] :
% 8.58/4.81 ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 8.58/4.81 = ( divide_divide_int @ A @ B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4487_div__minus__minus,axiom,
% 8.58/4.81 ! [A: code_integer,B: code_integer] :
% 8.58/4.81 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 8.58/4.81 = ( divide6298287555418463151nteger @ A @ B ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4488_dvd__minus__iff,axiom,
% 8.58/4.81 ! [X: real,Y: real] :
% 8.58/4.81 ( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y ) )
% 8.58/4.81 = ( dvd_dvd_real @ X @ Y ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4489_dvd__minus__iff,axiom,
% 8.58/4.81 ! [X: int,Y: int] :
% 8.58/4.81 ( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y ) )
% 8.58/4.81 = ( dvd_dvd_int @ X @ Y ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4490_dvd__minus__iff,axiom,
% 8.58/4.81 ! [X: complex,Y: complex] :
% 8.58/4.81 ( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y ) )
% 8.58/4.81 = ( dvd_dvd_complex @ X @ Y ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4491_dvd__minus__iff,axiom,
% 8.58/4.81 ! [X: rat,Y: rat] :
% 8.58/4.81 ( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y ) )
% 8.58/4.81 = ( dvd_dvd_rat @ X @ Y ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4492_dvd__minus__iff,axiom,
% 8.58/4.81 ! [X: code_integer,Y: code_integer] :
% 8.58/4.81 ( ( dvd_dvd_Code_integer @ X @ ( uminus1351360451143612070nteger @ Y ) )
% 8.58/4.81 = ( dvd_dvd_Code_integer @ X @ Y ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4493_minus__dvd__iff,axiom,
% 8.58/4.81 ! [X: real,Y: real] :
% 8.58/4.81 ( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y )
% 8.58/4.81 = ( dvd_dvd_real @ X @ Y ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4494_minus__dvd__iff,axiom,
% 8.58/4.81 ! [X: int,Y: int] :
% 8.58/4.81 ( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y )
% 8.58/4.81 = ( dvd_dvd_int @ X @ Y ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4495_minus__dvd__iff,axiom,
% 8.58/4.81 ! [X: complex,Y: complex] :
% 8.58/4.81 ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y )
% 8.58/4.81 = ( dvd_dvd_complex @ X @ Y ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4496_minus__dvd__iff,axiom,
% 8.58/4.81 ! [X: rat,Y: rat] :
% 8.58/4.81 ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y )
% 8.58/4.81 = ( dvd_dvd_rat @ X @ Y ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4497_minus__dvd__iff,axiom,
% 8.58/4.81 ! [X: code_integer,Y: code_integer] :
% 8.58/4.81 ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ Y )
% 8.58/4.81 = ( dvd_dvd_Code_integer @ X @ Y ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4498_mod__minus__minus,axiom,
% 8.58/4.81 ! [A: int,B: int] :
% 8.58/4.81 ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 8.58/4.81 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4499_mod__minus__minus,axiom,
% 8.58/4.81 ! [A: code_integer,B: code_integer] :
% 8.58/4.81 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 8.58/4.81 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4500_idom__abs__sgn__class_Osgn__minus,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( sgn_sgn_real @ ( uminus_uminus_real @ A ) )
% 8.58/4.81 = ( uminus_uminus_real @ ( sgn_sgn_real @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4501_idom__abs__sgn__class_Osgn__minus,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( sgn_sgn_int @ ( uminus_uminus_int @ A ) )
% 8.58/4.81 = ( uminus_uminus_int @ ( sgn_sgn_int @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4502_idom__abs__sgn__class_Osgn__minus,axiom,
% 8.58/4.81 ! [A: complex] :
% 8.58/4.81 ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ A ) )
% 8.58/4.81 = ( uminus1482373934393186551omplex @ ( sgn_sgn_complex @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4503_idom__abs__sgn__class_Osgn__minus,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ A ) )
% 8.58/4.81 = ( uminus_uminus_rat @ ( sgn_sgn_rat @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4504_idom__abs__sgn__class_Osgn__minus,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 8.58/4.81 = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ A ) ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4505_tanh__0,axiom,
% 8.58/4.81 ( ( tanh_real @ zero_zero_real )
% 8.58/4.81 = zero_zero_real ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4506_neg__less__eq__nonneg,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 8.58/4.81 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4507_neg__less__eq__nonneg,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 8.58/4.81 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4508_neg__less__eq__nonneg,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 8.58/4.81 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4509_neg__less__eq__nonneg,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 8.58/4.81 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4510_less__eq__neg__nonpos,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 8.58/4.81 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4511_less__eq__neg__nonpos,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 8.58/4.81 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4512_less__eq__neg__nonpos,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 8.58/4.81 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4513_less__eq__neg__nonpos,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 8.58/4.81 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4514_neg__le__0__iff__le,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 8.58/4.81 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4515_neg__le__0__iff__le,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 8.58/4.81 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4516_neg__le__0__iff__le,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 8.58/4.81 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4517_neg__le__0__iff__le,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 8.58/4.81 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4518_neg__0__le__iff__le,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 8.58/4.81 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4519_neg__0__le__iff__le,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 8.58/4.81 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4520_neg__0__le__iff__le,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 8.58/4.81 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4521_neg__0__le__iff__le,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 8.58/4.81 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4522_neg__less__0__iff__less,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 8.58/4.81 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4523_neg__less__0__iff__less,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 8.58/4.81 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4524_neg__less__0__iff__less,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 8.58/4.81 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4525_neg__less__0__iff__less,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 8.58/4.81 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4526_neg__0__less__iff__less,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 8.58/4.81 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4527_neg__0__less__iff__less,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 8.58/4.81 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4528_neg__0__less__iff__less,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 8.58/4.81 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4529_neg__0__less__iff__less,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 8.58/4.81 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4530_neg__less__pos,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 8.58/4.81 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4531_neg__less__pos,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 8.58/4.81 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4532_neg__less__pos,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 8.58/4.81 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4533_neg__less__pos,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 8.58/4.81 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4534_less__neg__neg,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 8.58/4.81 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4535_less__neg__neg,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 8.58/4.81 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4536_less__neg__neg,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 8.58/4.81 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4537_less__neg__neg,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 8.58/4.81 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4538_ab__left__minus,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 8.58/4.81 = zero_zero_real ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4539_ab__left__minus,axiom,
% 8.58/4.81 ! [A: int] :
% 8.58/4.81 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 8.58/4.81 = zero_zero_int ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4540_ab__left__minus,axiom,
% 8.58/4.81 ! [A: complex] :
% 8.58/4.81 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 8.58/4.81 = zero_zero_complex ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4541_ab__left__minus,axiom,
% 8.58/4.81 ! [A: rat] :
% 8.58/4.81 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 8.58/4.81 = zero_zero_rat ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4542_ab__left__minus,axiom,
% 8.58/4.81 ! [A: code_integer] :
% 8.58/4.81 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 8.58/4.81 = zero_z3403309356797280102nteger ) ).
% 8.58/4.81
% 8.58/4.81 thf(fact_4543_add_Oright__inverse,axiom,
% 8.58/4.81 ! [A: real] :
% 8.58/4.81 ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 8.58/4.82 = zero_zero_real ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4544_add_Oright__inverse,axiom,
% 8.58/4.82 ! [A: int] :
% 8.58/4.82 ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 8.58/4.82 = zero_zero_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4545_add_Oright__inverse,axiom,
% 8.58/4.82 ! [A: complex] :
% 8.58/4.82 ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 8.58/4.82 = zero_zero_complex ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4546_add_Oright__inverse,axiom,
% 8.58/4.82 ! [A: rat] :
% 8.58/4.82 ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 8.58/4.82 = zero_zero_rat ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4547_add_Oright__inverse,axiom,
% 8.58/4.82 ! [A: code_integer] :
% 8.58/4.82 ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 8.58/4.82 = zero_z3403309356797280102nteger ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4548_diff__0,axiom,
% 8.58/4.82 ! [A: real] :
% 8.58/4.82 ( ( minus_minus_real @ zero_zero_real @ A )
% 8.58/4.82 = ( uminus_uminus_real @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4549_diff__0,axiom,
% 8.58/4.82 ! [A: int] :
% 8.58/4.82 ( ( minus_minus_int @ zero_zero_int @ A )
% 8.58/4.82 = ( uminus_uminus_int @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4550_diff__0,axiom,
% 8.58/4.82 ! [A: complex] :
% 8.58/4.82 ( ( minus_minus_complex @ zero_zero_complex @ A )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4551_diff__0,axiom,
% 8.58/4.82 ! [A: rat] :
% 8.58/4.82 ( ( minus_minus_rat @ zero_zero_rat @ A )
% 8.58/4.82 = ( uminus_uminus_rat @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4552_diff__0,axiom,
% 8.58/4.82 ! [A: code_integer] :
% 8.58/4.82 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4553_verit__minus__simplify_I3_J,axiom,
% 8.58/4.82 ! [B: real] :
% 8.58/4.82 ( ( minus_minus_real @ zero_zero_real @ B )
% 8.58/4.82 = ( uminus_uminus_real @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4554_verit__minus__simplify_I3_J,axiom,
% 8.58/4.82 ! [B: int] :
% 8.58/4.82 ( ( minus_minus_int @ zero_zero_int @ B )
% 8.58/4.82 = ( uminus_uminus_int @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4555_verit__minus__simplify_I3_J,axiom,
% 8.58/4.82 ! [B: complex] :
% 8.58/4.82 ( ( minus_minus_complex @ zero_zero_complex @ B )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4556_verit__minus__simplify_I3_J,axiom,
% 8.58/4.82 ! [B: rat] :
% 8.58/4.82 ( ( minus_minus_rat @ zero_zero_rat @ B )
% 8.58/4.82 = ( uminus_uminus_rat @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4557_verit__minus__simplify_I3_J,axiom,
% 8.58/4.82 ! [B: code_integer] :
% 8.58/4.82 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4558_add__neg__numeral__simps_I3_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.58/4.82 = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4559_add__neg__numeral__simps_I3_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.82 = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4560_add__neg__numeral__simps_I3_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4561_add__neg__numeral__simps_I3_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.58/4.82 = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4562_add__neg__numeral__simps_I3_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4563_mult__minus1__right,axiom,
% 8.58/4.82 ! [Z2: real] :
% 8.58/4.82 ( ( times_times_real @ Z2 @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.82 = ( uminus_uminus_real @ Z2 ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4564_mult__minus1__right,axiom,
% 8.58/4.82 ! [Z2: int] :
% 8.58/4.82 ( ( times_times_int @ Z2 @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.82 = ( uminus_uminus_int @ Z2 ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4565_mult__minus1__right,axiom,
% 8.58/4.82 ! [Z2: complex] :
% 8.58/4.82 ( ( times_times_complex @ Z2 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ Z2 ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4566_mult__minus1__right,axiom,
% 8.58/4.82 ! [Z2: rat] :
% 8.58/4.82 ( ( times_times_rat @ Z2 @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.82 = ( uminus_uminus_rat @ Z2 ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4567_mult__minus1__right,axiom,
% 8.58/4.82 ! [Z2: code_integer] :
% 8.58/4.82 ( ( times_3573771949741848930nteger @ Z2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ Z2 ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4568_mult__minus1,axiom,
% 8.58/4.82 ! [Z2: real] :
% 8.58/4.82 ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z2 )
% 8.58/4.82 = ( uminus_uminus_real @ Z2 ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4569_mult__minus1,axiom,
% 8.58/4.82 ! [Z2: int] :
% 8.58/4.82 ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z2 )
% 8.58/4.82 = ( uminus_uminus_int @ Z2 ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4570_mult__minus1,axiom,
% 8.58/4.82 ! [Z2: complex] :
% 8.58/4.82 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z2 )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ Z2 ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4571_mult__minus1,axiom,
% 8.58/4.82 ! [Z2: rat] :
% 8.58/4.82 ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z2 )
% 8.58/4.82 = ( uminus_uminus_rat @ Z2 ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4572_mult__minus1,axiom,
% 8.58/4.82 ! [Z2: code_integer] :
% 8.58/4.82 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z2 )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ Z2 ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4573_diff__minus__eq__add,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
% 8.58/4.82 = ( plus_plus_real @ A @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4574_diff__minus__eq__add,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 8.58/4.82 = ( plus_plus_int @ A @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4575_diff__minus__eq__add,axiom,
% 8.58/4.82 ! [A: complex,B: complex] :
% 8.58/4.82 ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 8.58/4.82 = ( plus_plus_complex @ A @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4576_diff__minus__eq__add,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
% 8.58/4.82 = ( plus_plus_rat @ A @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4577_diff__minus__eq__add,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 8.58/4.82 = ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4578_uminus__add__conv__diff,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
% 8.58/4.82 = ( minus_minus_real @ B @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4579_uminus__add__conv__diff,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 8.58/4.82 = ( minus_minus_int @ B @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4580_uminus__add__conv__diff,axiom,
% 8.58/4.82 ! [A: complex,B: complex] :
% 8.58/4.82 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 8.58/4.82 = ( minus_minus_complex @ B @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4581_uminus__add__conv__diff,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
% 8.58/4.82 = ( minus_minus_rat @ B @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4582_uminus__add__conv__diff,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 8.58/4.82 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4583_divide__minus1,axiom,
% 8.58/4.82 ! [X: real] :
% 8.58/4.82 ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.82 = ( uminus_uminus_real @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4584_divide__minus1,axiom,
% 8.58/4.82 ! [X: complex] :
% 8.58/4.82 ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4585_divide__minus1,axiom,
% 8.58/4.82 ! [X: rat] :
% 8.58/4.82 ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.82 = ( uminus_uminus_rat @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4586_div__minus1__right,axiom,
% 8.58/4.82 ! [A: int] :
% 8.58/4.82 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.82 = ( uminus_uminus_int @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4587_div__minus1__right,axiom,
% 8.58/4.82 ! [A: code_integer] :
% 8.58/4.82 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4588_minus__mod__self1,axiom,
% 8.58/4.82 ! [B: int,A: int] :
% 8.58/4.82 ( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
% 8.58/4.82 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4589_minus__mod__self1,axiom,
% 8.58/4.82 ! [B: code_integer,A: code_integer] :
% 8.58/4.82 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
% 8.58/4.82 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4590_ceiling__zero,axiom,
% 8.58/4.82 ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
% 8.58/4.82 = zero_zero_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4591_ceiling__zero,axiom,
% 8.58/4.82 ( ( archim7802044766580827645g_real @ zero_zero_real )
% 8.58/4.82 = zero_zero_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4592_signed__take__bit__of__minus__1,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4593_signed__take__bit__of__minus__1,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.82 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4594_real__add__minus__iff,axiom,
% 8.58/4.82 ! [X: real,A: real] :
% 8.58/4.82 ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
% 8.58/4.82 = zero_zero_real )
% 8.58/4.82 = ( X = A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4595_ceiling__numeral,axiom,
% 8.58/4.82 ! [V: num] :
% 8.58/4.82 ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
% 8.58/4.82 = ( numeral_numeral_int @ V ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4596_ceiling__numeral,axiom,
% 8.58/4.82 ! [V: num] :
% 8.58/4.82 ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
% 8.58/4.82 = ( numeral_numeral_int @ V ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4597_ceiling__one,axiom,
% 8.58/4.82 ( ( archim2889992004027027881ng_rat @ one_one_rat )
% 8.58/4.82 = one_one_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4598_ceiling__one,axiom,
% 8.58/4.82 ( ( archim7802044766580827645g_real @ one_one_real )
% 8.58/4.82 = one_one_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4599_ceiling__of__nat,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( archim7802044766580827645g_real @ ( semiri5074537144036343181t_real @ N ) )
% 8.58/4.82 = ( semiri1314217659103216013at_int @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4600_ceiling__of__nat,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( archim2889992004027027881ng_rat @ ( semiri681578069525770553at_rat @ N ) )
% 8.58/4.82 = ( semiri1314217659103216013at_int @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4601_dbl__simps_I1_J,axiom,
% 8.58/4.82 ! [K: num] :
% 8.58/4.82 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 8.58/4.82 = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4602_dbl__simps_I1_J,axiom,
% 8.58/4.82 ! [K: num] :
% 8.58/4.82 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.58/4.82 = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4603_dbl__simps_I1_J,axiom,
% 8.58/4.82 ! [K: num] :
% 8.58/4.82 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4604_dbl__simps_I1_J,axiom,
% 8.58/4.82 ! [K: num] :
% 8.58/4.82 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 8.58/4.82 = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4605_dbl__simps_I1_J,axiom,
% 8.58/4.82 ! [K: num] :
% 8.58/4.82 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4606_add__neg__numeral__special_I7_J,axiom,
% 8.58/4.82 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.82 = zero_zero_real ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4607_add__neg__numeral__special_I7_J,axiom,
% 8.58/4.82 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.82 = zero_zero_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4608_add__neg__numeral__special_I7_J,axiom,
% 8.58/4.82 ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.82 = zero_zero_complex ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4609_add__neg__numeral__special_I7_J,axiom,
% 8.58/4.82 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.82 = zero_zero_rat ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4610_add__neg__numeral__special_I7_J,axiom,
% 8.58/4.82 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.82 = zero_z3403309356797280102nteger ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4611_add__neg__numeral__special_I8_J,axiom,
% 8.58/4.82 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 8.58/4.82 = zero_zero_real ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4612_add__neg__numeral__special_I8_J,axiom,
% 8.58/4.82 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 8.58/4.82 = zero_zero_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4613_add__neg__numeral__special_I8_J,axiom,
% 8.58/4.82 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 8.58/4.82 = zero_zero_complex ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4614_add__neg__numeral__special_I8_J,axiom,
% 8.58/4.82 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 8.58/4.82 = zero_zero_rat ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4615_add__neg__numeral__special_I8_J,axiom,
% 8.58/4.82 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 8.58/4.82 = zero_z3403309356797280102nteger ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4616_diff__numeral__special_I12_J,axiom,
% 8.58/4.82 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.82 = zero_zero_real ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4617_diff__numeral__special_I12_J,axiom,
% 8.58/4.82 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.82 = zero_zero_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4618_diff__numeral__special_I12_J,axiom,
% 8.58/4.82 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.82 = zero_zero_complex ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4619_diff__numeral__special_I12_J,axiom,
% 8.58/4.82 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.82 = zero_zero_rat ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4620_diff__numeral__special_I12_J,axiom,
% 8.58/4.82 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.82 = zero_z3403309356797280102nteger ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4621_numeral__eq__neg__one__iff,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
% 8.58/4.82 = ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.82 = ( N = one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4622_numeral__eq__neg__one__iff,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
% 8.58/4.82 = ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.82 = ( N = one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4623_numeral__eq__neg__one__iff,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.82 = ( N = one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4624_numeral__eq__neg__one__iff,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
% 8.58/4.82 = ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.82 = ( N = one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4625_numeral__eq__neg__one__iff,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.82 = ( N = one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4626_neg__one__eq__numeral__iff,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( ( uminus_uminus_real @ one_one_real )
% 8.58/4.82 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.58/4.82 = ( N = one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4627_neg__one__eq__numeral__iff,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( ( uminus_uminus_int @ one_one_int )
% 8.58/4.82 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.82 = ( N = one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4628_neg__one__eq__numeral__iff,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( ( uminus1482373934393186551omplex @ one_one_complex )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.58/4.82 = ( N = one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4629_neg__one__eq__numeral__iff,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( ( uminus_uminus_rat @ one_one_rat )
% 8.58/4.82 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.58/4.82 = ( N = one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4630_neg__one__eq__numeral__iff,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.58/4.82 = ( N = one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4631_minus__one__mult__self,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
% 8.58/4.82 = one_one_real ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4632_minus__one__mult__self,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
% 8.58/4.82 = one_one_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4633_minus__one__mult__self,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
% 8.58/4.82 = one_one_complex ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4634_minus__one__mult__self,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
% 8.58/4.82 = one_one_rat ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4635_minus__one__mult__self,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
% 8.58/4.82 = one_one_Code_integer ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4636_left__minus__one__mult__self,axiom,
% 8.58/4.82 ! [N: nat,A: real] :
% 8.58/4.82 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
% 8.58/4.82 = A ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4637_left__minus__one__mult__self,axiom,
% 8.58/4.82 ! [N: nat,A: int] :
% 8.58/4.82 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
% 8.58/4.82 = A ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4638_left__minus__one__mult__self,axiom,
% 8.58/4.82 ! [N: nat,A: complex] :
% 8.58/4.82 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
% 8.58/4.82 = A ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4639_left__minus__one__mult__self,axiom,
% 8.58/4.82 ! [N: nat,A: rat] :
% 8.58/4.82 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
% 8.58/4.82 = A ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4640_left__minus__one__mult__self,axiom,
% 8.58/4.82 ! [N: nat,A: code_integer] :
% 8.58/4.82 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
% 8.58/4.82 = A ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4641_mod__minus1__right,axiom,
% 8.58/4.82 ! [A: int] :
% 8.58/4.82 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.82 = zero_zero_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4642_mod__minus1__right,axiom,
% 8.58/4.82 ! [A: code_integer] :
% 8.58/4.82 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.82 = zero_z3403309356797280102nteger ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4643_norm__neg__numeral,axiom,
% 8.58/4.82 ! [W: num] :
% 8.58/4.82 ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.58/4.82 = ( numeral_numeral_real @ W ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4644_norm__neg__numeral,axiom,
% 8.58/4.82 ! [W: num] :
% 8.58/4.82 ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 8.58/4.82 = ( numeral_numeral_real @ W ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4645_semiring__norm_I168_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: real] :
% 8.58/4.82 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 8.58/4.82 = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4646_semiring__norm_I168_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: int] :
% 8.58/4.82 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 8.58/4.82 = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4647_semiring__norm_I168_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: complex] :
% 8.58/4.82 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 8.58/4.82 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4648_semiring__norm_I168_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: rat] :
% 8.58/4.82 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 8.58/4.82 = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4649_semiring__norm_I168_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: code_integer] :
% 8.58/4.82 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 8.58/4.82 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4650_diff__numeral__simps_I3_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 8.58/4.82 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4651_diff__numeral__simps_I3_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.82 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4652_diff__numeral__simps_I3_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4653_diff__numeral__simps_I3_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 8.58/4.82 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4654_diff__numeral__simps_I3_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4655_diff__numeral__simps_I2_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.58/4.82 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4656_diff__numeral__simps_I2_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.82 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4657_diff__numeral__simps_I2_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.58/4.82 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4658_diff__numeral__simps_I2_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.58/4.82 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4659_diff__numeral__simps_I2_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.58/4.82 = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4660_ceiling__neg__numeral,axiom,
% 8.58/4.82 ! [V: num] :
% 8.58/4.82 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.58/4.82 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4661_ceiling__neg__numeral,axiom,
% 8.58/4.82 ! [V: num] :
% 8.58/4.82 ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.58/4.82 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4662_semiring__norm_I172_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: real] :
% 8.58/4.82 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 8.58/4.82 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4663_semiring__norm_I172_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: int] :
% 8.58/4.82 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 8.58/4.82 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4664_semiring__norm_I172_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: complex] :
% 8.58/4.82 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 8.58/4.82 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4665_semiring__norm_I172_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: rat] :
% 8.58/4.82 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 8.58/4.82 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4666_semiring__norm_I172_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: code_integer] :
% 8.58/4.82 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 8.58/4.82 = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4667_semiring__norm_I171_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: real] :
% 8.58/4.82 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 8.58/4.82 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4668_semiring__norm_I171_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: int] :
% 8.58/4.82 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 8.58/4.82 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4669_semiring__norm_I171_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: complex] :
% 8.58/4.82 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 8.58/4.82 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4670_semiring__norm_I171_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: rat] :
% 8.58/4.82 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 8.58/4.82 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4671_semiring__norm_I171_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: code_integer] :
% 8.58/4.82 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 8.58/4.82 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4672_semiring__norm_I170_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: real] :
% 8.58/4.82 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y ) )
% 8.58/4.82 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4673_semiring__norm_I170_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: int] :
% 8.58/4.82 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
% 8.58/4.82 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4674_semiring__norm_I170_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: complex] :
% 8.58/4.82 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y ) )
% 8.58/4.82 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4675_semiring__norm_I170_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: rat] :
% 8.58/4.82 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
% 8.58/4.82 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4676_semiring__norm_I170_J,axiom,
% 8.58/4.82 ! [V: num,W: num,Y: code_integer] :
% 8.58/4.82 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
% 8.58/4.82 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4677_mult__neg__numeral__simps_I3_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.58/4.82 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4678_mult__neg__numeral__simps_I3_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.82 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4679_mult__neg__numeral__simps_I3_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4680_mult__neg__numeral__simps_I3_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.58/4.82 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4681_mult__neg__numeral__simps_I3_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4682_mult__neg__numeral__simps_I2_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 8.58/4.82 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4683_mult__neg__numeral__simps_I2_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.82 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4684_mult__neg__numeral__simps_I2_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4685_mult__neg__numeral__simps_I2_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 8.58/4.82 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4686_mult__neg__numeral__simps_I2_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4687_mult__neg__numeral__simps_I1_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.58/4.82 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4688_mult__neg__numeral__simps_I1_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.82 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4689_mult__neg__numeral__simps_I1_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.58/4.82 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4690_mult__neg__numeral__simps_I1_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.58/4.82 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4691_mult__neg__numeral__simps_I1_J,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.58/4.82 = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4692_neg__numeral__le__iff,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.58/4.82 = ( ord_less_eq_num @ N @ M ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4693_neg__numeral__le__iff,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.58/4.82 = ( ord_less_eq_num @ N @ M ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4694_neg__numeral__le__iff,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.58/4.82 = ( ord_less_eq_num @ N @ M ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4695_neg__numeral__le__iff,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.82 = ( ord_less_eq_num @ N @ M ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4696_neg__numeral__less__iff,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.58/4.82 = ( ord_less_num @ N @ M ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4697_neg__numeral__less__iff,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.82 = ( ord_less_num @ N @ M ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4698_neg__numeral__less__iff,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.58/4.82 = ( ord_less_num @ N @ M ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4699_neg__numeral__less__iff,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.58/4.82 = ( ord_less_num @ N @ M ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4700_not__neg__one__le__neg__numeral__iff,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 8.58/4.82 = ( M != one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4701_not__neg__one__le__neg__numeral__iff,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 8.58/4.82 = ( M != one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4702_not__neg__one__le__neg__numeral__iff,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 8.58/4.82 = ( M != one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4703_not__neg__one__le__neg__numeral__iff,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 8.58/4.82 = ( M != one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4704_le__divide__eq__numeral1_I2_J,axiom,
% 8.58/4.82 ! [A: real,B: real,W: num] :
% 8.58/4.82 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 8.58/4.82 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4705_le__divide__eq__numeral1_I2_J,axiom,
% 8.58/4.82 ! [A: rat,B: rat,W: num] :
% 8.58/4.82 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 8.58/4.82 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4706_divide__le__eq__numeral1_I2_J,axiom,
% 8.58/4.82 ! [B: real,W: num,A: real] :
% 8.58/4.82 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 8.58/4.82 = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4707_divide__le__eq__numeral1_I2_J,axiom,
% 8.58/4.82 ! [B: rat,W: num,A: rat] :
% 8.58/4.82 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 8.58/4.82 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4708_eq__divide__eq__numeral1_I2_J,axiom,
% 8.58/4.82 ! [A: real,B: real,W: num] :
% 8.58/4.82 ( ( A
% 8.58/4.82 = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 8.58/4.82 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 8.58/4.82 != zero_zero_real )
% 8.58/4.82 => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.58/4.82 = B ) )
% 8.58/4.82 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 8.58/4.82 = zero_zero_real )
% 8.58/4.82 => ( A = zero_zero_real ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4709_eq__divide__eq__numeral1_I2_J,axiom,
% 8.58/4.82 ! [A: complex,B: complex,W: num] :
% 8.58/4.82 ( ( A
% 8.58/4.82 = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
% 8.58/4.82 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.58/4.82 != zero_zero_complex )
% 8.58/4.82 => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 8.58/4.82 = B ) )
% 8.58/4.82 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.58/4.82 = zero_zero_complex )
% 8.58/4.82 => ( A = zero_zero_complex ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4710_eq__divide__eq__numeral1_I2_J,axiom,
% 8.58/4.82 ! [A: rat,B: rat,W: num] :
% 8.58/4.82 ( ( A
% 8.58/4.82 = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 8.58/4.82 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 8.58/4.82 != zero_zero_rat )
% 8.58/4.82 => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 8.58/4.82 = B ) )
% 8.58/4.82 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 8.58/4.82 = zero_zero_rat )
% 8.58/4.82 => ( A = zero_zero_rat ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4711_divide__eq__eq__numeral1_I2_J,axiom,
% 8.58/4.82 ! [B: real,W: num,A: real] :
% 8.58/4.82 ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.58/4.82 = A )
% 8.58/4.82 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 8.58/4.82 != zero_zero_real )
% 8.58/4.82 => ( B
% 8.58/4.82 = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
% 8.58/4.82 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 8.58/4.82 = zero_zero_real )
% 8.58/4.82 => ( A = zero_zero_real ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4712_divide__eq__eq__numeral1_I2_J,axiom,
% 8.58/4.82 ! [B: complex,W: num,A: complex] :
% 8.58/4.82 ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 8.58/4.82 = A )
% 8.58/4.82 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.58/4.82 != zero_zero_complex )
% 8.58/4.82 => ( B
% 8.58/4.82 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
% 8.58/4.82 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.58/4.82 = zero_zero_complex )
% 8.58/4.82 => ( A = zero_zero_complex ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4713_divide__eq__eq__numeral1_I2_J,axiom,
% 8.58/4.82 ! [B: rat,W: num,A: rat] :
% 8.58/4.82 ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 8.58/4.82 = A )
% 8.58/4.82 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 8.58/4.82 != zero_zero_rat )
% 8.58/4.82 => ( B
% 8.58/4.82 = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
% 8.58/4.82 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 8.58/4.82 = zero_zero_rat )
% 8.58/4.82 => ( A = zero_zero_rat ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4714_neg__numeral__less__neg__one__iff,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.82 = ( M != one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4715_neg__numeral__less__neg__one__iff,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.82 = ( M != one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4716_neg__numeral__less__neg__one__iff,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.82 = ( M != one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4717_neg__numeral__less__neg__one__iff,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.82 = ( M != one ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4718_less__divide__eq__numeral1_I2_J,axiom,
% 8.58/4.82 ! [A: real,B: real,W: num] :
% 8.58/4.82 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 8.58/4.82 = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4719_less__divide__eq__numeral1_I2_J,axiom,
% 8.58/4.82 ! [A: rat,B: rat,W: num] :
% 8.58/4.82 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 8.58/4.82 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4720_divide__less__eq__numeral1_I2_J,axiom,
% 8.58/4.82 ! [B: real,W: num,A: real] :
% 8.58/4.82 ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 8.58/4.82 = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4721_divide__less__eq__numeral1_I2_J,axiom,
% 8.58/4.82 ! [B: rat,W: num,A: rat] :
% 8.58/4.82 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 8.58/4.82 = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4722_power2__minus,axiom,
% 8.58/4.82 ! [A: real] :
% 8.58/4.82 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.82 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4723_power2__minus,axiom,
% 8.58/4.82 ! [A: int] :
% 8.58/4.82 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.82 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4724_power2__minus,axiom,
% 8.58/4.82 ! [A: complex] :
% 8.58/4.82 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.82 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4725_power2__minus,axiom,
% 8.58/4.82 ! [A: rat] :
% 8.58/4.82 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.82 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4726_power2__minus,axiom,
% 8.58/4.82 ! [A: code_integer] :
% 8.58/4.82 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.82 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4727_sgn__neg,axiom,
% 8.58/4.82 ! [A: real] :
% 8.58/4.82 ( ( ord_less_real @ A @ zero_zero_real )
% 8.58/4.82 => ( ( sgn_sgn_real @ A )
% 8.58/4.82 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4728_sgn__neg,axiom,
% 8.58/4.82 ! [A: int] :
% 8.58/4.82 ( ( ord_less_int @ A @ zero_zero_int )
% 8.58/4.82 => ( ( sgn_sgn_int @ A )
% 8.58/4.82 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4729_sgn__neg,axiom,
% 8.58/4.82 ! [A: rat] :
% 8.58/4.82 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.58/4.82 => ( ( sgn_sgn_rat @ A )
% 8.58/4.82 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4730_sgn__neg,axiom,
% 8.58/4.82 ! [A: code_integer] :
% 8.58/4.82 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 8.58/4.82 => ( ( sgn_sgn_Code_integer @ A )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4731_push__bit__Suc__minus__numeral,axiom,
% 8.58/4.82 ! [N: nat,K: num] :
% 8.58/4.82 ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.58/4.82 = ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4732_push__bit__Suc__minus__numeral,axiom,
% 8.58/4.82 ! [N: nat,K: num] :
% 8.58/4.82 ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 8.58/4.82 = ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4733_ceiling__le__zero,axiom,
% 8.58/4.82 ! [X: real] :
% 8.58/4.82 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
% 8.58/4.82 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4734_ceiling__le__zero,axiom,
% 8.58/4.82 ! [X: rat] :
% 8.58/4.82 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
% 8.58/4.82 = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4735_zero__less__ceiling,axiom,
% 8.58/4.82 ! [X: rat] :
% 8.58/4.82 ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
% 8.58/4.82 = ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4736_zero__less__ceiling,axiom,
% 8.58/4.82 ! [X: real] :
% 8.58/4.82 ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
% 8.58/4.82 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4737_ceiling__le__numeral,axiom,
% 8.58/4.82 ! [X: real,V: num] :
% 8.58/4.82 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 8.58/4.82 = ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4738_ceiling__le__numeral,axiom,
% 8.58/4.82 ! [X: rat,V: num] :
% 8.58/4.82 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 8.58/4.82 = ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4739_ceiling__less__one,axiom,
% 8.58/4.82 ! [X: real] :
% 8.58/4.82 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
% 8.58/4.82 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4740_ceiling__less__one,axiom,
% 8.58/4.82 ! [X: rat] :
% 8.58/4.82 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
% 8.58/4.82 = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4741_one__le__ceiling,axiom,
% 8.58/4.82 ! [X: rat] :
% 8.58/4.82 ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
% 8.58/4.82 = ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4742_one__le__ceiling,axiom,
% 8.58/4.82 ! [X: real] :
% 8.58/4.82 ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
% 8.58/4.82 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4743_numeral__less__ceiling,axiom,
% 8.58/4.82 ! [V: num,X: real] :
% 8.58/4.82 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
% 8.58/4.82 = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4744_numeral__less__ceiling,axiom,
% 8.58/4.82 ! [V: num,X: rat] :
% 8.58/4.82 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
% 8.58/4.82 = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4745_ceiling__le__one,axiom,
% 8.58/4.82 ! [X: real] :
% 8.58/4.82 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
% 8.58/4.82 = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4746_ceiling__le__one,axiom,
% 8.58/4.82 ! [X: rat] :
% 8.58/4.82 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
% 8.58/4.82 = ( ord_less_eq_rat @ X @ one_one_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4747_one__less__ceiling,axiom,
% 8.58/4.82 ! [X: rat] :
% 8.58/4.82 ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
% 8.58/4.82 = ( ord_less_rat @ one_one_rat @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4748_one__less__ceiling,axiom,
% 8.58/4.82 ! [X: real] :
% 8.58/4.82 ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
% 8.58/4.82 = ( ord_less_real @ one_one_real @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4749_ceiling__add__numeral,axiom,
% 8.58/4.82 ! [X: real,V: num] :
% 8.58/4.82 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 8.58/4.82 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4750_ceiling__add__numeral,axiom,
% 8.58/4.82 ! [X: rat,V: num] :
% 8.58/4.82 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 8.58/4.82 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4751_ceiling__add__one,axiom,
% 8.58/4.82 ! [X: rat] :
% 8.58/4.82 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
% 8.58/4.82 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4752_ceiling__add__one,axiom,
% 8.58/4.82 ! [X: real] :
% 8.58/4.82 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ one_one_real ) )
% 8.58/4.82 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4753_ceiling__diff__numeral,axiom,
% 8.58/4.82 ! [X: real,V: num] :
% 8.58/4.82 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 8.58/4.82 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4754_ceiling__diff__numeral,axiom,
% 8.58/4.82 ! [X: rat,V: num] :
% 8.58/4.82 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 8.58/4.82 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4755_ceiling__diff__one,axiom,
% 8.58/4.82 ! [X: rat] :
% 8.58/4.82 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
% 8.58/4.82 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4756_ceiling__diff__one,axiom,
% 8.58/4.82 ! [X: real] :
% 8.58/4.82 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ one_one_real ) )
% 8.58/4.82 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4757_ceiling__numeral__power,axiom,
% 8.58/4.82 ! [X: num,N: nat] :
% 8.58/4.82 ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 8.58/4.82 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4758_ceiling__numeral__power,axiom,
% 8.58/4.82 ! [X: num,N: nat] :
% 8.58/4.82 ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 8.58/4.82 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4759_add__neg__numeral__special_I9_J,axiom,
% 8.58/4.82 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.82 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4760_add__neg__numeral__special_I9_J,axiom,
% 8.58/4.82 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.82 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4761_add__neg__numeral__special_I9_J,axiom,
% 8.58/4.82 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4762_add__neg__numeral__special_I9_J,axiom,
% 8.58/4.82 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.82 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4763_add__neg__numeral__special_I9_J,axiom,
% 8.58/4.82 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4764_diff__numeral__special_I10_J,axiom,
% 8.58/4.82 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 8.58/4.82 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4765_diff__numeral__special_I10_J,axiom,
% 8.58/4.82 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 8.58/4.82 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4766_diff__numeral__special_I10_J,axiom,
% 8.58/4.82 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4767_diff__numeral__special_I10_J,axiom,
% 8.58/4.82 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 8.58/4.82 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4768_diff__numeral__special_I10_J,axiom,
% 8.58/4.82 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4769_diff__numeral__special_I11_J,axiom,
% 8.58/4.82 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.82 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4770_diff__numeral__special_I11_J,axiom,
% 8.58/4.82 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.82 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4771_diff__numeral__special_I11_J,axiom,
% 8.58/4.82 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.82 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4772_diff__numeral__special_I11_J,axiom,
% 8.58/4.82 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.82 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4773_diff__numeral__special_I11_J,axiom,
% 8.58/4.82 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.82 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4774_minus__1__div__2__eq,axiom,
% 8.58/4.82 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.82 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4775_minus__1__div__2__eq,axiom,
% 8.58/4.82 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4776_minus__1__mod__2__eq,axiom,
% 8.58/4.82 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.82 = one_one_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4777_minus__1__mod__2__eq,axiom,
% 8.58/4.82 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.82 = one_one_Code_integer ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4778_bits__minus__1__mod__2__eq,axiom,
% 8.58/4.82 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.82 = one_one_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4779_bits__minus__1__mod__2__eq,axiom,
% 8.58/4.82 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.82 = one_one_Code_integer ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4780_Power_Oring__1__class_Opower__minus__even,axiom,
% 8.58/4.82 ! [A: real,N: nat] :
% 8.58/4.82 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.82 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4781_Power_Oring__1__class_Opower__minus__even,axiom,
% 8.58/4.82 ! [A: int,N: nat] :
% 8.58/4.82 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.82 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4782_Power_Oring__1__class_Opower__minus__even,axiom,
% 8.58/4.82 ! [A: complex,N: nat] :
% 8.58/4.82 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.82 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4783_Power_Oring__1__class_Opower__minus__even,axiom,
% 8.58/4.82 ! [A: rat,N: nat] :
% 8.58/4.82 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.82 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4784_Power_Oring__1__class_Opower__minus__even,axiom,
% 8.58/4.82 ! [A: code_integer,N: nat] :
% 8.58/4.82 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.82 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4785_Parity_Oring__1__class_Opower__minus__even,axiom,
% 8.58/4.82 ! [N: nat,A: real] :
% 8.58/4.82 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 8.58/4.82 = ( power_power_real @ A @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4786_Parity_Oring__1__class_Opower__minus__even,axiom,
% 8.58/4.82 ! [N: nat,A: int] :
% 8.58/4.82 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 8.58/4.82 = ( power_power_int @ A @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4787_Parity_Oring__1__class_Opower__minus__even,axiom,
% 8.58/4.82 ! [N: nat,A: complex] :
% 8.58/4.82 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 8.58/4.82 = ( power_power_complex @ A @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4788_Parity_Oring__1__class_Opower__minus__even,axiom,
% 8.58/4.82 ! [N: nat,A: rat] :
% 8.58/4.82 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 8.58/4.82 = ( power_power_rat @ A @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4789_Parity_Oring__1__class_Opower__minus__even,axiom,
% 8.58/4.82 ! [N: nat,A: code_integer] :
% 8.58/4.82 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 8.58/4.82 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4790_power__minus__odd,axiom,
% 8.58/4.82 ! [N: nat,A: real] :
% 8.58/4.82 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 8.58/4.82 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4791_power__minus__odd,axiom,
% 8.58/4.82 ! [N: nat,A: int] :
% 8.58/4.82 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 8.58/4.82 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4792_power__minus__odd,axiom,
% 8.58/4.82 ! [N: nat,A: complex] :
% 8.58/4.82 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4793_power__minus__odd,axiom,
% 8.58/4.82 ! [N: nat,A: rat] :
% 8.58/4.82 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 8.58/4.82 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4794_power__minus__odd,axiom,
% 8.58/4.82 ! [N: nat,A: code_integer] :
% 8.58/4.82 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4795_diff__numeral__special_I4_J,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 8.58/4.82 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4796_diff__numeral__special_I4_J,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 8.58/4.82 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4797_diff__numeral__special_I4_J,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4798_diff__numeral__special_I4_J,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 8.58/4.82 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4799_diff__numeral__special_I4_J,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4800_diff__numeral__special_I3_J,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.58/4.82 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4801_diff__numeral__special_I3_J,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.82 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4802_diff__numeral__special_I3_J,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.58/4.82 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4803_diff__numeral__special_I3_J,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.58/4.82 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4804_diff__numeral__special_I3_J,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.58/4.82 = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4805_ceiling__le__neg__numeral,axiom,
% 8.58/4.82 ! [X: real,V: num] :
% 8.58/4.82 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.58/4.82 = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4806_ceiling__le__neg__numeral,axiom,
% 8.58/4.82 ! [X: rat,V: num] :
% 8.58/4.82 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.58/4.82 = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4807_ceiling__less__zero,axiom,
% 8.58/4.82 ! [X: real] :
% 8.58/4.82 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
% 8.58/4.82 = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4808_ceiling__less__zero,axiom,
% 8.58/4.82 ! [X: rat] :
% 8.58/4.82 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
% 8.58/4.82 = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4809_zero__le__ceiling,axiom,
% 8.58/4.82 ! [X: real] :
% 8.58/4.82 ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
% 8.58/4.82 = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4810_zero__le__ceiling,axiom,
% 8.58/4.82 ! [X: rat] :
% 8.58/4.82 ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
% 8.58/4.82 = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4811_neg__numeral__less__ceiling,axiom,
% 8.58/4.82 ! [V: num,X: real] :
% 8.58/4.82 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
% 8.58/4.82 = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4812_neg__numeral__less__ceiling,axiom,
% 8.58/4.82 ! [V: num,X: rat] :
% 8.58/4.82 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
% 8.58/4.82 = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4813_dbl__simps_I4_J,axiom,
% 8.58/4.82 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.82 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4814_dbl__simps_I4_J,axiom,
% 8.58/4.82 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.82 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4815_dbl__simps_I4_J,axiom,
% 8.58/4.82 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4816_dbl__simps_I4_J,axiom,
% 8.58/4.82 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.82 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4817_dbl__simps_I4_J,axiom,
% 8.58/4.82 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4818_power__minus1__even,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.82 = one_one_real ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4819_power__minus1__even,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.82 = one_one_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4820_power__minus1__even,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.82 = one_one_complex ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4821_power__minus1__even,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.82 = one_one_rat ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4822_power__minus1__even,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.82 = one_one_Code_integer ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4823_neg__one__even__power,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 8.58/4.82 = one_one_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4824_neg__one__even__power,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 8.58/4.82 = one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4825_neg__one__even__power,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 8.58/4.82 = one_one_complex ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4826_neg__one__even__power,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 8.58/4.82 = one_one_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4827_neg__one__even__power,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 8.58/4.82 = one_one_Code_integer ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4828_neg__one__odd__power,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 8.58/4.82 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4829_neg__one__odd__power,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 8.58/4.82 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4830_neg__one__odd__power,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4831_neg__one__odd__power,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 8.58/4.82 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4832_neg__one__odd__power,axiom,
% 8.58/4.82 ! [N: nat] :
% 8.58/4.82 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.82 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4833_ceiling__less__numeral,axiom,
% 8.58/4.82 ! [X: real,V: num] :
% 8.58/4.82 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 8.58/4.82 = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4834_ceiling__less__numeral,axiom,
% 8.58/4.82 ! [X: rat,V: num] :
% 8.58/4.82 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 8.58/4.82 = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4835_numeral__le__ceiling,axiom,
% 8.58/4.82 ! [V: num,X: real] :
% 8.58/4.82 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
% 8.58/4.82 = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4836_numeral__le__ceiling,axiom,
% 8.58/4.82 ! [V: num,X: rat] :
% 8.58/4.82 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
% 8.58/4.82 = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4837_signed__take__bit__0,axiom,
% 8.58/4.82 ! [A: code_integer] :
% 8.58/4.82 ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4838_signed__take__bit__0,axiom,
% 8.58/4.82 ! [A: int] :
% 8.58/4.82 ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 8.58/4.82 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4839_push__bit__numeral__minus__1,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.82 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4840_push__bit__numeral__minus__1,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4841_ceiling__less__neg__numeral,axiom,
% 8.58/4.82 ! [X: real,V: num] :
% 8.58/4.82 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.58/4.82 = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4842_ceiling__less__neg__numeral,axiom,
% 8.58/4.82 ! [X: rat,V: num] :
% 8.58/4.82 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.58/4.82 = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4843_neg__numeral__le__ceiling,axiom,
% 8.58/4.82 ! [V: num,X: real] :
% 8.58/4.82 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
% 8.58/4.82 = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4844_neg__numeral__le__ceiling,axiom,
% 8.58/4.82 ! [V: num,X: rat] :
% 8.58/4.82 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
% 8.58/4.82 = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4845_verit__negate__coefficient_I3_J,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( A = B )
% 8.58/4.82 => ( ( uminus_uminus_real @ A )
% 8.58/4.82 = ( uminus_uminus_real @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4846_verit__negate__coefficient_I3_J,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( A = B )
% 8.58/4.82 => ( ( uminus_uminus_int @ A )
% 8.58/4.82 = ( uminus_uminus_int @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4847_verit__negate__coefficient_I3_J,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( A = B )
% 8.58/4.82 => ( ( uminus_uminus_rat @ A )
% 8.58/4.82 = ( uminus_uminus_rat @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4848_verit__negate__coefficient_I3_J,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( A = B )
% 8.58/4.82 => ( ( uminus1351360451143612070nteger @ A )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4849_minus__equation__iff,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( ( uminus_uminus_real @ A )
% 8.58/4.82 = B )
% 8.58/4.82 = ( ( uminus_uminus_real @ B )
% 8.58/4.82 = A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4850_minus__equation__iff,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( ( uminus_uminus_int @ A )
% 8.58/4.82 = B )
% 8.58/4.82 = ( ( uminus_uminus_int @ B )
% 8.58/4.82 = A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4851_minus__equation__iff,axiom,
% 8.58/4.82 ! [A: complex,B: complex] :
% 8.58/4.82 ( ( ( uminus1482373934393186551omplex @ A )
% 8.58/4.82 = B )
% 8.58/4.82 = ( ( uminus1482373934393186551omplex @ B )
% 8.58/4.82 = A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4852_minus__equation__iff,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( ( uminus_uminus_rat @ A )
% 8.58/4.82 = B )
% 8.58/4.82 = ( ( uminus_uminus_rat @ B )
% 8.58/4.82 = A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4853_minus__equation__iff,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( ( uminus1351360451143612070nteger @ A )
% 8.58/4.82 = B )
% 8.58/4.82 = ( ( uminus1351360451143612070nteger @ B )
% 8.58/4.82 = A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4854_equation__minus__iff,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( A
% 8.58/4.82 = ( uminus_uminus_real @ B ) )
% 8.58/4.82 = ( B
% 8.58/4.82 = ( uminus_uminus_real @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4855_equation__minus__iff,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( A
% 8.58/4.82 = ( uminus_uminus_int @ B ) )
% 8.58/4.82 = ( B
% 8.58/4.82 = ( uminus_uminus_int @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4856_equation__minus__iff,axiom,
% 8.58/4.82 ! [A: complex,B: complex] :
% 8.58/4.82 ( ( A
% 8.58/4.82 = ( uminus1482373934393186551omplex @ B ) )
% 8.58/4.82 = ( B
% 8.58/4.82 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4857_equation__minus__iff,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( A
% 8.58/4.82 = ( uminus_uminus_rat @ B ) )
% 8.58/4.82 = ( B
% 8.58/4.82 = ( uminus_uminus_rat @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4858_equation__minus__iff,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( A
% 8.58/4.82 = ( uminus1351360451143612070nteger @ B ) )
% 8.58/4.82 = ( B
% 8.58/4.82 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4859_tanh__real__gt__neg1,axiom,
% 8.58/4.82 ! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4860_le__imp__neg__le,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( ord_less_eq_real @ A @ B )
% 8.58/4.82 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4861_le__imp__neg__le,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( ord_le3102999989581377725nteger @ A @ B )
% 8.58/4.82 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4862_le__imp__neg__le,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( ord_less_eq_rat @ A @ B )
% 8.58/4.82 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4863_le__imp__neg__le,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( ord_less_eq_int @ A @ B )
% 8.58/4.82 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4864_minus__le__iff,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 8.58/4.82 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4865_minus__le__iff,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 8.58/4.82 = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4866_minus__le__iff,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 8.58/4.82 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4867_minus__le__iff,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 8.58/4.82 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4868_le__minus__iff,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 8.58/4.82 = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4869_le__minus__iff,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 8.58/4.82 = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4870_le__minus__iff,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
% 8.58/4.82 = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4871_le__minus__iff,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 8.58/4.82 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4872_verit__negate__coefficient_I2_J,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( ord_less_real @ A @ B )
% 8.58/4.82 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4873_verit__negate__coefficient_I2_J,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( ord_less_int @ A @ B )
% 8.58/4.82 => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4874_verit__negate__coefficient_I2_J,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( ord_less_rat @ A @ B )
% 8.58/4.82 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4875_verit__negate__coefficient_I2_J,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( ord_le6747313008572928689nteger @ A @ B )
% 8.58/4.82 => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4876_minus__less__iff,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 8.58/4.82 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4877_minus__less__iff,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
% 8.58/4.82 = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4878_minus__less__iff,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
% 8.58/4.82 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4879_minus__less__iff,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 8.58/4.82 = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4880_less__minus__iff,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 8.58/4.82 = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4881_less__minus__iff,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
% 8.58/4.82 = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4882_less__minus__iff,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
% 8.58/4.82 = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4883_less__minus__iff,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 8.58/4.82 = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4884_numeral__neq__neg__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( numeral_numeral_real @ M )
% 8.58/4.82 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4885_numeral__neq__neg__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( numeral_numeral_int @ M )
% 8.58/4.82 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4886_numeral__neq__neg__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( numera6690914467698888265omplex @ M )
% 8.58/4.82 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4887_numeral__neq__neg__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( numeral_numeral_rat @ M )
% 8.58/4.82 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4888_numeral__neq__neg__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( numera6620942414471956472nteger @ M )
% 8.58/4.82 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4889_neg__numeral__neq__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 8.58/4.82 != ( numeral_numeral_real @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4890_neg__numeral__neq__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 8.58/4.82 != ( numeral_numeral_int @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4891_neg__numeral__neq__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 8.58/4.82 != ( numera6690914467698888265omplex @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4892_neg__numeral__neq__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 8.58/4.82 != ( numeral_numeral_rat @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4893_neg__numeral__neq__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 8.58/4.82 != ( numera6620942414471956472nteger @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4894_minus__mult__commute,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 8.58/4.82 = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4895_minus__mult__commute,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 8.58/4.82 = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4896_minus__mult__commute,axiom,
% 8.58/4.82 ! [A: complex,B: complex] :
% 8.58/4.82 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 8.58/4.82 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4897_minus__mult__commute,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 8.58/4.82 = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4898_minus__mult__commute,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 8.58/4.82 = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4899_square__eq__iff,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( ( times_times_real @ A @ A )
% 8.58/4.82 = ( times_times_real @ B @ B ) )
% 8.58/4.82 = ( ( A = B )
% 8.58/4.82 | ( A
% 8.58/4.82 = ( uminus_uminus_real @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4900_square__eq__iff,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( ( times_times_int @ A @ A )
% 8.58/4.82 = ( times_times_int @ B @ B ) )
% 8.58/4.82 = ( ( A = B )
% 8.58/4.82 | ( A
% 8.58/4.82 = ( uminus_uminus_int @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4901_square__eq__iff,axiom,
% 8.58/4.82 ! [A: complex,B: complex] :
% 8.58/4.82 ( ( ( times_times_complex @ A @ A )
% 8.58/4.82 = ( times_times_complex @ B @ B ) )
% 8.58/4.82 = ( ( A = B )
% 8.58/4.82 | ( A
% 8.58/4.82 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4902_square__eq__iff,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( ( times_times_rat @ A @ A )
% 8.58/4.82 = ( times_times_rat @ B @ B ) )
% 8.58/4.82 = ( ( A = B )
% 8.58/4.82 | ( A
% 8.58/4.82 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4903_square__eq__iff,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( ( times_3573771949741848930nteger @ A @ A )
% 8.58/4.82 = ( times_3573771949741848930nteger @ B @ B ) )
% 8.58/4.82 = ( ( A = B )
% 8.58/4.82 | ( A
% 8.58/4.82 = ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4904_one__neq__neg__one,axiom,
% 8.58/4.82 ( one_one_real
% 8.58/4.82 != ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4905_one__neq__neg__one,axiom,
% 8.58/4.82 ( one_one_int
% 8.58/4.82 != ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4906_one__neq__neg__one,axiom,
% 8.58/4.82 ( one_one_complex
% 8.58/4.82 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4907_one__neq__neg__one,axiom,
% 8.58/4.82 ( one_one_rat
% 8.58/4.82 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4908_one__neq__neg__one,axiom,
% 8.58/4.82 ( one_one_Code_integer
% 8.58/4.82 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4909_is__num__normalize_I8_J,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 8.58/4.82 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4910_is__num__normalize_I8_J,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 8.58/4.82 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4911_is__num__normalize_I8_J,axiom,
% 8.58/4.82 ! [A: complex,B: complex] :
% 8.58/4.82 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 8.58/4.82 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4912_is__num__normalize_I8_J,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 8.58/4.82 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4913_is__num__normalize_I8_J,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 8.58/4.82 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4914_add_Oinverse__distrib__swap,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 8.58/4.82 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4915_add_Oinverse__distrib__swap,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 8.58/4.82 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4916_add_Oinverse__distrib__swap,axiom,
% 8.58/4.82 ! [A: complex,B: complex] :
% 8.58/4.82 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 8.58/4.82 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4917_add_Oinverse__distrib__swap,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 8.58/4.82 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4918_add_Oinverse__distrib__swap,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 8.58/4.82 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4919_group__cancel_Oneg1,axiom,
% 8.58/4.82 ! [A3: real,K: real,A: real] :
% 8.58/4.82 ( ( A3
% 8.58/4.82 = ( plus_plus_real @ K @ A ) )
% 8.58/4.82 => ( ( uminus_uminus_real @ A3 )
% 8.58/4.82 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4920_group__cancel_Oneg1,axiom,
% 8.58/4.82 ! [A3: int,K: int,A: int] :
% 8.58/4.82 ( ( A3
% 8.58/4.82 = ( plus_plus_int @ K @ A ) )
% 8.58/4.82 => ( ( uminus_uminus_int @ A3 )
% 8.58/4.82 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4921_group__cancel_Oneg1,axiom,
% 8.58/4.82 ! [A3: complex,K: complex,A: complex] :
% 8.58/4.82 ( ( A3
% 8.58/4.82 = ( plus_plus_complex @ K @ A ) )
% 8.58/4.82 => ( ( uminus1482373934393186551omplex @ A3 )
% 8.58/4.82 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4922_group__cancel_Oneg1,axiom,
% 8.58/4.82 ! [A3: rat,K: rat,A: rat] :
% 8.58/4.82 ( ( A3
% 8.58/4.82 = ( plus_plus_rat @ K @ A ) )
% 8.58/4.82 => ( ( uminus_uminus_rat @ A3 )
% 8.58/4.82 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4923_group__cancel_Oneg1,axiom,
% 8.58/4.82 ! [A3: code_integer,K: code_integer,A: code_integer] :
% 8.58/4.82 ( ( A3
% 8.58/4.82 = ( plus_p5714425477246183910nteger @ K @ A ) )
% 8.58/4.82 => ( ( uminus1351360451143612070nteger @ A3 )
% 8.58/4.82 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4924_minus__diff__commute,axiom,
% 8.58/4.82 ! [B: real,A: real] :
% 8.58/4.82 ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 8.58/4.82 = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4925_minus__diff__commute,axiom,
% 8.58/4.82 ! [B: int,A: int] :
% 8.58/4.82 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 8.58/4.82 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4926_minus__diff__commute,axiom,
% 8.58/4.82 ! [B: complex,A: complex] :
% 8.58/4.82 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
% 8.58/4.82 = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4927_minus__diff__commute,axiom,
% 8.58/4.82 ! [B: rat,A: rat] :
% 8.58/4.82 ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
% 8.58/4.82 = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4928_minus__diff__commute,axiom,
% 8.58/4.82 ! [B: code_integer,A: code_integer] :
% 8.58/4.82 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
% 8.58/4.82 = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4929_minus__diff__minus,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 8.58/4.82 = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4930_minus__diff__minus,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 8.58/4.82 = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4931_minus__diff__minus,axiom,
% 8.58/4.82 ! [A: complex,B: complex] :
% 8.58/4.82 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4932_minus__diff__minus,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 8.58/4.82 = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4933_minus__diff__minus,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4934_minus__divide__left,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 8.58/4.82 = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4935_minus__divide__left,axiom,
% 8.58/4.82 ! [A: complex,B: complex] :
% 8.58/4.82 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.58/4.82 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4936_minus__divide__left,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 8.58/4.82 = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4937_minus__divide__divide,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 8.58/4.82 = ( divide_divide_real @ A @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4938_minus__divide__divide,axiom,
% 8.58/4.82 ! [A: complex,B: complex] :
% 8.58/4.82 ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 8.58/4.82 = ( divide1717551699836669952omplex @ A @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4939_minus__divide__divide,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 8.58/4.82 = ( divide_divide_rat @ A @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4940_minus__divide__right,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 8.58/4.82 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4941_minus__divide__right,axiom,
% 8.58/4.82 ! [A: complex,B: complex] :
% 8.58/4.82 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.58/4.82 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4942_minus__divide__right,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 8.58/4.82 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4943_div__minus__right,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 8.58/4.82 = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4944_div__minus__right,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 8.58/4.82 = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4945_mod__minus__right,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 8.58/4.82 = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4946_mod__minus__right,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4947_mod__minus__cong,axiom,
% 8.58/4.82 ! [A: int,B: int,A5: int] :
% 8.58/4.82 ( ( ( modulo_modulo_int @ A @ B )
% 8.58/4.82 = ( modulo_modulo_int @ A5 @ B ) )
% 8.58/4.82 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 8.58/4.82 = ( modulo_modulo_int @ ( uminus_uminus_int @ A5 ) @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4948_mod__minus__cong,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer,A5: code_integer] :
% 8.58/4.82 ( ( ( modulo364778990260209775nteger @ A @ B )
% 8.58/4.82 = ( modulo364778990260209775nteger @ A5 @ B ) )
% 8.58/4.82 => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 8.58/4.82 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A5 ) @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4949_mod__minus__eq,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
% 8.58/4.82 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4950_mod__minus__eq,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
% 8.58/4.82 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4951_push__bit__minus,axiom,
% 8.58/4.82 ! [N: nat,A: int] :
% 8.58/4.82 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ A ) )
% 8.58/4.82 = ( uminus_uminus_int @ ( bit_se545348938243370406it_int @ N @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4952_push__bit__minus,axiom,
% 8.58/4.82 ! [N: nat,A: code_integer] :
% 8.58/4.82 ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ A ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( bit_se7788150548672797655nteger @ N @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4953_sgn__real__def,axiom,
% 8.58/4.82 ( sgn_sgn_real
% 8.58/4.82 = ( ^ [A2: real] : ( if_real @ ( A2 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A2 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4954_ceiling__mono,axiom,
% 8.58/4.82 ! [Y: real,X: real] :
% 8.58/4.82 ( ( ord_less_eq_real @ Y @ X )
% 8.58/4.82 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4955_ceiling__mono,axiom,
% 8.58/4.82 ! [Y: rat,X: rat] :
% 8.58/4.82 ( ( ord_less_eq_rat @ Y @ X )
% 8.58/4.82 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4956_ceiling__less__cancel,axiom,
% 8.58/4.82 ! [X: rat,Y: rat] :
% 8.58/4.82 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) )
% 8.58/4.82 => ( ord_less_rat @ X @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4957_ceiling__less__cancel,axiom,
% 8.58/4.82 ! [X: real,Y: real] :
% 8.58/4.82 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
% 8.58/4.82 => ( ord_less_real @ X @ Y ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4958_not__numeral__le__neg__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4959_not__numeral__le__neg__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4960_not__numeral__le__neg__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4961_not__numeral__le__neg__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4962_neg__numeral__le__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4963_neg__numeral__le__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4964_neg__numeral__le__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4965_neg__numeral__le__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4966_zero__neq__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( zero_zero_real
% 8.58/4.82 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4967_zero__neq__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( zero_zero_int
% 8.58/4.82 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4968_zero__neq__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( zero_zero_complex
% 8.58/4.82 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4969_zero__neq__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( zero_zero_rat
% 8.58/4.82 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4970_zero__neq__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( zero_z3403309356797280102nteger
% 8.58/4.82 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4971_not__numeral__less__neg__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4972_not__numeral__less__neg__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4973_not__numeral__less__neg__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4974_not__numeral__less__neg__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] :
% 8.58/4.82 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4975_neg__numeral__less__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4976_neg__numeral__less__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4977_neg__numeral__less__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4978_neg__numeral__less__numeral,axiom,
% 8.58/4.82 ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4979_le__minus__one__simps_I2_J,axiom,
% 8.58/4.82 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4980_le__minus__one__simps_I2_J,axiom,
% 8.58/4.82 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4981_le__minus__one__simps_I2_J,axiom,
% 8.58/4.82 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4982_le__minus__one__simps_I2_J,axiom,
% 8.58/4.82 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4983_le__minus__one__simps_I4_J,axiom,
% 8.58/4.82 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4984_le__minus__one__simps_I4_J,axiom,
% 8.58/4.82 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4985_le__minus__one__simps_I4_J,axiom,
% 8.58/4.82 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4986_le__minus__one__simps_I4_J,axiom,
% 8.58/4.82 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4987_zero__neq__neg__one,axiom,
% 8.58/4.82 ( zero_zero_real
% 8.58/4.82 != ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4988_zero__neq__neg__one,axiom,
% 8.58/4.82 ( zero_zero_int
% 8.58/4.82 != ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4989_zero__neq__neg__one,axiom,
% 8.58/4.82 ( zero_zero_complex
% 8.58/4.82 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4990_zero__neq__neg__one,axiom,
% 8.58/4.82 ( zero_zero_rat
% 8.58/4.82 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4991_zero__neq__neg__one,axiom,
% 8.58/4.82 ( zero_z3403309356797280102nteger
% 8.58/4.82 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4992_neg__eq__iff__add__eq__0,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( ( uminus_uminus_real @ A )
% 8.58/4.82 = B )
% 8.58/4.82 = ( ( plus_plus_real @ A @ B )
% 8.58/4.82 = zero_zero_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4993_neg__eq__iff__add__eq__0,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( ( uminus_uminus_int @ A )
% 8.58/4.82 = B )
% 8.58/4.82 = ( ( plus_plus_int @ A @ B )
% 8.58/4.82 = zero_zero_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4994_neg__eq__iff__add__eq__0,axiom,
% 8.58/4.82 ! [A: complex,B: complex] :
% 8.58/4.82 ( ( ( uminus1482373934393186551omplex @ A )
% 8.58/4.82 = B )
% 8.58/4.82 = ( ( plus_plus_complex @ A @ B )
% 8.58/4.82 = zero_zero_complex ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4995_neg__eq__iff__add__eq__0,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( ( uminus_uminus_rat @ A )
% 8.58/4.82 = B )
% 8.58/4.82 = ( ( plus_plus_rat @ A @ B )
% 8.58/4.82 = zero_zero_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4996_neg__eq__iff__add__eq__0,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( ( uminus1351360451143612070nteger @ A )
% 8.58/4.82 = B )
% 8.58/4.82 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 8.58/4.82 = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4997_eq__neg__iff__add__eq__0,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( A
% 8.58/4.82 = ( uminus_uminus_real @ B ) )
% 8.58/4.82 = ( ( plus_plus_real @ A @ B )
% 8.58/4.82 = zero_zero_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4998_eq__neg__iff__add__eq__0,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( A
% 8.58/4.82 = ( uminus_uminus_int @ B ) )
% 8.58/4.82 = ( ( plus_plus_int @ A @ B )
% 8.58/4.82 = zero_zero_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_4999_eq__neg__iff__add__eq__0,axiom,
% 8.58/4.82 ! [A: complex,B: complex] :
% 8.58/4.82 ( ( A
% 8.58/4.82 = ( uminus1482373934393186551omplex @ B ) )
% 8.58/4.82 = ( ( plus_plus_complex @ A @ B )
% 8.58/4.82 = zero_zero_complex ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5000_eq__neg__iff__add__eq__0,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( A
% 8.58/4.82 = ( uminus_uminus_rat @ B ) )
% 8.58/4.82 = ( ( plus_plus_rat @ A @ B )
% 8.58/4.82 = zero_zero_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5001_eq__neg__iff__add__eq__0,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( A
% 8.58/4.82 = ( uminus1351360451143612070nteger @ B ) )
% 8.58/4.82 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 8.58/4.82 = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5002_add_Oinverse__unique,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( ( plus_plus_real @ A @ B )
% 8.58/4.82 = zero_zero_real )
% 8.58/4.82 => ( ( uminus_uminus_real @ A )
% 8.58/4.82 = B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5003_add_Oinverse__unique,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( ( plus_plus_int @ A @ B )
% 8.58/4.82 = zero_zero_int )
% 8.58/4.82 => ( ( uminus_uminus_int @ A )
% 8.58/4.82 = B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5004_add_Oinverse__unique,axiom,
% 8.58/4.82 ! [A: complex,B: complex] :
% 8.58/4.82 ( ( ( plus_plus_complex @ A @ B )
% 8.58/4.82 = zero_zero_complex )
% 8.58/4.82 => ( ( uminus1482373934393186551omplex @ A )
% 8.58/4.82 = B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5005_add_Oinverse__unique,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( ( plus_plus_rat @ A @ B )
% 8.58/4.82 = zero_zero_rat )
% 8.58/4.82 => ( ( uminus_uminus_rat @ A )
% 8.58/4.82 = B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5006_add_Oinverse__unique,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 8.58/4.82 = zero_z3403309356797280102nteger )
% 8.58/4.82 => ( ( uminus1351360451143612070nteger @ A )
% 8.58/4.82 = B ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5007_ab__group__add__class_Oab__left__minus,axiom,
% 8.58/4.82 ! [A: real] :
% 8.58/4.82 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 8.58/4.82 = zero_zero_real ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5008_ab__group__add__class_Oab__left__minus,axiom,
% 8.58/4.82 ! [A: int] :
% 8.58/4.82 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 8.58/4.82 = zero_zero_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5009_ab__group__add__class_Oab__left__minus,axiom,
% 8.58/4.82 ! [A: complex] :
% 8.58/4.82 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 8.58/4.82 = zero_zero_complex ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5010_ab__group__add__class_Oab__left__minus,axiom,
% 8.58/4.82 ! [A: rat] :
% 8.58/4.82 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 8.58/4.82 = zero_zero_rat ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5011_ab__group__add__class_Oab__left__minus,axiom,
% 8.58/4.82 ! [A: code_integer] :
% 8.58/4.82 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 8.58/4.82 = zero_z3403309356797280102nteger ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5012_add__eq__0__iff,axiom,
% 8.58/4.82 ! [A: real,B: real] :
% 8.58/4.82 ( ( ( plus_plus_real @ A @ B )
% 8.58/4.82 = zero_zero_real )
% 8.58/4.82 = ( B
% 8.58/4.82 = ( uminus_uminus_real @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5013_add__eq__0__iff,axiom,
% 8.58/4.82 ! [A: int,B: int] :
% 8.58/4.82 ( ( ( plus_plus_int @ A @ B )
% 8.58/4.82 = zero_zero_int )
% 8.58/4.82 = ( B
% 8.58/4.82 = ( uminus_uminus_int @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5014_add__eq__0__iff,axiom,
% 8.58/4.82 ! [A: complex,B: complex] :
% 8.58/4.82 ( ( ( plus_plus_complex @ A @ B )
% 8.58/4.82 = zero_zero_complex )
% 8.58/4.82 = ( B
% 8.58/4.82 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5015_add__eq__0__iff,axiom,
% 8.58/4.82 ! [A: rat,B: rat] :
% 8.58/4.82 ( ( ( plus_plus_rat @ A @ B )
% 8.58/4.82 = zero_zero_rat )
% 8.58/4.82 = ( B
% 8.58/4.82 = ( uminus_uminus_rat @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5016_add__eq__0__iff,axiom,
% 8.58/4.82 ! [A: code_integer,B: code_integer] :
% 8.58/4.82 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 8.58/4.82 = zero_z3403309356797280102nteger )
% 8.58/4.82 = ( B
% 8.58/4.82 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5017_less__minus__one__simps_I4_J,axiom,
% 8.58/4.82 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5018_less__minus__one__simps_I4_J,axiom,
% 8.58/4.82 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5019_less__minus__one__simps_I4_J,axiom,
% 8.58/4.82 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5020_less__minus__one__simps_I4_J,axiom,
% 8.58/4.82 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5021_less__minus__one__simps_I2_J,axiom,
% 8.58/4.82 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5022_less__minus__one__simps_I2_J,axiom,
% 8.58/4.82 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5023_less__minus__one__simps_I2_J,axiom,
% 8.58/4.82 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5024_less__minus__one__simps_I2_J,axiom,
% 8.58/4.82 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5025_numeral__times__minus__swap,axiom,
% 8.58/4.82 ! [W: num,X: real] :
% 8.58/4.82 ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
% 8.58/4.82 = ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5026_numeral__times__minus__swap,axiom,
% 8.58/4.82 ! [W: num,X: int] :
% 8.58/4.82 ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
% 8.58/4.82 = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5027_numeral__times__minus__swap,axiom,
% 8.58/4.82 ! [W: num,X: complex] :
% 8.58/4.82 ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
% 8.58/4.82 = ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5028_numeral__times__minus__swap,axiom,
% 8.58/4.82 ! [W: num,X: rat] :
% 8.58/4.82 ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
% 8.58/4.82 = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5029_numeral__times__minus__swap,axiom,
% 8.58/4.82 ! [W: num,X: code_integer] :
% 8.58/4.82 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
% 8.58/4.82 = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5030_nonzero__minus__divide__divide,axiom,
% 8.58/4.82 ! [B: real,A: real] :
% 8.58/4.82 ( ( B != zero_zero_real )
% 8.58/4.82 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 8.58/4.82 = ( divide_divide_real @ A @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5031_nonzero__minus__divide__divide,axiom,
% 8.58/4.82 ! [B: complex,A: complex] :
% 8.58/4.82 ( ( B != zero_zero_complex )
% 8.58/4.82 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 8.58/4.82 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5032_nonzero__minus__divide__divide,axiom,
% 8.58/4.82 ! [B: rat,A: rat] :
% 8.58/4.82 ( ( B != zero_zero_rat )
% 8.58/4.82 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 8.58/4.82 = ( divide_divide_rat @ A @ B ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5033_nonzero__minus__divide__right,axiom,
% 8.58/4.82 ! [B: real,A: real] :
% 8.58/4.82 ( ( B != zero_zero_real )
% 8.58/4.82 => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 8.58/4.82 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5034_nonzero__minus__divide__right,axiom,
% 8.58/4.82 ! [B: complex,A: complex] :
% 8.58/4.82 ( ( B != zero_zero_complex )
% 8.58/4.82 => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.58/4.82 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5035_nonzero__minus__divide__right,axiom,
% 8.58/4.82 ! [B: rat,A: rat] :
% 8.58/4.82 ( ( B != zero_zero_rat )
% 8.58/4.82 => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 8.58/4.82 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5036_numeral__neq__neg__one,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( numeral_numeral_real @ N )
% 8.58/4.82 != ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5037_numeral__neq__neg__one,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( numeral_numeral_int @ N )
% 8.58/4.82 != ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5038_numeral__neq__neg__one,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( numera6690914467698888265omplex @ N )
% 8.58/4.82 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5039_numeral__neq__neg__one,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( numeral_numeral_rat @ N )
% 8.58/4.82 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5040_numeral__neq__neg__one,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( ( numera6620942414471956472nteger @ N )
% 8.58/4.82 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5041_one__neq__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( one_one_real
% 8.58/4.82 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5042_one__neq__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( one_one_int
% 8.58/4.82 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5043_one__neq__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( one_one_complex
% 8.58/4.82 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5044_one__neq__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( one_one_rat
% 8.58/4.82 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5045_one__neq__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ( one_one_Code_integer
% 8.58/4.82 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5046_square__eq__1__iff,axiom,
% 8.58/4.82 ! [X: real] :
% 8.58/4.82 ( ( ( times_times_real @ X @ X )
% 8.58/4.82 = one_one_real )
% 8.58/4.82 = ( ( X = one_one_real )
% 8.58/4.82 | ( X
% 8.58/4.82 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5047_square__eq__1__iff,axiom,
% 8.58/4.82 ! [X: int] :
% 8.58/4.82 ( ( ( times_times_int @ X @ X )
% 8.58/4.82 = one_one_int )
% 8.58/4.82 = ( ( X = one_one_int )
% 8.58/4.82 | ( X
% 8.58/4.82 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5048_square__eq__1__iff,axiom,
% 8.58/4.82 ! [X: complex] :
% 8.58/4.82 ( ( ( times_times_complex @ X @ X )
% 8.58/4.82 = one_one_complex )
% 8.58/4.82 = ( ( X = one_one_complex )
% 8.58/4.82 | ( X
% 8.58/4.82 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5049_square__eq__1__iff,axiom,
% 8.58/4.82 ! [X: rat] :
% 8.58/4.82 ( ( ( times_times_rat @ X @ X )
% 8.58/4.82 = one_one_rat )
% 8.58/4.82 = ( ( X = one_one_rat )
% 8.58/4.82 | ( X
% 8.58/4.82 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5050_square__eq__1__iff,axiom,
% 8.58/4.82 ! [X: code_integer] :
% 8.58/4.82 ( ( ( times_3573771949741848930nteger @ X @ X )
% 8.58/4.82 = one_one_Code_integer )
% 8.58/4.82 = ( ( X = one_one_Code_integer )
% 8.58/4.82 | ( X
% 8.58/4.82 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5051_group__cancel_Osub2,axiom,
% 8.58/4.82 ! [B5: real,K: real,B: real,A: real] :
% 8.58/4.82 ( ( B5
% 8.58/4.82 = ( plus_plus_real @ K @ B ) )
% 8.58/4.82 => ( ( minus_minus_real @ A @ B5 )
% 8.58/4.82 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5052_group__cancel_Osub2,axiom,
% 8.58/4.82 ! [B5: int,K: int,B: int,A: int] :
% 8.58/4.82 ( ( B5
% 8.58/4.82 = ( plus_plus_int @ K @ B ) )
% 8.58/4.82 => ( ( minus_minus_int @ A @ B5 )
% 8.58/4.82 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5053_group__cancel_Osub2,axiom,
% 8.58/4.82 ! [B5: complex,K: complex,B: complex,A: complex] :
% 8.58/4.82 ( ( B5
% 8.58/4.82 = ( plus_plus_complex @ K @ B ) )
% 8.58/4.82 => ( ( minus_minus_complex @ A @ B5 )
% 8.58/4.82 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5054_group__cancel_Osub2,axiom,
% 8.58/4.82 ! [B5: rat,K: rat,B: rat,A: rat] :
% 8.58/4.82 ( ( B5
% 8.58/4.82 = ( plus_plus_rat @ K @ B ) )
% 8.58/4.82 => ( ( minus_minus_rat @ A @ B5 )
% 8.58/4.82 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5055_group__cancel_Osub2,axiom,
% 8.58/4.82 ! [B5: code_integer,K: code_integer,B: code_integer,A: code_integer] :
% 8.58/4.82 ( ( B5
% 8.58/4.82 = ( plus_p5714425477246183910nteger @ K @ B ) )
% 8.58/4.82 => ( ( minus_8373710615458151222nteger @ A @ B5 )
% 8.58/4.82 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5056_diff__conv__add__uminus,axiom,
% 8.58/4.82 ( minus_minus_real
% 8.58/4.82 = ( ^ [A2: real,B2: real] : ( plus_plus_real @ A2 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5057_diff__conv__add__uminus,axiom,
% 8.58/4.82 ( minus_minus_int
% 8.58/4.82 = ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5058_diff__conv__add__uminus,axiom,
% 8.58/4.82 ( minus_minus_complex
% 8.58/4.82 = ( ^ [A2: complex,B2: complex] : ( plus_plus_complex @ A2 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5059_diff__conv__add__uminus,axiom,
% 8.58/4.82 ( minus_minus_rat
% 8.58/4.82 = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5060_diff__conv__add__uminus,axiom,
% 8.58/4.82 ( minus_8373710615458151222nteger
% 8.58/4.82 = ( ^ [A2: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5061_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 8.58/4.82 ( minus_minus_real
% 8.58/4.82 = ( ^ [A2: real,B2: real] : ( plus_plus_real @ A2 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5062_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 8.58/4.82 ( minus_minus_int
% 8.58/4.82 = ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5063_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 8.58/4.82 ( minus_minus_complex
% 8.58/4.82 = ( ^ [A2: complex,B2: complex] : ( plus_plus_complex @ A2 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5064_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 8.58/4.82 ( minus_minus_rat
% 8.58/4.82 = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5065_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 8.58/4.82 ( minus_8373710615458151222nteger
% 8.58/4.82 = ( ^ [A2: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5066_dvd__neg__div,axiom,
% 8.58/4.82 ! [B: real,A: real] :
% 8.58/4.82 ( ( dvd_dvd_real @ B @ A )
% 8.58/4.82 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
% 8.58/4.82 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5067_dvd__neg__div,axiom,
% 8.58/4.82 ! [B: int,A: int] :
% 8.58/4.82 ( ( dvd_dvd_int @ B @ A )
% 8.58/4.82 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 8.58/4.82 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5068_dvd__neg__div,axiom,
% 8.58/4.82 ! [B: complex,A: complex] :
% 8.58/4.82 ( ( dvd_dvd_complex @ B @ A )
% 8.58/4.82 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5069_dvd__neg__div,axiom,
% 8.58/4.82 ! [B: rat,A: rat] :
% 8.58/4.82 ( ( dvd_dvd_rat @ B @ A )
% 8.58/4.82 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
% 8.58/4.82 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5070_dvd__neg__div,axiom,
% 8.58/4.82 ! [B: code_integer,A: code_integer] :
% 8.58/4.82 ( ( dvd_dvd_Code_integer @ B @ A )
% 8.58/4.82 => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5071_dvd__div__neg,axiom,
% 8.58/4.82 ! [B: real,A: real] :
% 8.58/4.82 ( ( dvd_dvd_real @ B @ A )
% 8.58/4.82 => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
% 8.58/4.82 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5072_dvd__div__neg,axiom,
% 8.58/4.82 ! [B: int,A: int] :
% 8.58/4.82 ( ( dvd_dvd_int @ B @ A )
% 8.58/4.82 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 8.58/4.82 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5073_dvd__div__neg,axiom,
% 8.58/4.82 ! [B: complex,A: complex] :
% 8.58/4.82 ( ( dvd_dvd_complex @ B @ A )
% 8.58/4.82 => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5074_dvd__div__neg,axiom,
% 8.58/4.82 ! [B: rat,A: rat] :
% 8.58/4.82 ( ( dvd_dvd_rat @ B @ A )
% 8.58/4.82 => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
% 8.58/4.82 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5075_dvd__div__neg,axiom,
% 8.58/4.82 ! [B: code_integer,A: code_integer] :
% 8.58/4.82 ( ( dvd_dvd_Code_integer @ B @ A )
% 8.58/4.82 => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5076_sgn__not__eq__imp,axiom,
% 8.58/4.82 ! [B: real,A: real] :
% 8.58/4.82 ( ( ( sgn_sgn_real @ B )
% 8.58/4.82 != ( sgn_sgn_real @ A ) )
% 8.58/4.82 => ( ( ( sgn_sgn_real @ A )
% 8.58/4.82 != zero_zero_real )
% 8.58/4.82 => ( ( ( sgn_sgn_real @ B )
% 8.58/4.82 != zero_zero_real )
% 8.58/4.82 => ( ( sgn_sgn_real @ A )
% 8.58/4.82 = ( uminus_uminus_real @ ( sgn_sgn_real @ B ) ) ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5077_sgn__not__eq__imp,axiom,
% 8.58/4.82 ! [B: int,A: int] :
% 8.58/4.82 ( ( ( sgn_sgn_int @ B )
% 8.58/4.82 != ( sgn_sgn_int @ A ) )
% 8.58/4.82 => ( ( ( sgn_sgn_int @ A )
% 8.58/4.82 != zero_zero_int )
% 8.58/4.82 => ( ( ( sgn_sgn_int @ B )
% 8.58/4.82 != zero_zero_int )
% 8.58/4.82 => ( ( sgn_sgn_int @ A )
% 8.58/4.82 = ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5078_sgn__not__eq__imp,axiom,
% 8.58/4.82 ! [B: rat,A: rat] :
% 8.58/4.82 ( ( ( sgn_sgn_rat @ B )
% 8.58/4.82 != ( sgn_sgn_rat @ A ) )
% 8.58/4.82 => ( ( ( sgn_sgn_rat @ A )
% 8.58/4.82 != zero_zero_rat )
% 8.58/4.82 => ( ( ( sgn_sgn_rat @ B )
% 8.58/4.82 != zero_zero_rat )
% 8.58/4.82 => ( ( sgn_sgn_rat @ A )
% 8.58/4.82 = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B ) ) ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5079_sgn__not__eq__imp,axiom,
% 8.58/4.82 ! [B: code_integer,A: code_integer] :
% 8.58/4.82 ( ( ( sgn_sgn_Code_integer @ B )
% 8.58/4.82 != ( sgn_sgn_Code_integer @ A ) )
% 8.58/4.82 => ( ( ( sgn_sgn_Code_integer @ A )
% 8.58/4.82 != zero_z3403309356797280102nteger )
% 8.58/4.82 => ( ( ( sgn_sgn_Code_integer @ B )
% 8.58/4.82 != zero_z3403309356797280102nteger )
% 8.58/4.82 => ( ( sgn_sgn_Code_integer @ A )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B ) ) ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5080_sgn__minus__1,axiom,
% 8.58/4.82 ( ( sgn_sgn_real @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.82 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5081_sgn__minus__1,axiom,
% 8.58/4.82 ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.82 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5082_sgn__minus__1,axiom,
% 8.58/4.82 ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.82 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5083_sgn__minus__1,axiom,
% 8.58/4.82 ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.82 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5084_sgn__minus__1,axiom,
% 8.58/4.82 ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.82 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5085_real__minus__mult__self__le,axiom,
% 8.58/4.82 ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5086_minus__real__def,axiom,
% 8.58/4.82 ( minus_minus_real
% 8.58/4.82 = ( ^ [X3: real,Y6: real] : ( plus_plus_real @ X3 @ ( uminus_uminus_real @ Y6 ) ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5087_tanh__real__lt__1,axiom,
% 8.58/4.82 ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5088_ceiling__add__le,axiom,
% 8.58/4.82 ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5089_ceiling__add__le,axiom,
% 8.58/4.82 ! [X: real,Y: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5090_not__zero__le__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5091_not__zero__le__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5092_not__zero__le__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5093_not__zero__le__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5094_neg__numeral__le__zero,axiom,
% 8.58/4.82 ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5095_neg__numeral__le__zero,axiom,
% 8.58/4.82 ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5096_neg__numeral__le__zero,axiom,
% 8.58/4.82 ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5097_neg__numeral__le__zero,axiom,
% 8.58/4.82 ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5098_not__zero__less__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5099_not__zero__less__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5100_not__zero__less__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5101_not__zero__less__neg__numeral,axiom,
% 8.58/4.82 ! [N: num] :
% 8.58/4.82 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5102_neg__numeral__less__zero,axiom,
% 8.58/4.82 ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5103_neg__numeral__less__zero,axiom,
% 8.58/4.82 ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5104_neg__numeral__less__zero,axiom,
% 8.58/4.82 ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5105_neg__numeral__less__zero,axiom,
% 8.58/4.82 ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5106_le__minus__one__simps_I1_J,axiom,
% 8.58/4.82 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5107_le__minus__one__simps_I1_J,axiom,
% 8.58/4.82 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5108_le__minus__one__simps_I1_J,axiom,
% 8.58/4.82 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5109_le__minus__one__simps_I1_J,axiom,
% 8.58/4.82 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5110_le__minus__one__simps_I3_J,axiom,
% 8.58/4.82 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5111_le__minus__one__simps_I3_J,axiom,
% 8.58/4.82 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5112_le__minus__one__simps_I3_J,axiom,
% 8.58/4.82 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5113_le__minus__one__simps_I3_J,axiom,
% 8.58/4.82 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5114_less__minus__one__simps_I1_J,axiom,
% 8.58/4.82 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5115_less__minus__one__simps_I1_J,axiom,
% 8.58/4.82 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5116_less__minus__one__simps_I1_J,axiom,
% 8.58/4.82 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5117_less__minus__one__simps_I1_J,axiom,
% 8.58/4.82 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5118_less__minus__one__simps_I3_J,axiom,
% 8.58/4.82 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5119_less__minus__one__simps_I3_J,axiom,
% 8.58/4.82 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5120_less__minus__one__simps_I3_J,axiom,
% 8.58/4.82 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5121_less__minus__one__simps_I3_J,axiom,
% 8.58/4.82 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5122_neg__numeral__le__one,axiom,
% 8.58/4.82 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5123_neg__numeral__le__one,axiom,
% 8.58/4.82 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5124_neg__numeral__le__one,axiom,
% 8.58/4.82 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5125_neg__numeral__le__one,axiom,
% 8.58/4.82 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5126_neg__one__le__numeral,axiom,
% 8.58/4.82 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5127_neg__one__le__numeral,axiom,
% 8.58/4.82 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5128_neg__one__le__numeral,axiom,
% 8.58/4.82 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5129_neg__one__le__numeral,axiom,
% 8.58/4.82 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5130_neg__numeral__le__neg__one,axiom,
% 8.58/4.82 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5131_neg__numeral__le__neg__one,axiom,
% 8.58/4.82 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5132_neg__numeral__le__neg__one,axiom,
% 8.58/4.82 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5133_neg__numeral__le__neg__one,axiom,
% 8.58/4.82 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5134_not__numeral__le__neg__one,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5135_not__numeral__le__neg__one,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5136_not__numeral__le__neg__one,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5137_not__numeral__le__neg__one,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5138_not__one__le__neg__numeral,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5139_not__one__le__neg__numeral,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5140_not__one__le__neg__numeral,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5141_not__one__le__neg__numeral,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5142_neg__numeral__less__one,axiom,
% 8.58/4.82 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5143_neg__numeral__less__one,axiom,
% 8.58/4.82 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5144_neg__numeral__less__one,axiom,
% 8.58/4.82 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5145_neg__numeral__less__one,axiom,
% 8.58/4.82 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5146_neg__one__less__numeral,axiom,
% 8.58/4.82 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5147_neg__one__less__numeral,axiom,
% 8.58/4.82 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5148_neg__one__less__numeral,axiom,
% 8.58/4.82 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5149_neg__one__less__numeral,axiom,
% 8.58/4.82 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5150_not__numeral__less__neg__one,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5151_not__numeral__less__neg__one,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5152_not__numeral__less__neg__one,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5153_not__numeral__less__neg__one,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5154_not__one__less__neg__numeral,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5155_not__one__less__neg__numeral,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5156_not__one__less__neg__numeral,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5157_not__one__less__neg__numeral,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5158_not__neg__one__less__neg__numeral,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5159_not__neg__one__less__neg__numeral,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 8.58/4.82
% 8.58/4.82 thf(fact_5160_not__neg__one__less__neg__numeral,axiom,
% 8.58/4.82 ! [M: num] :
% 8.58/4.82 ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5161_not__neg__one__less__neg__numeral,axiom,
% 8.58/4.83 ! [M: num] :
% 8.58/4.83 ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5162_nonzero__neg__divide__eq__eq2,axiom,
% 8.58/4.83 ! [B: real,C: real,A: real] :
% 8.58/4.83 ( ( B != zero_zero_real )
% 8.58/4.83 => ( ( C
% 8.58/4.83 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
% 8.58/4.83 = ( ( times_times_real @ C @ B )
% 8.58/4.83 = ( uminus_uminus_real @ A ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5163_nonzero__neg__divide__eq__eq2,axiom,
% 8.58/4.83 ! [B: complex,C: complex,A: complex] :
% 8.58/4.83 ( ( B != zero_zero_complex )
% 8.58/4.83 => ( ( C
% 8.58/4.83 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 8.58/4.83 = ( ( times_times_complex @ C @ B )
% 8.58/4.83 = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5164_nonzero__neg__divide__eq__eq2,axiom,
% 8.58/4.83 ! [B: rat,C: rat,A: rat] :
% 8.58/4.83 ( ( B != zero_zero_rat )
% 8.58/4.83 => ( ( C
% 8.58/4.83 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
% 8.58/4.83 = ( ( times_times_rat @ C @ B )
% 8.58/4.83 = ( uminus_uminus_rat @ A ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5165_nonzero__neg__divide__eq__eq,axiom,
% 8.58/4.83 ! [B: real,A: real,C: real] :
% 8.58/4.83 ( ( B != zero_zero_real )
% 8.58/4.83 => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 8.58/4.83 = C )
% 8.58/4.83 = ( ( uminus_uminus_real @ A )
% 8.58/4.83 = ( times_times_real @ C @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5166_nonzero__neg__divide__eq__eq,axiom,
% 8.58/4.83 ! [B: complex,A: complex,C: complex] :
% 8.58/4.83 ( ( B != zero_zero_complex )
% 8.58/4.83 => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.58/4.83 = C )
% 8.58/4.83 = ( ( uminus1482373934393186551omplex @ A )
% 8.58/4.83 = ( times_times_complex @ C @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5167_nonzero__neg__divide__eq__eq,axiom,
% 8.58/4.83 ! [B: rat,A: rat,C: rat] :
% 8.58/4.83 ( ( B != zero_zero_rat )
% 8.58/4.83 => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 8.58/4.83 = C )
% 8.58/4.83 = ( ( uminus_uminus_rat @ A )
% 8.58/4.83 = ( times_times_rat @ C @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5168_minus__divide__eq__eq,axiom,
% 8.58/4.83 ! [B: real,C: real,A: real] :
% 8.58/4.83 ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
% 8.58/4.83 = A )
% 8.58/4.83 = ( ( ( C != zero_zero_real )
% 8.58/4.83 => ( ( uminus_uminus_real @ B )
% 8.58/4.83 = ( times_times_real @ A @ C ) ) )
% 8.58/4.83 & ( ( C = zero_zero_real )
% 8.58/4.83 => ( A = zero_zero_real ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5169_minus__divide__eq__eq,axiom,
% 8.58/4.83 ! [B: complex,C: complex,A: complex] :
% 8.58/4.83 ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
% 8.58/4.83 = A )
% 8.58/4.83 = ( ( ( C != zero_zero_complex )
% 8.58/4.83 => ( ( uminus1482373934393186551omplex @ B )
% 8.58/4.83 = ( times_times_complex @ A @ C ) ) )
% 8.58/4.83 & ( ( C = zero_zero_complex )
% 8.58/4.83 => ( A = zero_zero_complex ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5170_minus__divide__eq__eq,axiom,
% 8.58/4.83 ! [B: rat,C: rat,A: rat] :
% 8.58/4.83 ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
% 8.58/4.83 = A )
% 8.58/4.83 = ( ( ( C != zero_zero_rat )
% 8.58/4.83 => ( ( uminus_uminus_rat @ B )
% 8.58/4.83 = ( times_times_rat @ A @ C ) ) )
% 8.58/4.83 & ( ( C = zero_zero_rat )
% 8.58/4.83 => ( A = zero_zero_rat ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5171_eq__minus__divide__eq,axiom,
% 8.58/4.83 ! [A: real,B: real,C: real] :
% 8.58/4.83 ( ( A
% 8.58/4.83 = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 8.58/4.83 = ( ( ( C != zero_zero_real )
% 8.58/4.83 => ( ( times_times_real @ A @ C )
% 8.58/4.83 = ( uminus_uminus_real @ B ) ) )
% 8.58/4.83 & ( ( C = zero_zero_real )
% 8.58/4.83 => ( A = zero_zero_real ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5172_eq__minus__divide__eq,axiom,
% 8.58/4.83 ! [A: complex,B: complex,C: complex] :
% 8.58/4.83 ( ( A
% 8.58/4.83 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
% 8.58/4.83 = ( ( ( C != zero_zero_complex )
% 8.58/4.83 => ( ( times_times_complex @ A @ C )
% 8.58/4.83 = ( uminus1482373934393186551omplex @ B ) ) )
% 8.58/4.83 & ( ( C = zero_zero_complex )
% 8.58/4.83 => ( A = zero_zero_complex ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5173_eq__minus__divide__eq,axiom,
% 8.58/4.83 ! [A: rat,B: rat,C: rat] :
% 8.58/4.83 ( ( A
% 8.58/4.83 = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 8.58/4.83 = ( ( ( C != zero_zero_rat )
% 8.58/4.83 => ( ( times_times_rat @ A @ C )
% 8.58/4.83 = ( uminus_uminus_rat @ B ) ) )
% 8.58/4.83 & ( ( C = zero_zero_rat )
% 8.58/4.83 => ( A = zero_zero_rat ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5174_divide__eq__minus__1__iff,axiom,
% 8.58/4.83 ! [A: real,B: real] :
% 8.58/4.83 ( ( ( divide_divide_real @ A @ B )
% 8.58/4.83 = ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.83 = ( ( B != zero_zero_real )
% 8.58/4.83 & ( A
% 8.58/4.83 = ( uminus_uminus_real @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5175_divide__eq__minus__1__iff,axiom,
% 8.58/4.83 ! [A: complex,B: complex] :
% 8.58/4.83 ( ( ( divide1717551699836669952omplex @ A @ B )
% 8.58/4.83 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.83 = ( ( B != zero_zero_complex )
% 8.58/4.83 & ( A
% 8.58/4.83 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5176_divide__eq__minus__1__iff,axiom,
% 8.58/4.83 ! [A: rat,B: rat] :
% 8.58/4.83 ( ( ( divide_divide_rat @ A @ B )
% 8.58/4.83 = ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.83 = ( ( B != zero_zero_rat )
% 8.58/4.83 & ( A
% 8.58/4.83 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5177_mult__1s__ring__1_I1_J,axiom,
% 8.58/4.83 ! [B: real] :
% 8.58/4.83 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
% 8.58/4.83 = ( uminus_uminus_real @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5178_mult__1s__ring__1_I1_J,axiom,
% 8.58/4.83 ! [B: int] :
% 8.58/4.83 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
% 8.58/4.83 = ( uminus_uminus_int @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5179_mult__1s__ring__1_I1_J,axiom,
% 8.58/4.83 ! [B: complex] :
% 8.58/4.83 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
% 8.58/4.83 = ( uminus1482373934393186551omplex @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5180_mult__1s__ring__1_I1_J,axiom,
% 8.58/4.83 ! [B: rat] :
% 8.58/4.83 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
% 8.58/4.83 = ( uminus_uminus_rat @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5181_mult__1s__ring__1_I1_J,axiom,
% 8.58/4.83 ! [B: code_integer] :
% 8.58/4.83 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
% 8.58/4.83 = ( uminus1351360451143612070nteger @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5182_mult__1s__ring__1_I2_J,axiom,
% 8.58/4.83 ! [B: real] :
% 8.58/4.83 ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
% 8.58/4.83 = ( uminus_uminus_real @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5183_mult__1s__ring__1_I2_J,axiom,
% 8.58/4.83 ! [B: int] :
% 8.58/4.83 ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
% 8.58/4.83 = ( uminus_uminus_int @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5184_mult__1s__ring__1_I2_J,axiom,
% 8.58/4.83 ! [B: complex] :
% 8.58/4.83 ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
% 8.58/4.83 = ( uminus1482373934393186551omplex @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5185_mult__1s__ring__1_I2_J,axiom,
% 8.58/4.83 ! [B: rat] :
% 8.58/4.83 ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
% 8.58/4.83 = ( uminus_uminus_rat @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5186_mult__1s__ring__1_I2_J,axiom,
% 8.58/4.83 ! [B: code_integer] :
% 8.58/4.83 ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
% 8.58/4.83 = ( uminus1351360451143612070nteger @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5187_uminus__numeral__One,axiom,
% 8.58/4.83 ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
% 8.58/4.83 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5188_uminus__numeral__One,axiom,
% 8.58/4.83 ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
% 8.58/4.83 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5189_uminus__numeral__One,axiom,
% 8.58/4.83 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
% 8.58/4.83 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5190_uminus__numeral__One,axiom,
% 8.58/4.83 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
% 8.58/4.83 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5191_uminus__numeral__One,axiom,
% 8.58/4.83 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
% 8.58/4.83 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5192_power__minus,axiom,
% 8.58/4.83 ! [A: real,N: nat] :
% 8.58/4.83 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 8.58/4.83 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5193_power__minus,axiom,
% 8.58/4.83 ! [A: int,N: nat] :
% 8.58/4.83 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 8.58/4.83 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5194_power__minus,axiom,
% 8.58/4.83 ! [A: complex,N: nat] :
% 8.58/4.83 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 8.58/4.83 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5195_power__minus,axiom,
% 8.58/4.83 ! [A: rat,N: nat] :
% 8.58/4.83 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 8.58/4.83 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5196_power__minus,axiom,
% 8.58/4.83 ! [A: code_integer,N: nat] :
% 8.58/4.83 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 8.58/4.83 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5197_power__minus__Bit0,axiom,
% 8.58/4.83 ! [X: real,K: num] :
% 8.58/4.83 ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 8.58/4.83 = ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5198_power__minus__Bit0,axiom,
% 8.58/4.83 ! [X: int,K: num] :
% 8.58/4.83 ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 8.58/4.83 = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5199_power__minus__Bit0,axiom,
% 8.58/4.83 ! [X: complex,K: num] :
% 8.58/4.83 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 8.58/4.83 = ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5200_power__minus__Bit0,axiom,
% 8.58/4.83 ! [X: rat,K: num] :
% 8.58/4.83 ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 8.58/4.83 = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5201_power__minus__Bit0,axiom,
% 8.58/4.83 ! [X: code_integer,K: num] :
% 8.58/4.83 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 8.58/4.83 = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5202_norm__uminus__minus,axiom,
% 8.58/4.83 ! [X: real,Y: real] :
% 8.58/4.83 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y ) )
% 8.58/4.83 = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5203_norm__uminus__minus,axiom,
% 8.58/4.83 ! [X: complex,Y: complex] :
% 8.58/4.83 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y ) )
% 8.58/4.83 = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5204_real__0__less__add__iff,axiom,
% 8.58/4.83 ! [X: real,Y: real] :
% 8.58/4.83 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.83 = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5205_real__add__less__0__iff,axiom,
% 8.58/4.83 ! [X: real,Y: real] :
% 8.58/4.83 ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 8.58/4.83 = ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5206_real__0__le__add__iff,axiom,
% 8.58/4.83 ! [X: real,Y: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.83 = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5207_real__add__le__0__iff,axiom,
% 8.58/4.83 ! [X: real,Y: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 8.58/4.83 = ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5208_pos__minus__divide__less__eq,axiom,
% 8.58/4.83 ! [C: real,B: real,A: real] :
% 8.58/4.83 ( ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 8.58/4.83 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5209_pos__minus__divide__less__eq,axiom,
% 8.58/4.83 ! [C: rat,B: rat,A: rat] :
% 8.58/4.83 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 8.58/4.83 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5210_pos__less__minus__divide__eq,axiom,
% 8.58/4.83 ! [C: real,A: real,B: real] :
% 8.58/4.83 ( ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 8.58/4.83 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5211_pos__less__minus__divide__eq,axiom,
% 8.58/4.83 ! [C: rat,A: rat,B: rat] :
% 8.58/4.83 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 8.58/4.83 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5212_neg__minus__divide__less__eq,axiom,
% 8.58/4.83 ! [C: real,B: real,A: real] :
% 8.58/4.83 ( ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 8.58/4.83 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5213_neg__minus__divide__less__eq,axiom,
% 8.58/4.83 ! [C: rat,B: rat,A: rat] :
% 8.58/4.83 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 8.58/4.83 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5214_neg__less__minus__divide__eq,axiom,
% 8.58/4.83 ! [C: real,A: real,B: real] :
% 8.58/4.83 ( ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 8.58/4.83 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5215_neg__less__minus__divide__eq,axiom,
% 8.58/4.83 ! [C: rat,A: rat,B: rat] :
% 8.58/4.83 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 8.58/4.83 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5216_minus__divide__less__eq,axiom,
% 8.58/4.83 ! [B: real,C: real,A: real] :
% 8.58/4.83 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 8.58/4.83 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 8.58/4.83 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 8.58/4.83 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5217_minus__divide__less__eq,axiom,
% 8.58/4.83 ! [B: rat,C: rat,A: rat] :
% 8.58/4.83 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 8.58/4.83 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 8.58/4.83 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 8.58/4.83 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5218_less__minus__divide__eq,axiom,
% 8.58/4.83 ! [A: real,B: real,C: real] :
% 8.58/4.83 ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 8.58/4.83 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 8.58/4.83 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 8.58/4.83 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5219_less__minus__divide__eq,axiom,
% 8.58/4.83 ! [A: rat,B: rat,C: rat] :
% 8.58/4.83 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 8.58/4.83 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 8.58/4.83 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 8.58/4.83 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5220_eq__divide__eq__numeral_I2_J,axiom,
% 8.58/4.83 ! [W: num,B: real,C: real] :
% 8.58/4.83 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 8.58/4.83 = ( divide_divide_real @ B @ C ) )
% 8.58/4.83 = ( ( ( C != zero_zero_real )
% 8.58/4.83 => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
% 8.58/4.83 = B ) )
% 8.58/4.83 & ( ( C = zero_zero_real )
% 8.58/4.83 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 8.58/4.83 = zero_zero_real ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5221_eq__divide__eq__numeral_I2_J,axiom,
% 8.58/4.83 ! [W: num,B: complex,C: complex] :
% 8.58/4.83 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.58/4.83 = ( divide1717551699836669952omplex @ B @ C ) )
% 8.58/4.83 = ( ( ( C != zero_zero_complex )
% 8.58/4.83 => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
% 8.58/4.83 = B ) )
% 8.58/4.83 & ( ( C = zero_zero_complex )
% 8.58/4.83 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.58/4.83 = zero_zero_complex ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5222_eq__divide__eq__numeral_I2_J,axiom,
% 8.58/4.83 ! [W: num,B: rat,C: rat] :
% 8.58/4.83 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 8.58/4.83 = ( divide_divide_rat @ B @ C ) )
% 8.58/4.83 = ( ( ( C != zero_zero_rat )
% 8.58/4.83 => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
% 8.58/4.83 = B ) )
% 8.58/4.83 & ( ( C = zero_zero_rat )
% 8.58/4.83 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 8.58/4.83 = zero_zero_rat ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5223_divide__eq__eq__numeral_I2_J,axiom,
% 8.58/4.83 ! [B: real,C: real,W: num] :
% 8.58/4.83 ( ( ( divide_divide_real @ B @ C )
% 8.58/4.83 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.58/4.83 = ( ( ( C != zero_zero_real )
% 8.58/4.83 => ( B
% 8.58/4.83 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 8.58/4.83 & ( ( C = zero_zero_real )
% 8.58/4.83 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 8.58/4.83 = zero_zero_real ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5224_divide__eq__eq__numeral_I2_J,axiom,
% 8.58/4.83 ! [B: complex,C: complex,W: num] :
% 8.58/4.83 ( ( ( divide1717551699836669952omplex @ B @ C )
% 8.58/4.83 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 8.58/4.83 = ( ( ( C != zero_zero_complex )
% 8.58/4.83 => ( B
% 8.58/4.83 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
% 8.58/4.83 & ( ( C = zero_zero_complex )
% 8.58/4.83 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.58/4.83 = zero_zero_complex ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5225_divide__eq__eq__numeral_I2_J,axiom,
% 8.58/4.83 ! [B: rat,C: rat,W: num] :
% 8.58/4.83 ( ( ( divide_divide_rat @ B @ C )
% 8.58/4.83 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 8.58/4.83 = ( ( ( C != zero_zero_rat )
% 8.58/4.83 => ( B
% 8.58/4.83 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 8.58/4.83 & ( ( C = zero_zero_rat )
% 8.58/4.83 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 8.58/4.83 = zero_zero_rat ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5226_minus__divide__add__eq__iff,axiom,
% 8.58/4.83 ! [Z2: real,X: real,Y: real] :
% 8.58/4.83 ( ( Z2 != zero_zero_real )
% 8.58/4.83 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z2 ) ) @ Y )
% 8.58/4.83 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5227_minus__divide__add__eq__iff,axiom,
% 8.58/4.83 ! [Z2: complex,X: complex,Y: complex] :
% 8.58/4.83 ( ( Z2 != zero_zero_complex )
% 8.58/4.83 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z2 ) ) @ Y )
% 8.58/4.83 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5228_minus__divide__add__eq__iff,axiom,
% 8.58/4.83 ! [Z2: rat,X: rat,Y: rat] :
% 8.58/4.83 ( ( Z2 != zero_zero_rat )
% 8.58/4.83 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y )
% 8.58/4.83 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5229_add__divide__eq__if__simps_I3_J,axiom,
% 8.58/4.83 ! [Z2: real,A: real,B: real] :
% 8.58/4.83 ( ( ( Z2 = zero_zero_real )
% 8.58/4.83 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 8.58/4.83 = B ) )
% 8.58/4.83 & ( ( Z2 != zero_zero_real )
% 8.58/4.83 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 8.58/4.83 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5230_add__divide__eq__if__simps_I3_J,axiom,
% 8.58/4.83 ! [Z2: complex,A: complex,B: complex] :
% 8.58/4.83 ( ( ( Z2 = zero_zero_complex )
% 8.58/4.83 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 8.58/4.83 = B ) )
% 8.58/4.83 & ( ( Z2 != zero_zero_complex )
% 8.58/4.83 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 8.58/4.83 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5231_add__divide__eq__if__simps_I3_J,axiom,
% 8.58/4.83 ! [Z2: rat,A: rat,B: rat] :
% 8.58/4.83 ( ( ( Z2 = zero_zero_rat )
% 8.58/4.83 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 8.58/4.83 = B ) )
% 8.58/4.83 & ( ( Z2 != zero_zero_rat )
% 8.58/4.83 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 8.58/4.83 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5232_minus__divide__diff__eq__iff,axiom,
% 8.58/4.83 ! [Z2: real,X: real,Y: real] :
% 8.58/4.83 ( ( Z2 != zero_zero_real )
% 8.58/4.83 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z2 ) ) @ Y )
% 8.58/4.83 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5233_minus__divide__diff__eq__iff,axiom,
% 8.58/4.83 ! [Z2: complex,X: complex,Y: complex] :
% 8.58/4.83 ( ( Z2 != zero_zero_complex )
% 8.58/4.83 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z2 ) ) @ Y )
% 8.58/4.83 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5234_minus__divide__diff__eq__iff,axiom,
% 8.58/4.83 ! [Z2: rat,X: rat,Y: rat] :
% 8.58/4.83 ( ( Z2 != zero_zero_rat )
% 8.58/4.83 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y )
% 8.58/4.83 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5235_add__divide__eq__if__simps_I5_J,axiom,
% 8.58/4.83 ! [Z2: real,A: real,B: real] :
% 8.58/4.83 ( ( ( Z2 = zero_zero_real )
% 8.58/4.83 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 8.58/4.83 = ( uminus_uminus_real @ B ) ) )
% 8.58/4.83 & ( ( Z2 != zero_zero_real )
% 8.58/4.83 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 8.58/4.83 = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5236_add__divide__eq__if__simps_I5_J,axiom,
% 8.58/4.83 ! [Z2: complex,A: complex,B: complex] :
% 8.58/4.83 ( ( ( Z2 = zero_zero_complex )
% 8.58/4.83 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 8.58/4.83 = ( uminus1482373934393186551omplex @ B ) ) )
% 8.58/4.83 & ( ( Z2 != zero_zero_complex )
% 8.58/4.83 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 8.58/4.83 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5237_add__divide__eq__if__simps_I5_J,axiom,
% 8.58/4.83 ! [Z2: rat,A: rat,B: rat] :
% 8.58/4.83 ( ( ( Z2 = zero_zero_rat )
% 8.58/4.83 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 8.58/4.83 = ( uminus_uminus_rat @ B ) ) )
% 8.58/4.83 & ( ( Z2 != zero_zero_rat )
% 8.58/4.83 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 8.58/4.83 = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5238_add__divide__eq__if__simps_I6_J,axiom,
% 8.58/4.83 ! [Z2: real,A: real,B: real] :
% 8.58/4.83 ( ( ( Z2 = zero_zero_real )
% 8.58/4.83 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 8.58/4.83 = ( uminus_uminus_real @ B ) ) )
% 8.58/4.83 & ( ( Z2 != zero_zero_real )
% 8.58/4.83 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 8.58/4.83 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5239_add__divide__eq__if__simps_I6_J,axiom,
% 8.58/4.83 ! [Z2: complex,A: complex,B: complex] :
% 8.58/4.83 ( ( ( Z2 = zero_zero_complex )
% 8.58/4.83 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 8.58/4.83 = ( uminus1482373934393186551omplex @ B ) ) )
% 8.58/4.83 & ( ( Z2 != zero_zero_complex )
% 8.58/4.83 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 8.58/4.83 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5240_add__divide__eq__if__simps_I6_J,axiom,
% 8.58/4.83 ! [Z2: rat,A: rat,B: rat] :
% 8.58/4.83 ( ( ( Z2 = zero_zero_rat )
% 8.58/4.83 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 8.58/4.83 = ( uminus_uminus_rat @ B ) ) )
% 8.58/4.83 & ( ( Z2 != zero_zero_rat )
% 8.58/4.83 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 8.58/4.83 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5241_even__minus,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
% 8.58/4.83 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5242_even__minus,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
% 8.58/4.83 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5243_power2__eq__iff,axiom,
% 8.58/4.83 ! [X: real,Y: real] :
% 8.58/4.83 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.83 = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.83 = ( ( X = Y )
% 8.58/4.83 | ( X
% 8.58/4.83 = ( uminus_uminus_real @ Y ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5244_power2__eq__iff,axiom,
% 8.58/4.83 ! [X: int,Y: int] :
% 8.58/4.83 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.83 = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.83 = ( ( X = Y )
% 8.58/4.83 | ( X
% 8.58/4.83 = ( uminus_uminus_int @ Y ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5245_power2__eq__iff,axiom,
% 8.58/4.83 ! [X: complex,Y: complex] :
% 8.58/4.83 ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.83 = ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.83 = ( ( X = Y )
% 8.58/4.83 | ( X
% 8.58/4.83 = ( uminus1482373934393186551omplex @ Y ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5246_power2__eq__iff,axiom,
% 8.58/4.83 ! [X: rat,Y: rat] :
% 8.58/4.83 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.83 = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.83 = ( ( X = Y )
% 8.58/4.83 | ( X
% 8.58/4.83 = ( uminus_uminus_rat @ Y ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5247_power2__eq__iff,axiom,
% 8.58/4.83 ! [X: code_integer,Y: code_integer] :
% 8.58/4.83 ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.83 = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.83 = ( ( X = Y )
% 8.58/4.83 | ( X
% 8.58/4.83 = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5248_sgn__if,axiom,
% 8.58/4.83 ( sgn_sgn_real
% 8.58/4.83 = ( ^ [X3: real] : ( if_real @ ( X3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5249_sgn__if,axiom,
% 8.58/4.83 ( sgn_sgn_int
% 8.58/4.83 = ( ^ [X3: int] : ( if_int @ ( X3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5250_sgn__if,axiom,
% 8.58/4.83 ( sgn_sgn_rat
% 8.58/4.83 = ( ^ [X3: rat] : ( if_rat @ ( X3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5251_sgn__if,axiom,
% 8.58/4.83 ( sgn_sgn_Code_integer
% 8.58/4.83 = ( ^ [X3: code_integer] : ( if_Code_integer @ ( X3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X3 ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5252_sgn__1__neg,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( ( sgn_sgn_real @ A )
% 8.58/4.83 = ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.83 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5253_sgn__1__neg,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( ( sgn_sgn_int @ A )
% 8.58/4.83 = ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.83 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5254_sgn__1__neg,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( ( sgn_sgn_rat @ A )
% 8.58/4.83 = ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.83 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5255_sgn__1__neg,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( ( sgn_sgn_Code_integer @ A )
% 8.58/4.83 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.83 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5256_pochhammer__of__nat__eq__0__lemma,axiom,
% 8.58/4.83 ! [N: nat,K: nat] :
% 8.58/4.83 ( ( ord_less_nat @ N @ K )
% 8.58/4.83 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 8.58/4.83 = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5257_pochhammer__of__nat__eq__0__lemma,axiom,
% 8.58/4.83 ! [N: nat,K: nat] :
% 8.58/4.83 ( ( ord_less_nat @ N @ K )
% 8.58/4.83 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 8.58/4.83 = zero_zero_complex ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5258_pochhammer__of__nat__eq__0__lemma,axiom,
% 8.58/4.83 ! [N: nat,K: nat] :
% 8.58/4.83 ( ( ord_less_nat @ N @ K )
% 8.58/4.83 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 8.58/4.83 = zero_zero_real ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5259_pochhammer__of__nat__eq__0__lemma,axiom,
% 8.58/4.83 ! [N: nat,K: nat] :
% 8.58/4.83 ( ( ord_less_nat @ N @ K )
% 8.58/4.83 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 8.58/4.83 = zero_zero_rat ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5260_pochhammer__of__nat__eq__0__lemma,axiom,
% 8.58/4.83 ! [N: nat,K: nat] :
% 8.58/4.83 ( ( ord_less_nat @ N @ K )
% 8.58/4.83 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 8.58/4.83 = zero_zero_int ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5261_pochhammer__of__nat__eq__0__iff,axiom,
% 8.58/4.83 ! [N: nat,K: nat] :
% 8.58/4.83 ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 8.58/4.83 = zero_z3403309356797280102nteger )
% 8.58/4.83 = ( ord_less_nat @ N @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5262_pochhammer__of__nat__eq__0__iff,axiom,
% 8.58/4.83 ! [N: nat,K: nat] :
% 8.58/4.83 ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 8.58/4.83 = zero_zero_complex )
% 8.58/4.83 = ( ord_less_nat @ N @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5263_pochhammer__of__nat__eq__0__iff,axiom,
% 8.58/4.83 ! [N: nat,K: nat] :
% 8.58/4.83 ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 8.58/4.83 = zero_zero_real )
% 8.58/4.83 = ( ord_less_nat @ N @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5264_pochhammer__of__nat__eq__0__iff,axiom,
% 8.58/4.83 ! [N: nat,K: nat] :
% 8.58/4.83 ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 8.58/4.83 = zero_zero_rat )
% 8.58/4.83 = ( ord_less_nat @ N @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5265_pochhammer__of__nat__eq__0__iff,axiom,
% 8.58/4.83 ! [N: nat,K: nat] :
% 8.58/4.83 ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 8.58/4.83 = zero_zero_int )
% 8.58/4.83 = ( ord_less_nat @ N @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5266_pochhammer__eq__0__iff,axiom,
% 8.58/4.83 ! [A: complex,N: nat] :
% 8.58/4.83 ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 8.58/4.83 = zero_zero_complex )
% 8.58/4.83 = ( ? [K2: nat] :
% 8.58/4.83 ( ( ord_less_nat @ K2 @ N )
% 8.58/4.83 & ( A
% 8.58/4.83 = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5267_pochhammer__eq__0__iff,axiom,
% 8.58/4.83 ! [A: real,N: nat] :
% 8.58/4.83 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 8.58/4.83 = zero_zero_real )
% 8.58/4.83 = ( ? [K2: nat] :
% 8.58/4.83 ( ( ord_less_nat @ K2 @ N )
% 8.58/4.83 & ( A
% 8.58/4.83 = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5268_pochhammer__eq__0__iff,axiom,
% 8.58/4.83 ! [A: rat,N: nat] :
% 8.58/4.83 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 8.58/4.83 = zero_zero_rat )
% 8.58/4.83 = ( ? [K2: nat] :
% 8.58/4.83 ( ( ord_less_nat @ K2 @ N )
% 8.58/4.83 & ( A
% 8.58/4.83 = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K2 ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5269_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 8.58/4.83 ! [K: nat,N: nat] :
% 8.58/4.83 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.83 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 8.58/4.83 != zero_z3403309356797280102nteger ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5270_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 8.58/4.83 ! [K: nat,N: nat] :
% 8.58/4.83 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.83 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 8.58/4.83 != zero_zero_complex ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5271_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 8.58/4.83 ! [K: nat,N: nat] :
% 8.58/4.83 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.83 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 8.58/4.83 != zero_zero_real ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5272_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 8.58/4.83 ! [K: nat,N: nat] :
% 8.58/4.83 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.83 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 8.58/4.83 != zero_zero_rat ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5273_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 8.58/4.83 ! [K: nat,N: nat] :
% 8.58/4.83 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.83 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 8.58/4.83 != zero_zero_int ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5274_ln__add__one__self__le__self2,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.83 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5275_mult__ceiling__le,axiom,
% 8.58/4.83 ! [A: real,B: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.83 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.58/4.83 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5276_mult__ceiling__le,axiom,
% 8.58/4.83 ! [A: rat,B: rat] :
% 8.58/4.83 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.58/4.83 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.58/4.83 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5277_fold__atLeastAtMost__nat_Oelims,axiom,
% 8.58/4.83 ! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
% 8.58/4.83 ( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
% 8.58/4.83 = Y )
% 8.58/4.83 => ( ( ( ord_less_nat @ Xb @ Xa )
% 8.58/4.83 => ( Y = Xc ) )
% 8.58/4.83 & ( ~ ( ord_less_nat @ Xb @ Xa )
% 8.58/4.83 => ( Y
% 8.58/4.83 = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5278_fold__atLeastAtMost__nat_Osimps,axiom,
% 8.58/4.83 ( set_fo2584398358068434914at_nat
% 8.58/4.83 = ( ^ [F2: nat > nat > nat,A2: nat,B2: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B2 @ A2 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F2 @ ( plus_plus_nat @ A2 @ one_one_nat ) @ B2 @ ( F2 @ A2 @ Acc ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5279_pos__minus__divide__le__eq,axiom,
% 8.58/4.83 ! [C: real,B: real,A: real] :
% 8.58/4.83 ( ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 8.58/4.83 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5280_pos__minus__divide__le__eq,axiom,
% 8.58/4.83 ! [C: rat,B: rat,A: rat] :
% 8.58/4.83 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 8.58/4.83 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5281_pos__le__minus__divide__eq,axiom,
% 8.58/4.83 ! [C: real,A: real,B: real] :
% 8.58/4.83 ( ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 8.58/4.83 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5282_pos__le__minus__divide__eq,axiom,
% 8.58/4.83 ! [C: rat,A: rat,B: rat] :
% 8.58/4.83 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 8.58/4.83 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5283_neg__minus__divide__le__eq,axiom,
% 8.58/4.83 ! [C: real,B: real,A: real] :
% 8.58/4.83 ( ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 8.58/4.83 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5284_neg__minus__divide__le__eq,axiom,
% 8.58/4.83 ! [C: rat,B: rat,A: rat] :
% 8.58/4.83 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 8.58/4.83 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5285_neg__le__minus__divide__eq,axiom,
% 8.58/4.83 ! [C: real,A: real,B: real] :
% 8.58/4.83 ( ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 8.58/4.83 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5286_neg__le__minus__divide__eq,axiom,
% 8.58/4.83 ! [C: rat,A: rat,B: rat] :
% 8.58/4.83 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 8.58/4.83 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5287_minus__divide__le__eq,axiom,
% 8.58/4.83 ! [B: real,C: real,A: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 8.58/4.83 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 8.58/4.83 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 8.58/4.83 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5288_minus__divide__le__eq,axiom,
% 8.58/4.83 ! [B: rat,C: rat,A: rat] :
% 8.58/4.83 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 8.58/4.83 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 8.58/4.83 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 8.58/4.83 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5289_le__minus__divide__eq,axiom,
% 8.58/4.83 ! [A: real,B: real,C: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 8.58/4.83 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 8.58/4.83 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 8.58/4.83 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5290_le__minus__divide__eq,axiom,
% 8.58/4.83 ! [A: rat,B: rat,C: rat] :
% 8.58/4.83 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 8.58/4.83 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 8.58/4.83 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 8.58/4.83 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5291_divide__less__eq__numeral_I2_J,axiom,
% 8.58/4.83 ! [B: real,C: real,W: num] :
% 8.58/4.83 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.58/4.83 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 8.58/4.83 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 8.58/4.83 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5292_divide__less__eq__numeral_I2_J,axiom,
% 8.58/4.83 ! [B: rat,C: rat,W: num] :
% 8.58/4.83 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 8.58/4.83 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 8.58/4.83 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 8.58/4.83 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5293_less__divide__eq__numeral_I2_J,axiom,
% 8.58/4.83 ! [W: num,B: real,C: real] :
% 8.58/4.83 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 8.58/4.83 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 8.58/4.83 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 8.58/4.83 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5294_less__divide__eq__numeral_I2_J,axiom,
% 8.58/4.83 ! [W: num,B: rat,C: rat] :
% 8.58/4.83 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 8.58/4.83 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 8.58/4.83 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 8.58/4.83 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5295_power2__eq__1__iff,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.83 = one_one_real )
% 8.58/4.83 = ( ( A = one_one_real )
% 8.58/4.83 | ( A
% 8.58/4.83 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5296_power2__eq__1__iff,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.83 = one_one_int )
% 8.58/4.83 = ( ( A = one_one_int )
% 8.58/4.83 | ( A
% 8.58/4.83 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5297_power2__eq__1__iff,axiom,
% 8.58/4.83 ! [A: complex] :
% 8.58/4.83 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.83 = one_one_complex )
% 8.58/4.83 = ( ( A = one_one_complex )
% 8.58/4.83 | ( A
% 8.58/4.83 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5298_power2__eq__1__iff,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.83 = one_one_rat )
% 8.58/4.83 = ( ( A = one_one_rat )
% 8.58/4.83 | ( A
% 8.58/4.83 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5299_power2__eq__1__iff,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.83 = one_one_Code_integer )
% 8.58/4.83 = ( ( A = one_one_Code_integer )
% 8.58/4.83 | ( A
% 8.58/4.83 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5300_uminus__power__if,axiom,
% 8.58/4.83 ! [N: nat,A: real] :
% 8.58/4.83 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 8.58/4.83 = ( power_power_real @ A @ N ) ) )
% 8.58/4.83 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 8.58/4.83 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5301_uminus__power__if,axiom,
% 8.58/4.83 ! [N: nat,A: int] :
% 8.58/4.83 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 8.58/4.83 = ( power_power_int @ A @ N ) ) )
% 8.58/4.83 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 8.58/4.83 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5302_uminus__power__if,axiom,
% 8.58/4.83 ! [N: nat,A: complex] :
% 8.58/4.83 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 8.58/4.83 = ( power_power_complex @ A @ N ) ) )
% 8.58/4.83 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 8.58/4.83 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5303_uminus__power__if,axiom,
% 8.58/4.83 ! [N: nat,A: rat] :
% 8.58/4.83 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 8.58/4.83 = ( power_power_rat @ A @ N ) ) )
% 8.58/4.83 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 8.58/4.83 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5304_uminus__power__if,axiom,
% 8.58/4.83 ! [N: nat,A: code_integer] :
% 8.58/4.83 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 8.58/4.83 = ( power_8256067586552552935nteger @ A @ N ) ) )
% 8.58/4.83 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 8.58/4.83 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5305_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 8.58/4.83 ! [K: nat,N: nat] :
% 8.58/4.83 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.83 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 8.58/4.83 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5306_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 8.58/4.83 ! [K: nat,N: nat] :
% 8.58/4.83 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.83 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 8.58/4.83 = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5307_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 8.58/4.83 ! [K: nat,N: nat] :
% 8.58/4.83 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.83 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 8.58/4.83 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5308_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 8.58/4.83 ! [K: nat,N: nat] :
% 8.58/4.83 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.83 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 8.58/4.83 = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5309_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 8.58/4.83 ! [K: nat,N: nat] :
% 8.58/4.83 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.83 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 8.58/4.83 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5310_realpow__square__minus__le,axiom,
% 8.58/4.83 ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5311_ln__one__minus__pos__upper__bound,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.83 => ( ( ord_less_real @ X @ one_one_real )
% 8.58/4.83 => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5312_le__divide__eq__numeral_I2_J,axiom,
% 8.58/4.83 ! [W: num,B: real,C: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 8.58/4.83 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 8.58/4.83 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 8.58/4.83 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5313_le__divide__eq__numeral_I2_J,axiom,
% 8.58/4.83 ! [W: num,B: rat,C: rat] :
% 8.58/4.83 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 8.58/4.83 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 8.58/4.83 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 8.58/4.83 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5314_divide__le__eq__numeral_I2_J,axiom,
% 8.58/4.83 ! [B: real,C: real,W: num] :
% 8.58/4.83 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.58/4.83 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 8.58/4.83 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.83 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 8.58/4.83 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.58/4.83 => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5315_divide__le__eq__numeral_I2_J,axiom,
% 8.58/4.83 ! [B: rat,C: rat,W: num] :
% 8.58/4.83 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 8.58/4.83 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 8.58/4.83 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.58/4.83 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 8.58/4.83 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.58/4.83 => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5316_square__le__1,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.83 => ( ( ord_less_eq_real @ X @ one_one_real )
% 8.58/4.83 => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5317_square__le__1,axiom,
% 8.58/4.83 ! [X: code_integer] :
% 8.58/4.83 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 8.58/4.83 => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
% 8.58/4.83 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5318_square__le__1,axiom,
% 8.58/4.83 ! [X: rat] :
% 8.58/4.83 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
% 8.58/4.83 => ( ( ord_less_eq_rat @ X @ one_one_rat )
% 8.58/4.83 => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5319_square__le__1,axiom,
% 8.58/4.83 ! [X: int] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 8.58/4.83 => ( ( ord_less_eq_int @ X @ one_one_int )
% 8.58/4.83 => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5320_minus__power__mult__self,axiom,
% 8.58/4.83 ! [A: real,N: nat] :
% 8.58/4.83 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 8.58/4.83 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5321_minus__power__mult__self,axiom,
% 8.58/4.83 ! [A: int,N: nat] :
% 8.58/4.83 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 8.58/4.83 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5322_minus__power__mult__self,axiom,
% 8.58/4.83 ! [A: complex,N: nat] :
% 8.58/4.83 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
% 8.58/4.83 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5323_minus__power__mult__self,axiom,
% 8.58/4.83 ! [A: rat,N: nat] :
% 8.58/4.83 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 8.58/4.83 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5324_minus__power__mult__self,axiom,
% 8.58/4.83 ! [A: code_integer,N: nat] :
% 8.58/4.83 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 8.58/4.83 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5325_minus__one__power__iff,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 8.58/4.83 = one_one_real ) )
% 8.58/4.83 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 8.58/4.83 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5326_minus__one__power__iff,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 8.58/4.83 = one_one_int ) )
% 8.58/4.83 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 8.58/4.83 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5327_minus__one__power__iff,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 8.58/4.83 = one_one_complex ) )
% 8.58/4.83 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 8.58/4.83 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5328_minus__one__power__iff,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 8.58/4.83 = one_one_rat ) )
% 8.58/4.83 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 8.58/4.83 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5329_minus__one__power__iff,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 8.58/4.83 = one_one_Code_integer ) )
% 8.58/4.83 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.83 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 8.58/4.83 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5330_pochhammer__absorb__comp,axiom,
% 8.58/4.83 ! [R: code_integer,K: nat] :
% 8.58/4.83 ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R ) @ K ) )
% 8.58/4.83 = ( times_3573771949741848930nteger @ R @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R ) @ one_one_Code_integer ) @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5331_pochhammer__absorb__comp,axiom,
% 8.58/4.83 ! [R: complex,K: nat] :
% 8.58/4.83 ( ( times_times_complex @ ( minus_minus_complex @ R @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R ) @ K ) )
% 8.58/4.83 = ( times_times_complex @ R @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R ) @ one_one_complex ) @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5332_pochhammer__absorb__comp,axiom,
% 8.58/4.83 ! [R: real,K: nat] :
% 8.58/4.83 ( ( times_times_real @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R ) @ K ) )
% 8.58/4.83 = ( times_times_real @ R @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R ) @ one_one_real ) @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5333_pochhammer__absorb__comp,axiom,
% 8.58/4.83 ! [R: rat,K: nat] :
% 8.58/4.83 ( ( times_times_rat @ ( minus_minus_rat @ R @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R ) @ K ) )
% 8.58/4.83 = ( times_times_rat @ R @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R ) @ one_one_rat ) @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5334_pochhammer__absorb__comp,axiom,
% 8.58/4.83 ! [R: int,K: nat] :
% 8.58/4.83 ( ( times_times_int @ ( minus_minus_int @ R @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R ) @ K ) )
% 8.58/4.83 = ( times_times_int @ R @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R ) @ one_one_int ) @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5335_pochhammer__same,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
% 8.58/4.83 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5336_pochhammer__same,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
% 8.58/4.83 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5337_pochhammer__same,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
% 8.58/4.83 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5338_pochhammer__same,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
% 8.58/4.83 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5339_pochhammer__same,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
% 8.58/4.83 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5340_Bernoulli__inequality,axiom,
% 8.58/4.83 ! [X: real,N: nat] :
% 8.58/4.83 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.83 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5341_power__minus1__odd,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.58/4.83 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5342_power__minus1__odd,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.58/4.83 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5343_power__minus1__odd,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.58/4.83 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5344_power__minus1__odd,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.58/4.83 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5345_power__minus1__odd,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.58/4.83 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5346_pochhammer__minus,axiom,
% 8.58/4.83 ! [B: code_integer,K: nat] :
% 8.58/4.83 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
% 8.58/4.83 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5347_pochhammer__minus,axiom,
% 8.58/4.83 ! [B: complex,K: nat] :
% 8.58/4.83 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
% 8.58/4.83 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5348_pochhammer__minus,axiom,
% 8.58/4.83 ! [B: real,K: nat] :
% 8.58/4.83 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
% 8.58/4.83 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5349_pochhammer__minus,axiom,
% 8.58/4.83 ! [B: rat,K: nat] :
% 8.58/4.83 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
% 8.58/4.83 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5350_pochhammer__minus,axiom,
% 8.58/4.83 ! [B: int,K: nat] :
% 8.58/4.83 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
% 8.58/4.83 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5351_pochhammer__minus_H,axiom,
% 8.58/4.83 ! [B: code_integer,K: nat] :
% 8.58/4.83 ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
% 8.58/4.83 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5352_pochhammer__minus_H,axiom,
% 8.58/4.83 ! [B: complex,K: nat] :
% 8.58/4.83 ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
% 8.58/4.83 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5353_pochhammer__minus_H,axiom,
% 8.58/4.83 ! [B: real,K: nat] :
% 8.58/4.83 ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
% 8.58/4.83 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5354_pochhammer__minus_H,axiom,
% 8.58/4.83 ! [B: rat,K: nat] :
% 8.58/4.83 ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
% 8.58/4.83 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5355_pochhammer__minus_H,axiom,
% 8.58/4.83 ! [B: int,K: nat] :
% 8.58/4.83 ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
% 8.58/4.83 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5356_eq__diff__eq_H,axiom,
% 8.58/4.83 ! [X: real,Y: real,Z2: real] :
% 8.58/4.83 ( ( X
% 8.58/4.83 = ( minus_minus_real @ Y @ Z2 ) )
% 8.58/4.83 = ( Y
% 8.58/4.83 = ( plus_plus_real @ X @ Z2 ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5357_take__bit__Suc__minus__1__eq,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.83 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5358_take__bit__Suc__minus__1__eq,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.83 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5359_take__bit__numeral__minus__1__eq,axiom,
% 8.58/4.83 ! [K: num] :
% 8.58/4.83 ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.83 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5360_take__bit__numeral__minus__1__eq,axiom,
% 8.58/4.83 ! [K: num] :
% 8.58/4.83 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.83 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5361_fact__code,axiom,
% 8.58/4.83 ( semiri5044797733671781792omplex
% 8.58/4.83 = ( ^ [N3: nat] : ( semiri8010041392384452111omplex @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5362_fact__code,axiom,
% 8.58/4.83 ( semiri773545260158071498ct_rat
% 8.58/4.83 = ( ^ [N3: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5363_fact__code,axiom,
% 8.58/4.83 ( semiri1406184849735516958ct_int
% 8.58/4.83 = ( ^ [N3: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5364_fact__code,axiom,
% 8.58/4.83 ( semiri1408675320244567234ct_nat
% 8.58/4.83 = ( ^ [N3: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5365_fact__code,axiom,
% 8.58/4.83 ( semiri2265585572941072030t_real
% 8.58/4.83 = ( ^ [N3: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5366_signed__take__bit__rec,axiom,
% 8.58/4.83 ( bit_ri6519982836138164636nteger
% 8.58/4.83 = ( ^ [N3: nat,A2: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5367_signed__take__bit__rec,axiom,
% 8.58/4.83 ( bit_ri631733984087533419it_int
% 8.58/4.83 = ( ^ [N3: nat,A2: int] : ( if_int @ ( N3 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5368_sin__coeff__def,axiom,
% 8.58/4.83 ( sin_coeff
% 8.58/4.83 = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5369_cos__coeff__def,axiom,
% 8.58/4.83 ( cos_coeff
% 8.58/4.83 = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ zero_zero_real ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5370_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 8.58/4.83 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 8.58/4.83 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5371_abs__ln__one__plus__x__minus__x__bound,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.83 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5372_ceiling__log__eq__powr__iff,axiom,
% 8.58/4.83 ! [X: real,B: real,K: nat] :
% 8.58/4.83 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.83 => ( ( ord_less_real @ one_one_real @ B )
% 8.58/4.83 => ( ( ( archim7802044766580827645g_real @ ( log2 @ B @ X ) )
% 8.58/4.83 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 8.58/4.83 = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
% 8.58/4.83 & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5373_floor__log__nat__eq__powr__iff,axiom,
% 8.58/4.83 ! [B: nat,K: nat,N: nat] :
% 8.58/4.83 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 8.58/4.83 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.58/4.83 => ( ( ( archim6058952711729229775r_real @ ( log2 @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 8.58/4.83 = ( semiri1314217659103216013at_int @ N ) )
% 8.58/4.83 = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 8.58/4.83 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5374_divide__int__unfold,axiom,
% 8.58/4.83 ! [L: int,K: int,N: nat,M: nat] :
% 8.58/4.83 ( ( ( ( ( sgn_sgn_int @ L )
% 8.58/4.83 = zero_zero_int )
% 8.58/4.83 | ( ( sgn_sgn_int @ K )
% 8.58/4.83 = zero_zero_int )
% 8.58/4.83 | ( N = zero_zero_nat ) )
% 8.58/4.83 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.58/4.83 = zero_zero_int ) )
% 8.58/4.83 & ( ~ ( ( ( sgn_sgn_int @ L )
% 8.58/4.83 = zero_zero_int )
% 8.58/4.83 | ( ( sgn_sgn_int @ K )
% 8.58/4.83 = zero_zero_int )
% 8.58/4.83 | ( N = zero_zero_nat ) )
% 8.58/4.83 => ( ( ( ( sgn_sgn_int @ K )
% 8.58/4.83 = ( sgn_sgn_int @ L ) )
% 8.58/4.83 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.58/4.83 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 8.58/4.83 & ( ( ( sgn_sgn_int @ K )
% 8.58/4.83 != ( sgn_sgn_int @ L ) )
% 8.58/4.83 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.58/4.83 = ( uminus_uminus_int
% 8.58/4.83 @ ( semiri1314217659103216013at_int
% 8.58/4.83 @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 8.58/4.83 @ ( zero_n2687167440665602831ol_nat
% 8.58/4.83 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5375_abs__idempotent,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 8.58/4.83 = ( abs_abs_real @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5376_abs__idempotent,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 8.58/4.83 = ( abs_abs_int @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5377_abs__idempotent,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 8.58/4.83 = ( abs_abs_Code_integer @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5378_abs__idempotent,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 8.58/4.83 = ( abs_abs_rat @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5379_abs__abs,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 8.58/4.83 = ( abs_abs_real @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5380_abs__abs,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 8.58/4.83 = ( abs_abs_int @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5381_abs__abs,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 8.58/4.83 = ( abs_abs_Code_integer @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5382_abs__abs,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 8.58/4.83 = ( abs_abs_rat @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5383_abs__0,axiom,
% 8.58/4.83 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 8.58/4.83 = zero_z3403309356797280102nteger ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5384_abs__0,axiom,
% 8.58/4.83 ( ( abs_abs_real @ zero_zero_real )
% 8.58/4.83 = zero_zero_real ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5385_abs__0,axiom,
% 8.58/4.83 ( ( abs_abs_rat @ zero_zero_rat )
% 8.58/4.83 = zero_zero_rat ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5386_abs__0,axiom,
% 8.58/4.83 ( ( abs_abs_int @ zero_zero_int )
% 8.58/4.83 = zero_zero_int ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5387_abs__zero,axiom,
% 8.58/4.83 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 8.58/4.83 = zero_z3403309356797280102nteger ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5388_abs__zero,axiom,
% 8.58/4.83 ( ( abs_abs_real @ zero_zero_real )
% 8.58/4.83 = zero_zero_real ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5389_abs__zero,axiom,
% 8.58/4.83 ( ( abs_abs_rat @ zero_zero_rat )
% 8.58/4.83 = zero_zero_rat ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5390_abs__zero,axiom,
% 8.58/4.83 ( ( abs_abs_int @ zero_zero_int )
% 8.58/4.83 = zero_zero_int ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5391_abs__eq__0,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( ( abs_abs_Code_integer @ A )
% 8.58/4.83 = zero_z3403309356797280102nteger )
% 8.58/4.83 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5392_abs__eq__0,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( ( abs_abs_real @ A )
% 8.58/4.83 = zero_zero_real )
% 8.58/4.83 = ( A = zero_zero_real ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5393_abs__eq__0,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( ( abs_abs_rat @ A )
% 8.58/4.83 = zero_zero_rat )
% 8.58/4.83 = ( A = zero_zero_rat ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5394_abs__eq__0,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( ( abs_abs_int @ A )
% 8.58/4.83 = zero_zero_int )
% 8.58/4.83 = ( A = zero_zero_int ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5395_abs__0__eq,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( zero_z3403309356797280102nteger
% 8.58/4.83 = ( abs_abs_Code_integer @ A ) )
% 8.58/4.83 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5396_abs__0__eq,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( zero_zero_real
% 8.58/4.83 = ( abs_abs_real @ A ) )
% 8.58/4.83 = ( A = zero_zero_real ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5397_abs__0__eq,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( zero_zero_rat
% 8.58/4.83 = ( abs_abs_rat @ A ) )
% 8.58/4.83 = ( A = zero_zero_rat ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5398_abs__0__eq,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( zero_zero_int
% 8.58/4.83 = ( abs_abs_int @ A ) )
% 8.58/4.83 = ( A = zero_zero_int ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5399_abs__numeral,axiom,
% 8.58/4.83 ! [N: num] :
% 8.58/4.83 ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
% 8.58/4.83 = ( numera6620942414471956472nteger @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5400_abs__numeral,axiom,
% 8.58/4.83 ! [N: num] :
% 8.58/4.83 ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
% 8.58/4.83 = ( numeral_numeral_real @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5401_abs__numeral,axiom,
% 8.58/4.83 ! [N: num] :
% 8.58/4.83 ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
% 8.58/4.83 = ( numeral_numeral_rat @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5402_abs__numeral,axiom,
% 8.58/4.83 ! [N: num] :
% 8.58/4.83 ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
% 8.58/4.83 = ( numeral_numeral_int @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5403_abs__mult__self__eq,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 8.58/4.83 = ( times_3573771949741848930nteger @ A @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5404_abs__mult__self__eq,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 8.58/4.83 = ( times_times_real @ A @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5405_abs__mult__self__eq,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
% 8.58/4.83 = ( times_times_rat @ A @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5406_abs__mult__self__eq,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
% 8.58/4.83 = ( times_times_int @ A @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5407_abs__1,axiom,
% 8.58/4.83 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 8.58/4.83 = one_one_Code_integer ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5408_abs__1,axiom,
% 8.58/4.83 ( ( abs_abs_complex @ one_one_complex )
% 8.58/4.83 = one_one_complex ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5409_abs__1,axiom,
% 8.58/4.83 ( ( abs_abs_real @ one_one_real )
% 8.58/4.83 = one_one_real ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5410_abs__1,axiom,
% 8.58/4.83 ( ( abs_abs_rat @ one_one_rat )
% 8.58/4.83 = one_one_rat ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5411_abs__1,axiom,
% 8.58/4.83 ( ( abs_abs_int @ one_one_int )
% 8.58/4.83 = one_one_int ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5412_abs__add__abs,axiom,
% 8.58/4.83 ! [A: code_integer,B: code_integer] :
% 8.58/4.83 ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
% 8.58/4.83 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5413_abs__add__abs,axiom,
% 8.58/4.83 ! [A: real,B: real] :
% 8.58/4.83 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 8.58/4.83 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5414_abs__add__abs,axiom,
% 8.58/4.83 ! [A: rat,B: rat] :
% 8.58/4.83 ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
% 8.58/4.83 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5415_abs__add__abs,axiom,
% 8.58/4.83 ! [A: int,B: int] :
% 8.58/4.83 ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
% 8.58/4.83 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5416_abs__minus__cancel,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 8.58/4.83 = ( abs_abs_real @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5417_abs__minus__cancel,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 8.58/4.83 = ( abs_abs_int @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5418_abs__minus__cancel,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 8.58/4.83 = ( abs_abs_rat @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5419_abs__minus__cancel,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 8.58/4.83 = ( abs_abs_Code_integer @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5420_abs__minus,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 8.58/4.83 = ( abs_abs_real @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5421_abs__minus,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 8.58/4.83 = ( abs_abs_int @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5422_abs__minus,axiom,
% 8.58/4.83 ! [A: complex] :
% 8.58/4.83 ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
% 8.58/4.83 = ( abs_abs_complex @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5423_abs__minus,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 8.58/4.83 = ( abs_abs_rat @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5424_abs__minus,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 8.58/4.83 = ( abs_abs_Code_integer @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5425_abs__divide,axiom,
% 8.58/4.83 ! [A: complex,B: complex] :
% 8.58/4.83 ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.58/4.83 = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5426_abs__divide,axiom,
% 8.58/4.83 ! [A: real,B: real] :
% 8.58/4.83 ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 8.58/4.83 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5427_abs__divide,axiom,
% 8.58/4.83 ! [A: rat,B: rat] :
% 8.58/4.83 ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 8.58/4.83 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5428_dvd__abs__iff,axiom,
% 8.58/4.83 ! [M: real,K: real] :
% 8.58/4.83 ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
% 8.58/4.83 = ( dvd_dvd_real @ M @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5429_dvd__abs__iff,axiom,
% 8.58/4.83 ! [M: int,K: int] :
% 8.58/4.83 ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
% 8.58/4.83 = ( dvd_dvd_int @ M @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5430_dvd__abs__iff,axiom,
% 8.58/4.83 ! [M: code_integer,K: code_integer] :
% 8.58/4.83 ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
% 8.58/4.83 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5431_dvd__abs__iff,axiom,
% 8.58/4.83 ! [M: rat,K: rat] :
% 8.58/4.83 ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
% 8.58/4.83 = ( dvd_dvd_rat @ M @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5432_abs__dvd__iff,axiom,
% 8.58/4.83 ! [M: real,K: real] :
% 8.58/4.83 ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
% 8.58/4.83 = ( dvd_dvd_real @ M @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5433_abs__dvd__iff,axiom,
% 8.58/4.83 ! [M: int,K: int] :
% 8.58/4.83 ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
% 8.58/4.83 = ( dvd_dvd_int @ M @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5434_abs__dvd__iff,axiom,
% 8.58/4.83 ! [M: code_integer,K: code_integer] :
% 8.58/4.83 ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
% 8.58/4.83 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5435_abs__dvd__iff,axiom,
% 8.58/4.83 ! [M: rat,K: rat] :
% 8.58/4.83 ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
% 8.58/4.83 = ( dvd_dvd_rat @ M @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5436_abs__of__nat,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 8.58/4.83 = ( semiri4939895301339042750nteger @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5437_abs__of__nat,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 8.58/4.83 = ( semiri5074537144036343181t_real @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5438_abs__of__nat,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 8.58/4.83 = ( semiri681578069525770553at_rat @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5439_abs__of__nat,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 8.58/4.83 = ( semiri1314217659103216013at_int @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5440_powr__eq__0__iff,axiom,
% 8.58/4.83 ! [W: real,Z2: real] :
% 8.58/4.83 ( ( ( powr_real @ W @ Z2 )
% 8.58/4.83 = zero_zero_real )
% 8.58/4.83 = ( W = zero_zero_real ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5441_powr__0,axiom,
% 8.58/4.83 ! [Z2: real] :
% 8.58/4.83 ( ( powr_real @ zero_zero_real @ Z2 )
% 8.58/4.83 = zero_zero_real ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5442_powr__one__eq__one,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( powr_real @ one_one_real @ A )
% 8.58/4.83 = one_one_real ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5443_abs__bool__eq,axiom,
% 8.58/4.83 ! [P: $o] :
% 8.58/4.83 ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
% 8.58/4.83 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5444_abs__bool__eq,axiom,
% 8.58/4.83 ! [P: $o] :
% 8.58/4.83 ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 8.58/4.83 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5445_abs__bool__eq,axiom,
% 8.58/4.83 ! [P: $o] :
% 8.58/4.83 ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 8.58/4.83 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5446_abs__bool__eq,axiom,
% 8.58/4.83 ! [P: $o] :
% 8.58/4.83 ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
% 8.58/4.83 = ( zero_n356916108424825756nteger @ P ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5447_abs__of__nonneg,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 8.58/4.83 => ( ( abs_abs_Code_integer @ A )
% 8.58/4.83 = A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5448_abs__of__nonneg,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.83 => ( ( abs_abs_real @ A )
% 8.58/4.83 = A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5449_abs__of__nonneg,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.58/4.83 => ( ( abs_abs_rat @ A )
% 8.58/4.83 = A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5450_abs__of__nonneg,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.58/4.83 => ( ( abs_abs_int @ A )
% 8.58/4.83 = A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5451_abs__le__self__iff,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 8.58/4.83 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5452_abs__le__self__iff,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 8.58/4.83 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5453_abs__le__self__iff,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 8.58/4.83 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5454_abs__le__self__iff,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 8.58/4.83 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5455_abs__le__zero__iff,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 8.58/4.83 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5456_abs__le__zero__iff,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 8.58/4.83 = ( A = zero_zero_real ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5457_abs__le__zero__iff,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 8.58/4.83 = ( A = zero_zero_rat ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5458_abs__le__zero__iff,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 8.58/4.83 = ( A = zero_zero_int ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5459_zero__less__abs__iff,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 8.58/4.83 = ( A != zero_z3403309356797280102nteger ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5460_zero__less__abs__iff,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 8.58/4.83 = ( A != zero_zero_real ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5461_zero__less__abs__iff,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 8.58/4.83 = ( A != zero_zero_rat ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5462_zero__less__abs__iff,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 8.58/4.83 = ( A != zero_zero_int ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5463_abs__neg__numeral,axiom,
% 8.58/4.83 ! [N: num] :
% 8.58/4.83 ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.58/4.83 = ( numeral_numeral_real @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5464_abs__neg__numeral,axiom,
% 8.58/4.83 ! [N: num] :
% 8.58/4.83 ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.83 = ( numeral_numeral_int @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5465_abs__neg__numeral,axiom,
% 8.58/4.83 ! [N: num] :
% 8.58/4.83 ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.58/4.83 = ( numeral_numeral_rat @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5466_abs__neg__numeral,axiom,
% 8.58/4.83 ! [N: num] :
% 8.58/4.83 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.58/4.83 = ( numera6620942414471956472nteger @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5467_abs__neg__one,axiom,
% 8.58/4.83 ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.83 = one_one_real ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5468_abs__neg__one,axiom,
% 8.58/4.83 ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.83 = one_one_int ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5469_abs__neg__one,axiom,
% 8.58/4.83 ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.83 = one_one_rat ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5470_abs__neg__one,axiom,
% 8.58/4.83 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.83 = one_one_Code_integer ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5471_abs__power__minus,axiom,
% 8.58/4.83 ! [A: real,N: nat] :
% 8.58/4.83 ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 8.58/4.83 = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5472_abs__power__minus,axiom,
% 8.58/4.83 ! [A: int,N: nat] :
% 8.58/4.83 ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 8.58/4.83 = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5473_abs__power__minus,axiom,
% 8.58/4.83 ! [A: rat,N: nat] :
% 8.58/4.83 ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 8.58/4.83 = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5474_abs__power__minus,axiom,
% 8.58/4.83 ! [A: code_integer,N: nat] :
% 8.58/4.83 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 8.58/4.83 = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5475_powr__zero__eq__one,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ( X = zero_zero_real )
% 8.58/4.83 => ( ( powr_real @ X @ zero_zero_real )
% 8.58/4.83 = zero_zero_real ) )
% 8.58/4.83 & ( ( X != zero_zero_real )
% 8.58/4.83 => ( ( powr_real @ X @ zero_zero_real )
% 8.58/4.83 = one_one_real ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5476_floor__zero,axiom,
% 8.58/4.83 ( ( archim6058952711729229775r_real @ zero_zero_real )
% 8.58/4.83 = zero_zero_int ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5477_floor__zero,axiom,
% 8.58/4.83 ( ( archim3151403230148437115or_rat @ zero_zero_rat )
% 8.58/4.83 = zero_zero_int ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5478_floor__numeral,axiom,
% 8.58/4.83 ! [V: num] :
% 8.58/4.83 ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
% 8.58/4.83 = ( numeral_numeral_int @ V ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5479_floor__numeral,axiom,
% 8.58/4.83 ! [V: num] :
% 8.58/4.83 ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
% 8.58/4.83 = ( numeral_numeral_int @ V ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5480_negative__eq__positive,axiom,
% 8.58/4.83 ! [N: nat,M: nat] :
% 8.58/4.83 ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
% 8.58/4.83 = ( semiri1314217659103216013at_int @ M ) )
% 8.58/4.83 = ( ( N = zero_zero_nat )
% 8.58/4.83 & ( M = zero_zero_nat ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5481_floor__one,axiom,
% 8.58/4.83 ( ( archim6058952711729229775r_real @ one_one_real )
% 8.58/4.83 = one_one_int ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5482_floor__one,axiom,
% 8.58/4.83 ( ( archim3151403230148437115or_rat @ one_one_rat )
% 8.58/4.83 = one_one_int ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5483_powr__less__cancel__iff,axiom,
% 8.58/4.83 ! [X: real,A: real,B: real] :
% 8.58/4.83 ( ( ord_less_real @ one_one_real @ X )
% 8.58/4.83 => ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 8.58/4.83 = ( ord_less_real @ A @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5484_negative__zle,axiom,
% 8.58/4.83 ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5485_floor__of__nat,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( archim6058952711729229775r_real @ ( semiri5074537144036343181t_real @ N ) )
% 8.58/4.83 = ( semiri1314217659103216013at_int @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5486_floor__of__nat,axiom,
% 8.58/4.83 ! [N: nat] :
% 8.58/4.83 ( ( archim3151403230148437115or_rat @ ( semiri681578069525770553at_rat @ N ) )
% 8.58/4.83 = ( semiri1314217659103216013at_int @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5487_sin__coeff__0,axiom,
% 8.58/4.83 ( ( sin_coeff @ zero_zero_nat )
% 8.58/4.83 = zero_zero_real ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5488_cos__coeff__0,axiom,
% 8.58/4.83 ( ( cos_coeff @ zero_zero_nat )
% 8.58/4.83 = one_one_real ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5489_abs__of__nonpos,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.58/4.83 => ( ( abs_abs_real @ A )
% 8.58/4.83 = ( uminus_uminus_real @ A ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5490_abs__of__nonpos,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 8.58/4.83 => ( ( abs_abs_Code_integer @ A )
% 8.58/4.83 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5491_abs__of__nonpos,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.58/4.83 => ( ( abs_abs_rat @ A )
% 8.58/4.83 = ( uminus_uminus_rat @ A ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5492_abs__of__nonpos,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.58/4.83 => ( ( abs_abs_int @ A )
% 8.58/4.83 = ( uminus_uminus_int @ A ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5493_divide__le__0__abs__iff,axiom,
% 8.58/4.83 ! [A: real,B: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 8.58/4.83 = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.58/4.83 | ( B = zero_zero_real ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5494_divide__le__0__abs__iff,axiom,
% 8.58/4.83 ! [A: rat,B: rat] :
% 8.58/4.83 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
% 8.58/4.83 = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.58/4.83 | ( B = zero_zero_rat ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5495_zero__le__divide__abs__iff,axiom,
% 8.58/4.83 ! [A: real,B: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 8.58/4.83 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.83 | ( B = zero_zero_real ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5496_zero__le__divide__abs__iff,axiom,
% 8.58/4.83 ! [A: rat,B: rat] :
% 8.58/4.83 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
% 8.58/4.83 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.58/4.83 | ( B = zero_zero_rat ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5497_abs__sgn__eq__1,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( A != zero_z3403309356797280102nteger )
% 8.58/4.83 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 8.58/4.83 = one_one_Code_integer ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5498_abs__sgn__eq__1,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( A != zero_zero_real )
% 8.58/4.83 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 8.58/4.83 = one_one_real ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5499_abs__sgn__eq__1,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( A != zero_zero_rat )
% 8.58/4.83 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 8.58/4.83 = one_one_rat ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5500_abs__sgn__eq__1,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( A != zero_zero_int )
% 8.58/4.83 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 8.58/4.83 = one_one_int ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5501_powr__eq__one__iff,axiom,
% 8.58/4.83 ! [A: real,X: real] :
% 8.58/4.83 ( ( ord_less_real @ one_one_real @ A )
% 8.58/4.83 => ( ( ( powr_real @ A @ X )
% 8.58/4.83 = one_one_real )
% 8.58/4.83 = ( X = zero_zero_real ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5502_powr__one__gt__zero__iff,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ( powr_real @ X @ one_one_real )
% 8.58/4.83 = X )
% 8.58/4.83 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5503_powr__one,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.83 => ( ( powr_real @ X @ one_one_real )
% 8.58/4.83 = X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5504_powr__le__cancel__iff,axiom,
% 8.58/4.83 ! [X: real,A: real,B: real] :
% 8.58/4.83 ( ( ord_less_real @ one_one_real @ X )
% 8.58/4.83 => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 8.58/4.83 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5505_negative__zless,axiom,
% 8.58/4.83 ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5506_artanh__minus__real,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.83 => ( ( artanh_real @ ( uminus_uminus_real @ X ) )
% 8.58/4.83 = ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5507_idom__abs__sgn__class_Oabs__sgn,axiom,
% 8.58/4.83 ! [A: complex] :
% 8.58/4.83 ( ( sgn_sgn_complex @ ( abs_abs_complex @ A ) )
% 8.58/4.83 = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5508_idom__abs__sgn__class_Oabs__sgn,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( sgn_sgn_real @ ( abs_abs_real @ A ) )
% 8.58/4.83 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5509_idom__abs__sgn__class_Oabs__sgn,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( sgn_sgn_rat @ ( abs_abs_rat @ A ) )
% 8.58/4.83 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5510_idom__abs__sgn__class_Oabs__sgn,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( sgn_sgn_int @ ( abs_abs_int @ A ) )
% 8.58/4.83 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5511_idom__abs__sgn__class_Oabs__sgn,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( sgn_sgn_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 8.58/4.83 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5512_sgn__abs,axiom,
% 8.58/4.83 ! [A: complex] :
% 8.58/4.83 ( ( abs_abs_complex @ ( sgn_sgn_complex @ A ) )
% 8.58/4.83 = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5513_sgn__abs,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 8.58/4.83 = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5514_sgn__abs,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 8.58/4.83 = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5515_sgn__abs,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 8.58/4.83 = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5516_sgn__abs,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 8.58/4.83 = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5517_numeral__powr__numeral__real,axiom,
% 8.58/4.83 ! [M: num,N: num] :
% 8.58/4.83 ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 8.58/4.83 = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5518_zero__less__power__abs__iff,axiom,
% 8.58/4.83 ! [A: code_integer,N: nat] :
% 8.58/4.83 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
% 8.58/4.83 = ( ( A != zero_z3403309356797280102nteger )
% 8.58/4.83 | ( N = zero_zero_nat ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5519_zero__less__power__abs__iff,axiom,
% 8.58/4.83 ! [A: real,N: nat] :
% 8.58/4.83 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 8.58/4.83 = ( ( A != zero_zero_real )
% 8.58/4.83 | ( N = zero_zero_nat ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5520_zero__less__power__abs__iff,axiom,
% 8.58/4.83 ! [A: rat,N: nat] :
% 8.58/4.83 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
% 8.58/4.83 = ( ( A != zero_zero_rat )
% 8.58/4.83 | ( N = zero_zero_nat ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5521_zero__less__power__abs__iff,axiom,
% 8.58/4.83 ! [A: int,N: nat] :
% 8.58/4.83 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 8.58/4.83 = ( ( A != zero_zero_int )
% 8.58/4.83 | ( N = zero_zero_nat ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5522_abs__power2,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.83 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5523_abs__power2,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.83 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5524_abs__power2,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.83 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5525_abs__power2,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.83 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5526_power2__abs,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.83 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5527_power2__abs,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.83 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5528_power2__abs,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.83 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5529_power2__abs,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.83 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5530_zero__le__floor,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
% 8.58/4.83 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5531_zero__le__floor,axiom,
% 8.58/4.83 ! [X: rat] :
% 8.58/4.83 ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
% 8.58/4.83 = ( ord_less_eq_rat @ zero_zero_rat @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5532_floor__less__zero,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
% 8.58/4.83 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5533_floor__less__zero,axiom,
% 8.58/4.83 ! [X: rat] :
% 8.58/4.83 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
% 8.58/4.83 = ( ord_less_rat @ X @ zero_zero_rat ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5534_numeral__le__floor,axiom,
% 8.58/4.83 ! [V: num,X: real] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
% 8.58/4.83 = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5535_numeral__le__floor,axiom,
% 8.58/4.83 ! [V: num,X: rat] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
% 8.58/4.83 = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5536_zero__less__floor,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
% 8.58/4.83 = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5537_zero__less__floor,axiom,
% 8.58/4.83 ! [X: rat] :
% 8.58/4.83 ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
% 8.58/4.83 = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5538_floor__le__zero,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
% 8.58/4.83 = ( ord_less_real @ X @ one_one_real ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5539_floor__le__zero,axiom,
% 8.58/4.83 ! [X: rat] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
% 8.58/4.83 = ( ord_less_rat @ X @ one_one_rat ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5540_floor__less__numeral,axiom,
% 8.58/4.83 ! [X: real,V: num] :
% 8.58/4.83 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
% 8.58/4.83 = ( ord_less_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5541_floor__less__numeral,axiom,
% 8.58/4.83 ! [X: rat,V: num] :
% 8.58/4.83 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 8.58/4.83 = ( ord_less_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5542_one__le__floor,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
% 8.58/4.83 = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5543_one__le__floor,axiom,
% 8.58/4.83 ! [X: rat] :
% 8.58/4.83 ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
% 8.58/4.83 = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5544_floor__less__one,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 8.58/4.83 = ( ord_less_real @ X @ one_one_real ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5545_floor__less__one,axiom,
% 8.58/4.83 ! [X: rat] :
% 8.58/4.83 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 8.58/4.83 = ( ord_less_rat @ X @ one_one_rat ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5546_floor__neg__numeral,axiom,
% 8.58/4.83 ! [V: num] :
% 8.58/4.83 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.58/4.83 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5547_floor__neg__numeral,axiom,
% 8.58/4.83 ! [V: num] :
% 8.58/4.83 ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.58/4.83 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5548_floor__diff__numeral,axiom,
% 8.58/4.83 ! [X: real,V: num] :
% 8.58/4.83 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
% 8.58/4.83 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5549_floor__diff__numeral,axiom,
% 8.58/4.83 ! [X: rat,V: num] :
% 8.58/4.83 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
% 8.58/4.83 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5550_floor__diff__one,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ one_one_real ) )
% 8.58/4.83 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5551_floor__diff__one,axiom,
% 8.58/4.83 ! [X: rat] :
% 8.58/4.83 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
% 8.58/4.83 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5552_floor__numeral__power,axiom,
% 8.58/4.83 ! [X: num,N: nat] :
% 8.58/4.83 ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 8.58/4.83 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5553_floor__numeral__power,axiom,
% 8.58/4.83 ! [X: num,N: nat] :
% 8.58/4.83 ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 8.58/4.83 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5554_log__powr__cancel,axiom,
% 8.58/4.83 ! [A: real,Y: real] :
% 8.58/4.83 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.83 => ( ( A != one_one_real )
% 8.58/4.83 => ( ( log2 @ A @ ( powr_real @ A @ Y ) )
% 8.58/4.83 = Y ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5555_powr__log__cancel,axiom,
% 8.58/4.83 ! [A: real,X: real] :
% 8.58/4.83 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.83 => ( ( A != one_one_real )
% 8.58/4.83 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.83 => ( ( powr_real @ A @ ( log2 @ A @ X ) )
% 8.58/4.83 = X ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5556_floor__divide__eq__div__numeral,axiom,
% 8.58/4.83 ! [A: num,B: num] :
% 8.58/4.83 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 8.58/4.83 = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5557_power__even__abs__numeral,axiom,
% 8.58/4.83 ! [W: num,A: code_integer] :
% 8.58/4.83 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.58/4.83 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
% 8.58/4.83 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5558_power__even__abs__numeral,axiom,
% 8.58/4.83 ! [W: num,A: rat] :
% 8.58/4.83 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.58/4.83 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
% 8.58/4.83 = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5559_power__even__abs__numeral,axiom,
% 8.58/4.83 ! [W: num,A: real] :
% 8.58/4.83 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.58/4.83 => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
% 8.58/4.83 = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5560_power__even__abs__numeral,axiom,
% 8.58/4.83 ! [W: num,A: int] :
% 8.58/4.83 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.58/4.83 => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
% 8.58/4.83 = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5561_powr__numeral,axiom,
% 8.58/4.83 ! [X: real,N: num] :
% 8.58/4.83 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.83 => ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
% 8.58/4.83 = ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5562_ceiling__divide__eq__div__numeral,axiom,
% 8.58/4.83 ! [A: num,B: num] :
% 8.58/4.83 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 8.58/4.83 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5563_numeral__less__floor,axiom,
% 8.58/4.83 ! [V: num,X: real] :
% 8.58/4.83 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
% 8.58/4.83 = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5564_numeral__less__floor,axiom,
% 8.58/4.83 ! [V: num,X: rat] :
% 8.58/4.83 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
% 8.58/4.83 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5565_floor__le__numeral,axiom,
% 8.58/4.83 ! [X: real,V: num] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
% 8.58/4.83 = ( ord_less_real @ X @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5566_floor__le__numeral,axiom,
% 8.58/4.83 ! [X: rat,V: num] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 8.58/4.83 = ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5567_one__less__floor,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
% 8.58/4.83 = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5568_one__less__floor,axiom,
% 8.58/4.83 ! [X: rat] :
% 8.58/4.83 ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
% 8.58/4.83 = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5569_floor__le__one,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 8.58/4.83 = ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5570_floor__le__one,axiom,
% 8.58/4.83 ! [X: rat] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 8.58/4.83 = ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5571_neg__numeral__le__floor,axiom,
% 8.58/4.83 ! [V: num,X: real] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
% 8.58/4.83 = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5572_neg__numeral__le__floor,axiom,
% 8.58/4.83 ! [V: num,X: rat] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
% 8.58/4.83 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5573_floor__less__neg__numeral,axiom,
% 8.58/4.83 ! [X: real,V: num] :
% 8.58/4.83 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.58/4.83 = ( ord_less_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5574_floor__less__neg__numeral,axiom,
% 8.58/4.83 ! [X: rat,V: num] :
% 8.58/4.83 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.58/4.83 = ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5575_signed__take__bit__Suc__minus__bit0,axiom,
% 8.58/4.83 ! [N: nat,K: num] :
% 8.58/4.83 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 8.58/4.83 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5576_floor__one__divide__eq__div__numeral,axiom,
% 8.58/4.83 ! [B: num] :
% 8.58/4.83 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
% 8.58/4.83 = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5577_floor__minus__divide__eq__div__numeral,axiom,
% 8.58/4.83 ! [A: num,B: num] :
% 8.58/4.83 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 8.58/4.83 = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5578_ceiling__minus__divide__eq__div__numeral,axiom,
% 8.58/4.83 ! [A: num,B: num] :
% 8.58/4.83 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 8.58/4.83 = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5579_neg__numeral__less__floor,axiom,
% 8.58/4.83 ! [V: num,X: real] :
% 8.58/4.83 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
% 8.58/4.83 = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5580_neg__numeral__less__floor,axiom,
% 8.58/4.83 ! [V: num,X: rat] :
% 8.58/4.83 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
% 8.58/4.83 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5581_floor__le__neg__numeral,axiom,
% 8.58/4.83 ! [X: real,V: num] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.58/4.83 = ( ord_less_real @ X @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5582_floor__le__neg__numeral,axiom,
% 8.58/4.83 ! [X: rat,V: num] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.58/4.83 = ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5583_square__powr__half,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.83 = ( abs_abs_real @ X ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5584_floor__minus__one__divide__eq__div__numeral,axiom,
% 8.58/4.83 ! [B: num] :
% 8.58/4.83 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
% 8.58/4.83 = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5585_abs__le__D1,axiom,
% 8.58/4.83 ! [A: real,B: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 8.58/4.83 => ( ord_less_eq_real @ A @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5586_abs__le__D1,axiom,
% 8.58/4.83 ! [A: code_integer,B: code_integer] :
% 8.58/4.83 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 8.58/4.83 => ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5587_abs__le__D1,axiom,
% 8.58/4.83 ! [A: rat,B: rat] :
% 8.58/4.83 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 8.58/4.83 => ( ord_less_eq_rat @ A @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5588_abs__le__D1,axiom,
% 8.58/4.83 ! [A: int,B: int] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 8.58/4.83 => ( ord_less_eq_int @ A @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5589_abs__ge__self,axiom,
% 8.58/4.83 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5590_abs__ge__self,axiom,
% 8.58/4.83 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5591_abs__ge__self,axiom,
% 8.58/4.83 ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5592_abs__ge__self,axiom,
% 8.58/4.83 ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5593_abs__eq__0__iff,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( ( abs_abs_Code_integer @ A )
% 8.58/4.83 = zero_z3403309356797280102nteger )
% 8.58/4.83 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5594_abs__eq__0__iff,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( ( abs_abs_real @ A )
% 8.58/4.83 = zero_zero_real )
% 8.58/4.83 = ( A = zero_zero_real ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5595_abs__eq__0__iff,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( ( abs_abs_rat @ A )
% 8.58/4.83 = zero_zero_rat )
% 8.58/4.83 = ( A = zero_zero_rat ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5596_abs__eq__0__iff,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( ( abs_abs_int @ A )
% 8.58/4.83 = zero_zero_int )
% 8.58/4.83 = ( A = zero_zero_int ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5597_abs__mult,axiom,
% 8.58/4.83 ! [A: code_integer,B: code_integer] :
% 8.58/4.83 ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 8.58/4.83 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5598_abs__mult,axiom,
% 8.58/4.83 ! [A: real,B: real] :
% 8.58/4.83 ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 8.58/4.83 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5599_abs__mult,axiom,
% 8.58/4.83 ! [A: rat,B: rat] :
% 8.58/4.83 ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 8.58/4.83 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5600_abs__mult,axiom,
% 8.58/4.83 ! [A: int,B: int] :
% 8.58/4.83 ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 8.58/4.83 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5601_abs__one,axiom,
% 8.58/4.83 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 8.58/4.83 = one_one_Code_integer ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5602_abs__one,axiom,
% 8.58/4.83 ( ( abs_abs_real @ one_one_real )
% 8.58/4.83 = one_one_real ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5603_abs__one,axiom,
% 8.58/4.83 ( ( abs_abs_rat @ one_one_rat )
% 8.58/4.83 = one_one_rat ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5604_abs__one,axiom,
% 8.58/4.83 ( ( abs_abs_int @ one_one_int )
% 8.58/4.83 = one_one_int ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5605_abs__minus__commute,axiom,
% 8.58/4.83 ! [A: code_integer,B: code_integer] :
% 8.58/4.83 ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
% 8.58/4.83 = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5606_abs__minus__commute,axiom,
% 8.58/4.83 ! [A: real,B: real] :
% 8.58/4.83 ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 8.58/4.83 = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5607_abs__minus__commute,axiom,
% 8.58/4.83 ! [A: rat,B: rat] :
% 8.58/4.83 ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
% 8.58/4.83 = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5608_abs__minus__commute,axiom,
% 8.58/4.83 ! [A: int,B: int] :
% 8.58/4.83 ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 8.58/4.83 = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5609_abs__eq__iff,axiom,
% 8.58/4.83 ! [X: real,Y: real] :
% 8.58/4.83 ( ( ( abs_abs_real @ X )
% 8.58/4.83 = ( abs_abs_real @ Y ) )
% 8.58/4.83 = ( ( X = Y )
% 8.58/4.83 | ( X
% 8.58/4.83 = ( uminus_uminus_real @ Y ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5610_abs__eq__iff,axiom,
% 8.58/4.83 ! [X: int,Y: int] :
% 8.58/4.83 ( ( ( abs_abs_int @ X )
% 8.58/4.83 = ( abs_abs_int @ Y ) )
% 8.58/4.83 = ( ( X = Y )
% 8.58/4.83 | ( X
% 8.58/4.83 = ( uminus_uminus_int @ Y ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5611_abs__eq__iff,axiom,
% 8.58/4.83 ! [X: rat,Y: rat] :
% 8.58/4.83 ( ( ( abs_abs_rat @ X )
% 8.58/4.83 = ( abs_abs_rat @ Y ) )
% 8.58/4.83 = ( ( X = Y )
% 8.58/4.83 | ( X
% 8.58/4.83 = ( uminus_uminus_rat @ Y ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5612_abs__eq__iff,axiom,
% 8.58/4.83 ! [X: code_integer,Y: code_integer] :
% 8.58/4.83 ( ( ( abs_abs_Code_integer @ X )
% 8.58/4.83 = ( abs_abs_Code_integer @ Y ) )
% 8.58/4.83 = ( ( X = Y )
% 8.58/4.83 | ( X
% 8.58/4.83 = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5613_power__abs,axiom,
% 8.58/4.83 ! [A: code_integer,N: nat] :
% 8.58/4.83 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 8.58/4.83 = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5614_power__abs,axiom,
% 8.58/4.83 ! [A: rat,N: nat] :
% 8.58/4.83 ( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
% 8.58/4.83 = ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5615_power__abs,axiom,
% 8.58/4.83 ! [A: real,N: nat] :
% 8.58/4.83 ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
% 8.58/4.83 = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5616_power__abs,axiom,
% 8.58/4.83 ! [A: int,N: nat] :
% 8.58/4.83 ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
% 8.58/4.83 = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5617_dvd__if__abs__eq,axiom,
% 8.58/4.83 ! [L: real,K: real] :
% 8.58/4.83 ( ( ( abs_abs_real @ L )
% 8.58/4.83 = ( abs_abs_real @ K ) )
% 8.58/4.83 => ( dvd_dvd_real @ L @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5618_dvd__if__abs__eq,axiom,
% 8.58/4.83 ! [L: int,K: int] :
% 8.58/4.83 ( ( ( abs_abs_int @ L )
% 8.58/4.83 = ( abs_abs_int @ K ) )
% 8.58/4.83 => ( dvd_dvd_int @ L @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5619_dvd__if__abs__eq,axiom,
% 8.58/4.83 ! [L: code_integer,K: code_integer] :
% 8.58/4.83 ( ( ( abs_abs_Code_integer @ L )
% 8.58/4.83 = ( abs_abs_Code_integer @ K ) )
% 8.58/4.83 => ( dvd_dvd_Code_integer @ L @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5620_dvd__if__abs__eq,axiom,
% 8.58/4.83 ! [L: rat,K: rat] :
% 8.58/4.83 ( ( ( abs_abs_rat @ L )
% 8.58/4.83 = ( abs_abs_rat @ K ) )
% 8.58/4.83 => ( dvd_dvd_rat @ L @ K ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5621_powr__powr,axiom,
% 8.58/4.83 ! [X: real,A: real,B: real] :
% 8.58/4.83 ( ( powr_real @ ( powr_real @ X @ A ) @ B )
% 8.58/4.83 = ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5622_real__sgn__eq,axiom,
% 8.58/4.83 ( sgn_sgn_real
% 8.58/4.83 = ( ^ [X3: real] : ( divide_divide_real @ X3 @ ( abs_abs_real @ X3 ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5623_uminus__int__code_I1_J,axiom,
% 8.58/4.83 ( ( uminus_uminus_int @ zero_zero_int )
% 8.58/4.83 = zero_zero_int ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5624_int__cases2,axiom,
% 8.58/4.83 ! [Z2: int] :
% 8.58/4.83 ( ! [N2: nat] :
% 8.58/4.83 ( Z2
% 8.58/4.83 != ( semiri1314217659103216013at_int @ N2 ) )
% 8.58/4.83 => ~ ! [N2: nat] :
% 8.58/4.83 ( Z2
% 8.58/4.83 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5625_uminus__dvd__conv_I1_J,axiom,
% 8.58/4.83 ( dvd_dvd_int
% 8.58/4.83 = ( ^ [D4: int] : ( dvd_dvd_int @ ( uminus_uminus_int @ D4 ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5626_uminus__dvd__conv_I2_J,axiom,
% 8.58/4.83 ( dvd_dvd_int
% 8.58/4.83 = ( ^ [D4: int,T2: int] : ( dvd_dvd_int @ D4 @ ( uminus_uminus_int @ T2 ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5627_take__bit__minus,axiom,
% 8.58/4.83 ! [N: nat,K: int] :
% 8.58/4.83 ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 8.58/4.83 = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5628_signed__take__bit__minus,axiom,
% 8.58/4.83 ! [N: nat,K: int] :
% 8.58/4.83 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
% 8.58/4.83 = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5629_abs__ge__zero,axiom,
% 8.58/4.83 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5630_abs__ge__zero,axiom,
% 8.58/4.83 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5631_abs__ge__zero,axiom,
% 8.58/4.83 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5632_abs__ge__zero,axiom,
% 8.58/4.83 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5633_floor__mono,axiom,
% 8.58/4.83 ! [X: real,Y: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ X @ Y )
% 8.58/4.83 => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5634_floor__mono,axiom,
% 8.58/4.83 ! [X: rat,Y: rat] :
% 8.58/4.83 ( ( ord_less_eq_rat @ X @ Y )
% 8.58/4.83 => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5635_abs__not__less__zero,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5636_abs__not__less__zero,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5637_abs__not__less__zero,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5638_abs__not__less__zero,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5639_abs__of__pos,axiom,
% 8.58/4.83 ! [A: code_integer] :
% 8.58/4.83 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 8.58/4.83 => ( ( abs_abs_Code_integer @ A )
% 8.58/4.83 = A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5640_abs__of__pos,axiom,
% 8.58/4.83 ! [A: real] :
% 8.58/4.83 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.83 => ( ( abs_abs_real @ A )
% 8.58/4.83 = A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5641_abs__of__pos,axiom,
% 8.58/4.83 ! [A: rat] :
% 8.58/4.83 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.58/4.83 => ( ( abs_abs_rat @ A )
% 8.58/4.83 = A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5642_abs__of__pos,axiom,
% 8.58/4.83 ! [A: int] :
% 8.58/4.83 ( ( ord_less_int @ zero_zero_int @ A )
% 8.58/4.83 => ( ( abs_abs_int @ A )
% 8.58/4.83 = A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5643_floor__less__cancel,axiom,
% 8.58/4.83 ! [X: real,Y: real] :
% 8.58/4.83 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) )
% 8.58/4.83 => ( ord_less_real @ X @ Y ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5644_floor__less__cancel,axiom,
% 8.58/4.83 ! [X: rat,Y: rat] :
% 8.58/4.83 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) )
% 8.58/4.83 => ( ord_less_rat @ X @ Y ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5645_abs__triangle__ineq,axiom,
% 8.58/4.83 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5646_abs__triangle__ineq,axiom,
% 8.58/4.83 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5647_abs__triangle__ineq,axiom,
% 8.58/4.83 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5648_abs__triangle__ineq,axiom,
% 8.58/4.83 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5649_abs__mult__less,axiom,
% 8.58/4.83 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 8.58/4.83 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 8.58/4.83 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
% 8.58/4.83 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5650_abs__mult__less,axiom,
% 8.58/4.83 ! [A: real,C: real,B: real,D: real] :
% 8.58/4.83 ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 8.58/4.83 => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
% 8.58/4.83 => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5651_abs__mult__less,axiom,
% 8.58/4.83 ! [A: rat,C: rat,B: rat,D: rat] :
% 8.58/4.83 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 8.58/4.83 => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
% 8.58/4.83 => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5652_abs__mult__less,axiom,
% 8.58/4.83 ! [A: int,C: int,B: int,D: int] :
% 8.58/4.83 ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 8.58/4.83 => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
% 8.58/4.83 => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5653_abs__triangle__ineq2,axiom,
% 8.58/4.83 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5654_abs__triangle__ineq2,axiom,
% 8.58/4.83 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5655_abs__triangle__ineq2,axiom,
% 8.58/4.83 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5656_abs__triangle__ineq2,axiom,
% 8.58/4.83 ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5657_abs__triangle__ineq3,axiom,
% 8.58/4.83 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5658_abs__triangle__ineq3,axiom,
% 8.58/4.83 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5659_abs__triangle__ineq3,axiom,
% 8.58/4.83 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5660_abs__triangle__ineq3,axiom,
% 8.58/4.83 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5661_abs__triangle__ineq2__sym,axiom,
% 8.58/4.83 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5662_abs__triangle__ineq2__sym,axiom,
% 8.58/4.83 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5663_abs__triangle__ineq2__sym,axiom,
% 8.58/4.83 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5664_abs__triangle__ineq2__sym,axiom,
% 8.58/4.83 ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5665_abs__leI,axiom,
% 8.58/4.83 ! [A: real,B: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ A @ B )
% 8.58/4.83 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 8.58/4.83 => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5666_abs__leI,axiom,
% 8.58/4.83 ! [A: code_integer,B: code_integer] :
% 8.58/4.83 ( ( ord_le3102999989581377725nteger @ A @ B )
% 8.58/4.83 => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 8.58/4.83 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5667_abs__leI,axiom,
% 8.58/4.83 ! [A: rat,B: rat] :
% 8.58/4.83 ( ( ord_less_eq_rat @ A @ B )
% 8.58/4.83 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 8.58/4.83 => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5668_abs__leI,axiom,
% 8.58/4.83 ! [A: int,B: int] :
% 8.58/4.83 ( ( ord_less_eq_int @ A @ B )
% 8.58/4.83 => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 8.58/4.83 => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5669_abs__le__D2,axiom,
% 8.58/4.83 ! [A: real,B: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 8.58/4.83 => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5670_abs__le__D2,axiom,
% 8.58/4.83 ! [A: code_integer,B: code_integer] :
% 8.58/4.83 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 8.58/4.83 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5671_abs__le__D2,axiom,
% 8.58/4.83 ! [A: rat,B: rat] :
% 8.58/4.83 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 8.58/4.83 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5672_abs__le__D2,axiom,
% 8.58/4.83 ! [A: int,B: int] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 8.58/4.83 => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5673_abs__le__iff,axiom,
% 8.58/4.83 ! [A: real,B: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 8.58/4.83 = ( ( ord_less_eq_real @ A @ B )
% 8.58/4.83 & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5674_abs__le__iff,axiom,
% 8.58/4.83 ! [A: code_integer,B: code_integer] :
% 8.58/4.83 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 8.58/4.83 = ( ( ord_le3102999989581377725nteger @ A @ B )
% 8.58/4.83 & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5675_abs__le__iff,axiom,
% 8.58/4.83 ! [A: rat,B: rat] :
% 8.58/4.83 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 8.58/4.83 = ( ( ord_less_eq_rat @ A @ B )
% 8.58/4.83 & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5676_abs__le__iff,axiom,
% 8.58/4.83 ! [A: int,B: int] :
% 8.58/4.83 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 8.58/4.83 = ( ( ord_less_eq_int @ A @ B )
% 8.58/4.83 & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5677_abs__ge__minus__self,axiom,
% 8.58/4.83 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5678_abs__ge__minus__self,axiom,
% 8.58/4.83 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5679_abs__ge__minus__self,axiom,
% 8.58/4.83 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5680_abs__ge__minus__self,axiom,
% 8.58/4.83 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5681_nonzero__abs__divide,axiom,
% 8.58/4.83 ! [B: real,A: real] :
% 8.58/4.83 ( ( B != zero_zero_real )
% 8.58/4.83 => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 8.58/4.83 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5682_nonzero__abs__divide,axiom,
% 8.58/4.83 ! [B: rat,A: rat] :
% 8.58/4.83 ( ( B != zero_zero_rat )
% 8.58/4.83 => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 8.58/4.83 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5683_abs__less__iff,axiom,
% 8.58/4.83 ! [A: real,B: real] :
% 8.58/4.83 ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 8.58/4.83 = ( ( ord_less_real @ A @ B )
% 8.58/4.83 & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5684_abs__less__iff,axiom,
% 8.58/4.83 ! [A: int,B: int] :
% 8.58/4.83 ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
% 8.58/4.83 = ( ( ord_less_int @ A @ B )
% 8.58/4.83 & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5685_abs__less__iff,axiom,
% 8.58/4.83 ! [A: rat,B: rat] :
% 8.58/4.83 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
% 8.58/4.83 = ( ( ord_less_rat @ A @ B )
% 8.58/4.83 & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5686_abs__less__iff,axiom,
% 8.58/4.83 ! [A: code_integer,B: code_integer] :
% 8.58/4.83 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 8.58/4.83 = ( ( ord_le6747313008572928689nteger @ A @ B )
% 8.58/4.83 & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5687_powr__less__cancel,axiom,
% 8.58/4.83 ! [X: real,A: real,B: real] :
% 8.58/4.83 ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 8.58/4.83 => ( ( ord_less_real @ one_one_real @ X )
% 8.58/4.83 => ( ord_less_real @ A @ B ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5688_powr__less__mono,axiom,
% 8.58/4.83 ! [A: real,B: real,X: real] :
% 8.58/4.83 ( ( ord_less_real @ A @ B )
% 8.58/4.83 => ( ( ord_less_real @ one_one_real @ X )
% 8.58/4.83 => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5689_powr__mono,axiom,
% 8.58/4.83 ! [A: real,B: real,X: real] :
% 8.58/4.83 ( ( ord_less_eq_real @ A @ B )
% 8.58/4.83 => ( ( ord_less_eq_real @ one_one_real @ X )
% 8.58/4.83 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5690_ceiling__def,axiom,
% 8.58/4.83 ( archim7802044766580827645g_real
% 8.58/4.83 = ( ^ [X3: real] : ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5691_ceiling__def,axiom,
% 8.58/4.83 ( archim2889992004027027881ng_rat
% 8.58/4.83 = ( ^ [X3: rat] : ( uminus_uminus_int @ ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ X3 ) ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5692_floor__minus,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X ) )
% 8.58/4.83 = ( uminus_uminus_int @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5693_floor__minus,axiom,
% 8.58/4.83 ! [X: rat] :
% 8.58/4.83 ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ X ) )
% 8.58/4.83 = ( uminus_uminus_int @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5694_ceiling__minus,axiom,
% 8.58/4.83 ! [X: real] :
% 8.58/4.83 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ X ) )
% 8.58/4.83 = ( uminus_uminus_int @ ( archim6058952711729229775r_real @ X ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5695_ceiling__minus,axiom,
% 8.58/4.83 ! [X: rat] :
% 8.58/4.83 ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ X ) )
% 8.58/4.83 = ( uminus_uminus_int @ ( archim3151403230148437115or_rat @ X ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5696_linordered__idom__class_Oabs__sgn,axiom,
% 8.58/4.83 ( abs_abs_Code_integer
% 8.58/4.83 = ( ^ [K2: code_integer] : ( times_3573771949741848930nteger @ K2 @ ( sgn_sgn_Code_integer @ K2 ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5697_linordered__idom__class_Oabs__sgn,axiom,
% 8.58/4.83 ( abs_abs_real
% 8.58/4.83 = ( ^ [K2: real] : ( times_times_real @ K2 @ ( sgn_sgn_real @ K2 ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5698_linordered__idom__class_Oabs__sgn,axiom,
% 8.58/4.83 ( abs_abs_rat
% 8.58/4.83 = ( ^ [K2: rat] : ( times_times_rat @ K2 @ ( sgn_sgn_rat @ K2 ) ) ) ) ).
% 8.58/4.83
% 8.58/4.83 thf(fact_5699_linordered__idom__class_Oabs__sgn,axiom,
% 8.58/4.83 ( abs_abs_int
% 8.58/4.83 = ( ^ [K2: int] : ( times_times_int @ K2 @ ( sgn_sgn_int @ K2 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5700_abs__mult__sgn,axiom,
% 8.58/4.84 ! [A: complex] :
% 8.58/4.84 ( ( times_times_complex @ ( abs_abs_complex @ A ) @ ( sgn_sgn_complex @ A ) )
% 8.58/4.84 = A ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5701_abs__mult__sgn,axiom,
% 8.58/4.84 ! [A: code_integer] :
% 8.58/4.84 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
% 8.58/4.84 = A ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5702_abs__mult__sgn,axiom,
% 8.58/4.84 ! [A: real] :
% 8.58/4.84 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( sgn_sgn_real @ A ) )
% 8.58/4.84 = A ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5703_abs__mult__sgn,axiom,
% 8.58/4.84 ! [A: rat] :
% 8.58/4.84 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( sgn_sgn_rat @ A ) )
% 8.58/4.84 = A ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5704_abs__mult__sgn,axiom,
% 8.58/4.84 ! [A: int] :
% 8.58/4.84 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
% 8.58/4.84 = A ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5705_sgn__mult__abs,axiom,
% 8.58/4.84 ! [A: complex] :
% 8.58/4.84 ( ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( abs_abs_complex @ A ) )
% 8.58/4.84 = A ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5706_sgn__mult__abs,axiom,
% 8.58/4.84 ! [A: code_integer] :
% 8.58/4.84 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 8.58/4.84 = A ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5707_sgn__mult__abs,axiom,
% 8.58/4.84 ! [A: real] :
% 8.58/4.84 ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( abs_abs_real @ A ) )
% 8.58/4.84 = A ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5708_sgn__mult__abs,axiom,
% 8.58/4.84 ! [A: rat] :
% 8.58/4.84 ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( abs_abs_rat @ A ) )
% 8.58/4.84 = A ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5709_sgn__mult__abs,axiom,
% 8.58/4.84 ! [A: int] :
% 8.58/4.84 ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
% 8.58/4.84 = A ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5710_mult__sgn__abs,axiom,
% 8.58/4.84 ! [X: code_integer] :
% 8.58/4.84 ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ X ) @ ( abs_abs_Code_integer @ X ) )
% 8.58/4.84 = X ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5711_mult__sgn__abs,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( times_times_real @ ( sgn_sgn_real @ X ) @ ( abs_abs_real @ X ) )
% 8.58/4.84 = X ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5712_mult__sgn__abs,axiom,
% 8.58/4.84 ! [X: rat] :
% 8.58/4.84 ( ( times_times_rat @ ( sgn_sgn_rat @ X ) @ ( abs_abs_rat @ X ) )
% 8.58/4.84 = X ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5713_mult__sgn__abs,axiom,
% 8.58/4.84 ! [X: int] :
% 8.58/4.84 ( ( times_times_int @ ( sgn_sgn_int @ X ) @ ( abs_abs_int @ X ) )
% 8.58/4.84 = X ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5714_same__sgn__abs__add,axiom,
% 8.58/4.84 ! [B: code_integer,A: code_integer] :
% 8.58/4.84 ( ( ( sgn_sgn_Code_integer @ B )
% 8.58/4.84 = ( sgn_sgn_Code_integer @ A ) )
% 8.58/4.84 => ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 8.58/4.84 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5715_same__sgn__abs__add,axiom,
% 8.58/4.84 ! [B: real,A: real] :
% 8.58/4.84 ( ( ( sgn_sgn_real @ B )
% 8.58/4.84 = ( sgn_sgn_real @ A ) )
% 8.58/4.84 => ( ( abs_abs_real @ ( plus_plus_real @ A @ B ) )
% 8.58/4.84 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5716_same__sgn__abs__add,axiom,
% 8.58/4.84 ! [B: rat,A: rat] :
% 8.58/4.84 ( ( ( sgn_sgn_rat @ B )
% 8.58/4.84 = ( sgn_sgn_rat @ A ) )
% 8.58/4.84 => ( ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) )
% 8.58/4.84 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5717_same__sgn__abs__add,axiom,
% 8.58/4.84 ! [B: int,A: int] :
% 8.58/4.84 ( ( ( sgn_sgn_int @ B )
% 8.58/4.84 = ( sgn_sgn_int @ A ) )
% 8.58/4.84 => ( ( abs_abs_int @ ( plus_plus_int @ A @ B ) )
% 8.58/4.84 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5718_floor__le__ceiling,axiom,
% 8.58/4.84 ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5719_floor__le__ceiling,axiom,
% 8.58/4.84 ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim2889992004027027881ng_rat @ X ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5720_int__of__nat__induct,axiom,
% 8.58/4.84 ! [P: int > $o,Z2: int] :
% 8.58/4.84 ( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
% 8.58/4.84 => ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
% 8.58/4.84 => ( P @ Z2 ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5721_int__cases,axiom,
% 8.58/4.84 ! [Z2: int] :
% 8.58/4.84 ( ! [N2: nat] :
% 8.58/4.84 ( Z2
% 8.58/4.84 != ( semiri1314217659103216013at_int @ N2 ) )
% 8.58/4.84 => ~ ! [N2: nat] :
% 8.58/4.84 ( Z2
% 8.58/4.84 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5722_pos__zmult__eq__1__iff__lemma,axiom,
% 8.58/4.84 ! [M: int,N: int] :
% 8.58/4.84 ( ( ( times_times_int @ M @ N )
% 8.58/4.84 = one_one_int )
% 8.58/4.84 => ( ( M = one_one_int )
% 8.58/4.84 | ( M
% 8.58/4.84 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5723_zmult__eq__1__iff,axiom,
% 8.58/4.84 ! [M: int,N: int] :
% 8.58/4.84 ( ( ( times_times_int @ M @ N )
% 8.58/4.84 = one_one_int )
% 8.58/4.84 = ( ( ( M = one_one_int )
% 8.58/4.84 & ( N = one_one_int ) )
% 8.58/4.84 | ( ( M
% 8.58/4.84 = ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.84 & ( N
% 8.58/4.84 = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5724_minus__int__code_I2_J,axiom,
% 8.58/4.84 ! [L: int] :
% 8.58/4.84 ( ( minus_minus_int @ zero_zero_int @ L )
% 8.58/4.84 = ( uminus_uminus_int @ L ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5725_not__int__zless__negative,axiom,
% 8.58/4.84 ! [N: nat,M: nat] :
% 8.58/4.84 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5726_zmod__zminus1__not__zero,axiom,
% 8.58/4.84 ! [K: int,L: int] :
% 8.58/4.84 ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 8.58/4.84 != zero_zero_int )
% 8.58/4.84 => ( ( modulo_modulo_int @ K @ L )
% 8.58/4.84 != zero_zero_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5727_zmod__zminus2__not__zero,axiom,
% 8.58/4.84 ! [K: int,L: int] :
% 8.58/4.84 ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
% 8.58/4.84 != zero_zero_int )
% 8.58/4.84 => ( ( modulo_modulo_int @ K @ L )
% 8.58/4.84 != zero_zero_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5728_sin__coeff__Suc,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( sin_coeff @ ( suc @ N ) )
% 8.58/4.84 = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5729_dense__eq0__I,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ! [E2: real] :
% 8.58/4.84 ( ( ord_less_real @ zero_zero_real @ E2 )
% 8.58/4.84 => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E2 ) )
% 8.58/4.84 => ( X = zero_zero_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5730_dense__eq0__I,axiom,
% 8.58/4.84 ! [X: rat] :
% 8.58/4.84 ( ! [E2: rat] :
% 8.58/4.84 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 8.58/4.84 => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E2 ) )
% 8.58/4.84 => ( X = zero_zero_rat ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5731_abs__eq__mult,axiom,
% 8.58/4.84 ! [A: code_integer,B: code_integer] :
% 8.58/4.84 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 8.58/4.84 | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 8.58/4.84 & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 8.58/4.84 | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 8.58/4.84 => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 8.58/4.84 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5732_abs__eq__mult,axiom,
% 8.58/4.84 ! [A: real,B: real] :
% 8.58/4.84 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.84 | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 8.58/4.84 & ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.58/4.84 | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 8.58/4.84 => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 8.58/4.84 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5733_abs__eq__mult,axiom,
% 8.58/4.84 ! [A: rat,B: rat] :
% 8.58/4.84 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.58/4.84 | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 8.58/4.84 & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.58/4.84 | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 8.58/4.84 => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 8.58/4.84 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5734_abs__eq__mult,axiom,
% 8.58/4.84 ! [A: int,B: int] :
% 8.58/4.84 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.58/4.84 | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 8.58/4.84 & ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.58/4.84 | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 8.58/4.84 => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 8.58/4.84 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5735_abs__mult__pos,axiom,
% 8.58/4.84 ! [X: code_integer,Y: code_integer] :
% 8.58/4.84 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 8.58/4.84 => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
% 8.58/4.84 = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5736_abs__mult__pos,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
% 8.58/4.84 = ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5737_abs__mult__pos,axiom,
% 8.58/4.84 ! [X: rat,Y: rat] :
% 8.58/4.84 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 8.58/4.84 => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
% 8.58/4.84 = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5738_abs__mult__pos,axiom,
% 8.58/4.84 ! [X: int,Y: int] :
% 8.58/4.84 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.58/4.84 => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
% 8.58/4.84 = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5739_abs__eq__iff_H,axiom,
% 8.58/4.84 ! [A: real,B: real] :
% 8.58/4.84 ( ( ( abs_abs_real @ A )
% 8.58/4.84 = B )
% 8.58/4.84 = ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.58/4.84 & ( ( A = B )
% 8.58/4.84 | ( A
% 8.58/4.84 = ( uminus_uminus_real @ B ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5740_abs__eq__iff_H,axiom,
% 8.58/4.84 ! [A: code_integer,B: code_integer] :
% 8.58/4.84 ( ( ( abs_abs_Code_integer @ A )
% 8.58/4.84 = B )
% 8.58/4.84 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 8.58/4.84 & ( ( A = B )
% 8.58/4.84 | ( A
% 8.58/4.84 = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5741_abs__eq__iff_H,axiom,
% 8.58/4.84 ! [A: rat,B: rat] :
% 8.58/4.84 ( ( ( abs_abs_rat @ A )
% 8.58/4.84 = B )
% 8.58/4.84 = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.58/4.84 & ( ( A = B )
% 8.58/4.84 | ( A
% 8.58/4.84 = ( uminus_uminus_rat @ B ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5742_abs__eq__iff_H,axiom,
% 8.58/4.84 ! [A: int,B: int] :
% 8.58/4.84 ( ( ( abs_abs_int @ A )
% 8.58/4.84 = B )
% 8.58/4.84 = ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.58/4.84 & ( ( A = B )
% 8.58/4.84 | ( A
% 8.58/4.84 = ( uminus_uminus_int @ B ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5743_eq__abs__iff_H,axiom,
% 8.58/4.84 ! [A: real,B: real] :
% 8.58/4.84 ( ( A
% 8.58/4.84 = ( abs_abs_real @ B ) )
% 8.58/4.84 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.84 & ( ( B = A )
% 8.58/4.84 | ( B
% 8.58/4.84 = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5744_eq__abs__iff_H,axiom,
% 8.58/4.84 ! [A: code_integer,B: code_integer] :
% 8.58/4.84 ( ( A
% 8.58/4.84 = ( abs_abs_Code_integer @ B ) )
% 8.58/4.84 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 8.58/4.84 & ( ( B = A )
% 8.58/4.84 | ( B
% 8.58/4.84 = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5745_eq__abs__iff_H,axiom,
% 8.58/4.84 ! [A: rat,B: rat] :
% 8.58/4.84 ( ( A
% 8.58/4.84 = ( abs_abs_rat @ B ) )
% 8.58/4.84 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.58/4.84 & ( ( B = A )
% 8.58/4.84 | ( B
% 8.58/4.84 = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5746_eq__abs__iff_H,axiom,
% 8.58/4.84 ! [A: int,B: int] :
% 8.58/4.84 ( ( A
% 8.58/4.84 = ( abs_abs_int @ B ) )
% 8.58/4.84 = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.58/4.84 & ( ( B = A )
% 8.58/4.84 | ( B
% 8.58/4.84 = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5747_abs__minus__le__zero,axiom,
% 8.58/4.84 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5748_abs__minus__le__zero,axiom,
% 8.58/4.84 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5749_abs__minus__le__zero,axiom,
% 8.58/4.84 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5750_abs__minus__le__zero,axiom,
% 8.58/4.84 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5751_zero__le__power__abs,axiom,
% 8.58/4.84 ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5752_zero__le__power__abs,axiom,
% 8.58/4.84 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5753_zero__le__power__abs,axiom,
% 8.58/4.84 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5754_zero__le__power__abs,axiom,
% 8.58/4.84 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5755_abs__if,axiom,
% 8.58/4.84 ( abs_abs_real
% 8.58/4.84 = ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5756_abs__if,axiom,
% 8.58/4.84 ( abs_abs_int
% 8.58/4.84 = ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5757_abs__if,axiom,
% 8.58/4.84 ( abs_abs_rat
% 8.58/4.84 = ( ^ [A2: rat] : ( if_rat @ ( ord_less_rat @ A2 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A2 ) @ A2 ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5758_abs__if,axiom,
% 8.58/4.84 ( abs_abs_Code_integer
% 8.58/4.84 = ( ^ [A2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A2 ) @ A2 ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5759_abs__of__neg,axiom,
% 8.58/4.84 ! [A: real] :
% 8.58/4.84 ( ( ord_less_real @ A @ zero_zero_real )
% 8.58/4.84 => ( ( abs_abs_real @ A )
% 8.58/4.84 = ( uminus_uminus_real @ A ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5760_abs__of__neg,axiom,
% 8.58/4.84 ! [A: int] :
% 8.58/4.84 ( ( ord_less_int @ A @ zero_zero_int )
% 8.58/4.84 => ( ( abs_abs_int @ A )
% 8.58/4.84 = ( uminus_uminus_int @ A ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5761_abs__of__neg,axiom,
% 8.58/4.84 ! [A: rat] :
% 8.58/4.84 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.58/4.84 => ( ( abs_abs_rat @ A )
% 8.58/4.84 = ( uminus_uminus_rat @ A ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5762_abs__of__neg,axiom,
% 8.58/4.84 ! [A: code_integer] :
% 8.58/4.84 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 8.58/4.84 => ( ( abs_abs_Code_integer @ A )
% 8.58/4.84 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5763_abs__if__raw,axiom,
% 8.58/4.84 ( abs_abs_real
% 8.58/4.84 = ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5764_abs__if__raw,axiom,
% 8.58/4.84 ( abs_abs_int
% 8.58/4.84 = ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5765_abs__if__raw,axiom,
% 8.58/4.84 ( abs_abs_rat
% 8.58/4.84 = ( ^ [A2: rat] : ( if_rat @ ( ord_less_rat @ A2 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A2 ) @ A2 ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5766_abs__if__raw,axiom,
% 8.58/4.84 ( abs_abs_Code_integer
% 8.58/4.84 = ( ^ [A2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A2 ) @ A2 ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5767_abs__div__pos,axiom,
% 8.58/4.84 ! [Y: real,X: real] :
% 8.58/4.84 ( ( ord_less_real @ zero_zero_real @ Y )
% 8.58/4.84 => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
% 8.58/4.84 = ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5768_abs__div__pos,axiom,
% 8.58/4.84 ! [Y: rat,X: rat] :
% 8.58/4.84 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 8.58/4.84 => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
% 8.58/4.84 = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5769_abs__diff__triangle__ineq,axiom,
% 8.58/4.84 ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5770_abs__diff__triangle__ineq,axiom,
% 8.58/4.84 ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5771_abs__diff__triangle__ineq,axiom,
% 8.58/4.84 ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5772_abs__diff__triangle__ineq,axiom,
% 8.58/4.84 ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5773_abs__triangle__ineq4,axiom,
% 8.58/4.84 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5774_abs__triangle__ineq4,axiom,
% 8.58/4.84 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5775_abs__triangle__ineq4,axiom,
% 8.58/4.84 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5776_abs__triangle__ineq4,axiom,
% 8.58/4.84 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5777_abs__diff__le__iff,axiom,
% 8.58/4.84 ! [X: code_integer,A: code_integer,R: code_integer] :
% 8.58/4.84 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R )
% 8.58/4.84 = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X )
% 8.58/4.84 & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5778_abs__diff__le__iff,axiom,
% 8.58/4.84 ! [X: real,A: real,R: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
% 8.58/4.84 = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X )
% 8.58/4.84 & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5779_abs__diff__le__iff,axiom,
% 8.58/4.84 ! [X: rat,A: rat,R: rat] :
% 8.58/4.84 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R )
% 8.58/4.84 = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R ) @ X )
% 8.58/4.84 & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5780_abs__diff__le__iff,axiom,
% 8.58/4.84 ! [X: int,A: int,R: int] :
% 8.58/4.84 ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
% 8.58/4.84 = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R ) @ X )
% 8.58/4.84 & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5781_abs__diff__less__iff,axiom,
% 8.58/4.84 ! [X: code_integer,A: code_integer,R: code_integer] :
% 8.58/4.84 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R )
% 8.58/4.84 = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X )
% 8.58/4.84 & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5782_abs__diff__less__iff,axiom,
% 8.58/4.84 ! [X: real,A: real,R: real] :
% 8.58/4.84 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
% 8.58/4.84 = ( ( ord_less_real @ ( minus_minus_real @ A @ R ) @ X )
% 8.58/4.84 & ( ord_less_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5783_abs__diff__less__iff,axiom,
% 8.58/4.84 ! [X: rat,A: rat,R: rat] :
% 8.58/4.84 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R )
% 8.58/4.84 = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R ) @ X )
% 8.58/4.84 & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5784_abs__diff__less__iff,axiom,
% 8.58/4.84 ! [X: int,A: int,R: int] :
% 8.58/4.84 ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
% 8.58/4.84 = ( ( ord_less_int @ ( minus_minus_int @ A @ R ) @ X )
% 8.58/4.84 & ( ord_less_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5785_le__floor__add,axiom,
% 8.58/4.84 ! [X: real,Y: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5786_le__floor__add,axiom,
% 8.58/4.84 ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5787_gr__one__powr,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_real @ one_one_real @ X )
% 8.58/4.84 => ( ( ord_less_real @ zero_zero_real @ Y )
% 8.58/4.84 => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5788_powr__inj,axiom,
% 8.58/4.84 ! [A: real,X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.84 => ( ( A != one_one_real )
% 8.58/4.84 => ( ( ( powr_real @ A @ X )
% 8.58/4.84 = ( powr_real @ A @ Y ) )
% 8.58/4.84 = ( X = Y ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5789_ge__one__powr__ge__zero,axiom,
% 8.58/4.84 ! [X: real,A: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ one_one_real @ X )
% 8.58/4.84 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.84 => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5790_powr__mono__both,axiom,
% 8.58/4.84 ! [A: real,B: real,X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.84 => ( ( ord_less_eq_real @ A @ B )
% 8.58/4.84 => ( ( ord_less_eq_real @ one_one_real @ X )
% 8.58/4.84 => ( ( ord_less_eq_real @ X @ Y )
% 8.58/4.84 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5791_powr__le1,axiom,
% 8.58/4.84 ! [A: real,X: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.84 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( ord_less_eq_real @ X @ one_one_real )
% 8.58/4.84 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5792_powr__divide,axiom,
% 8.58/4.84 ! [X: real,Y: real,A: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.58/4.84 => ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A )
% 8.58/4.84 = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5793_powr__mult,axiom,
% 8.58/4.84 ! [X: real,Y: real,A: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.58/4.84 => ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
% 8.58/4.84 = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5794_abs__sgn__eq,axiom,
% 8.58/4.84 ! [A: code_integer] :
% 8.58/4.84 ( ( ( A = zero_z3403309356797280102nteger )
% 8.58/4.84 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 8.58/4.84 = zero_z3403309356797280102nteger ) )
% 8.58/4.84 & ( ( A != zero_z3403309356797280102nteger )
% 8.58/4.84 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 8.58/4.84 = one_one_Code_integer ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5795_abs__sgn__eq,axiom,
% 8.58/4.84 ! [A: real] :
% 8.58/4.84 ( ( ( A = zero_zero_real )
% 8.58/4.84 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 8.58/4.84 = zero_zero_real ) )
% 8.58/4.84 & ( ( A != zero_zero_real )
% 8.58/4.84 => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 8.58/4.84 = one_one_real ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5796_abs__sgn__eq,axiom,
% 8.58/4.84 ! [A: rat] :
% 8.58/4.84 ( ( ( A = zero_zero_rat )
% 8.58/4.84 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 8.58/4.84 = zero_zero_rat ) )
% 8.58/4.84 & ( ( A != zero_zero_rat )
% 8.58/4.84 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 8.58/4.84 = one_one_rat ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5797_abs__sgn__eq,axiom,
% 8.58/4.84 ! [A: int] :
% 8.58/4.84 ( ( ( A = zero_zero_int )
% 8.58/4.84 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 8.58/4.84 = zero_zero_int ) )
% 8.58/4.84 & ( ( A != zero_zero_int )
% 8.58/4.84 => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 8.58/4.84 = one_one_int ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5798_divide__powr__uminus,axiom,
% 8.58/4.84 ! [A: real,B: real,C: real] :
% 8.58/4.84 ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
% 8.58/4.84 = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5799_log__base__powr,axiom,
% 8.58/4.84 ! [A: real,B: real,X: real] :
% 8.58/4.84 ( ( A != zero_zero_real )
% 8.58/4.84 => ( ( log2 @ ( powr_real @ A @ B ) @ X )
% 8.58/4.84 = ( divide_divide_real @ ( log2 @ A @ X ) @ B ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5800_ln__powr,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( X != zero_zero_real )
% 8.58/4.84 => ( ( ln_ln_real @ ( powr_real @ X @ Y ) )
% 8.58/4.84 = ( times_times_real @ Y @ ( ln_ln_real @ X ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5801_log__powr,axiom,
% 8.58/4.84 ! [X: real,B: real,Y: real] :
% 8.58/4.84 ( ( X != zero_zero_real )
% 8.58/4.84 => ( ( log2 @ B @ ( powr_real @ X @ Y ) )
% 8.58/4.84 = ( times_times_real @ Y @ ( log2 @ B @ X ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5802_abs__real__def,axiom,
% 8.58/4.84 ( abs_abs_real
% 8.58/4.84 = ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5803_lemma__interval__lt,axiom,
% 8.58/4.84 ! [A: real,X: real,B: real] :
% 8.58/4.84 ( ( ord_less_real @ A @ X )
% 8.58/4.84 => ( ( ord_less_real @ X @ B )
% 8.58/4.84 => ? [D2: real] :
% 8.58/4.84 ( ( ord_less_real @ zero_zero_real @ D2 )
% 8.58/4.84 & ! [Y5: real] :
% 8.58/4.84 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D2 )
% 8.58/4.84 => ( ( ord_less_real @ A @ Y5 )
% 8.58/4.84 & ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5804_powr__add,axiom,
% 8.58/4.84 ! [X: real,A: real,B: real] :
% 8.58/4.84 ( ( powr_real @ X @ ( plus_plus_real @ A @ B ) )
% 8.58/4.84 = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5805_powr__diff,axiom,
% 8.58/4.84 ! [W: real,Z1: real,Z22: real] :
% 8.58/4.84 ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
% 8.58/4.84 = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5806_sin__bound__lemma,axiom,
% 8.58/4.84 ! [X: real,Y: real,U: real,V: real] :
% 8.58/4.84 ( ( X = Y )
% 8.58/4.84 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 8.58/4.84 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5807_cos__coeff__Suc,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( cos_coeff @ ( suc @ N ) )
% 8.58/4.84 = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5808_int__cases4,axiom,
% 8.58/4.84 ! [M: int] :
% 8.58/4.84 ( ! [N2: nat] :
% 8.58/4.84 ( M
% 8.58/4.84 != ( semiri1314217659103216013at_int @ N2 ) )
% 8.58/4.84 => ~ ! [N2: nat] :
% 8.58/4.84 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 8.58/4.84 => ( M
% 8.58/4.84 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5809_int__zle__neg,axiom,
% 8.58/4.84 ! [N: nat,M: nat] :
% 8.58/4.84 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 8.58/4.84 = ( ( N = zero_zero_nat )
% 8.58/4.84 & ( M = zero_zero_nat ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5810_negative__zle__0,axiom,
% 8.58/4.84 ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5811_nonpos__int__cases,axiom,
% 8.58/4.84 ! [K: int] :
% 8.58/4.84 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 8.58/4.84 => ~ ! [N2: nat] :
% 8.58/4.84 ( K
% 8.58/4.84 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5812_zmod__zminus2__eq__if,axiom,
% 8.58/4.84 ! [A: int,B: int] :
% 8.58/4.84 ( ( ( ( modulo_modulo_int @ A @ B )
% 8.58/4.84 = zero_zero_int )
% 8.58/4.84 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 8.58/4.84 = zero_zero_int ) )
% 8.58/4.84 & ( ( ( modulo_modulo_int @ A @ B )
% 8.58/4.84 != zero_zero_int )
% 8.58/4.84 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 8.58/4.84 = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5813_zmod__zminus1__eq__if,axiom,
% 8.58/4.84 ! [A: int,B: int] :
% 8.58/4.84 ( ( ( ( modulo_modulo_int @ A @ B )
% 8.58/4.84 = zero_zero_int )
% 8.58/4.84 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 8.58/4.84 = zero_zero_int ) )
% 8.58/4.84 & ( ( ( modulo_modulo_int @ A @ B )
% 8.58/4.84 != zero_zero_int )
% 8.58/4.84 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 8.58/4.84 = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5814_abs__add__one__gt__zero,axiom,
% 8.58/4.84 ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5815_abs__add__one__gt__zero,axiom,
% 8.58/4.84 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5816_abs__add__one__gt__zero,axiom,
% 8.58/4.84 ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5817_abs__add__one__gt__zero,axiom,
% 8.58/4.84 ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5818_one__add__floor,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 8.58/4.84 = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5819_one__add__floor,axiom,
% 8.58/4.84 ! [X: rat] :
% 8.58/4.84 ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 8.58/4.84 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ one_one_rat ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5820_floor__divide__of__nat__eq,axiom,
% 8.58/4.84 ! [M: nat,N: nat] :
% 8.58/4.84 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 8.58/4.84 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5821_floor__divide__of__nat__eq,axiom,
% 8.58/4.84 ! [M: nat,N: nat] :
% 8.58/4.84 ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) )
% 8.58/4.84 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5822_powr__realpow,axiom,
% 8.58/4.84 ! [X: real,N: nat] :
% 8.58/4.84 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 8.58/4.84 = ( power_power_real @ X @ N ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5823_less__log__iff,axiom,
% 8.58/4.84 ! [B: real,X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_real @ one_one_real @ B )
% 8.58/4.84 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( ord_less_real @ Y @ ( log2 @ B @ X ) )
% 8.58/4.84 = ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5824_log__less__iff,axiom,
% 8.58/4.84 ! [B: real,X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_real @ one_one_real @ B )
% 8.58/4.84 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( ord_less_real @ ( log2 @ B @ X ) @ Y )
% 8.58/4.84 = ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5825_less__powr__iff,axiom,
% 8.58/4.84 ! [B: real,X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_real @ one_one_real @ B )
% 8.58/4.84 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( ord_less_real @ X @ ( powr_real @ B @ Y ) )
% 8.58/4.84 = ( ord_less_real @ ( log2 @ B @ X ) @ Y ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5826_powr__less__iff,axiom,
% 8.58/4.84 ! [B: real,X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_real @ one_one_real @ B )
% 8.58/4.84 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X )
% 8.58/4.84 = ( ord_less_real @ Y @ ( log2 @ B @ X ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5827_ceiling__diff__floor__le__1,axiom,
% 8.58/4.84 ! [X: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) @ one_one_int ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5828_ceiling__diff__floor__le__1,axiom,
% 8.58/4.84 ! [X: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) @ one_one_int ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5829_sgn__power__injE,axiom,
% 8.58/4.84 ! [A: real,N: nat,X: real,B: real] :
% 8.58/4.84 ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 8.58/4.84 = X )
% 8.58/4.84 => ( ( X
% 8.58/4.84 = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
% 8.58/4.84 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.84 => ( A = B ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5830_lemma__interval,axiom,
% 8.58/4.84 ! [A: real,X: real,B: real] :
% 8.58/4.84 ( ( ord_less_real @ A @ X )
% 8.58/4.84 => ( ( ord_less_real @ X @ B )
% 8.58/4.84 => ? [D2: real] :
% 8.58/4.84 ( ( ord_less_real @ zero_zero_real @ D2 )
% 8.58/4.84 & ! [Y5: real] :
% 8.58/4.84 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D2 )
% 8.58/4.84 => ( ( ord_less_eq_real @ A @ Y5 )
% 8.58/4.84 & ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5831_norm__triangle__ineq3,axiom,
% 8.58/4.84 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5832_norm__triangle__ineq3,axiom,
% 8.58/4.84 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5833_int__cases3,axiom,
% 8.58/4.84 ! [K: int] :
% 8.58/4.84 ( ( K != zero_zero_int )
% 8.58/4.84 => ( ! [N2: nat] :
% 8.58/4.84 ( ( K
% 8.58/4.84 = ( semiri1314217659103216013at_int @ N2 ) )
% 8.58/4.84 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 8.58/4.84 => ~ ! [N2: nat] :
% 8.58/4.84 ( ( K
% 8.58/4.84 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 8.58/4.84 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5834_not__zle__0__negative,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5835_negD,axiom,
% 8.58/4.84 ! [X: int] :
% 8.58/4.84 ( ( ord_less_int @ X @ zero_zero_int )
% 8.58/4.84 => ? [N2: nat] :
% 8.58/4.84 ( X
% 8.58/4.84 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5836_negative__zless__0,axiom,
% 8.58/4.84 ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5837_verit__less__mono__div__int2,axiom,
% 8.58/4.84 ! [A3: int,B5: int,N: int] :
% 8.58/4.84 ( ( ord_less_eq_int @ A3 @ B5 )
% 8.58/4.84 => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 8.58/4.84 => ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A3 @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5838_div__eq__minus1,axiom,
% 8.58/4.84 ! [B: int] :
% 8.58/4.84 ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.84 => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 8.58/4.84 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5839_zsgn__def,axiom,
% 8.58/4.84 ( sgn_sgn_int
% 8.58/4.84 = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5840_powr__minus__divide,axiom,
% 8.58/4.84 ! [X: real,A: real] :
% 8.58/4.84 ( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
% 8.58/4.84 = ( divide_divide_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5841_le__mult__floor,axiom,
% 8.58/4.84 ! [A: real,B: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.84 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.58/4.84 => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5842_le__mult__floor,axiom,
% 8.58/4.84 ! [A: rat,B: rat] :
% 8.58/4.84 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.58/4.84 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.58/4.84 => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5843_abs__le__square__iff,axiom,
% 8.58/4.84 ! [X: code_integer,Y: code_integer] :
% 8.58/4.84 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
% 8.58/4.84 = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5844_abs__le__square__iff,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
% 8.58/4.84 = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5845_abs__le__square__iff,axiom,
% 8.58/4.84 ! [X: rat,Y: rat] :
% 8.58/4.84 ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
% 8.58/4.84 = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5846_abs__le__square__iff,axiom,
% 8.58/4.84 ! [X: int,Y: int] :
% 8.58/4.84 ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
% 8.58/4.84 = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5847_abs__square__eq__1,axiom,
% 8.58/4.84 ! [X: code_integer] :
% 8.58/4.84 ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.84 = one_one_Code_integer )
% 8.58/4.84 = ( ( abs_abs_Code_integer @ X )
% 8.58/4.84 = one_one_Code_integer ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5848_abs__square__eq__1,axiom,
% 8.58/4.84 ! [X: rat] :
% 8.58/4.84 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.84 = one_one_rat )
% 8.58/4.84 = ( ( abs_abs_rat @ X )
% 8.58/4.84 = one_one_rat ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5849_abs__square__eq__1,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.84 = one_one_real )
% 8.58/4.84 = ( ( abs_abs_real @ X )
% 8.58/4.84 = one_one_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5850_abs__square__eq__1,axiom,
% 8.58/4.84 ! [X: int] :
% 8.58/4.84 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.84 = one_one_int )
% 8.58/4.84 = ( ( abs_abs_int @ X )
% 8.58/4.84 = one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5851_power__even__abs,axiom,
% 8.58/4.84 ! [N: nat,A: code_integer] :
% 8.58/4.84 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.84 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
% 8.58/4.84 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5852_power__even__abs,axiom,
% 8.58/4.84 ! [N: nat,A: rat] :
% 8.58/4.84 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.84 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
% 8.58/4.84 = ( power_power_rat @ A @ N ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5853_power__even__abs,axiom,
% 8.58/4.84 ! [N: nat,A: real] :
% 8.58/4.84 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.84 => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
% 8.58/4.84 = ( power_power_real @ A @ N ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5854_power__even__abs,axiom,
% 8.58/4.84 ! [N: nat,A: int] :
% 8.58/4.84 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.84 => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
% 8.58/4.84 = ( power_power_int @ A @ N ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5855_powr__neg__one,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.84 = ( divide_divide_real @ one_one_real @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5856_powr__mult__base,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
% 8.58/4.84 = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5857_powr__le__iff,axiom,
% 8.58/4.84 ! [B: real,X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_real @ one_one_real @ B )
% 8.58/4.84 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
% 8.58/4.84 = ( ord_less_eq_real @ Y @ ( log2 @ B @ X ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5858_le__powr__iff,axiom,
% 8.58/4.84 ! [B: real,X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_real @ one_one_real @ B )
% 8.58/4.84 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
% 8.58/4.84 = ( ord_less_eq_real @ ( log2 @ B @ X ) @ Y ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5859_log__le__iff,axiom,
% 8.58/4.84 ! [B: real,X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_real @ one_one_real @ B )
% 8.58/4.84 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( ord_less_eq_real @ ( log2 @ B @ X ) @ Y )
% 8.58/4.84 = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5860_le__log__iff,axiom,
% 8.58/4.84 ! [B: real,X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_real @ one_one_real @ B )
% 8.58/4.84 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( ord_less_eq_real @ Y @ ( log2 @ B @ X ) )
% 8.58/4.84 = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5861_neg__int__cases,axiom,
% 8.58/4.84 ! [K: int] :
% 8.58/4.84 ( ( ord_less_int @ K @ zero_zero_int )
% 8.58/4.84 => ~ ! [N2: nat] :
% 8.58/4.84 ( ( K
% 8.58/4.84 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 8.58/4.84 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5862_minus__mod__int__eq,axiom,
% 8.58/4.84 ! [L: int,K: int] :
% 8.58/4.84 ( ( ord_less_eq_int @ zero_zero_int @ L )
% 8.58/4.84 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 8.58/4.84 = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5863_zmod__minus1,axiom,
% 8.58/4.84 ! [B: int] :
% 8.58/4.84 ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.84 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 8.58/4.84 = ( minus_minus_int @ B @ one_one_int ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5864_zdiv__zminus1__eq__if,axiom,
% 8.58/4.84 ! [B: int,A: int] :
% 8.58/4.84 ( ( B != zero_zero_int )
% 8.58/4.84 => ( ( ( ( modulo_modulo_int @ A @ B )
% 8.58/4.84 = zero_zero_int )
% 8.58/4.84 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 8.58/4.84 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 8.58/4.84 & ( ( ( modulo_modulo_int @ A @ B )
% 8.58/4.84 != zero_zero_int )
% 8.58/4.84 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 8.58/4.84 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5865_zdiv__zminus2__eq__if,axiom,
% 8.58/4.84 ! [B: int,A: int] :
% 8.58/4.84 ( ( B != zero_zero_int )
% 8.58/4.84 => ( ( ( ( modulo_modulo_int @ A @ B )
% 8.58/4.84 = zero_zero_int )
% 8.58/4.84 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 8.58/4.84 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 8.58/4.84 & ( ( ( modulo_modulo_int @ A @ B )
% 8.58/4.84 != zero_zero_int )
% 8.58/4.84 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 8.58/4.84 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5866_power2__le__iff__abs__le,axiom,
% 8.58/4.84 ! [Y: code_integer,X: code_integer] :
% 8.58/4.84 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
% 8.58/4.84 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5867_power2__le__iff__abs__le,axiom,
% 8.58/4.84 ! [Y: real,X: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.58/4.84 => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5868_power2__le__iff__abs__le,axiom,
% 8.58/4.84 ! [Y: rat,X: rat] :
% 8.58/4.84 ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 8.58/4.84 => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5869_power2__le__iff__abs__le,axiom,
% 8.58/4.84 ! [Y: int,X: int] :
% 8.58/4.84 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.58/4.84 => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5870_abs__square__le__1,axiom,
% 8.58/4.84 ! [X: code_integer] :
% 8.58/4.84 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 8.58/4.84 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5871_abs__square__le__1,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 8.58/4.84 = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5872_abs__square__le__1,axiom,
% 8.58/4.84 ! [X: rat] :
% 8.58/4.84 ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 8.58/4.84 = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5873_abs__square__le__1,axiom,
% 8.58/4.84 ! [X: int] :
% 8.58/4.84 ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 8.58/4.84 = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5874_abs__square__less__1,axiom,
% 8.58/4.84 ! [X: code_integer] :
% 8.58/4.84 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 8.58/4.84 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5875_abs__square__less__1,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 8.58/4.84 = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5876_abs__square__less__1,axiom,
% 8.58/4.84 ! [X: rat] :
% 8.58/4.84 ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 8.58/4.84 = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5877_abs__square__less__1,axiom,
% 8.58/4.84 ! [X: int] :
% 8.58/4.84 ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 8.58/4.84 = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5878_power__mono__even,axiom,
% 8.58/4.84 ! [N: nat,A: code_integer,B: code_integer] :
% 8.58/4.84 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.84 => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
% 8.58/4.84 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5879_power__mono__even,axiom,
% 8.58/4.84 ! [N: nat,A: real,B: real] :
% 8.58/4.84 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.84 => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
% 8.58/4.84 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5880_power__mono__even,axiom,
% 8.58/4.84 ! [N: nat,A: rat,B: rat] :
% 8.58/4.84 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.84 => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
% 8.58/4.84 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5881_power__mono__even,axiom,
% 8.58/4.84 ! [N: nat,A: int,B: int] :
% 8.58/4.84 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.84 => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
% 8.58/4.84 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5882_ln__powr__bound,axiom,
% 8.58/4.84 ! [X: real,A: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ one_one_real @ X )
% 8.58/4.84 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.84 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5883_ln__powr__bound2,axiom,
% 8.58/4.84 ! [X: real,A: real] :
% 8.58/4.84 ( ( ord_less_real @ one_one_real @ X )
% 8.58/4.84 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.84 => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5884_add__log__eq__powr,axiom,
% 8.58/4.84 ! [B: real,X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_real @ zero_zero_real @ B )
% 8.58/4.84 => ( ( B != one_one_real )
% 8.58/4.84 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( plus_plus_real @ Y @ ( log2 @ B @ X ) )
% 8.58/4.84 = ( log2 @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5885_log__add__eq__powr,axiom,
% 8.58/4.84 ! [B: real,X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_real @ zero_zero_real @ B )
% 8.58/4.84 => ( ( B != one_one_real )
% 8.58/4.84 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( plus_plus_real @ ( log2 @ B @ X ) @ Y )
% 8.58/4.84 = ( log2 @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5886_minus__log__eq__powr,axiom,
% 8.58/4.84 ! [B: real,X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_real @ zero_zero_real @ B )
% 8.58/4.84 => ( ( B != one_one_real )
% 8.58/4.84 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( minus_minus_real @ Y @ ( log2 @ B @ X ) )
% 8.58/4.84 = ( log2 @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5887_take__bit__Suc__minus__bit0,axiom,
% 8.58/4.84 ! [N: nat,K: num] :
% 8.58/4.84 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 8.58/4.84 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5888_minus__1__div__exp__eq__int,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.84 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5889_div__pos__neg__trivial,axiom,
% 8.58/4.84 ! [K: int,L: int] :
% 8.58/4.84 ( ( ord_less_int @ zero_zero_int @ K )
% 8.58/4.84 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 8.58/4.84 => ( ( divide_divide_int @ K @ L )
% 8.58/4.84 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5890_signed__take__bit__int__less__eq__self__iff,axiom,
% 8.58/4.84 ! [N: nat,K: int] :
% 8.58/4.84 ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 8.58/4.84 = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5891_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 8.58/4.84 ! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5892_signed__take__bit__int__greater__self__iff,axiom,
% 8.58/4.84 ! [K: int,N: nat] :
% 8.58/4.84 ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 8.58/4.84 = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5893_push__bit__minus__one,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.84 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5894_log__minus__eq__powr,axiom,
% 8.58/4.84 ! [B: real,X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_real @ zero_zero_real @ B )
% 8.58/4.84 => ( ( B != one_one_real )
% 8.58/4.84 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( minus_minus_real @ ( log2 @ B @ X ) @ Y )
% 8.58/4.84 = ( log2 @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5895_int__bit__induct,axiom,
% 8.58/4.84 ! [P: int > $o,K: int] :
% 8.58/4.84 ( ( P @ zero_zero_int )
% 8.58/4.84 => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.84 => ( ! [K3: int] :
% 8.58/4.84 ( ( P @ K3 )
% 8.58/4.84 => ( ( K3 != zero_zero_int )
% 8.58/4.84 => ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.84 => ( ! [K3: int] :
% 8.58/4.84 ( ( P @ K3 )
% 8.58/4.84 => ( ( K3
% 8.58/4.84 != ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.84 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 8.58/4.84 => ( P @ K ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5896_signed__take__bit__int__eq__self,axiom,
% 8.58/4.84 ! [N: nat,K: int] :
% 8.58/4.84 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 8.58/4.84 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.84 => ( ( bit_ri631733984087533419it_int @ N @ K )
% 8.58/4.84 = K ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5897_signed__take__bit__int__eq__self__iff,axiom,
% 8.58/4.84 ! [N: nat,K: int] :
% 8.58/4.84 ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 8.58/4.84 = K )
% 8.58/4.84 = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 8.58/4.84 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5898_powr__neg__numeral,axiom,
% 8.58/4.84 ! [X: real,N: num] :
% 8.58/4.84 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.58/4.84 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5899_floor__log2__div2,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.84 => ( ( archim6058952711729229775r_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 8.58/4.84 = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5900_signed__take__bit__int__greater__eq,axiom,
% 8.58/4.84 ! [K: int,N: nat] :
% 8.58/4.84 ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 8.58/4.84 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5901_take__bit__minus__small__eq,axiom,
% 8.58/4.84 ! [K: int,N: nat] :
% 8.58/4.84 ( ( ord_less_int @ zero_zero_int @ K )
% 8.58/4.84 => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.84 => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 8.58/4.84 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5902_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( ord_less_eq_real @ X @ one_one_real )
% 8.58/4.84 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5903_floor__log__nat__eq__if,axiom,
% 8.58/4.84 ! [B: nat,N: nat,K: nat] :
% 8.58/4.84 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 8.58/4.84 => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 8.58/4.84 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 8.58/4.84 => ( ( archim6058952711729229775r_real @ ( log2 @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 8.58/4.84 = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5904_abs__sqrt__wlog,axiom,
% 8.58/4.84 ! [P: code_integer > code_integer > $o,X: code_integer] :
% 8.58/4.84 ( ! [X4: code_integer] :
% 8.58/4.84 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
% 8.58/4.84 => ( P @ X4 @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.84 => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5905_abs__sqrt__wlog,axiom,
% 8.58/4.84 ! [P: real > real > $o,X: real] :
% 8.58/4.84 ( ! [X4: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 8.58/4.84 => ( P @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.84 => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5906_abs__sqrt__wlog,axiom,
% 8.58/4.84 ! [P: rat > rat > $o,X: rat] :
% 8.58/4.84 ( ! [X4: rat] :
% 8.58/4.84 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 8.58/4.84 => ( P @ X4 @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.84 => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5907_abs__sqrt__wlog,axiom,
% 8.58/4.84 ! [P: int > int > $o,X: int] :
% 8.58/4.84 ( ! [X4: int] :
% 8.58/4.84 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 8.58/4.84 => ( P @ X4 @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.84 => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5908_signed__take__bit__Suc__minus__bit1,axiom,
% 8.58/4.84 ! [N: nat,K: num] :
% 8.58/4.84 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 8.58/4.84 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5909_arcosh__def,axiom,
% 8.58/4.84 ( arcosh_real
% 8.58/4.84 = ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5910_arctan__double,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.84 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
% 8.58/4.84 = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5911_arsinh__def,axiom,
% 8.58/4.84 ( arsinh_real
% 8.58/4.84 = ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5912_signed__take__bit__Suc__bit1,axiom,
% 8.58/4.84 ! [N: nat,K: num] :
% 8.58/4.84 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 8.58/4.84 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5913_verit__eq__simplify_I9_J,axiom,
% 8.58/4.84 ! [X32: num,Y32: num] :
% 8.58/4.84 ( ( ( bit1 @ X32 )
% 8.58/4.84 = ( bit1 @ Y32 ) )
% 8.58/4.84 = ( X32 = Y32 ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5914_semiring__norm_I90_J,axiom,
% 8.58/4.84 ! [M: num,N: num] :
% 8.58/4.84 ( ( ( bit1 @ M )
% 8.58/4.84 = ( bit1 @ N ) )
% 8.58/4.84 = ( M = N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5915_semiring__norm_I88_J,axiom,
% 8.58/4.84 ! [M: num,N: num] :
% 8.58/4.84 ( ( bit0 @ M )
% 8.58/4.84 != ( bit1 @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5916_semiring__norm_I89_J,axiom,
% 8.58/4.84 ! [M: num,N: num] :
% 8.58/4.84 ( ( bit1 @ M )
% 8.58/4.84 != ( bit0 @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5917_semiring__norm_I84_J,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( one
% 8.58/4.84 != ( bit1 @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5918_semiring__norm_I86_J,axiom,
% 8.58/4.84 ! [M: num] :
% 8.58/4.84 ( ( bit1 @ M )
% 8.58/4.84 != one ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5919_zdvd1__eq,axiom,
% 8.58/4.84 ! [X: int] :
% 8.58/4.84 ( ( dvd_dvd_int @ X @ one_one_int )
% 8.58/4.84 = ( ( abs_abs_int @ X )
% 8.58/4.84 = one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5920_semiring__norm_I80_J,axiom,
% 8.58/4.84 ! [M: num,N: num] :
% 8.58/4.84 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.58/4.84 = ( ord_less_num @ M @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5921_semiring__norm_I73_J,axiom,
% 8.58/4.84 ! [M: num,N: num] :
% 8.58/4.84 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.58/4.84 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5922_of__real__0,axiom,
% 8.58/4.84 ( ( real_V1803761363581548252l_real @ zero_zero_real )
% 8.58/4.84 = zero_zero_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5923_of__real__0,axiom,
% 8.58/4.84 ( ( real_V4546457046886955230omplex @ zero_zero_real )
% 8.58/4.84 = zero_zero_complex ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5924_of__real__eq__0__iff,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ( real_V1803761363581548252l_real @ X )
% 8.58/4.84 = zero_zero_real )
% 8.58/4.84 = ( X = zero_zero_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5925_of__real__eq__0__iff,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ( real_V4546457046886955230omplex @ X )
% 8.58/4.84 = zero_zero_complex )
% 8.58/4.84 = ( X = zero_zero_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5926_of__real__1,axiom,
% 8.58/4.84 ( ( real_V1803761363581548252l_real @ one_one_real )
% 8.58/4.84 = one_one_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5927_of__real__1,axiom,
% 8.58/4.84 ( ( real_V4546457046886955230omplex @ one_one_real )
% 8.58/4.84 = one_one_complex ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5928_of__real__eq__1__iff,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ( real_V1803761363581548252l_real @ X )
% 8.58/4.84 = one_one_real )
% 8.58/4.84 = ( X = one_one_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5929_of__real__eq__1__iff,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ( real_V4546457046886955230omplex @ X )
% 8.58/4.84 = one_one_complex )
% 8.58/4.84 = ( X = one_one_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5930_of__real__mult,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( real_V1803761363581548252l_real @ ( times_times_real @ X @ Y ) )
% 8.58/4.84 = ( times_times_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5931_of__real__mult,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( real_V4546457046886955230omplex @ ( times_times_real @ X @ Y ) )
% 8.58/4.84 = ( times_times_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5932_of__real__numeral,axiom,
% 8.58/4.84 ! [W: num] :
% 8.58/4.84 ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
% 8.58/4.84 = ( numeral_numeral_real @ W ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5933_of__real__numeral,axiom,
% 8.58/4.84 ! [W: num] :
% 8.58/4.84 ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
% 8.58/4.84 = ( numera6690914467698888265omplex @ W ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5934_zabs__less__one__iff,axiom,
% 8.58/4.84 ! [Z2: int] :
% 8.58/4.84 ( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
% 8.58/4.84 = ( Z2 = zero_zero_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5935_of__real__divide,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y ) )
% 8.58/4.84 = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5936_of__real__divide,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y ) )
% 8.58/4.84 = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5937_of__real__add,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.84 = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5938_of__real__add,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.84 = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5939_of__real__power,axiom,
% 8.58/4.84 ! [X: real,N: nat] :
% 8.58/4.84 ( ( real_V1803761363581548252l_real @ ( power_power_real @ X @ N ) )
% 8.58/4.84 = ( power_power_real @ ( real_V1803761363581548252l_real @ X ) @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5940_of__real__power,axiom,
% 8.58/4.84 ! [X: real,N: nat] :
% 8.58/4.84 ( ( real_V4546457046886955230omplex @ ( power_power_real @ X @ N ) )
% 8.58/4.84 = ( power_power_complex @ ( real_V4546457046886955230omplex @ X ) @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5941_of__real__diff,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( real_V1803761363581548252l_real @ ( minus_minus_real @ X @ Y ) )
% 8.58/4.84 = ( minus_minus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5942_of__real__diff,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( real_V4546457046886955230omplex @ ( minus_minus_real @ X @ Y ) )
% 8.58/4.84 = ( minus_minus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5943_semiring__norm_I9_J,axiom,
% 8.58/4.84 ! [M: num,N: num] :
% 8.58/4.84 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.58/4.84 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5944_semiring__norm_I7_J,axiom,
% 8.58/4.84 ! [M: num,N: num] :
% 8.58/4.84 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.58/4.84 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5945_of__real__of__nat__eq,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( real_V4546457046886955230omplex @ ( semiri5074537144036343181t_real @ N ) )
% 8.58/4.84 = ( semiri8010041392384452111omplex @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5946_of__real__of__nat__eq,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( real_V1803761363581548252l_real @ ( semiri5074537144036343181t_real @ N ) )
% 8.58/4.84 = ( semiri5074537144036343181t_real @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5947_semiring__norm_I15_J,axiom,
% 8.58/4.84 ! [M: num,N: num] :
% 8.58/4.84 ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.58/4.84 = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5948_semiring__norm_I14_J,axiom,
% 8.58/4.84 ! [M: num,N: num] :
% 8.58/4.84 ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.58/4.84 = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5949_semiring__norm_I81_J,axiom,
% 8.58/4.84 ! [M: num,N: num] :
% 8.58/4.84 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.58/4.84 = ( ord_less_num @ M @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5950_semiring__norm_I72_J,axiom,
% 8.58/4.84 ! [M: num,N: num] :
% 8.58/4.84 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.58/4.84 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5951_semiring__norm_I77_J,axiom,
% 8.58/4.84 ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5952_semiring__norm_I70_J,axiom,
% 8.58/4.84 ! [M: num] :
% 8.58/4.84 ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5953_zdiv__numeral__Bit1,axiom,
% 8.58/4.84 ! [V: num,W: num] :
% 8.58/4.84 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 8.58/4.84 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5954_semiring__norm_I3_J,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( plus_plus_num @ one @ ( bit0 @ N ) )
% 8.58/4.84 = ( bit1 @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5955_semiring__norm_I4_J,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( plus_plus_num @ one @ ( bit1 @ N ) )
% 8.58/4.84 = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5956_semiring__norm_I5_J,axiom,
% 8.58/4.84 ! [M: num] :
% 8.58/4.84 ( ( plus_plus_num @ ( bit0 @ M ) @ one )
% 8.58/4.84 = ( bit1 @ M ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5957_semiring__norm_I8_J,axiom,
% 8.58/4.84 ! [M: num] :
% 8.58/4.84 ( ( plus_plus_num @ ( bit1 @ M ) @ one )
% 8.58/4.84 = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5958_semiring__norm_I10_J,axiom,
% 8.58/4.84 ! [M: num,N: num] :
% 8.58/4.84 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.58/4.84 = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5959_semiring__norm_I16_J,axiom,
% 8.58/4.84 ! [M: num,N: num] :
% 8.58/4.84 ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.58/4.84 = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5960_semiring__norm_I74_J,axiom,
% 8.58/4.84 ! [M: num,N: num] :
% 8.58/4.84 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.58/4.84 = ( ord_less_num @ M @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5961_semiring__norm_I79_J,axiom,
% 8.58/4.84 ! [M: num,N: num] :
% 8.58/4.84 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.58/4.84 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5962_of__real__neg__numeral,axiom,
% 8.58/4.84 ! [W: num] :
% 8.58/4.84 ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.58/4.84 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5963_of__real__neg__numeral,axiom,
% 8.58/4.84 ! [W: num] :
% 8.58/4.84 ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.58/4.84 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5964_Suc__div__eq__add3__div__numeral,axiom,
% 8.58/4.84 ! [M: nat,V: num] :
% 8.58/4.84 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 8.58/4.84 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5965_div__Suc__eq__div__add3,axiom,
% 8.58/4.84 ! [M: nat,N: nat] :
% 8.58/4.84 ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 8.58/4.84 = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5966_Suc__mod__eq__add3__mod__numeral,axiom,
% 8.58/4.84 ! [M: nat,V: num] :
% 8.58/4.84 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 8.58/4.84 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5967_mod__Suc__eq__mod__add3,axiom,
% 8.58/4.84 ! [M: nat,N: nat] :
% 8.58/4.84 ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 8.58/4.84 = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5968_norm__of__real__add1,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ one_one_real ) )
% 8.58/4.84 = ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5969_norm__of__real__add1,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ one_one_complex ) )
% 8.58/4.84 = ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5970_norm__of__real__addn,axiom,
% 8.58/4.84 ! [X: real,B: num] :
% 8.58/4.84 ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( numeral_numeral_real @ B ) ) )
% 8.58/4.84 = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5971_norm__of__real__addn,axiom,
% 8.58/4.84 ! [X: real,B: num] :
% 8.58/4.84 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( numera6690914467698888265omplex @ B ) ) )
% 8.58/4.84 = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5972_zmod__numeral__Bit1,axiom,
% 8.58/4.84 ! [V: num,W: num] :
% 8.58/4.84 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 8.58/4.84 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5973_verit__eq__simplify_I14_J,axiom,
% 8.58/4.84 ! [X2: num,X32: num] :
% 8.58/4.84 ( ( bit0 @ X2 )
% 8.58/4.84 != ( bit1 @ X32 ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5974_verit__eq__simplify_I12_J,axiom,
% 8.58/4.84 ! [X32: num] :
% 8.58/4.84 ( one
% 8.58/4.84 != ( bit1 @ X32 ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5975_zdvd__antisym__abs,axiom,
% 8.58/4.84 ! [A: int,B: int] :
% 8.58/4.84 ( ( dvd_dvd_int @ A @ B )
% 8.58/4.84 => ( ( dvd_dvd_int @ B @ A )
% 8.58/4.84 => ( ( abs_abs_int @ A )
% 8.58/4.84 = ( abs_abs_int @ B ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5976_num_Oexhaust,axiom,
% 8.58/4.84 ! [Y: num] :
% 8.58/4.84 ( ( Y != one )
% 8.58/4.84 => ( ! [X22: num] :
% 8.58/4.84 ( Y
% 8.58/4.84 != ( bit0 @ X22 ) )
% 8.58/4.84 => ~ ! [X33: num] :
% 8.58/4.84 ( Y
% 8.58/4.84 != ( bit1 @ X33 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5977_abs__zmult__eq__1,axiom,
% 8.58/4.84 ! [M: int,N: int] :
% 8.58/4.84 ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
% 8.58/4.84 = one_one_int )
% 8.58/4.84 => ( ( abs_abs_int @ M )
% 8.58/4.84 = one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5978_abs__div,axiom,
% 8.58/4.84 ! [Y: int,X: int] :
% 8.58/4.84 ( ( dvd_dvd_int @ Y @ X )
% 8.58/4.84 => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y ) )
% 8.58/4.84 = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5979_div__eq__sgn__abs,axiom,
% 8.58/4.84 ! [K: int,L: int] :
% 8.58/4.84 ( ( ( sgn_sgn_int @ K )
% 8.58/4.84 = ( sgn_sgn_int @ L ) )
% 8.58/4.84 => ( ( divide_divide_int @ K @ L )
% 8.58/4.84 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5980_nonzero__of__real__divide,axiom,
% 8.58/4.84 ! [Y: real,X: real] :
% 8.58/4.84 ( ( Y != zero_zero_real )
% 8.58/4.84 => ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y ) )
% 8.58/4.84 = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5981_nonzero__of__real__divide,axiom,
% 8.58/4.84 ! [Y: real,X: real] :
% 8.58/4.84 ( ( Y != zero_zero_real )
% 8.58/4.84 => ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y ) )
% 8.58/4.84 = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5982_numeral__Bit1,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 8.58/4.84 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5983_numeral__Bit1,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 8.58/4.84 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5984_numeral__Bit1,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 8.58/4.84 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5985_numeral__Bit1,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 8.58/4.84 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5986_numeral__Bit1,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 8.58/4.84 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5987_eval__nat__numeral_I3_J,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 8.58/4.84 = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5988_cong__exp__iff__simps_I10_J,axiom,
% 8.58/4.84 ! [M: num,Q2: num,N: num] :
% 8.58/4.84 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5989_cong__exp__iff__simps_I10_J,axiom,
% 8.58/4.84 ! [M: num,Q2: num,N: num] :
% 8.58/4.84 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5990_cong__exp__iff__simps_I10_J,axiom,
% 8.58/4.84 ! [M: num,Q2: num,N: num] :
% 8.58/4.84 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5991_cong__exp__iff__simps_I12_J,axiom,
% 8.58/4.84 ! [M: num,Q2: num,N: num] :
% 8.58/4.84 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5992_cong__exp__iff__simps_I12_J,axiom,
% 8.58/4.84 ! [M: num,Q2: num,N: num] :
% 8.58/4.84 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5993_cong__exp__iff__simps_I12_J,axiom,
% 8.58/4.84 ! [M: num,Q2: num,N: num] :
% 8.58/4.84 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5994_cong__exp__iff__simps_I13_J,axiom,
% 8.58/4.84 ! [M: num,Q2: num,N: num] :
% 8.58/4.84 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 8.58/4.84 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 8.58/4.84 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5995_cong__exp__iff__simps_I13_J,axiom,
% 8.58/4.84 ! [M: num,Q2: num,N: num] :
% 8.58/4.84 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 8.58/4.84 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 8.58/4.84 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5996_cong__exp__iff__simps_I13_J,axiom,
% 8.58/4.84 ! [M: num,Q2: num,N: num] :
% 8.58/4.84 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 8.58/4.84 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 8.58/4.84 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5997_power__minus__Bit1,axiom,
% 8.58/4.84 ! [X: real,K: num] :
% 8.58/4.84 ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 8.58/4.84 = ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5998_power__minus__Bit1,axiom,
% 8.58/4.84 ! [X: int,K: num] :
% 8.58/4.84 ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 8.58/4.84 = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_5999_power__minus__Bit1,axiom,
% 8.58/4.84 ! [X: complex,K: num] :
% 8.58/4.84 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 8.58/4.84 = ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6000_power__minus__Bit1,axiom,
% 8.58/4.84 ! [X: rat,K: num] :
% 8.58/4.84 ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 8.58/4.84 = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6001_power__minus__Bit1,axiom,
% 8.58/4.84 ! [X: code_integer,K: num] :
% 8.58/4.84 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 8.58/4.84 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6002_zabs__def,axiom,
% 8.58/4.84 ( abs_abs_int
% 8.58/4.84 = ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6003_dvd__imp__le__int,axiom,
% 8.58/4.84 ! [I: int,D: int] :
% 8.58/4.84 ( ( I != zero_zero_int )
% 8.58/4.84 => ( ( dvd_dvd_int @ D @ I )
% 8.58/4.84 => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6004_abs__mod__less,axiom,
% 8.58/4.84 ! [L: int,K: int] :
% 8.58/4.84 ( ( L != zero_zero_int )
% 8.58/4.84 => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6005_norm__less__p1,axiom,
% 8.58/4.84 ! [X: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X ) ) @ one_one_real ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6006_norm__less__p1,axiom,
% 8.58/4.84 ! [X: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X ) ) @ one_one_complex ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6007_numeral__Bit1__div__2,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.84 = ( numeral_numeral_nat @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6008_numeral__Bit1__div__2,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.84 = ( numeral_numeral_int @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6009_odd__numeral,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6010_odd__numeral,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6011_odd__numeral,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6012_cong__exp__iff__simps_I3_J,axiom,
% 8.58/4.84 ! [N: num,Q2: num] :
% 8.58/4.84 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 != zero_zero_nat ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6013_cong__exp__iff__simps_I3_J,axiom,
% 8.58/4.84 ! [N: num,Q2: num] :
% 8.58/4.84 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 != zero_zero_int ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6014_cong__exp__iff__simps_I3_J,axiom,
% 8.58/4.84 ! [N: num,Q2: num] :
% 8.58/4.84 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 != zero_z3403309356797280102nteger ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6015_power3__eq__cube,axiom,
% 8.58/4.84 ! [A: complex] :
% 8.58/4.84 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 8.58/4.84 = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6016_power3__eq__cube,axiom,
% 8.58/4.84 ! [A: real] :
% 8.58/4.84 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 8.58/4.84 = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6017_power3__eq__cube,axiom,
% 8.58/4.84 ! [A: rat] :
% 8.58/4.84 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 8.58/4.84 = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6018_power3__eq__cube,axiom,
% 8.58/4.84 ! [A: nat] :
% 8.58/4.84 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 8.58/4.84 = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6019_power3__eq__cube,axiom,
% 8.58/4.84 ! [A: int] :
% 8.58/4.84 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 8.58/4.84 = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6020_numeral__3__eq__3,axiom,
% 8.58/4.84 ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 8.58/4.84 = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6021_Suc3__eq__add__3,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( suc @ ( suc @ ( suc @ N ) ) )
% 8.58/4.84 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6022_zdvd__mult__cancel1,axiom,
% 8.58/4.84 ! [M: int,N: int] :
% 8.58/4.84 ( ( M != zero_zero_int )
% 8.58/4.84 => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
% 8.58/4.84 = ( ( abs_abs_int @ N )
% 8.58/4.84 = one_one_int ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6023_div__sgn__abs__cancel,axiom,
% 8.58/4.84 ! [V: int,K: int,L: int] :
% 8.58/4.84 ( ( V != zero_zero_int )
% 8.58/4.84 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
% 8.58/4.84 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6024_div__dvd__sgn__abs,axiom,
% 8.58/4.84 ! [L: int,K: int] :
% 8.58/4.84 ( ( dvd_dvd_int @ L @ K )
% 8.58/4.84 => ( ( divide_divide_int @ K @ L )
% 8.58/4.84 = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6025_num_Osize__gen_I3_J,axiom,
% 8.58/4.84 ! [X32: num] :
% 8.58/4.84 ( ( size_num @ ( bit1 @ X32 ) )
% 8.58/4.84 = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6026_norm__of__real__diff,axiom,
% 8.58/4.84 ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( real_V1803761363581548252l_real @ B ) @ ( real_V1803761363581548252l_real @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6027_norm__of__real__diff,axiom,
% 8.58/4.84 ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( real_V4546457046886955230omplex @ B ) @ ( real_V4546457046886955230omplex @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6028_cong__exp__iff__simps_I7_J,axiom,
% 8.58/4.84 ! [Q2: num,N: num] :
% 8.58/4.84 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 8.58/4.84 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 8.58/4.84 = zero_zero_nat ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6029_cong__exp__iff__simps_I7_J,axiom,
% 8.58/4.84 ! [Q2: num,N: num] :
% 8.58/4.84 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 8.58/4.84 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 8.58/4.84 = zero_zero_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6030_cong__exp__iff__simps_I7_J,axiom,
% 8.58/4.84 ! [Q2: num,N: num] :
% 8.58/4.84 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 8.58/4.84 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 8.58/4.84 = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6031_cong__exp__iff__simps_I11_J,axiom,
% 8.58/4.84 ! [M: num,Q2: num] :
% 8.58/4.84 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 8.58/4.84 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 8.58/4.84 = zero_zero_nat ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6032_cong__exp__iff__simps_I11_J,axiom,
% 8.58/4.84 ! [M: num,Q2: num] :
% 8.58/4.84 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 8.58/4.84 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 8.58/4.84 = zero_zero_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6033_cong__exp__iff__simps_I11_J,axiom,
% 8.58/4.84 ! [M: num,Q2: num] :
% 8.58/4.84 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 8.58/4.84 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 8.58/4.84 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 8.58/4.84 = zero_z3403309356797280102nteger ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6034_even__abs__add__iff,axiom,
% 8.58/4.84 ! [K: int,L: int] :
% 8.58/4.84 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
% 8.58/4.84 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6035_even__add__abs__iff,axiom,
% 8.58/4.84 ! [K: int,L: int] :
% 8.58/4.84 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
% 8.58/4.84 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6036_Suc__div__eq__add3__div,axiom,
% 8.58/4.84 ! [M: nat,N: nat] :
% 8.58/4.84 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 8.58/4.84 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6037_Suc__mod__eq__add3__mod,axiom,
% 8.58/4.84 ! [M: nat,N: nat] :
% 8.58/4.84 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 8.58/4.84 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6038_nat__intermed__int__val,axiom,
% 8.58/4.84 ! [M: nat,N: nat,F: nat > int,K: int] :
% 8.58/4.84 ( ! [I2: nat] :
% 8.58/4.84 ( ( ( ord_less_eq_nat @ M @ I2 )
% 8.58/4.84 & ( ord_less_nat @ I2 @ N ) )
% 8.58/4.84 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 8.58/4.84 => ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.84 => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 8.58/4.84 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 8.58/4.84 => ? [I2: nat] :
% 8.58/4.84 ( ( ord_less_eq_nat @ M @ I2 )
% 8.58/4.84 & ( ord_less_eq_nat @ I2 @ N )
% 8.58/4.84 & ( ( F @ I2 )
% 8.58/4.84 = K ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6039_incr__lemma,axiom,
% 8.58/4.84 ! [D: int,Z2: int,X: int] :
% 8.58/4.84 ( ( ord_less_int @ zero_zero_int @ D )
% 8.58/4.84 => ( ord_less_int @ Z2 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6040_decr__lemma,axiom,
% 8.58/4.84 ! [D: int,X: int,Z2: int] :
% 8.58/4.84 ( ( ord_less_int @ zero_zero_int @ D )
% 8.58/4.84 => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) @ Z2 ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6041_mod__exhaust__less__4,axiom,
% 8.58/4.84 ! [M: nat] :
% 8.58/4.84 ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = zero_zero_nat )
% 8.58/4.84 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = one_one_nat )
% 8.58/4.84 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.84 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6042_div__noneq__sgn__abs,axiom,
% 8.58/4.84 ! [L: int,K: int] :
% 8.58/4.84 ( ( L != zero_zero_int )
% 8.58/4.84 => ( ( ( sgn_sgn_int @ K )
% 8.58/4.84 != ( sgn_sgn_int @ L ) )
% 8.58/4.84 => ( ( divide_divide_int @ K @ L )
% 8.58/4.84 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
% 8.58/4.84 @ ( zero_n2684676970156552555ol_int
% 8.58/4.84 @ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6043_nat__ivt__aux,axiom,
% 8.58/4.84 ! [N: nat,F: nat > int,K: int] :
% 8.58/4.84 ( ! [I2: nat] :
% 8.58/4.84 ( ( ord_less_nat @ I2 @ N )
% 8.58/4.84 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 8.58/4.84 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 8.58/4.84 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 8.58/4.84 => ? [I2: nat] :
% 8.58/4.84 ( ( ord_less_eq_nat @ I2 @ N )
% 8.58/4.84 & ( ( F @ I2 )
% 8.58/4.84 = K ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6044_complex__mod__triangle__ineq2,axiom,
% 8.58/4.84 ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6045_take__bit__Suc__bit1,axiom,
% 8.58/4.84 ! [N: nat,K: num] :
% 8.58/4.84 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 8.58/4.84 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6046_take__bit__Suc__bit1,axiom,
% 8.58/4.84 ! [N: nat,K: num] :
% 8.58/4.84 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 8.58/4.84 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6047_nat0__intermed__int__val,axiom,
% 8.58/4.84 ! [N: nat,F: nat > int,K: int] :
% 8.58/4.84 ( ! [I2: nat] :
% 8.58/4.84 ( ( ord_less_nat @ I2 @ N )
% 8.58/4.84 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 8.58/4.84 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 8.58/4.84 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 8.58/4.84 => ? [I2: nat] :
% 8.58/4.84 ( ( ord_less_eq_nat @ I2 @ N )
% 8.58/4.84 & ( ( F @ I2 )
% 8.58/4.84 = K ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6048_arctan__add,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.84 => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 8.58/4.84 => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 8.58/4.84 = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6049_odd__mod__4__div__2,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 8.58/4.84 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6050_signed__take__bit__numeral__minus__bit1,axiom,
% 8.58/4.84 ! [L: num,K: num] :
% 8.58/4.84 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 8.58/4.84 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6051_dbl__dec__simps_I4_J,axiom,
% 8.58/4.84 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.84 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6052_dbl__dec__simps_I4_J,axiom,
% 8.58/4.84 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.84 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6053_dbl__dec__simps_I4_J,axiom,
% 8.58/4.84 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.84 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6054_dbl__dec__simps_I4_J,axiom,
% 8.58/4.84 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.84 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6055_dbl__dec__simps_I4_J,axiom,
% 8.58/4.84 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.84 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6056_take__bit__Suc__minus__bit1,axiom,
% 8.58/4.84 ! [N: nat,K: num] :
% 8.58/4.84 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 8.58/4.84 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6057_signed__take__bit__numeral__bit1,axiom,
% 8.58/4.84 ! [L: num,K: num] :
% 8.58/4.84 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 8.58/4.84 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6058_dbl__inc__simps_I3_J,axiom,
% 8.58/4.84 ( ( neg_nu8557863876264182079omplex @ one_one_complex )
% 8.58/4.84 = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6059_dbl__inc__simps_I3_J,axiom,
% 8.58/4.84 ( ( neg_nu8295874005876285629c_real @ one_one_real )
% 8.58/4.84 = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6060_dbl__inc__simps_I3_J,axiom,
% 8.58/4.84 ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
% 8.58/4.84 = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6061_dbl__inc__simps_I3_J,axiom,
% 8.58/4.84 ( ( neg_nu5851722552734809277nc_int @ one_one_int )
% 8.58/4.84 = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6062_arctan__inverse,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( X != zero_zero_real )
% 8.58/4.84 => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
% 8.58/4.84 = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6063_dbl__dec__simps_I3_J,axiom,
% 8.58/4.84 ( ( neg_nu6511756317524482435omplex @ one_one_complex )
% 8.58/4.84 = one_one_complex ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6064_dbl__dec__simps_I3_J,axiom,
% 8.58/4.84 ( ( neg_nu6075765906172075777c_real @ one_one_real )
% 8.58/4.84 = one_one_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6065_dbl__dec__simps_I3_J,axiom,
% 8.58/4.84 ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
% 8.58/4.84 = one_one_rat ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6066_dbl__dec__simps_I3_J,axiom,
% 8.58/4.84 ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 8.58/4.84 = one_one_int ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6067_pred__numeral__simps_I1_J,axiom,
% 8.58/4.84 ( ( pred_numeral @ one )
% 8.58/4.84 = zero_zero_nat ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6068_Suc__eq__numeral,axiom,
% 8.58/4.84 ! [N: nat,K: num] :
% 8.58/4.84 ( ( ( suc @ N )
% 8.58/4.84 = ( numeral_numeral_nat @ K ) )
% 8.58/4.84 = ( N
% 8.58/4.84 = ( pred_numeral @ K ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6069_eq__numeral__Suc,axiom,
% 8.58/4.84 ! [K: num,N: nat] :
% 8.58/4.84 ( ( ( numeral_numeral_nat @ K )
% 8.58/4.84 = ( suc @ N ) )
% 8.58/4.84 = ( ( pred_numeral @ K )
% 8.58/4.84 = N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6070_dbl__inc__simps_I2_J,axiom,
% 8.58/4.84 ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 8.58/4.84 = one_one_complex ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6071_dbl__inc__simps_I2_J,axiom,
% 8.58/4.84 ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 8.58/4.84 = one_one_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6072_dbl__inc__simps_I2_J,axiom,
% 8.58/4.84 ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 8.58/4.84 = one_one_rat ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6073_dbl__inc__simps_I2_J,axiom,
% 8.58/4.84 ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 8.58/4.84 = one_one_int ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6074_pred__numeral__inc,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( pred_numeral @ ( inc @ K ) )
% 8.58/4.84 = ( numeral_numeral_nat @ K ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6075_dbl__inc__simps_I4_J,axiom,
% 8.58/4.84 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.84 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6076_dbl__inc__simps_I4_J,axiom,
% 8.58/4.84 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.84 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6077_dbl__inc__simps_I4_J,axiom,
% 8.58/4.84 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.84 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6078_dbl__inc__simps_I4_J,axiom,
% 8.58/4.84 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.84 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6079_dbl__inc__simps_I4_J,axiom,
% 8.58/4.84 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.84 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6080_dbl__inc__simps_I5_J,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
% 8.58/4.84 = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6081_dbl__inc__simps_I5_J,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
% 8.58/4.84 = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6082_dbl__inc__simps_I5_J,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
% 8.58/4.84 = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6083_dbl__inc__simps_I5_J,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
% 8.58/4.84 = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6084_pred__numeral__simps_I3_J,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( pred_numeral @ ( bit1 @ K ) )
% 8.58/4.84 = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6085_less__Suc__numeral,axiom,
% 8.58/4.84 ! [N: nat,K: num] :
% 8.58/4.84 ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 8.58/4.84 = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6086_less__numeral__Suc,axiom,
% 8.58/4.84 ! [K: num,N: nat] :
% 8.58/4.84 ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 8.58/4.84 = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6087_le__Suc__numeral,axiom,
% 8.58/4.84 ! [N: nat,K: num] :
% 8.58/4.84 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 8.58/4.84 = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6088_le__numeral__Suc,axiom,
% 8.58/4.84 ! [K: num,N: nat] :
% 8.58/4.84 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 8.58/4.84 = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6089_diff__Suc__numeral,axiom,
% 8.58/4.84 ! [N: nat,K: num] :
% 8.58/4.84 ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 8.58/4.84 = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6090_diff__numeral__Suc,axiom,
% 8.58/4.84 ! [K: num,N: nat] :
% 8.58/4.84 ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 8.58/4.84 = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6091_dbl__dec__simps_I2_J,axiom,
% 8.58/4.84 ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 8.58/4.84 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6092_dbl__dec__simps_I2_J,axiom,
% 8.58/4.84 ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 8.58/4.84 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6093_dbl__dec__simps_I2_J,axiom,
% 8.58/4.84 ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 8.58/4.84 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6094_dbl__dec__simps_I2_J,axiom,
% 8.58/4.84 ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 8.58/4.84 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6095_dbl__dec__simps_I2_J,axiom,
% 8.58/4.84 ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 8.58/4.84 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6096_dbl__dec__simps_I1_J,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 8.58/4.84 = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6097_dbl__dec__simps_I1_J,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.58/4.84 = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6098_dbl__dec__simps_I1_J,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 8.58/4.84 = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6099_dbl__dec__simps_I1_J,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 8.58/4.84 = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6100_dbl__dec__simps_I1_J,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 8.58/4.84 = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6101_dbl__inc__simps_I1_J,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 8.58/4.84 = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6102_dbl__inc__simps_I1_J,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.58/4.84 = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6103_dbl__inc__simps_I1_J,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 8.58/4.84 = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6104_dbl__inc__simps_I1_J,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 8.58/4.84 = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6105_dbl__inc__simps_I1_J,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 8.58/4.84 = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6106_add__neg__numeral__special_I5_J,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.58/4.84 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6107_add__neg__numeral__special_I5_J,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.84 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6108_add__neg__numeral__special_I5_J,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.58/4.84 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6109_add__neg__numeral__special_I5_J,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.58/4.84 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6110_add__neg__numeral__special_I5_J,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.58/4.84 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6111_add__neg__numeral__special_I6_J,axiom,
% 8.58/4.84 ! [M: num] :
% 8.58/4.84 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.84 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6112_add__neg__numeral__special_I6_J,axiom,
% 8.58/4.84 ! [M: num] :
% 8.58/4.84 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.84 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6113_add__neg__numeral__special_I6_J,axiom,
% 8.58/4.84 ! [M: num] :
% 8.58/4.84 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.84 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6114_add__neg__numeral__special_I6_J,axiom,
% 8.58/4.84 ! [M: num] :
% 8.58/4.84 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.84 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6115_add__neg__numeral__special_I6_J,axiom,
% 8.58/4.84 ! [M: num] :
% 8.58/4.84 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.84 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6116_diff__numeral__special_I5_J,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
% 8.58/4.84 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6117_diff__numeral__special_I5_J,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.84 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6118_diff__numeral__special_I5_J,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
% 8.58/4.84 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6119_diff__numeral__special_I5_J,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
% 8.58/4.84 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6120_diff__numeral__special_I5_J,axiom,
% 8.58/4.84 ! [N: num] :
% 8.58/4.84 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
% 8.58/4.84 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6121_diff__numeral__special_I6_J,axiom,
% 8.58/4.84 ! [M: num] :
% 8.58/4.84 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.84 = ( numeral_numeral_real @ ( inc @ M ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6122_diff__numeral__special_I6_J,axiom,
% 8.58/4.84 ! [M: num] :
% 8.58/4.84 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.84 = ( numeral_numeral_int @ ( inc @ M ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6123_diff__numeral__special_I6_J,axiom,
% 8.58/4.84 ! [M: num] :
% 8.58/4.84 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.84 = ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6124_diff__numeral__special_I6_J,axiom,
% 8.58/4.84 ! [M: num] :
% 8.58/4.84 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.84 = ( numeral_numeral_rat @ ( inc @ M ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6125_diff__numeral__special_I6_J,axiom,
% 8.58/4.84 ! [M: num] :
% 8.58/4.84 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.84 = ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6126_push__bit__numeral,axiom,
% 8.58/4.84 ! [L: num,K: num] :
% 8.58/4.84 ( ( bit_se547839408752420682it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ K ) )
% 8.58/4.84 = ( bit_se547839408752420682it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6127_push__bit__numeral,axiom,
% 8.58/4.84 ! [L: num,K: num] :
% 8.58/4.84 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ K ) )
% 8.58/4.84 = ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6128_push__bit__numeral,axiom,
% 8.58/4.84 ! [L: num,K: num] :
% 8.58/4.84 ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ L ) @ ( numera6620942414471956472nteger @ K ) )
% 8.58/4.84 = ( bit_se7788150548672797655nteger @ ( pred_numeral @ L ) @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6129_push__bit__numeral,axiom,
% 8.58/4.84 ! [L: num,K: num] :
% 8.58/4.84 ( ( bit_se6611745700429515170atural @ ( numeral_numeral_nat @ L ) @ ( numera5444537566228673987atural @ K ) )
% 8.58/4.84 = ( bit_se6611745700429515170atural @ ( pred_numeral @ L ) @ ( numera5444537566228673987atural @ ( bit0 @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6130_push__bit__minus__numeral,axiom,
% 8.58/4.84 ! [L: num,K: num] :
% 8.58/4.84 ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.58/4.84 = ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6131_push__bit__minus__numeral,axiom,
% 8.58/4.84 ! [L: num,K: num] :
% 8.58/4.84 ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 8.58/4.84 = ( bit_se7788150548672797655nteger @ ( pred_numeral @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6132_signed__take__bit__numeral__bit0,axiom,
% 8.58/4.84 ! [L: num,K: num] :
% 8.58/4.84 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 8.58/4.84 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6133_signed__take__bit__numeral__minus__bit0,axiom,
% 8.58/4.84 ! [L: num,K: num] :
% 8.58/4.84 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 8.58/4.84 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6134_num__induct,axiom,
% 8.58/4.84 ! [P: num > $o,X: num] :
% 8.58/4.84 ( ( P @ one )
% 8.58/4.84 => ( ! [X4: num] :
% 8.58/4.84 ( ( P @ X4 )
% 8.58/4.84 => ( P @ ( inc @ X4 ) ) )
% 8.58/4.84 => ( P @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6135_add__inc,axiom,
% 8.58/4.84 ! [X: num,Y: num] :
% 8.58/4.84 ( ( plus_plus_num @ X @ ( inc @ Y ) )
% 8.58/4.84 = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6136_numeral__eq__Suc,axiom,
% 8.58/4.84 ( numeral_numeral_nat
% 8.58/4.84 = ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6137_inc_Osimps_I1_J,axiom,
% 8.58/4.84 ( ( inc @ one )
% 8.58/4.84 = ( bit0 @ one ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6138_inc_Osimps_I3_J,axiom,
% 8.58/4.84 ! [X: num] :
% 8.58/4.84 ( ( inc @ ( bit1 @ X ) )
% 8.58/4.84 = ( bit0 @ ( inc @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6139_inc_Osimps_I2_J,axiom,
% 8.58/4.84 ! [X: num] :
% 8.58/4.84 ( ( inc @ ( bit0 @ X ) )
% 8.58/4.84 = ( bit1 @ X ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6140_add__One,axiom,
% 8.58/4.84 ! [X: num] :
% 8.58/4.84 ( ( plus_plus_num @ X @ one )
% 8.58/4.84 = ( inc @ X ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6141_mult__inc,axiom,
% 8.58/4.84 ! [X: num,Y: num] :
% 8.58/4.84 ( ( times_times_num @ X @ ( inc @ Y ) )
% 8.58/4.84 = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6142_pred__numeral__def,axiom,
% 8.58/4.84 ( pred_numeral
% 8.58/4.84 = ( ^ [K2: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ one_one_nat ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6143_numeral__inc,axiom,
% 8.58/4.84 ! [X: num] :
% 8.58/4.84 ( ( numera6690914467698888265omplex @ ( inc @ X ) )
% 8.58/4.84 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6144_numeral__inc,axiom,
% 8.58/4.84 ! [X: num] :
% 8.58/4.84 ( ( numeral_numeral_real @ ( inc @ X ) )
% 8.58/4.84 = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6145_numeral__inc,axiom,
% 8.58/4.84 ! [X: num] :
% 8.58/4.84 ( ( numeral_numeral_rat @ ( inc @ X ) )
% 8.58/4.84 = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6146_numeral__inc,axiom,
% 8.58/4.84 ! [X: num] :
% 8.58/4.84 ( ( numeral_numeral_nat @ ( inc @ X ) )
% 8.58/4.84 = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6147_numeral__inc,axiom,
% 8.58/4.84 ! [X: num] :
% 8.58/4.84 ( ( numeral_numeral_int @ ( inc @ X ) )
% 8.58/4.84 = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6148_pi__less__4,axiom,
% 8.58/4.84 ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6149_pi__ge__two,axiom,
% 8.58/4.84 ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6150_pi__half__neq__two,axiom,
% 8.58/4.84 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.58/4.84 != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6151_dbl__inc__def,axiom,
% 8.58/4.84 ( neg_nu8557863876264182079omplex
% 8.58/4.84 = ( ^ [X3: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6152_dbl__inc__def,axiom,
% 8.58/4.84 ( neg_nu8295874005876285629c_real
% 8.58/4.84 = ( ^ [X3: real] : ( plus_plus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6153_dbl__inc__def,axiom,
% 8.58/4.84 ( neg_nu5219082963157363817nc_rat
% 8.58/4.84 = ( ^ [X3: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6154_dbl__inc__def,axiom,
% 8.58/4.84 ( neg_nu5851722552734809277nc_int
% 8.58/4.84 = ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6155_fact__numeral,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
% 8.58/4.84 = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6156_fact__numeral,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
% 8.58/4.84 = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6157_fact__numeral,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
% 8.58/4.84 = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6158_fact__numeral,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
% 8.58/4.84 = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6159_fact__numeral,axiom,
% 8.58/4.84 ! [K: num] :
% 8.58/4.84 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
% 8.58/4.84 = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6160_pi__half__neq__zero,axiom,
% 8.58/4.84 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.58/4.84 != zero_zero_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6161_pi__half__less__two,axiom,
% 8.58/4.84 ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6162_pi__half__le__two,axiom,
% 8.58/4.84 ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6163_take__bit__numeral__minus__bit1,axiom,
% 8.58/4.84 ! [L: num,K: num] :
% 8.58/4.84 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 8.58/4.84 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6164_dbl__dec__def,axiom,
% 8.58/4.84 ( neg_nu6511756317524482435omplex
% 8.58/4.84 = ( ^ [X3: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6165_dbl__dec__def,axiom,
% 8.58/4.84 ( neg_nu6075765906172075777c_real
% 8.58/4.84 = ( ^ [X3: real] : ( minus_minus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6166_dbl__dec__def,axiom,
% 8.58/4.84 ( neg_nu3179335615603231917ec_rat
% 8.58/4.84 = ( ^ [X3: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6167_dbl__dec__def,axiom,
% 8.58/4.84 ( neg_nu3811975205180677377ec_int
% 8.58/4.84 = ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6168_pi__half__gt__zero,axiom,
% 8.58/4.84 ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6169_pi__half__ge__zero,axiom,
% 8.58/4.84 ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6170_m2pi__less__pi,axiom,
% 8.58/4.84 ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6171_arctan__ubound,axiom,
% 8.58/4.84 ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6172_arctan__one,axiom,
% 8.58/4.84 ( ( arctan @ one_one_real )
% 8.58/4.84 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6173_take__bit__numeral__bit0,axiom,
% 8.58/4.84 ! [L: num,K: num] :
% 8.58/4.84 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 8.58/4.84 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6174_take__bit__numeral__bit0,axiom,
% 8.58/4.84 ! [L: num,K: num] :
% 8.58/4.84 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 8.58/4.84 = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6175_minus__pi__half__less__zero,axiom,
% 8.58/4.84 ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6176_arctan__bounded,axiom,
% 8.58/4.84 ! [Y: real] :
% 8.58/4.84 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
% 8.58/4.84 & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6177_arctan__lbound,axiom,
% 8.58/4.84 ! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6178_take__bit__numeral__minus__bit0,axiom,
% 8.58/4.84 ! [L: num,K: num] :
% 8.58/4.84 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 8.58/4.84 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6179_machin__Euler,axiom,
% 8.58/4.84 ( ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 8.58/4.84 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6180_machin,axiom,
% 8.58/4.84 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6181_take__bit__numeral__bit1,axiom,
% 8.58/4.84 ! [L: num,K: num] :
% 8.58/4.84 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 8.58/4.84 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6182_take__bit__numeral__bit1,axiom,
% 8.58/4.84 ! [L: num,K: num] :
% 8.58/4.84 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 8.58/4.84 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6183_sin__cos__npi,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6184_cos__pi__eq__zero,axiom,
% 8.58/4.84 ! [M: nat] :
% 8.58/4.84 ( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = zero_zero_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6185_cos__zero__iff,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ( cos_real @ X )
% 8.58/4.84 = zero_zero_real )
% 8.58/4.84 = ( ? [N3: nat] :
% 8.58/4.84 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 8.58/4.84 & ( X
% 8.58/4.84 = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.84 | ? [N3: nat] :
% 8.58/4.84 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 8.58/4.84 & ( X
% 8.58/4.84 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6186_cot__less__zero,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 8.58/4.84 => ( ( ord_less_real @ X @ zero_zero_real )
% 8.58/4.84 => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6187_cos__zero__lemma,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.84 => ( ( ( cos_real @ X )
% 8.58/4.84 = zero_zero_real )
% 8.58/4.84 => ? [N2: nat] :
% 8.58/4.84 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 8.58/4.84 & ( X
% 8.58/4.84 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6188_sin__zero__iff,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ( sin_real @ X )
% 8.58/4.84 = zero_zero_real )
% 8.58/4.84 = ( ? [N3: nat] :
% 8.58/4.84 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 8.58/4.84 & ( X
% 8.58/4.84 = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.84 | ? [N3: nat] :
% 8.58/4.84 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 8.58/4.84 & ( X
% 8.58/4.84 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6189_sin__zero,axiom,
% 8.58/4.84 ( ( sin_real @ zero_zero_real )
% 8.58/4.84 = zero_zero_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6190_cot__zero,axiom,
% 8.58/4.84 ( ( cot_real @ zero_zero_real )
% 8.58/4.84 = zero_zero_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6191_cos__zero,axiom,
% 8.58/4.84 ( ( cos_complex @ zero_zero_complex )
% 8.58/4.84 = one_one_complex ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6192_cos__zero,axiom,
% 8.58/4.84 ( ( cos_real @ zero_zero_real )
% 8.58/4.84 = one_one_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6193_sin__pi__minus,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( sin_real @ ( minus_minus_real @ pi @ X ) )
% 8.58/4.84 = ( sin_real @ X ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6194_sin__of__real__pi,axiom,
% 8.58/4.84 ( ( sin_real @ ( real_V1803761363581548252l_real @ pi ) )
% 8.58/4.84 = zero_zero_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6195_sin__of__real__pi,axiom,
% 8.58/4.84 ( ( sin_complex @ ( real_V4546457046886955230omplex @ pi ) )
% 8.58/4.84 = zero_zero_complex ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6196_cos__pi,axiom,
% 8.58/4.84 ( ( cos_real @ pi )
% 8.58/4.84 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6197_cos__periodic__pi,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( cos_real @ ( plus_plus_real @ X @ pi ) )
% 8.58/4.84 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6198_cos__periodic__pi2,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( cos_real @ ( plus_plus_real @ pi @ X ) )
% 8.58/4.84 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6199_sin__periodic__pi,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( sin_real @ ( plus_plus_real @ X @ pi ) )
% 8.58/4.84 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6200_sin__periodic__pi2,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( sin_real @ ( plus_plus_real @ pi @ X ) )
% 8.58/4.84 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6201_cos__minus__pi,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( cos_real @ ( minus_minus_real @ X @ pi ) )
% 8.58/4.84 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6202_cos__pi__minus,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( cos_real @ ( minus_minus_real @ pi @ X ) )
% 8.58/4.84 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6203_sin__minus__pi,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( sin_real @ ( minus_minus_real @ X @ pi ) )
% 8.58/4.84 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6204_sin__cos__squared__add3,axiom,
% 8.58/4.84 ! [X: complex] :
% 8.58/4.84 ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
% 8.58/4.84 = one_one_complex ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6205_sin__cos__squared__add3,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
% 8.58/4.84 = one_one_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6206_cos__of__real__pi,axiom,
% 8.58/4.84 ( ( cos_real @ ( real_V1803761363581548252l_real @ pi ) )
% 8.58/4.84 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6207_cos__of__real__pi,axiom,
% 8.58/4.84 ( ( cos_complex @ ( real_V4546457046886955230omplex @ pi ) )
% 8.58/4.84 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6208_sin__npi2,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 8.58/4.84 = zero_zero_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6209_sin__npi,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 8.58/4.84 = zero_zero_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6210_cot__npi,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 8.58/4.84 = zero_zero_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6211_cos__pi__half,axiom,
% 8.58/4.84 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = zero_zero_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6212_sin__two__pi,axiom,
% 8.58/4.84 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 8.58/4.84 = zero_zero_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6213_sin__pi__half,axiom,
% 8.58/4.84 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = one_one_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6214_cos__two__pi,axiom,
% 8.58/4.84 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 8.58/4.84 = one_one_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6215_cos__periodic,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 8.58/4.84 = ( cos_real @ X ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6216_sin__periodic,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 8.58/4.84 = ( sin_real @ X ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6217_cos__2pi__minus,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 8.58/4.84 = ( cos_real @ X ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6218_cos__npi2,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 8.58/4.84 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6219_cos__npi,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 8.58/4.84 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6220_cot__periodic,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 8.58/4.84 = ( cot_real @ X ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6221_sin__cos__squared__add,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = one_one_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6222_sin__cos__squared__add,axiom,
% 8.58/4.84 ! [X: complex] :
% 8.58/4.84 ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = one_one_complex ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6223_sin__cos__squared__add2,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = one_one_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6224_sin__cos__squared__add2,axiom,
% 8.58/4.84 ! [X: complex] :
% 8.58/4.84 ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = one_one_complex ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6225_cos__of__real__pi__half,axiom,
% 8.58/4.84 ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = zero_zero_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6226_cos__of__real__pi__half,axiom,
% 8.58/4.84 ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = zero_zero_complex ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6227_sin__of__real__pi__half,axiom,
% 8.58/4.84 ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = one_one_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6228_sin__of__real__pi__half,axiom,
% 8.58/4.84 ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 8.58/4.84 = one_one_complex ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6229_sin__2npi,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 8.58/4.84 = zero_zero_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6230_cos__2npi,axiom,
% 8.58/4.84 ! [N: nat] :
% 8.58/4.84 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 8.58/4.84 = one_one_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6231_sin__2pi__minus,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 8.58/4.84 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6232_cos__3over2__pi,axiom,
% 8.58/4.84 ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 8.58/4.84 = zero_zero_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6233_sin__3over2__pi,axiom,
% 8.58/4.84 ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 8.58/4.84 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6234_sin__add,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( sin_real @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.84 = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6235_cos__one__sin__zero,axiom,
% 8.58/4.84 ! [X: complex] :
% 8.58/4.84 ( ( ( cos_complex @ X )
% 8.58/4.84 = one_one_complex )
% 8.58/4.84 => ( ( sin_complex @ X )
% 8.58/4.84 = zero_zero_complex ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6236_cos__one__sin__zero,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ( cos_real @ X )
% 8.58/4.84 = one_one_real )
% 8.58/4.84 => ( ( sin_real @ X )
% 8.58/4.84 = zero_zero_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6237_cot__def,axiom,
% 8.58/4.84 ( cot_complex
% 8.58/4.84 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( cos_complex @ X3 ) @ ( sin_complex @ X3 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6238_cot__def,axiom,
% 8.58/4.84 ( cot_real
% 8.58/4.84 = ( ^ [X3: real] : ( divide_divide_real @ ( cos_real @ X3 ) @ ( sin_real @ X3 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6239_polar__Ex,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ? [R2: real,A4: real] :
% 8.58/4.84 ( ( X
% 8.58/4.84 = ( times_times_real @ R2 @ ( cos_real @ A4 ) ) )
% 8.58/4.84 & ( Y
% 8.58/4.84 = ( times_times_real @ R2 @ ( sin_real @ A4 ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6240_sin__diff,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( sin_real @ ( minus_minus_real @ X @ Y ) )
% 8.58/4.84 = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6241_cos__add,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.84 = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6242_cos__diff,axiom,
% 8.58/4.84 ! [X: real,Y: real] :
% 8.58/4.84 ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
% 8.58/4.84 = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6243_sin__zero__norm__cos__one,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ( sin_real @ X )
% 8.58/4.84 = zero_zero_real )
% 8.58/4.84 => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
% 8.58/4.84 = one_one_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6244_sin__zero__norm__cos__one,axiom,
% 8.58/4.84 ! [X: complex] :
% 8.58/4.84 ( ( ( sin_complex @ X )
% 8.58/4.84 = zero_zero_complex )
% 8.58/4.84 => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
% 8.58/4.84 = one_one_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6245_sin__zero__abs__cos__one,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( ( sin_real @ X )
% 8.58/4.84 = zero_zero_real )
% 8.58/4.84 => ( ( abs_abs_real @ ( cos_real @ X ) )
% 8.58/4.84 = one_one_real ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6246_sin__double,axiom,
% 8.58/4.84 ! [X: complex] :
% 8.58/4.84 ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 8.58/4.84 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X ) ) @ ( cos_complex @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6247_sin__double,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 8.58/4.84 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X ) ) @ ( cos_real @ X ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6248_sin__le__one,axiom,
% 8.58/4.84 ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6249_cos__le__one,axiom,
% 8.58/4.84 ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6250_sin__cos__le1,axiom,
% 8.58/4.84 ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6251_cos__squared__eq,axiom,
% 8.58/4.84 ! [X: complex] :
% 8.58/4.84 ( ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.84 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6252_cos__squared__eq,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.84 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6253_sin__squared__eq,axiom,
% 8.58/4.84 ! [X: complex] :
% 8.58/4.84 ( ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.84 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6254_sin__squared__eq,axiom,
% 8.58/4.84 ! [X: real] :
% 8.58/4.84 ( ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.84 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6255_sin__ge__minus__one,axiom,
% 8.58/4.84 ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6256_cos__ge__minus__one,axiom,
% 8.58/4.84 ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6257_abs__sin__le__one,axiom,
% 8.58/4.84 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6258_abs__cos__le__one,axiom,
% 8.58/4.84 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6259_sin__times__sin,axiom,
% 8.58/4.84 ! [W: complex,Z2: complex] :
% 8.58/4.84 ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z2 ) )
% 8.58/4.84 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6260_sin__times__sin,axiom,
% 8.58/4.84 ! [W: real,Z2: real] :
% 8.58/4.84 ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z2 ) )
% 8.58/4.84 = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6261_sin__times__cos,axiom,
% 8.58/4.84 ! [W: complex,Z2: complex] :
% 8.58/4.84 ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z2 ) )
% 8.58/4.84 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6262_sin__times__cos,axiom,
% 8.58/4.84 ! [W: real,Z2: real] :
% 8.58/4.84 ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z2 ) )
% 8.58/4.84 = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6263_cos__times__sin,axiom,
% 8.58/4.84 ! [W: complex,Z2: complex] :
% 8.58/4.84 ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z2 ) )
% 8.58/4.84 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6264_cos__times__sin,axiom,
% 8.58/4.84 ! [W: real,Z2: real] :
% 8.58/4.84 ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z2 ) )
% 8.58/4.84 = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6265_sin__plus__sin,axiom,
% 8.58/4.84 ! [W: complex,Z2: complex] :
% 8.58/4.84 ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z2 ) )
% 8.58/4.84 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6266_sin__plus__sin,axiom,
% 8.58/4.84 ! [W: real,Z2: real] :
% 8.58/4.84 ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z2 ) )
% 8.58/4.84 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6267_sin__diff__sin,axiom,
% 8.58/4.84 ! [W: complex,Z2: complex] :
% 8.58/4.84 ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z2 ) )
% 8.58/4.84 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6268_sin__diff__sin,axiom,
% 8.58/4.84 ! [W: real,Z2: real] :
% 8.58/4.84 ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z2 ) )
% 8.58/4.84 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6269_cos__diff__cos,axiom,
% 8.58/4.84 ! [W: complex,Z2: complex] :
% 8.58/4.84 ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z2 ) )
% 8.58/4.84 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z2 @ W ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6270_cos__diff__cos,axiom,
% 8.58/4.84 ! [W: real,Z2: real] :
% 8.58/4.84 ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z2 ) )
% 8.58/4.84 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z2 @ W ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.84
% 8.58/4.84 thf(fact_6271_cos__double,axiom,
% 8.58/4.84 ! [X: complex] :
% 8.58/4.84 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 8.58/4.85 = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6272_cos__double,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 8.58/4.85 = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6273_sin__cos__eq,axiom,
% 8.58/4.85 ( sin_real
% 8.58/4.85 = ( ^ [X3: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6274_sin__cos__eq,axiom,
% 8.58/4.85 ( sin_complex
% 8.58/4.85 = ( ^ [X3: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6275_cos__sin__eq,axiom,
% 8.58/4.85 ( cos_real
% 8.58/4.85 = ( ^ [X3: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6276_cos__sin__eq,axiom,
% 8.58/4.85 ( cos_complex
% 8.58/4.85 = ( ^ [X3: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6277_cos__double__sin,axiom,
% 8.58/4.85 ! [W: complex] :
% 8.58/4.85 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 8.58/4.85 = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6278_cos__double__sin,axiom,
% 8.58/4.85 ! [W: real] :
% 8.58/4.85 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 8.58/4.85 = ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6279_minus__sin__cos__eq,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( uminus_uminus_real @ ( sin_real @ X ) )
% 8.58/4.85 = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6280_minus__sin__cos__eq,axiom,
% 8.58/4.85 ! [X: complex] :
% 8.58/4.85 ( ( uminus1482373934393186551omplex @ ( sin_complex @ X ) )
% 8.58/4.85 = ( cos_complex @ ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6281_cos__two__neq__zero,axiom,
% 8.58/4.85 ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.58/4.85 != zero_zero_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6282_sincos__total__pi,axiom,
% 8.58/4.85 ! [Y: real,X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.58/4.85 => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.85 = one_one_real )
% 8.58/4.85 => ? [T3: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 8.58/4.85 & ( ord_less_eq_real @ T3 @ pi )
% 8.58/4.85 & ( X
% 8.58/4.85 = ( cos_real @ T3 ) )
% 8.58/4.85 & ( Y
% 8.58/4.85 = ( sin_real @ T3 ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6283_sin__expansion__lemma,axiom,
% 8.58/4.85 ! [X: real,M: nat] :
% 8.58/4.85 ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 8.58/4.85 = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6284_cos__expansion__lemma,axiom,
% 8.58/4.85 ! [X: real,M: nat] :
% 8.58/4.85 ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 8.58/4.85 = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6285_sin__gt__zero__02,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.58/4.85 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6286_cos__two__less__zero,axiom,
% 8.58/4.85 ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6287_cos__is__zero,axiom,
% 8.58/4.85 ? [X4: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 8.58/4.85 & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.58/4.85 & ( ( cos_real @ X4 )
% 8.58/4.85 = zero_zero_real )
% 8.58/4.85 & ! [Y5: real] :
% 8.58/4.85 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 8.58/4.85 & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.58/4.85 & ( ( cos_real @ Y5 )
% 8.58/4.85 = zero_zero_real ) )
% 8.58/4.85 => ( Y5 = X4 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6288_cos__two__le__zero,axiom,
% 8.58/4.85 ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6289_cos__total,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ? [X4: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 8.58/4.85 & ( ord_less_eq_real @ X4 @ pi )
% 8.58/4.85 & ( ( cos_real @ X4 )
% 8.58/4.85 = Y )
% 8.58/4.85 & ! [Y5: real] :
% 8.58/4.85 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 8.58/4.85 & ( ord_less_eq_real @ Y5 @ pi )
% 8.58/4.85 & ( ( cos_real @ Y5 )
% 8.58/4.85 = Y ) )
% 8.58/4.85 => ( Y5 = X4 ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6290_sincos__total__pi__half,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.58/4.85 => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.85 = one_one_real )
% 8.58/4.85 => ? [T3: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 8.58/4.85 & ( ord_less_eq_real @ T3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 & ( X
% 8.58/4.85 = ( cos_real @ T3 ) )
% 8.58/4.85 & ( Y
% 8.58/4.85 = ( sin_real @ T3 ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6291_sincos__total__2pi__le,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.85 = one_one_real )
% 8.58/4.85 => ? [T3: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 8.58/4.85 & ( ord_less_eq_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 8.58/4.85 & ( X
% 8.58/4.85 = ( cos_real @ T3 ) )
% 8.58/4.85 & ( Y
% 8.58/4.85 = ( sin_real @ T3 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6292_sincos__total__2pi,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.85 = one_one_real )
% 8.58/4.85 => ~ ! [T3: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 8.58/4.85 => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 8.58/4.85 => ( ( X
% 8.58/4.85 = ( cos_real @ T3 ) )
% 8.58/4.85 => ( Y
% 8.58/4.85 != ( sin_real @ T3 ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6293_sin__pi__divide__n__ge__0,axiom,
% 8.58/4.85 ! [N: nat] :
% 8.58/4.85 ( ( N != zero_zero_nat )
% 8.58/4.85 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6294_cos__times__cos,axiom,
% 8.58/4.85 ! [W: complex,Z2: complex] :
% 8.58/4.85 ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z2 ) )
% 8.58/4.85 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6295_cos__times__cos,axiom,
% 8.58/4.85 ! [W: real,Z2: real] :
% 8.58/4.85 ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z2 ) )
% 8.58/4.85 = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6296_cos__plus__cos,axiom,
% 8.58/4.85 ! [W: complex,Z2: complex] :
% 8.58/4.85 ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z2 ) )
% 8.58/4.85 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6297_cos__plus__cos,axiom,
% 8.58/4.85 ! [W: real,Z2: real] :
% 8.58/4.85 ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z2 ) )
% 8.58/4.85 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6298_sin__gt__zero2,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6299_sin__lt__zero,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ pi @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 8.58/4.85 => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6300_cos__double__less__one,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.58/4.85 => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6301_sin__30,axiom,
% 8.58/4.85 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 8.58/4.85 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6302_cos__gt__zero,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6303_sin__inj__pi,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ( sin_real @ X )
% 8.58/4.85 = ( sin_real @ Y ) )
% 8.58/4.85 => ( X = Y ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6304_sin__mono__le__eq,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
% 8.58/4.85 = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6305_sin__monotone__2pi__le,axiom,
% 8.58/4.85 ! [Y: real,X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6306_cos__60,axiom,
% 8.58/4.85 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 8.58/4.85 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6307_cos__double__cos,axiom,
% 8.58/4.85 ! [W: complex] :
% 8.58/4.85 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 8.58/4.85 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6308_cos__double__cos,axiom,
% 8.58/4.85 ! [W: real] :
% 8.58/4.85 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 8.58/4.85 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6309_cos__treble__cos,axiom,
% 8.58/4.85 ! [X: complex] :
% 8.58/4.85 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X ) )
% 8.58/4.85 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6310_cos__treble__cos,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X ) )
% 8.58/4.85 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6311_sin__le__zero,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ pi @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 8.58/4.85 => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6312_sin__less__zero,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ zero_zero_real )
% 8.58/4.85 => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6313_sin__monotone__2pi,axiom,
% 8.58/4.85 ! [Y: real,X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 8.58/4.85 => ( ( ord_less_real @ Y @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6314_sin__mono__less__eq,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
% 8.58/4.85 = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6315_sin__total,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ? [X4: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 8.58/4.85 & ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 & ( ( sin_real @ X4 )
% 8.58/4.85 = Y )
% 8.58/4.85 & ! [Y5: real] :
% 8.58/4.85 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 8.58/4.85 & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 & ( ( sin_real @ Y5 )
% 8.58/4.85 = Y ) )
% 8.58/4.85 => ( Y5 = X4 ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6316_cos__gt__zero__pi,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6317_cos__ge__zero,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6318_cos__one__2pi,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ( cos_real @ X )
% 8.58/4.85 = one_one_real )
% 8.58/4.85 = ( ? [X3: nat] :
% 8.58/4.85 ( X
% 8.58/4.85 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 8.58/4.85 | ? [X3: nat] :
% 8.58/4.85 ( X
% 8.58/4.85 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6319_sin__pi__divide__n__gt__0,axiom,
% 8.58/4.85 ! [N: nat] :
% 8.58/4.85 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.85 => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6320_cot__gt__zero,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6321_sin__zero__lemma,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ( sin_real @ X )
% 8.58/4.85 = zero_zero_real )
% 8.58/4.85 => ? [N2: nat] :
% 8.58/4.85 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 8.58/4.85 & ( X
% 8.58/4.85 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6322_tan__double,axiom,
% 8.58/4.85 ! [X: complex] :
% 8.58/4.85 ( ( ( cos_complex @ X )
% 8.58/4.85 != zero_zero_complex )
% 8.58/4.85 => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 8.58/4.85 != zero_zero_complex )
% 8.58/4.85 => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 8.58/4.85 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6323_tan__double,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ( cos_real @ X )
% 8.58/4.85 != zero_zero_real )
% 8.58/4.85 => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 8.58/4.85 != zero_zero_real )
% 8.58/4.85 => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 8.58/4.85 = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6324_complex__unimodular__polar,axiom,
% 8.58/4.85 ! [Z2: complex] :
% 8.58/4.85 ( ( ( real_V1022390504157884413omplex @ Z2 )
% 8.58/4.85 = one_one_real )
% 8.58/4.85 => ~ ! [T3: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ T3 )
% 8.58/4.85 => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 8.58/4.85 => ( Z2
% 8.58/4.85 != ( complex2 @ ( cos_real @ T3 ) @ ( sin_real @ T3 ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6325_cos__npi__int,axiom,
% 8.58/4.85 ! [N: int] :
% 8.58/4.85 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 8.58/4.85 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 8.58/4.85 = one_one_real ) )
% 8.58/4.85 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 8.58/4.85 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 8.58/4.85 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6326_le__arcsin__iff,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ one_one_real )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ ( arcsin @ X ) )
% 8.58/4.85 = ( ord_less_eq_real @ ( sin_real @ Y ) @ X ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6327_arcsin__le__iff,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ one_one_real )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y )
% 8.58/4.85 = ( ord_less_eq_real @ X @ ( sin_real @ Y ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6328_arcsin__pi,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 8.58/4.85 & ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
% 8.58/4.85 & ( ( sin_real @ ( arcsin @ Y ) )
% 8.58/4.85 = Y ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6329_of__int__eq__iff,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ( ring_1_of_int_real @ W )
% 8.58/4.85 = ( ring_1_of_int_real @ Z2 ) )
% 8.58/4.85 = ( W = Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6330_of__int__eq__iff,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ( ring_1_of_int_rat @ W )
% 8.58/4.85 = ( ring_1_of_int_rat @ Z2 ) )
% 8.58/4.85 = ( W = Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6331_tan__zero,axiom,
% 8.58/4.85 ( ( tan_real @ zero_zero_real )
% 8.58/4.85 = zero_zero_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6332_tan__periodic__pi,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( tan_real @ ( plus_plus_real @ X @ pi ) )
% 8.58/4.85 = ( tan_real @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6333_of__int__floor__cancel,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) )
% 8.58/4.85 = X )
% 8.58/4.85 = ( ? [N3: int] :
% 8.58/4.85 ( X
% 8.58/4.85 = ( ring_1_of_int_real @ N3 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6334_of__int__floor__cancel,axiom,
% 8.58/4.85 ! [X: rat] :
% 8.58/4.85 ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) )
% 8.58/4.85 = X )
% 8.58/4.85 = ( ? [N3: int] :
% 8.58/4.85 ( X
% 8.58/4.85 = ( ring_1_of_int_rat @ N3 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6335_floor__of__int,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( archim6058952711729229775r_real @ ( ring_1_of_int_real @ Z2 ) )
% 8.58/4.85 = Z2 ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6336_floor__of__int,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( archim3151403230148437115or_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 8.58/4.85 = Z2 ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6337_of__int__ceiling__cancel,axiom,
% 8.58/4.85 ! [X: rat] :
% 8.58/4.85 ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) )
% 8.58/4.85 = X )
% 8.58/4.85 = ( ? [N3: int] :
% 8.58/4.85 ( X
% 8.58/4.85 = ( ring_1_of_int_rat @ N3 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6338_of__int__ceiling__cancel,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
% 8.58/4.85 = X )
% 8.58/4.85 = ( ? [N3: int] :
% 8.58/4.85 ( X
% 8.58/4.85 = ( ring_1_of_int_real @ N3 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6339_ceiling__of__int,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( archim2889992004027027881ng_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 8.58/4.85 = Z2 ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6340_ceiling__of__int,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( archim7802044766580827645g_real @ ( ring_1_of_int_real @ Z2 ) )
% 8.58/4.85 = Z2 ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6341_of__int__eq__0__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ( ring_1_of_int_int @ Z2 )
% 8.58/4.85 = zero_zero_int )
% 8.58/4.85 = ( Z2 = zero_zero_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6342_of__int__eq__0__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ( ring_1_of_int_real @ Z2 )
% 8.58/4.85 = zero_zero_real )
% 8.58/4.85 = ( Z2 = zero_zero_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6343_of__int__eq__0__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ( ring_1_of_int_rat @ Z2 )
% 8.58/4.85 = zero_zero_rat )
% 8.58/4.85 = ( Z2 = zero_zero_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6344_of__int__0__eq__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( zero_zero_int
% 8.58/4.85 = ( ring_1_of_int_int @ Z2 ) )
% 8.58/4.85 = ( Z2 = zero_zero_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6345_of__int__0__eq__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( zero_zero_real
% 8.58/4.85 = ( ring_1_of_int_real @ Z2 ) )
% 8.58/4.85 = ( Z2 = zero_zero_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6346_of__int__0__eq__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( zero_zero_rat
% 8.58/4.85 = ( ring_1_of_int_rat @ Z2 ) )
% 8.58/4.85 = ( Z2 = zero_zero_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6347_of__int__0,axiom,
% 8.58/4.85 ( ( ring_1_of_int_int @ zero_zero_int )
% 8.58/4.85 = zero_zero_int ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6348_of__int__0,axiom,
% 8.58/4.85 ( ( ring_1_of_int_real @ zero_zero_int )
% 8.58/4.85 = zero_zero_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6349_of__int__0,axiom,
% 8.58/4.85 ( ( ring_1_of_int_rat @ zero_zero_int )
% 8.58/4.85 = zero_zero_rat ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6350_of__int__le__iff,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) )
% 8.58/4.85 = ( ord_less_eq_int @ W @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6351_of__int__le__iff,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z2 ) )
% 8.58/4.85 = ( ord_less_eq_int @ W @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6352_of__int__le__iff,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) )
% 8.58/4.85 = ( ord_less_eq_int @ W @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6353_of__int__numeral,axiom,
% 8.58/4.85 ! [K: num] :
% 8.58/4.85 ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
% 8.58/4.85 = ( numera6690914467698888265omplex @ K ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6354_of__int__numeral,axiom,
% 8.58/4.85 ! [K: num] :
% 8.58/4.85 ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 8.58/4.85 = ( numeral_numeral_real @ K ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6355_of__int__numeral,axiom,
% 8.58/4.85 ! [K: num] :
% 8.58/4.85 ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 8.58/4.85 = ( numeral_numeral_rat @ K ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6356_of__int__numeral,axiom,
% 8.58/4.85 ! [K: num] :
% 8.58/4.85 ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 8.58/4.85 = ( numeral_numeral_int @ K ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6357_of__int__eq__numeral__iff,axiom,
% 8.58/4.85 ! [Z2: int,N: num] :
% 8.58/4.85 ( ( ( ring_17405671764205052669omplex @ Z2 )
% 8.58/4.85 = ( numera6690914467698888265omplex @ N ) )
% 8.58/4.85 = ( Z2
% 8.58/4.85 = ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6358_of__int__eq__numeral__iff,axiom,
% 8.58/4.85 ! [Z2: int,N: num] :
% 8.58/4.85 ( ( ( ring_1_of_int_real @ Z2 )
% 8.58/4.85 = ( numeral_numeral_real @ N ) )
% 8.58/4.85 = ( Z2
% 8.58/4.85 = ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6359_of__int__eq__numeral__iff,axiom,
% 8.58/4.85 ! [Z2: int,N: num] :
% 8.58/4.85 ( ( ( ring_1_of_int_rat @ Z2 )
% 8.58/4.85 = ( numeral_numeral_rat @ N ) )
% 8.58/4.85 = ( Z2
% 8.58/4.85 = ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6360_of__int__eq__numeral__iff,axiom,
% 8.58/4.85 ! [Z2: int,N: num] :
% 8.58/4.85 ( ( ( ring_1_of_int_int @ Z2 )
% 8.58/4.85 = ( numeral_numeral_int @ N ) )
% 8.58/4.85 = ( Z2
% 8.58/4.85 = ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6361_of__int__less__iff,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) )
% 8.58/4.85 = ( ord_less_int @ W @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6362_of__int__less__iff,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z2 ) )
% 8.58/4.85 = ( ord_less_int @ W @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6363_of__int__less__iff,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) )
% 8.58/4.85 = ( ord_less_int @ W @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6364_of__int__1,axiom,
% 8.58/4.85 ( ( ring_17405671764205052669omplex @ one_one_int )
% 8.58/4.85 = one_one_complex ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6365_of__int__1,axiom,
% 8.58/4.85 ( ( ring_1_of_int_int @ one_one_int )
% 8.58/4.85 = one_one_int ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6366_of__int__1,axiom,
% 8.58/4.85 ( ( ring_1_of_int_real @ one_one_int )
% 8.58/4.85 = one_one_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6367_of__int__1,axiom,
% 8.58/4.85 ( ( ring_1_of_int_rat @ one_one_int )
% 8.58/4.85 = one_one_rat ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6368_of__int__eq__1__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ( ring_17405671764205052669omplex @ Z2 )
% 8.58/4.85 = one_one_complex )
% 8.58/4.85 = ( Z2 = one_one_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6369_of__int__eq__1__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ( ring_1_of_int_int @ Z2 )
% 8.58/4.85 = one_one_int )
% 8.58/4.85 = ( Z2 = one_one_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6370_of__int__eq__1__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ( ring_1_of_int_real @ Z2 )
% 8.58/4.85 = one_one_real )
% 8.58/4.85 = ( Z2 = one_one_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6371_of__int__eq__1__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ( ring_1_of_int_rat @ Z2 )
% 8.58/4.85 = one_one_rat )
% 8.58/4.85 = ( Z2 = one_one_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6372_of__int__mult,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z2 ) )
% 8.58/4.85 = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6373_of__int__mult,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z2 ) )
% 8.58/4.85 = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6374_of__int__mult,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z2 ) )
% 8.58/4.85 = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6375_of__int__add,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z2 ) )
% 8.58/4.85 = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6376_of__int__add,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z2 ) )
% 8.58/4.85 = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6377_of__int__add,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z2 ) )
% 8.58/4.85 = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6378_of__int__minus,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z2 ) )
% 8.58/4.85 = ( uminus_uminus_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6379_of__int__minus,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z2 ) )
% 8.58/4.85 = ( uminus_uminus_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6380_of__int__minus,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ Z2 ) )
% 8.58/4.85 = ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6381_of__int__minus,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z2 ) )
% 8.58/4.85 = ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6382_of__int__minus,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ Z2 ) )
% 8.58/4.85 = ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6383_of__int__diff,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z2 ) )
% 8.58/4.85 = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6384_of__int__diff,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z2 ) )
% 8.58/4.85 = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6385_of__int__diff,axiom,
% 8.58/4.85 ! [W: int,Z2: int] :
% 8.58/4.85 ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z2 ) )
% 8.58/4.85 = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6386_of__int__of__nat__eq,axiom,
% 8.58/4.85 ! [N: nat] :
% 8.58/4.85 ( ( ring_17405671764205052669omplex @ ( semiri1314217659103216013at_int @ N ) )
% 8.58/4.85 = ( semiri8010041392384452111omplex @ N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6387_of__int__of__nat__eq,axiom,
% 8.58/4.85 ! [N: nat] :
% 8.58/4.85 ( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
% 8.58/4.85 = ( semiri5074537144036343181t_real @ N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6388_of__int__of__nat__eq,axiom,
% 8.58/4.85 ! [N: nat] :
% 8.58/4.85 ( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N ) )
% 8.58/4.85 = ( semiri681578069525770553at_rat @ N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6389_of__int__of__nat__eq,axiom,
% 8.58/4.85 ! [N: nat] :
% 8.58/4.85 ( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
% 8.58/4.85 = ( semiri1314217659103216013at_int @ N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6390_of__int__abs,axiom,
% 8.58/4.85 ! [X: int] :
% 8.58/4.85 ( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
% 8.58/4.85 = ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6391_of__int__abs,axiom,
% 8.58/4.85 ! [X: int] :
% 8.58/4.85 ( ( ring_18347121197199848620nteger @ ( abs_abs_int @ X ) )
% 8.58/4.85 = ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6392_of__int__abs,axiom,
% 8.58/4.85 ! [X: int] :
% 8.58/4.85 ( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
% 8.58/4.85 = ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6393_of__int__abs,axiom,
% 8.58/4.85 ! [X: int] :
% 8.58/4.85 ( ( ring_1_of_int_rat @ ( abs_abs_int @ X ) )
% 8.58/4.85 = ( abs_abs_rat @ ( ring_1_of_int_rat @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6394_of__int__power,axiom,
% 8.58/4.85 ! [Z2: int,N: nat] :
% 8.58/4.85 ( ( ring_1_of_int_rat @ ( power_power_int @ Z2 @ N ) )
% 8.58/4.85 = ( power_power_rat @ ( ring_1_of_int_rat @ Z2 ) @ N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6395_of__int__power,axiom,
% 8.58/4.85 ! [Z2: int,N: nat] :
% 8.58/4.85 ( ( ring_1_of_int_real @ ( power_power_int @ Z2 @ N ) )
% 8.58/4.85 = ( power_power_real @ ( ring_1_of_int_real @ Z2 ) @ N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6396_of__int__power,axiom,
% 8.58/4.85 ! [Z2: int,N: nat] :
% 8.58/4.85 ( ( ring_1_of_int_int @ ( power_power_int @ Z2 @ N ) )
% 8.58/4.85 = ( power_power_int @ ( ring_1_of_int_int @ Z2 ) @ N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6397_of__int__power,axiom,
% 8.58/4.85 ! [Z2: int,N: nat] :
% 8.58/4.85 ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z2 @ N ) )
% 8.58/4.85 = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z2 ) @ N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6398_of__int__eq__of__int__power__cancel__iff,axiom,
% 8.58/4.85 ! [B: int,W: nat,X: int] :
% 8.58/4.85 ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
% 8.58/4.85 = ( ring_1_of_int_rat @ X ) )
% 8.58/4.85 = ( ( power_power_int @ B @ W )
% 8.58/4.85 = X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6399_of__int__eq__of__int__power__cancel__iff,axiom,
% 8.58/4.85 ! [B: int,W: nat,X: int] :
% 8.58/4.85 ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 8.58/4.85 = ( ring_1_of_int_real @ X ) )
% 8.58/4.85 = ( ( power_power_int @ B @ W )
% 8.58/4.85 = X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6400_of__int__eq__of__int__power__cancel__iff,axiom,
% 8.58/4.85 ! [B: int,W: nat,X: int] :
% 8.58/4.85 ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 8.58/4.85 = ( ring_1_of_int_int @ X ) )
% 8.58/4.85 = ( ( power_power_int @ B @ W )
% 8.58/4.85 = X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6401_of__int__eq__of__int__power__cancel__iff,axiom,
% 8.58/4.85 ! [B: int,W: nat,X: int] :
% 8.58/4.85 ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
% 8.58/4.85 = ( ring_17405671764205052669omplex @ X ) )
% 8.58/4.85 = ( ( power_power_int @ B @ W )
% 8.58/4.85 = X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6402_of__int__power__eq__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: int,B: int,W: nat] :
% 8.58/4.85 ( ( ( ring_1_of_int_rat @ X )
% 8.58/4.85 = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 8.58/4.85 = ( X
% 8.58/4.85 = ( power_power_int @ B @ W ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6403_of__int__power__eq__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: int,B: int,W: nat] :
% 8.58/4.85 ( ( ( ring_1_of_int_real @ X )
% 8.58/4.85 = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 8.58/4.85 = ( X
% 8.58/4.85 = ( power_power_int @ B @ W ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6404_of__int__power__eq__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: int,B: int,W: nat] :
% 8.58/4.85 ( ( ( ring_1_of_int_int @ X )
% 8.58/4.85 = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 8.58/4.85 = ( X
% 8.58/4.85 = ( power_power_int @ B @ W ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6405_of__int__power__eq__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: int,B: int,W: nat] :
% 8.58/4.85 ( ( ( ring_17405671764205052669omplex @ X )
% 8.58/4.85 = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
% 8.58/4.85 = ( X
% 8.58/4.85 = ( power_power_int @ B @ W ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6406_of__int__of__bool,axiom,
% 8.58/4.85 ! [P: $o] :
% 8.58/4.85 ( ( ring_1_of_int_real @ ( zero_n2684676970156552555ol_int @ P ) )
% 8.58/4.85 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6407_of__int__of__bool,axiom,
% 8.58/4.85 ! [P: $o] :
% 8.58/4.85 ( ( ring_1_of_int_rat @ ( zero_n2684676970156552555ol_int @ P ) )
% 8.58/4.85 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6408_of__int__of__bool,axiom,
% 8.58/4.85 ! [P: $o] :
% 8.58/4.85 ( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 8.58/4.85 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6409_of__int__of__bool,axiom,
% 8.58/4.85 ! [P: $o] :
% 8.58/4.85 ( ( ring_18347121197199848620nteger @ ( zero_n2684676970156552555ol_int @ P ) )
% 8.58/4.85 = ( zero_n356916108424825756nteger @ P ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6410_tan__npi,axiom,
% 8.58/4.85 ! [N: nat] :
% 8.58/4.85 ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 8.58/4.85 = zero_zero_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6411_floor__uminus__of__int,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( ring_1_of_int_real @ Z2 ) ) )
% 8.58/4.85 = ( uminus_uminus_int @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6412_floor__uminus__of__int,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z2 ) ) )
% 8.58/4.85 = ( uminus_uminus_int @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6413_tan__periodic__n,axiom,
% 8.58/4.85 ! [X: real,N: num] :
% 8.58/4.85 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
% 8.58/4.85 = ( tan_real @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6414_ceiling__add__of__int,axiom,
% 8.58/4.85 ! [X: rat,Z2: int] :
% 8.58/4.85 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) )
% 8.58/4.85 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6415_ceiling__add__of__int,axiom,
% 8.58/4.85 ! [X: real,Z2: int] :
% 8.58/4.85 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z2 ) ) )
% 8.58/4.85 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6416_floor__diff__of__int,axiom,
% 8.58/4.85 ! [X: real,Z2: int] :
% 8.58/4.85 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z2 ) ) )
% 8.58/4.85 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6417_floor__diff__of__int,axiom,
% 8.58/4.85 ! [X: rat,Z2: int] :
% 8.58/4.85 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) )
% 8.58/4.85 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6418_ceiling__diff__of__int,axiom,
% 8.58/4.85 ! [X: rat,Z2: int] :
% 8.58/4.85 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) )
% 8.58/4.85 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6419_ceiling__diff__of__int,axiom,
% 8.58/4.85 ! [X: real,Z2: int] :
% 8.58/4.85 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z2 ) ) )
% 8.58/4.85 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6420_tan__periodic__nat,axiom,
% 8.58/4.85 ! [X: real,N: nat] :
% 8.58/4.85 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
% 8.58/4.85 = ( tan_real @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6421_tan__periodic__int,axiom,
% 8.58/4.85 ! [X: real,I: int] :
% 8.58/4.85 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
% 8.58/4.85 = ( tan_real @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6422_of__int__le__0__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
% 8.58/4.85 = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6423_of__int__le__0__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
% 8.58/4.85 = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6424_of__int__le__0__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
% 8.58/4.85 = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6425_of__int__0__le__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
% 8.58/4.85 = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6426_of__int__0__le__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 8.58/4.85 = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6427_of__int__0__le__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
% 8.58/4.85 = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6428_of__int__less__0__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
% 8.58/4.85 = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6429_of__int__less__0__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
% 8.58/4.85 = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6430_of__int__less__0__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
% 8.58/4.85 = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6431_of__int__0__less__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
% 8.58/4.85 = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6432_of__int__0__less__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 8.58/4.85 = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6433_of__int__0__less__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
% 8.58/4.85 = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6434_of__int__le__numeral__iff,axiom,
% 8.58/4.85 ! [Z2: int,N: num] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
% 8.58/4.85 = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6435_of__int__le__numeral__iff,axiom,
% 8.58/4.85 ! [Z2: int,N: num] :
% 8.58/4.85 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
% 8.58/4.85 = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6436_of__int__le__numeral__iff,axiom,
% 8.58/4.85 ! [Z2: int,N: num] :
% 8.58/4.85 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.85 = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6437_of__int__numeral__le__iff,axiom,
% 8.58/4.85 ! [N: num,Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
% 8.58/4.85 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6438_of__int__numeral__le__iff,axiom,
% 8.58/4.85 ! [N: num,Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
% 8.58/4.85 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6439_of__int__numeral__le__iff,axiom,
% 8.58/4.85 ! [N: num,Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
% 8.58/4.85 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6440_of__int__less__numeral__iff,axiom,
% 8.58/4.85 ! [Z2: int,N: num] :
% 8.58/4.85 ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
% 8.58/4.85 = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6441_of__int__less__numeral__iff,axiom,
% 8.58/4.85 ! [Z2: int,N: num] :
% 8.58/4.85 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
% 8.58/4.85 = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6442_of__int__less__numeral__iff,axiom,
% 8.58/4.85 ! [Z2: int,N: num] :
% 8.58/4.85 ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.85 = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6443_of__int__numeral__less__iff,axiom,
% 8.58/4.85 ! [N: num,Z2: int] :
% 8.58/4.85 ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
% 8.58/4.85 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6444_of__int__numeral__less__iff,axiom,
% 8.58/4.85 ! [N: num,Z2: int] :
% 8.58/4.85 ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
% 8.58/4.85 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6445_of__int__numeral__less__iff,axiom,
% 8.58/4.85 ! [N: num,Z2: int] :
% 8.58/4.85 ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
% 8.58/4.85 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6446_of__int__le__1__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
% 8.58/4.85 = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6447_of__int__le__1__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
% 8.58/4.85 = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6448_of__int__le__1__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
% 8.58/4.85 = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6449_of__int__1__le__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
% 8.58/4.85 = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6450_of__int__1__le__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 8.58/4.85 = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6451_of__int__1__le__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
% 8.58/4.85 = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6452_of__int__less__1__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
% 8.58/4.85 = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6453_of__int__less__1__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
% 8.58/4.85 = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6454_of__int__less__1__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
% 8.58/4.85 = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6455_of__int__1__less__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
% 8.58/4.85 = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6456_of__int__1__less__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 8.58/4.85 = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6457_of__int__1__less__iff,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
% 8.58/4.85 = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6458_of__int__power__le__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: int,B: int,W: nat] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 8.58/4.85 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6459_of__int__power__le__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: int,B: int,W: nat] :
% 8.58/4.85 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 8.58/4.85 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6460_of__int__power__le__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: int,B: int,W: nat] :
% 8.58/4.85 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 8.58/4.85 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6461_of__int__le__of__int__power__cancel__iff,axiom,
% 8.58/4.85 ! [B: int,W: nat,X: int] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 8.58/4.85 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6462_of__int__le__of__int__power__cancel__iff,axiom,
% 8.58/4.85 ! [B: int,W: nat,X: int] :
% 8.58/4.85 ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 8.58/4.85 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6463_of__int__le__of__int__power__cancel__iff,axiom,
% 8.58/4.85 ! [B: int,W: nat,X: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 8.58/4.85 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6464_numeral__power__eq__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,Y: int] :
% 8.58/4.85 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
% 8.58/4.85 = ( ring_17405671764205052669omplex @ Y ) )
% 8.58/4.85 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 8.58/4.85 = Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6465_numeral__power__eq__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,Y: int] :
% 8.58/4.85 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 8.58/4.85 = ( ring_1_of_int_real @ Y ) )
% 8.58/4.85 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 8.58/4.85 = Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6466_numeral__power__eq__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,Y: int] :
% 8.58/4.85 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
% 8.58/4.85 = ( ring_1_of_int_rat @ Y ) )
% 8.58/4.85 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 8.58/4.85 = Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6467_numeral__power__eq__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,Y: int] :
% 8.58/4.85 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 8.58/4.85 = ( ring_1_of_int_int @ Y ) )
% 8.58/4.85 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 8.58/4.85 = Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6468_of__int__eq__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [Y: int,X: num,N: nat] :
% 8.58/4.85 ( ( ( ring_17405671764205052669omplex @ Y )
% 8.58/4.85 = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
% 8.58/4.85 = ( Y
% 8.58/4.85 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6469_of__int__eq__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [Y: int,X: num,N: nat] :
% 8.58/4.85 ( ( ( ring_1_of_int_real @ Y )
% 8.58/4.85 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 8.58/4.85 = ( Y
% 8.58/4.85 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6470_of__int__eq__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [Y: int,X: num,N: nat] :
% 8.58/4.85 ( ( ( ring_1_of_int_rat @ Y )
% 8.58/4.85 = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 8.58/4.85 = ( Y
% 8.58/4.85 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6471_of__int__eq__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [Y: int,X: num,N: nat] :
% 8.58/4.85 ( ( ( ring_1_of_int_int @ Y )
% 8.58/4.85 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 8.58/4.85 = ( Y
% 8.58/4.85 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6472_of__int__power__less__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: int,B: int,W: nat] :
% 8.58/4.85 ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 8.58/4.85 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6473_of__int__power__less__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: int,B: int,W: nat] :
% 8.58/4.85 ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 8.58/4.85 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6474_of__int__power__less__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: int,B: int,W: nat] :
% 8.58/4.85 ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 8.58/4.85 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6475_of__int__less__of__int__power__cancel__iff,axiom,
% 8.58/4.85 ! [B: int,W: nat,X: int] :
% 8.58/4.85 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 8.58/4.85 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6476_of__int__less__of__int__power__cancel__iff,axiom,
% 8.58/4.85 ! [B: int,W: nat,X: int] :
% 8.58/4.85 ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 8.58/4.85 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6477_of__int__less__of__int__power__cancel__iff,axiom,
% 8.58/4.85 ! [B: int,W: nat,X: int] :
% 8.58/4.85 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 8.58/4.85 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6478_sin__npi__int,axiom,
% 8.58/4.85 ! [N: int] :
% 8.58/4.85 ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 8.58/4.85 = zero_zero_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6479_sin__arcsin,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ( ( sin_real @ ( arcsin @ Y ) )
% 8.58/4.85 = Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6480_norm__cos__sin,axiom,
% 8.58/4.85 ! [T: real] :
% 8.58/4.85 ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
% 8.58/4.85 = one_one_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6481_tan__periodic,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 8.58/4.85 = ( tan_real @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6482_numeral__power__le__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,A: int] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 8.58/4.85 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6483_numeral__power__le__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,A: int] :
% 8.58/4.85 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 8.58/4.85 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6484_numeral__power__le__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,A: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 8.58/4.85 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6485_of__int__le__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [A: int,X: num,N: nat] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 8.58/4.85 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6486_of__int__le__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [A: int,X: num,N: nat] :
% 8.58/4.85 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 8.58/4.85 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6487_of__int__le__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [A: int,X: num,N: nat] :
% 8.58/4.85 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 8.58/4.85 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6488_numeral__power__less__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,A: int] :
% 8.58/4.85 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 8.58/4.85 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6489_numeral__power__less__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,A: int] :
% 8.58/4.85 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 8.58/4.85 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6490_numeral__power__less__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,A: int] :
% 8.58/4.85 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 8.58/4.85 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6491_of__int__less__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [A: int,X: num,N: nat] :
% 8.58/4.85 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 8.58/4.85 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6492_of__int__less__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [A: int,X: num,N: nat] :
% 8.58/4.85 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 8.58/4.85 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6493_of__int__less__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [A: int,X: num,N: nat] :
% 8.58/4.85 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 8.58/4.85 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6494_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,Y: int] :
% 8.58/4.85 ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N )
% 8.58/4.85 = ( ring_1_of_int_real @ Y ) )
% 8.58/4.85 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 8.58/4.85 = Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6495_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,Y: int] :
% 8.58/4.85 ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 8.58/4.85 = ( ring_1_of_int_int @ Y ) )
% 8.58/4.85 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 8.58/4.85 = Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6496_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,Y: int] :
% 8.58/4.85 ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N )
% 8.58/4.85 = ( ring_17405671764205052669omplex @ Y ) )
% 8.58/4.85 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 8.58/4.85 = Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6497_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,Y: int] :
% 8.58/4.85 ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
% 8.58/4.85 = ( ring_1_of_int_rat @ Y ) )
% 8.58/4.85 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 8.58/4.85 = Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6498_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,Y: int] :
% 8.58/4.85 ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
% 8.58/4.85 = ( ring_18347121197199848620nteger @ Y ) )
% 8.58/4.85 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 8.58/4.85 = Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6499_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [Y: int,X: num,N: nat] :
% 8.58/4.85 ( ( ( ring_1_of_int_real @ Y )
% 8.58/4.85 = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 8.58/4.85 = ( Y
% 8.58/4.85 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6500_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [Y: int,X: num,N: nat] :
% 8.58/4.85 ( ( ( ring_1_of_int_int @ Y )
% 8.58/4.85 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 8.58/4.85 = ( Y
% 8.58/4.85 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6501_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [Y: int,X: num,N: nat] :
% 8.58/4.85 ( ( ( ring_17405671764205052669omplex @ Y )
% 8.58/4.85 = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N ) )
% 8.58/4.85 = ( Y
% 8.58/4.85 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6502_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [Y: int,X: num,N: nat] :
% 8.58/4.85 ( ( ( ring_1_of_int_rat @ Y )
% 8.58/4.85 = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 8.58/4.85 = ( Y
% 8.58/4.85 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6503_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [Y: int,X: num,N: nat] :
% 8.58/4.85 ( ( ( ring_18347121197199848620nteger @ Y )
% 8.58/4.85 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 8.58/4.85 = ( Y
% 8.58/4.85 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6504_arcsin__1,axiom,
% 8.58/4.85 ( ( arcsin @ one_one_real )
% 8.58/4.85 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6505_sin__int__2pin,axiom,
% 8.58/4.85 ! [N: int] :
% 8.58/4.85 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 8.58/4.85 = zero_zero_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6506_cos__int__2pin,axiom,
% 8.58/4.85 ! [N: int] :
% 8.58/4.85 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 8.58/4.85 = one_one_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6507_arcsin__minus__1,axiom,
% 8.58/4.85 ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.85 = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6508_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,A: int] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 8.58/4.85 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6509_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,A: int] :
% 8.58/4.85 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 8.58/4.85 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6510_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,A: int] :
% 8.58/4.85 ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 8.58/4.85 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6511_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,A: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 8.58/4.85 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6512_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [A: int,X: num,N: nat] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 8.58/4.85 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6513_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [A: int,X: num,N: nat] :
% 8.58/4.85 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 8.58/4.85 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6514_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [A: int,X: num,N: nat] :
% 8.58/4.85 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 8.58/4.85 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6515_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [A: int,X: num,N: nat] :
% 8.58/4.85 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 8.58/4.85 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6516_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,A: int] :
% 8.58/4.85 ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 8.58/4.85 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6517_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,A: int] :
% 8.58/4.85 ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 8.58/4.85 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6518_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,A: int] :
% 8.58/4.85 ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 8.58/4.85 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6519_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 8.58/4.85 ! [X: num,N: nat,A: int] :
% 8.58/4.85 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 8.58/4.85 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6520_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [A: int,X: num,N: nat] :
% 8.58/4.85 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 8.58/4.85 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6521_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [A: int,X: num,N: nat] :
% 8.58/4.85 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 8.58/4.85 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6522_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [A: int,X: num,N: nat] :
% 8.58/4.85 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 8.58/4.85 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6523_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 8.58/4.85 ! [A: int,X: num,N: nat] :
% 8.58/4.85 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 8.58/4.85 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6524_ex__le__of__int,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ? [Z4: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z4 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6525_ex__le__of__int,axiom,
% 8.58/4.85 ! [X: rat] :
% 8.58/4.85 ? [Z4: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z4 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6526_ex__less__of__int,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ? [Z4: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z4 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6527_ex__less__of__int,axiom,
% 8.58/4.85 ! [X: rat] :
% 8.58/4.85 ? [Z4: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z4 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6528_ex__of__int__less,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ? [Z4: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z4 ) @ X ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6529_ex__of__int__less,axiom,
% 8.58/4.85 ! [X: rat] :
% 8.58/4.85 ? [Z4: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z4 ) @ X ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6530_mult__of__int__commute,axiom,
% 8.58/4.85 ! [X: int,Y: real] :
% 8.58/4.85 ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
% 8.58/4.85 = ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6531_mult__of__int__commute,axiom,
% 8.58/4.85 ! [X: int,Y: rat] :
% 8.58/4.85 ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y )
% 8.58/4.85 = ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6532_mult__of__int__commute,axiom,
% 8.58/4.85 ! [X: int,Y: int] :
% 8.58/4.85 ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
% 8.58/4.85 = ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6533_Complex__mult__complex__of__real,axiom,
% 8.58/4.85 ! [X: real,Y: real,R: real] :
% 8.58/4.85 ( ( times_times_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R ) )
% 8.58/4.85 = ( complex2 @ ( times_times_real @ X @ R ) @ ( times_times_real @ Y @ R ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6534_complex__of__real__mult__Complex,axiom,
% 8.58/4.85 ! [R: real,X: real,Y: real] :
% 8.58/4.85 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X @ Y ) )
% 8.58/4.85 = ( complex2 @ ( times_times_real @ R @ X ) @ ( times_times_real @ R @ Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6535_complex__diff,axiom,
% 8.58/4.85 ! [A: real,B: real,C: real,D: real] :
% 8.58/4.85 ( ( minus_minus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 8.58/4.85 = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6536_of__int__floor__le,axiom,
% 8.58/4.85 ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6537_of__int__floor__le,axiom,
% 8.58/4.85 ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6538_Complex__eq__1,axiom,
% 8.58/4.85 ! [A: real,B: real] :
% 8.58/4.85 ( ( ( complex2 @ A @ B )
% 8.58/4.85 = one_one_complex )
% 8.58/4.85 = ( ( A = one_one_real )
% 8.58/4.85 & ( B = zero_zero_real ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6539_one__complex_Ocode,axiom,
% 8.58/4.85 ( one_one_complex
% 8.58/4.85 = ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6540_le__of__int__ceiling,axiom,
% 8.58/4.85 ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6541_le__of__int__ceiling,axiom,
% 8.58/4.85 ! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6542_take__bit__of__int,axiom,
% 8.58/4.85 ! [N: nat,K: int] :
% 8.58/4.85 ( ( bit_se2923211474154528505it_int @ N @ ( ring_1_of_int_int @ K ) )
% 8.58/4.85 = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6543_push__bit__of__int,axiom,
% 8.58/4.85 ! [N: nat,K: int] :
% 8.58/4.85 ( ( bit_se545348938243370406it_int @ N @ ( ring_1_of_int_int @ K ) )
% 8.58/4.85 = ( ring_1_of_int_int @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6544_push__bit__of__int,axiom,
% 8.58/4.85 ! [N: nat,K: int] :
% 8.58/4.85 ( ( bit_se7788150548672797655nteger @ N @ ( ring_18347121197199848620nteger @ K ) )
% 8.58/4.85 = ( ring_18347121197199848620nteger @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6545_complex__add,axiom,
% 8.58/4.85 ! [A: real,B: real,C: real,D: real] :
% 8.58/4.85 ( ( plus_plus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 8.58/4.85 = ( complex2 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6546_Complex__add__complex__of__real,axiom,
% 8.58/4.85 ! [X: real,Y: real,R: real] :
% 8.58/4.85 ( ( plus_plus_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R ) )
% 8.58/4.85 = ( complex2 @ ( plus_plus_real @ X @ R ) @ Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6547_complex__of__real__add__Complex,axiom,
% 8.58/4.85 ! [R: real,X: real,Y: real] :
% 8.58/4.85 ( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X @ Y ) )
% 8.58/4.85 = ( complex2 @ ( plus_plus_real @ R @ X ) @ Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6548_cos__int__times__real,axiom,
% 8.58/4.85 ! [M: int,X: real] :
% 8.58/4.85 ( ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
% 8.58/4.85 = ( real_V1803761363581548252l_real @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6549_cos__int__times__real,axiom,
% 8.58/4.85 ! [M: int,X: real] :
% 8.58/4.85 ( ( cos_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
% 8.58/4.85 = ( real_V4546457046886955230omplex @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6550_sin__int__times__real,axiom,
% 8.58/4.85 ! [M: int,X: real] :
% 8.58/4.85 ( ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
% 8.58/4.85 = ( real_V1803761363581548252l_real @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6551_sin__int__times__real,axiom,
% 8.58/4.85 ! [M: int,X: real] :
% 8.58/4.85 ( ( sin_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
% 8.58/4.85 = ( real_V4546457046886955230omplex @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6552_le__floor__iff,axiom,
% 8.58/4.85 ! [Z2: int,X: real] :
% 8.58/4.85 ( ( ord_less_eq_int @ Z2 @ ( archim6058952711729229775r_real @ X ) )
% 8.58/4.85 = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6553_le__floor__iff,axiom,
% 8.58/4.85 ! [Z2: int,X: rat] :
% 8.58/4.85 ( ( ord_less_eq_int @ Z2 @ ( archim3151403230148437115or_rat @ X ) )
% 8.58/4.85 = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6554_floor__less__iff,axiom,
% 8.58/4.85 ! [X: real,Z2: int] :
% 8.58/4.85 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z2 )
% 8.58/4.85 = ( ord_less_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6555_floor__less__iff,axiom,
% 8.58/4.85 ! [X: rat,Z2: int] :
% 8.58/4.85 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 )
% 8.58/4.85 = ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6556_ceiling__le__iff,axiom,
% 8.58/4.85 ! [X: real,Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z2 )
% 8.58/4.85 = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6557_ceiling__le__iff,axiom,
% 8.58/4.85 ! [X: rat,Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 )
% 8.58/4.85 = ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6558_ceiling__le,axiom,
% 8.58/4.85 ! [X: real,A: int] :
% 8.58/4.85 ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
% 8.58/4.85 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6559_ceiling__le,axiom,
% 8.58/4.85 ! [X: rat,A: int] :
% 8.58/4.85 ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) )
% 8.58/4.85 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6560_less__ceiling__iff,axiom,
% 8.58/4.85 ! [Z2: int,X: rat] :
% 8.58/4.85 ( ( ord_less_int @ Z2 @ ( archim2889992004027027881ng_rat @ X ) )
% 8.58/4.85 = ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6561_less__ceiling__iff,axiom,
% 8.58/4.85 ! [Z2: int,X: real] :
% 8.58/4.85 ( ( ord_less_int @ Z2 @ ( archim7802044766580827645g_real @ X ) )
% 8.58/4.85 = ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6562_int__add__floor,axiom,
% 8.58/4.85 ! [Z2: int,X: real] :
% 8.58/4.85 ( ( plus_plus_int @ Z2 @ ( archim6058952711729229775r_real @ X ) )
% 8.58/4.85 = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6563_int__add__floor,axiom,
% 8.58/4.85 ! [Z2: int,X: rat] :
% 8.58/4.85 ( ( plus_plus_int @ Z2 @ ( archim3151403230148437115or_rat @ X ) )
% 8.58/4.85 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6564_floor__add__int,axiom,
% 8.58/4.85 ! [X: real,Z2: int] :
% 8.58/4.85 ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ Z2 )
% 8.58/4.85 = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6565_floor__add__int,axiom,
% 8.58/4.85 ! [X: rat,Z2: int] :
% 8.58/4.85 ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 )
% 8.58/4.85 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6566_Complex__eq__neg__1,axiom,
% 8.58/4.85 ! [A: real,B: real] :
% 8.58/4.85 ( ( ( complex2 @ A @ B )
% 8.58/4.85 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.85 = ( ( A
% 8.58/4.85 = ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.85 & ( B = zero_zero_real ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6567_real__of__int__div4,axiom,
% 8.58/4.85 ! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6568_floor__divide__of__int__eq,axiom,
% 8.58/4.85 ! [K: int,L: int] :
% 8.58/4.85 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L ) ) )
% 8.58/4.85 = ( divide_divide_int @ K @ L ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6569_floor__divide__of__int__eq,axiom,
% 8.58/4.85 ! [K: int,L: int] :
% 8.58/4.85 ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L ) ) )
% 8.58/4.85 = ( divide_divide_int @ K @ L ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6570_floor__power,axiom,
% 8.58/4.85 ! [X: real,N: nat] :
% 8.58/4.85 ( ( X
% 8.58/4.85 = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
% 8.58/4.85 => ( ( archim6058952711729229775r_real @ ( power_power_real @ X @ N ) )
% 8.58/4.85 = ( power_power_int @ ( archim6058952711729229775r_real @ X ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6571_floor__power,axiom,
% 8.58/4.85 ! [X: rat,N: nat] :
% 8.58/4.85 ( ( X
% 8.58/4.85 = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
% 8.58/4.85 => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X @ N ) )
% 8.58/4.85 = ( power_power_int @ ( archim3151403230148437115or_rat @ X ) @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6572_complex__mult,axiom,
% 8.58/4.85 ! [A: real,B: real,C: real,D: real] :
% 8.58/4.85 ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 8.58/4.85 = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6573_real__of__int__div,axiom,
% 8.58/4.85 ! [D: int,N: int] :
% 8.58/4.85 ( ( dvd_dvd_int @ D @ N )
% 8.58/4.85 => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
% 8.58/4.85 = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6574_arcsin__le__arcsin,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6575_arcsin__minus,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ one_one_real )
% 8.58/4.85 => ( ( arcsin @ ( uminus_uminus_real @ X ) )
% 8.58/4.85 = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6576_arcsin__le__mono,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
% 8.58/4.85 = ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6577_arcsin__eq__iff,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 8.58/4.85 => ( ( ( arcsin @ X )
% 8.58/4.85 = ( arcsin @ Y ) )
% 8.58/4.85 = ( X = Y ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6578_tan__def,axiom,
% 8.58/4.85 ( tan_complex
% 8.58/4.85 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X3 ) @ ( cos_complex @ X3 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6579_tan__def,axiom,
% 8.58/4.85 ( tan_real
% 8.58/4.85 = ( ^ [X3: real] : ( divide_divide_real @ ( sin_real @ X3 ) @ ( cos_real @ X3 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6580_of__int__nonneg,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.85 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6581_of__int__nonneg,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.85 => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6582_of__int__nonneg,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.85 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6583_of__int__leD,axiom,
% 8.58/4.85 ! [N: int,X: code_integer] :
% 8.58/4.85 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 8.58/4.85 => ( ( N = zero_zero_int )
% 8.58/4.85 | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6584_of__int__leD,axiom,
% 8.58/4.85 ! [N: int,X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 8.58/4.85 => ( ( N = zero_zero_int )
% 8.58/4.85 | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6585_of__int__leD,axiom,
% 8.58/4.85 ! [N: int,X: rat] :
% 8.58/4.85 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 8.58/4.85 => ( ( N = zero_zero_int )
% 8.58/4.85 | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6586_of__int__leD,axiom,
% 8.58/4.85 ! [N: int,X: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 8.58/4.85 => ( ( N = zero_zero_int )
% 8.58/4.85 | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6587_of__int__pos,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_int @ zero_zero_int @ Z2 )
% 8.58/4.85 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6588_of__int__pos,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_int @ zero_zero_int @ Z2 )
% 8.58/4.85 => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6589_of__int__pos,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( ord_less_int @ zero_zero_int @ Z2 )
% 8.58/4.85 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6590_of__int__lessD,axiom,
% 8.58/4.85 ! [N: int,X: code_integer] :
% 8.58/4.85 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 8.58/4.85 => ( ( N = zero_zero_int )
% 8.58/4.85 | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6591_of__int__lessD,axiom,
% 8.58/4.85 ! [N: int,X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 8.58/4.85 => ( ( N = zero_zero_int )
% 8.58/4.85 | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6592_of__int__lessD,axiom,
% 8.58/4.85 ! [N: int,X: rat] :
% 8.58/4.85 ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 8.58/4.85 => ( ( N = zero_zero_int )
% 8.58/4.85 | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6593_of__int__lessD,axiom,
% 8.58/4.85 ! [N: int,X: int] :
% 8.58/4.85 ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 8.58/4.85 => ( ( N = zero_zero_int )
% 8.58/4.85 | ( ord_less_int @ one_one_int @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6594_floor__exists1,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ? [X4: int] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X )
% 8.58/4.85 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 8.58/4.85 & ! [Y5: int] :
% 8.58/4.85 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X )
% 8.58/4.85 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 8.58/4.85 => ( Y5 = X4 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6595_floor__exists1,axiom,
% 8.58/4.85 ! [X: rat] :
% 8.58/4.85 ? [X4: int] :
% 8.58/4.85 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X )
% 8.58/4.85 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 8.58/4.85 & ! [Y5: int] :
% 8.58/4.85 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X )
% 8.58/4.85 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 8.58/4.85 => ( Y5 = X4 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6596_floor__exists,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ? [Z4: int] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z4 ) @ X )
% 8.58/4.85 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6597_floor__exists,axiom,
% 8.58/4.85 ! [X: rat] :
% 8.58/4.85 ? [Z4: int] :
% 8.58/4.85 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z4 ) @ X )
% 8.58/4.85 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6598_of__int__ceiling__le__add__one,axiom,
% 8.58/4.85 ! [R: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R ) ) @ ( plus_plus_real @ R @ one_one_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6599_of__int__ceiling__le__add__one,axiom,
% 8.58/4.85 ! [R: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R ) ) @ ( plus_plus_rat @ R @ one_one_rat ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6600_of__int__neg__numeral,axiom,
% 8.58/4.85 ! [K: num] :
% 8.58/4.85 ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.58/4.85 = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6601_of__int__neg__numeral,axiom,
% 8.58/4.85 ! [K: num] :
% 8.58/4.85 ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.58/4.85 = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6602_of__int__neg__numeral,axiom,
% 8.58/4.85 ! [K: num] :
% 8.58/4.85 ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.58/4.85 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6603_of__int__neg__numeral,axiom,
% 8.58/4.85 ! [K: num] :
% 8.58/4.85 ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.58/4.85 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6604_of__int__neg__numeral,axiom,
% 8.58/4.85 ! [K: num] :
% 8.58/4.85 ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.58/4.85 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6605_of__int__ceiling__diff__one__le,axiom,
% 8.58/4.85 ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R ) ) @ one_one_real ) @ R ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6606_of__int__ceiling__diff__one__le,axiom,
% 8.58/4.85 ! [R: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R ) ) @ one_one_rat ) @ R ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6607_of__nat__less__of__int__iff,axiom,
% 8.58/4.85 ! [N: nat,X: int] :
% 8.58/4.85 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
% 8.58/4.85 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6608_of__nat__less__of__int__iff,axiom,
% 8.58/4.85 ! [N: nat,X: int] :
% 8.58/4.85 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
% 8.58/4.85 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6609_of__nat__less__of__int__iff,axiom,
% 8.58/4.85 ! [N: nat,X: int] :
% 8.58/4.85 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
% 8.58/4.85 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6610_int__le__real__less,axiom,
% 8.58/4.85 ( ord_less_eq_int
% 8.58/4.85 = ( ^ [N3: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6611_int__less__real__le,axiom,
% 8.58/4.85 ( ord_less_int
% 8.58/4.85 = ( ^ [N3: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6612_ceiling__divide__eq__div,axiom,
% 8.58/4.85 ! [A: int,B: int] :
% 8.58/4.85 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
% 8.58/4.85 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6613_ceiling__divide__eq__div,axiom,
% 8.58/4.85 ! [A: int,B: int] :
% 8.58/4.85 ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) )
% 8.58/4.85 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6614_sin__zero__iff__int2,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ( sin_real @ X )
% 8.58/4.85 = zero_zero_real )
% 8.58/4.85 = ( ? [I4: int] :
% 8.58/4.85 ( X
% 8.58/4.85 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ pi ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6615_ceiling__altdef,axiom,
% 8.58/4.85 ( archim7802044766580827645g_real
% 8.58/4.85 = ( ^ [X3: real] :
% 8.58/4.85 ( if_int
% 8.58/4.85 @ ( X3
% 8.58/4.85 = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X3 ) ) )
% 8.58/4.85 @ ( archim6058952711729229775r_real @ X3 )
% 8.58/4.85 @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6616_ceiling__altdef,axiom,
% 8.58/4.85 ( archim2889992004027027881ng_rat
% 8.58/4.85 = ( ^ [X3: rat] :
% 8.58/4.85 ( if_int
% 8.58/4.85 @ ( X3
% 8.58/4.85 = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) )
% 8.58/4.85 @ ( archim3151403230148437115or_rat @ X3 )
% 8.58/4.85 @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6617_real__of__int__div__aux,axiom,
% 8.58/4.85 ! [X: int,D: int] :
% 8.58/4.85 ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
% 8.58/4.85 = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6618_real__of__int__floor__add__one__gt,axiom,
% 8.58/4.85 ! [R: real] : ( ord_less_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6619_floor__eq,axiom,
% 8.58/4.85 ! [N: int,X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 8.58/4.85 => ( ( archim6058952711729229775r_real @ X )
% 8.58/4.85 = N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6620_real__of__int__floor__add__one__ge,axiom,
% 8.58/4.85 ! [R: real] : ( ord_less_eq_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6621_real__of__int__floor__gt__diff__one,axiom,
% 8.58/4.85 ! [R: real] : ( ord_less_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6622_real__of__int__floor__ge__diff__one,axiom,
% 8.58/4.85 ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6623_arcsin__less__arcsin,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6624_arcsin__less__mono,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 8.58/4.85 => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
% 8.58/4.85 = ( ord_less_real @ X @ Y ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6625_floor__split,axiom,
% 8.58/4.85 ! [P: int > $o,T: real] :
% 8.58/4.85 ( ( P @ ( archim6058952711729229775r_real @ T ) )
% 8.58/4.85 = ( ! [I4: int] :
% 8.58/4.85 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I4 ) @ T )
% 8.58/4.85 & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) ) )
% 8.58/4.85 => ( P @ I4 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6626_floor__split,axiom,
% 8.58/4.85 ! [P: int > $o,T: rat] :
% 8.58/4.85 ( ( P @ ( archim3151403230148437115or_rat @ T ) )
% 8.58/4.85 = ( ! [I4: int] :
% 8.58/4.85 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I4 ) @ T )
% 8.58/4.85 & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) ) )
% 8.58/4.85 => ( P @ I4 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6627_floor__eq__iff,axiom,
% 8.58/4.85 ! [X: real,A: int] :
% 8.58/4.85 ( ( ( archim6058952711729229775r_real @ X )
% 8.58/4.85 = A )
% 8.58/4.85 = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X )
% 8.58/4.85 & ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6628_floor__eq__iff,axiom,
% 8.58/4.85 ! [X: rat,A: int] :
% 8.58/4.85 ( ( ( archim3151403230148437115or_rat @ X )
% 8.58/4.85 = A )
% 8.58/4.85 = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
% 8.58/4.85 & ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6629_floor__unique,axiom,
% 8.58/4.85 ! [Z2: int,X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) )
% 8.58/4.85 => ( ( archim6058952711729229775r_real @ X )
% 8.58/4.85 = Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6630_floor__unique,axiom,
% 8.58/4.85 ! [Z2: int,X: rat] :
% 8.58/4.85 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
% 8.58/4.85 => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) )
% 8.58/4.85 => ( ( archim3151403230148437115or_rat @ X )
% 8.58/4.85 = Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6631_ceiling__correct,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) @ one_one_real ) @ X )
% 8.58/4.85 & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6632_ceiling__correct,axiom,
% 8.58/4.85 ! [X: rat] :
% 8.58/4.85 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
% 8.58/4.85 & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6633_ceiling__unique,axiom,
% 8.58/4.85 ! [Z2: int,X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) )
% 8.58/4.85 => ( ( archim7802044766580827645g_real @ X )
% 8.58/4.85 = Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6634_ceiling__unique,axiom,
% 8.58/4.85 ! [Z2: int,X: rat] :
% 8.58/4.85 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) )
% 8.58/4.85 => ( ( archim2889992004027027881ng_rat @ X )
% 8.58/4.85 = Z2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6635_ceiling__eq__iff,axiom,
% 8.58/4.85 ! [X: real,A: int] :
% 8.58/4.85 ( ( ( archim7802044766580827645g_real @ X )
% 8.58/4.85 = A )
% 8.58/4.85 = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
% 8.58/4.85 & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6636_ceiling__eq__iff,axiom,
% 8.58/4.85 ! [X: rat,A: int] :
% 8.58/4.85 ( ( ( archim2889992004027027881ng_rat @ X )
% 8.58/4.85 = A )
% 8.58/4.85 = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
% 8.58/4.85 & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6637_ceiling__split,axiom,
% 8.58/4.85 ! [P: int > $o,T: real] :
% 8.58/4.85 ( ( P @ ( archim7802044766580827645g_real @ T ) )
% 8.58/4.85 = ( ! [I4: int] :
% 8.58/4.85 ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) @ T )
% 8.58/4.85 & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I4 ) ) )
% 8.58/4.85 => ( P @ I4 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6638_ceiling__split,axiom,
% 8.58/4.85 ! [P: int > $o,T: rat] :
% 8.58/4.85 ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
% 8.58/4.85 = ( ! [I4: int] :
% 8.58/4.85 ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) @ T )
% 8.58/4.85 & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I4 ) ) )
% 8.58/4.85 => ( P @ I4 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6639_less__floor__iff,axiom,
% 8.58/4.85 ! [Z2: int,X: real] :
% 8.58/4.85 ( ( ord_less_int @ Z2 @ ( archim6058952711729229775r_real @ X ) )
% 8.58/4.85 = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6640_less__floor__iff,axiom,
% 8.58/4.85 ! [Z2: int,X: rat] :
% 8.58/4.85 ( ( ord_less_int @ Z2 @ ( archim3151403230148437115or_rat @ X ) )
% 8.58/4.85 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6641_floor__le__iff,axiom,
% 8.58/4.85 ! [X: real,Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z2 )
% 8.58/4.85 = ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6642_floor__le__iff,axiom,
% 8.58/4.85 ! [X: rat,Z2: int] :
% 8.58/4.85 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 )
% 8.58/4.85 = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6643_ceiling__less__iff,axiom,
% 8.58/4.85 ! [X: real,Z2: int] :
% 8.58/4.85 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z2 )
% 8.58/4.85 = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6644_ceiling__less__iff,axiom,
% 8.58/4.85 ! [X: rat,Z2: int] :
% 8.58/4.85 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 )
% 8.58/4.85 = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6645_le__ceiling__iff,axiom,
% 8.58/4.85 ! [Z2: int,X: rat] :
% 8.58/4.85 ( ( ord_less_eq_int @ Z2 @ ( archim2889992004027027881ng_rat @ X ) )
% 8.58/4.85 = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6646_le__ceiling__iff,axiom,
% 8.58/4.85 ! [Z2: int,X: real] :
% 8.58/4.85 ( ( ord_less_eq_int @ Z2 @ ( archim7802044766580827645g_real @ X ) )
% 8.58/4.85 = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6647_floor__correct,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X )
% 8.58/4.85 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6648_floor__correct,axiom,
% 8.58/4.85 ! [X: rat] :
% 8.58/4.85 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
% 8.58/4.85 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6649_real__of__int__div2,axiom,
% 8.58/4.85 ! [N: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6650_real__of__int__div3,axiom,
% 8.58/4.85 ! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) @ one_one_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6651_floor__eq2,axiom,
% 8.58/4.85 ! [N: int,X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 8.58/4.85 => ( ( archim6058952711729229775r_real @ X )
% 8.58/4.85 = N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6652_floor__divide__real__eq__div,axiom,
% 8.58/4.85 ! [B: int,A: real] :
% 8.58/4.85 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.58/4.85 => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
% 8.58/4.85 = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6653_floor__divide__lower,axiom,
% 8.58/4.85 ! [Q2: real,P2: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 8.58/4.85 => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ Q2 ) @ P2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6654_floor__divide__lower,axiom,
% 8.58/4.85 ! [Q2: rat,P2: rat] :
% 8.58/4.85 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 8.58/4.85 => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ Q2 ) @ P2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6655_ceiling__divide__upper,axiom,
% 8.58/4.85 ! [Q2: real,P2: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 8.58/4.85 => ( ord_less_eq_real @ P2 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ Q2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6656_ceiling__divide__upper,axiom,
% 8.58/4.85 ! [Q2: rat,P2: rat] :
% 8.58/4.85 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 8.58/4.85 => ( ord_less_eq_rat @ P2 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ Q2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6657_tan__45,axiom,
% 8.58/4.85 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 8.58/4.85 = one_one_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6658_even__of__int__iff,axiom,
% 8.58/4.85 ! [K: int] :
% 8.58/4.85 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
% 8.58/4.85 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6659_even__of__int__iff,axiom,
% 8.58/4.85 ! [K: int] :
% 8.58/4.85 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
% 8.58/4.85 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6660_cos__arcsin__nonzero,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ one_one_real )
% 8.58/4.85 => ( ( cos_real @ ( arcsin @ X ) )
% 8.58/4.85 != zero_zero_real ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6661_lemma__tan__total,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ Y )
% 8.58/4.85 => ? [X4: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ X4 )
% 8.58/4.85 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 & ( ord_less_real @ Y @ ( tan_real @ X4 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6662_tan__gt__zero,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6663_floor__divide__upper,axiom,
% 8.58/4.85 ! [Q2: real,P2: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 8.58/4.85 => ( ord_less_real @ P2 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6664_floor__divide__upper,axiom,
% 8.58/4.85 ! [Q2: rat,P2: rat] :
% 8.58/4.85 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 8.58/4.85 => ( ord_less_rat @ P2 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6665_ceiling__divide__lower,axiom,
% 8.58/4.85 ! [Q2: real,P2: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ Q2 )
% 8.58/4.85 => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) @ P2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6666_ceiling__divide__lower,axiom,
% 8.58/4.85 ! [Q2: rat,P2: rat] :
% 8.58/4.85 ( ( ord_less_rat @ zero_zero_rat @ Q2 )
% 8.58/4.85 => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) @ P2 ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6667_lemma__tan__total1,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ? [X4: real] :
% 8.58/4.85 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 8.58/4.85 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 & ( ( tan_real @ X4 )
% 8.58/4.85 = Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6668_tan__mono__lt__eq,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 8.58/4.85 => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 8.58/4.85 = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6669_tan__monotone_H,axiom,
% 8.58/4.85 ! [Y: real,X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 8.58/4.85 => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ord_less_real @ Y @ X )
% 8.58/4.85 = ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6670_tan__monotone,axiom,
% 8.58/4.85 ! [Y: real,X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 8.58/4.85 => ( ( ord_less_real @ Y @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6671_tan__total,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ? [X4: real] :
% 8.58/4.85 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 8.58/4.85 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 & ( ( tan_real @ X4 )
% 8.58/4.85 = Y )
% 8.58/4.85 & ! [Y5: real] :
% 8.58/4.85 ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 8.58/4.85 & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 & ( ( tan_real @ Y5 )
% 8.58/4.85 = Y ) )
% 8.58/4.85 => ( Y5 = X4 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6672_tan__minus__45,axiom,
% 8.58/4.85 ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.85 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6673_tan__inverse,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y ) )
% 8.58/4.85 = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6674_ceiling__eq,axiom,
% 8.58/4.85 ! [N: int,X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 8.58/4.85 => ( ( archim7802044766580827645g_real @ X )
% 8.58/4.85 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6675_ceiling__eq,axiom,
% 8.58/4.85 ! [N: int,X: rat] :
% 8.58/4.85 ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
% 8.58/4.85 => ( ( archim2889992004027027881ng_rat @ X )
% 8.58/4.85 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6676_cos__one__2pi__int,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ( cos_real @ X )
% 8.58/4.85 = one_one_real )
% 8.58/4.85 = ( ? [X3: int] :
% 8.58/4.85 ( X
% 8.58/4.85 = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6677_add__tan__eq,axiom,
% 8.58/4.85 ! [X: complex,Y: complex] :
% 8.58/4.85 ( ( ( cos_complex @ X )
% 8.58/4.85 != zero_zero_complex )
% 8.58/4.85 => ( ( ( cos_complex @ Y )
% 8.58/4.85 != zero_zero_complex )
% 8.58/4.85 => ( ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) )
% 8.58/4.85 = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6678_add__tan__eq,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ( cos_real @ X )
% 8.58/4.85 != zero_zero_real )
% 8.58/4.85 => ( ( ( cos_real @ Y )
% 8.58/4.85 != zero_zero_real )
% 8.58/4.85 => ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 8.58/4.85 = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6679_tan__cot_H,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 8.58/4.85 = ( cot_real @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6680_tan__pos__pi2__le,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6681_tan__total__pos,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.58/4.85 => ? [X4: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 8.58/4.85 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 & ( ( tan_real @ X4 )
% 8.58/4.85 = Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6682_tan__less__zero,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ zero_zero_real )
% 8.58/4.85 => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6683_tan__mono__le,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ Y )
% 8.58/4.85 => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6684_tan__mono__le__eq,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 8.58/4.85 => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 8.58/4.85 = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6685_tan__bound__pi2,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 8.58/4.85 => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6686_arctan,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
% 8.58/4.85 & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 & ( ( tan_real @ ( arctan @ Y ) )
% 8.58/4.85 = Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6687_arctan__tan,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( arctan @ ( tan_real @ X ) )
% 8.58/4.85 = X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6688_arctan__unique,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ( tan_real @ X )
% 8.58/4.85 = Y )
% 8.58/4.85 => ( ( arctan @ Y )
% 8.58/4.85 = X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6689_tan__add,axiom,
% 8.58/4.85 ! [X: complex,Y: complex] :
% 8.58/4.85 ( ( ( cos_complex @ X )
% 8.58/4.85 != zero_zero_complex )
% 8.58/4.85 => ( ( ( cos_complex @ Y )
% 8.58/4.85 != zero_zero_complex )
% 8.58/4.85 => ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
% 8.58/4.85 != zero_zero_complex )
% 8.58/4.85 => ( ( tan_complex @ ( plus_plus_complex @ X @ Y ) )
% 8.58/4.85 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6690_tan__add,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ( cos_real @ X )
% 8.58/4.85 != zero_zero_real )
% 8.58/4.85 => ( ( ( cos_real @ Y )
% 8.58/4.85 != zero_zero_real )
% 8.58/4.85 => ( ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.85 != zero_zero_real )
% 8.58/4.85 => ( ( tan_real @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.85 = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6691_tan__diff,axiom,
% 8.58/4.85 ! [X: complex,Y: complex] :
% 8.58/4.85 ( ( ( cos_complex @ X )
% 8.58/4.85 != zero_zero_complex )
% 8.58/4.85 => ( ( ( cos_complex @ Y )
% 8.58/4.85 != zero_zero_complex )
% 8.58/4.85 => ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
% 8.58/4.85 != zero_zero_complex )
% 8.58/4.85 => ( ( tan_complex @ ( minus_minus_complex @ X @ Y ) )
% 8.58/4.85 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6692_tan__diff,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ( cos_real @ X )
% 8.58/4.85 != zero_zero_real )
% 8.58/4.85 => ( ( ( cos_real @ Y )
% 8.58/4.85 != zero_zero_real )
% 8.58/4.85 => ( ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
% 8.58/4.85 != zero_zero_real )
% 8.58/4.85 => ( ( tan_real @ ( minus_minus_real @ X @ Y ) )
% 8.58/4.85 = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6693_lemma__tan__add1,axiom,
% 8.58/4.85 ! [X: complex,Y: complex] :
% 8.58/4.85 ( ( ( cos_complex @ X )
% 8.58/4.85 != zero_zero_complex )
% 8.58/4.85 => ( ( ( cos_complex @ Y )
% 8.58/4.85 != zero_zero_complex )
% 8.58/4.85 => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) )
% 8.58/4.85 = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6694_lemma__tan__add1,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ( cos_real @ X )
% 8.58/4.85 != zero_zero_real )
% 8.58/4.85 => ( ( ( cos_real @ Y )
% 8.58/4.85 != zero_zero_real )
% 8.58/4.85 => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) )
% 8.58/4.85 = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6695_tan__total__pi4,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.85 => ? [Z4: real] :
% 8.58/4.85 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z4 )
% 8.58/4.85 & ( ord_less_real @ Z4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 8.58/4.85 & ( ( tan_real @ Z4 )
% 8.58/4.85 = X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6696_floor__log__eq__powr__iff,axiom,
% 8.58/4.85 ! [X: real,B: real,K: int] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ord_less_real @ one_one_real @ B )
% 8.58/4.85 => ( ( ( archim6058952711729229775r_real @ ( log2 @ B @ X ) )
% 8.58/4.85 = K )
% 8.58/4.85 = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
% 8.58/4.85 & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6697_arcsin__lt__bounded,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 8.58/4.85 => ( ( ord_less_real @ Y @ one_one_real )
% 8.58/4.85 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 8.58/4.85 & ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6698_arcsin__lbound,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6699_arcsin__ubound,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6700_arcsin__bounded,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 8.58/4.85 & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6701_arcsin__sin,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( arcsin @ ( sin_real @ X ) )
% 8.58/4.85 = X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6702_cos__zero__iff__int,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ( cos_real @ X )
% 8.58/4.85 = zero_zero_real )
% 8.58/4.85 = ( ? [I4: int] :
% 8.58/4.85 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
% 8.58/4.85 & ( X
% 8.58/4.85 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6703_sin__zero__iff__int,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ( sin_real @ X )
% 8.58/4.85 = zero_zero_real )
% 8.58/4.85 = ( ? [I4: int] :
% 8.58/4.85 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
% 8.58/4.85 & ( X
% 8.58/4.85 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6704_tan__half,axiom,
% 8.58/4.85 ( tan_complex
% 8.58/4.85 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_complex ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6705_tan__half,axiom,
% 8.58/4.85 ( tan_real
% 8.58/4.85 = ( ^ [X3: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_real ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6706_arcsin,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 8.58/4.85 & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 & ( ( sin_real @ ( arcsin @ Y ) )
% 8.58/4.85 = Y ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6707_round__unique,axiom,
% 8.58/4.85 ! [X: real,Y: int] :
% 8.58/4.85 ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 8.58/4.85 => ( ( archim8280529875227126926d_real @ X )
% 8.58/4.85 = Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6708_round__unique,axiom,
% 8.58/4.85 ! [X: rat,Y: int] :
% 8.58/4.85 ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
% 8.58/4.85 => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.85 => ( ( archim7778729529865785530nd_rat @ X )
% 8.58/4.85 = Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6709_round__unique_H,axiom,
% 8.58/4.85 ! [X: real,N: int] :
% 8.58/4.85 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( archim8280529875227126926d_real @ X )
% 8.58/4.85 = N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6710_round__unique_H,axiom,
% 8.58/4.85 ! [X: rat,N: int] :
% 8.58/4.85 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( archim7778729529865785530nd_rat @ X )
% 8.58/4.85 = N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6711_of__int__round__abs__le,axiom,
% 8.58/4.85 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6712_of__int__round__abs__le,axiom,
% 8.58/4.85 ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6713_sin__tan,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( sin_real @ X )
% 8.58/4.85 = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6714_cos__tan,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( cos_real @ X )
% 8.58/4.85 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6715_of__int__round__gt,axiom,
% 8.58/4.85 ! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6716_of__int__round__gt,axiom,
% 8.58/4.85 ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6717_real__sqrt__eq__iff,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ( sqrt @ X )
% 8.58/4.85 = ( sqrt @ Y ) )
% 8.58/4.85 = ( X = Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6718_real__sqrt__zero,axiom,
% 8.58/4.85 ( ( sqrt @ zero_zero_real )
% 8.58/4.85 = zero_zero_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6719_real__sqrt__eq__zero__cancel__iff,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ( sqrt @ X )
% 8.58/4.85 = zero_zero_real )
% 8.58/4.85 = ( X = zero_zero_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6720_real__sqrt__less__iff,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
% 8.58/4.85 = ( ord_less_real @ X @ Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6721_real__sqrt__le__iff,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
% 8.58/4.85 = ( ord_less_eq_real @ X @ Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6722_real__sqrt__one,axiom,
% 8.58/4.85 ( ( sqrt @ one_one_real )
% 8.58/4.85 = one_one_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6723_real__sqrt__eq__1__iff,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ( sqrt @ X )
% 8.58/4.85 = one_one_real )
% 8.58/4.85 = ( X = one_one_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6724_round__of__int,axiom,
% 8.58/4.85 ! [N: int] :
% 8.58/4.85 ( ( archim8280529875227126926d_real @ ( ring_1_of_int_real @ N ) )
% 8.58/4.85 = N ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6725_round__of__int,axiom,
% 8.58/4.85 ! [N: int] :
% 8.58/4.85 ( ( archim7778729529865785530nd_rat @ ( ring_1_of_int_rat @ N ) )
% 8.58/4.85 = N ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6726_real__sqrt__lt__0__iff,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
% 8.58/4.85 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6727_real__sqrt__gt__0__iff,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
% 8.58/4.85 = ( ord_less_real @ zero_zero_real @ Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6728_real__sqrt__ge__0__iff,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
% 8.58/4.85 = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6729_real__sqrt__le__0__iff,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
% 8.58/4.85 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6730_real__sqrt__gt__1__iff,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
% 8.58/4.85 = ( ord_less_real @ one_one_real @ Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6731_real__sqrt__lt__1__iff,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
% 8.58/4.85 = ( ord_less_real @ X @ one_one_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6732_real__sqrt__ge__1__iff,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
% 8.58/4.85 = ( ord_less_eq_real @ one_one_real @ Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6733_real__sqrt__le__1__iff,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
% 8.58/4.85 = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6734_real__sqrt__mult__self,axiom,
% 8.58/4.85 ! [A: real] :
% 8.58/4.85 ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
% 8.58/4.85 = ( abs_abs_real @ A ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6735_real__sqrt__abs2,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( sqrt @ ( times_times_real @ X @ X ) )
% 8.58/4.85 = ( abs_abs_real @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6736_round__0,axiom,
% 8.58/4.85 ( ( archim8280529875227126926d_real @ zero_zero_real )
% 8.58/4.85 = zero_zero_int ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6737_round__0,axiom,
% 8.58/4.85 ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
% 8.58/4.85 = zero_zero_int ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6738_round__numeral,axiom,
% 8.58/4.85 ! [N: num] :
% 8.58/4.85 ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
% 8.58/4.85 = ( numeral_numeral_int @ N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6739_round__numeral,axiom,
% 8.58/4.85 ! [N: num] :
% 8.58/4.85 ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
% 8.58/4.85 = ( numeral_numeral_int @ N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6740_round__1,axiom,
% 8.58/4.85 ( ( archim8280529875227126926d_real @ one_one_real )
% 8.58/4.85 = one_one_int ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6741_round__1,axiom,
% 8.58/4.85 ( ( archim7778729529865785530nd_rat @ one_one_rat )
% 8.58/4.85 = one_one_int ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6742_round__of__nat,axiom,
% 8.58/4.85 ! [N: nat] :
% 8.58/4.85 ( ( archim8280529875227126926d_real @ ( semiri5074537144036343181t_real @ N ) )
% 8.58/4.85 = ( semiri1314217659103216013at_int @ N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6743_round__of__nat,axiom,
% 8.58/4.85 ! [N: nat] :
% 8.58/4.85 ( ( archim7778729529865785530nd_rat @ ( semiri681578069525770553at_rat @ N ) )
% 8.58/4.85 = ( semiri1314217659103216013at_int @ N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6744_real__sqrt__four,axiom,
% 8.58/4.85 ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.58/4.85 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6745_round__neg__numeral,axiom,
% 8.58/4.85 ! [N: num] :
% 8.58/4.85 ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.58/4.85 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6746_round__neg__numeral,axiom,
% 8.58/4.85 ! [N: num] :
% 8.58/4.85 ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.58/4.85 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6747_real__sqrt__abs,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.85 = ( abs_abs_real @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6748_real__sqrt__pow2__iff,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.85 = X )
% 8.58/4.85 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6749_real__sqrt__pow2,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.85 = X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6750_real__sqrt__sum__squares__mult__squared__eq,axiom,
% 8.58/4.85 ! [X: real,Y: real,Xa: real,Ya: real] :
% 8.58/4.85 ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.85 = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6751_real__sqrt__less__mono,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_real @ X @ Y )
% 8.58/4.85 => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6752_real__sqrt__le__mono,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ X @ Y )
% 8.58/4.85 => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6753_real__sqrt__divide,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( sqrt @ ( divide_divide_real @ X @ Y ) )
% 8.58/4.85 = ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6754_real__sqrt__mult,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( sqrt @ ( times_times_real @ X @ Y ) )
% 8.58/4.85 = ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6755_real__sqrt__power,axiom,
% 8.58/4.85 ! [X: real,K: nat] :
% 8.58/4.85 ( ( sqrt @ ( power_power_real @ X @ K ) )
% 8.58/4.85 = ( power_power_real @ ( sqrt @ X ) @ K ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6756_real__sqrt__minus,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( sqrt @ ( uminus_uminus_real @ X ) )
% 8.58/4.85 = ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6757_real__sqrt__gt__zero,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6758_real__sqrt__ge__zero,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6759_real__sqrt__eq__zero__cancel,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ( sqrt @ X )
% 8.58/4.85 = zero_zero_real )
% 8.58/4.85 => ( X = zero_zero_real ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6760_real__sqrt__ge__one,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ one_one_real @ X )
% 8.58/4.85 => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6761_real__div__sqrt,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
% 8.58/4.85 = ( sqrt @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6762_sqrt__add__le__add__sqrt,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.58/4.85 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6763_le__real__sqrt__sumsq,axiom,
% 8.58/4.85 ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6764_round__mono,axiom,
% 8.58/4.85 ! [X: rat,Y: rat] :
% 8.58/4.85 ( ( ord_less_eq_rat @ X @ Y )
% 8.58/4.85 => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6765_floor__le__round,axiom,
% 8.58/4.85 ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim8280529875227126926d_real @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6766_floor__le__round,axiom,
% 8.58/4.85 ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim7778729529865785530nd_rat @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6767_ceiling__ge__round,axiom,
% 8.58/4.85 ! [X: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6768_sqrt2__less__2,axiom,
% 8.58/4.85 ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6769_real__less__rsqrt,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
% 8.58/4.85 => ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6770_real__le__rsqrt,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
% 8.58/4.85 => ( ord_less_eq_real @ X @ ( sqrt @ Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6771_sqrt__le__D,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y )
% 8.58/4.85 => ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6772_tan__60,axiom,
% 8.58/4.85 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 8.58/4.85 = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6773_real__sqrt__unique,axiom,
% 8.58/4.85 ! [Y: real,X: real] :
% 8.58/4.85 ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.85 = X )
% 8.58/4.85 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.58/4.85 => ( ( sqrt @ X )
% 8.58/4.85 = Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6774_real__le__lsqrt,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ord_less_eq_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6775_lemma__real__divide__sqrt__less,axiom,
% 8.58/4.85 ! [U: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ U )
% 8.58/4.85 => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6776_real__sqrt__sum__squares__eq__cancel2,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.85 = Y )
% 8.58/4.85 => ( X = zero_zero_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6777_real__sqrt__sum__squares__eq__cancel,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.85 = X )
% 8.58/4.85 => ( Y = zero_zero_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6778_real__sqrt__sum__squares__triangle__ineq,axiom,
% 8.58/4.85 ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6779_real__sqrt__sum__squares__ge2,axiom,
% 8.58/4.85 ! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6780_real__sqrt__sum__squares__ge1,axiom,
% 8.58/4.85 ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6781_sqrt__ge__absD,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y ) )
% 8.58/4.85 => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6782_cos__45,axiom,
% 8.58/4.85 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 8.58/4.85 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6783_sin__45,axiom,
% 8.58/4.85 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 8.58/4.85 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6784_round__diff__minimal,axiom,
% 8.58/4.85 ! [Z2: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z2 ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ M ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6785_round__diff__minimal,axiom,
% 8.58/4.85 ! [Z2: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z2 ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ M ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6786_real__less__lsqrt,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.58/4.85 => ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6787_sqrt__sum__squares__le__sum,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.58/4.85 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6788_tan__30,axiom,
% 8.58/4.85 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 8.58/4.85 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6789_ln__sqrt,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ln_ln_real @ ( sqrt @ X ) )
% 8.58/4.85 = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6790_real__sqrt__ge__abs1,axiom,
% 8.58/4.85 ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6791_real__sqrt__ge__abs2,axiom,
% 8.58/4.85 ! [Y: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6792_sqrt__sum__squares__le__sum__abs,axiom,
% 8.58/4.85 ! [X: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6793_sqrt__even__pow2,axiom,
% 8.58/4.85 ! [N: nat] :
% 8.58/4.85 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.85 => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.85 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6794_cos__30,axiom,
% 8.58/4.85 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 8.58/4.85 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6795_sin__60,axiom,
% 8.58/4.85 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 8.58/4.85 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6796_arsinh__real__def,axiom,
% 8.58/4.85 ( arsinh_real
% 8.58/4.85 = ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6797_complex__norm,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y ) )
% 8.58/4.85 = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6798_arsinh__real__aux,axiom,
% 8.58/4.85 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6799_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 8.58/4.85 ! [X: real,Y: real,Xa: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6800_real__sqrt__power__even,axiom,
% 8.58/4.85 ! [N: nat,X: real] :
% 8.58/4.85 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.85 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( power_power_real @ ( sqrt @ X ) @ N )
% 8.58/4.85 = ( power_power_real @ X @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6801_arith__geo__mean__sqrt,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.58/4.85 => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6802_powr__half__sqrt,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 = ( sqrt @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6803_cos__x__y__le__one,axiom,
% 8.58/4.85 ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6804_real__sqrt__sum__squares__less,axiom,
% 8.58/4.85 ! [X: real,U: real,Y: real] :
% 8.58/4.85 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 8.58/4.85 => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 8.58/4.85 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6805_arcosh__real__def,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ one_one_real @ X )
% 8.58/4.85 => ( ( arcosh_real @ X )
% 8.58/4.85 = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6806_cos__arctan,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( cos_real @ ( arctan @ X ) )
% 8.58/4.85 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6807_sin__arctan,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( sin_real @ ( arctan @ X ) )
% 8.58/4.85 = ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6808_sqrt__sum__squares__half__less,axiom,
% 8.58/4.85 ! [X: real,U: real,Y: real] :
% 8.58/4.85 ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.58/4.85 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6809_sin__cos__sqrt,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
% 8.58/4.85 => ( ( sin_real @ X )
% 8.58/4.85 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6810_arctan__half,axiom,
% 8.58/4.85 ( arctan
% 8.58/4.85 = ( ^ [X3: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X3 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6811_cos__arcsin,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ one_one_real )
% 8.58/4.85 => ( ( cos_real @ ( arcsin @ X ) )
% 8.58/4.85 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6812_round__def,axiom,
% 8.58/4.85 ( archim8280529875227126926d_real
% 8.58/4.85 = ( ^ [X3: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6813_round__def,axiom,
% 8.58/4.85 ( archim7778729529865785530nd_rat
% 8.58/4.85 = ( ^ [X3: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6814_of__int__round__le,axiom,
% 8.58/4.85 ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6815_of__int__round__le,axiom,
% 8.58/4.85 ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6816_of__int__round__ge,axiom,
% 8.58/4.85 ! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6817_of__int__round__ge,axiom,
% 8.58/4.85 ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6818_round__altdef,axiom,
% 8.58/4.85 ( archim8280529875227126926d_real
% 8.58/4.85 = ( ^ [X3: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X3 ) ) @ ( archim7802044766580827645g_real @ X3 ) @ ( archim6058952711729229775r_real @ X3 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6819_round__altdef,axiom,
% 8.58/4.85 ( archim7778729529865785530nd_rat
% 8.58/4.85 = ( ^ [X3: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X3 ) ) @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim3151403230148437115or_rat @ X3 ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6820_sin__arccos__abs,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 8.58/4.85 => ( ( sin_real @ ( arccos @ Y ) )
% 8.58/4.85 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6821_sin__arccos,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ one_one_real )
% 8.58/4.85 => ( ( sin_real @ ( arccos @ X ) )
% 8.58/4.85 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6822_arccos__cos__eq__abs__2pi,axiom,
% 8.58/4.85 ! [Theta: real] :
% 8.58/4.85 ~ ! [K3: int] :
% 8.58/4.85 ( ( arccos @ ( cos_real @ Theta ) )
% 8.58/4.85 != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6823_log__base__10__eq1,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( log2 @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 8.58/4.85 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6824_exp__zero,axiom,
% 8.58/4.85 ( ( exp_complex @ zero_zero_complex )
% 8.58/4.85 = one_one_complex ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6825_exp__zero,axiom,
% 8.58/4.85 ( ( exp_real @ zero_zero_real )
% 8.58/4.85 = one_one_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6826_exp__eq__one__iff,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ( exp_real @ X )
% 8.58/4.85 = one_one_real )
% 8.58/4.85 = ( X = zero_zero_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6827_arccos__1,axiom,
% 8.58/4.85 ( ( arccos @ one_one_real )
% 8.58/4.85 = zero_zero_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6828_frac__of__int,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z2 ) )
% 8.58/4.85 = zero_zero_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6829_frac__of__int,axiom,
% 8.58/4.85 ! [Z2: int] :
% 8.58/4.85 ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 8.58/4.85 = zero_zero_rat ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6830_one__less__exp__iff,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
% 8.58/4.85 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6831_exp__less__one__iff,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
% 8.58/4.85 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6832_one__le__exp__iff,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
% 8.58/4.85 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6833_exp__le__one__iff,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
% 8.58/4.85 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6834_arccos__minus__1,axiom,
% 8.58/4.85 ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.85 = pi ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6835_cos__arccos,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ( ( cos_real @ ( arccos @ Y ) )
% 8.58/4.85 = Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6836_arccos__0,axiom,
% 8.58/4.85 ( ( arccos @ zero_zero_real )
% 8.58/4.85 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6837_exp__not__eq__zero,axiom,
% 8.58/4.85 ! [X: complex] :
% 8.58/4.85 ( ( exp_complex @ X )
% 8.58/4.85 != zero_zero_complex ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6838_exp__not__eq__zero,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( exp_real @ X )
% 8.58/4.85 != zero_zero_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6839_exp__times__arg__commute,axiom,
% 8.58/4.85 ! [A3: complex] :
% 8.58/4.85 ( ( times_times_complex @ ( exp_complex @ A3 ) @ A3 )
% 8.58/4.85 = ( times_times_complex @ A3 @ ( exp_complex @ A3 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6840_exp__times__arg__commute,axiom,
% 8.58/4.85 ! [A3: real] :
% 8.58/4.85 ( ( times_times_real @ ( exp_real @ A3 ) @ A3 )
% 8.58/4.85 = ( times_times_real @ A3 @ ( exp_real @ A3 ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6841_mult__exp__exp,axiom,
% 8.58/4.85 ! [X: complex,Y: complex] :
% 8.58/4.85 ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) )
% 8.58/4.85 = ( exp_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6842_mult__exp__exp,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 8.58/4.85 = ( exp_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6843_exp__add__commuting,axiom,
% 8.58/4.85 ! [X: complex,Y: complex] :
% 8.58/4.85 ( ( ( times_times_complex @ X @ Y )
% 8.58/4.85 = ( times_times_complex @ Y @ X ) )
% 8.58/4.85 => ( ( exp_complex @ ( plus_plus_complex @ X @ Y ) )
% 8.58/4.85 = ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6844_exp__add__commuting,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ( times_times_real @ X @ Y )
% 8.58/4.85 = ( times_times_real @ Y @ X ) )
% 8.58/4.85 => ( ( exp_real @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.85 = ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6845_exp__diff,axiom,
% 8.58/4.85 ! [X: complex,Y: complex] :
% 8.58/4.85 ( ( exp_complex @ ( minus_minus_complex @ X @ Y ) )
% 8.58/4.85 = ( divide1717551699836669952omplex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6846_exp__diff,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( exp_real @ ( minus_minus_real @ X @ Y ) )
% 8.58/4.85 = ( divide_divide_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6847_frac__ge__0,axiom,
% 8.58/4.85 ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6848_frac__ge__0,axiom,
% 8.58/4.85 ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6849_frac__lt__1,axiom,
% 8.58/4.85 ! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6850_frac__lt__1,axiom,
% 8.58/4.85 ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6851_frac__1__eq,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ one_one_real ) )
% 8.58/4.85 = ( archim2898591450579166408c_real @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6852_frac__1__eq,axiom,
% 8.58/4.85 ! [X: rat] :
% 8.58/4.85 ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
% 8.58/4.85 = ( archimedean_frac_rat @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6853_exp__gt__one,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6854_exp__ge__add__one__self,axiom,
% 8.58/4.85 ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6855_exp__minus__inverse,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
% 8.58/4.85 = one_one_real ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6856_exp__minus__inverse,axiom,
% 8.58/4.85 ! [X: complex] :
% 8.58/4.85 ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
% 8.58/4.85 = one_one_complex ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6857_exp__of__nat2__mult,axiom,
% 8.58/4.85 ! [X: complex,N: nat] :
% 8.58/4.85 ( ( exp_complex @ ( times_times_complex @ X @ ( semiri8010041392384452111omplex @ N ) ) )
% 8.58/4.85 = ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6858_exp__of__nat2__mult,axiom,
% 8.58/4.85 ! [X: real,N: nat] :
% 8.58/4.85 ( ( exp_real @ ( times_times_real @ X @ ( semiri5074537144036343181t_real @ N ) ) )
% 8.58/4.85 = ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6859_exp__of__nat__mult,axiom,
% 8.58/4.85 ! [N: nat,X: complex] :
% 8.58/4.85 ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X ) )
% 8.58/4.85 = ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6860_exp__of__nat__mult,axiom,
% 8.58/4.85 ! [N: nat,X: real] :
% 8.58/4.85 ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) )
% 8.58/4.85 = ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6861_log__ln,axiom,
% 8.58/4.85 ( ln_ln_real
% 8.58/4.85 = ( log2 @ ( exp_real @ one_one_real ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6862_arccos__le__arccos,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6863_arccos__le__mono,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y ) )
% 8.58/4.85 = ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6864_arccos__eq__iff,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.85 & ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
% 8.58/4.85 => ( ( ( arccos @ X )
% 8.58/4.85 = ( arccos @ Y ) )
% 8.58/4.85 = ( X = Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6865_exp__ge__add__one__self__aux,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6866_lemma__exp__total,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ one_one_real @ Y )
% 8.58/4.85 => ? [X4: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 8.58/4.85 & ( ord_less_eq_real @ X4 @ ( minus_minus_real @ Y @ one_one_real ) )
% 8.58/4.85 & ( ( exp_real @ X4 )
% 8.58/4.85 = Y ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6867_ln__x__over__x__mono,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ Y )
% 8.58/4.85 => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6868_frac__def,axiom,
% 8.58/4.85 ( archim2898591450579166408c_real
% 8.58/4.85 = ( ^ [X3: real] : ( minus_minus_real @ X3 @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X3 ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6869_frac__def,axiom,
% 8.58/4.85 ( archimedean_frac_rat
% 8.58/4.85 = ( ^ [X3: rat] : ( minus_minus_rat @ X3 @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6870_powr__def,axiom,
% 8.58/4.85 ( powr_real
% 8.58/4.85 = ( ^ [X3: real,A2: real] : ( if_real @ ( X3 = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A2 @ ( ln_ln_real @ X3 ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6871_arccos__lbound,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6872_arccos__less__arccos,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6873_arccos__less__mono,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.85 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 8.58/4.85 => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
% 8.58/4.85 = ( ord_less_real @ Y @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6874_arccos__ubound,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6875_exp__le,axiom,
% 8.58/4.85 ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6876_cos__arccos__abs,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 8.58/4.85 => ( ( cos_real @ ( arccos @ Y ) )
% 8.58/4.85 = Y ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6877_exp__divide__power__eq,axiom,
% 8.58/4.85 ! [N: nat,X: complex] :
% 8.58/4.85 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.85 => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
% 8.58/4.85 = ( exp_complex @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6878_exp__divide__power__eq,axiom,
% 8.58/4.85 ! [N: nat,X: real] :
% 8.58/4.85 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.85 => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 8.58/4.85 = ( exp_real @ X ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6879_frac__eq,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ( archim2898591450579166408c_real @ X )
% 8.58/4.85 = X )
% 8.58/4.85 = ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 & ( ord_less_real @ X @ one_one_real ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6880_frac__eq,axiom,
% 8.58/4.85 ! [X: rat] :
% 8.58/4.85 ( ( ( archimedean_frac_rat @ X )
% 8.58/4.85 = X )
% 8.58/4.85 = ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 8.58/4.85 & ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6881_frac__add,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 8.58/4.85 => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.85 = ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) ) )
% 8.58/4.85 & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 8.58/4.85 => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.85 = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6882_frac__add,axiom,
% 8.58/4.85 ! [X: rat,Y: rat] :
% 8.58/4.85 ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 8.58/4.85 => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
% 8.58/4.85 = ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) ) )
% 8.58/4.85 & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 8.58/4.85 => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
% 8.58/4.85 = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6883_tanh__altdef,axiom,
% 8.58/4.85 ( tanh_real
% 8.58/4.85 = ( ^ [X3: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) @ ( plus_plus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6884_tanh__altdef,axiom,
% 8.58/4.85 ( tanh_complex
% 8.58/4.85 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6885_exp__half__le2,axiom,
% 8.58/4.85 ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6886_arccos__lt__bounded,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 8.58/4.85 => ( ( ord_less_real @ Y @ one_one_real )
% 8.58/4.85 => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
% 8.58/4.85 & ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6887_arccos__bounded,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
% 8.58/4.85 & ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6888_sin__arccos__nonzero,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.85 => ( ( ord_less_real @ X @ one_one_real )
% 8.58/4.85 => ( ( sin_real @ ( arccos @ X ) )
% 8.58/4.85 != zero_zero_real ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6889_exp__double,axiom,
% 8.58/4.85 ! [Z2: complex] :
% 8.58/4.85 ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 ) )
% 8.58/4.85 = ( power_power_complex @ ( exp_complex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6890_exp__double,axiom,
% 8.58/4.85 ! [Z2: real] :
% 8.58/4.85 ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 ) )
% 8.58/4.85 = ( power_power_real @ ( exp_real @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6891_arccos__minus,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ one_one_real )
% 8.58/4.85 => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 8.58/4.85 = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6892_arccos,axiom,
% 8.58/4.85 ! [Y: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 8.58/4.85 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.85 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
% 8.58/4.85 & ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
% 8.58/4.85 & ( ( cos_real @ ( arccos @ Y ) )
% 8.58/4.85 = Y ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6893_exp__bound__half,axiom,
% 8.58/4.85 ! [Z2: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6894_exp__bound__half,axiom,
% 8.58/4.85 ! [Z2: complex] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.85 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6895_arccos__minus__abs,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.85 => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 8.58/4.85 = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6896_exp__bound,axiom,
% 8.58/4.85 ! [X: real] :
% 8.58/4.85 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.85 => ( ( ord_less_eq_real @ X @ one_one_real )
% 8.58/4.85 => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6897_floor__add,axiom,
% 8.58/4.85 ! [X: real,Y: real] :
% 8.58/4.85 ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 8.58/4.85 => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.85 = ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) )
% 8.58/4.85 & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 8.58/4.85 => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.85 = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ one_one_int ) ) ) ) ).
% 8.58/4.85
% 8.58/4.85 thf(fact_6898_floor__add,axiom,
% 8.58/4.85 ! [X: rat,Y: rat] :
% 8.58/4.85 ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 8.58/4.85 => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
% 8.58/4.85 = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) )
% 8.58/4.85 & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 8.58/4.85 => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
% 8.58/4.86 = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ one_one_int ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6899_real__exp__bound__lemma,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.86 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.86 => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6900_exp__ge__one__plus__x__over__n__power__n,axiom,
% 8.58/4.86 ! [N: nat,X: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
% 8.58/4.86 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.86 => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6901_exp__ge__one__minus__x__over__n__power__n,axiom,
% 8.58/4.86 ! [X: real,N: nat] :
% 8.58/4.86 ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 8.58/4.86 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.86 => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6902_arccos__le__pi2,axiom,
% 8.58/4.86 ! [Y: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.58/4.86 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 8.58/4.86 => ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6903_exp__bound__lemma,axiom,
% 8.58/4.86 ! [Z2: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.86 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z2 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6904_exp__bound__lemma,axiom,
% 8.58/4.86 ! [Z2: complex] :
% 8.58/4.86 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.86 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z2 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6905_exp__lower__Taylor__quadratic,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.86 => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6906_log__base__10__eq2,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.86 => ( ( log2 @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 8.58/4.86 = ( times_times_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6907_tanh__real__altdef,axiom,
% 8.58/4.86 ( tanh_real
% 8.58/4.86 = ( ^ [X3: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6908_modulo__int__def,axiom,
% 8.58/4.86 ( modulo_modulo_int
% 8.58/4.86 = ( ^ [K2: int,L2: int] :
% 8.58/4.86 ( if_int @ ( L2 = zero_zero_int ) @ K2
% 8.58/4.86 @ ( if_int
% 8.58/4.86 @ ( ( sgn_sgn_int @ K2 )
% 8.58/4.86 = ( sgn_sgn_int @ L2 ) )
% 8.58/4.86 @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
% 8.58/4.86 @ ( times_times_int @ ( sgn_sgn_int @ L2 )
% 8.58/4.86 @ ( minus_minus_int
% 8.58/4.86 @ ( times_times_int @ ( abs_abs_int @ L2 )
% 8.58/4.86 @ ( zero_n2684676970156552555ol_int
% 8.58/4.86 @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) )
% 8.58/4.86 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6909_divide__int__def,axiom,
% 8.58/4.86 ( divide_divide_int
% 8.58/4.86 = ( ^ [K2: int,L2: int] :
% 8.58/4.86 ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
% 8.58/4.86 @ ( if_int
% 8.58/4.86 @ ( ( sgn_sgn_int @ K2 )
% 8.58/4.86 = ( sgn_sgn_int @ L2 ) )
% 8.58/4.86 @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
% 8.58/4.86 @ ( uminus_uminus_int
% 8.58/4.86 @ ( semiri1314217659103216013at_int
% 8.58/4.86 @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
% 8.58/4.86 @ ( zero_n2687167440665602831ol_nat
% 8.58/4.86 @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) ) ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6910_signed__take__bit__eq__take__bit__minus,axiom,
% 8.58/4.86 ( bit_ri631733984087533419it_int
% 8.58/4.86 = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6911_mask__numeral,axiom,
% 8.58/4.86 ! [N: num] :
% 8.58/4.86 ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
% 8.58/4.86 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6912_mask__numeral,axiom,
% 8.58/4.86 ! [N: num] :
% 8.58/4.86 ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
% 8.58/4.86 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6913_powr__int,axiom,
% 8.58/4.86 ! [X: real,I: int] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.86 => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 8.58/4.86 => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 8.58/4.86 = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
% 8.58/4.86 & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 8.58/4.86 => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 8.58/4.86 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6914_mask__nat__positive__iff,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 8.58/4.86 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6915_nat__int,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
% 8.58/4.86 = N ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6916_nat__numeral,axiom,
% 8.58/4.86 ! [K: num] :
% 8.58/4.86 ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 8.58/4.86 = ( numeral_numeral_nat @ K ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6917_mask__0,axiom,
% 8.58/4.86 ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
% 8.58/4.86 = zero_zero_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6918_mask__0,axiom,
% 8.58/4.86 ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
% 8.58/4.86 = zero_zero_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6919_mask__eq__0__iff,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( ( bit_se2002935070580805687sk_nat @ N )
% 8.58/4.86 = zero_zero_nat )
% 8.58/4.86 = ( N = zero_zero_nat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6920_mask__eq__0__iff,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( ( bit_se2000444600071755411sk_int @ N )
% 8.58/4.86 = zero_zero_int )
% 8.58/4.86 = ( N = zero_zero_nat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6921_nat__of__bool,axiom,
% 8.58/4.86 ! [P: $o] :
% 8.58/4.86 ( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
% 8.58/4.86 = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6922_bit__numeral__Bit0__Suc__iff,axiom,
% 8.58/4.86 ! [M: num,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 8.58/4.86 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6923_bit__numeral__Bit0__Suc__iff,axiom,
% 8.58/4.86 ! [M: num,N: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 8.58/4.86 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6924_bit__numeral__Bit1__Suc__iff,axiom,
% 8.58/4.86 ! [M: num,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 8.58/4.86 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6925_bit__numeral__Bit1__Suc__iff,axiom,
% 8.58/4.86 ! [M: num,N: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 8.58/4.86 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6926_nat__1,axiom,
% 8.58/4.86 ( ( nat2 @ one_one_int )
% 8.58/4.86 = ( suc @ zero_zero_nat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6927_mask__Suc__0,axiom,
% 8.58/4.86 ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
% 8.58/4.86 = one_one_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6928_mask__Suc__0,axiom,
% 8.58/4.86 ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
% 8.58/4.86 = one_one_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6929_nat__0__iff,axiom,
% 8.58/4.86 ! [I: int] :
% 8.58/4.86 ( ( ( nat2 @ I )
% 8.58/4.86 = zero_zero_nat )
% 8.58/4.86 = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6930_nat__le__0,axiom,
% 8.58/4.86 ! [Z2: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
% 8.58/4.86 => ( ( nat2 @ Z2 )
% 8.58/4.86 = zero_zero_nat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6931_zless__nat__conj,axiom,
% 8.58/4.86 ! [W: int,Z2: int] :
% 8.58/4.86 ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
% 8.58/4.86 = ( ( ord_less_int @ zero_zero_int @ Z2 )
% 8.58/4.86 & ( ord_less_int @ W @ Z2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6932_nat__neg__numeral,axiom,
% 8.58/4.86 ! [K: num] :
% 8.58/4.86 ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.58/4.86 = zero_zero_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6933_nat__zminus__int,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.58/4.86 = zero_zero_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6934_int__nat__eq,axiom,
% 8.58/4.86 ! [Z2: int] :
% 8.58/4.86 ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.86 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
% 8.58/4.86 = Z2 ) )
% 8.58/4.86 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.86 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
% 8.58/4.86 = zero_zero_int ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6935_take__bit__minus__one__eq__mask,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( bit_se1745604003318907178nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.86 = ( bit_se2119862282449309892nteger @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6936_take__bit__minus__one__eq__mask,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.86 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6937_of__nat__nat__take__bit__eq,axiom,
% 8.58/4.86 ! [N: nat,K: int] :
% 8.58/4.86 ( ( semiri8010041392384452111omplex @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 8.58/4.86 = ( ring_17405671764205052669omplex @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6938_of__nat__nat__take__bit__eq,axiom,
% 8.58/4.86 ! [N: nat,K: int] :
% 8.58/4.86 ( ( semiri5074537144036343181t_real @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 8.58/4.86 = ( ring_1_of_int_real @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6939_of__nat__nat__take__bit__eq,axiom,
% 8.58/4.86 ! [N: nat,K: int] :
% 8.58/4.86 ( ( semiri681578069525770553at_rat @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 8.58/4.86 = ( ring_1_of_int_rat @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6940_of__nat__nat__take__bit__eq,axiom,
% 8.58/4.86 ! [N: nat,K: int] :
% 8.58/4.86 ( ( semiri1314217659103216013at_int @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 8.58/4.86 = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6941_signed__take__bit__nonnegative__iff,axiom,
% 8.58/4.86 ! [N: nat,K: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 8.58/4.86 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6942_signed__take__bit__negative__iff,axiom,
% 8.58/4.86 ! [N: nat,K: int] :
% 8.58/4.86 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 8.58/4.86 = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6943_zero__less__nat__eq,axiom,
% 8.58/4.86 ! [Z2: int] :
% 8.58/4.86 ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
% 8.58/4.86 = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6944_of__nat__nat,axiom,
% 8.58/4.86 ! [Z2: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.86 => ( ( semiri8010041392384452111omplex @ ( nat2 @ Z2 ) )
% 8.58/4.86 = ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6945_of__nat__nat,axiom,
% 8.58/4.86 ! [Z2: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.86 => ( ( semiri5074537144036343181t_real @ ( nat2 @ Z2 ) )
% 8.58/4.86 = ( ring_1_of_int_real @ Z2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6946_of__nat__nat,axiom,
% 8.58/4.86 ! [Z2: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.86 => ( ( semiri681578069525770553at_rat @ ( nat2 @ Z2 ) )
% 8.58/4.86 = ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6947_of__nat__nat,axiom,
% 8.58/4.86 ! [Z2: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.86 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
% 8.58/4.86 = ( ring_1_of_int_int @ Z2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6948_bit__numeral__simps_I2_J,axiom,
% 8.58/4.86 ! [W: num,N: num] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 8.58/4.86 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6949_bit__numeral__simps_I2_J,axiom,
% 8.58/4.86 ! [W: num,N: num] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 8.58/4.86 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6950_diff__nat__numeral,axiom,
% 8.58/4.86 ! [V: num,V2: num] :
% 8.58/4.86 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V2 ) )
% 8.58/4.86 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6951_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 8.58/4.86 ! [W: num,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
% 8.58/4.86 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6952_bit__numeral__simps_I3_J,axiom,
% 8.58/4.86 ! [W: num,N: num] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 8.58/4.86 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6953_bit__numeral__simps_I3_J,axiom,
% 8.58/4.86 ! [W: num,N: num] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 8.58/4.86 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6954_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 8.58/4.86 ! [W: num,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
% 8.58/4.86 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6955_numeral__power__eq__nat__cancel__iff,axiom,
% 8.58/4.86 ! [X: num,N: nat,Y: int] :
% 8.58/4.86 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 8.58/4.86 = ( nat2 @ Y ) )
% 8.58/4.86 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 8.58/4.86 = Y ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6956_nat__eq__numeral__power__cancel__iff,axiom,
% 8.58/4.86 ! [Y: int,X: num,N: nat] :
% 8.58/4.86 ( ( ( nat2 @ Y )
% 8.58/4.86 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 8.58/4.86 = ( Y
% 8.58/4.86 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6957_dvd__nat__abs__iff,axiom,
% 8.58/4.86 ! [N: nat,K: int] :
% 8.58/4.86 ( ( dvd_dvd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 8.58/4.86 = ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6958_nat__abs__dvd__iff,axiom,
% 8.58/4.86 ! [K: int,N: nat] :
% 8.58/4.86 ( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 8.58/4.86 = ( dvd_dvd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6959_nat__ceiling__le__eq,axiom,
% 8.58/4.86 ! [X: real,A: nat] :
% 8.58/4.86 ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
% 8.58/4.86 = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6960_bit__0,axiom,
% 8.58/4.86 ! [A: code_integer] :
% 8.58/4.86 ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
% 8.58/4.86 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6961_bit__0,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
% 8.58/4.86 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6962_bit__0,axiom,
% 8.58/4.86 ! [A: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
% 8.58/4.86 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6963_one__less__nat__eq,axiom,
% 8.58/4.86 ! [Z2: int] :
% 8.58/4.86 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
% 8.58/4.86 = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6964_bit__minus__numeral__int_I1_J,axiom,
% 8.58/4.86 ! [W: num,N: num] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 8.58/4.86 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6965_bit__minus__numeral__int_I2_J,axiom,
% 8.58/4.86 ! [W: num,N: num] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 8.58/4.86 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6966_bit__mod__2__iff,axiom,
% 8.58/4.86 ! [A: code_integer,N: nat] :
% 8.58/4.86 ( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N )
% 8.58/4.86 = ( ( N = zero_zero_nat )
% 8.58/4.86 & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6967_bit__mod__2__iff,axiom,
% 8.58/4.86 ! [A: code_natural,N: nat] :
% 8.58/4.86 ( ( bit_se8040316288895769887atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ N )
% 8.58/4.86 = ( ( N = zero_zero_nat )
% 8.58/4.86 & ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6968_bit__mod__2__iff,axiom,
% 8.58/4.86 ! [A: int,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N )
% 8.58/4.86 = ( ( N = zero_zero_nat )
% 8.58/4.86 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6969_bit__mod__2__iff,axiom,
% 8.58/4.86 ! [A: nat,N: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 8.58/4.86 = ( ( N = zero_zero_nat )
% 8.58/4.86 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6970_nat__numeral__diff__1,axiom,
% 8.58/4.86 ! [V: num] :
% 8.58/4.86 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 8.58/4.86 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6971_numeral__power__less__nat__cancel__iff,axiom,
% 8.58/4.86 ! [X: num,N: nat,A: int] :
% 8.58/4.86 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 8.58/4.86 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6972_nat__less__numeral__power__cancel__iff,axiom,
% 8.58/4.86 ! [A: int,X: num,N: nat] :
% 8.58/4.86 ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 8.58/4.86 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6973_numeral__power__le__nat__cancel__iff,axiom,
% 8.58/4.86 ! [X: num,N: nat,A: int] :
% 8.58/4.86 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 8.58/4.86 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6974_nat__le__numeral__power__cancel__iff,axiom,
% 8.58/4.86 ! [A: int,X: num,N: nat] :
% 8.58/4.86 ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 8.58/4.86 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6975_bit__numeral__iff,axiom,
% 8.58/4.86 ! [M: num,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N )
% 8.58/4.86 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6976_bit__numeral__iff,axiom,
% 8.58/4.86 ! [M: num,N: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N )
% 8.58/4.86 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6977_of__nat__mask__eq,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 8.58/4.86 = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6978_of__nat__mask__eq,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
% 8.58/4.86 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6979_nat__mask__eq,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
% 8.58/4.86 = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6980_bit__of__nat__iff__bit,axiom,
% 8.58/4.86 ! [M: nat,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( semiri1314217659103216013at_int @ M ) @ N )
% 8.58/4.86 = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6981_bit__of__nat__iff__bit,axiom,
% 8.58/4.86 ! [M: nat,N: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ ( semiri1316708129612266289at_nat @ M ) @ N )
% 8.58/4.86 = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6982_bit__disjunctive__add__iff,axiom,
% 8.58/4.86 ! [A: int,B: int,N: nat] :
% 8.58/4.86 ( ! [N2: nat] :
% 8.58/4.86 ( ~ ( bit_se1146084159140164899it_int @ A @ N2 )
% 8.58/4.86 | ~ ( bit_se1146084159140164899it_int @ B @ N2 ) )
% 8.58/4.86 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N )
% 8.58/4.86 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 8.58/4.86 | ( bit_se1146084159140164899it_int @ B @ N ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6983_bit__disjunctive__add__iff,axiom,
% 8.58/4.86 ! [A: nat,B: nat,N: nat] :
% 8.58/4.86 ( ! [N2: nat] :
% 8.58/4.86 ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 )
% 8.58/4.86 | ~ ( bit_se1148574629649215175it_nat @ B @ N2 ) )
% 8.58/4.86 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 8.58/4.86 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 8.58/4.86 | ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6984_of__int__mask__eq,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( ring_1_of_int_int @ ( bit_se2000444600071755411sk_int @ N ) )
% 8.58/4.86 = ( bit_se2000444600071755411sk_int @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6985_bit__unset__bit__iff,axiom,
% 8.58/4.86 ! [M: nat,A: int,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( bit_se4203085406695923979it_int @ M @ A ) @ N )
% 8.58/4.86 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 8.58/4.86 & ( M != N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6986_bit__unset__bit__iff,axiom,
% 8.58/4.86 ! [M: nat,A: nat,N: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ ( bit_se4205575877204974255it_nat @ M @ A ) @ N )
% 8.58/4.86 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 8.58/4.86 & ( M != N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6987_less__eq__mask,axiom,
% 8.58/4.86 ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6988_not__bit__1__Suc,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6989_not__bit__1__Suc,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6990_bit__1__iff,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ one_one_int @ N )
% 8.58/4.86 = ( N = zero_zero_nat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6991_bit__1__iff,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N )
% 8.58/4.86 = ( N = zero_zero_nat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6992_bit__numeral__simps_I1_J,axiom,
% 8.58/4.86 ! [N: num] :
% 8.58/4.86 ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6993_bit__numeral__simps_I1_J,axiom,
% 8.58/4.86 ! [N: num] :
% 8.58/4.86 ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6994_bit__take__bit__iff,axiom,
% 8.58/4.86 ! [M: nat,A: int,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N )
% 8.58/4.86 = ( ( ord_less_nat @ N @ M )
% 8.58/4.86 & ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6995_bit__take__bit__iff,axiom,
% 8.58/4.86 ! [M: nat,A: nat,N: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N )
% 8.58/4.86 = ( ( ord_less_nat @ N @ M )
% 8.58/4.86 & ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6996_bit__of__bool__iff,axiom,
% 8.58/4.86 ! [B: $o,N: nat] :
% 8.58/4.86 ( ( bit_se9216721137139052372nteger @ ( zero_n356916108424825756nteger @ B ) @ N )
% 8.58/4.86 = ( B
% 8.58/4.86 & ( N = zero_zero_nat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6997_bit__of__bool__iff,axiom,
% 8.58/4.86 ! [B: $o,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B ) @ N )
% 8.58/4.86 = ( B
% 8.58/4.86 & ( N = zero_zero_nat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6998_bit__of__bool__iff,axiom,
% 8.58/4.86 ! [B: $o,N: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ N )
% 8.58/4.86 = ( B
% 8.58/4.86 & ( N = zero_zero_nat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_6999_nat__zero__as__int,axiom,
% 8.58/4.86 ( zero_zero_nat
% 8.58/4.86 = ( nat2 @ zero_zero_int ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7000_nat__numeral__as__int,axiom,
% 8.58/4.86 ( numeral_numeral_nat
% 8.58/4.86 = ( ^ [I4: num] : ( nat2 @ ( numeral_numeral_int @ I4 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7001_nat__mono,axiom,
% 8.58/4.86 ! [X: int,Y: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ X @ Y )
% 8.58/4.86 => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7002_ex__nat,axiom,
% 8.58/4.86 ( ( ^ [P5: nat > $o] :
% 8.58/4.86 ? [X7: nat] : ( P5 @ X7 ) )
% 8.58/4.86 = ( ^ [P6: nat > $o] :
% 8.58/4.86 ? [X3: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 8.58/4.86 & ( P6 @ ( nat2 @ X3 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7003_all__nat,axiom,
% 8.58/4.86 ( ( ^ [P5: nat > $o] :
% 8.58/4.86 ! [X7: nat] : ( P5 @ X7 ) )
% 8.58/4.86 = ( ^ [P6: nat > $o] :
% 8.58/4.86 ! [X3: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 8.58/4.86 => ( P6 @ ( nat2 @ X3 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7004_eq__nat__nat__iff,axiom,
% 8.58/4.86 ! [Z2: int,Z6: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.86 => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 8.58/4.86 => ( ( ( nat2 @ Z2 )
% 8.58/4.86 = ( nat2 @ Z6 ) )
% 8.58/4.86 = ( Z2 = Z6 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7005_nat__one__as__int,axiom,
% 8.58/4.86 ( one_one_nat
% 8.58/4.86 = ( nat2 @ one_one_int ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7006_signed__take__bit__eq__if__positive,axiom,
% 8.58/4.86 ! [A: int,N: nat] :
% 8.58/4.86 ( ~ ( bit_se1146084159140164899it_int @ A @ N )
% 8.58/4.86 => ( ( bit_ri631733984087533419it_int @ N @ A )
% 8.58/4.86 = ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7007_complex__exp__exists,axiom,
% 8.58/4.86 ! [Z2: complex] :
% 8.58/4.86 ? [A4: complex,R2: real] :
% 8.58/4.86 ( Z2
% 8.58/4.86 = ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( exp_complex @ A4 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7008_unset__bit__nat__def,axiom,
% 8.58/4.86 ( bit_se4205575877204974255it_nat
% 8.58/4.86 = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M3 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7009_mask__nonnegative__int,axiom,
% 8.58/4.86 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7010_not__mask__negative__int,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7011_push__bit__nat__eq,axiom,
% 8.58/4.86 ! [N: nat,K: int] :
% 8.58/4.86 ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
% 8.58/4.86 = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7012_nat__mono__iff,axiom,
% 8.58/4.86 ! [Z2: int,W: int] :
% 8.58/4.86 ( ( ord_less_int @ zero_zero_int @ Z2 )
% 8.58/4.86 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
% 8.58/4.86 = ( ord_less_int @ W @ Z2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7013_of__nat__ceiling,axiom,
% 8.58/4.86 ! [R: real] : ( ord_less_eq_real @ R @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7014_of__nat__ceiling,axiom,
% 8.58/4.86 ! [R: rat] : ( ord_less_eq_rat @ R @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7015_zless__nat__eq__int__zless,axiom,
% 8.58/4.86 ! [M: nat,Z2: int] :
% 8.58/4.86 ( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
% 8.58/4.86 = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7016_nat__le__iff,axiom,
% 8.58/4.86 ! [X: int,N: nat] :
% 8.58/4.86 ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
% 8.58/4.86 = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7017_int__eq__iff,axiom,
% 8.58/4.86 ! [M: nat,Z2: int] :
% 8.58/4.86 ( ( ( semiri1314217659103216013at_int @ M )
% 8.58/4.86 = Z2 )
% 8.58/4.86 = ( ( M
% 8.58/4.86 = ( nat2 @ Z2 ) )
% 8.58/4.86 & ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7018_nat__0__le,axiom,
% 8.58/4.86 ! [Z2: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.86 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
% 8.58/4.86 = Z2 ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7019_bit__not__int__iff_H,axiom,
% 8.58/4.86 ! [K: int,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
% 8.58/4.86 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7020_nat__int__add,axiom,
% 8.58/4.86 ! [A: nat,B: nat] :
% 8.58/4.86 ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
% 8.58/4.86 = ( plus_plus_nat @ A @ B ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7021_int__minus,axiom,
% 8.58/4.86 ! [N: nat,M: nat] :
% 8.58/4.86 ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
% 8.58/4.86 = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7022_nat__abs__mult__distrib,axiom,
% 8.58/4.86 ! [W: int,Z2: int] :
% 8.58/4.86 ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z2 ) ) )
% 8.58/4.86 = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7023_bit__push__bit__iff__int,axiom,
% 8.58/4.86 ! [M: nat,K: int,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 8.58/4.86 = ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.86 & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7024_flip__bit__eq__if,axiom,
% 8.58/4.86 ( bit_se2159334234014336723it_int
% 8.58/4.86 = ( ^ [N3: nat,A2: int] : ( if_nat_int_int @ ( bit_se1146084159140164899it_int @ A2 @ N3 ) @ bit_se4203085406695923979it_int @ bit_se7879613467334960850it_int @ N3 @ A2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7025_flip__bit__eq__if,axiom,
% 8.58/4.86 ( bit_se2161824704523386999it_nat
% 8.58/4.86 = ( ^ [N3: nat,A2: nat] : ( if_nat_nat_nat @ ( bit_se1148574629649215175it_nat @ A2 @ N3 ) @ bit_se4205575877204974255it_nat @ bit_se7882103937844011126it_nat @ N3 @ A2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7026_real__nat__ceiling__ge,axiom,
% 8.58/4.86 ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7027_less__mask,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.58/4.86 => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7028_of__nat__floor,axiom,
% 8.58/4.86 ! [R: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ zero_zero_real @ R )
% 8.58/4.86 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R ) ) ) @ R ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7029_of__nat__floor,axiom,
% 8.58/4.86 ! [R: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ zero_zero_rat @ R )
% 8.58/4.86 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R ) ) ) @ R ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7030_nat__less__eq__zless,axiom,
% 8.58/4.86 ! [W: int,Z2: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 8.58/4.86 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
% 8.58/4.86 = ( ord_less_int @ W @ Z2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7031_nat__le__eq__zle,axiom,
% 8.58/4.86 ! [W: int,Z2: int] :
% 8.58/4.86 ( ( ( ord_less_int @ zero_zero_int @ W )
% 8.58/4.86 | ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
% 8.58/4.86 => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
% 8.58/4.86 = ( ord_less_eq_int @ W @ Z2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7032_nat__eq__iff2,axiom,
% 8.58/4.86 ! [M: nat,W: int] :
% 8.58/4.86 ( ( M
% 8.58/4.86 = ( nat2 @ W ) )
% 8.58/4.86 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 8.58/4.86 => ( W
% 8.58/4.86 = ( semiri1314217659103216013at_int @ M ) ) )
% 8.58/4.86 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 8.58/4.86 => ( M = zero_zero_nat ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7033_nat__eq__iff,axiom,
% 8.58/4.86 ! [W: int,M: nat] :
% 8.58/4.86 ( ( ( nat2 @ W )
% 8.58/4.86 = M )
% 8.58/4.86 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 8.58/4.86 => ( W
% 8.58/4.86 = ( semiri1314217659103216013at_int @ M ) ) )
% 8.58/4.86 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 8.58/4.86 => ( M = zero_zero_nat ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7034_split__nat,axiom,
% 8.58/4.86 ! [P: nat > $o,I: int] :
% 8.58/4.86 ( ( P @ ( nat2 @ I ) )
% 8.58/4.86 = ( ! [N3: nat] :
% 8.58/4.86 ( ( I
% 8.58/4.86 = ( semiri1314217659103216013at_int @ N3 ) )
% 8.58/4.86 => ( P @ N3 ) )
% 8.58/4.86 & ( ( ord_less_int @ I @ zero_zero_int )
% 8.58/4.86 => ( P @ zero_zero_nat ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7035_le__mult__nat__floor,axiom,
% 8.58/4.86 ! [A: real,B: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7036_le__mult__nat__floor,axiom,
% 8.58/4.86 ! [A: rat,B: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7037_le__nat__iff,axiom,
% 8.58/4.86 ! [K: int,N: nat] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.58/4.86 => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 8.58/4.86 = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7038_nat__add__distrib,axiom,
% 8.58/4.86 ! [Z2: int,Z6: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.86 => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 8.58/4.86 => ( ( nat2 @ ( plus_plus_int @ Z2 @ Z6 ) )
% 8.58/4.86 = ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7039_bit__imp__take__bit__positive,axiom,
% 8.58/4.86 ! [N: nat,M: nat,K: int] :
% 8.58/4.86 ( ( ord_less_nat @ N @ M )
% 8.58/4.86 => ( ( bit_se1146084159140164899it_int @ K @ N )
% 8.58/4.86 => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7040_nat__mult__distrib,axiom,
% 8.58/4.86 ! [Z2: int,Z6: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.86 => ( ( nat2 @ ( times_times_int @ Z2 @ Z6 ) )
% 8.58/4.86 = ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7041_Suc__as__int,axiom,
% 8.58/4.86 ( suc
% 8.58/4.86 = ( ^ [A2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ one_one_int ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7042_nat__diff__distrib,axiom,
% 8.58/4.86 ! [Z6: int,Z2: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 8.58/4.86 => ( ( ord_less_eq_int @ Z6 @ Z2 )
% 8.58/4.86 => ( ( nat2 @ ( minus_minus_int @ Z2 @ Z6 ) )
% 8.58/4.86 = ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7043_nat__diff__distrib_H,axiom,
% 8.58/4.86 ! [X: int,Y: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.58/4.86 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.58/4.86 => ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
% 8.58/4.86 = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7044_nat__abs__triangle__ineq,axiom,
% 8.58/4.86 ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7045_nat__div__distrib_H,axiom,
% 8.58/4.86 ! [Y: int,X: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.58/4.86 => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
% 8.58/4.86 = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7046_nat__div__distrib,axiom,
% 8.58/4.86 ! [X: int,Y: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.58/4.86 => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
% 8.58/4.86 = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7047_bit__concat__bit__iff,axiom,
% 8.58/4.86 ! [M: nat,K: int,L: int,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
% 8.58/4.86 = ( ( ( ord_less_nat @ N @ M )
% 8.58/4.86 & ( bit_se1146084159140164899it_int @ K @ N ) )
% 8.58/4.86 | ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.86 & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7048_nat__floor__neg,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 8.58/4.86 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 8.58/4.86 = zero_zero_nat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7049_nat__power__eq,axiom,
% 8.58/4.86 ! [Z2: int,N: nat] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.86 => ( ( nat2 @ ( power_power_int @ Z2 @ N ) )
% 8.58/4.86 = ( power_power_nat @ ( nat2 @ Z2 ) @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7050_nat__mod__distrib,axiom,
% 8.58/4.86 ! [X: int,Y: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.58/4.86 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.58/4.86 => ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
% 8.58/4.86 = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7051_div__abs__eq__div__nat,axiom,
% 8.58/4.86 ! [K: int,L: int] :
% 8.58/4.86 ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
% 8.58/4.86 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7052_floor__eq3,axiom,
% 8.58/4.86 ! [N: nat,X: real] :
% 8.58/4.86 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 8.58/4.86 => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 8.58/4.86 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 8.58/4.86 = N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7053_le__nat__floor,axiom,
% 8.58/4.86 ! [X: nat,A: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
% 8.58/4.86 => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7054_mod__abs__eq__div__nat,axiom,
% 8.58/4.86 ! [K: int,L: int] :
% 8.58/4.86 ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
% 8.58/4.86 = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7055_nat__take__bit__eq,axiom,
% 8.58/4.86 ! [K: int,N: nat] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.58/4.86 => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 8.58/4.86 = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7056_take__bit__nat__eq,axiom,
% 8.58/4.86 ! [K: int,N: nat] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.58/4.86 => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
% 8.58/4.86 = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7057_signed__take__bit__eq__concat__bit,axiom,
% 8.58/4.86 ( bit_ri631733984087533419it_int
% 8.58/4.86 = ( ^ [N3: nat,K2: int] : ( bit_concat_bit @ N3 @ K2 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7058_exp__eq__0__imp__not__bit,axiom,
% 8.58/4.86 ! [N: nat,A: int] :
% 8.58/4.86 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 8.58/4.86 = zero_zero_int )
% 8.58/4.86 => ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7059_exp__eq__0__imp__not__bit,axiom,
% 8.58/4.86 ! [N: nat,A: nat] :
% 8.58/4.86 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.86 = zero_zero_nat )
% 8.58/4.86 => ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7060_bit__Suc,axiom,
% 8.58/4.86 ! [A: int,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ A @ ( suc @ N ) )
% 8.58/4.86 = ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7061_bit__Suc,axiom,
% 8.58/4.86 ! [A: nat,N: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N ) )
% 8.58/4.86 = ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7062_stable__imp__bit__iff__odd,axiom,
% 8.58/4.86 ! [A: code_integer,N: nat] :
% 8.58/4.86 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.86 = A )
% 8.58/4.86 => ( ( bit_se9216721137139052372nteger @ A @ N )
% 8.58/4.86 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7063_stable__imp__bit__iff__odd,axiom,
% 8.58/4.86 ! [A: int,N: nat] :
% 8.58/4.86 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.86 = A )
% 8.58/4.86 => ( ( bit_se1146084159140164899it_int @ A @ N )
% 8.58/4.86 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7064_stable__imp__bit__iff__odd,axiom,
% 8.58/4.86 ! [A: nat,N: nat] :
% 8.58/4.86 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.86 = A )
% 8.58/4.86 => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 8.58/4.86 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7065_bit__iff__idd__imp__stable,axiom,
% 8.58/4.86 ! [A: code_integer] :
% 8.58/4.86 ( ! [N2: nat] :
% 8.58/4.86 ( ( bit_se9216721137139052372nteger @ A @ N2 )
% 8.58/4.86 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 8.58/4.86 => ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.86 = A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7066_bit__iff__idd__imp__stable,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ! [N2: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ A @ N2 )
% 8.58/4.86 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 8.58/4.86 => ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.86 = A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7067_bit__iff__idd__imp__stable,axiom,
% 8.58/4.86 ! [A: nat] :
% 8.58/4.86 ( ! [N2: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ A @ N2 )
% 8.58/4.86 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 8.58/4.86 => ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.86 = A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7068_nat__2,axiom,
% 8.58/4.86 ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.86 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7069_take__bit__eq__mask__iff,axiom,
% 8.58/4.86 ! [N: nat,K: int] :
% 8.58/4.86 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 8.58/4.86 = ( bit_se2000444600071755411sk_int @ N ) )
% 8.58/4.86 = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 8.58/4.86 = zero_zero_int ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7070_int__bit__bound,axiom,
% 8.58/4.86 ! [K: int] :
% 8.58/4.86 ~ ! [N2: nat] :
% 8.58/4.86 ( ! [M4: nat] :
% 8.58/4.86 ( ( ord_less_eq_nat @ N2 @ M4 )
% 8.58/4.86 => ( ( bit_se1146084159140164899it_int @ K @ M4 )
% 8.58/4.86 = ( bit_se1146084159140164899it_int @ K @ N2 ) ) )
% 8.58/4.86 => ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 8.58/4.86 => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 8.58/4.86 = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7071_Suc__nat__eq__nat__zadd1,axiom,
% 8.58/4.86 ! [Z2: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.86 => ( ( suc @ ( nat2 @ Z2 ) )
% 8.58/4.86 = ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7072_nat__less__iff,axiom,
% 8.58/4.86 ! [W: int,M: nat] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 8.58/4.86 => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
% 8.58/4.86 = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7073_nat__mult__distrib__neg,axiom,
% 8.58/4.86 ! [Z2: int,Z6: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
% 8.58/4.86 => ( ( nat2 @ ( times_times_int @ Z2 @ Z6 ) )
% 8.58/4.86 = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7074_nat__abs__int__diff,axiom,
% 8.58/4.86 ! [A: nat,B: nat] :
% 8.58/4.86 ( ( ( ord_less_eq_nat @ A @ B )
% 8.58/4.86 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 8.58/4.86 = ( minus_minus_nat @ B @ A ) ) )
% 8.58/4.86 & ( ~ ( ord_less_eq_nat @ A @ B )
% 8.58/4.86 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 8.58/4.86 = ( minus_minus_nat @ A @ B ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7075_floor__eq4,axiom,
% 8.58/4.86 ! [N: nat,X: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 8.58/4.86 => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 8.58/4.86 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 8.58/4.86 = N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7076_bit__iff__odd,axiom,
% 8.58/4.86 ( bit_se9216721137139052372nteger
% 8.58/4.86 = ( ^ [A2: code_integer,N3: nat] :
% 8.58/4.86 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7077_bit__iff__odd,axiom,
% 8.58/4.86 ( bit_se1146084159140164899it_int
% 8.58/4.86 = ( ^ [A2: int,N3: nat] :
% 8.58/4.86 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7078_bit__iff__odd,axiom,
% 8.58/4.86 ( bit_se1148574629649215175it_nat
% 8.58/4.86 = ( ^ [A2: nat,N3: nat] :
% 8.58/4.86 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7079_Suc__mask__eq__exp,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
% 8.58/4.86 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7080_mask__nat__less__exp,axiom,
% 8.58/4.86 ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7081_of__int__of__nat,axiom,
% 8.58/4.86 ( ring_18347121197199848620nteger
% 8.58/4.86 = ( ^ [K2: int] : ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri4939895301339042750nteger @ ( nat2 @ K2 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7082_of__int__of__nat,axiom,
% 8.58/4.86 ( ring_17405671764205052669omplex
% 8.58/4.86 = ( ^ [K2: int] : ( if_complex @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri8010041392384452111omplex @ ( nat2 @ K2 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7083_of__int__of__nat,axiom,
% 8.58/4.86 ( ring_1_of_int_real
% 8.58/4.86 = ( ^ [K2: int] : ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K2 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7084_of__int__of__nat,axiom,
% 8.58/4.86 ( ring_1_of_int_rat
% 8.58/4.86 = ( ^ [K2: int] : ( if_rat @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri681578069525770553at_rat @ ( nat2 @ K2 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7085_of__int__of__nat,axiom,
% 8.58/4.86 ( ring_1_of_int_int
% 8.58/4.86 = ( ^ [K2: int] : ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K2 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7086_bit__int__def,axiom,
% 8.58/4.86 ( bit_se1146084159140164899it_int
% 8.58/4.86 = ( ^ [K2: int,N3: nat] :
% 8.58/4.86 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7087_nat__dvd__iff,axiom,
% 8.58/4.86 ! [Z2: int,M: nat] :
% 8.58/4.86 ( ( dvd_dvd_nat @ ( nat2 @ Z2 ) @ M )
% 8.58/4.86 = ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.86 => ( dvd_dvd_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) )
% 8.58/4.86 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 8.58/4.86 => ( M = zero_zero_nat ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7088_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N ) )
% 8.58/4.86 = ( N = zero_zero_nat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7089_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) )
% 8.58/4.86 = ( N = zero_zero_nat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7090_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) )
% 8.58/4.86 = ( N = zero_zero_nat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7091_even__bit__succ__iff,axiom,
% 8.58/4.86 ! [A: code_integer,N: nat] :
% 8.58/4.86 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.58/4.86 => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N )
% 8.58/4.86 = ( ( bit_se9216721137139052372nteger @ A @ N )
% 8.58/4.86 | ( N = zero_zero_nat ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7092_even__bit__succ__iff,axiom,
% 8.58/4.86 ! [A: int,N: nat] :
% 8.58/4.86 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.58/4.86 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N )
% 8.58/4.86 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 8.58/4.86 | ( N = zero_zero_nat ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7093_even__bit__succ__iff,axiom,
% 8.58/4.86 ! [A: nat,N: nat] :
% 8.58/4.86 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.58/4.86 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N )
% 8.58/4.86 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 8.58/4.86 | ( N = zero_zero_nat ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7094_odd__bit__iff__bit__pred,axiom,
% 8.58/4.86 ! [A: code_integer,N: nat] :
% 8.58/4.86 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.58/4.86 => ( ( bit_se9216721137139052372nteger @ A @ N )
% 8.58/4.86 = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N )
% 8.58/4.86 | ( N = zero_zero_nat ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7095_odd__bit__iff__bit__pred,axiom,
% 8.58/4.86 ! [A: int,N: nat] :
% 8.58/4.86 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.58/4.86 => ( ( bit_se1146084159140164899it_int @ A @ N )
% 8.58/4.86 = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N )
% 8.58/4.86 | ( N = zero_zero_nat ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7096_odd__bit__iff__bit__pred,axiom,
% 8.58/4.86 ! [A: nat,N: nat] :
% 8.58/4.86 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.58/4.86 => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 8.58/4.86 = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N )
% 8.58/4.86 | ( N = zero_zero_nat ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7097_mask__half__int,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.86 = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7098_mask__nat__def,axiom,
% 8.58/4.86 ( bit_se2002935070580805687sk_nat
% 8.58/4.86 = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7099_mask__int__def,axiom,
% 8.58/4.86 ( bit_se2000444600071755411sk_int
% 8.58/4.86 = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7100_bit__sum__mult__2__cases,axiom,
% 8.58/4.86 ! [A: code_integer,B: code_integer,N: nat] :
% 8.58/4.86 ( ! [J2: nat] :
% 8.58/4.86 ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J2 ) )
% 8.58/4.86 => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N )
% 8.58/4.86 = ( ( ( N = zero_zero_nat )
% 8.58/4.86 => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 8.58/4.86 & ( ( N != zero_zero_nat )
% 8.58/4.86 => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7101_bit__sum__mult__2__cases,axiom,
% 8.58/4.86 ! [A: int,B: int,N: nat] :
% 8.58/4.86 ( ! [J2: nat] :
% 8.58/4.86 ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J2 ) )
% 8.58/4.86 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ N )
% 8.58/4.86 = ( ( ( N = zero_zero_nat )
% 8.58/4.86 => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 8.58/4.86 & ( ( N != zero_zero_nat )
% 8.58/4.86 => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7102_bit__sum__mult__2__cases,axiom,
% 8.58/4.86 ! [A: nat,B: nat,N: nat] :
% 8.58/4.86 ( ! [J2: nat] :
% 8.58/4.86 ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J2 ) )
% 8.58/4.86 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ N )
% 8.58/4.86 = ( ( ( N = zero_zero_nat )
% 8.58/4.86 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 8.58/4.86 & ( ( N != zero_zero_nat )
% 8.58/4.86 => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7103_bit__rec,axiom,
% 8.58/4.86 ( bit_se9216721137139052372nteger
% 8.58/4.86 = ( ^ [A2: code_integer,N3: nat] :
% 8.58/4.86 ( ( ( N3 = zero_zero_nat )
% 8.58/4.86 => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) )
% 8.58/4.86 & ( ( N3 != zero_zero_nat )
% 8.58/4.86 => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7104_bit__rec,axiom,
% 8.58/4.86 ( bit_se1146084159140164899it_int
% 8.58/4.86 = ( ^ [A2: int,N3: nat] :
% 8.58/4.86 ( ( ( N3 = zero_zero_nat )
% 8.58/4.86 => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 8.58/4.86 & ( ( N3 != zero_zero_nat )
% 8.58/4.86 => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7105_bit__rec,axiom,
% 8.58/4.86 ( bit_se1148574629649215175it_nat
% 8.58/4.86 = ( ^ [A2: nat,N3: nat] :
% 8.58/4.86 ( ( ( N3 = zero_zero_nat )
% 8.58/4.86 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
% 8.58/4.86 & ( ( N3 != zero_zero_nat )
% 8.58/4.86 => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7106_mask__eq__exp__minus__1,axiom,
% 8.58/4.86 ( bit_se2002935070580805687sk_nat
% 8.58/4.86 = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7107_mask__eq__exp__minus__1,axiom,
% 8.58/4.86 ( bit_se2000444600071755411sk_int
% 8.58/4.86 = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7108_even__nat__iff,axiom,
% 8.58/4.86 ! [K: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.58/4.86 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 8.58/4.86 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7109_set__bit__eq,axiom,
% 8.58/4.86 ( bit_se7879613467334960850it_int
% 8.58/4.86 = ( ^ [N3: nat,K2: int] :
% 8.58/4.86 ( plus_plus_int @ K2
% 8.58/4.86 @ ( times_times_int
% 8.58/4.86 @ ( zero_n2684676970156552555ol_int
% 8.58/4.86 @ ~ ( bit_se1146084159140164899it_int @ K2 @ N3 ) )
% 8.58/4.86 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7110_unset__bit__eq,axiom,
% 8.58/4.86 ( bit_se4203085406695923979it_int
% 8.58/4.86 = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ K2 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7111_take__bit__Suc__from__most,axiom,
% 8.58/4.86 ! [N: nat,K: int] :
% 8.58/4.86 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
% 8.58/4.86 = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7112_take__bit__eq__mask__iff__exp__dvd,axiom,
% 8.58/4.86 ! [N: nat,K: int] :
% 8.58/4.86 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 8.58/4.86 = ( bit_se2000444600071755411sk_int @ N ) )
% 8.58/4.86 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7113_and__int__unfold,axiom,
% 8.58/4.86 ( bit_se725231765392027082nd_int
% 8.58/4.86 = ( ^ [K2: int,L2: int] :
% 8.58/4.86 ( if_int
% 8.58/4.86 @ ( ( K2 = zero_zero_int )
% 8.58/4.86 | ( L2 = zero_zero_int ) )
% 8.58/4.86 @ zero_zero_int
% 8.58/4.86 @ ( if_int
% 8.58/4.86 @ ( K2
% 8.58/4.86 = ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.86 @ L2
% 8.58/4.86 @ ( if_int
% 8.58/4.86 @ ( L2
% 8.58/4.86 = ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.86 @ K2
% 8.58/4.86 @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7114_powr__real__of__int,axiom,
% 8.58/4.86 ! [X: real,N: int] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.86 => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 8.58/4.86 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 8.58/4.86 = ( power_power_real @ X @ ( nat2 @ N ) ) ) )
% 8.58/4.86 & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 8.58/4.86 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 8.58/4.86 = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7115_cis__2pi,axiom,
% 8.58/4.86 ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 8.58/4.86 = one_one_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7116_exp__two__pi__i,axiom,
% 8.58/4.86 ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
% 8.58/4.86 = one_one_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7117_exp__two__pi__i_H,axiom,
% 8.58/4.86 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
% 8.58/4.86 = one_one_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7118_inverse__eq__iff__eq,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ( inverse_inverse_real @ A )
% 8.58/4.86 = ( inverse_inverse_real @ B ) )
% 8.58/4.86 = ( A = B ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7119_inverse__eq__iff__eq,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( ( invers8013647133539491842omplex @ A )
% 8.58/4.86 = ( invers8013647133539491842omplex @ B ) )
% 8.58/4.86 = ( A = B ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7120_inverse__eq__iff__eq,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ( inverse_inverse_rat @ A )
% 8.58/4.86 = ( inverse_inverse_rat @ B ) )
% 8.58/4.86 = ( A = B ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7121_inverse__inverse__eq,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7122_inverse__inverse__eq,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7123_inverse__inverse__eq,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7124_and_Oidem,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ A @ A )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7125_and_Oidem,axiom,
% 8.58/4.86 ! [A: nat] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ A @ A )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7126_and_Oleft__idem,axiom,
% 8.58/4.86 ! [A: int,B: int] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 8.58/4.86 = ( bit_se725231765392027082nd_int @ A @ B ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7127_and_Oleft__idem,axiom,
% 8.58/4.86 ! [A: nat,B: nat] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 8.58/4.86 = ( bit_se727722235901077358nd_nat @ A @ B ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7128_and_Oright__idem,axiom,
% 8.58/4.86 ! [A: int,B: int] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ B )
% 8.58/4.86 = ( bit_se725231765392027082nd_int @ A @ B ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7129_and_Oright__idem,axiom,
% 8.58/4.86 ! [A: nat,B: nat] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ B )
% 8.58/4.86 = ( bit_se727722235901077358nd_nat @ A @ B ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7130_inverse__nonzero__iff__nonzero,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( ( inverse_inverse_real @ A )
% 8.58/4.86 = zero_zero_real )
% 8.58/4.86 = ( A = zero_zero_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7131_inverse__nonzero__iff__nonzero,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( ( invers8013647133539491842omplex @ A )
% 8.58/4.86 = zero_zero_complex )
% 8.58/4.86 = ( A = zero_zero_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7132_inverse__nonzero__iff__nonzero,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( ( inverse_inverse_rat @ A )
% 8.58/4.86 = zero_zero_rat )
% 8.58/4.86 = ( A = zero_zero_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7133_inverse__zero,axiom,
% 8.58/4.86 ( ( inverse_inverse_real @ zero_zero_real )
% 8.58/4.86 = zero_zero_real ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7134_inverse__zero,axiom,
% 8.58/4.86 ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 8.58/4.86 = zero_zero_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7135_inverse__zero,axiom,
% 8.58/4.86 ( ( inverse_inverse_rat @ zero_zero_rat )
% 8.58/4.86 = zero_zero_rat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7136_inverse__mult__distrib,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
% 8.58/4.86 = ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7137_inverse__mult__distrib,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
% 8.58/4.86 = ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7138_inverse__mult__distrib,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
% 8.58/4.86 = ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7139_inverse__eq__1__iff,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ( inverse_inverse_real @ X )
% 8.58/4.86 = one_one_real )
% 8.58/4.86 = ( X = one_one_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7140_inverse__eq__1__iff,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( ( invers8013647133539491842omplex @ X )
% 8.58/4.86 = one_one_complex )
% 8.58/4.86 = ( X = one_one_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7141_inverse__eq__1__iff,axiom,
% 8.58/4.86 ! [X: rat] :
% 8.58/4.86 ( ( ( inverse_inverse_rat @ X )
% 8.58/4.86 = one_one_rat )
% 8.58/4.86 = ( X = one_one_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7142_inverse__1,axiom,
% 8.58/4.86 ( ( inverse_inverse_real @ one_one_real )
% 8.58/4.86 = one_one_real ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7143_inverse__1,axiom,
% 8.58/4.86 ( ( invers8013647133539491842omplex @ one_one_complex )
% 8.58/4.86 = one_one_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7144_inverse__1,axiom,
% 8.58/4.86 ( ( inverse_inverse_rat @ one_one_rat )
% 8.58/4.86 = one_one_rat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7145_bit_Oconj__zero__right,axiom,
% 8.58/4.86 ! [X: int] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
% 8.58/4.86 = zero_zero_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7146_bit_Oconj__zero__left,axiom,
% 8.58/4.86 ! [X: int] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
% 8.58/4.86 = zero_zero_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7147_zero__and__eq,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
% 8.58/4.86 = zero_zero_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7148_zero__and__eq,axiom,
% 8.58/4.86 ! [A: nat] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
% 8.58/4.86 = zero_zero_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7149_and__zero__eq,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
% 8.58/4.86 = zero_zero_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7150_and__zero__eq,axiom,
% 8.58/4.86 ! [A: nat] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
% 8.58/4.86 = zero_zero_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7151_inverse__minus__eq,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 8.58/4.86 = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7152_inverse__minus__eq,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
% 8.58/4.86 = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7153_inverse__minus__eq,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
% 8.58/4.86 = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7154_inverse__divide,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
% 8.58/4.86 = ( divide_divide_real @ B @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7155_inverse__divide,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.58/4.86 = ( divide1717551699836669952omplex @ B @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7156_inverse__divide,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B ) )
% 8.58/4.86 = ( divide_divide_rat @ B @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7157_abs__inverse,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
% 8.58/4.86 = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7158_abs__inverse,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( abs_abs_complex @ ( invers8013647133539491842omplex @ A ) )
% 8.58/4.86 = ( invers8013647133539491842omplex @ ( abs_abs_complex @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7159_abs__inverse,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
% 8.58/4.86 = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7160_sgn__inverse,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( sgn_sgn_real @ ( inverse_inverse_real @ A ) )
% 8.58/4.86 = ( inverse_inverse_real @ ( sgn_sgn_real @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7161_sgn__inverse,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( sgn_sgn_complex @ ( invers8013647133539491842omplex @ A ) )
% 8.58/4.86 = ( invers8013647133539491842omplex @ ( sgn_sgn_complex @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7162_sgn__inverse,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( sgn_sgn_rat @ ( inverse_inverse_rat @ A ) )
% 8.58/4.86 = ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7163_inverse__sgn,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( inverse_inverse_real @ ( sgn_sgn_real @ A ) )
% 8.58/4.86 = ( sgn_sgn_real @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7164_inverse__sgn,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) )
% 8.58/4.86 = ( sgn_sgn_rat @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7165_take__bit__and,axiom,
% 8.58/4.86 ! [N: nat,A: int,B: int] :
% 8.58/4.86 ( ( bit_se2923211474154528505it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 8.58/4.86 = ( bit_se725231765392027082nd_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7166_take__bit__and,axiom,
% 8.58/4.86 ! [N: nat,A: nat,B: nat] :
% 8.58/4.86 ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 8.58/4.86 = ( bit_se727722235901077358nd_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7167_push__bit__and,axiom,
% 8.58/4.86 ! [N: nat,A: nat,B: nat] :
% 8.58/4.86 ( ( bit_se547839408752420682it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 8.58/4.86 = ( bit_se727722235901077358nd_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7168_push__bit__and,axiom,
% 8.58/4.86 ! [N: nat,A: int,B: int] :
% 8.58/4.86 ( ( bit_se545348938243370406it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 8.58/4.86 = ( bit_se725231765392027082nd_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7169_push__bit__and,axiom,
% 8.58/4.86 ! [N: nat,A: code_integer,B: code_integer] :
% 8.58/4.86 ( ( bit_se7788150548672797655nteger @ N @ ( bit_se3949692690581998587nteger @ A @ B ) )
% 8.58/4.86 = ( bit_se3949692690581998587nteger @ ( bit_se7788150548672797655nteger @ N @ A ) @ ( bit_se7788150548672797655nteger @ N @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7170_push__bit__and,axiom,
% 8.58/4.86 ! [N: nat,A: code_natural,B: code_natural] :
% 8.58/4.86 ( ( bit_se6611745700429515170atural @ N @ ( bit_se2773287842338716102atural @ A @ B ) )
% 8.58/4.86 = ( bit_se2773287842338716102atural @ ( bit_se6611745700429515170atural @ N @ A ) @ ( bit_se6611745700429515170atural @ N @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7171_inverse__nonnegative__iff__nonnegative,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 8.58/4.86 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7172_inverse__nonnegative__iff__nonnegative,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 8.58/4.86 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7173_inverse__nonpositive__iff__nonpositive,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 8.58/4.86 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7174_inverse__nonpositive__iff__nonpositive,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 8.58/4.86 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7175_inverse__less__iff__less,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.86 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.58/4.86 => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 8.58/4.86 = ( ord_less_real @ B @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7176_inverse__less__iff__less,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.58/4.86 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 8.58/4.86 => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 8.58/4.86 = ( ord_less_rat @ B @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7177_inverse__less__iff__less__neg,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_real @ A @ zero_zero_real )
% 8.58/4.86 => ( ( ord_less_real @ B @ zero_zero_real )
% 8.58/4.86 => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 8.58/4.86 = ( ord_less_real @ B @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7178_inverse__less__iff__less__neg,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.58/4.86 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 8.58/4.86 => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 8.58/4.86 = ( ord_less_rat @ B @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7179_inverse__negative__iff__negative,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 8.58/4.86 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7180_inverse__negative__iff__negative,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 8.58/4.86 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7181_inverse__positive__iff__positive,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 8.58/4.86 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7182_inverse__positive__iff__positive,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 8.58/4.86 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7183_and_Oleft__neutral,axiom,
% 8.58/4.86 ! [A: code_integer] :
% 8.58/4.86 ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7184_and_Oleft__neutral,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7185_and_Oright__neutral,axiom,
% 8.58/4.86 ! [A: code_integer] :
% 8.58/4.86 ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7186_and_Oright__neutral,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7187_bit_Oconj__one__right,axiom,
% 8.58/4.86 ! [X: code_integer] :
% 8.58/4.86 ( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.86 = X ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7188_bit_Oconj__one__right,axiom,
% 8.58/4.86 ! [X: int] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.86 = X ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7189_and__nonnegative__int__iff,axiom,
% 8.58/4.86 ! [K: int,L: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 8.58/4.86 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.58/4.86 | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7190_and__negative__int__iff,axiom,
% 8.58/4.86 ! [K: int,L: int] :
% 8.58/4.86 ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
% 8.58/4.86 = ( ( ord_less_int @ K @ zero_zero_int )
% 8.58/4.86 & ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7191_norm__ii,axiom,
% 8.58/4.86 ( ( real_V1022390504157884413omplex @ imaginary_unit )
% 8.58/4.86 = one_one_real ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7192_norm__cis,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
% 8.58/4.86 = one_one_real ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7193_cis__zero,axiom,
% 8.58/4.86 ( ( cis @ zero_zero_real )
% 8.58/4.86 = one_one_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7194_divide__i,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( divide1717551699836669952omplex @ X @ imaginary_unit )
% 8.58/4.86 = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7195_complex__i__mult__minus,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X ) )
% 8.58/4.86 = ( uminus1482373934393186551omplex @ X ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7196_inverse__le__iff__le,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.86 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.58/4.86 => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 8.58/4.86 = ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7197_inverse__le__iff__le,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.58/4.86 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 8.58/4.86 => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 8.58/4.86 = ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7198_inverse__le__iff__le__neg,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_real @ A @ zero_zero_real )
% 8.58/4.86 => ( ( ord_less_real @ B @ zero_zero_real )
% 8.58/4.86 => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 8.58/4.86 = ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7199_inverse__le__iff__le__neg,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.58/4.86 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 8.58/4.86 => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 8.58/4.86 = ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7200_right__inverse,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( A != zero_zero_real )
% 8.58/4.86 => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
% 8.58/4.86 = one_one_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7201_right__inverse,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( A != zero_zero_complex )
% 8.58/4.86 => ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
% 8.58/4.86 = one_one_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7202_right__inverse,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( A != zero_zero_rat )
% 8.58/4.86 => ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
% 8.58/4.86 = one_one_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7203_left__inverse,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( A != zero_zero_real )
% 8.58/4.86 => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 8.58/4.86 = one_one_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7204_left__inverse,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( A != zero_zero_complex )
% 8.58/4.86 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 8.58/4.86 = one_one_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7205_left__inverse,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( A != zero_zero_rat )
% 8.58/4.86 => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 8.58/4.86 = one_one_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7206_inverse__eq__divide__numeral,axiom,
% 8.58/4.86 ! [W: num] :
% 8.58/4.86 ( ( inverse_inverse_real @ ( numeral_numeral_real @ W ) )
% 8.58/4.86 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7207_inverse__eq__divide__numeral,axiom,
% 8.58/4.86 ! [W: num] :
% 8.58/4.86 ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.58/4.86 = ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7208_inverse__eq__divide__numeral,axiom,
% 8.58/4.86 ! [W: num] :
% 8.58/4.86 ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W ) )
% 8.58/4.86 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7209_and__numerals_I2_J,axiom,
% 8.58/4.86 ! [Y: num] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 8.58/4.86 = one_one_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7210_and__numerals_I2_J,axiom,
% 8.58/4.86 ! [Y: num] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 8.58/4.86 = one_one_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7211_and__numerals_I8_J,axiom,
% 8.58/4.86 ! [X: num] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
% 8.58/4.86 = one_one_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7212_and__numerals_I8_J,axiom,
% 8.58/4.86 ! [X: num] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
% 8.58/4.86 = one_one_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7213_cis__pi,axiom,
% 8.58/4.86 ( ( cis @ pi )
% 8.58/4.86 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7214_i__squared,axiom,
% 8.58/4.86 ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
% 8.58/4.86 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7215_divide__numeral__i,axiom,
% 8.58/4.86 ! [Z2: complex,N: num] :
% 8.58/4.86 ( ( divide1717551699836669952omplex @ Z2 @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
% 8.58/4.86 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z2 ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7216_and__numerals_I5_J,axiom,
% 8.58/4.86 ! [X: num] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 8.58/4.86 = zero_zero_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7217_and__numerals_I5_J,axiom,
% 8.58/4.86 ! [X: num] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 8.58/4.86 = zero_zero_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7218_and__numerals_I1_J,axiom,
% 8.58/4.86 ! [Y: num] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 8.58/4.86 = zero_zero_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7219_and__numerals_I1_J,axiom,
% 8.58/4.86 ! [Y: num] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 8.58/4.86 = zero_zero_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7220_inverse__eq__divide__neg__numeral,axiom,
% 8.58/4.86 ! [W: num] :
% 8.58/4.86 ( ( inverse_inverse_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.58/4.86 = ( divide_divide_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7221_inverse__eq__divide__neg__numeral,axiom,
% 8.58/4.86 ! [W: num] :
% 8.58/4.86 ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 8.58/4.86 = ( divide1717551699836669952omplex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7222_inverse__eq__divide__neg__numeral,axiom,
% 8.58/4.86 ! [W: num] :
% 8.58/4.86 ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 8.58/4.86 = ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7223_and__numerals_I3_J,axiom,
% 8.58/4.86 ! [X: num,Y: num] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 8.58/4.86 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7224_and__numerals_I3_J,axiom,
% 8.58/4.86 ! [X: num,Y: num] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 8.58/4.86 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7225_and__minus__numerals_I2_J,axiom,
% 8.58/4.86 ! [N: num] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.58/4.86 = one_one_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7226_and__minus__numerals_I6_J,axiom,
% 8.58/4.86 ! [N: num] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 8.58/4.86 = one_one_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7227_and__numerals_I6_J,axiom,
% 8.58/4.86 ! [X: num,Y: num] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 8.58/4.86 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7228_and__numerals_I6_J,axiom,
% 8.58/4.86 ! [X: num,Y: num] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 8.58/4.86 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7229_and__numerals_I4_J,axiom,
% 8.58/4.86 ! [X: num,Y: num] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 8.58/4.86 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7230_and__numerals_I4_J,axiom,
% 8.58/4.86 ! [X: num,Y: num] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 8.58/4.86 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7231_and__minus__numerals_I5_J,axiom,
% 8.58/4.86 ! [N: num] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 8.58/4.86 = zero_zero_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7232_and__minus__numerals_I1_J,axiom,
% 8.58/4.86 ! [N: num] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.58/4.86 = zero_zero_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7233_power2__i,axiom,
% 8.58/4.86 ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.86 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7234_cis__pi__half,axiom,
% 8.58/4.86 ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.86 = imaginary_unit ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7235_i__even__power,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.86 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7236_and__numerals_I7_J,axiom,
% 8.58/4.86 ! [X: num,Y: num] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 8.58/4.86 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7237_and__numerals_I7_J,axiom,
% 8.58/4.86 ! [X: num,Y: num] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 8.58/4.86 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7238_exp__pi__i,axiom,
% 8.58/4.86 ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
% 8.58/4.86 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7239_exp__pi__i_H,axiom,
% 8.58/4.86 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
% 8.58/4.86 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7240_cis__minus__pi__half,axiom,
% 8.58/4.86 ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 8.58/4.86 = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7241_bit__and__iff,axiom,
% 8.58/4.86 ! [A: int,B: int,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ N )
% 8.58/4.86 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 8.58/4.86 & ( bit_se1146084159140164899it_int @ B @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7242_bit__and__iff,axiom,
% 8.58/4.86 ! [A: nat,B: nat,N: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ N )
% 8.58/4.86 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 8.58/4.86 & ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7243_of__int__and__eq,axiom,
% 8.58/4.86 ! [K: int,L: int] :
% 8.58/4.86 ( ( ring_1_of_int_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 8.58/4.86 = ( bit_se725231765392027082nd_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7244_field__class_Ofield__inverse__zero,axiom,
% 8.58/4.86 ( ( inverse_inverse_real @ zero_zero_real )
% 8.58/4.86 = zero_zero_real ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7245_field__class_Ofield__inverse__zero,axiom,
% 8.58/4.86 ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 8.58/4.86 = zero_zero_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7246_field__class_Ofield__inverse__zero,axiom,
% 8.58/4.86 ( ( inverse_inverse_rat @ zero_zero_rat )
% 8.58/4.86 = zero_zero_rat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7247_inverse__zero__imp__zero,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( ( inverse_inverse_real @ A )
% 8.58/4.86 = zero_zero_real )
% 8.58/4.86 => ( A = zero_zero_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7248_inverse__zero__imp__zero,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( ( invers8013647133539491842omplex @ A )
% 8.58/4.86 = zero_zero_complex )
% 8.58/4.86 => ( A = zero_zero_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7249_inverse__zero__imp__zero,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( ( inverse_inverse_rat @ A )
% 8.58/4.86 = zero_zero_rat )
% 8.58/4.86 => ( A = zero_zero_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7250_nonzero__inverse__eq__imp__eq,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ( inverse_inverse_real @ A )
% 8.58/4.86 = ( inverse_inverse_real @ B ) )
% 8.58/4.86 => ( ( A != zero_zero_real )
% 8.58/4.86 => ( ( B != zero_zero_real )
% 8.58/4.86 => ( A = B ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7251_nonzero__inverse__eq__imp__eq,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( ( invers8013647133539491842omplex @ A )
% 8.58/4.86 = ( invers8013647133539491842omplex @ B ) )
% 8.58/4.86 => ( ( A != zero_zero_complex )
% 8.58/4.86 => ( ( B != zero_zero_complex )
% 8.58/4.86 => ( A = B ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7252_nonzero__inverse__eq__imp__eq,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ( inverse_inverse_rat @ A )
% 8.58/4.86 = ( inverse_inverse_rat @ B ) )
% 8.58/4.86 => ( ( A != zero_zero_rat )
% 8.58/4.86 => ( ( B != zero_zero_rat )
% 8.58/4.86 => ( A = B ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7253_nonzero__inverse__inverse__eq,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( A != zero_zero_real )
% 8.58/4.86 => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 8.58/4.86 = A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7254_nonzero__inverse__inverse__eq,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( A != zero_zero_complex )
% 8.58/4.86 => ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
% 8.58/4.86 = A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7255_nonzero__inverse__inverse__eq,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( A != zero_zero_rat )
% 8.58/4.86 => ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
% 8.58/4.86 = A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7256_nonzero__imp__inverse__nonzero,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( A != zero_zero_real )
% 8.58/4.86 => ( ( inverse_inverse_real @ A )
% 8.58/4.86 != zero_zero_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7257_nonzero__imp__inverse__nonzero,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( A != zero_zero_complex )
% 8.58/4.86 => ( ( invers8013647133539491842omplex @ A )
% 8.58/4.86 != zero_zero_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7258_nonzero__imp__inverse__nonzero,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( A != zero_zero_rat )
% 8.58/4.86 => ( ( inverse_inverse_rat @ A )
% 8.58/4.86 != zero_zero_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7259_mult__commute__imp__mult__inverse__commute,axiom,
% 8.58/4.86 ! [Y: real,X: real] :
% 8.58/4.86 ( ( ( times_times_real @ Y @ X )
% 8.58/4.86 = ( times_times_real @ X @ Y ) )
% 8.58/4.86 => ( ( times_times_real @ ( inverse_inverse_real @ Y ) @ X )
% 8.58/4.86 = ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7260_mult__commute__imp__mult__inverse__commute,axiom,
% 8.58/4.86 ! [Y: complex,X: complex] :
% 8.58/4.86 ( ( ( times_times_complex @ Y @ X )
% 8.58/4.86 = ( times_times_complex @ X @ Y ) )
% 8.58/4.86 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ Y ) @ X )
% 8.58/4.86 = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7261_mult__commute__imp__mult__inverse__commute,axiom,
% 8.58/4.86 ! [Y: rat,X: rat] :
% 8.58/4.86 ( ( ( times_times_rat @ Y @ X )
% 8.58/4.86 = ( times_times_rat @ X @ Y ) )
% 8.58/4.86 => ( ( times_times_rat @ ( inverse_inverse_rat @ Y ) @ X )
% 8.58/4.86 = ( times_times_rat @ X @ ( inverse_inverse_rat @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7262_of__nat__and__eq,axiom,
% 8.58/4.86 ! [M: nat,N: nat] :
% 8.58/4.86 ( ( semiri1314217659103216013at_int @ ( bit_se727722235901077358nd_nat @ M @ N ) )
% 8.58/4.86 = ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7263_of__nat__and__eq,axiom,
% 8.58/4.86 ! [M: nat,N: nat] :
% 8.58/4.86 ( ( semiri1316708129612266289at_nat @ ( bit_se727722235901077358nd_nat @ M @ N ) )
% 8.58/4.86 = ( bit_se727722235901077358nd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7264_inverse__eq__imp__eq,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ( inverse_inverse_real @ A )
% 8.58/4.86 = ( inverse_inverse_real @ B ) )
% 8.58/4.86 => ( A = B ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7265_inverse__eq__imp__eq,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( ( invers8013647133539491842omplex @ A )
% 8.58/4.86 = ( invers8013647133539491842omplex @ B ) )
% 8.58/4.86 => ( A = B ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7266_inverse__eq__imp__eq,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ( inverse_inverse_rat @ A )
% 8.58/4.86 = ( inverse_inverse_rat @ B ) )
% 8.58/4.86 => ( A = B ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7267_and_Oassoc,axiom,
% 8.58/4.86 ! [A: int,B: int,C: int] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ C )
% 8.58/4.86 = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7268_and_Oassoc,axiom,
% 8.58/4.86 ! [A: nat,B: nat,C: nat] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ C )
% 8.58/4.86 = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7269_and_Ocommute,axiom,
% 8.58/4.86 ( bit_se725231765392027082nd_int
% 8.58/4.86 = ( ^ [A2: int,B2: int] : ( bit_se725231765392027082nd_int @ B2 @ A2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7270_and_Ocommute,axiom,
% 8.58/4.86 ( bit_se727722235901077358nd_nat
% 8.58/4.86 = ( ^ [A2: nat,B2: nat] : ( bit_se727722235901077358nd_nat @ B2 @ A2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7271_and_Oleft__commute,axiom,
% 8.58/4.86 ! [B: int,A: int,C: int] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ B @ ( bit_se725231765392027082nd_int @ A @ C ) )
% 8.58/4.86 = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7272_and_Oleft__commute,axiom,
% 8.58/4.86 ! [B: nat,A: nat,C: nat] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ B @ ( bit_se727722235901077358nd_nat @ A @ C ) )
% 8.58/4.86 = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7273_nonzero__norm__inverse,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( A != zero_zero_real )
% 8.58/4.86 => ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
% 8.58/4.86 = ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7274_nonzero__norm__inverse,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( A != zero_zero_complex )
% 8.58/4.86 => ( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
% 8.58/4.86 = ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7275_power__inverse,axiom,
% 8.58/4.86 ! [A: real,N: nat] :
% 8.58/4.86 ( ( power_power_real @ ( inverse_inverse_real @ A ) @ N )
% 8.58/4.86 = ( inverse_inverse_real @ ( power_power_real @ A @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7276_power__inverse,axiom,
% 8.58/4.86 ! [A: complex,N: nat] :
% 8.58/4.86 ( ( power_power_complex @ ( invers8013647133539491842omplex @ A ) @ N )
% 8.58/4.86 = ( invers8013647133539491842omplex @ ( power_power_complex @ A @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7277_power__inverse,axiom,
% 8.58/4.86 ! [A: rat,N: nat] :
% 8.58/4.86 ( ( power_power_rat @ ( inverse_inverse_rat @ A ) @ N )
% 8.58/4.86 = ( inverse_inverse_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7278_real__sqrt__inverse,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( sqrt @ ( inverse_inverse_real @ X ) )
% 8.58/4.86 = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7279_bit__and__int__iff,axiom,
% 8.58/4.86 ! [K: int,L: int,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ N )
% 8.58/4.86 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 8.58/4.86 & ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7280_cis__conv__exp,axiom,
% 8.58/4.86 ( cis
% 8.58/4.86 = ( ^ [B2: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B2 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7281_cis__divide,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
% 8.58/4.86 = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7282_complex__i__not__one,axiom,
% 8.58/4.86 imaginary_unit != one_one_complex ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7283_and__eq__minus__1__iff,axiom,
% 8.58/4.86 ! [A: code_integer,B: code_integer] :
% 8.58/4.86 ( ( ( bit_se3949692690581998587nteger @ A @ B )
% 8.58/4.86 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.86 = ( ( A
% 8.58/4.86 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.86 & ( B
% 8.58/4.86 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7284_and__eq__minus__1__iff,axiom,
% 8.58/4.86 ! [A: int,B: int] :
% 8.58/4.86 ( ( ( bit_se725231765392027082nd_int @ A @ B )
% 8.58/4.86 = ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.86 = ( ( A
% 8.58/4.86 = ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.86 & ( B
% 8.58/4.86 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7285_positive__imp__inverse__positive,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.86 => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7286_positive__imp__inverse__positive,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.58/4.86 => ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7287_negative__imp__inverse__negative,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( ord_less_real @ A @ zero_zero_real )
% 8.58/4.86 => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7288_negative__imp__inverse__negative,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.58/4.86 => ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7289_inverse__positive__imp__positive,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 8.58/4.86 => ( ( A != zero_zero_real )
% 8.58/4.86 => ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7290_inverse__positive__imp__positive,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 8.58/4.86 => ( ( A != zero_zero_rat )
% 8.58/4.86 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7291_inverse__negative__imp__negative,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 8.58/4.86 => ( ( A != zero_zero_real )
% 8.58/4.86 => ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7292_inverse__negative__imp__negative,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 8.58/4.86 => ( ( A != zero_zero_rat )
% 8.58/4.86 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7293_less__imp__inverse__less__neg,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_real @ A @ B )
% 8.58/4.86 => ( ( ord_less_real @ B @ zero_zero_real )
% 8.58/4.86 => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7294_less__imp__inverse__less__neg,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_rat @ A @ B )
% 8.58/4.86 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 8.58/4.86 => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7295_inverse__less__imp__less__neg,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 8.58/4.86 => ( ( ord_less_real @ B @ zero_zero_real )
% 8.58/4.86 => ( ord_less_real @ B @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7296_inverse__less__imp__less__neg,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 8.58/4.86 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 8.58/4.86 => ( ord_less_rat @ B @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7297_less__imp__inverse__less,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_real @ A @ B )
% 8.58/4.86 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.86 => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7298_less__imp__inverse__less,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_rat @ A @ B )
% 8.58/4.86 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.58/4.86 => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7299_inverse__less__imp__less,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 8.58/4.86 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.86 => ( ord_less_real @ B @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7300_inverse__less__imp__less,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 8.58/4.86 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.58/4.86 => ( ord_less_rat @ B @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7301_nonzero__inverse__mult__distrib,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( A != zero_zero_real )
% 8.58/4.86 => ( ( B != zero_zero_real )
% 8.58/4.86 => ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
% 8.58/4.86 = ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7302_nonzero__inverse__mult__distrib,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( A != zero_zero_complex )
% 8.58/4.86 => ( ( B != zero_zero_complex )
% 8.58/4.86 => ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
% 8.58/4.86 = ( times_times_complex @ ( invers8013647133539491842omplex @ B ) @ ( invers8013647133539491842omplex @ A ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7303_nonzero__inverse__mult__distrib,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( A != zero_zero_rat )
% 8.58/4.86 => ( ( B != zero_zero_rat )
% 8.58/4.86 => ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
% 8.58/4.86 = ( times_times_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7304_not__bit__Suc__0__Suc,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7305_bit__Suc__0__iff,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.58/4.86 = ( N = zero_zero_nat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7306_nonzero__inverse__minus__eq,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( A != zero_zero_real )
% 8.58/4.86 => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 8.58/4.86 = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7307_nonzero__inverse__minus__eq,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( A != zero_zero_complex )
% 8.58/4.86 => ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
% 8.58/4.86 = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7308_nonzero__inverse__minus__eq,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( A != zero_zero_rat )
% 8.58/4.86 => ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
% 8.58/4.86 = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7309_inverse__unique,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ( times_times_real @ A @ B )
% 8.58/4.86 = one_one_real )
% 8.58/4.86 => ( ( inverse_inverse_real @ A )
% 8.58/4.86 = B ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7310_inverse__unique,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( ( times_times_complex @ A @ B )
% 8.58/4.86 = one_one_complex )
% 8.58/4.86 => ( ( invers8013647133539491842omplex @ A )
% 8.58/4.86 = B ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7311_inverse__unique,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ( times_times_rat @ A @ B )
% 8.58/4.86 = one_one_rat )
% 8.58/4.86 => ( ( inverse_inverse_rat @ A )
% 8.58/4.86 = B ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7312_inverse__numeral__1,axiom,
% 8.58/4.86 ( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
% 8.58/4.86 = ( numeral_numeral_real @ one ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7313_inverse__numeral__1,axiom,
% 8.58/4.86 ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ one ) )
% 8.58/4.86 = ( numera6690914467698888265omplex @ one ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7314_inverse__numeral__1,axiom,
% 8.58/4.86 ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ one ) )
% 8.58/4.86 = ( numeral_numeral_rat @ one ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7315_divide__inverse__commute,axiom,
% 8.58/4.86 ( divide_divide_real
% 8.58/4.86 = ( ^ [A2: real,B2: real] : ( times_times_real @ ( inverse_inverse_real @ B2 ) @ A2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7316_divide__inverse__commute,axiom,
% 8.58/4.86 ( divide1717551699836669952omplex
% 8.58/4.86 = ( ^ [A2: complex,B2: complex] : ( times_times_complex @ ( invers8013647133539491842omplex @ B2 ) @ A2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7317_divide__inverse__commute,axiom,
% 8.58/4.86 ( divide_divide_rat
% 8.58/4.86 = ( ^ [A2: rat,B2: rat] : ( times_times_rat @ ( inverse_inverse_rat @ B2 ) @ A2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7318_divide__inverse,axiom,
% 8.58/4.86 ( divide_divide_real
% 8.58/4.86 = ( ^ [A2: real,B2: real] : ( times_times_real @ A2 @ ( inverse_inverse_real @ B2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7319_divide__inverse,axiom,
% 8.58/4.86 ( divide1717551699836669952omplex
% 8.58/4.86 = ( ^ [A2: complex,B2: complex] : ( times_times_complex @ A2 @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7320_divide__inverse,axiom,
% 8.58/4.86 ( divide_divide_rat
% 8.58/4.86 = ( ^ [A2: rat,B2: rat] : ( times_times_rat @ A2 @ ( inverse_inverse_rat @ B2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7321_field__class_Ofield__divide__inverse,axiom,
% 8.58/4.86 ( divide_divide_real
% 8.58/4.86 = ( ^ [A2: real,B2: real] : ( times_times_real @ A2 @ ( inverse_inverse_real @ B2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7322_field__class_Ofield__divide__inverse,axiom,
% 8.58/4.86 ( divide1717551699836669952omplex
% 8.58/4.86 = ( ^ [A2: complex,B2: complex] : ( times_times_complex @ A2 @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7323_field__class_Ofield__divide__inverse,axiom,
% 8.58/4.86 ( divide_divide_rat
% 8.58/4.86 = ( ^ [A2: rat,B2: rat] : ( times_times_rat @ A2 @ ( inverse_inverse_rat @ B2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7324_inverse__eq__divide,axiom,
% 8.58/4.86 ( inverse_inverse_real
% 8.58/4.86 = ( divide_divide_real @ one_one_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7325_inverse__eq__divide,axiom,
% 8.58/4.86 ( invers8013647133539491842omplex
% 8.58/4.86 = ( divide1717551699836669952omplex @ one_one_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7326_inverse__eq__divide,axiom,
% 8.58/4.86 ( inverse_inverse_rat
% 8.58/4.86 = ( divide_divide_rat @ one_one_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7327_power__mult__inverse__distrib,axiom,
% 8.58/4.86 ! [X: real,M: nat] :
% 8.58/4.86 ( ( times_times_real @ ( power_power_real @ X @ M ) @ ( inverse_inverse_real @ X ) )
% 8.58/4.86 = ( times_times_real @ ( inverse_inverse_real @ X ) @ ( power_power_real @ X @ M ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7328_power__mult__inverse__distrib,axiom,
% 8.58/4.86 ! [X: complex,M: nat] :
% 8.58/4.86 ( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( invers8013647133539491842omplex @ X ) )
% 8.58/4.86 = ( times_times_complex @ ( invers8013647133539491842omplex @ X ) @ ( power_power_complex @ X @ M ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7329_power__mult__inverse__distrib,axiom,
% 8.58/4.86 ! [X: rat,M: nat] :
% 8.58/4.86 ( ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( inverse_inverse_rat @ X ) )
% 8.58/4.86 = ( times_times_rat @ ( inverse_inverse_rat @ X ) @ ( power_power_rat @ X @ M ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7330_power__mult__power__inverse__commute,axiom,
% 8.58/4.86 ! [X: real,M: nat,N: nat] :
% 8.58/4.86 ( ( times_times_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) )
% 8.58/4.86 = ( times_times_real @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) @ ( power_power_real @ X @ M ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7331_power__mult__power__inverse__commute,axiom,
% 8.58/4.86 ! [X: complex,M: nat,N: nat] :
% 8.58/4.86 ( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) )
% 8.58/4.86 = ( times_times_complex @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) @ ( power_power_complex @ X @ M ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7332_power__mult__power__inverse__commute,axiom,
% 8.58/4.86 ! [X: rat,M: nat,N: nat] :
% 8.58/4.86 ( ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) )
% 8.58/4.86 = ( times_times_rat @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) @ ( power_power_rat @ X @ M ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7333_mult__inverse__of__nat__commute,axiom,
% 8.58/4.86 ! [Xa: nat,X: real] :
% 8.58/4.86 ( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) @ X )
% 8.58/4.86 = ( times_times_real @ X @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7334_mult__inverse__of__nat__commute,axiom,
% 8.58/4.86 ! [Xa: nat,X: complex] :
% 8.58/4.86 ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) @ X )
% 8.58/4.86 = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7335_mult__inverse__of__nat__commute,axiom,
% 8.58/4.86 ! [Xa: nat,X: rat] :
% 8.58/4.86 ( ( times_times_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) @ X )
% 8.58/4.86 = ( times_times_rat @ X @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7336_nonzero__abs__inverse,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( A != zero_zero_real )
% 8.58/4.86 => ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
% 8.58/4.86 = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7337_nonzero__abs__inverse,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( A != zero_zero_rat )
% 8.58/4.86 => ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
% 8.58/4.86 = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7338_mult__inverse__of__int__commute,axiom,
% 8.58/4.86 ! [Xa: int,X: real] :
% 8.58/4.86 ( ( times_times_real @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) @ X )
% 8.58/4.86 = ( times_times_real @ X @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7339_mult__inverse__of__int__commute,axiom,
% 8.58/4.86 ! [Xa: int,X: complex] :
% 8.58/4.86 ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) @ X )
% 8.58/4.86 = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7340_mult__inverse__of__int__commute,axiom,
% 8.58/4.86 ! [Xa: int,X: rat] :
% 8.58/4.86 ( ( times_times_rat @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) @ X )
% 8.58/4.86 = ( times_times_rat @ X @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7341_AND__lower,axiom,
% 8.58/4.86 ! [X: int,Y: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.58/4.86 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7342_AND__upper1,axiom,
% 8.58/4.86 ! [X: int,Y: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.58/4.86 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7343_AND__upper2,axiom,
% 8.58/4.86 ! [Y: int,X: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.58/4.86 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7344_AND__upper1_H,axiom,
% 8.58/4.86 ! [Y: int,Z2: int,Ya: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.58/4.86 => ( ( ord_less_eq_int @ Y @ Z2 )
% 8.58/4.86 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7345_AND__upper2_H,axiom,
% 8.58/4.86 ! [Y: int,Z2: int,X: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.58/4.86 => ( ( ord_less_eq_int @ Y @ Z2 )
% 8.58/4.86 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7346_divide__real__def,axiom,
% 8.58/4.86 ( divide_divide_real
% 8.58/4.86 = ( ^ [X3: real,Y6: real] : ( times_times_real @ X3 @ ( inverse_inverse_real @ Y6 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7347_take__bit__eq__mask,axiom,
% 8.58/4.86 ( bit_se2923211474154528505it_int
% 8.58/4.86 = ( ^ [N3: nat,A2: int] : ( bit_se725231765392027082nd_int @ A2 @ ( bit_se2000444600071755411sk_int @ N3 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7348_take__bit__eq__mask,axiom,
% 8.58/4.86 ( bit_se2925701944663578781it_nat
% 8.58/4.86 = ( ^ [N3: nat,A2: nat] : ( bit_se727722235901077358nd_nat @ A2 @ ( bit_se2002935070580805687sk_nat @ N3 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7349_DeMoivre,axiom,
% 8.58/4.86 ! [A: real,N: nat] :
% 8.58/4.86 ( ( power_power_complex @ ( cis @ A ) @ N )
% 8.58/4.86 = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7350_cis__mult,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
% 8.58/4.86 = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7351_i__times__eq__iff,axiom,
% 8.58/4.86 ! [W: complex,Z2: complex] :
% 8.58/4.86 ( ( ( times_times_complex @ imaginary_unit @ W )
% 8.58/4.86 = Z2 )
% 8.58/4.86 = ( W
% 8.58/4.86 = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7352_inverse__le__imp__le,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 8.58/4.86 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.86 => ( ord_less_eq_real @ B @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7353_inverse__le__imp__le,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 8.58/4.86 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.58/4.86 => ( ord_less_eq_rat @ B @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7354_le__imp__inverse__le,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ A @ B )
% 8.58/4.86 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.86 => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7355_le__imp__inverse__le,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ A @ B )
% 8.58/4.86 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.58/4.86 => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7356_inverse__le__imp__le__neg,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 8.58/4.86 => ( ( ord_less_real @ B @ zero_zero_real )
% 8.58/4.86 => ( ord_less_eq_real @ B @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7357_inverse__le__imp__le__neg,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 8.58/4.86 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 8.58/4.86 => ( ord_less_eq_rat @ B @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7358_le__imp__inverse__le__neg,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ A @ B )
% 8.58/4.86 => ( ( ord_less_real @ B @ zero_zero_real )
% 8.58/4.86 => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7359_le__imp__inverse__le__neg,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ A @ B )
% 8.58/4.86 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 8.58/4.86 => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7360_inverse__le__1__iff,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ ( inverse_inverse_real @ X ) @ one_one_real )
% 8.58/4.86 = ( ( ord_less_eq_real @ X @ zero_zero_real )
% 8.58/4.86 | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7361_inverse__le__1__iff,axiom,
% 8.58/4.86 ! [X: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
% 8.58/4.86 = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 8.58/4.86 | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7362_one__less__inverse__iff,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X ) )
% 8.58/4.86 = ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.86 & ( ord_less_real @ X @ one_one_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7363_one__less__inverse__iff,axiom,
% 8.58/4.86 ! [X: rat] :
% 8.58/4.86 ( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
% 8.58/4.86 = ( ( ord_less_rat @ zero_zero_rat @ X )
% 8.58/4.86 & ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7364_one__less__inverse,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.86 => ( ( ord_less_real @ A @ one_one_real )
% 8.58/4.86 => ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7365_one__less__inverse,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.58/4.86 => ( ( ord_less_rat @ A @ one_one_rat )
% 8.58/4.86 => ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7366_field__class_Ofield__inverse,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( A != zero_zero_real )
% 8.58/4.86 => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 8.58/4.86 = one_one_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7367_field__class_Ofield__inverse,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( A != zero_zero_complex )
% 8.58/4.86 => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 8.58/4.86 = one_one_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7368_field__class_Ofield__inverse,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( A != zero_zero_rat )
% 8.58/4.86 => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 8.58/4.86 = one_one_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7369_division__ring__inverse__add,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( A != zero_zero_real )
% 8.58/4.86 => ( ( B != zero_zero_real )
% 8.58/4.86 => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 8.58/4.86 = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7370_division__ring__inverse__add,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( A != zero_zero_complex )
% 8.58/4.86 => ( ( B != zero_zero_complex )
% 8.58/4.86 => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 8.58/4.86 = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( plus_plus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7371_division__ring__inverse__add,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( A != zero_zero_rat )
% 8.58/4.86 => ( ( B != zero_zero_rat )
% 8.58/4.86 => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 8.58/4.86 = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7372_inverse__add,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( A != zero_zero_real )
% 8.58/4.86 => ( ( B != zero_zero_real )
% 8.58/4.86 => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 8.58/4.86 = ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7373_inverse__add,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( A != zero_zero_complex )
% 8.58/4.86 => ( ( B != zero_zero_complex )
% 8.58/4.86 => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 8.58/4.86 = ( times_times_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( invers8013647133539491842omplex @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7374_inverse__add,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( A != zero_zero_rat )
% 8.58/4.86 => ( ( B != zero_zero_rat )
% 8.58/4.86 => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 8.58/4.86 = ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7375_division__ring__inverse__diff,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( A != zero_zero_real )
% 8.58/4.86 => ( ( B != zero_zero_real )
% 8.58/4.86 => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 8.58/4.86 = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7376_division__ring__inverse__diff,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( A != zero_zero_complex )
% 8.58/4.86 => ( ( B != zero_zero_complex )
% 8.58/4.86 => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 8.58/4.86 = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ B @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7377_division__ring__inverse__diff,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( A != zero_zero_rat )
% 8.58/4.86 => ( ( B != zero_zero_rat )
% 8.58/4.86 => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 8.58/4.86 = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ B @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7378_nonzero__inverse__eq__divide,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( A != zero_zero_real )
% 8.58/4.86 => ( ( inverse_inverse_real @ A )
% 8.58/4.86 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7379_nonzero__inverse__eq__divide,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( A != zero_zero_complex )
% 8.58/4.86 => ( ( invers8013647133539491842omplex @ A )
% 8.58/4.86 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7380_nonzero__inverse__eq__divide,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( A != zero_zero_rat )
% 8.58/4.86 => ( ( inverse_inverse_rat @ A )
% 8.58/4.86 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7381_not__bit__Suc__0__numeral,axiom,
% 8.58/4.86 ! [N: num] :
% 8.58/4.86 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7382_and__less__eq,axiom,
% 8.58/4.86 ! [L: int,K: int] :
% 8.58/4.86 ( ( ord_less_int @ L @ zero_zero_int )
% 8.58/4.86 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7383_AND__upper1_H_H,axiom,
% 8.58/4.86 ! [Y: int,Z2: int,Ya: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.58/4.86 => ( ( ord_less_int @ Y @ Z2 )
% 8.58/4.86 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7384_AND__upper2_H_H,axiom,
% 8.58/4.86 ! [Y: int,Z2: int,X: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.58/4.86 => ( ( ord_less_int @ Y @ Z2 )
% 8.58/4.86 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7385_bit__push__bit__iff__nat,axiom,
% 8.58/4.86 ! [M: nat,Q2: nat,N: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
% 8.58/4.86 = ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.86 & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7386_imaginary__unit_Ocode,axiom,
% 8.58/4.86 ( imaginary_unit
% 8.58/4.86 = ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7387_Complex__eq__i,axiom,
% 8.58/4.86 ! [X: real,Y: real] :
% 8.58/4.86 ( ( ( complex2 @ X @ Y )
% 8.58/4.86 = imaginary_unit )
% 8.58/4.86 = ( ( X = zero_zero_real )
% 8.58/4.86 & ( Y = one_one_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7388_even__and__iff,axiom,
% 8.58/4.86 ! [A: code_integer,B: code_integer] :
% 8.58/4.86 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ A @ B ) )
% 8.58/4.86 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.58/4.86 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7389_even__and__iff,axiom,
% 8.58/4.86 ! [A: int,B: int] :
% 8.58/4.86 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 8.58/4.86 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.58/4.86 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7390_even__and__iff,axiom,
% 8.58/4.86 ! [A: nat,B: nat] :
% 8.58/4.86 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 8.58/4.86 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.58/4.86 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7391_inverse__less__iff,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 8.58/4.86 = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 8.58/4.86 => ( ord_less_real @ B @ A ) )
% 8.58/4.86 & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 8.58/4.86 => ( ord_less_real @ A @ B ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7392_inverse__less__iff,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 8.58/4.86 = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 8.58/4.86 => ( ord_less_rat @ B @ A ) )
% 8.58/4.86 & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 8.58/4.86 => ( ord_less_rat @ A @ B ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7393_inverse__le__iff,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 8.58/4.86 = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 8.58/4.86 => ( ord_less_eq_real @ B @ A ) )
% 8.58/4.86 & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 8.58/4.86 => ( ord_less_eq_real @ A @ B ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7394_inverse__le__iff,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 8.58/4.86 = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 8.58/4.86 => ( ord_less_eq_rat @ B @ A ) )
% 8.58/4.86 & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 8.58/4.86 => ( ord_less_eq_rat @ A @ B ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7395_one__le__inverse__iff,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X ) )
% 8.58/4.86 = ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.86 & ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7396_one__le__inverse__iff,axiom,
% 8.58/4.86 ! [X: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
% 8.58/4.86 = ( ( ord_less_rat @ zero_zero_rat @ X )
% 8.58/4.86 & ( ord_less_eq_rat @ X @ one_one_rat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7397_inverse__less__1__iff,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_real @ ( inverse_inverse_real @ X ) @ one_one_real )
% 8.58/4.86 = ( ( ord_less_eq_real @ X @ zero_zero_real )
% 8.58/4.86 | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7398_inverse__less__1__iff,axiom,
% 8.58/4.86 ! [X: rat] :
% 8.58/4.86 ( ( ord_less_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
% 8.58/4.86 = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 8.58/4.86 | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7399_one__le__inverse,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.86 => ( ( ord_less_eq_real @ A @ one_one_real )
% 8.58/4.86 => ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7400_one__le__inverse,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.58/4.86 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 8.58/4.86 => ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7401_inverse__diff__inverse,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( A != zero_zero_real )
% 8.58/4.86 => ( ( B != zero_zero_real )
% 8.58/4.86 => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 8.58/4.86 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7402_inverse__diff__inverse,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( A != zero_zero_complex )
% 8.58/4.86 => ( ( B != zero_zero_complex )
% 8.58/4.86 => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 8.58/4.86 = ( uminus1482373934393186551omplex @ ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7403_inverse__diff__inverse,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( A != zero_zero_rat )
% 8.58/4.86 => ( ( B != zero_zero_rat )
% 8.58/4.86 => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 8.58/4.86 = ( uminus_uminus_rat @ ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7404_reals__Archimedean,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.86 => ? [N2: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ X ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7405_reals__Archimedean,axiom,
% 8.58/4.86 ! [X: rat] :
% 8.58/4.86 ( ( ord_less_rat @ zero_zero_rat @ X )
% 8.58/4.86 => ? [N2: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ X ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7406_i__mult__Complex,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B ) )
% 8.58/4.86 = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7407_Complex__mult__i,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( times_times_complex @ ( complex2 @ A @ B ) @ imaginary_unit )
% 8.58/4.86 = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7408_bit__iff__and__push__bit__not__eq__0,axiom,
% 8.58/4.86 ( bit_se1148574629649215175it_nat
% 8.58/4.86 = ( ^ [A2: nat,N3: nat] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ A2 @ ( bit_se547839408752420682it_nat @ N3 @ one_one_nat ) )
% 8.58/4.86 != zero_zero_nat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7409_bit__iff__and__push__bit__not__eq__0,axiom,
% 8.58/4.86 ( bit_se1146084159140164899it_int
% 8.58/4.86 = ( ^ [A2: int,N3: nat] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ A2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) )
% 8.58/4.86 != zero_zero_int ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7410_bit__iff__and__push__bit__not__eq__0,axiom,
% 8.58/4.86 ( bit_se9216721137139052372nteger
% 8.58/4.86 = ( ^ [A2: code_integer,N3: nat] :
% 8.58/4.86 ( ( bit_se3949692690581998587nteger @ A2 @ ( bit_se7788150548672797655nteger @ N3 @ one_one_Code_integer ) )
% 8.58/4.86 != zero_z3403309356797280102nteger ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7411_bit__iff__and__push__bit__not__eq__0,axiom,
% 8.58/4.86 ( bit_se8040316288895769887atural
% 8.58/4.86 = ( ^ [A2: code_natural,N3: nat] :
% 8.58/4.86 ( ( bit_se2773287842338716102atural @ A2 @ ( bit_se6611745700429515170atural @ N3 @ one_one_Code_natural ) )
% 8.58/4.86 != zero_z2226904508553997617atural ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7412_even__and__iff__int,axiom,
% 8.58/4.86 ! [K: int,L: int] :
% 8.58/4.86 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 8.58/4.86 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 8.58/4.86 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7413_forall__pos__mono__1,axiom,
% 8.58/4.86 ! [P: real > $o,E: real] :
% 8.58/4.86 ( ! [D2: real,E2: real] :
% 8.58/4.86 ( ( ord_less_real @ D2 @ E2 )
% 8.58/4.86 => ( ( P @ D2 )
% 8.58/4.86 => ( P @ E2 ) ) )
% 8.58/4.86 => ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
% 8.58/4.86 => ( ( ord_less_real @ zero_zero_real @ E )
% 8.58/4.86 => ( P @ E ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7414_forall__pos__mono,axiom,
% 8.58/4.86 ! [P: real > $o,E: real] :
% 8.58/4.86 ( ! [D2: real,E2: real] :
% 8.58/4.86 ( ( ord_less_real @ D2 @ E2 )
% 8.58/4.86 => ( ( P @ D2 )
% 8.58/4.86 => ( P @ E2 ) ) )
% 8.58/4.86 => ( ! [N2: nat] :
% 8.58/4.86 ( ( N2 != zero_zero_nat )
% 8.58/4.86 => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
% 8.58/4.86 => ( ( ord_less_real @ zero_zero_real @ E )
% 8.58/4.86 => ( P @ E ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7415_real__arch__inverse,axiom,
% 8.58/4.86 ! [E: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ E )
% 8.58/4.86 = ( ? [N3: nat] :
% 8.58/4.86 ( ( N3 != zero_zero_nat )
% 8.58/4.86 & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 8.58/4.86 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7416_bit__nat__iff,axiom,
% 8.58/4.86 ! [K: int,N: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
% 8.58/4.86 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.58/4.86 & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7417_sqrt__divide__self__eq,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.86 => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
% 8.58/4.86 = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7418_and__one__eq,axiom,
% 8.58/4.86 ! [A: code_integer] :
% 8.58/4.86 ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
% 8.58/4.86 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7419_and__one__eq,axiom,
% 8.58/4.86 ! [A: code_natural] :
% 8.58/4.86 ( ( bit_se2773287842338716102atural @ A @ one_one_Code_natural )
% 8.58/4.86 = ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7420_and__one__eq,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ A @ one_one_int )
% 8.58/4.86 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7421_and__one__eq,axiom,
% 8.58/4.86 ! [A: nat] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
% 8.58/4.86 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7422_one__and__eq,axiom,
% 8.58/4.86 ! [A: code_integer] :
% 8.58/4.86 ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
% 8.58/4.86 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7423_one__and__eq,axiom,
% 8.58/4.86 ! [A: code_natural] :
% 8.58/4.86 ( ( bit_se2773287842338716102atural @ one_one_Code_natural @ A )
% 8.58/4.86 = ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7424_one__and__eq,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ one_one_int @ A )
% 8.58/4.86 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7425_one__and__eq,axiom,
% 8.58/4.86 ! [A: nat] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
% 8.58/4.86 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7426_ex__inverse__of__nat__less,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.86 => ? [N2: nat] :
% 8.58/4.86 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 8.58/4.86 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7427_ex__inverse__of__nat__less,axiom,
% 8.58/4.86 ! [X: rat] :
% 8.58/4.86 ( ( ord_less_rat @ zero_zero_rat @ X )
% 8.58/4.86 => ? [N2: nat] :
% 8.58/4.86 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 8.58/4.86 & ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7428_power__diff__conv__inverse,axiom,
% 8.58/4.86 ! [X: real,M: nat,N: nat] :
% 8.58/4.86 ( ( X != zero_zero_real )
% 8.58/4.86 => ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.86 => ( ( power_power_real @ X @ ( minus_minus_nat @ N @ M ) )
% 8.58/4.86 = ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ M ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7429_power__diff__conv__inverse,axiom,
% 8.58/4.86 ! [X: complex,M: nat,N: nat] :
% 8.58/4.86 ( ( X != zero_zero_complex )
% 8.58/4.86 => ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.86 => ( ( power_power_complex @ X @ ( minus_minus_nat @ N @ M ) )
% 8.58/4.86 = ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ M ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7430_power__diff__conv__inverse,axiom,
% 8.58/4.86 ! [X: rat,M: nat,N: nat] :
% 8.58/4.86 ( ( X != zero_zero_rat )
% 8.58/4.86 => ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.86 => ( ( power_power_rat @ X @ ( minus_minus_nat @ N @ M ) )
% 8.58/4.86 = ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ M ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7431_complex__of__real__i,axiom,
% 8.58/4.86 ! [R: real] :
% 8.58/4.86 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ imaginary_unit )
% 8.58/4.86 = ( complex2 @ zero_zero_real @ R ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7432_i__complex__of__real,axiom,
% 8.58/4.86 ! [R: real] :
% 8.58/4.86 ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R ) )
% 8.58/4.86 = ( complex2 @ zero_zero_real @ R ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7433_log__inverse,axiom,
% 8.58/4.86 ! [A: real,X: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.86 => ( ( A != one_one_real )
% 8.58/4.86 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.86 => ( ( log2 @ A @ ( inverse_inverse_real @ X ) )
% 8.58/4.86 = ( uminus_uminus_real @ ( log2 @ A @ X ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7434_Complex__eq,axiom,
% 8.58/4.86 ( complex2
% 8.58/4.86 = ( ^ [A2: real,B2: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A2 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B2 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7435_and__exp__eq__0__iff__not__bit,axiom,
% 8.58/4.86 ! [A: int,N: nat] :
% 8.58/4.86 ( ( ( bit_se725231765392027082nd_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.86 = zero_zero_int )
% 8.58/4.86 = ( ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7436_and__exp__eq__0__iff__not__bit,axiom,
% 8.58/4.86 ! [A: nat,N: nat] :
% 8.58/4.86 ( ( ( bit_se727722235901077358nd_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.86 = zero_zero_nat )
% 8.58/4.86 = ( ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7437_bit__nat__def,axiom,
% 8.58/4.86 ( bit_se1148574629649215175it_nat
% 8.58/4.86 = ( ^ [M3: nat,N3: nat] :
% 8.58/4.86 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7438_exp__plus__inverse__exp,axiom,
% 8.58/4.86 ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7439_complex__split__polar,axiom,
% 8.58/4.86 ! [Z2: complex] :
% 8.58/4.86 ? [R2: real,A4: real] :
% 8.58/4.86 ( Z2
% 8.58/4.86 = ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A4 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A4 ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7440_plus__inverse__ge__2,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.86 => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7441_real__inv__sqrt__pow2,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.86 => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.86 = ( inverse_inverse_real @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7442_tan__cot,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 8.58/4.86 = ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7443_and__int__rec,axiom,
% 8.58/4.86 ( bit_se725231765392027082nd_int
% 8.58/4.86 = ( ^ [K2: int,L2: int] :
% 8.58/4.86 ( plus_plus_int
% 8.58/4.86 @ ( zero_n2684676970156552555ol_int
% 8.58/4.86 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 8.58/4.86 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 8.58/4.86 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7444_real__le__x__sinh,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.86 => ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7445_real__le__abs__sinh,axiom,
% 8.58/4.86 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7446_tan__sec,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ( cos_real @ X )
% 8.58/4.86 != zero_zero_real )
% 8.58/4.86 => ( ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.86 = ( power_power_real @ ( inverse_inverse_real @ ( cos_real @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7447_tan__sec,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( ( cos_complex @ X )
% 8.58/4.86 != zero_zero_complex )
% 8.58/4.86 => ( ( plus_plus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.86 = ( power_power_complex @ ( invers8013647133539491842omplex @ ( cos_complex @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7448_cmod__unit__one,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
% 8.58/4.86 = one_one_real ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7449_cmod__complex__polar,axiom,
% 8.58/4.86 ! [R: real,A: real] :
% 8.58/4.86 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
% 8.58/4.86 = ( abs_abs_real @ R ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7450_Arg__minus__ii,axiom,
% 8.58/4.86 ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
% 8.58/4.86 = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7451_csqrt__ii,axiom,
% 8.58/4.86 ( ( csqrt @ imaginary_unit )
% 8.58/4.86 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7452_Arg__ii,axiom,
% 8.58/4.86 ( ( arg @ imaginary_unit )
% 8.58/4.86 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7453_sinh__ln__real,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.86 => ( ( sinh_real @ ( ln_ln_real @ X ) )
% 8.58/4.86 = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7454_cis__multiple__2pi,axiom,
% 8.58/4.86 ! [N: real] :
% 8.58/4.86 ( ( member_real @ N @ ring_1_Ints_real )
% 8.58/4.86 => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 8.58/4.86 = one_one_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7455_sinh__0,axiom,
% 8.58/4.86 ( ( sinh_real @ zero_zero_real )
% 8.58/4.86 = zero_zero_real ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7456_frac__in__Ints__iff,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( member_real @ ( archim2898591450579166408c_real @ X ) @ ring_1_Ints_real )
% 8.58/4.86 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7457_csqrt__eq__1,axiom,
% 8.58/4.86 ! [Z2: complex] :
% 8.58/4.86 ( ( ( csqrt @ Z2 )
% 8.58/4.86 = one_one_complex )
% 8.58/4.86 = ( Z2 = one_one_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7458_csqrt__1,axiom,
% 8.58/4.86 ( ( csqrt @ one_one_complex )
% 8.58/4.86 = one_one_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7459_frac__eq__0__iff,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ( archim2898591450579166408c_real @ X )
% 8.58/4.86 = zero_zero_real )
% 8.58/4.86 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7460_frac__eq__0__iff,axiom,
% 8.58/4.86 ! [X: rat] :
% 8.58/4.86 ( ( ( archimedean_frac_rat @ X )
% 8.58/4.86 = zero_zero_rat )
% 8.58/4.86 = ( member_rat @ X @ ring_1_Ints_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7461_floor__add2,axiom,
% 8.58/4.86 ! [X: real,Y: real] :
% 8.58/4.86 ( ( ( member_real @ X @ ring_1_Ints_real )
% 8.58/4.86 | ( member_real @ Y @ ring_1_Ints_real ) )
% 8.58/4.86 => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.86 = ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7462_floor__add2,axiom,
% 8.58/4.86 ! [X: rat,Y: rat] :
% 8.58/4.86 ( ( ( member_rat @ X @ ring_1_Ints_rat )
% 8.58/4.86 | ( member_rat @ Y @ ring_1_Ints_rat ) )
% 8.58/4.86 => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
% 8.58/4.86 = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7463_frac__gt__0__iff,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) )
% 8.58/4.86 = ( ~ ( member_real @ X @ ring_1_Ints_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7464_frac__gt__0__iff,axiom,
% 8.58/4.86 ! [X: rat] :
% 8.58/4.86 ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) )
% 8.58/4.86 = ( ~ ( member_rat @ X @ ring_1_Ints_rat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7465_and__nat__numerals_I1_J,axiom,
% 8.58/4.86 ! [Y: num] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 8.58/4.86 = zero_zero_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7466_and__nat__numerals_I3_J,axiom,
% 8.58/4.86 ! [X: num] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 8.58/4.86 = zero_zero_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7467_power2__csqrt,axiom,
% 8.58/4.86 ! [Z2: complex] :
% 8.58/4.86 ( ( power_power_complex @ ( csqrt @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.86 = Z2 ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7468_and__nat__numerals_I2_J,axiom,
% 8.58/4.86 ! [Y: num] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 8.58/4.86 = one_one_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7469_and__nat__numerals_I4_J,axiom,
% 8.58/4.86 ! [X: num] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 8.58/4.86 = one_one_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7470_and__Suc__0__eq,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 8.58/4.86 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7471_Suc__0__and__eq,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.58/4.86 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7472_Ints__0,axiom,
% 8.58/4.86 member_complex @ zero_zero_complex @ ring_1_Ints_complex ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7473_Ints__0,axiom,
% 8.58/4.86 member_real @ zero_zero_real @ ring_1_Ints_real ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7474_Ints__0,axiom,
% 8.58/4.86 member_rat @ zero_zero_rat @ ring_1_Ints_rat ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7475_Ints__0,axiom,
% 8.58/4.86 member_int @ zero_zero_int @ ring_1_Ints_int ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7476_Ints__numeral,axiom,
% 8.58/4.86 ! [N: num] : ( member_complex @ ( numera6690914467698888265omplex @ N ) @ ring_1_Ints_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7477_Ints__numeral,axiom,
% 8.58/4.86 ! [N: num] : ( member_real @ ( numeral_numeral_real @ N ) @ ring_1_Ints_real ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7478_Ints__numeral,axiom,
% 8.58/4.86 ! [N: num] : ( member_rat @ ( numeral_numeral_rat @ N ) @ ring_1_Ints_rat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7479_Ints__numeral,axiom,
% 8.58/4.86 ! [N: num] : ( member_int @ ( numeral_numeral_int @ N ) @ ring_1_Ints_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7480_Ints__mult,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( member_complex @ A @ ring_1_Ints_complex )
% 8.58/4.86 => ( ( member_complex @ B @ ring_1_Ints_complex )
% 8.58/4.86 => ( member_complex @ ( times_times_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7481_Ints__mult,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( member_real @ A @ ring_1_Ints_real )
% 8.58/4.86 => ( ( member_real @ B @ ring_1_Ints_real )
% 8.58/4.86 => ( member_real @ ( times_times_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7482_Ints__mult,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( member_rat @ A @ ring_1_Ints_rat )
% 8.58/4.86 => ( ( member_rat @ B @ ring_1_Ints_rat )
% 8.58/4.86 => ( member_rat @ ( times_times_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7483_Ints__mult,axiom,
% 8.58/4.86 ! [A: int,B: int] :
% 8.58/4.86 ( ( member_int @ A @ ring_1_Ints_int )
% 8.58/4.86 => ( ( member_int @ B @ ring_1_Ints_int )
% 8.58/4.86 => ( member_int @ ( times_times_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7484_Ints__1,axiom,
% 8.58/4.86 member_complex @ one_one_complex @ ring_1_Ints_complex ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7485_Ints__1,axiom,
% 8.58/4.86 member_rat @ one_one_rat @ ring_1_Ints_rat ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7486_Ints__1,axiom,
% 8.58/4.86 member_int @ one_one_int @ ring_1_Ints_int ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7487_Ints__1,axiom,
% 8.58/4.86 member_real @ one_one_real @ ring_1_Ints_real ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7488_Ints__add,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( member_complex @ A @ ring_1_Ints_complex )
% 8.58/4.86 => ( ( member_complex @ B @ ring_1_Ints_complex )
% 8.58/4.86 => ( member_complex @ ( plus_plus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7489_Ints__add,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( member_real @ A @ ring_1_Ints_real )
% 8.58/4.86 => ( ( member_real @ B @ ring_1_Ints_real )
% 8.58/4.86 => ( member_real @ ( plus_plus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7490_Ints__add,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( member_rat @ A @ ring_1_Ints_rat )
% 8.58/4.86 => ( ( member_rat @ B @ ring_1_Ints_rat )
% 8.58/4.86 => ( member_rat @ ( plus_plus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7491_Ints__add,axiom,
% 8.58/4.86 ! [A: int,B: int] :
% 8.58/4.86 ( ( member_int @ A @ ring_1_Ints_int )
% 8.58/4.86 => ( ( member_int @ B @ ring_1_Ints_int )
% 8.58/4.86 => ( member_int @ ( plus_plus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7492_Ints__diff,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( member_complex @ A @ ring_1_Ints_complex )
% 8.58/4.86 => ( ( member_complex @ B @ ring_1_Ints_complex )
% 8.58/4.86 => ( member_complex @ ( minus_minus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7493_Ints__diff,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( member_real @ A @ ring_1_Ints_real )
% 8.58/4.86 => ( ( member_real @ B @ ring_1_Ints_real )
% 8.58/4.86 => ( member_real @ ( minus_minus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7494_Ints__diff,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( member_rat @ A @ ring_1_Ints_rat )
% 8.58/4.86 => ( ( member_rat @ B @ ring_1_Ints_rat )
% 8.58/4.86 => ( member_rat @ ( minus_minus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7495_Ints__diff,axiom,
% 8.58/4.86 ! [A: int,B: int] :
% 8.58/4.86 ( ( member_int @ A @ ring_1_Ints_int )
% 8.58/4.86 => ( ( member_int @ B @ ring_1_Ints_int )
% 8.58/4.86 => ( member_int @ ( minus_minus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7496_minus__in__Ints__iff,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( member_real @ ( uminus_uminus_real @ X ) @ ring_1_Ints_real )
% 8.58/4.86 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7497_minus__in__Ints__iff,axiom,
% 8.58/4.86 ! [X: int] :
% 8.58/4.86 ( ( member_int @ ( uminus_uminus_int @ X ) @ ring_1_Ints_int )
% 8.58/4.86 = ( member_int @ X @ ring_1_Ints_int ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7498_minus__in__Ints__iff,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( member_complex @ ( uminus1482373934393186551omplex @ X ) @ ring_1_Ints_complex )
% 8.58/4.86 = ( member_complex @ X @ ring_1_Ints_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7499_minus__in__Ints__iff,axiom,
% 8.58/4.86 ! [X: rat] :
% 8.58/4.86 ( ( member_rat @ ( uminus_uminus_rat @ X ) @ ring_1_Ints_rat )
% 8.58/4.86 = ( member_rat @ X @ ring_1_Ints_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7500_minus__in__Ints__iff,axiom,
% 8.58/4.86 ! [X: code_integer] :
% 8.58/4.86 ( ( member_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ ring_11222124179247155820nteger )
% 8.58/4.86 = ( member_Code_integer @ X @ ring_11222124179247155820nteger ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7501_Ints__minus,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( member_real @ A @ ring_1_Ints_real )
% 8.58/4.86 => ( member_real @ ( uminus_uminus_real @ A ) @ ring_1_Ints_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7502_Ints__minus,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( member_int @ A @ ring_1_Ints_int )
% 8.58/4.86 => ( member_int @ ( uminus_uminus_int @ A ) @ ring_1_Ints_int ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7503_Ints__minus,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( member_complex @ A @ ring_1_Ints_complex )
% 8.58/4.86 => ( member_complex @ ( uminus1482373934393186551omplex @ A ) @ ring_1_Ints_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7504_Ints__minus,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( member_rat @ A @ ring_1_Ints_rat )
% 8.58/4.86 => ( member_rat @ ( uminus_uminus_rat @ A ) @ ring_1_Ints_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7505_Ints__minus,axiom,
% 8.58/4.86 ! [A: code_integer] :
% 8.58/4.86 ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
% 8.58/4.86 => ( member_Code_integer @ ( uminus1351360451143612070nteger @ A ) @ ring_11222124179247155820nteger ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7506_divide__complex__def,axiom,
% 8.58/4.86 ( divide1717551699836669952omplex
% 8.58/4.86 = ( ^ [X3: complex,Y6: complex] : ( times_times_complex @ X3 @ ( invers8013647133539491842omplex @ Y6 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7507_Ints__power,axiom,
% 8.58/4.86 ! [A: real,N: nat] :
% 8.58/4.86 ( ( member_real @ A @ ring_1_Ints_real )
% 8.58/4.86 => ( member_real @ ( power_power_real @ A @ N ) @ ring_1_Ints_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7508_Ints__power,axiom,
% 8.58/4.86 ! [A: int,N: nat] :
% 8.58/4.86 ( ( member_int @ A @ ring_1_Ints_int )
% 8.58/4.86 => ( member_int @ ( power_power_int @ A @ N ) @ ring_1_Ints_int ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7509_Ints__power,axiom,
% 8.58/4.86 ! [A: complex,N: nat] :
% 8.58/4.86 ( ( member_complex @ A @ ring_1_Ints_complex )
% 8.58/4.86 => ( member_complex @ ( power_power_complex @ A @ N ) @ ring_1_Ints_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7510_Ints__of__nat,axiom,
% 8.58/4.86 ! [N: nat] : ( member_complex @ ( semiri8010041392384452111omplex @ N ) @ ring_1_Ints_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7511_Ints__of__nat,axiom,
% 8.58/4.86 ! [N: nat] : ( member_real @ ( semiri5074537144036343181t_real @ N ) @ ring_1_Ints_real ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7512_Ints__of__nat,axiom,
% 8.58/4.86 ! [N: nat] : ( member_rat @ ( semiri681578069525770553at_rat @ N ) @ ring_1_Ints_rat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7513_Ints__of__nat,axiom,
% 8.58/4.86 ! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ring_1_Ints_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7514_Ints__abs,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( member_int @ A @ ring_1_Ints_int )
% 8.58/4.86 => ( member_int @ ( abs_abs_int @ A ) @ ring_1_Ints_int ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7515_Ints__abs,axiom,
% 8.58/4.86 ! [A: code_integer] :
% 8.58/4.86 ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
% 8.58/4.86 => ( member_Code_integer @ ( abs_abs_Code_integer @ A ) @ ring_11222124179247155820nteger ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7516_Ints__abs,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( member_rat @ A @ ring_1_Ints_rat )
% 8.58/4.86 => ( member_rat @ ( abs_abs_rat @ A ) @ ring_1_Ints_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7517_Ints__abs,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( member_real @ A @ ring_1_Ints_real )
% 8.58/4.86 => ( member_real @ ( abs_abs_real @ A ) @ ring_1_Ints_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7518_Ints__of__int,axiom,
% 8.58/4.86 ! [Z2: int] : ( member_complex @ ( ring_17405671764205052669omplex @ Z2 ) @ ring_1_Ints_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7519_Ints__of__int,axiom,
% 8.58/4.86 ! [Z2: int] : ( member_int @ ( ring_1_of_int_int @ Z2 ) @ ring_1_Ints_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7520_Ints__of__int,axiom,
% 8.58/4.86 ! [Z2: int] : ( member_real @ ( ring_1_of_int_real @ Z2 ) @ ring_1_Ints_real ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7521_Ints__of__int,axiom,
% 8.58/4.86 ! [Z2: int] : ( member_rat @ ( ring_1_of_int_rat @ Z2 ) @ ring_1_Ints_rat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7522_Ints__induct,axiom,
% 8.58/4.86 ! [Q2: complex,P: complex > $o] :
% 8.58/4.86 ( ( member_complex @ Q2 @ ring_1_Ints_complex )
% 8.58/4.86 => ( ! [Z4: int] : ( P @ ( ring_17405671764205052669omplex @ Z4 ) )
% 8.58/4.86 => ( P @ Q2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7523_Ints__induct,axiom,
% 8.58/4.86 ! [Q2: int,P: int > $o] :
% 8.58/4.86 ( ( member_int @ Q2 @ ring_1_Ints_int )
% 8.58/4.86 => ( ! [Z4: int] : ( P @ ( ring_1_of_int_int @ Z4 ) )
% 8.58/4.86 => ( P @ Q2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7524_Ints__induct,axiom,
% 8.58/4.86 ! [Q2: real,P: real > $o] :
% 8.58/4.86 ( ( member_real @ Q2 @ ring_1_Ints_real )
% 8.58/4.86 => ( ! [Z4: int] : ( P @ ( ring_1_of_int_real @ Z4 ) )
% 8.58/4.86 => ( P @ Q2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7525_Ints__induct,axiom,
% 8.58/4.86 ! [Q2: rat,P: rat > $o] :
% 8.58/4.86 ( ( member_rat @ Q2 @ ring_1_Ints_rat )
% 8.58/4.86 => ( ! [Z4: int] : ( P @ ( ring_1_of_int_rat @ Z4 ) )
% 8.58/4.86 => ( P @ Q2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7526_Ints__cases,axiom,
% 8.58/4.86 ! [Q2: complex] :
% 8.58/4.86 ( ( member_complex @ Q2 @ ring_1_Ints_complex )
% 8.58/4.86 => ~ ! [Z4: int] :
% 8.58/4.86 ( Q2
% 8.58/4.86 != ( ring_17405671764205052669omplex @ Z4 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7527_Ints__cases,axiom,
% 8.58/4.86 ! [Q2: int] :
% 8.58/4.86 ( ( member_int @ Q2 @ ring_1_Ints_int )
% 8.58/4.86 => ~ ! [Z4: int] :
% 8.58/4.86 ( Q2
% 8.58/4.86 != ( ring_1_of_int_int @ Z4 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7528_Ints__cases,axiom,
% 8.58/4.86 ! [Q2: real] :
% 8.58/4.86 ( ( member_real @ Q2 @ ring_1_Ints_real )
% 8.58/4.86 => ~ ! [Z4: int] :
% 8.58/4.86 ( Q2
% 8.58/4.86 != ( ring_1_of_int_real @ Z4 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7529_Ints__cases,axiom,
% 8.58/4.86 ! [Q2: rat] :
% 8.58/4.86 ( ( member_rat @ Q2 @ ring_1_Ints_rat )
% 8.58/4.86 => ~ ! [Z4: int] :
% 8.58/4.86 ( Q2
% 8.58/4.86 != ( ring_1_of_int_rat @ Z4 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7530_Ints__double__eq__0__iff,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( member_complex @ A @ ring_1_Ints_complex )
% 8.58/4.86 => ( ( ( plus_plus_complex @ A @ A )
% 8.58/4.86 = zero_zero_complex )
% 8.58/4.86 = ( A = zero_zero_complex ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7531_Ints__double__eq__0__iff,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( member_real @ A @ ring_1_Ints_real )
% 8.58/4.86 => ( ( ( plus_plus_real @ A @ A )
% 8.58/4.86 = zero_zero_real )
% 8.58/4.86 = ( A = zero_zero_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7532_Ints__double__eq__0__iff,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( member_rat @ A @ ring_1_Ints_rat )
% 8.58/4.86 => ( ( ( plus_plus_rat @ A @ A )
% 8.58/4.86 = zero_zero_rat )
% 8.58/4.86 = ( A = zero_zero_rat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7533_Ints__double__eq__0__iff,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( member_int @ A @ ring_1_Ints_int )
% 8.58/4.86 => ( ( ( plus_plus_int @ A @ A )
% 8.58/4.86 = zero_zero_int )
% 8.58/4.86 = ( A = zero_zero_int ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7534_and__nat__def,axiom,
% 8.58/4.86 ( bit_se727722235901077358nd_nat
% 8.58/4.86 = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7535_Ints__odd__nonzero,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( member_complex @ A @ ring_1_Ints_complex )
% 8.58/4.86 => ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A ) @ A )
% 8.58/4.86 != zero_zero_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7536_Ints__odd__nonzero,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( member_real @ A @ ring_1_Ints_real )
% 8.58/4.86 => ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
% 8.58/4.86 != zero_zero_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7537_Ints__odd__nonzero,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( member_rat @ A @ ring_1_Ints_rat )
% 8.58/4.86 => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A )
% 8.58/4.86 != zero_zero_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7538_Ints__odd__nonzero,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( member_int @ A @ ring_1_Ints_int )
% 8.58/4.86 => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
% 8.58/4.86 != zero_zero_int ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7539_of__int__divide__in__Ints,axiom,
% 8.58/4.86 ! [B: int,A: int] :
% 8.58/4.86 ( ( dvd_dvd_int @ B @ A )
% 8.58/4.86 => ( member_complex @ ( divide1717551699836669952omplex @ ( ring_17405671764205052669omplex @ A ) @ ( ring_17405671764205052669omplex @ B ) ) @ ring_1_Ints_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7540_of__int__divide__in__Ints,axiom,
% 8.58/4.86 ! [B: int,A: int] :
% 8.58/4.86 ( ( dvd_dvd_int @ B @ A )
% 8.58/4.86 => ( member_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) @ ring_1_Ints_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7541_of__int__divide__in__Ints,axiom,
% 8.58/4.86 ! [B: int,A: int] :
% 8.58/4.86 ( ( dvd_dvd_int @ B @ A )
% 8.58/4.86 => ( member_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) @ ring_1_Ints_rat ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7542_of__int__divide__in__Ints,axiom,
% 8.58/4.86 ! [B: int,A: int] :
% 8.58/4.86 ( ( dvd_dvd_int @ B @ A )
% 8.58/4.86 => ( member_int @ ( divide_divide_int @ ( ring_1_of_int_int @ A ) @ ( ring_1_of_int_int @ B ) ) @ ring_1_Ints_int ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7543_Ints__odd__less__0,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( member_real @ A @ ring_1_Ints_real )
% 8.58/4.86 => ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A ) @ zero_zero_real )
% 8.58/4.86 = ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7544_Ints__odd__less__0,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( member_rat @ A @ ring_1_Ints_rat )
% 8.58/4.86 => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A ) @ zero_zero_rat )
% 8.58/4.86 = ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7545_Ints__odd__less__0,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( member_int @ A @ ring_1_Ints_int )
% 8.58/4.86 => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A ) @ zero_zero_int )
% 8.58/4.86 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7546_Ints__nonzero__abs__ge1,axiom,
% 8.58/4.86 ! [X: code_integer] :
% 8.58/4.86 ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 8.58/4.86 => ( ( X != zero_z3403309356797280102nteger )
% 8.58/4.86 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7547_Ints__nonzero__abs__ge1,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( member_real @ X @ ring_1_Ints_real )
% 8.58/4.86 => ( ( X != zero_zero_real )
% 8.58/4.86 => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7548_Ints__nonzero__abs__ge1,axiom,
% 8.58/4.86 ! [X: rat] :
% 8.58/4.86 ( ( member_rat @ X @ ring_1_Ints_rat )
% 8.58/4.86 => ( ( X != zero_zero_rat )
% 8.58/4.86 => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7549_Ints__nonzero__abs__ge1,axiom,
% 8.58/4.86 ! [X: int] :
% 8.58/4.86 ( ( member_int @ X @ ring_1_Ints_int )
% 8.58/4.86 => ( ( X != zero_zero_int )
% 8.58/4.86 => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7550_Ints__nonzero__abs__less1,axiom,
% 8.58/4.86 ! [X: code_integer] :
% 8.58/4.86 ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 8.58/4.86 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer )
% 8.58/4.86 => ( X = zero_z3403309356797280102nteger ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7551_Ints__nonzero__abs__less1,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( member_real @ X @ ring_1_Ints_real )
% 8.58/4.86 => ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.86 => ( X = zero_zero_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7552_Ints__nonzero__abs__less1,axiom,
% 8.58/4.86 ! [X: rat] :
% 8.58/4.86 ( ( member_rat @ X @ ring_1_Ints_rat )
% 8.58/4.86 => ( ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat )
% 8.58/4.86 => ( X = zero_zero_rat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7553_Ints__nonzero__abs__less1,axiom,
% 8.58/4.86 ! [X: int] :
% 8.58/4.86 ( ( member_int @ X @ ring_1_Ints_int )
% 8.58/4.86 => ( ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int )
% 8.58/4.86 => ( X = zero_zero_int ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7554_Ints__eq__abs__less1,axiom,
% 8.58/4.86 ! [X: code_integer,Y: code_integer] :
% 8.58/4.86 ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
% 8.58/4.86 => ( ( member_Code_integer @ Y @ ring_11222124179247155820nteger )
% 8.58/4.86 => ( ( X = Y )
% 8.58/4.86 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ Y ) ) @ one_one_Code_integer ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7555_Ints__eq__abs__less1,axiom,
% 8.58/4.86 ! [X: real,Y: real] :
% 8.58/4.86 ( ( member_real @ X @ ring_1_Ints_real )
% 8.58/4.86 => ( ( member_real @ Y @ ring_1_Ints_real )
% 8.58/4.86 => ( ( X = Y )
% 8.58/4.86 = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ one_one_real ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7556_Ints__eq__abs__less1,axiom,
% 8.58/4.86 ! [X: rat,Y: rat] :
% 8.58/4.86 ( ( member_rat @ X @ ring_1_Ints_rat )
% 8.58/4.86 => ( ( member_rat @ Y @ ring_1_Ints_rat )
% 8.58/4.86 => ( ( X = Y )
% 8.58/4.86 = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ Y ) ) @ one_one_rat ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7557_Ints__eq__abs__less1,axiom,
% 8.58/4.86 ! [X: int,Y: int] :
% 8.58/4.86 ( ( member_int @ X @ ring_1_Ints_int )
% 8.58/4.86 => ( ( member_int @ Y @ ring_1_Ints_int )
% 8.58/4.86 => ( ( X = Y )
% 8.58/4.86 = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y ) ) @ one_one_int ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7558_sin__times__pi__eq__0,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
% 8.58/4.86 = zero_zero_real )
% 8.58/4.86 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7559_frac__neg,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ( member_real @ X @ ring_1_Ints_real )
% 8.58/4.86 => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
% 8.58/4.86 = zero_zero_real ) )
% 8.58/4.86 & ( ~ ( member_real @ X @ ring_1_Ints_real )
% 8.58/4.86 => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
% 8.58/4.86 = ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7560_frac__neg,axiom,
% 8.58/4.86 ! [X: rat] :
% 8.58/4.86 ( ( ( member_rat @ X @ ring_1_Ints_rat )
% 8.58/4.86 => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
% 8.58/4.86 = zero_zero_rat ) )
% 8.58/4.86 & ( ~ ( member_rat @ X @ ring_1_Ints_rat )
% 8.58/4.86 => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
% 8.58/4.86 = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7561_le__mult__floor__Ints,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.86 => ( ( member_real @ A @ ring_1_Ints_real )
% 8.58/4.86 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7562_le__mult__floor__Ints,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.86 => ( ( member_real @ A @ ring_1_Ints_real )
% 8.58/4.86 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7563_le__mult__floor__Ints,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.86 => ( ( member_real @ A @ ring_1_Ints_real )
% 8.58/4.86 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_int @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7564_le__mult__floor__Ints,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.58/4.86 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 8.58/4.86 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_real @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7565_le__mult__floor__Ints,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.58/4.86 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 8.58/4.86 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7566_le__mult__floor__Ints,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.58/4.86 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 8.58/4.86 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_int @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7567_frac__unique__iff,axiom,
% 8.58/4.86 ! [X: real,A: real] :
% 8.58/4.86 ( ( ( archim2898591450579166408c_real @ X )
% 8.58/4.86 = A )
% 8.58/4.86 = ( ( member_real @ ( minus_minus_real @ X @ A ) @ ring_1_Ints_real )
% 8.58/4.86 & ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.86 & ( ord_less_real @ A @ one_one_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7568_frac__unique__iff,axiom,
% 8.58/4.86 ! [X: rat,A: rat] :
% 8.58/4.86 ( ( ( archimedean_frac_rat @ X )
% 8.58/4.86 = A )
% 8.58/4.86 = ( ( member_rat @ ( minus_minus_rat @ X @ A ) @ ring_1_Ints_rat )
% 8.58/4.86 & ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.58/4.86 & ( ord_less_rat @ A @ one_one_rat ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7569_mult__ceiling__le__Ints,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.86 => ( ( member_real @ A @ ring_1_Ints_real )
% 8.58/4.86 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7570_mult__ceiling__le__Ints,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.86 => ( ( member_real @ A @ ring_1_Ints_real )
% 8.58/4.86 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7571_mult__ceiling__le__Ints,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.86 => ( ( member_real @ A @ ring_1_Ints_real )
% 8.58/4.86 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7572_mult__ceiling__le__Ints,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.58/4.86 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 8.58/4.86 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7573_mult__ceiling__le__Ints,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.58/4.86 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 8.58/4.86 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7574_mult__ceiling__le__Ints,axiom,
% 8.58/4.86 ! [A: rat,B: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.58/4.86 => ( ( member_rat @ A @ ring_1_Ints_rat )
% 8.58/4.86 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7575_sin__integer__2pi,axiom,
% 8.58/4.86 ! [N: real] :
% 8.58/4.86 ( ( member_real @ N @ ring_1_Ints_real )
% 8.58/4.86 => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 8.58/4.86 = zero_zero_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7576_cos__integer__2pi,axiom,
% 8.58/4.86 ! [N: real] :
% 8.58/4.86 ( ( member_real @ N @ ring_1_Ints_real )
% 8.58/4.86 => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 8.58/4.86 = one_one_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7577_and__nat__unfold,axiom,
% 8.58/4.86 ( bit_se727722235901077358nd_nat
% 8.58/4.86 = ( ^ [M3: nat,N3: nat] :
% 8.58/4.86 ( if_nat
% 8.58/4.86 @ ( ( M3 = zero_zero_nat )
% 8.58/4.86 | ( N3 = zero_zero_nat ) )
% 8.58/4.86 @ zero_zero_nat
% 8.58/4.86 @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7578_and__nat__rec,axiom,
% 8.58/4.86 ( bit_se727722235901077358nd_nat
% 8.58/4.86 = ( ^ [M3: nat,N3: nat] :
% 8.58/4.86 ( plus_plus_nat
% 8.58/4.86 @ ( zero_n2687167440665602831ol_nat
% 8.58/4.86 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 8.58/4.86 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 8.58/4.86 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7579_complex__inverse,axiom,
% 8.58/4.86 ! [A: real,B: real] :
% 8.58/4.86 ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
% 8.58/4.86 = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7580_sinh__field__def,axiom,
% 8.58/4.86 ( sinh_real
% 8.58/4.86 = ( ^ [Z5: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7581_sinh__field__def,axiom,
% 8.58/4.86 ( sinh_complex
% 8.58/4.86 = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7582_cosh__ln__real,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.86 => ( ( cosh_real @ ( ln_ln_real @ X ) )
% 8.58/4.86 = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7583_xor__Suc__0__eq,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 8.58/4.86 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.58/4.86 @ ( zero_n2687167440665602831ol_nat
% 8.58/4.86 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7584_Suc__0__xor__eq,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.58/4.86 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.58/4.86 @ ( zero_n2687167440665602831ol_nat
% 8.58/4.86 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7585_gbinomial__absorption_H,axiom,
% 8.58/4.86 ! [K: nat,A: complex] :
% 8.58/4.86 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.58/4.86 => ( ( gbinomial_complex @ A @ K )
% 8.58/4.86 = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7586_gbinomial__absorption_H,axiom,
% 8.58/4.86 ! [K: nat,A: real] :
% 8.58/4.86 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.58/4.86 => ( ( gbinomial_real @ A @ K )
% 8.58/4.86 = ( times_times_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7587_gbinomial__absorption_H,axiom,
% 8.58/4.86 ! [K: nat,A: rat] :
% 8.58/4.86 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.58/4.86 => ( ( gbinomial_rat @ A @ K )
% 8.58/4.86 = ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7588_cosh__double,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( cosh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 8.58/4.86 = ( plus_plus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7589_cosh__double,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( cosh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 8.58/4.86 = ( plus_plus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7590_bit_Oxor__left__self,axiom,
% 8.58/4.86 ! [X: int,Y: int] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ X @ ( bit_se6526347334894502574or_int @ X @ Y ) )
% 8.58/4.86 = Y ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7591_xor_Oright__neutral,axiom,
% 8.58/4.86 ! [A: nat] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7592_xor_Oright__neutral,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7593_xor_Oleft__neutral,axiom,
% 8.58/4.86 ! [A: nat] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7594_xor_Oleft__neutral,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7595_xor__self__eq,axiom,
% 8.58/4.86 ! [A: nat] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ A @ A )
% 8.58/4.86 = zero_zero_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7596_xor__self__eq,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ A @ A )
% 8.58/4.86 = zero_zero_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7597_bit_Oxor__self,axiom,
% 8.58/4.86 ! [X: int] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ X @ X )
% 8.58/4.86 = zero_zero_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7598_take__bit__xor,axiom,
% 8.58/4.86 ! [N: nat,A: int,B: int] :
% 8.58/4.86 ( ( bit_se2923211474154528505it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 8.58/4.86 = ( bit_se6526347334894502574or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7599_take__bit__xor,axiom,
% 8.58/4.86 ! [N: nat,A: nat,B: nat] :
% 8.58/4.86 ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 8.58/4.86 = ( bit_se6528837805403552850or_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7600_push__bit__xor,axiom,
% 8.58/4.86 ! [N: nat,A: nat,B: nat] :
% 8.58/4.86 ( ( bit_se547839408752420682it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 8.58/4.86 = ( bit_se6528837805403552850or_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7601_push__bit__xor,axiom,
% 8.58/4.86 ! [N: nat,A: int,B: int] :
% 8.58/4.86 ( ( bit_se545348938243370406it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 8.58/4.86 = ( bit_se6526347334894502574or_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7602_push__bit__xor,axiom,
% 8.58/4.86 ! [N: nat,A: code_integer,B: code_integer] :
% 8.58/4.86 ( ( bit_se7788150548672797655nteger @ N @ ( bit_se3222712562003087583nteger @ A @ B ) )
% 8.58/4.86 = ( bit_se3222712562003087583nteger @ ( bit_se7788150548672797655nteger @ N @ A ) @ ( bit_se7788150548672797655nteger @ N @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7603_push__bit__xor,axiom,
% 8.58/4.86 ! [N: nat,A: code_natural,B: code_natural] :
% 8.58/4.86 ( ( bit_se6611745700429515170atural @ N @ ( bit_se2046307713759805098atural @ A @ B ) )
% 8.58/4.86 = ( bit_se2046307713759805098atural @ ( bit_se6611745700429515170atural @ N @ A ) @ ( bit_se6611745700429515170atural @ N @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7604_cosh__0,axiom,
% 8.58/4.86 ( ( cosh_complex @ zero_zero_complex )
% 8.58/4.86 = one_one_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7605_cosh__0,axiom,
% 8.58/4.86 ( ( cosh_real @ zero_zero_real )
% 8.58/4.86 = one_one_real ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7606_gbinomial__0_I2_J,axiom,
% 8.58/4.86 ! [K: nat] :
% 8.58/4.86 ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
% 8.58/4.86 = zero_zero_real ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7607_gbinomial__0_I2_J,axiom,
% 8.58/4.86 ! [K: nat] :
% 8.58/4.86 ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
% 8.58/4.86 = zero_zero_rat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7608_gbinomial__0_I2_J,axiom,
% 8.58/4.86 ! [K: nat] :
% 8.58/4.86 ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
% 8.58/4.86 = zero_zero_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7609_gbinomial__0_I2_J,axiom,
% 8.58/4.86 ! [K: nat] :
% 8.58/4.86 ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
% 8.58/4.86 = zero_zero_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7610_gbinomial__0_I1_J,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( gbinomial_complex @ A @ zero_zero_nat )
% 8.58/4.86 = one_one_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7611_gbinomial__0_I1_J,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( gbinomial_real @ A @ zero_zero_nat )
% 8.58/4.86 = one_one_real ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7612_gbinomial__0_I1_J,axiom,
% 8.58/4.86 ! [A: rat] :
% 8.58/4.86 ( ( gbinomial_rat @ A @ zero_zero_nat )
% 8.58/4.86 = one_one_rat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7613_gbinomial__0_I1_J,axiom,
% 8.58/4.86 ! [A: nat] :
% 8.58/4.86 ( ( gbinomial_nat @ A @ zero_zero_nat )
% 8.58/4.86 = one_one_nat ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7614_gbinomial__0_I1_J,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( gbinomial_int @ A @ zero_zero_nat )
% 8.58/4.86 = one_one_int ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7615_xor__numerals_I3_J,axiom,
% 8.58/4.86 ! [X: num,Y: num] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 8.58/4.86 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7616_xor__numerals_I3_J,axiom,
% 8.58/4.86 ! [X: num,Y: num] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 8.58/4.86 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7617_xor__numerals_I8_J,axiom,
% 8.58/4.86 ! [X: num] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
% 8.58/4.86 = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7618_xor__numerals_I8_J,axiom,
% 8.58/4.86 ! [X: num] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
% 8.58/4.86 = ( numeral_numeral_int @ ( bit0 @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7619_xor__numerals_I5_J,axiom,
% 8.58/4.86 ! [X: num] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 8.58/4.86 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7620_xor__numerals_I5_J,axiom,
% 8.58/4.86 ! [X: num] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 8.58/4.86 = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7621_xor__numerals_I2_J,axiom,
% 8.58/4.86 ! [Y: num] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 8.58/4.86 = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7622_xor__numerals_I2_J,axiom,
% 8.58/4.86 ! [Y: num] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 8.58/4.86 = ( numeral_numeral_int @ ( bit0 @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7623_xor__numerals_I1_J,axiom,
% 8.58/4.86 ! [Y: num] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 8.58/4.86 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7624_xor__numerals_I1_J,axiom,
% 8.58/4.86 ! [Y: num] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 8.58/4.86 = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7625_xor__numerals_I7_J,axiom,
% 8.58/4.86 ! [X: num,Y: num] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 8.58/4.86 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7626_xor__numerals_I7_J,axiom,
% 8.58/4.86 ! [X: num,Y: num] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 8.58/4.86 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7627_xor__nat__numerals_I4_J,axiom,
% 8.58/4.86 ! [X: num] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 8.58/4.86 = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7628_xor__nat__numerals_I3_J,axiom,
% 8.58/4.86 ! [X: num] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 8.58/4.86 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7629_xor__nat__numerals_I2_J,axiom,
% 8.58/4.86 ! [Y: num] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 8.58/4.86 = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7630_xor__nat__numerals_I1_J,axiom,
% 8.58/4.86 ! [Y: num] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 8.58/4.86 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7631_xor__numerals_I6_J,axiom,
% 8.58/4.86 ! [X: num,Y: num] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 8.58/4.86 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7632_xor__numerals_I6_J,axiom,
% 8.58/4.86 ! [X: num,Y: num] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 8.58/4.86 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7633_xor__numerals_I4_J,axiom,
% 8.58/4.86 ! [X: num,Y: num] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 8.58/4.86 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7634_xor__numerals_I4_J,axiom,
% 8.58/4.86 ! [X: num,Y: num] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 8.58/4.86 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7635_of__int__xor__eq,axiom,
% 8.58/4.86 ! [K: int,L: int] :
% 8.58/4.86 ( ( ring_1_of_int_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 8.58/4.86 = ( bit_se6526347334894502574or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7636_of__nat__gbinomial,axiom,
% 8.58/4.86 ! [N: nat,K: nat] :
% 8.58/4.86 ( ( semiri8010041392384452111omplex @ ( gbinomial_nat @ N @ K ) )
% 8.58/4.86 = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7637_of__nat__gbinomial,axiom,
% 8.58/4.86 ! [N: nat,K: nat] :
% 8.58/4.86 ( ( semiri5074537144036343181t_real @ ( gbinomial_nat @ N @ K ) )
% 8.58/4.86 = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7638_of__nat__gbinomial,axiom,
% 8.58/4.86 ! [N: nat,K: nat] :
% 8.58/4.86 ( ( semiri681578069525770553at_rat @ ( gbinomial_nat @ N @ K ) )
% 8.58/4.86 = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7639_xor_Oassoc,axiom,
% 8.58/4.86 ! [A: nat,B: nat,C: nat] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ ( bit_se6528837805403552850or_nat @ A @ B ) @ C )
% 8.58/4.86 = ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B @ C ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7640_xor_Oassoc,axiom,
% 8.58/4.86 ! [A: int,B: int,C: int] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ ( bit_se6526347334894502574or_int @ A @ B ) @ C )
% 8.58/4.86 = ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B @ C ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7641_xor_Ocommute,axiom,
% 8.58/4.86 ( bit_se6528837805403552850or_nat
% 8.58/4.86 = ( ^ [A2: nat,B2: nat] : ( bit_se6528837805403552850or_nat @ B2 @ A2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7642_xor_Ocommute,axiom,
% 8.58/4.86 ( bit_se6526347334894502574or_int
% 8.58/4.86 = ( ^ [A2: int,B2: int] : ( bit_se6526347334894502574or_int @ B2 @ A2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7643_xor_Oleft__commute,axiom,
% 8.58/4.86 ! [B: nat,A: nat,C: nat] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ B @ ( bit_se6528837805403552850or_nat @ A @ C ) )
% 8.58/4.86 = ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B @ C ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7644_xor_Oleft__commute,axiom,
% 8.58/4.86 ! [B: int,A: int,C: int] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ B @ ( bit_se6526347334894502574or_int @ A @ C ) )
% 8.58/4.86 = ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B @ C ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7645_of__nat__xor__eq,axiom,
% 8.58/4.86 ! [M: nat,N: nat] :
% 8.58/4.86 ( ( semiri1316708129612266289at_nat @ ( bit_se6528837805403552850or_nat @ M @ N ) )
% 8.58/4.86 = ( bit_se6528837805403552850or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7646_of__nat__xor__eq,axiom,
% 8.58/4.86 ! [M: nat,N: nat] :
% 8.58/4.86 ( ( semiri1314217659103216013at_int @ ( bit_se6528837805403552850or_nat @ M @ N ) )
% 8.58/4.86 = ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7647_bit_Oconj__xor__distrib,axiom,
% 8.58/4.86 ! [X: int,Y: int,Z2: int] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ X @ ( bit_se6526347334894502574or_int @ Y @ Z2 ) )
% 8.58/4.86 = ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se725231765392027082nd_int @ X @ Z2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7648_bit_Oconj__xor__distrib2,axiom,
% 8.58/4.86 ! [Y: int,Z2: int,X: int] :
% 8.58/4.86 ( ( bit_se725231765392027082nd_int @ ( bit_se6526347334894502574or_int @ Y @ Z2 ) @ X )
% 8.58/4.86 = ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ Y @ X ) @ ( bit_se725231765392027082nd_int @ Z2 @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7649_bit__xor__iff,axiom,
% 8.58/4.86 ! [A: nat,B: nat,N: nat] :
% 8.58/4.86 ( ( bit_se1148574629649215175it_nat @ ( bit_se6528837805403552850or_nat @ A @ B ) @ N )
% 8.58/4.86 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 8.58/4.86 != ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7650_bit__xor__iff,axiom,
% 8.58/4.86 ! [A: int,B: int,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ A @ B ) @ N )
% 8.58/4.86 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 8.58/4.86 != ( bit_se1146084159140164899it_int @ B @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7651_cosh__real__ge__1,axiom,
% 8.58/4.86 ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7652_binomial__gbinomial,axiom,
% 8.58/4.86 ! [N: nat,K: nat] :
% 8.58/4.86 ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 8.58/4.86 = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7653_binomial__gbinomial,axiom,
% 8.58/4.86 ! [N: nat,K: nat] :
% 8.58/4.86 ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 8.58/4.86 = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7654_binomial__gbinomial,axiom,
% 8.58/4.86 ! [N: nat,K: nat] :
% 8.58/4.86 ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 8.58/4.86 = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7655_flip__bit__nat__def,axiom,
% 8.58/4.86 ( bit_se2161824704523386999it_nat
% 8.58/4.86 = ( ^ [M3: nat,N3: nat] : ( bit_se6528837805403552850or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7656_gbinomial__Suc__Suc,axiom,
% 8.58/4.86 ! [A: complex,K: nat] :
% 8.58/4.86 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 8.58/4.86 = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7657_gbinomial__Suc__Suc,axiom,
% 8.58/4.86 ! [A: real,K: nat] :
% 8.58/4.86 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 8.58/4.86 = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7658_gbinomial__Suc__Suc,axiom,
% 8.58/4.86 ! [A: rat,K: nat] :
% 8.58/4.86 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 8.58/4.86 = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7659_gbinomial__of__nat__symmetric,axiom,
% 8.58/4.86 ! [K: nat,N: nat] :
% 8.58/4.86 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.86 => ( ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K )
% 8.58/4.86 = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7660_gbinomial__of__nat__symmetric,axiom,
% 8.58/4.86 ! [K: nat,N: nat] :
% 8.58/4.86 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.86 => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
% 8.58/4.86 = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7661_gbinomial__of__nat__symmetric,axiom,
% 8.58/4.86 ! [K: nat,N: nat] :
% 8.58/4.86 ( ( ord_less_eq_nat @ K @ N )
% 8.58/4.86 => ( ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K )
% 8.58/4.86 = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7662_flip__bit__eq__xor,axiom,
% 8.58/4.86 ( bit_se2161824704523386999it_nat
% 8.58/4.86 = ( ^ [N3: nat,A2: nat] : ( bit_se6528837805403552850or_nat @ A2 @ ( bit_se547839408752420682it_nat @ N3 @ one_one_nat ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7663_flip__bit__eq__xor,axiom,
% 8.58/4.86 ( bit_se2159334234014336723it_int
% 8.58/4.86 = ( ^ [N3: nat,A2: int] : ( bit_se6526347334894502574or_int @ A2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7664_flip__bit__eq__xor,axiom,
% 8.58/4.86 ( bit_se1345352211410354436nteger
% 8.58/4.86 = ( ^ [N3: nat,A2: code_integer] : ( bit_se3222712562003087583nteger @ A2 @ ( bit_se7788150548672797655nteger @ N3 @ one_one_Code_integer ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7665_flip__bit__eq__xor,axiom,
% 8.58/4.86 ( bit_se168947363167071951atural
% 8.58/4.86 = ( ^ [N3: nat,A2: code_natural] : ( bit_se2046307713759805098atural @ A2 @ ( bit_se6611745700429515170atural @ N3 @ one_one_Code_natural ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7666_sinh__add,axiom,
% 8.58/4.86 ! [X: real,Y: real] :
% 8.58/4.86 ( ( sinh_real @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.86 = ( plus_plus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7667_cosh__add,axiom,
% 8.58/4.86 ! [X: real,Y: real] :
% 8.58/4.86 ( ( cosh_real @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.86 = ( plus_plus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7668_sinh__diff,axiom,
% 8.58/4.86 ! [X: real,Y: real] :
% 8.58/4.86 ( ( sinh_real @ ( minus_minus_real @ X @ Y ) )
% 8.58/4.86 = ( minus_minus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7669_cosh__diff,axiom,
% 8.58/4.86 ! [X: real,Y: real] :
% 8.58/4.86 ( ( cosh_real @ ( minus_minus_real @ X @ Y ) )
% 8.58/4.86 = ( minus_minus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7670_sinh__plus__cosh,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( plus_plus_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) )
% 8.58/4.86 = ( exp_complex @ X ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7671_sinh__plus__cosh,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( plus_plus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
% 8.58/4.86 = ( exp_real @ X ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7672_cosh__plus__sinh,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( plus_plus_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ X ) )
% 8.58/4.86 = ( exp_complex @ X ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7673_cosh__plus__sinh,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( plus_plus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
% 8.58/4.86 = ( exp_real @ X ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7674_tanh__def,axiom,
% 8.58/4.86 ( tanh_complex
% 8.58/4.86 = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sinh_complex @ X3 ) @ ( cosh_complex @ X3 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7675_tanh__def,axiom,
% 8.58/4.86 ( tanh_real
% 8.58/4.86 = ( ^ [X3: real] : ( divide_divide_real @ ( sinh_real @ X3 ) @ ( cosh_real @ X3 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7676_gbinomial__addition__formula,axiom,
% 8.58/4.86 ! [A: complex,K: nat] :
% 8.58/4.86 ( ( gbinomial_complex @ A @ ( suc @ K ) )
% 8.58/4.86 = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7677_gbinomial__addition__formula,axiom,
% 8.58/4.86 ! [A: real,K: nat] :
% 8.58/4.86 ( ( gbinomial_real @ A @ ( suc @ K ) )
% 8.58/4.86 = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7678_gbinomial__addition__formula,axiom,
% 8.58/4.86 ! [A: rat,K: nat] :
% 8.58/4.86 ( ( gbinomial_rat @ A @ ( suc @ K ) )
% 8.58/4.86 = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7679_gbinomial__absorb__comp,axiom,
% 8.58/4.86 ! [A: complex,K: nat] :
% 8.58/4.86 ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
% 8.58/4.86 = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7680_gbinomial__absorb__comp,axiom,
% 8.58/4.86 ! [A: real,K: nat] :
% 8.58/4.86 ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
% 8.58/4.86 = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7681_gbinomial__absorb__comp,axiom,
% 8.58/4.86 ! [A: rat,K: nat] :
% 8.58/4.86 ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
% 8.58/4.86 = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7682_gbinomial__ge__n__over__k__pow__k,axiom,
% 8.58/4.86 ! [K: nat,A: real] :
% 8.58/4.86 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
% 8.58/4.86 => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7683_gbinomial__ge__n__over__k__pow__k,axiom,
% 8.58/4.86 ! [K: nat,A: rat] :
% 8.58/4.86 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
% 8.58/4.86 => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7684_gbinomial__mult__1_H,axiom,
% 8.58/4.86 ! [A: complex,K: nat] :
% 8.58/4.86 ( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ A )
% 8.58/4.86 = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7685_gbinomial__mult__1_H,axiom,
% 8.58/4.86 ! [A: real,K: nat] :
% 8.58/4.86 ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
% 8.58/4.86 = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7686_gbinomial__mult__1_H,axiom,
% 8.58/4.86 ! [A: rat,K: nat] :
% 8.58/4.86 ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
% 8.58/4.86 = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7687_gbinomial__mult__1,axiom,
% 8.58/4.86 ! [A: complex,K: nat] :
% 8.58/4.86 ( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K ) )
% 8.58/4.86 = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7688_gbinomial__mult__1,axiom,
% 8.58/4.86 ! [A: real,K: nat] :
% 8.58/4.86 ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
% 8.58/4.86 = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7689_gbinomial__mult__1,axiom,
% 8.58/4.86 ! [A: rat,K: nat] :
% 8.58/4.86 ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
% 8.58/4.86 = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7690_even__xor__iff,axiom,
% 8.58/4.86 ! [A: code_integer,B: code_integer] :
% 8.58/4.86 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ A @ B ) )
% 8.58/4.86 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.58/4.86 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7691_even__xor__iff,axiom,
% 8.58/4.86 ! [A: nat,B: nat] :
% 8.58/4.86 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 8.58/4.86 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.58/4.86 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7692_even__xor__iff,axiom,
% 8.58/4.86 ! [A: int,B: int] :
% 8.58/4.86 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 8.58/4.86 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.58/4.86 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7693_cosh__minus__sinh,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( minus_minus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
% 8.58/4.86 = ( exp_real @ ( uminus_uminus_real @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7694_cosh__minus__sinh,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( minus_minus_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ X ) )
% 8.58/4.86 = ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7695_sinh__minus__cosh,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( minus_minus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
% 8.58/4.86 = ( uminus_uminus_real @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7696_sinh__minus__cosh,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( minus_minus_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) )
% 8.58/4.86 = ( uminus1482373934393186551omplex @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7697_Suc__times__gbinomial,axiom,
% 8.58/4.86 ! [K: nat,A: complex] :
% 8.58/4.86 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
% 8.58/4.86 = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7698_Suc__times__gbinomial,axiom,
% 8.58/4.86 ! [K: nat,A: real] :
% 8.58/4.86 ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
% 8.58/4.86 = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7699_Suc__times__gbinomial,axiom,
% 8.58/4.86 ! [K: nat,A: rat] :
% 8.58/4.86 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
% 8.58/4.86 = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7700_gbinomial__absorption,axiom,
% 8.58/4.86 ! [K: nat,A: complex] :
% 8.58/4.86 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
% 8.58/4.86 = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7701_gbinomial__absorption,axiom,
% 8.58/4.86 ! [K: nat,A: real] :
% 8.58/4.86 ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
% 8.58/4.86 = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7702_gbinomial__absorption,axiom,
% 8.58/4.86 ! [K: nat,A: rat] :
% 8.58/4.86 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
% 8.58/4.86 = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7703_gbinomial__trinomial__revision,axiom,
% 8.58/4.86 ! [K: nat,M: nat,A: complex] :
% 8.58/4.86 ( ( ord_less_eq_nat @ K @ M )
% 8.58/4.86 => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
% 8.58/4.86 = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7704_gbinomial__trinomial__revision,axiom,
% 8.58/4.86 ! [K: nat,M: nat,A: real] :
% 8.58/4.86 ( ( ord_less_eq_nat @ K @ M )
% 8.58/4.86 => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
% 8.58/4.86 = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7705_gbinomial__trinomial__revision,axiom,
% 8.58/4.86 ! [K: nat,M: nat,A: rat] :
% 8.58/4.86 ( ( ord_less_eq_nat @ K @ M )
% 8.58/4.86 => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
% 8.58/4.86 = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7706_sinh__double,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( sinh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 8.58/4.86 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sinh_complex @ X ) ) @ ( cosh_complex @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7707_sinh__double,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( sinh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 8.58/4.86 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sinh_real @ X ) ) @ ( cosh_real @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7708_gbinomial__rec,axiom,
% 8.58/4.86 ! [A: complex,K: nat] :
% 8.58/4.86 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 8.58/4.86 = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7709_gbinomial__rec,axiom,
% 8.58/4.86 ! [A: real,K: nat] :
% 8.58/4.86 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 8.58/4.86 = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7710_gbinomial__rec,axiom,
% 8.58/4.86 ! [A: rat,K: nat] :
% 8.58/4.86 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 8.58/4.86 = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7711_gbinomial__factors,axiom,
% 8.58/4.86 ! [A: complex,K: nat] :
% 8.58/4.86 ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 8.58/4.86 = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7712_gbinomial__factors,axiom,
% 8.58/4.86 ! [A: real,K: nat] :
% 8.58/4.86 ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 8.58/4.86 = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7713_gbinomial__factors,axiom,
% 8.58/4.86 ! [A: rat,K: nat] :
% 8.58/4.86 ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 8.58/4.86 = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7714_gbinomial__negated__upper,axiom,
% 8.58/4.86 ( gbinomial_complex
% 8.58/4.86 = ( ^ [A2: complex,K2: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K2 ) @ A2 ) @ one_one_complex ) @ K2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7715_gbinomial__negated__upper,axiom,
% 8.58/4.86 ( gbinomial_real
% 8.58/4.86 = ( ^ [A2: real,K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K2 ) @ A2 ) @ one_one_real ) @ K2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7716_gbinomial__negated__upper,axiom,
% 8.58/4.86 ( gbinomial_rat
% 8.58/4.86 = ( ^ [A2: rat,K2: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K2 ) @ A2 ) @ one_one_rat ) @ K2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7717_gbinomial__index__swap,axiom,
% 8.58/4.86 ! [K: nat,N: nat] :
% 8.58/4.86 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
% 8.58/4.86 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7718_gbinomial__index__swap,axiom,
% 8.58/4.86 ! [K: nat,N: nat] :
% 8.58/4.86 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ K ) )
% 8.58/4.86 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7719_gbinomial__index__swap,axiom,
% 8.58/4.86 ! [K: nat,N: nat] :
% 8.58/4.86 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ K ) )
% 8.58/4.86 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7720_tanh__add,axiom,
% 8.58/4.86 ! [X: complex,Y: complex] :
% 8.58/4.86 ( ( ( cosh_complex @ X )
% 8.58/4.86 != zero_zero_complex )
% 8.58/4.86 => ( ( ( cosh_complex @ Y )
% 8.58/4.86 != zero_zero_complex )
% 8.58/4.86 => ( ( tanh_complex @ ( plus_plus_complex @ X @ Y ) )
% 8.58/4.86 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y ) ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7721_tanh__add,axiom,
% 8.58/4.86 ! [X: real,Y: real] :
% 8.58/4.86 ( ( ( cosh_real @ X )
% 8.58/4.86 != zero_zero_real )
% 8.58/4.86 => ( ( ( cosh_real @ Y )
% 8.58/4.86 != zero_zero_real )
% 8.58/4.86 => ( ( tanh_real @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.86 = ( divide_divide_real @ ( plus_plus_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7722_gbinomial__minus,axiom,
% 8.58/4.86 ! [A: complex,K: nat] :
% 8.58/4.86 ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
% 8.58/4.86 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7723_gbinomial__minus,axiom,
% 8.58/4.86 ! [A: real,K: nat] :
% 8.58/4.86 ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
% 8.58/4.86 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7724_gbinomial__minus,axiom,
% 8.58/4.86 ! [A: rat,K: nat] :
% 8.58/4.86 ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
% 8.58/4.86 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7725_gbinomial__reduce__nat,axiom,
% 8.58/4.86 ! [K: nat,A: complex] :
% 8.58/4.86 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.58/4.86 => ( ( gbinomial_complex @ A @ K )
% 8.58/4.86 = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7726_gbinomial__reduce__nat,axiom,
% 8.58/4.86 ! [K: nat,A: real] :
% 8.58/4.86 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.58/4.86 => ( ( gbinomial_real @ A @ K )
% 8.58/4.86 = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7727_gbinomial__reduce__nat,axiom,
% 8.58/4.86 ! [K: nat,A: rat] :
% 8.58/4.86 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.58/4.86 => ( ( gbinomial_rat @ A @ K )
% 8.58/4.86 = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7728_gbinomial__pochhammer,axiom,
% 8.58/4.86 ( gbinomial_complex
% 8.58/4.86 = ( ^ [A2: complex,K2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A2 ) @ K2 ) ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7729_gbinomial__pochhammer,axiom,
% 8.58/4.86 ( gbinomial_rat
% 8.58/4.86 = ( ^ [A2: rat,K2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A2 ) @ K2 ) ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7730_gbinomial__pochhammer,axiom,
% 8.58/4.86 ( gbinomial_real
% 8.58/4.86 = ( ^ [A2: real,K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A2 ) @ K2 ) ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7731_gbinomial__pochhammer_H,axiom,
% 8.58/4.86 ( gbinomial_complex
% 8.58/4.86 = ( ^ [A2: complex,K2: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A2 @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7732_gbinomial__pochhammer_H,axiom,
% 8.58/4.86 ( gbinomial_rat
% 8.58/4.86 = ( ^ [A2: rat,K2: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7733_gbinomial__pochhammer_H,axiom,
% 8.58/4.86 ( gbinomial_real
% 8.58/4.86 = ( ^ [A2: real,K2: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A2 @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7734_cosh__field__def,axiom,
% 8.58/4.86 ( cosh_real
% 8.58/4.86 = ( ^ [Z5: real] : ( divide_divide_real @ ( plus_plus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7735_cosh__field__def,axiom,
% 8.58/4.86 ( cosh_complex
% 8.58/4.86 = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7736_xor__nat__unfold,axiom,
% 8.58/4.86 ( bit_se6528837805403552850or_nat
% 8.58/4.86 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7737_cosh__square__eq,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.86 = ( plus_plus_real @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7738_cosh__square__eq,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.86 = ( plus_plus_complex @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7739_sinh__square__eq,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.86 = ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7740_sinh__square__eq,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.86 = ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7741_hyperbolic__pythagoras,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.86 = one_one_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7742_hyperbolic__pythagoras,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.86 = one_one_real ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7743_xor__nat__rec,axiom,
% 8.58/4.86 ( bit_se6528837805403552850or_nat
% 8.58/4.86 = ( ^ [M3: nat,N3: nat] :
% 8.58/4.86 ( plus_plus_nat
% 8.58/4.86 @ ( zero_n2687167440665602831ol_nat
% 8.58/4.86 @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 8.58/4.86 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 8.58/4.86 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7744_xor__one__eq,axiom,
% 8.58/4.86 ! [A: code_integer] :
% 8.58/4.86 ( ( bit_se3222712562003087583nteger @ A @ one_one_Code_integer )
% 8.58/4.86 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 8.58/4.86 @ ( zero_n356916108424825756nteger
% 8.58/4.86 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7745_xor__one__eq,axiom,
% 8.58/4.86 ! [A: nat] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
% 8.58/4.86 = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 8.58/4.86 @ ( zero_n2687167440665602831ol_nat
% 8.58/4.86 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7746_xor__one__eq,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ A @ one_one_int )
% 8.58/4.86 = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 8.58/4.86 @ ( zero_n2684676970156552555ol_int
% 8.58/4.86 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7747_one__xor__eq,axiom,
% 8.58/4.86 ! [A: code_integer] :
% 8.58/4.86 ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A )
% 8.58/4.86 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 8.58/4.86 @ ( zero_n356916108424825756nteger
% 8.58/4.86 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7748_one__xor__eq,axiom,
% 8.58/4.86 ! [A: nat] :
% 8.58/4.86 ( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
% 8.58/4.86 = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 8.58/4.86 @ ( zero_n2687167440665602831ol_nat
% 8.58/4.86 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7749_one__xor__eq,axiom,
% 8.58/4.86 ! [A: int] :
% 8.58/4.86 ( ( bit_se6526347334894502574or_int @ one_one_int @ A )
% 8.58/4.86 = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 8.58/4.86 @ ( zero_n2684676970156552555ol_int
% 8.58/4.86 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7750_cosh__zero__iff,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( ( cosh_real @ X )
% 8.58/4.86 = zero_zero_real )
% 8.58/4.86 = ( ( power_power_real @ ( exp_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.86 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7751_cosh__zero__iff,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( ( cosh_complex @ X )
% 8.58/4.86 = zero_zero_complex )
% 8.58/4.86 = ( ( power_power_complex @ ( exp_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.86 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7752_Cauchy__iff2,axiom,
% 8.58/4.86 ( topolo4055970368930404560y_real
% 8.58/4.86 = ( ^ [X8: nat > real] :
% 8.58/4.86 ! [J3: nat] :
% 8.58/4.86 ? [M7: nat] :
% 8.58/4.86 ! [M3: nat] :
% 8.58/4.86 ( ( ord_less_eq_nat @ M7 @ M3 )
% 8.58/4.86 => ! [N3: nat] :
% 8.58/4.86 ( ( ord_less_eq_nat @ M7 @ N3 )
% 8.58/4.86 => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7753_csqrt_Osimps_I1_J,axiom,
% 8.58/4.86 ! [Z2: complex] :
% 8.58/4.86 ( ( re @ ( csqrt @ Z2 ) )
% 8.58/4.86 = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7754_sinh__def,axiom,
% 8.58/4.86 ( sinh_real
% 8.58/4.86 = ( ^ [X3: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7755_sinh__def,axiom,
% 8.58/4.86 ( sinh_complex
% 8.58/4.86 = ( ^ [X3: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( minus_minus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7756_cosh__def,axiom,
% 8.58/4.86 ( cosh_real
% 8.58/4.86 = ( ^ [X3: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7757_cosh__def,axiom,
% 8.58/4.86 ( cosh_complex
% 8.58/4.86 = ( ^ [X3: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7758_complex__div__cnj,axiom,
% 8.58/4.86 ( divide1717551699836669952omplex
% 8.58/4.86 = ( ^ [A2: complex,B2: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ ( cnj @ B2 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7759_scaleR__cancel__right,axiom,
% 8.58/4.86 ! [A: real,X: real,B: real] :
% 8.58/4.86 ( ( ( real_V1485227260804924795R_real @ A @ X )
% 8.58/4.86 = ( real_V1485227260804924795R_real @ B @ X ) )
% 8.58/4.86 = ( ( A = B )
% 8.58/4.86 | ( X = zero_zero_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7760_scaleR__cancel__right,axiom,
% 8.58/4.86 ! [A: real,X: complex,B: real] :
% 8.58/4.86 ( ( ( real_V2046097035970521341omplex @ A @ X )
% 8.58/4.86 = ( real_V2046097035970521341omplex @ B @ X ) )
% 8.58/4.86 = ( ( A = B )
% 8.58/4.86 | ( X = zero_zero_complex ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7761_scaleR__zero__right,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( real_V1485227260804924795R_real @ A @ zero_zero_real )
% 8.58/4.86 = zero_zero_real ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7762_scaleR__zero__right,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( real_V2046097035970521341omplex @ A @ zero_zero_complex )
% 8.58/4.86 = zero_zero_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7763_mult__scaleR__right,axiom,
% 8.58/4.86 ! [X: real,A: real,Y: real] :
% 8.58/4.86 ( ( times_times_real @ X @ ( real_V1485227260804924795R_real @ A @ Y ) )
% 8.58/4.86 = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7764_mult__scaleR__right,axiom,
% 8.58/4.86 ! [X: complex,A: real,Y: complex] :
% 8.58/4.86 ( ( times_times_complex @ X @ ( real_V2046097035970521341omplex @ A @ Y ) )
% 8.58/4.86 = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7765_mult__scaleR__left,axiom,
% 8.58/4.86 ! [A: real,X: real,Y: real] :
% 8.58/4.86 ( ( times_times_real @ ( real_V1485227260804924795R_real @ A @ X ) @ Y )
% 8.58/4.86 = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7766_mult__scaleR__left,axiom,
% 8.58/4.86 ! [A: real,X: complex,Y: complex] :
% 8.58/4.86 ( ( times_times_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ Y )
% 8.58/4.86 = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7767_scaleR__one,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( real_V1485227260804924795R_real @ one_one_real @ X )
% 8.58/4.86 = X ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7768_scaleR__one,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( real_V2046097035970521341omplex @ one_one_real @ X )
% 8.58/4.86 = X ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7769_scaleR__scaleR,axiom,
% 8.58/4.86 ! [A: real,B: real,X: real] :
% 8.58/4.86 ( ( real_V1485227260804924795R_real @ A @ ( real_V1485227260804924795R_real @ B @ X ) )
% 8.58/4.86 = ( real_V1485227260804924795R_real @ ( times_times_real @ A @ B ) @ X ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7770_scaleR__scaleR,axiom,
% 8.58/4.86 ! [A: real,B: real,X: complex] :
% 8.58/4.86 ( ( real_V2046097035970521341omplex @ A @ ( real_V2046097035970521341omplex @ B @ X ) )
% 8.58/4.86 = ( real_V2046097035970521341omplex @ ( times_times_real @ A @ B ) @ X ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7771_complex__Re__of__nat,axiom,
% 8.58/4.86 ! [N: nat] :
% 8.58/4.86 ( ( re @ ( semiri8010041392384452111omplex @ N ) )
% 8.58/4.86 = ( semiri5074537144036343181t_real @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7772_complex__cnj__mult,axiom,
% 8.58/4.86 ! [X: complex,Y: complex] :
% 8.58/4.86 ( ( cnj @ ( times_times_complex @ X @ Y ) )
% 8.58/4.86 = ( times_times_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7773_complex__cnj__one__iff,axiom,
% 8.58/4.86 ! [Z2: complex] :
% 8.58/4.86 ( ( ( cnj @ Z2 )
% 8.58/4.86 = one_one_complex )
% 8.58/4.86 = ( Z2 = one_one_complex ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7774_complex__cnj__one,axiom,
% 8.58/4.86 ( ( cnj @ one_one_complex )
% 8.58/4.86 = one_one_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7775_complex__cnj__power,axiom,
% 8.58/4.86 ! [X: complex,N: nat] :
% 8.58/4.86 ( ( cnj @ ( power_power_complex @ X @ N ) )
% 8.58/4.86 = ( power_power_complex @ ( cnj @ X ) @ N ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7776_complex__cnj__add,axiom,
% 8.58/4.86 ! [X: complex,Y: complex] :
% 8.58/4.86 ( ( cnj @ ( plus_plus_complex @ X @ Y ) )
% 8.58/4.86 = ( plus_plus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7777_complex__cnj__diff,axiom,
% 8.58/4.86 ! [X: complex,Y: complex] :
% 8.58/4.86 ( ( cnj @ ( minus_minus_complex @ X @ Y ) )
% 8.58/4.86 = ( minus_minus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7778_scaleR__eq__0__iff,axiom,
% 8.58/4.86 ! [A: real,X: real] :
% 8.58/4.86 ( ( ( real_V1485227260804924795R_real @ A @ X )
% 8.58/4.86 = zero_zero_real )
% 8.58/4.86 = ( ( A = zero_zero_real )
% 8.58/4.86 | ( X = zero_zero_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7779_scaleR__eq__0__iff,axiom,
% 8.58/4.86 ! [A: real,X: complex] :
% 8.58/4.86 ( ( ( real_V2046097035970521341omplex @ A @ X )
% 8.58/4.86 = zero_zero_complex )
% 8.58/4.86 = ( ( A = zero_zero_real )
% 8.58/4.86 | ( X = zero_zero_complex ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7780_scaleR__zero__left,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
% 8.58/4.86 = zero_zero_real ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7781_scaleR__zero__left,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( real_V2046097035970521341omplex @ zero_zero_real @ X )
% 8.58/4.86 = zero_zero_complex ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7782_scaleR__eq__iff,axiom,
% 8.58/4.86 ! [B: real,U: real,A: real] :
% 8.58/4.86 ( ( ( plus_plus_real @ B @ ( real_V1485227260804924795R_real @ U @ A ) )
% 8.58/4.86 = ( plus_plus_real @ A @ ( real_V1485227260804924795R_real @ U @ B ) ) )
% 8.58/4.86 = ( ( A = B )
% 8.58/4.86 | ( U = one_one_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7783_scaleR__eq__iff,axiom,
% 8.58/4.86 ! [B: complex,U: real,A: complex] :
% 8.58/4.86 ( ( ( plus_plus_complex @ B @ ( real_V2046097035970521341omplex @ U @ A ) )
% 8.58/4.86 = ( plus_plus_complex @ A @ ( real_V2046097035970521341omplex @ U @ B ) ) )
% 8.58/4.86 = ( ( A = B )
% 8.58/4.86 | ( U = one_one_real ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7784_scaleR__power,axiom,
% 8.58/4.86 ! [X: real,Y: real,N: nat] :
% 8.58/4.86 ( ( power_power_real @ ( real_V1485227260804924795R_real @ X @ Y ) @ N )
% 8.58/4.86 = ( real_V1485227260804924795R_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7785_scaleR__power,axiom,
% 8.58/4.86 ! [X: real,Y: complex,N: nat] :
% 8.58/4.86 ( ( power_power_complex @ ( real_V2046097035970521341omplex @ X @ Y ) @ N )
% 8.58/4.86 = ( real_V2046097035970521341omplex @ ( power_power_real @ X @ N ) @ ( power_power_complex @ Y @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7786_Re__sgn,axiom,
% 8.58/4.86 ! [Z2: complex] :
% 8.58/4.86 ( ( re @ ( sgn_sgn_complex @ Z2 ) )
% 8.58/4.86 = ( divide_divide_real @ ( re @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7787_xor__nonnegative__int__iff,axiom,
% 8.58/4.86 ! [K: int,L: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 8.58/4.86 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.58/4.86 = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7788_xor__negative__int__iff,axiom,
% 8.58/4.86 ! [K: int,L: int] :
% 8.58/4.86 ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
% 8.58/4.86 = ( ( ord_less_int @ K @ zero_zero_int )
% 8.58/4.86 != ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7789_scaleR__minus1__left,axiom,
% 8.58/4.86 ! [X: real] :
% 8.58/4.86 ( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.86 = ( uminus_uminus_real @ X ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7790_scaleR__minus1__left,axiom,
% 8.58/4.86 ! [X: complex] :
% 8.58/4.86 ( ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.86 = ( uminus1482373934393186551omplex @ X ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7791_scaleR__collapse,axiom,
% 8.58/4.86 ! [U: real,A: real] :
% 8.58/4.86 ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V1485227260804924795R_real @ U @ A ) )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7792_scaleR__collapse,axiom,
% 8.58/4.86 ! [U: real,A: complex] :
% 8.58/4.86 ( ( plus_plus_complex @ ( real_V2046097035970521341omplex @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V2046097035970521341omplex @ U @ A ) )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7793_norm__scaleR,axiom,
% 8.58/4.86 ! [A: real,X: real] :
% 8.58/4.86 ( ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ A @ X ) )
% 8.58/4.86 = ( times_times_real @ ( abs_abs_real @ A ) @ ( real_V7735802525324610683m_real @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7794_norm__scaleR,axiom,
% 8.58/4.86 ! [A: real,X: complex] :
% 8.58/4.86 ( ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
% 8.58/4.86 = ( times_times_real @ ( abs_abs_real @ A ) @ ( real_V1022390504157884413omplex @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7795_Re__divide__of__nat,axiom,
% 8.58/4.86 ! [Z2: complex,N: nat] :
% 8.58/4.86 ( ( re @ ( divide1717551699836669952omplex @ Z2 @ ( semiri8010041392384452111omplex @ N ) ) )
% 8.58/4.86 = ( divide_divide_real @ ( re @ Z2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7796_Re__divide__of__real,axiom,
% 8.58/4.86 ! [Z2: complex,R: real] :
% 8.58/4.86 ( ( re @ ( divide1717551699836669952omplex @ Z2 @ ( real_V4546457046886955230omplex @ R ) ) )
% 8.58/4.86 = ( divide_divide_real @ ( re @ Z2 ) @ R ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7797_scaleR__times,axiom,
% 8.58/4.86 ! [U: num,W: num,A: real] :
% 8.58/4.86 ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ U ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 8.58/4.86 = ( real_V1485227260804924795R_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7798_scaleR__times,axiom,
% 8.58/4.86 ! [U: num,W: num,A: complex] :
% 8.58/4.86 ( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ U ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 8.58/4.86 = ( real_V2046097035970521341omplex @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7799_Re__divide__numeral,axiom,
% 8.58/4.86 ! [Z2: complex,W: num] :
% 8.58/4.86 ( ( re @ ( divide1717551699836669952omplex @ Z2 @ ( numera6690914467698888265omplex @ W ) ) )
% 8.58/4.86 = ( divide_divide_real @ ( re @ Z2 ) @ ( numeral_numeral_real @ W ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7800_inverse__scaleR__times,axiom,
% 8.58/4.86 ! [V: num,W: num,A: real] :
% 8.58/4.86 ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 8.58/4.86 = ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7801_inverse__scaleR__times,axiom,
% 8.58/4.86 ! [V: num,W: num,A: complex] :
% 8.58/4.86 ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 8.58/4.86 = ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7802_fraction__scaleR__times,axiom,
% 8.58/4.86 ! [U: num,V: num,W: num,A: real] :
% 8.58/4.86 ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 8.58/4.86 = ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7803_fraction__scaleR__times,axiom,
% 8.58/4.86 ! [U: num,V: num,W: num,A: complex] :
% 8.58/4.86 ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 8.58/4.86 = ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7804_scaleR__half__double,axiom,
% 8.58/4.86 ! [A: real] :
% 8.58/4.86 ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ A @ A ) )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7805_scaleR__half__double,axiom,
% 8.58/4.86 ! [A: complex] :
% 8.58/4.86 ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_complex @ A @ A ) )
% 8.58/4.86 = A ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7806_bit__xor__int__iff,axiom,
% 8.58/4.86 ! [K: int,L: int,N: nat] :
% 8.58/4.86 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ N )
% 8.58/4.86 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 8.58/4.86 != ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7807_real__scaleR__def,axiom,
% 8.58/4.86 real_V1485227260804924795R_real = times_times_real ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7808_scaleR__right__imp__eq,axiom,
% 8.58/4.86 ! [X: real,A: real,B: real] :
% 8.58/4.86 ( ( X != zero_zero_real )
% 8.58/4.86 => ( ( ( real_V1485227260804924795R_real @ A @ X )
% 8.58/4.86 = ( real_V1485227260804924795R_real @ B @ X ) )
% 8.58/4.86 => ( A = B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7809_scaleR__right__imp__eq,axiom,
% 8.58/4.86 ! [X: complex,A: real,B: real] :
% 8.58/4.86 ( ( X != zero_zero_complex )
% 8.58/4.86 => ( ( ( real_V2046097035970521341omplex @ A @ X )
% 8.58/4.86 = ( real_V2046097035970521341omplex @ B @ X ) )
% 8.58/4.86 => ( A = B ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7810_scaleR__right__distrib,axiom,
% 8.58/4.86 ! [A: real,X: real,Y: real] :
% 8.58/4.86 ( ( real_V1485227260804924795R_real @ A @ ( plus_plus_real @ X @ Y ) )
% 8.58/4.86 = ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7811_scaleR__right__distrib,axiom,
% 8.58/4.86 ! [A: real,X: complex,Y: complex] :
% 8.58/4.86 ( ( real_V2046097035970521341omplex @ A @ ( plus_plus_complex @ X @ Y ) )
% 8.58/4.86 = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ A @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7812_scaleR__right__diff__distrib,axiom,
% 8.58/4.86 ! [A: real,X: real,Y: real] :
% 8.58/4.86 ( ( real_V1485227260804924795R_real @ A @ ( minus_minus_real @ X @ Y ) )
% 8.58/4.86 = ( minus_minus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7813_scaleR__right__diff__distrib,axiom,
% 8.58/4.86 ! [A: real,X: complex,Y: complex] :
% 8.58/4.86 ( ( real_V2046097035970521341omplex @ A @ ( minus_minus_complex @ X @ Y ) )
% 8.58/4.86 = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ A @ Y ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7814_scaleR__complex_Osimps_I1_J,axiom,
% 8.58/4.86 ! [R: real,X: complex] :
% 8.58/4.86 ( ( re @ ( real_V2046097035970521341omplex @ R @ X ) )
% 8.58/4.86 = ( times_times_real @ R @ ( re @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7815_Re__complex__div__eq__0,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.58/4.86 = zero_zero_real )
% 8.58/4.86 = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 8.58/4.86 = zero_zero_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7816_complex__mod__sqrt__Re__mult__cnj,axiom,
% 8.58/4.86 ( real_V1022390504157884413omplex
% 8.58/4.86 = ( ^ [Z5: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z5 @ ( cnj @ Z5 ) ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7817_Re__complex__div__lt__0,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 8.58/4.86 = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7818_Re__complex__div__gt__0,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 8.58/4.86 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7819_Re__complex__div__le__0,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 8.58/4.86 = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7820_Re__complex__div__ge__0,axiom,
% 8.58/4.86 ! [A: complex,B: complex] :
% 8.58/4.86 ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 8.58/4.86 = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7821_XOR__lower,axiom,
% 8.58/4.86 ! [X: int,Y: int] :
% 8.58/4.86 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.58/4.86 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.58/4.86 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7822_scaleR__left_Oadd,axiom,
% 8.58/4.86 ! [X: real,Y: real,Xa: real] :
% 8.58/4.86 ( ( real_V1485227260804924795R_real @ ( plus_plus_real @ X @ Y ) @ Xa )
% 8.58/4.86 = ( plus_plus_real @ ( real_V1485227260804924795R_real @ X @ Xa ) @ ( real_V1485227260804924795R_real @ Y @ Xa ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7823_scaleR__left_Oadd,axiom,
% 8.58/4.86 ! [X: real,Y: real,Xa: complex] :
% 8.58/4.86 ( ( real_V2046097035970521341omplex @ ( plus_plus_real @ X @ Y ) @ Xa )
% 8.58/4.86 = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ X @ Xa ) @ ( real_V2046097035970521341omplex @ Y @ Xa ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7824_scaleR__left__distrib,axiom,
% 8.58/4.86 ! [A: real,B: real,X: real] :
% 8.58/4.86 ( ( real_V1485227260804924795R_real @ ( plus_plus_real @ A @ B ) @ X )
% 8.58/4.86 = ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7825_scaleR__left__distrib,axiom,
% 8.58/4.86 ! [A: real,B: real,X: complex] :
% 8.58/4.86 ( ( real_V2046097035970521341omplex @ ( plus_plus_real @ A @ B ) @ X )
% 8.58/4.86 = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ B @ X ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7826_scaleR__conv__of__real,axiom,
% 8.58/4.86 ( real_V1485227260804924795R_real
% 8.58/4.86 = ( ^ [R5: real] : ( times_times_real @ ( real_V1803761363581548252l_real @ R5 ) ) ) ) ).
% 8.58/4.86
% 8.58/4.86 thf(fact_7827_scaleR__conv__of__real,axiom,
% 8.58/4.86 ( real_V2046097035970521341omplex
% 8.58/4.86 = ( ^ [R5: real] : ( times_times_complex @ ( real_V4546457046886955230omplex @ R5 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7828_of__real__def,axiom,
% 8.58/4.87 ( real_V1803761363581548252l_real
% 8.58/4.87 = ( ^ [R5: real] : ( real_V1485227260804924795R_real @ R5 @ one_one_real ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7829_of__real__def,axiom,
% 8.58/4.87 ( real_V4546457046886955230omplex
% 8.58/4.87 = ( ^ [R5: real] : ( real_V2046097035970521341omplex @ R5 @ one_one_complex ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7830_scaleR__left_Odiff,axiom,
% 8.58/4.87 ! [X: real,Y: real,Xa: real] :
% 8.58/4.87 ( ( real_V1485227260804924795R_real @ ( minus_minus_real @ X @ Y ) @ Xa )
% 8.58/4.87 = ( minus_minus_real @ ( real_V1485227260804924795R_real @ X @ Xa ) @ ( real_V1485227260804924795R_real @ Y @ Xa ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7831_scaleR__left_Odiff,axiom,
% 8.58/4.87 ! [X: real,Y: real,Xa: complex] :
% 8.58/4.87 ( ( real_V2046097035970521341omplex @ ( minus_minus_real @ X @ Y ) @ Xa )
% 8.58/4.87 = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ X @ Xa ) @ ( real_V2046097035970521341omplex @ Y @ Xa ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7832_scaleR__left__diff__distrib,axiom,
% 8.58/4.87 ! [A: real,B: real,X: real] :
% 8.58/4.87 ( ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ X )
% 8.58/4.87 = ( minus_minus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7833_scaleR__left__diff__distrib,axiom,
% 8.58/4.87 ! [A: real,B: real,X: complex] :
% 8.58/4.87 ( ( real_V2046097035970521341omplex @ ( minus_minus_real @ A @ B ) @ X )
% 8.58/4.87 = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ B @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7834_complex__scaleR,axiom,
% 8.58/4.87 ! [R: real,A: real,B: real] :
% 8.58/4.87 ( ( real_V2046097035970521341omplex @ R @ ( complex2 @ A @ B ) )
% 8.58/4.87 = ( complex2 @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7835_one__complex_Osimps_I1_J,axiom,
% 8.58/4.87 ( ( re @ one_one_complex )
% 8.58/4.87 = one_one_real ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7836_plus__complex_Osimps_I1_J,axiom,
% 8.58/4.87 ! [X: complex,Y: complex] :
% 8.58/4.87 ( ( re @ ( plus_plus_complex @ X @ Y ) )
% 8.58/4.87 = ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7837_minus__complex_Osimps_I1_J,axiom,
% 8.58/4.87 ! [X: complex,Y: complex] :
% 8.58/4.87 ( ( re @ ( minus_minus_complex @ X @ Y ) )
% 8.58/4.87 = ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7838_scaleR__right__mono__neg,axiom,
% 8.58/4.87 ! [B: real,A: real,C: real] :
% 8.58/4.87 ( ( ord_less_eq_real @ B @ A )
% 8.58/4.87 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.58/4.87 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ C ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7839_scaleR__right__mono,axiom,
% 8.58/4.87 ! [A: real,B: real,X: real] :
% 8.58/4.87 ( ( ord_less_eq_real @ A @ B )
% 8.58/4.87 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.87 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7840_complex__add__cnj,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( plus_plus_complex @ Z2 @ ( cnj @ Z2 ) )
% 8.58/4.87 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7841_vector__fraction__eq__iff,axiom,
% 8.58/4.87 ! [U: real,V: real,A: real,X: real] :
% 8.58/4.87 ( ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A )
% 8.58/4.87 = X )
% 8.58/4.87 = ( ( ( V = zero_zero_real )
% 8.58/4.87 => ( X = zero_zero_real ) )
% 8.58/4.87 & ( ( V != zero_zero_real )
% 8.58/4.87 => ( ( real_V1485227260804924795R_real @ U @ A )
% 8.58/4.87 = ( real_V1485227260804924795R_real @ V @ X ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7842_vector__fraction__eq__iff,axiom,
% 8.58/4.87 ! [U: real,V: real,A: complex,X: complex] :
% 8.58/4.87 ( ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A )
% 8.58/4.87 = X )
% 8.58/4.87 = ( ( ( V = zero_zero_real )
% 8.58/4.87 => ( X = zero_zero_complex ) )
% 8.58/4.87 & ( ( V != zero_zero_real )
% 8.58/4.87 => ( ( real_V2046097035970521341omplex @ U @ A )
% 8.58/4.87 = ( real_V2046097035970521341omplex @ V @ X ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7843_eq__vector__fraction__iff,axiom,
% 8.58/4.87 ! [X: real,U: real,V: real,A: real] :
% 8.58/4.87 ( ( X
% 8.58/4.87 = ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A ) )
% 8.58/4.87 = ( ( ( V = zero_zero_real )
% 8.58/4.87 => ( X = zero_zero_real ) )
% 8.58/4.87 & ( ( V != zero_zero_real )
% 8.58/4.87 => ( ( real_V1485227260804924795R_real @ V @ X )
% 8.58/4.87 = ( real_V1485227260804924795R_real @ U @ A ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7844_eq__vector__fraction__iff,axiom,
% 8.58/4.87 ! [X: complex,U: real,V: real,A: complex] :
% 8.58/4.87 ( ( X
% 8.58/4.87 = ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A ) )
% 8.58/4.87 = ( ( ( V = zero_zero_real )
% 8.58/4.87 => ( X = zero_zero_complex ) )
% 8.58/4.87 & ( ( V != zero_zero_real )
% 8.58/4.87 => ( ( real_V2046097035970521341omplex @ V @ X )
% 8.58/4.87 = ( real_V2046097035970521341omplex @ U @ A ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7845_Real__Vector__Spaces_Ole__add__iff2,axiom,
% 8.58/4.87 ! [A: real,E: real,C: real,B: real,D: real] :
% 8.58/4.87 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E ) @ D ) )
% 8.58/4.87 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7846_Real__Vector__Spaces_Ole__add__iff1,axiom,
% 8.58/4.87 ! [A: real,E: real,C: real,B: real,D: real] :
% 8.58/4.87 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E ) @ D ) )
% 8.58/4.87 = ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7847_xor__nat__def,axiom,
% 8.58/4.87 ( bit_se6528837805403552850or_nat
% 8.58/4.87 = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7848_cnj__add__mult__eq__Re,axiom,
% 8.58/4.87 ! [Z2: complex,W: complex] :
% 8.58/4.87 ( ( plus_plus_complex @ ( times_times_complex @ Z2 @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z2 ) @ W ) )
% 8.58/4.87 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z2 @ ( cnj @ W ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7849_flip__bit__int__def,axiom,
% 8.58/4.87 ( bit_se2159334234014336723it_int
% 8.58/4.87 = ( ^ [N3: nat,K2: int] : ( bit_se6526347334894502574or_int @ K2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7850_zero__le__scaleR__iff,axiom,
% 8.58/4.87 ! [A: real,B: real] :
% 8.58/4.87 ( ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) )
% 8.58/4.87 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.87 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 8.58/4.87 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.58/4.87 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 8.58/4.87 | ( A = zero_zero_real ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7851_scaleR__le__0__iff,axiom,
% 8.58/4.87 ! [A: real,B: real] :
% 8.58/4.87 ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ B ) @ zero_zero_real )
% 8.58/4.87 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.87 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 8.58/4.87 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.58/4.87 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 8.58/4.87 | ( A = zero_zero_real ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7852_scaleR__nonpos__nonpos,axiom,
% 8.58/4.87 ! [A: real,B: real] :
% 8.58/4.87 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.58/4.87 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 8.58/4.87 => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7853_scaleR__nonpos__nonneg,axiom,
% 8.58/4.87 ! [A: real,X: real] :
% 8.58/4.87 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.58/4.87 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.87 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7854_scaleR__nonneg__nonpos,axiom,
% 8.58/4.87 ! [A: real,X: real] :
% 8.58/4.87 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.87 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 8.58/4.87 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7855_scaleR__nonneg__nonneg,axiom,
% 8.58/4.87 ! [A: real,X: real] :
% 8.58/4.87 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.87 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.87 => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7856_split__scaleR__pos__le,axiom,
% 8.58/4.87 ! [A: real,B: real] :
% 8.58/4.87 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.87 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 8.58/4.87 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.58/4.87 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 8.58/4.87 => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7857_split__scaleR__neg__le,axiom,
% 8.58/4.87 ! [A: real,X: real] :
% 8.58/4.87 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.87 & ( ord_less_eq_real @ X @ zero_zero_real ) )
% 8.58/4.87 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.58/4.87 & ( ord_less_eq_real @ zero_zero_real @ X ) ) )
% 8.58/4.87 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7858_scaleR__mono_H,axiom,
% 8.58/4.87 ! [A: real,B: real,C: real,D: real] :
% 8.58/4.87 ( ( ord_less_eq_real @ A @ B )
% 8.58/4.87 => ( ( ord_less_eq_real @ C @ D )
% 8.58/4.87 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.58/4.87 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.58/4.87 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ D ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7859_scaleR__mono,axiom,
% 8.58/4.87 ! [A: real,B: real,X: real,Y: real] :
% 8.58/4.87 ( ( ord_less_eq_real @ A @ B )
% 8.58/4.87 => ( ( ord_less_eq_real @ X @ Y )
% 8.58/4.87 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.58/4.87 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.87 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ Y ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7860_scaleR__left__le__one__le,axiom,
% 8.58/4.87 ! [X: real,A: real] :
% 8.58/4.87 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.87 => ( ( ord_less_eq_real @ A @ one_one_real )
% 8.58/4.87 => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7861_scaleR__2,axiom,
% 8.58/4.87 ! [X: real] :
% 8.58/4.87 ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
% 8.58/4.87 = ( plus_plus_real @ X @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7862_scaleR__2,axiom,
% 8.58/4.87 ! [X: complex] :
% 8.58/4.87 ( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
% 8.58/4.87 = ( plus_plus_complex @ X @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7863_real__vector__eq__affinity,axiom,
% 8.58/4.87 ! [M: real,Y: real,X: real,C: real] :
% 8.58/4.87 ( ( M != zero_zero_real )
% 8.58/4.87 => ( ( Y
% 8.58/4.87 = ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X ) @ C ) )
% 8.58/4.87 = ( ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) )
% 8.58/4.87 = X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7864_real__vector__eq__affinity,axiom,
% 8.58/4.87 ! [M: real,Y: complex,X: complex,C: complex] :
% 8.58/4.87 ( ( M != zero_zero_real )
% 8.58/4.87 => ( ( Y
% 8.58/4.87 = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ M @ X ) @ C ) )
% 8.58/4.87 = ( ( minus_minus_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ Y ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ C ) )
% 8.58/4.87 = X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7865_real__vector__affinity__eq,axiom,
% 8.58/4.87 ! [M: real,X: real,C: real,Y: real] :
% 8.58/4.87 ( ( M != zero_zero_real )
% 8.58/4.87 => ( ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X ) @ C )
% 8.58/4.87 = Y )
% 8.58/4.87 = ( X
% 8.58/4.87 = ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7866_real__vector__affinity__eq,axiom,
% 8.58/4.87 ! [M: real,X: complex,C: complex,Y: complex] :
% 8.58/4.87 ( ( M != zero_zero_real )
% 8.58/4.87 => ( ( ( plus_plus_complex @ ( real_V2046097035970521341omplex @ M @ X ) @ C )
% 8.58/4.87 = Y )
% 8.58/4.87 = ( X
% 8.58/4.87 = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ Y ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ C ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7867_nonzero__inverse__scaleR__distrib,axiom,
% 8.58/4.87 ! [A: real,X: real] :
% 8.58/4.87 ( ( A != zero_zero_real )
% 8.58/4.87 => ( ( X != zero_zero_real )
% 8.58/4.87 => ( ( inverse_inverse_real @ ( real_V1485227260804924795R_real @ A @ X ) )
% 8.58/4.87 = ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ X ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7868_nonzero__inverse__scaleR__distrib,axiom,
% 8.58/4.87 ! [A: real,X: complex] :
% 8.58/4.87 ( ( A != zero_zero_real )
% 8.58/4.87 => ( ( X != zero_zero_complex )
% 8.58/4.87 => ( ( invers8013647133539491842omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
% 8.58/4.87 = ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ A ) @ ( invers8013647133539491842omplex @ X ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7869_cmod__plus__Re__le__0__iff,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ zero_zero_real )
% 8.58/4.87 = ( ( re @ Z2 )
% 8.58/4.87 = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7870_cos__n__Re__cis__pow__n,axiom,
% 8.58/4.87 ! [N: nat,A: real] :
% 8.58/4.87 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 8.58/4.87 = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7871_complex__mod__mult__cnj,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) )
% 8.58/4.87 = ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7872_Cauchy__iff,axiom,
% 8.58/4.87 ( topolo6517432010174082258omplex
% 8.58/4.87 = ( ^ [X8: nat > complex] :
% 8.58/4.87 ! [E3: real] :
% 8.58/4.87 ( ( ord_less_real @ zero_zero_real @ E3 )
% 8.58/4.87 => ? [M7: nat] :
% 8.58/4.87 ! [M3: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ M7 @ M3 )
% 8.58/4.87 => ! [N3: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ M7 @ N3 )
% 8.58/4.87 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) ) @ E3 ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7873_Cauchy__iff,axiom,
% 8.58/4.87 ( topolo4055970368930404560y_real
% 8.58/4.87 = ( ^ [X8: nat > real] :
% 8.58/4.87 ! [E3: real] :
% 8.58/4.87 ( ( ord_less_real @ zero_zero_real @ E3 )
% 8.58/4.87 => ? [M7: nat] :
% 8.58/4.87 ! [M3: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ M7 @ M3 )
% 8.58/4.87 => ! [N3: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ M7 @ N3 )
% 8.58/4.87 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) ) @ E3 ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7874_CauchyI,axiom,
% 8.58/4.87 ! [X9: nat > complex] :
% 8.58/4.87 ( ! [E2: real] :
% 8.58/4.87 ( ( ord_less_real @ zero_zero_real @ E2 )
% 8.58/4.87 => ? [M8: nat] :
% 8.58/4.87 ! [M2: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ M8 @ M2 )
% 8.58/4.87 => ! [N2: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ M8 @ N2 )
% 8.58/4.87 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X9 @ M2 ) @ ( X9 @ N2 ) ) ) @ E2 ) ) ) )
% 8.58/4.87 => ( topolo6517432010174082258omplex @ X9 ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7875_CauchyI,axiom,
% 8.58/4.87 ! [X9: nat > real] :
% 8.58/4.87 ( ! [E2: real] :
% 8.58/4.87 ( ( ord_less_real @ zero_zero_real @ E2 )
% 8.58/4.87 => ? [M8: nat] :
% 8.58/4.87 ! [M2: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ M8 @ M2 )
% 8.58/4.87 => ! [N2: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ M8 @ N2 )
% 8.58/4.87 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X9 @ M2 ) @ ( X9 @ N2 ) ) ) @ E2 ) ) ) )
% 8.58/4.87 => ( topolo4055970368930404560y_real @ X9 ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7876_CauchyD,axiom,
% 8.58/4.87 ! [X9: nat > complex,E: real] :
% 8.58/4.87 ( ( topolo6517432010174082258omplex @ X9 )
% 8.58/4.87 => ( ( ord_less_real @ zero_zero_real @ E )
% 8.58/4.87 => ? [M9: nat] :
% 8.58/4.87 ! [M4: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ M9 @ M4 )
% 8.58/4.87 => ! [N6: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ M9 @ N6 )
% 8.58/4.87 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X9 @ M4 ) @ ( X9 @ N6 ) ) ) @ E ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7877_CauchyD,axiom,
% 8.58/4.87 ! [X9: nat > real,E: real] :
% 8.58/4.87 ( ( topolo4055970368930404560y_real @ X9 )
% 8.58/4.87 => ( ( ord_less_real @ zero_zero_real @ E )
% 8.58/4.87 => ? [M9: nat] :
% 8.58/4.87 ! [M4: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ M9 @ M4 )
% 8.58/4.87 => ! [N6: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ M9 @ N6 )
% 8.58/4.87 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X9 @ M4 ) @ ( X9 @ N6 ) ) ) @ E ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7878_XOR__upper,axiom,
% 8.58/4.87 ! [X: int,N: nat,Y: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.58/4.87 => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.87 => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.87 => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7879_complex__norm__square,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.87 = ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7880_xor__int__rec,axiom,
% 8.58/4.87 ( bit_se6526347334894502574or_int
% 8.58/4.87 = ( ^ [K2: int,L2: int] :
% 8.58/4.87 ( plus_plus_int
% 8.58/4.87 @ ( zero_n2684676970156552555ol_int
% 8.58/4.87 @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) )
% 8.58/4.87 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 8.58/4.87 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7881_csqrt_Ocode,axiom,
% 8.58/4.87 ( csqrt
% 8.58/4.87 = ( ^ [Z5: complex] :
% 8.58/4.87 ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.87 @ ( times_times_real
% 8.58/4.87 @ ( if_real
% 8.58/4.87 @ ( ( im @ Z5 )
% 8.58/4.87 = zero_zero_real )
% 8.58/4.87 @ one_one_real
% 8.58/4.87 @ ( sgn_sgn_real @ ( im @ Z5 ) ) )
% 8.58/4.87 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7882_csqrt_Osimps_I2_J,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( im @ ( csqrt @ Z2 ) )
% 8.58/4.87 = ( times_times_real
% 8.58/4.87 @ ( if_real
% 8.58/4.87 @ ( ( im @ Z2 )
% 8.58/4.87 = zero_zero_real )
% 8.58/4.87 @ one_one_real
% 8.58/4.87 @ ( sgn_sgn_real @ ( im @ Z2 ) ) )
% 8.58/4.87 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7883_xor__int__unfold,axiom,
% 8.58/4.87 ( bit_se6526347334894502574or_int
% 8.58/4.87 = ( ^ [K2: int,L2: int] :
% 8.58/4.87 ( if_int
% 8.58/4.87 @ ( K2
% 8.58/4.87 = ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.87 @ ( bit_ri7919022796975470100ot_int @ L2 )
% 8.58/4.87 @ ( if_int
% 8.58/4.87 @ ( L2
% 8.58/4.87 = ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.87 @ ( bit_ri7919022796975470100ot_int @ K2 )
% 8.58/4.87 @ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7884_csqrt__of__real__nonpos,axiom,
% 8.58/4.87 ! [X: complex] :
% 8.58/4.87 ( ( ( im @ X )
% 8.58/4.87 = zero_zero_real )
% 8.58/4.87 => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
% 8.58/4.87 => ( ( csqrt @ X )
% 8.58/4.87 = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7885_Complex__divide,axiom,
% 8.58/4.87 ( divide1717551699836669952omplex
% 8.58/4.87 = ( ^ [X3: complex,Y6: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7886_bit_Odouble__compl,axiom,
% 8.58/4.87 ! [X: int] :
% 8.58/4.87 ( ( bit_ri7919022796975470100ot_int @ ( bit_ri7919022796975470100ot_int @ X ) )
% 8.58/4.87 = X ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7887_bit_Ocompl__eq__compl__iff,axiom,
% 8.58/4.87 ! [X: int,Y: int] :
% 8.58/4.87 ( ( ( bit_ri7919022796975470100ot_int @ X )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ Y ) )
% 8.58/4.87 = ( X = Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7888_bit_Oxor__compl__left,axiom,
% 8.58/4.87 ! [X: int,Y: int] :
% 8.58/4.87 ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ Y )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7889_bit_Oxor__compl__right,axiom,
% 8.58/4.87 ! [X: int,Y: int] :
% 8.58/4.87 ( ( bit_se6526347334894502574or_int @ X @ ( bit_ri7919022796975470100ot_int @ Y ) )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7890_bit_Oconj__cancel__left,axiom,
% 8.58/4.87 ! [X: int] :
% 8.58/4.87 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
% 8.58/4.87 = zero_zero_int ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7891_bit_Oconj__cancel__right,axiom,
% 8.58/4.87 ! [X: int] :
% 8.58/4.87 ( ( bit_se725231765392027082nd_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
% 8.58/4.87 = zero_zero_int ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7892_Im__power__real,axiom,
% 8.58/4.87 ! [X: complex,N: nat] :
% 8.58/4.87 ( ( ( im @ X )
% 8.58/4.87 = zero_zero_real )
% 8.58/4.87 => ( ( im @ ( power_power_complex @ X @ N ) )
% 8.58/4.87 = zero_zero_real ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7893_bit_Ocompl__zero,axiom,
% 8.58/4.87 ( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7894_bit_Ocompl__zero,axiom,
% 8.58/4.87 ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
% 8.58/4.87 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7895_bit_Ocompl__one,axiom,
% 8.58/4.87 ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.87 = zero_z3403309356797280102nteger ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7896_bit_Ocompl__one,axiom,
% 8.58/4.87 ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.87 = zero_zero_int ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7897_bit_Oxor__cancel__right,axiom,
% 8.58/4.87 ! [X: code_integer] :
% 8.58/4.87 ( ( bit_se3222712562003087583nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7898_bit_Oxor__cancel__right,axiom,
% 8.58/4.87 ! [X: int] :
% 8.58/4.87 ( ( bit_se6526347334894502574or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
% 8.58/4.87 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7899_bit_Oxor__cancel__left,axiom,
% 8.58/4.87 ! [X: code_integer] :
% 8.58/4.87 ( ( bit_se3222712562003087583nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7900_bit_Oxor__cancel__left,axiom,
% 8.58/4.87 ! [X: int] :
% 8.58/4.87 ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
% 8.58/4.87 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7901_bit_Oxor__one__right,axiom,
% 8.58/4.87 ! [X: code_integer] :
% 8.58/4.87 ( ( bit_se3222712562003087583nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.87 = ( bit_ri7632146776885996613nteger @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7902_bit_Oxor__one__right,axiom,
% 8.58/4.87 ! [X: int] :
% 8.58/4.87 ( ( bit_se6526347334894502574or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7903_bit_Oxor__one__left,axiom,
% 8.58/4.87 ! [X: code_integer] :
% 8.58/4.87 ( ( bit_se3222712562003087583nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 8.58/4.87 = ( bit_ri7632146776885996613nteger @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7904_bit_Oxor__one__left,axiom,
% 8.58/4.87 ! [X: int] :
% 8.58/4.87 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7905_not__negative__int__iff,axiom,
% 8.58/4.87 ! [K: int] :
% 8.58/4.87 ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 8.58/4.87 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7906_not__nonnegative__int__iff,axiom,
% 8.58/4.87 ! [K: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 8.58/4.87 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7907_minus__not__numeral__eq,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( uminus1351360451143612070nteger @ ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.58/4.87 = ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7908_minus__not__numeral__eq,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( uminus_uminus_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.87 = ( numeral_numeral_int @ ( inc @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7909_complex__In__mult__cnj__zero,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( im @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) )
% 8.58/4.87 = zero_zero_real ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7910_Im__i__times,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( im @ ( times_times_complex @ imaginary_unit @ Z2 ) )
% 8.58/4.87 = ( re @ Z2 ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7911_Im__divide__of__real,axiom,
% 8.58/4.87 ! [Z2: complex,R: real] :
% 8.58/4.87 ( ( im @ ( divide1717551699836669952omplex @ Z2 @ ( real_V4546457046886955230omplex @ R ) ) )
% 8.58/4.87 = ( divide_divide_real @ ( im @ Z2 ) @ R ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7912_Im__sgn,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( im @ ( sgn_sgn_complex @ Z2 ) )
% 8.58/4.87 = ( divide_divide_real @ ( im @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7913_even__not__iff,axiom,
% 8.58/4.87 ! [A: code_integer] :
% 8.58/4.87 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri7632146776885996613nteger @ A ) )
% 8.58/4.87 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7914_even__not__iff,axiom,
% 8.58/4.87 ! [A: int] :
% 8.58/4.87 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ A ) )
% 8.58/4.87 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7915_push__bit__minus__one__eq__not__mask,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7916_push__bit__minus__one__eq__not__mask,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.87 = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7917_Re__power__real,axiom,
% 8.58/4.87 ! [X: complex,N: nat] :
% 8.58/4.87 ( ( ( im @ X )
% 8.58/4.87 = zero_zero_real )
% 8.58/4.87 => ( ( re @ ( power_power_complex @ X @ N ) )
% 8.58/4.87 = ( power_power_real @ ( re @ X ) @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7918_Re__i__times,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( re @ ( times_times_complex @ imaginary_unit @ Z2 ) )
% 8.58/4.87 = ( uminus_uminus_real @ ( im @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7919_Im__divide__numeral,axiom,
% 8.58/4.87 ! [Z2: complex,W: num] :
% 8.58/4.87 ( ( im @ ( divide1717551699836669952omplex @ Z2 @ ( numera6690914467698888265omplex @ W ) ) )
% 8.58/4.87 = ( divide_divide_real @ ( im @ Z2 ) @ ( numeral_numeral_real @ W ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7920_Im__divide__of__nat,axiom,
% 8.58/4.87 ! [Z2: complex,N: nat] :
% 8.58/4.87 ( ( im @ ( divide1717551699836669952omplex @ Z2 @ ( semiri8010041392384452111omplex @ N ) ) )
% 8.58/4.87 = ( divide_divide_real @ ( im @ Z2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7921_not__one__eq,axiom,
% 8.58/4.87 ( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7922_not__one__eq,axiom,
% 8.58/4.87 ( ( bit_ri7919022796975470100ot_int @ one_one_int )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7923_csqrt__minus,axiom,
% 8.58/4.87 ! [X: complex] :
% 8.58/4.87 ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
% 8.58/4.87 | ( ( ( im @ X )
% 8.58/4.87 = zero_zero_real )
% 8.58/4.87 & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
% 8.58/4.87 => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
% 8.58/4.87 = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7924_scaleR__complex_Osimps_I2_J,axiom,
% 8.58/4.87 ! [R: real,X: complex] :
% 8.58/4.87 ( ( im @ ( real_V2046097035970521341omplex @ R @ X ) )
% 8.58/4.87 = ( times_times_real @ R @ ( im @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7925_take__bit__not__take__bit,axiom,
% 8.58/4.87 ! [N: nat,A: int] :
% 8.58/4.87 ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 8.58/4.87 = ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7926_take__bit__not__iff,axiom,
% 8.58/4.87 ! [N: nat,A: int,B: int] :
% 8.58/4.87 ( ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) )
% 8.58/4.87 = ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ B ) ) )
% 8.58/4.87 = ( ( bit_se2923211474154528505it_int @ N @ A )
% 8.58/4.87 = ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7927_bit__not__int__iff,axiom,
% 8.58/4.87 ! [K: int,N: nat] :
% 8.58/4.87 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
% 8.58/4.87 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7928_of__int__not__eq,axiom,
% 8.58/4.87 ! [K: int] :
% 8.58/4.87 ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ ( ring_1_of_int_int @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7929_of__int__not__numeral,axiom,
% 8.58/4.87 ! [K: num] :
% 8.58/4.87 ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7930_not__add__distrib,axiom,
% 8.58/4.87 ! [A: int,B: int] :
% 8.58/4.87 ( ( bit_ri7919022796975470100ot_int @ ( plus_plus_int @ A @ B ) )
% 8.58/4.87 = ( minus_minus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7931_not__diff__distrib,axiom,
% 8.58/4.87 ! [A: int,B: int] :
% 8.58/4.87 ( ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A @ B ) )
% 8.58/4.87 = ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7932_imaginary__unit_Osimps_I2_J,axiom,
% 8.58/4.87 ( ( im @ imaginary_unit )
% 8.58/4.87 = one_one_real ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7933_one__complex_Osimps_I2_J,axiom,
% 8.58/4.87 ( ( im @ one_one_complex )
% 8.58/4.87 = zero_zero_real ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7934_plus__complex_Osimps_I2_J,axiom,
% 8.58/4.87 ! [X: complex,Y: complex] :
% 8.58/4.87 ( ( im @ ( plus_plus_complex @ X @ Y ) )
% 8.58/4.87 = ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7935_minus__complex_Osimps_I2_J,axiom,
% 8.58/4.87 ! [X: complex,Y: complex] :
% 8.58/4.87 ( ( im @ ( minus_minus_complex @ X @ Y ) )
% 8.58/4.87 = ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7936_minus__eq__not__plus__1,axiom,
% 8.58/4.87 ( uminus1351360451143612070nteger
% 8.58/4.87 = ( ^ [A2: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A2 ) @ one_one_Code_integer ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7937_minus__eq__not__plus__1,axiom,
% 8.58/4.87 ( uminus_uminus_int
% 8.58/4.87 = ( ^ [A2: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A2 ) @ one_one_int ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7938_minus__eq__not__minus__1,axiom,
% 8.58/4.87 ( uminus1351360451143612070nteger
% 8.58/4.87 = ( ^ [A2: code_integer] : ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A2 @ one_one_Code_integer ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7939_minus__eq__not__minus__1,axiom,
% 8.58/4.87 ( uminus_uminus_int
% 8.58/4.87 = ( ^ [A2: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A2 @ one_one_int ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7940_not__eq__complement,axiom,
% 8.58/4.87 ( bit_ri7632146776885996613nteger
% 8.58/4.87 = ( ^ [A2: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A2 ) @ one_one_Code_integer ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7941_not__eq__complement,axiom,
% 8.58/4.87 ( bit_ri7919022796975470100ot_int
% 8.58/4.87 = ( ^ [A2: int] : ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ one_one_int ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7942_not__int__def,axiom,
% 8.58/4.87 ( bit_ri7919022796975470100ot_int
% 8.58/4.87 = ( ^ [K2: int] : ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7943_and__not__numerals_I1_J,axiom,
% 8.58/4.87 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 8.58/4.87 = zero_zero_int ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7944_scaleR__complex_Ocode,axiom,
% 8.58/4.87 ( real_V2046097035970521341omplex
% 8.58/4.87 = ( ^ [R5: real,X3: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X3 ) ) @ ( times_times_real @ R5 @ ( im @ X3 ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7945_disjunctive__diff,axiom,
% 8.58/4.87 ! [B: int,A: int] :
% 8.58/4.87 ( ! [N2: nat] :
% 8.58/4.87 ( ( bit_se1146084159140164899it_int @ B @ N2 )
% 8.58/4.87 => ( bit_se1146084159140164899it_int @ A @ N2 ) )
% 8.58/4.87 => ( ( minus_minus_int @ A @ B )
% 8.58/4.87 = ( bit_se725231765392027082nd_int @ A @ ( bit_ri7919022796975470100ot_int @ B ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7946_take__bit__not__eq__mask__diff,axiom,
% 8.58/4.87 ! [N: nat,A: int] :
% 8.58/4.87 ( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) )
% 8.58/4.87 = ( minus_minus_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7947_minus__numeral__inc__eq,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) )
% 8.58/4.87 = ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7948_minus__numeral__inc__eq,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7949_not__int__div__2,axiom,
% 8.58/4.87 ! [K: int] :
% 8.58/4.87 ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7950_even__not__iff__int,axiom,
% 8.58/4.87 ! [K: int] :
% 8.58/4.87 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
% 8.58/4.87 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7951_and__not__numerals_I4_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 8.58/4.87 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7952_and__not__numerals_I2_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.58/4.87 = one_one_int ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7953_not__numeral__Bit0__eq,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7954_not__numeral__Bit0__eq,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7955_bit__minus__int__iff,axiom,
% 8.58/4.87 ! [K: int,N: nat] :
% 8.58/4.87 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
% 8.58/4.87 = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7956_times__complex_Osimps_I2_J,axiom,
% 8.58/4.87 ! [X: complex,Y: complex] :
% 8.58/4.87 ( ( im @ ( times_times_complex @ X @ Y ) )
% 8.58/4.87 = ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7957_times__complex_Osimps_I1_J,axiom,
% 8.58/4.87 ! [X: complex,Y: complex] :
% 8.58/4.87 ( ( re @ ( times_times_complex @ X @ Y ) )
% 8.58/4.87 = ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7958_take__bit__not__mask__eq__0,axiom,
% 8.58/4.87 ! [M: nat,N: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.87 => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) )
% 8.58/4.87 = zero_zero_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7959_Im__complex__div__eq__0,axiom,
% 8.58/4.87 ! [A: complex,B: complex] :
% 8.58/4.87 ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.58/4.87 = zero_zero_real )
% 8.58/4.87 = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 8.58/4.87 = zero_zero_real ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7960_push__bit__mask__eq,axiom,
% 8.58/4.87 ! [M: nat,N: nat] :
% 8.58/4.87 ( ( bit_se545348938243370406it_int @ M @ ( bit_se2000444600071755411sk_int @ N ) )
% 8.58/4.87 = ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ N @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ M ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7961_push__bit__mask__eq,axiom,
% 8.58/4.87 ! [M: nat,N: nat] :
% 8.58/4.87 ( ( bit_se7788150548672797655nteger @ M @ ( bit_se2119862282449309892nteger @ N ) )
% 8.58/4.87 = ( bit_se3949692690581998587nteger @ ( bit_se2119862282449309892nteger @ ( plus_plus_nat @ N @ M ) ) @ ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ M ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7962_plus__complex_Ocode,axiom,
% 8.58/4.87 ( plus_plus_complex
% 8.58/4.87 = ( ^ [X3: complex,Y6: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X3 ) @ ( re @ Y6 ) ) @ ( plus_plus_real @ ( im @ X3 ) @ ( im @ Y6 ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7963_unset__bit__eq__and__not,axiom,
% 8.58/4.87 ( bit_se8260200283734997820nteger
% 8.58/4.87 = ( ^ [N3: nat,A2: code_integer] : ( bit_se3949692690581998587nteger @ A2 @ ( bit_ri7632146776885996613nteger @ ( bit_se7788150548672797655nteger @ N3 @ one_one_Code_integer ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7964_unset__bit__eq__and__not,axiom,
% 8.58/4.87 ( bit_se4203085406695923979it_int
% 8.58/4.87 = ( ^ [N3: nat,A2: int] : ( bit_se725231765392027082nd_int @ A2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7965_minus__complex_Ocode,axiom,
% 8.58/4.87 ( minus_minus_complex
% 8.58/4.87 = ( ^ [X3: complex,Y6: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X3 ) @ ( re @ Y6 ) ) @ ( minus_minus_real @ ( im @ X3 ) @ ( im @ Y6 ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7966_unset__bit__int__def,axiom,
% 8.58/4.87 ( bit_se4203085406695923979it_int
% 8.58/4.87 = ( ^ [N3: nat,K2: int] : ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7967_and__not__numerals_I5_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.58/4.87 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7968_and__not__numerals_I7_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 8.58/4.87 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7969_and__not__numerals_I3_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.58/4.87 = zero_zero_int ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7970_cmod__le,axiom,
% 8.58/4.87 ! [Z2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z2 ) ) @ ( abs_abs_real @ ( im @ Z2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7971_Im__complex__div__gt__0,axiom,
% 8.58/4.87 ! [A: complex,B: complex] :
% 8.58/4.87 ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 8.58/4.87 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7972_Im__complex__div__lt__0,axiom,
% 8.58/4.87 ! [A: complex,B: complex] :
% 8.58/4.87 ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 8.58/4.87 = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7973_Im__complex__div__ge__0,axiom,
% 8.58/4.87 ! [A: complex,B: complex] :
% 8.58/4.87 ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 8.58/4.87 = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7974_Im__complex__div__le__0,axiom,
% 8.58/4.87 ! [A: complex,B: complex] :
% 8.58/4.87 ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 8.58/4.87 = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7975_sin__n__Im__cis__pow__n,axiom,
% 8.58/4.87 ! [N: nat,A: real] :
% 8.58/4.87 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 8.58/4.87 = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7976_Re__exp,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( re @ ( exp_complex @ Z2 ) )
% 8.58/4.87 = ( times_times_real @ ( exp_real @ ( re @ Z2 ) ) @ ( cos_real @ ( im @ Z2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7977_Im__exp,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( im @ ( exp_complex @ Z2 ) )
% 8.58/4.87 = ( times_times_real @ ( exp_real @ ( re @ Z2 ) ) @ ( sin_real @ ( im @ Z2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7978_complex__eq,axiom,
% 8.58/4.87 ! [A: complex] :
% 8.58/4.87 ( A
% 8.58/4.87 = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7979_and__not__numerals_I9_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.58/4.87 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7980_and__not__numerals_I6_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.58/4.87 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7981_bit__not__iff__eq,axiom,
% 8.58/4.87 ! [A: int,N: nat] :
% 8.58/4.87 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ A ) @ N )
% 8.58/4.87 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 8.58/4.87 != zero_zero_int )
% 8.58/4.87 & ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7982_minus__exp__eq__not__mask,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.87 = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7983_minus__exp__eq__not__mask,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7984_times__complex_Ocode,axiom,
% 8.58/4.87 ( times_times_complex
% 8.58/4.87 = ( ^ [X3: complex,Y6: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y6 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y6 ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7985_complex__div__gt__0,axiom,
% 8.58/4.87 ! [A: complex,B: complex] :
% 8.58/4.87 ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 8.58/4.87 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
% 8.58/4.87 & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 8.58/4.87 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7986_exp__eq__polar,axiom,
% 8.58/4.87 ( exp_complex
% 8.58/4.87 = ( ^ [Z5: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z5 ) ) ) @ ( cis @ ( im @ Z5 ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7987_complex__eq__0,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( Z2 = zero_zero_complex )
% 8.58/4.87 = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.87 = zero_zero_real ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7988_cmod__power2,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.87 = ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7989_Im__power2,axiom,
% 8.58/4.87 ! [X: complex] :
% 8.58/4.87 ( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.87 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7990_Re__power2,axiom,
% 8.58/4.87 ! [X: complex] :
% 8.58/4.87 ( ( re @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.87 = ( minus_minus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7991_complex__neq__0,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( Z2 != zero_zero_complex )
% 8.58/4.87 = ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7992_and__not__numerals_I8_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.58/4.87 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7993_norm__complex__def,axiom,
% 8.58/4.87 ( real_V1022390504157884413omplex
% 8.58/4.87 = ( ^ [Z5: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7994_not__int__rec,axiom,
% 8.58/4.87 ( bit_ri7919022796975470100ot_int
% 8.58/4.87 = ( ^ [K2: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7995_inverse__complex_Osimps_I1_J,axiom,
% 8.58/4.87 ! [X: complex] :
% 8.58/4.87 ( ( re @ ( invers8013647133539491842omplex @ X ) )
% 8.58/4.87 = ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7996_complex__unit__circle,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( Z2 != zero_zero_complex )
% 8.58/4.87 => ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.87 = one_one_real ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7997_Re__divide,axiom,
% 8.58/4.87 ! [X: complex,Y: complex] :
% 8.58/4.87 ( ( re @ ( divide1717551699836669952omplex @ X @ Y ) )
% 8.58/4.87 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7998_complex__mult__cnj,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( times_times_complex @ Z2 @ ( cnj @ Z2 ) )
% 8.58/4.87 = ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_7999_csqrt__unique,axiom,
% 8.58/4.87 ! [W: complex,Z2: complex] :
% 8.58/4.87 ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.87 = Z2 )
% 8.58/4.87 => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
% 8.58/4.87 | ( ( ( re @ W )
% 8.58/4.87 = zero_zero_real )
% 8.58/4.87 & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
% 8.58/4.87 => ( ( csqrt @ Z2 )
% 8.58/4.87 = W ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8000_csqrt__square,axiom,
% 8.58/4.87 ! [B: complex] :
% 8.58/4.87 ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
% 8.58/4.87 | ( ( ( re @ B )
% 8.58/4.87 = zero_zero_real )
% 8.58/4.87 & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
% 8.58/4.87 => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.87 = B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8001_inverse__complex_Osimps_I2_J,axiom,
% 8.58/4.87 ! [X: complex] :
% 8.58/4.87 ( ( im @ ( invers8013647133539491842omplex @ X ) )
% 8.58/4.87 = ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8002_complex__diff__cnj,axiom,
% 8.58/4.87 ! [Z2: complex] :
% 8.58/4.87 ( ( minus_minus_complex @ Z2 @ ( cnj @ Z2 ) )
% 8.58/4.87 = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z2 ) ) ) @ imaginary_unit ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8003_Im__divide,axiom,
% 8.58/4.87 ! [X: complex,Y: complex] :
% 8.58/4.87 ( ( im @ ( divide1717551699836669952omplex @ X @ Y ) )
% 8.58/4.87 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8004_complex__abs__le__norm,axiom,
% 8.58/4.87 ! [Z2: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z2 ) ) @ ( abs_abs_real @ ( im @ Z2 ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8005_inverse__complex_Ocode,axiom,
% 8.58/4.87 ( invers8013647133539491842omplex
% 8.58/4.87 = ( ^ [X3: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X3 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X3 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8006_Im__Reals__divide,axiom,
% 8.58/4.87 ! [R: complex,Z2: complex] :
% 8.58/4.87 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( ( im @ ( divide1717551699836669952omplex @ R @ Z2 ) )
% 8.58/4.87 = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R ) ) @ ( im @ Z2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8007_Re__Reals__divide,axiom,
% 8.58/4.87 ! [R: complex,Z2: complex] :
% 8.58/4.87 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( ( re @ ( divide1717551699836669952omplex @ R @ Z2 ) )
% 8.58/4.87 = ( divide_divide_real @ ( times_times_real @ ( re @ R ) @ ( re @ Z2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8008_num_Osize_I6_J,axiom,
% 8.58/4.87 ! [X32: num] :
% 8.58/4.87 ( ( size_size_num @ ( bit1 @ X32 ) )
% 8.58/4.87 = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8009_num_Osize_I5_J,axiom,
% 8.58/4.87 ! [X2: num] :
% 8.58/4.87 ( ( size_size_num @ ( bit0 @ X2 ) )
% 8.58/4.87 = ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8010_or__int__rec,axiom,
% 8.58/4.87 ( bit_se1409905431419307370or_int
% 8.58/4.87 = ( ^ [K2: int,L2: int] :
% 8.58/4.87 ( plus_plus_int
% 8.58/4.87 @ ( zero_n2684676970156552555ol_int
% 8.58/4.87 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 8.58/4.87 | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 8.58/4.87 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8011_or_Oright__idem,axiom,
% 8.58/4.87 ! [A: int,B: int] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ B )
% 8.58/4.87 = ( bit_se1409905431419307370or_int @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8012_or_Oright__idem,axiom,
% 8.58/4.87 ! [A: nat,B: nat] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ B )
% 8.58/4.87 = ( bit_se1412395901928357646or_nat @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8013_or_Oleft__idem,axiom,
% 8.58/4.87 ! [A: int,B: int] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 8.58/4.87 = ( bit_se1409905431419307370or_int @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8014_or_Oleft__idem,axiom,
% 8.58/4.87 ! [A: nat,B: nat] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 8.58/4.87 = ( bit_se1412395901928357646or_nat @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8015_or_Oidem,axiom,
% 8.58/4.87 ! [A: int] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ A @ A )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8016_or_Oidem,axiom,
% 8.58/4.87 ! [A: nat] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ A @ A )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8017_or_Oright__neutral,axiom,
% 8.58/4.87 ! [A: int] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ A @ zero_zero_int )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8018_or_Oright__neutral,axiom,
% 8.58/4.87 ! [A: nat] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ A @ zero_zero_nat )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8019_or_Oleft__neutral,axiom,
% 8.58/4.87 ! [A: int] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ zero_zero_int @ A )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8020_or_Oleft__neutral,axiom,
% 8.58/4.87 ! [A: nat] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ zero_zero_nat @ A )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8021_take__bit__or,axiom,
% 8.58/4.87 ! [N: nat,A: int,B: int] :
% 8.58/4.87 ( ( bit_se2923211474154528505it_int @ N @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 8.58/4.87 = ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8022_take__bit__or,axiom,
% 8.58/4.87 ! [N: nat,A: nat,B: nat] :
% 8.58/4.87 ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 8.58/4.87 = ( bit_se1412395901928357646or_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8023_push__bit__or,axiom,
% 8.58/4.87 ! [N: nat,A: nat,B: nat] :
% 8.58/4.87 ( ( bit_se547839408752420682it_nat @ N @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 8.58/4.87 = ( bit_se1412395901928357646or_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8024_push__bit__or,axiom,
% 8.58/4.87 ! [N: nat,A: int,B: int] :
% 8.58/4.87 ( ( bit_se545348938243370406it_int @ N @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 8.58/4.87 = ( bit_se1409905431419307370or_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8025_push__bit__or,axiom,
% 8.58/4.87 ! [N: nat,A: code_integer,B: code_integer] :
% 8.58/4.87 ( ( bit_se7788150548672797655nteger @ N @ ( bit_se1080825931792720795nteger @ A @ B ) )
% 8.58/4.87 = ( bit_se1080825931792720795nteger @ ( bit_se7788150548672797655nteger @ N @ A ) @ ( bit_se7788150548672797655nteger @ N @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8026_push__bit__or,axiom,
% 8.58/4.87 ! [N: nat,A: code_natural,B: code_natural] :
% 8.58/4.87 ( ( bit_se6611745700429515170atural @ N @ ( bit_se9127793120404214118atural @ A @ B ) )
% 8.58/4.87 = ( bit_se9127793120404214118atural @ ( bit_se6611745700429515170atural @ N @ A ) @ ( bit_se6611745700429515170atural @ N @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8027_bit_Odisj__one__right,axiom,
% 8.58/4.87 ! [X: code_integer] :
% 8.58/4.87 ( ( bit_se1080825931792720795nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8028_bit_Odisj__one__right,axiom,
% 8.58/4.87 ! [X: int] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.87 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8029_bit_Odisj__one__left,axiom,
% 8.58/4.87 ! [X: code_integer] :
% 8.58/4.87 ( ( bit_se1080825931792720795nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8030_bit_Odisj__one__left,axiom,
% 8.58/4.87 ! [X: int] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 8.58/4.87 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8031_or__nonnegative__int__iff,axiom,
% 8.58/4.87 ! [K: int,L: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
% 8.58/4.87 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.58/4.87 & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8032_or__negative__int__iff,axiom,
% 8.58/4.87 ! [K: int,L: int] :
% 8.58/4.87 ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
% 8.58/4.87 = ( ( ord_less_int @ K @ zero_zero_int )
% 8.58/4.87 | ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8033_bit_Ode__Morgan__disj,axiom,
% 8.58/4.87 ! [X: int,Y: int] :
% 8.58/4.87 ( ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ X @ Y ) )
% 8.58/4.87 = ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X ) @ ( bit_ri7919022796975470100ot_int @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8034_bit_Ode__Morgan__conj,axiom,
% 8.58/4.87 ! [X: int,Y: int] :
% 8.58/4.87 ( ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ X @ Y ) )
% 8.58/4.87 = ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ ( bit_ri7919022796975470100ot_int @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8035_or__numerals_I8_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
% 8.58/4.87 = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8036_or__numerals_I8_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
% 8.58/4.87 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8037_or__numerals_I2_J,axiom,
% 8.58/4.87 ! [Y: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 8.58/4.87 = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8038_or__numerals_I2_J,axiom,
% 8.58/4.87 ! [Y: num] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 8.58/4.87 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8039_bit_Odisj__cancel__right,axiom,
% 8.58/4.87 ! [X: code_integer] :
% 8.58/4.87 ( ( bit_se1080825931792720795nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8040_bit_Odisj__cancel__right,axiom,
% 8.58/4.87 ! [X: int] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
% 8.58/4.87 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8041_bit_Odisj__cancel__left,axiom,
% 8.58/4.87 ! [X: code_integer] :
% 8.58/4.87 ( ( bit_se1080825931792720795nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8042_bit_Odisj__cancel__left,axiom,
% 8.58/4.87 ! [X: int] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
% 8.58/4.87 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8043_Re__divide__Reals,axiom,
% 8.58/4.87 ! [R: complex,Z2: complex] :
% 8.58/4.87 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( ( re @ ( divide1717551699836669952omplex @ Z2 @ R ) )
% 8.58/4.87 = ( divide_divide_real @ ( re @ Z2 ) @ ( re @ R ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8044_imaginary__eq__real__iff,axiom,
% 8.58/4.87 ! [Y: complex,X: complex] :
% 8.58/4.87 ( ( member_complex @ Y @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( ( member_complex @ X @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( ( ( times_times_complex @ imaginary_unit @ Y )
% 8.58/4.87 = X )
% 8.58/4.87 = ( ( X = zero_zero_complex )
% 8.58/4.87 & ( Y = zero_zero_complex ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8045_real__eq__imaginary__iff,axiom,
% 8.58/4.87 ! [Y: complex,X: complex] :
% 8.58/4.87 ( ( member_complex @ Y @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( ( member_complex @ X @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( ( X
% 8.58/4.87 = ( times_times_complex @ imaginary_unit @ Y ) )
% 8.58/4.87 = ( ( X = zero_zero_complex )
% 8.58/4.87 & ( Y = zero_zero_complex ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8046_or__numerals_I3_J,axiom,
% 8.58/4.87 ! [X: num,Y: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 8.58/4.87 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8047_or__numerals_I3_J,axiom,
% 8.58/4.87 ! [X: num,Y: num] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 8.58/4.87 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8048_or__numerals_I1_J,axiom,
% 8.58/4.87 ! [Y: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 8.58/4.87 = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8049_or__numerals_I1_J,axiom,
% 8.58/4.87 ! [Y: num] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 8.58/4.87 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8050_or__numerals_I5_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 8.58/4.87 = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8051_or__numerals_I5_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 8.58/4.87 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8052_or__minus__numerals_I6_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8053_or__minus__numerals_I2_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8054_Im__divide__Reals,axiom,
% 8.58/4.87 ! [R: complex,Z2: complex] :
% 8.58/4.87 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( ( im @ ( divide1717551699836669952omplex @ Z2 @ R ) )
% 8.58/4.87 = ( divide_divide_real @ ( im @ Z2 ) @ ( re @ R ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8055_or__minus__minus__numerals,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8056_and__minus__minus__numerals,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8057_or__numerals_I4_J,axiom,
% 8.58/4.87 ! [X: num,Y: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 8.58/4.87 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8058_or__numerals_I4_J,axiom,
% 8.58/4.87 ! [X: num,Y: num] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 8.58/4.87 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8059_or__numerals_I6_J,axiom,
% 8.58/4.87 ! [X: num,Y: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 8.58/4.87 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8060_or__numerals_I6_J,axiom,
% 8.58/4.87 ! [X: num,Y: num] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 8.58/4.87 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8061_or__numerals_I7_J,axiom,
% 8.58/4.87 ! [X: num,Y: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
% 8.58/4.87 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8062_or__numerals_I7_J,axiom,
% 8.58/4.87 ! [X: num,Y: num] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 8.58/4.87 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8063_of__nat__or__eq,axiom,
% 8.58/4.87 ! [M: nat,N: nat] :
% 8.58/4.87 ( ( semiri1314217659103216013at_int @ ( bit_se1412395901928357646or_nat @ M @ N ) )
% 8.58/4.87 = ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8064_of__nat__or__eq,axiom,
% 8.58/4.87 ! [M: nat,N: nat] :
% 8.58/4.87 ( ( semiri1316708129612266289at_nat @ ( bit_se1412395901928357646or_nat @ M @ N ) )
% 8.58/4.87 = ( bit_se1412395901928357646or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8065_Reals__numeral,axiom,
% 8.58/4.87 ! [W: num] : ( member_complex @ ( numera6690914467698888265omplex @ W ) @ real_V2521375963428798218omplex ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8066_Reals__numeral,axiom,
% 8.58/4.87 ! [W: num] : ( member_real @ ( numeral_numeral_real @ W ) @ real_V470468836141973256s_real ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8067_Reals__mult,axiom,
% 8.58/4.87 ! [A: real,B: real] :
% 8.58/4.87 ( ( member_real @ A @ real_V470468836141973256s_real )
% 8.58/4.87 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 8.58/4.87 => ( member_real @ ( times_times_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8068_Reals__mult,axiom,
% 8.58/4.87 ! [A: complex,B: complex] :
% 8.58/4.87 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( member_complex @ ( times_times_complex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8069_Reals__1,axiom,
% 8.58/4.87 member_real @ one_one_real @ real_V470468836141973256s_real ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8070_Reals__1,axiom,
% 8.58/4.87 member_complex @ one_one_complex @ real_V2521375963428798218omplex ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8071_Reals__add,axiom,
% 8.58/4.87 ! [A: real,B: real] :
% 8.58/4.87 ( ( member_real @ A @ real_V470468836141973256s_real )
% 8.58/4.87 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 8.58/4.87 => ( member_real @ ( plus_plus_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8072_Reals__add,axiom,
% 8.58/4.87 ! [A: complex,B: complex] :
% 8.58/4.87 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( member_complex @ ( plus_plus_complex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8073_Reals__diff,axiom,
% 8.58/4.87 ! [A: real,B: real] :
% 8.58/4.87 ( ( member_real @ A @ real_V470468836141973256s_real )
% 8.58/4.87 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 8.58/4.87 => ( member_real @ ( minus_minus_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8074_Reals__diff,axiom,
% 8.58/4.87 ! [A: complex,B: complex] :
% 8.58/4.87 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( member_complex @ ( minus_minus_complex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8075_Reals__divide,axiom,
% 8.58/4.87 ! [A: complex,B: complex] :
% 8.58/4.87 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( member_complex @ ( divide1717551699836669952omplex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8076_Reals__divide,axiom,
% 8.58/4.87 ! [A: real,B: real] :
% 8.58/4.87 ( ( member_real @ A @ real_V470468836141973256s_real )
% 8.58/4.87 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 8.58/4.87 => ( member_real @ ( divide_divide_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8077_bit__or__iff,axiom,
% 8.58/4.87 ! [A: int,B: int,N: nat] :
% 8.58/4.87 ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ N )
% 8.58/4.87 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 8.58/4.87 | ( bit_se1146084159140164899it_int @ B @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8078_bit__or__iff,axiom,
% 8.58/4.87 ! [A: nat,B: nat,N: nat] :
% 8.58/4.87 ( ( bit_se1148574629649215175it_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ N )
% 8.58/4.87 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 8.58/4.87 | ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8079_size__neq__size__imp__neq,axiom,
% 8.58/4.87 ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
% 8.58/4.87 ( ( ( size_s6755466524823107622T_VEBT @ X )
% 8.58/4.87 != ( size_s6755466524823107622T_VEBT @ Y ) )
% 8.58/4.87 => ( X != Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8080_size__neq__size__imp__neq,axiom,
% 8.58/4.87 ! [X: list_o,Y: list_o] :
% 8.58/4.87 ( ( ( size_size_list_o @ X )
% 8.58/4.87 != ( size_size_list_o @ Y ) )
% 8.58/4.87 => ( X != Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8081_size__neq__size__imp__neq,axiom,
% 8.58/4.87 ! [X: list_nat,Y: list_nat] :
% 8.58/4.87 ( ( ( size_size_list_nat @ X )
% 8.58/4.87 != ( size_size_list_nat @ Y ) )
% 8.58/4.87 => ( X != Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8082_size__neq__size__imp__neq,axiom,
% 8.58/4.87 ! [X: list_int,Y: list_int] :
% 8.58/4.87 ( ( ( size_size_list_int @ X )
% 8.58/4.87 != ( size_size_list_int @ Y ) )
% 8.58/4.87 => ( X != Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8083_size__neq__size__imp__neq,axiom,
% 8.58/4.87 ! [X: num,Y: num] :
% 8.58/4.87 ( ( ( size_size_num @ X )
% 8.58/4.87 != ( size_size_num @ Y ) )
% 8.58/4.87 => ( X != Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8084_or_Oleft__commute,axiom,
% 8.58/4.87 ! [B: int,A: int,C: int] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ B @ ( bit_se1409905431419307370or_int @ A @ C ) )
% 8.58/4.87 = ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ B @ C ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8085_or_Oleft__commute,axiom,
% 8.58/4.87 ! [B: nat,A: nat,C: nat] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ B @ ( bit_se1412395901928357646or_nat @ A @ C ) )
% 8.58/4.87 = ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ B @ C ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8086_or_Ocommute,axiom,
% 8.58/4.87 ( bit_se1409905431419307370or_int
% 8.58/4.87 = ( ^ [A2: int,B2: int] : ( bit_se1409905431419307370or_int @ B2 @ A2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8087_or_Ocommute,axiom,
% 8.58/4.87 ( bit_se1412395901928357646or_nat
% 8.58/4.87 = ( ^ [A2: nat,B2: nat] : ( bit_se1412395901928357646or_nat @ B2 @ A2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8088_or_Oassoc,axiom,
% 8.58/4.87 ! [A: int,B: int,C: int] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ C )
% 8.58/4.87 = ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ B @ C ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8089_or_Oassoc,axiom,
% 8.58/4.87 ! [A: nat,B: nat,C: nat] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ C )
% 8.58/4.87 = ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ B @ C ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8090_bit_Odisj__zero__right,axiom,
% 8.58/4.87 ! [X: int] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ X @ zero_zero_int )
% 8.58/4.87 = X ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8091_or__eq__0__iff,axiom,
% 8.58/4.87 ! [A: int,B: int] :
% 8.58/4.87 ( ( ( bit_se1409905431419307370or_int @ A @ B )
% 8.58/4.87 = zero_zero_int )
% 8.58/4.87 = ( ( A = zero_zero_int )
% 8.58/4.87 & ( B = zero_zero_int ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8092_or__eq__0__iff,axiom,
% 8.58/4.87 ! [A: nat,B: nat] :
% 8.58/4.87 ( ( ( bit_se1412395901928357646or_nat @ A @ B )
% 8.58/4.87 = zero_zero_nat )
% 8.58/4.87 = ( ( A = zero_zero_nat )
% 8.58/4.87 & ( B = zero_zero_nat ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8093_of__int__or__eq,axiom,
% 8.58/4.87 ! [K: int,L: int] :
% 8.58/4.87 ( ( ring_1_of_int_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
% 8.58/4.87 = ( bit_se1409905431419307370or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8094_bit__or__int__iff,axiom,
% 8.58/4.87 ! [K: int,L: int,N: nat] :
% 8.58/4.87 ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ N )
% 8.58/4.87 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 8.58/4.87 | ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8095_bit_Odisj__conj__distrib2,axiom,
% 8.58/4.87 ! [Y: int,Z2: int,X: int] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ Y @ Z2 ) @ X )
% 8.58/4.87 = ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ Y @ X ) @ ( bit_se1409905431419307370or_int @ Z2 @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8096_bit_Oconj__disj__distrib2,axiom,
% 8.58/4.87 ! [Y: int,Z2: int,X: int] :
% 8.58/4.87 ( ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ Y @ Z2 ) @ X )
% 8.58/4.87 = ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ Y @ X ) @ ( bit_se725231765392027082nd_int @ Z2 @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8097_bit_Odisj__conj__distrib,axiom,
% 8.58/4.87 ! [X: int,Y: int,Z2: int] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ X @ ( bit_se725231765392027082nd_int @ Y @ Z2 ) )
% 8.58/4.87 = ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8098_bit_Oconj__disj__distrib,axiom,
% 8.58/4.87 ! [X: int,Y: int,Z2: int] :
% 8.58/4.87 ( ( bit_se725231765392027082nd_int @ X @ ( bit_se1409905431419307370or_int @ Y @ Z2 ) )
% 8.58/4.87 = ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se725231765392027082nd_int @ X @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8099_Reals__0,axiom,
% 8.58/4.87 member_real @ zero_zero_real @ real_V470468836141973256s_real ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8100_Reals__0,axiom,
% 8.58/4.87 member_complex @ zero_zero_complex @ real_V2521375963428798218omplex ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8101_Reals__power,axiom,
% 8.58/4.87 ! [A: real,N: nat] :
% 8.58/4.87 ( ( member_real @ A @ real_V470468836141973256s_real )
% 8.58/4.87 => ( member_real @ ( power_power_real @ A @ N ) @ real_V470468836141973256s_real ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8102_Reals__power,axiom,
% 8.58/4.87 ! [A: complex,N: nat] :
% 8.58/4.87 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( member_complex @ ( power_power_complex @ A @ N ) @ real_V2521375963428798218omplex ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8103_Reals__of__nat,axiom,
% 8.58/4.87 ! [N: nat] : ( member_complex @ ( semiri8010041392384452111omplex @ N ) @ real_V2521375963428798218omplex ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8104_Reals__of__nat,axiom,
% 8.58/4.87 ! [N: nat] : ( member_real @ ( semiri5074537144036343181t_real @ N ) @ real_V470468836141973256s_real ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8105_or__greater__eq,axiom,
% 8.58/4.87 ! [L: int,K: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ zero_zero_int @ L )
% 8.58/4.87 => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8106_OR__lower,axiom,
% 8.58/4.87 ! [X: int,Y: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.58/4.87 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.58/4.87 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8107_disjunctive__add,axiom,
% 8.58/4.87 ! [A: int,B: int] :
% 8.58/4.87 ( ! [N2: nat] :
% 8.58/4.87 ( ~ ( bit_se1146084159140164899it_int @ A @ N2 )
% 8.58/4.87 | ~ ( bit_se1146084159140164899it_int @ B @ N2 ) )
% 8.58/4.87 => ( ( plus_plus_int @ A @ B )
% 8.58/4.87 = ( bit_se1409905431419307370or_int @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8108_disjunctive__add,axiom,
% 8.58/4.87 ! [A: nat,B: nat] :
% 8.58/4.87 ( ! [N2: nat] :
% 8.58/4.87 ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 )
% 8.58/4.87 | ~ ( bit_se1148574629649215175it_nat @ B @ N2 ) )
% 8.58/4.87 => ( ( plus_plus_nat @ A @ B )
% 8.58/4.87 = ( bit_se1412395901928357646or_nat @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8109_plus__and__or,axiom,
% 8.58/4.87 ! [X: int,Y: int] :
% 8.58/4.87 ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Y ) )
% 8.58/4.87 = ( plus_plus_int @ X @ Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8110_or__eq__not__not__and,axiom,
% 8.58/4.87 ( bit_se1409905431419307370or_int
% 8.58/4.87 = ( ^ [A2: int,B2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ A2 ) @ ( bit_ri7919022796975470100ot_int @ B2 ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8111_and__eq__not__not__or,axiom,
% 8.58/4.87 ( bit_se725231765392027082nd_int
% 8.58/4.87 = ( ^ [A2: int,B2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ A2 ) @ ( bit_ri7919022796975470100ot_int @ B2 ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8112_or__int__def,axiom,
% 8.58/4.87 ( bit_se1409905431419307370or_int
% 8.58/4.87 = ( ^ [K2: int,L2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ ( bit_ri7919022796975470100ot_int @ L2 ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8113_nonzero__Reals__divide,axiom,
% 8.58/4.87 ! [A: complex,B: complex] :
% 8.58/4.87 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( ( member_complex @ B @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( ( B != zero_zero_complex )
% 8.58/4.87 => ( member_complex @ ( divide1717551699836669952omplex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8114_nonzero__Reals__divide,axiom,
% 8.58/4.87 ! [A: real,B: real] :
% 8.58/4.87 ( ( member_real @ A @ real_V470468836141973256s_real )
% 8.58/4.87 => ( ( member_real @ B @ real_V470468836141973256s_real )
% 8.58/4.87 => ( ( B != zero_zero_real )
% 8.58/4.87 => ( member_real @ ( divide_divide_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8115_nonzero__Reals__inverse,axiom,
% 8.58/4.87 ! [A: real] :
% 8.58/4.87 ( ( member_real @ A @ real_V470468836141973256s_real )
% 8.58/4.87 => ( ( A != zero_zero_real )
% 8.58/4.87 => ( member_real @ ( inverse_inverse_real @ A ) @ real_V470468836141973256s_real ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8116_nonzero__Reals__inverse,axiom,
% 8.58/4.87 ! [A: complex] :
% 8.58/4.87 ( ( member_complex @ A @ real_V2521375963428798218omplex )
% 8.58/4.87 => ( ( A != zero_zero_complex )
% 8.58/4.87 => ( member_complex @ ( invers8013647133539491842omplex @ A ) @ real_V2521375963428798218omplex ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8117_or__not__numerals_I1_J,axiom,
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8118_bit_Oxor__def2,axiom,
% 8.58/4.87 ( bit_se6526347334894502574or_int
% 8.58/4.87 = ( ^ [X3: int,Y6: int] : ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ X3 @ Y6 ) @ ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ X3 ) @ ( bit_ri7919022796975470100ot_int @ Y6 ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8119_bit_Oxor__def,axiom,
% 8.58/4.87 ( bit_se6526347334894502574or_int
% 8.58/4.87 = ( ^ [X3: int,Y6: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ X3 @ ( bit_ri7919022796975470100ot_int @ Y6 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X3 ) @ Y6 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8120_set__bit__eq__or,axiom,
% 8.58/4.87 ( bit_se7882103937844011126it_nat
% 8.58/4.87 = ( ^ [N3: nat,A2: nat] : ( bit_se1412395901928357646or_nat @ A2 @ ( bit_se547839408752420682it_nat @ N3 @ one_one_nat ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8121_set__bit__eq__or,axiom,
% 8.58/4.87 ( bit_se7879613467334960850it_int
% 8.58/4.87 = ( ^ [N3: nat,A2: int] : ( bit_se1409905431419307370or_int @ A2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8122_set__bit__eq__or,axiom,
% 8.58/4.87 ( bit_se2793503036327961859nteger
% 8.58/4.87 = ( ^ [N3: nat,A2: code_integer] : ( bit_se1080825931792720795nteger @ A2 @ ( bit_se7788150548672797655nteger @ N3 @ one_one_Code_integer ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8123_set__bit__eq__or,axiom,
% 8.58/4.87 ( bit_se1617098188084679374atural
% 8.58/4.87 = ( ^ [N3: nat,A2: code_natural] : ( bit_se9127793120404214118atural @ A2 @ ( bit_se6611745700429515170atural @ N3 @ one_one_Code_natural ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8124_xor__int__def,axiom,
% 8.58/4.87 ( bit_se6526347334894502574or_int
% 8.58/4.87 = ( ^ [K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ L2 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ L2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8125_concat__bit__def,axiom,
% 8.58/4.87 ( bit_concat_bit
% 8.58/4.87 = ( ^ [N3: nat,K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N3 @ K2 ) @ ( bit_se545348938243370406it_int @ N3 @ L2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8126_set__bit__int__def,axiom,
% 8.58/4.87 ( bit_se7879613467334960850it_int
% 8.58/4.87 = ( ^ [N3: nat,K2: int] : ( bit_se1409905431419307370or_int @ K2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8127_even__or__iff,axiom,
% 8.58/4.87 ! [A: code_integer,B: code_integer] :
% 8.58/4.87 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ A @ B ) )
% 8.58/4.87 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.58/4.87 & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8128_even__or__iff,axiom,
% 8.58/4.87 ! [A: int,B: int] :
% 8.58/4.87 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 8.58/4.87 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.58/4.87 & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8129_even__or__iff,axiom,
% 8.58/4.87 ! [A: nat,B: nat] :
% 8.58/4.87 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 8.58/4.87 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.58/4.87 & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8130_bit_Ocomplement__unique,axiom,
% 8.58/4.87 ! [A: code_integer,X: code_integer,Y: code_integer] :
% 8.58/4.87 ( ( ( bit_se3949692690581998587nteger @ A @ X )
% 8.58/4.87 = zero_z3403309356797280102nteger )
% 8.58/4.87 => ( ( ( bit_se1080825931792720795nteger @ A @ X )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.87 => ( ( ( bit_se3949692690581998587nteger @ A @ Y )
% 8.58/4.87 = zero_z3403309356797280102nteger )
% 8.58/4.87 => ( ( ( bit_se1080825931792720795nteger @ A @ Y )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.87 => ( X = Y ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8131_bit_Ocomplement__unique,axiom,
% 8.58/4.87 ! [A: int,X: int,Y: int] :
% 8.58/4.87 ( ( ( bit_se725231765392027082nd_int @ A @ X )
% 8.58/4.87 = zero_zero_int )
% 8.58/4.87 => ( ( ( bit_se1409905431419307370or_int @ A @ X )
% 8.58/4.87 = ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.87 => ( ( ( bit_se725231765392027082nd_int @ A @ Y )
% 8.58/4.87 = zero_zero_int )
% 8.58/4.87 => ( ( ( bit_se1409905431419307370or_int @ A @ Y )
% 8.58/4.87 = ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.87 => ( X = Y ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8132_or__not__numerals_I2_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8133_or__not__numerals_I4_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8134_num_Osize_I4_J,axiom,
% 8.58/4.87 ( ( size_size_num @ one )
% 8.58/4.87 = zero_zero_nat ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8135_bit_Ocompl__unique,axiom,
% 8.58/4.87 ! [X: code_integer,Y: code_integer] :
% 8.58/4.87 ( ( ( bit_se3949692690581998587nteger @ X @ Y )
% 8.58/4.87 = zero_z3403309356797280102nteger )
% 8.58/4.87 => ( ( ( bit_se1080825931792720795nteger @ X @ Y )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.87 => ( ( bit_ri7632146776885996613nteger @ X )
% 8.58/4.87 = Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8136_bit_Ocompl__unique,axiom,
% 8.58/4.87 ! [X: int,Y: int] :
% 8.58/4.87 ( ( ( bit_se725231765392027082nd_int @ X @ Y )
% 8.58/4.87 = zero_zero_int )
% 8.58/4.87 => ( ( ( bit_se1409905431419307370or_int @ X @ Y )
% 8.58/4.87 = ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.87 => ( ( bit_ri7919022796975470100ot_int @ X )
% 8.58/4.87 = Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8137_or__not__numerals_I3_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8138_or__not__numerals_I7_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8139_signed__take__bit__eq__if__negative,axiom,
% 8.58/4.87 ! [A: int,N: nat] :
% 8.58/4.87 ( ( bit_se1146084159140164899it_int @ A @ N )
% 8.58/4.87 => ( ( bit_ri631733984087533419it_int @ N @ A )
% 8.58/4.87 = ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8140_mask__Suc__exp,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
% 8.58/4.87 = ( bit_se1412395901928357646or_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8141_mask__Suc__exp,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
% 8.58/4.87 = ( bit_se1409905431419307370or_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8142_or__not__numerals_I6_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.58/4.87 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8143_or__one__eq,axiom,
% 8.58/4.87 ! [A: code_integer] :
% 8.58/4.87 ( ( bit_se1080825931792720795nteger @ A @ one_one_Code_integer )
% 8.58/4.87 = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8144_or__one__eq,axiom,
% 8.58/4.87 ! [A: int] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ A @ one_one_int )
% 8.58/4.87 = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8145_or__one__eq,axiom,
% 8.58/4.87 ! [A: nat] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ A @ one_one_nat )
% 8.58/4.87 = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8146_one__or__eq,axiom,
% 8.58/4.87 ! [A: code_integer] :
% 8.58/4.87 ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ A )
% 8.58/4.87 = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8147_one__or__eq,axiom,
% 8.58/4.87 ! [A: int] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ one_one_int @ A )
% 8.58/4.87 = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8148_one__or__eq,axiom,
% 8.58/4.87 ! [A: nat] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ one_one_nat @ A )
% 8.58/4.87 = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8149_mask__Suc__double,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
% 8.58/4.87 = ( bit_se1412395901928357646or_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8150_mask__Suc__double,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
% 8.58/4.87 = ( bit_se1409905431419307370or_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8151_OR__upper,axiom,
% 8.58/4.87 ! [X: int,N: nat,Y: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.58/4.87 => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.87 => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.58/4.87 => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8152_or__not__numerals_I5_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.58/4.87 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8153_signed__take__bit__def,axiom,
% 8.58/4.87 ( bit_ri6519982836138164636nteger
% 8.58/4.87 = ( ^ [N3: nat,A2: code_integer] : ( bit_se1080825931792720795nteger @ ( bit_se1745604003318907178nteger @ N3 @ A2 ) @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( bit_se9216721137139052372nteger @ A2 @ N3 ) ) @ ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N3 ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8154_signed__take__bit__def,axiom,
% 8.58/4.87 ( bit_ri631733984087533419it_int
% 8.58/4.87 = ( ^ [N3: nat,A2: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N3 @ A2 ) @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ A2 @ N3 ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N3 ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8155_or__not__numerals_I9_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.58/4.87 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8156_or__not__numerals_I8_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.58/4.87 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8157_or__int__unfold,axiom,
% 8.58/4.87 ( bit_se1409905431419307370or_int
% 8.58/4.87 = ( ^ [K2: int,L2: int] :
% 8.58/4.87 ( if_int
% 8.58/4.87 @ ( ( K2
% 8.58/4.87 = ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.87 | ( L2
% 8.58/4.87 = ( uminus_uminus_int @ one_one_int ) ) )
% 8.58/4.87 @ ( uminus_uminus_int @ one_one_int )
% 8.58/4.87 @ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8158_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
% 8.58/4.87 boolea2445317508997433345nteger @ bit_se3949692690581998587nteger @ bit_se1080825931792720795nteger @ bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ bit_se3222712562003087583nteger ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8159_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
% 8.58/4.87 boolea8527374999097803216ff_int @ bit_se725231765392027082nd_int @ bit_se1409905431419307370or_int @ bit_ri7919022796975470100ot_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) @ bit_se6526347334894502574or_int ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8160_or__minus__numerals_I4_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8161_or__minus__numerals_I8_J,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8162_max_Oidem,axiom,
% 8.58/4.87 ! [A: int] :
% 8.58/4.87 ( ( ord_max_int @ A @ A )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8163_max_Oidem,axiom,
% 8.58/4.87 ! [A: nat] :
% 8.58/4.87 ( ( ord_max_nat @ A @ A )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8164_max_Oleft__idem,axiom,
% 8.58/4.87 ! [A: int,B: int] :
% 8.58/4.87 ( ( ord_max_int @ A @ ( ord_max_int @ A @ B ) )
% 8.58/4.87 = ( ord_max_int @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8165_max_Oleft__idem,axiom,
% 8.58/4.87 ! [A: nat,B: nat] :
% 8.58/4.87 ( ( ord_max_nat @ A @ ( ord_max_nat @ A @ B ) )
% 8.58/4.87 = ( ord_max_nat @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8166_max_Oright__idem,axiom,
% 8.58/4.87 ! [A: int,B: int] :
% 8.58/4.87 ( ( ord_max_int @ ( ord_max_int @ A @ B ) @ B )
% 8.58/4.87 = ( ord_max_int @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8167_max_Oright__idem,axiom,
% 8.58/4.87 ! [A: nat,B: nat] :
% 8.58/4.87 ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ B )
% 8.58/4.87 = ( ord_max_nat @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8168_max_Obounded__iff,axiom,
% 8.58/4.87 ! [B: rat,C: rat,A: rat] :
% 8.58/4.87 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 8.58/4.87 = ( ( ord_less_eq_rat @ B @ A )
% 8.58/4.87 & ( ord_less_eq_rat @ C @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8169_max_Obounded__iff,axiom,
% 8.58/4.87 ! [B: num,C: num,A: num] :
% 8.58/4.87 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 8.58/4.87 = ( ( ord_less_eq_num @ B @ A )
% 8.58/4.87 & ( ord_less_eq_num @ C @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8170_max_Obounded__iff,axiom,
% 8.58/4.87 ! [B: nat,C: nat,A: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 8.58/4.87 = ( ( ord_less_eq_nat @ B @ A )
% 8.58/4.87 & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8171_max_Obounded__iff,axiom,
% 8.58/4.87 ! [B: int,C: int,A: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 8.58/4.87 = ( ( ord_less_eq_int @ B @ A )
% 8.58/4.87 & ( ord_less_eq_int @ C @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8172_max_Oabsorb2,axiom,
% 8.58/4.87 ! [A: rat,B: rat] :
% 8.58/4.87 ( ( ord_less_eq_rat @ A @ B )
% 8.58/4.87 => ( ( ord_max_rat @ A @ B )
% 8.58/4.87 = B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8173_max_Oabsorb2,axiom,
% 8.58/4.87 ! [A: num,B: num] :
% 8.58/4.87 ( ( ord_less_eq_num @ A @ B )
% 8.58/4.87 => ( ( ord_max_num @ A @ B )
% 8.58/4.87 = B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8174_max_Oabsorb2,axiom,
% 8.58/4.87 ! [A: nat,B: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ A @ B )
% 8.58/4.87 => ( ( ord_max_nat @ A @ B )
% 8.58/4.87 = B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8175_max_Oabsorb2,axiom,
% 8.58/4.87 ! [A: int,B: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ A @ B )
% 8.58/4.87 => ( ( ord_max_int @ A @ B )
% 8.58/4.87 = B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8176_max_Oabsorb1,axiom,
% 8.58/4.87 ! [B: rat,A: rat] :
% 8.58/4.87 ( ( ord_less_eq_rat @ B @ A )
% 8.58/4.87 => ( ( ord_max_rat @ A @ B )
% 8.58/4.87 = A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8177_max_Oabsorb1,axiom,
% 8.58/4.87 ! [B: num,A: num] :
% 8.58/4.87 ( ( ord_less_eq_num @ B @ A )
% 8.58/4.87 => ( ( ord_max_num @ A @ B )
% 8.58/4.87 = A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8178_max_Oabsorb1,axiom,
% 8.58/4.87 ! [B: nat,A: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ B @ A )
% 8.58/4.87 => ( ( ord_max_nat @ A @ B )
% 8.58/4.87 = A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8179_max_Oabsorb1,axiom,
% 8.58/4.87 ! [B: int,A: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ B @ A )
% 8.58/4.87 => ( ( ord_max_int @ A @ B )
% 8.58/4.87 = A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8180_max__less__iff__conj,axiom,
% 8.58/4.87 ! [X: real,Y: real,Z2: real] :
% 8.58/4.87 ( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z2 )
% 8.58/4.87 = ( ( ord_less_real @ X @ Z2 )
% 8.58/4.87 & ( ord_less_real @ Y @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8181_max__less__iff__conj,axiom,
% 8.58/4.87 ! [X: rat,Y: rat,Z2: rat] :
% 8.58/4.87 ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
% 8.58/4.87 = ( ( ord_less_rat @ X @ Z2 )
% 8.58/4.87 & ( ord_less_rat @ Y @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8182_max__less__iff__conj,axiom,
% 8.58/4.87 ! [X: num,Y: num,Z2: num] :
% 8.58/4.87 ( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z2 )
% 8.58/4.87 = ( ( ord_less_num @ X @ Z2 )
% 8.58/4.87 & ( ord_less_num @ Y @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8183_max__less__iff__conj,axiom,
% 8.58/4.87 ! [X: nat,Y: nat,Z2: nat] :
% 8.58/4.87 ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
% 8.58/4.87 = ( ( ord_less_nat @ X @ Z2 )
% 8.58/4.87 & ( ord_less_nat @ Y @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8184_max__less__iff__conj,axiom,
% 8.58/4.87 ! [X: int,Y: int,Z2: int] :
% 8.58/4.87 ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z2 )
% 8.58/4.87 = ( ( ord_less_int @ X @ Z2 )
% 8.58/4.87 & ( ord_less_int @ Y @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8185_max_Oabsorb4,axiom,
% 8.58/4.87 ! [A: real,B: real] :
% 8.58/4.87 ( ( ord_less_real @ A @ B )
% 8.58/4.87 => ( ( ord_max_real @ A @ B )
% 8.58/4.87 = B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8186_max_Oabsorb4,axiom,
% 8.58/4.87 ! [A: rat,B: rat] :
% 8.58/4.87 ( ( ord_less_rat @ A @ B )
% 8.58/4.87 => ( ( ord_max_rat @ A @ B )
% 8.58/4.87 = B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8187_max_Oabsorb4,axiom,
% 8.58/4.87 ! [A: num,B: num] :
% 8.58/4.87 ( ( ord_less_num @ A @ B )
% 8.58/4.87 => ( ( ord_max_num @ A @ B )
% 8.58/4.87 = B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8188_max_Oabsorb4,axiom,
% 8.58/4.87 ! [A: nat,B: nat] :
% 8.58/4.87 ( ( ord_less_nat @ A @ B )
% 8.58/4.87 => ( ( ord_max_nat @ A @ B )
% 8.58/4.87 = B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8189_max_Oabsorb4,axiom,
% 8.58/4.87 ! [A: int,B: int] :
% 8.58/4.87 ( ( ord_less_int @ A @ B )
% 8.58/4.87 => ( ( ord_max_int @ A @ B )
% 8.58/4.87 = B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8190_max_Oabsorb3,axiom,
% 8.58/4.87 ! [B: real,A: real] :
% 8.58/4.87 ( ( ord_less_real @ B @ A )
% 8.58/4.87 => ( ( ord_max_real @ A @ B )
% 8.58/4.87 = A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8191_max_Oabsorb3,axiom,
% 8.58/4.87 ! [B: rat,A: rat] :
% 8.58/4.87 ( ( ord_less_rat @ B @ A )
% 8.58/4.87 => ( ( ord_max_rat @ A @ B )
% 8.58/4.87 = A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8192_max_Oabsorb3,axiom,
% 8.58/4.87 ! [B: num,A: num] :
% 8.58/4.87 ( ( ord_less_num @ B @ A )
% 8.58/4.87 => ( ( ord_max_num @ A @ B )
% 8.58/4.87 = A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8193_max_Oabsorb3,axiom,
% 8.58/4.87 ! [B: nat,A: nat] :
% 8.58/4.87 ( ( ord_less_nat @ B @ A )
% 8.58/4.87 => ( ( ord_max_nat @ A @ B )
% 8.58/4.87 = A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8194_max_Oabsorb3,axiom,
% 8.58/4.87 ! [B: int,A: int] :
% 8.58/4.87 ( ( ord_less_int @ B @ A )
% 8.58/4.87 => ( ( ord_max_int @ A @ B )
% 8.58/4.87 = A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8195_of__bool__or__iff,axiom,
% 8.58/4.87 ! [P: $o,Q: $o] :
% 8.58/4.87 ( ( zero_n2687167440665602831ol_nat
% 8.58/4.87 @ ( P
% 8.58/4.87 | Q ) )
% 8.58/4.87 = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8196_of__bool__or__iff,axiom,
% 8.58/4.87 ! [P: $o,Q: $o] :
% 8.58/4.87 ( ( zero_n2684676970156552555ol_int
% 8.58/4.87 @ ( P
% 8.58/4.87 | Q ) )
% 8.58/4.87 = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8197_of__bool__or__iff,axiom,
% 8.58/4.87 ! [P: $o,Q: $o] :
% 8.58/4.87 ( ( zero_n356916108424825756nteger
% 8.58/4.87 @ ( P
% 8.58/4.87 | Q ) )
% 8.58/4.87 = ( ord_max_Code_integer @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8198_max__number__of_I1_J,axiom,
% 8.58/4.87 ! [U: num,V: num] :
% 8.58/4.87 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 8.58/4.87 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 8.58/4.87 = ( numeral_numeral_real @ V ) ) )
% 8.58/4.87 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 8.58/4.87 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 8.58/4.87 = ( numeral_numeral_real @ U ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8199_max__number__of_I1_J,axiom,
% 8.58/4.87 ! [U: num,V: num] :
% 8.58/4.87 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 8.58/4.87 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 8.58/4.87 = ( numeral_numeral_rat @ V ) ) )
% 8.58/4.87 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 8.58/4.87 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 8.58/4.87 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8200_max__number__of_I1_J,axiom,
% 8.58/4.87 ! [U: num,V: num] :
% 8.58/4.87 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 8.58/4.87 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 8.58/4.87 = ( numeral_numeral_nat @ V ) ) )
% 8.58/4.87 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 8.58/4.87 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 8.58/4.87 = ( numeral_numeral_nat @ U ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8201_max__number__of_I1_J,axiom,
% 8.58/4.87 ! [U: num,V: num] :
% 8.58/4.87 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 8.58/4.87 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 8.58/4.87 = ( numeral_numeral_int @ V ) ) )
% 8.58/4.87 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 8.58/4.87 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 8.58/4.87 = ( numeral_numeral_int @ U ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8202_max__0__1_I3_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
% 8.58/4.87 = ( numeral_numeral_real @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8203_max__0__1_I3_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
% 8.58/4.87 = ( numeral_numeral_rat @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8204_max__0__1_I3_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
% 8.58/4.87 = ( numeral_numeral_nat @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8205_max__0__1_I3_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
% 8.58/4.87 = ( numeral_numeral_int @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8206_max__0__1_I4_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
% 8.58/4.87 = ( numeral_numeral_real @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8207_max__0__1_I4_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
% 8.58/4.87 = ( numeral_numeral_rat @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8208_max__0__1_I4_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
% 8.58/4.87 = ( numeral_numeral_nat @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8209_max__0__1_I4_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
% 8.58/4.87 = ( numeral_numeral_int @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8210_max__0__1_I2_J,axiom,
% 8.58/4.87 ( ( ord_max_real @ one_one_real @ zero_zero_real )
% 8.58/4.87 = one_one_real ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8211_max__0__1_I2_J,axiom,
% 8.58/4.87 ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
% 8.58/4.87 = one_one_rat ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8212_max__0__1_I2_J,axiom,
% 8.58/4.87 ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 8.58/4.87 = one_one_int ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8213_max__0__1_I2_J,axiom,
% 8.58/4.87 ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 8.58/4.87 = one_one_nat ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8214_max__0__1_I1_J,axiom,
% 8.58/4.87 ( ( ord_max_real @ zero_zero_real @ one_one_real )
% 8.58/4.87 = one_one_real ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8215_max__0__1_I1_J,axiom,
% 8.58/4.87 ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
% 8.58/4.87 = one_one_rat ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8216_max__0__1_I1_J,axiom,
% 8.58/4.87 ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 8.58/4.87 = one_one_int ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8217_max__0__1_I1_J,axiom,
% 8.58/4.87 ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 8.58/4.87 = one_one_nat ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8218_max__0__1_I5_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 8.58/4.87 = ( numeral_numeral_real @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8219_max__0__1_I5_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 8.58/4.87 = ( numeral_numeral_rat @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8220_max__0__1_I5_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 8.58/4.87 = ( numeral_numeral_nat @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8221_max__0__1_I5_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 8.58/4.87 = ( numeral_numeral_int @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8222_max__0__1_I6_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
% 8.58/4.87 = ( numeral_numeral_real @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8223_max__0__1_I6_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
% 8.58/4.87 = ( numeral_numeral_rat @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8224_max__0__1_I6_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
% 8.58/4.87 = ( numeral_numeral_nat @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8225_max__0__1_I6_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
% 8.58/4.87 = ( numeral_numeral_int @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8226_max__number__of_I4_J,axiom,
% 8.58/4.87 ! [U: num,V: num] :
% 8.58/4.87 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.58/4.87 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.58/4.87 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 8.58/4.87 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.58/4.87 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.58/4.87 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8227_max__number__of_I4_J,axiom,
% 8.58/4.87 ! [U: num,V: num] :
% 8.58/4.87 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 8.58/4.87 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 8.58/4.87 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 8.58/4.87 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8228_max__number__of_I4_J,axiom,
% 8.58/4.87 ! [U: num,V: num] :
% 8.58/4.87 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.58/4.87 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.58/4.87 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 8.58/4.87 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.58/4.87 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.58/4.87 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8229_max__number__of_I4_J,axiom,
% 8.58/4.87 ! [U: num,V: num] :
% 8.58/4.87 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.58/4.87 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 8.58/4.87 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.58/4.87 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8230_max__number__of_I3_J,axiom,
% 8.58/4.87 ! [U: num,V: num] :
% 8.58/4.87 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 8.58/4.87 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 8.58/4.87 = ( numeral_numeral_real @ V ) ) )
% 8.58/4.87 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 8.58/4.87 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 8.58/4.87 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8231_max__number__of_I3_J,axiom,
% 8.58/4.87 ! [U: num,V: num] :
% 8.58/4.87 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 8.58/4.87 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 8.58/4.87 = ( numera6620942414471956472nteger @ V ) ) )
% 8.58/4.87 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 8.58/4.87 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8232_max__number__of_I3_J,axiom,
% 8.58/4.87 ! [U: num,V: num] :
% 8.58/4.87 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 8.58/4.87 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 8.58/4.87 = ( numeral_numeral_rat @ V ) ) )
% 8.58/4.87 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 8.58/4.87 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 8.58/4.87 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8233_max__number__of_I3_J,axiom,
% 8.58/4.87 ! [U: num,V: num] :
% 8.58/4.87 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 8.58/4.87 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 8.58/4.87 = ( numeral_numeral_int @ V ) ) )
% 8.58/4.87 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 8.58/4.87 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8234_max__number__of_I2_J,axiom,
% 8.58/4.87 ! [U: num,V: num] :
% 8.58/4.87 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.58/4.87 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.58/4.87 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 8.58/4.87 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.58/4.87 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.58/4.87 = ( numeral_numeral_real @ U ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8235_max__number__of_I2_J,axiom,
% 8.58/4.87 ! [U: num,V: num] :
% 8.58/4.87 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 8.58/4.87 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 8.58/4.87 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 8.58/4.87 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 8.58/4.87 = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8236_max__number__of_I2_J,axiom,
% 8.58/4.87 ! [U: num,V: num] :
% 8.58/4.87 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.58/4.87 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.58/4.87 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 8.58/4.87 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.58/4.87 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.58/4.87 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8237_max__number__of_I2_J,axiom,
% 8.58/4.87 ! [U: num,V: num] :
% 8.58/4.87 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.58/4.87 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 8.58/4.87 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.58/4.87 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.58/4.87 = ( numeral_numeral_int @ U ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8238_or__nat__numerals_I4_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 8.58/4.87 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8239_or__nat__numerals_I2_J,axiom,
% 8.58/4.87 ! [Y: num] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 8.58/4.87 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8240_or__nat__numerals_I3_J,axiom,
% 8.58/4.87 ! [X: num] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 8.58/4.87 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8241_or__nat__numerals_I1_J,axiom,
% 8.58/4.87 ! [Y: num] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 8.58/4.87 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8242_of__int__max,axiom,
% 8.58/4.87 ! [X: int,Y: int] :
% 8.58/4.87 ( ( ring_1_of_int_real @ ( ord_max_int @ X @ Y ) )
% 8.58/4.87 = ( ord_max_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8243_of__int__max,axiom,
% 8.58/4.87 ! [X: int,Y: int] :
% 8.58/4.87 ( ( ring_1_of_int_rat @ ( ord_max_int @ X @ Y ) )
% 8.58/4.87 = ( ord_max_rat @ ( ring_1_of_int_rat @ X ) @ ( ring_1_of_int_rat @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8244_of__int__max,axiom,
% 8.58/4.87 ! [X: int,Y: int] :
% 8.58/4.87 ( ( ring_1_of_int_int @ ( ord_max_int @ X @ Y ) )
% 8.58/4.87 = ( ord_max_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8245_max_OcoboundedI2,axiom,
% 8.58/4.87 ! [C: rat,B: rat,A: rat] :
% 8.58/4.87 ( ( ord_less_eq_rat @ C @ B )
% 8.58/4.87 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8246_max_OcoboundedI2,axiom,
% 8.58/4.87 ! [C: num,B: num,A: num] :
% 8.58/4.87 ( ( ord_less_eq_num @ C @ B )
% 8.58/4.87 => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8247_max_OcoboundedI2,axiom,
% 8.58/4.87 ! [C: nat,B: nat,A: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ C @ B )
% 8.58/4.87 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8248_max_OcoboundedI2,axiom,
% 8.58/4.87 ! [C: int,B: int,A: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ C @ B )
% 8.58/4.87 => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8249_max_OcoboundedI1,axiom,
% 8.58/4.87 ! [C: rat,A: rat,B: rat] :
% 8.58/4.87 ( ( ord_less_eq_rat @ C @ A )
% 8.58/4.87 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8250_max_OcoboundedI1,axiom,
% 8.58/4.87 ! [C: num,A: num,B: num] :
% 8.58/4.87 ( ( ord_less_eq_num @ C @ A )
% 8.58/4.87 => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8251_max_OcoboundedI1,axiom,
% 8.58/4.87 ! [C: nat,A: nat,B: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ C @ A )
% 8.58/4.87 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8252_max_OcoboundedI1,axiom,
% 8.58/4.87 ! [C: int,A: int,B: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ C @ A )
% 8.58/4.87 => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8253_max_Oabsorb__iff2,axiom,
% 8.58/4.87 ( ord_less_eq_rat
% 8.58/4.87 = ( ^ [A2: rat,B2: rat] :
% 8.58/4.87 ( ( ord_max_rat @ A2 @ B2 )
% 8.58/4.87 = B2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8254_max_Oabsorb__iff2,axiom,
% 8.58/4.87 ( ord_less_eq_num
% 8.58/4.87 = ( ^ [A2: num,B2: num] :
% 8.58/4.87 ( ( ord_max_num @ A2 @ B2 )
% 8.58/4.87 = B2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8255_max_Oabsorb__iff2,axiom,
% 8.58/4.87 ( ord_less_eq_nat
% 8.58/4.87 = ( ^ [A2: nat,B2: nat] :
% 8.58/4.87 ( ( ord_max_nat @ A2 @ B2 )
% 8.58/4.87 = B2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8256_max_Oabsorb__iff2,axiom,
% 8.58/4.87 ( ord_less_eq_int
% 8.58/4.87 = ( ^ [A2: int,B2: int] :
% 8.58/4.87 ( ( ord_max_int @ A2 @ B2 )
% 8.58/4.87 = B2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8257_max_Oabsorb__iff1,axiom,
% 8.58/4.87 ( ord_less_eq_rat
% 8.58/4.87 = ( ^ [B2: rat,A2: rat] :
% 8.58/4.87 ( ( ord_max_rat @ A2 @ B2 )
% 8.58/4.87 = A2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8258_max_Oabsorb__iff1,axiom,
% 8.58/4.87 ( ord_less_eq_num
% 8.58/4.87 = ( ^ [B2: num,A2: num] :
% 8.58/4.87 ( ( ord_max_num @ A2 @ B2 )
% 8.58/4.87 = A2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8259_max_Oabsorb__iff1,axiom,
% 8.58/4.87 ( ord_less_eq_nat
% 8.58/4.87 = ( ^ [B2: nat,A2: nat] :
% 8.58/4.87 ( ( ord_max_nat @ A2 @ B2 )
% 8.58/4.87 = A2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8260_max_Oabsorb__iff1,axiom,
% 8.58/4.87 ( ord_less_eq_int
% 8.58/4.87 = ( ^ [B2: int,A2: int] :
% 8.58/4.87 ( ( ord_max_int @ A2 @ B2 )
% 8.58/4.87 = A2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8261_le__max__iff__disj,axiom,
% 8.58/4.87 ! [Z2: rat,X: rat,Y: rat] :
% 8.58/4.87 ( ( ord_less_eq_rat @ Z2 @ ( ord_max_rat @ X @ Y ) )
% 8.58/4.87 = ( ( ord_less_eq_rat @ Z2 @ X )
% 8.58/4.87 | ( ord_less_eq_rat @ Z2 @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8262_le__max__iff__disj,axiom,
% 8.58/4.87 ! [Z2: num,X: num,Y: num] :
% 8.58/4.87 ( ( ord_less_eq_num @ Z2 @ ( ord_max_num @ X @ Y ) )
% 8.58/4.87 = ( ( ord_less_eq_num @ Z2 @ X )
% 8.58/4.87 | ( ord_less_eq_num @ Z2 @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8263_le__max__iff__disj,axiom,
% 8.58/4.87 ! [Z2: nat,X: nat,Y: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
% 8.58/4.87 = ( ( ord_less_eq_nat @ Z2 @ X )
% 8.58/4.87 | ( ord_less_eq_nat @ Z2 @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8264_le__max__iff__disj,axiom,
% 8.58/4.87 ! [Z2: int,X: int,Y: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ Z2 @ ( ord_max_int @ X @ Y ) )
% 8.58/4.87 = ( ( ord_less_eq_int @ Z2 @ X )
% 8.58/4.87 | ( ord_less_eq_int @ Z2 @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8265_max_Ocobounded2,axiom,
% 8.58/4.87 ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8266_max_Ocobounded2,axiom,
% 8.58/4.87 ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8267_max_Ocobounded2,axiom,
% 8.58/4.87 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8268_max_Ocobounded2,axiom,
% 8.58/4.87 ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8269_max_Ocobounded1,axiom,
% 8.58/4.87 ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8270_max_Ocobounded1,axiom,
% 8.58/4.87 ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8271_max_Ocobounded1,axiom,
% 8.58/4.87 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8272_max_Ocobounded1,axiom,
% 8.58/4.87 ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8273_max_Oorder__iff,axiom,
% 8.58/4.87 ( ord_less_eq_rat
% 8.58/4.87 = ( ^ [B2: rat,A2: rat] :
% 8.58/4.87 ( A2
% 8.58/4.87 = ( ord_max_rat @ A2 @ B2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8274_max_Oorder__iff,axiom,
% 8.58/4.87 ( ord_less_eq_num
% 8.58/4.87 = ( ^ [B2: num,A2: num] :
% 8.58/4.87 ( A2
% 8.58/4.87 = ( ord_max_num @ A2 @ B2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8275_max_Oorder__iff,axiom,
% 8.58/4.87 ( ord_less_eq_nat
% 8.58/4.87 = ( ^ [B2: nat,A2: nat] :
% 8.58/4.87 ( A2
% 8.58/4.87 = ( ord_max_nat @ A2 @ B2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8276_max_Oorder__iff,axiom,
% 8.58/4.87 ( ord_less_eq_int
% 8.58/4.87 = ( ^ [B2: int,A2: int] :
% 8.58/4.87 ( A2
% 8.58/4.87 = ( ord_max_int @ A2 @ B2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8277_max_OboundedI,axiom,
% 8.58/4.87 ! [B: rat,A: rat,C: rat] :
% 8.58/4.87 ( ( ord_less_eq_rat @ B @ A )
% 8.58/4.87 => ( ( ord_less_eq_rat @ C @ A )
% 8.58/4.87 => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8278_max_OboundedI,axiom,
% 8.58/4.87 ! [B: num,A: num,C: num] :
% 8.58/4.87 ( ( ord_less_eq_num @ B @ A )
% 8.58/4.87 => ( ( ord_less_eq_num @ C @ A )
% 8.58/4.87 => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8279_max_OboundedI,axiom,
% 8.58/4.87 ! [B: nat,A: nat,C: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ B @ A )
% 8.58/4.87 => ( ( ord_less_eq_nat @ C @ A )
% 8.58/4.87 => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8280_max_OboundedI,axiom,
% 8.58/4.87 ! [B: int,A: int,C: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ B @ A )
% 8.58/4.87 => ( ( ord_less_eq_int @ C @ A )
% 8.58/4.87 => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8281_max_OboundedE,axiom,
% 8.58/4.87 ! [B: rat,C: rat,A: rat] :
% 8.58/4.87 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 8.58/4.87 => ~ ( ( ord_less_eq_rat @ B @ A )
% 8.58/4.87 => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8282_max_OboundedE,axiom,
% 8.58/4.87 ! [B: num,C: num,A: num] :
% 8.58/4.87 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 8.58/4.87 => ~ ( ( ord_less_eq_num @ B @ A )
% 8.58/4.87 => ~ ( ord_less_eq_num @ C @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8283_max_OboundedE,axiom,
% 8.58/4.87 ! [B: nat,C: nat,A: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 8.58/4.87 => ~ ( ( ord_less_eq_nat @ B @ A )
% 8.58/4.87 => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8284_max_OboundedE,axiom,
% 8.58/4.87 ! [B: int,C: int,A: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 8.58/4.87 => ~ ( ( ord_less_eq_int @ B @ A )
% 8.58/4.87 => ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8285_max_OorderI,axiom,
% 8.58/4.87 ! [A: rat,B: rat] :
% 8.58/4.87 ( ( A
% 8.58/4.87 = ( ord_max_rat @ A @ B ) )
% 8.58/4.87 => ( ord_less_eq_rat @ B @ A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8286_max_OorderI,axiom,
% 8.58/4.87 ! [A: num,B: num] :
% 8.58/4.87 ( ( A
% 8.58/4.87 = ( ord_max_num @ A @ B ) )
% 8.58/4.87 => ( ord_less_eq_num @ B @ A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8287_max_OorderI,axiom,
% 8.58/4.87 ! [A: nat,B: nat] :
% 8.58/4.87 ( ( A
% 8.58/4.87 = ( ord_max_nat @ A @ B ) )
% 8.58/4.87 => ( ord_less_eq_nat @ B @ A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8288_max_OorderI,axiom,
% 8.58/4.87 ! [A: int,B: int] :
% 8.58/4.87 ( ( A
% 8.58/4.87 = ( ord_max_int @ A @ B ) )
% 8.58/4.87 => ( ord_less_eq_int @ B @ A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8289_max_OorderE,axiom,
% 8.58/4.87 ! [B: rat,A: rat] :
% 8.58/4.87 ( ( ord_less_eq_rat @ B @ A )
% 8.58/4.87 => ( A
% 8.58/4.87 = ( ord_max_rat @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8290_max_OorderE,axiom,
% 8.58/4.87 ! [B: num,A: num] :
% 8.58/4.87 ( ( ord_less_eq_num @ B @ A )
% 8.58/4.87 => ( A
% 8.58/4.87 = ( ord_max_num @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8291_max_OorderE,axiom,
% 8.58/4.87 ! [B: nat,A: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ B @ A )
% 8.58/4.87 => ( A
% 8.58/4.87 = ( ord_max_nat @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8292_max_OorderE,axiom,
% 8.58/4.87 ! [B: int,A: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ B @ A )
% 8.58/4.87 => ( A
% 8.58/4.87 = ( ord_max_int @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8293_max_Omono,axiom,
% 8.58/4.87 ! [C: rat,A: rat,D: rat,B: rat] :
% 8.58/4.87 ( ( ord_less_eq_rat @ C @ A )
% 8.58/4.87 => ( ( ord_less_eq_rat @ D @ B )
% 8.58/4.87 => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8294_max_Omono,axiom,
% 8.58/4.87 ! [C: num,A: num,D: num,B: num] :
% 8.58/4.87 ( ( ord_less_eq_num @ C @ A )
% 8.58/4.87 => ( ( ord_less_eq_num @ D @ B )
% 8.58/4.87 => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8295_max_Omono,axiom,
% 8.58/4.87 ! [C: nat,A: nat,D: nat,B: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ C @ A )
% 8.58/4.87 => ( ( ord_less_eq_nat @ D @ B )
% 8.58/4.87 => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8296_max_Omono,axiom,
% 8.58/4.87 ! [C: int,A: int,D: int,B: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ C @ A )
% 8.58/4.87 => ( ( ord_less_eq_int @ D @ B )
% 8.58/4.87 => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8297_max_Ostrict__coboundedI2,axiom,
% 8.58/4.87 ! [C: real,B: real,A: real] :
% 8.58/4.87 ( ( ord_less_real @ C @ B )
% 8.58/4.87 => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8298_max_Ostrict__coboundedI2,axiom,
% 8.58/4.87 ! [C: rat,B: rat,A: rat] :
% 8.58/4.87 ( ( ord_less_rat @ C @ B )
% 8.58/4.87 => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8299_max_Ostrict__coboundedI2,axiom,
% 8.58/4.87 ! [C: num,B: num,A: num] :
% 8.58/4.87 ( ( ord_less_num @ C @ B )
% 8.58/4.87 => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8300_max_Ostrict__coboundedI2,axiom,
% 8.58/4.87 ! [C: nat,B: nat,A: nat] :
% 8.58/4.87 ( ( ord_less_nat @ C @ B )
% 8.58/4.87 => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8301_max_Ostrict__coboundedI2,axiom,
% 8.58/4.87 ! [C: int,B: int,A: int] :
% 8.58/4.87 ( ( ord_less_int @ C @ B )
% 8.58/4.87 => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8302_max_Ostrict__coboundedI1,axiom,
% 8.58/4.87 ! [C: real,A: real,B: real] :
% 8.58/4.87 ( ( ord_less_real @ C @ A )
% 8.58/4.87 => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8303_max_Ostrict__coboundedI1,axiom,
% 8.58/4.87 ! [C: rat,A: rat,B: rat] :
% 8.58/4.87 ( ( ord_less_rat @ C @ A )
% 8.58/4.87 => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8304_max_Ostrict__coboundedI1,axiom,
% 8.58/4.87 ! [C: num,A: num,B: num] :
% 8.58/4.87 ( ( ord_less_num @ C @ A )
% 8.58/4.87 => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8305_max_Ostrict__coboundedI1,axiom,
% 8.58/4.87 ! [C: nat,A: nat,B: nat] :
% 8.58/4.87 ( ( ord_less_nat @ C @ A )
% 8.58/4.87 => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8306_max_Ostrict__coboundedI1,axiom,
% 8.58/4.87 ! [C: int,A: int,B: int] :
% 8.58/4.87 ( ( ord_less_int @ C @ A )
% 8.58/4.87 => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8307_max_Ostrict__order__iff,axiom,
% 8.58/4.87 ( ord_less_real
% 8.58/4.87 = ( ^ [B2: real,A2: real] :
% 8.58/4.87 ( ( A2
% 8.58/4.87 = ( ord_max_real @ A2 @ B2 ) )
% 8.58/4.87 & ( A2 != B2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8308_max_Ostrict__order__iff,axiom,
% 8.58/4.87 ( ord_less_rat
% 8.58/4.87 = ( ^ [B2: rat,A2: rat] :
% 8.58/4.87 ( ( A2
% 8.58/4.87 = ( ord_max_rat @ A2 @ B2 ) )
% 8.58/4.87 & ( A2 != B2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8309_max_Ostrict__order__iff,axiom,
% 8.58/4.87 ( ord_less_num
% 8.58/4.87 = ( ^ [B2: num,A2: num] :
% 8.58/4.87 ( ( A2
% 8.58/4.87 = ( ord_max_num @ A2 @ B2 ) )
% 8.58/4.87 & ( A2 != B2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8310_max_Ostrict__order__iff,axiom,
% 8.58/4.87 ( ord_less_nat
% 8.58/4.87 = ( ^ [B2: nat,A2: nat] :
% 8.58/4.87 ( ( A2
% 8.58/4.87 = ( ord_max_nat @ A2 @ B2 ) )
% 8.58/4.87 & ( A2 != B2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8311_max_Ostrict__order__iff,axiom,
% 8.58/4.87 ( ord_less_int
% 8.58/4.87 = ( ^ [B2: int,A2: int] :
% 8.58/4.87 ( ( A2
% 8.58/4.87 = ( ord_max_int @ A2 @ B2 ) )
% 8.58/4.87 & ( A2 != B2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8312_max_Ostrict__boundedE,axiom,
% 8.58/4.87 ! [B: real,C: real,A: real] :
% 8.58/4.87 ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
% 8.58/4.87 => ~ ( ( ord_less_real @ B @ A )
% 8.58/4.87 => ~ ( ord_less_real @ C @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8313_max_Ostrict__boundedE,axiom,
% 8.58/4.87 ! [B: rat,C: rat,A: rat] :
% 8.58/4.87 ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
% 8.58/4.87 => ~ ( ( ord_less_rat @ B @ A )
% 8.58/4.87 => ~ ( ord_less_rat @ C @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8314_max_Ostrict__boundedE,axiom,
% 8.58/4.87 ! [B: num,C: num,A: num] :
% 8.58/4.87 ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
% 8.58/4.87 => ~ ( ( ord_less_num @ B @ A )
% 8.58/4.87 => ~ ( ord_less_num @ C @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8315_max_Ostrict__boundedE,axiom,
% 8.58/4.87 ! [B: nat,C: nat,A: nat] :
% 8.58/4.87 ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
% 8.58/4.87 => ~ ( ( ord_less_nat @ B @ A )
% 8.58/4.87 => ~ ( ord_less_nat @ C @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8316_max_Ostrict__boundedE,axiom,
% 8.58/4.87 ! [B: int,C: int,A: int] :
% 8.58/4.87 ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
% 8.58/4.87 => ~ ( ( ord_less_int @ B @ A )
% 8.58/4.87 => ~ ( ord_less_int @ C @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8317_less__max__iff__disj,axiom,
% 8.58/4.87 ! [Z2: real,X: real,Y: real] :
% 8.58/4.87 ( ( ord_less_real @ Z2 @ ( ord_max_real @ X @ Y ) )
% 8.58/4.87 = ( ( ord_less_real @ Z2 @ X )
% 8.58/4.87 | ( ord_less_real @ Z2 @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8318_less__max__iff__disj,axiom,
% 8.58/4.87 ! [Z2: rat,X: rat,Y: rat] :
% 8.58/4.87 ( ( ord_less_rat @ Z2 @ ( ord_max_rat @ X @ Y ) )
% 8.58/4.87 = ( ( ord_less_rat @ Z2 @ X )
% 8.58/4.87 | ( ord_less_rat @ Z2 @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8319_less__max__iff__disj,axiom,
% 8.58/4.87 ! [Z2: num,X: num,Y: num] :
% 8.58/4.87 ( ( ord_less_num @ Z2 @ ( ord_max_num @ X @ Y ) )
% 8.58/4.87 = ( ( ord_less_num @ Z2 @ X )
% 8.58/4.87 | ( ord_less_num @ Z2 @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8320_less__max__iff__disj,axiom,
% 8.58/4.87 ! [Z2: nat,X: nat,Y: nat] :
% 8.58/4.87 ( ( ord_less_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
% 8.58/4.87 = ( ( ord_less_nat @ Z2 @ X )
% 8.58/4.87 | ( ord_less_nat @ Z2 @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8321_less__max__iff__disj,axiom,
% 8.58/4.87 ! [Z2: int,X: int,Y: int] :
% 8.58/4.87 ( ( ord_less_int @ Z2 @ ( ord_max_int @ X @ Y ) )
% 8.58/4.87 = ( ( ord_less_int @ Z2 @ X )
% 8.58/4.87 | ( ord_less_int @ Z2 @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8322_max__add__distrib__right,axiom,
% 8.58/4.87 ! [X: real,Y: real,Z2: real] :
% 8.58/4.87 ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z2 ) )
% 8.58/4.87 = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8323_max__add__distrib__right,axiom,
% 8.58/4.87 ! [X: rat,Y: rat,Z2: rat] :
% 8.58/4.87 ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z2 ) )
% 8.58/4.87 = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8324_max__add__distrib__right,axiom,
% 8.58/4.87 ! [X: int,Y: int,Z2: int] :
% 8.58/4.87 ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z2 ) )
% 8.58/4.87 = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8325_max__add__distrib__right,axiom,
% 8.58/4.87 ! [X: nat,Y: nat,Z2: nat] :
% 8.58/4.87 ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z2 ) )
% 8.58/4.87 = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8326_max__add__distrib__left,axiom,
% 8.58/4.87 ! [X: real,Y: real,Z2: real] :
% 8.58/4.87 ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
% 8.58/4.87 = ( ord_max_real @ ( plus_plus_real @ X @ Z2 ) @ ( plus_plus_real @ Y @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8327_max__add__distrib__left,axiom,
% 8.58/4.87 ! [X: rat,Y: rat,Z2: rat] :
% 8.58/4.87 ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
% 8.58/4.87 = ( ord_max_rat @ ( plus_plus_rat @ X @ Z2 ) @ ( plus_plus_rat @ Y @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8328_max__add__distrib__left,axiom,
% 8.58/4.87 ! [X: int,Y: int,Z2: int] :
% 8.58/4.87 ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z2 )
% 8.58/4.87 = ( ord_max_int @ ( plus_plus_int @ X @ Z2 ) @ ( plus_plus_int @ Y @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8329_max__add__distrib__left,axiom,
% 8.58/4.87 ! [X: nat,Y: nat,Z2: nat] :
% 8.58/4.87 ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
% 8.58/4.87 = ( ord_max_nat @ ( plus_plus_nat @ X @ Z2 ) @ ( plus_plus_nat @ Y @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8330_max__diff__distrib__left,axiom,
% 8.58/4.87 ! [X: real,Y: real,Z2: real] :
% 8.58/4.87 ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
% 8.58/4.87 = ( ord_max_real @ ( minus_minus_real @ X @ Z2 ) @ ( minus_minus_real @ Y @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8331_max__diff__distrib__left,axiom,
% 8.58/4.87 ! [X: rat,Y: rat,Z2: rat] :
% 8.58/4.87 ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
% 8.58/4.87 = ( ord_max_rat @ ( minus_minus_rat @ X @ Z2 ) @ ( minus_minus_rat @ Y @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8332_max__diff__distrib__left,axiom,
% 8.58/4.87 ! [X: int,Y: int,Z2: int] :
% 8.58/4.87 ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z2 )
% 8.58/4.87 = ( ord_max_int @ ( minus_minus_int @ X @ Z2 ) @ ( minus_minus_int @ Y @ Z2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8333_max_Oassoc,axiom,
% 8.58/4.87 ! [A: int,B: int,C: int] :
% 8.58/4.87 ( ( ord_max_int @ ( ord_max_int @ A @ B ) @ C )
% 8.58/4.87 = ( ord_max_int @ A @ ( ord_max_int @ B @ C ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8334_max_Oassoc,axiom,
% 8.58/4.87 ! [A: nat,B: nat,C: nat] :
% 8.58/4.87 ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ C )
% 8.58/4.87 = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8335_max_Ocommute,axiom,
% 8.58/4.87 ( ord_max_int
% 8.58/4.87 = ( ^ [A2: int,B2: int] : ( ord_max_int @ B2 @ A2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8336_max_Ocommute,axiom,
% 8.58/4.87 ( ord_max_nat
% 8.58/4.87 = ( ^ [A2: nat,B2: nat] : ( ord_max_nat @ B2 @ A2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8337_max_Oleft__commute,axiom,
% 8.58/4.87 ! [B: int,A: int,C: int] :
% 8.58/4.87 ( ( ord_max_int @ B @ ( ord_max_int @ A @ C ) )
% 8.58/4.87 = ( ord_max_int @ A @ ( ord_max_int @ B @ C ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8338_max_Oleft__commute,axiom,
% 8.58/4.87 ! [B: nat,A: nat,C: nat] :
% 8.58/4.87 ( ( ord_max_nat @ B @ ( ord_max_nat @ A @ C ) )
% 8.58/4.87 = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8339_of__nat__max,axiom,
% 8.58/4.87 ! [X: nat,Y: nat] :
% 8.58/4.87 ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
% 8.58/4.87 = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8340_of__nat__max,axiom,
% 8.58/4.87 ! [X: nat,Y: nat] :
% 8.58/4.87 ( ( semiri681578069525770553at_rat @ ( ord_max_nat @ X @ Y ) )
% 8.58/4.87 = ( ord_max_rat @ ( semiri681578069525770553at_rat @ X ) @ ( semiri681578069525770553at_rat @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8341_of__nat__max,axiom,
% 8.58/4.87 ! [X: nat,Y: nat] :
% 8.58/4.87 ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
% 8.58/4.87 = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8342_of__nat__max,axiom,
% 8.58/4.87 ! [X: nat,Y: nat] :
% 8.58/4.87 ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
% 8.58/4.87 = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8343_or__not__num__neg_Osimps_I1_J,axiom,
% 8.58/4.87 ( ( bit_or_not_num_neg @ one @ one )
% 8.58/4.87 = one ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8344_set__bit__nat__def,axiom,
% 8.58/4.87 ( bit_se7882103937844011126it_nat
% 8.58/4.87 = ( ^ [M3: nat,N3: nat] : ( bit_se1412395901928357646or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8345_or__not__num__neg_Osimps_I4_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
% 8.58/4.87 = ( bit0 @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8346_or__not__num__neg_Osimps_I6_J,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
% 8.58/4.87 = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8347_or__not__num__neg_Osimps_I7_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
% 8.58/4.87 = one ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8348_or__not__num__neg_Osimps_I3_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
% 8.58/4.87 = ( bit1 @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8349_or__nat__def,axiom,
% 8.58/4.87 ( bit_se1412395901928357646or_nat
% 8.58/4.87 = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8350_or__not__num__neg_Osimps_I2_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
% 8.58/4.87 = ( bit1 @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8351_int__numeral__or__not__num__neg,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8352_int__numeral__not__or__num__neg,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8353_numeral__or__not__num__eq,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8354_Suc__0__or__eq,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.58/4.87 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8355_or__Suc__0__eq,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 8.58/4.87 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8356_or__nat__rec,axiom,
% 8.58/4.87 ( bit_se1412395901928357646or_nat
% 8.58/4.87 = ( ^ [M3: nat,N3: nat] :
% 8.58/4.87 ( plus_plus_nat
% 8.58/4.87 @ ( zero_n2687167440665602831ol_nat
% 8.58/4.87 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 8.58/4.87 | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 8.58/4.87 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8357_or__minus__numerals_I5_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8358_or__minus__numerals_I1_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8359_or__nat__unfold,axiom,
% 8.58/4.87 ( bit_se1412395901928357646or_nat
% 8.58/4.87 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8360_horner__sum__of__bool__2__less,axiom,
% 8.58/4.87 ! [Bs: list_o] : ( ord_less_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_size_list_o @ Bs ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8361_max__Suc__Suc,axiom,
% 8.58/4.87 ! [M: nat,N: nat] :
% 8.58/4.87 ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 8.58/4.87 = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8362_max__0R,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( ord_max_nat @ N @ zero_zero_nat )
% 8.58/4.87 = N ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8363_max__0L,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( ord_max_nat @ zero_zero_nat @ N )
% 8.58/4.87 = N ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8364_max__nat_Oright__neutral,axiom,
% 8.58/4.87 ! [A: nat] :
% 8.58/4.87 ( ( ord_max_nat @ A @ zero_zero_nat )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8365_max__nat_Oneutr__eq__iff,axiom,
% 8.58/4.87 ! [A: nat,B: nat] :
% 8.58/4.87 ( ( zero_zero_nat
% 8.58/4.87 = ( ord_max_nat @ A @ B ) )
% 8.58/4.87 = ( ( A = zero_zero_nat )
% 8.58/4.87 & ( B = zero_zero_nat ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8366_max__nat_Oleft__neutral,axiom,
% 8.58/4.87 ! [A: nat] :
% 8.58/4.87 ( ( ord_max_nat @ zero_zero_nat @ A )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8367_max__nat_Oeq__neutr__iff,axiom,
% 8.58/4.87 ! [A: nat,B: nat] :
% 8.58/4.87 ( ( ( ord_max_nat @ A @ B )
% 8.58/4.87 = zero_zero_nat )
% 8.58/4.87 = ( ( A = zero_zero_nat )
% 8.58/4.87 & ( B = zero_zero_nat ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8368_dbl__dec__simps_I5_J,axiom,
% 8.58/4.87 ! [K: num] :
% 8.58/4.87 ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
% 8.58/4.87 = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8369_dbl__dec__simps_I5_J,axiom,
% 8.58/4.87 ! [K: num] :
% 8.58/4.87 ( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) )
% 8.58/4.87 = ( numeral_numeral_real @ ( bitM @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8370_dbl__dec__simps_I5_J,axiom,
% 8.58/4.87 ! [K: num] :
% 8.58/4.87 ( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) )
% 8.58/4.87 = ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8371_dbl__dec__simps_I5_J,axiom,
% 8.58/4.87 ! [K: num] :
% 8.58/4.87 ( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
% 8.58/4.87 = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8372_pred__numeral__simps_I2_J,axiom,
% 8.58/4.87 ! [K: num] :
% 8.58/4.87 ( ( pred_numeral @ ( bit0 @ K ) )
% 8.58/4.87 = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8373_max__numeral__Suc,axiom,
% 8.58/4.87 ! [K: num,N: nat] :
% 8.58/4.87 ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 8.58/4.87 = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8374_max__Suc__numeral,axiom,
% 8.58/4.87 ! [N: nat,K: num] :
% 8.58/4.87 ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 8.58/4.87 = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8375_or__minus__numerals_I7_J,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8376_or__minus__numerals_I3_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8377_nat__mult__max__left,axiom,
% 8.58/4.87 ! [M: nat,N: nat,Q2: nat] :
% 8.58/4.87 ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 8.58/4.87 = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8378_nat__mult__max__right,axiom,
% 8.58/4.87 ! [M: nat,N: nat,Q2: nat] :
% 8.58/4.87 ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 8.58/4.87 = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8379_nat__add__max__left,axiom,
% 8.58/4.87 ! [M: nat,N: nat,Q2: nat] :
% 8.58/4.87 ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 8.58/4.87 = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8380_nat__add__max__right,axiom,
% 8.58/4.87 ! [M: nat,N: nat,Q2: nat] :
% 8.58/4.87 ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 8.58/4.87 = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8381_semiring__norm_I26_J,axiom,
% 8.58/4.87 ( ( bitM @ one )
% 8.58/4.87 = one ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8382_nat__minus__add__max,axiom,
% 8.58/4.87 ! [N: nat,M: nat] :
% 8.58/4.87 ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 8.58/4.87 = ( ord_max_nat @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8383_semiring__norm_I27_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bitM @ ( bit0 @ N ) )
% 8.58/4.87 = ( bit1 @ ( bitM @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8384_semiring__norm_I28_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bitM @ ( bit1 @ N ) )
% 8.58/4.87 = ( bit1 @ ( bit0 @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8385_or__not__num__neg_Osimps_I5_J,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
% 8.58/4.87 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8386_inc__BitM__eq,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( inc @ ( bitM @ N ) )
% 8.58/4.87 = ( bit0 @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8387_or__not__num__neg_Osimps_I9_J,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
% 8.58/4.87 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8388_BitM__inc__eq,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bitM @ ( inc @ N ) )
% 8.58/4.87 = ( bit1 @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8389_eval__nat__numeral_I2_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 8.58/4.87 = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8390_one__plus__BitM,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( plus_plus_num @ one @ ( bitM @ N ) )
% 8.58/4.87 = ( bit0 @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8391_BitM__plus__one,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( plus_plus_num @ ( bitM @ N ) @ one )
% 8.58/4.87 = ( bit0 @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8392_or__not__num__neg_Osimps_I8_J,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
% 8.58/4.87 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8393_numeral__BitM,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( numera6690914467698888265omplex @ ( bitM @ N ) )
% 8.58/4.87 = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N ) ) @ one_one_complex ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8394_numeral__BitM,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( numeral_numeral_real @ ( bitM @ N ) )
% 8.58/4.87 = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N ) ) @ one_one_real ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8395_numeral__BitM,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( numeral_numeral_rat @ ( bitM @ N ) )
% 8.58/4.87 = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8396_numeral__BitM,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( numeral_numeral_int @ ( bitM @ N ) )
% 8.58/4.87 = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8397_odd__numeral__BitM,axiom,
% 8.58/4.87 ! [W: num] :
% 8.58/4.87 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8398_odd__numeral__BitM,axiom,
% 8.58/4.87 ! [W: num] :
% 8.58/4.87 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8399_odd__numeral__BitM,axiom,
% 8.58/4.87 ! [W: num] :
% 8.58/4.87 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8400_not__numeral__BitM__eq,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bitM @ N ) ) )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8401_not__numeral__BitM__eq,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bitM @ N ) ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8402_or__not__num__neg_Oelims,axiom,
% 8.58/4.87 ! [X: num,Xa: num,Y: num] :
% 8.58/4.87 ( ( ( bit_or_not_num_neg @ X @ Xa )
% 8.58/4.87 = Y )
% 8.58/4.87 => ( ( ( X = one )
% 8.58/4.87 => ( ( Xa = one )
% 8.58/4.87 => ( Y != one ) ) )
% 8.58/4.87 => ( ( ( X = one )
% 8.58/4.87 => ! [M2: num] :
% 8.58/4.87 ( ( Xa
% 8.58/4.87 = ( bit0 @ M2 ) )
% 8.58/4.87 => ( Y
% 8.58/4.87 != ( bit1 @ M2 ) ) ) )
% 8.58/4.87 => ( ( ( X = one )
% 8.58/4.87 => ! [M2: num] :
% 8.58/4.87 ( ( Xa
% 8.58/4.87 = ( bit1 @ M2 ) )
% 8.58/4.87 => ( Y
% 8.58/4.87 != ( bit1 @ M2 ) ) ) )
% 8.58/4.87 => ( ( ? [N2: num] :
% 8.58/4.87 ( X
% 8.58/4.87 = ( bit0 @ N2 ) )
% 8.58/4.87 => ( ( Xa = one )
% 8.58/4.87 => ( Y
% 8.58/4.87 != ( bit0 @ one ) ) ) )
% 8.58/4.87 => ( ! [N2: num] :
% 8.58/4.87 ( ( X
% 8.58/4.87 = ( bit0 @ N2 ) )
% 8.58/4.87 => ! [M2: num] :
% 8.58/4.87 ( ( Xa
% 8.58/4.87 = ( bit0 @ M2 ) )
% 8.58/4.87 => ( Y
% 8.58/4.87 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M2 ) ) ) ) )
% 8.58/4.87 => ( ! [N2: num] :
% 8.58/4.87 ( ( X
% 8.58/4.87 = ( bit0 @ N2 ) )
% 8.58/4.87 => ! [M2: num] :
% 8.58/4.87 ( ( Xa
% 8.58/4.87 = ( bit1 @ M2 ) )
% 8.58/4.87 => ( Y
% 8.58/4.87 != ( bit0 @ ( bit_or_not_num_neg @ N2 @ M2 ) ) ) ) )
% 8.58/4.87 => ( ( ? [N2: num] :
% 8.58/4.87 ( X
% 8.58/4.87 = ( bit1 @ N2 ) )
% 8.58/4.87 => ( ( Xa = one )
% 8.58/4.87 => ( Y != one ) ) )
% 8.58/4.87 => ( ! [N2: num] :
% 8.58/4.87 ( ( X
% 8.58/4.87 = ( bit1 @ N2 ) )
% 8.58/4.87 => ! [M2: num] :
% 8.58/4.87 ( ( Xa
% 8.58/4.87 = ( bit0 @ M2 ) )
% 8.58/4.87 => ( Y
% 8.58/4.87 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M2 ) ) ) ) )
% 8.58/4.87 => ~ ! [N2: num] :
% 8.58/4.87 ( ( X
% 8.58/4.87 = ( bit1 @ N2 ) )
% 8.58/4.87 => ! [M2: num] :
% 8.58/4.87 ( ( Xa
% 8.58/4.87 = ( bit1 @ M2 ) )
% 8.58/4.87 => ( Y
% 8.58/4.87 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8403_bit__horner__sum__bit__iff,axiom,
% 8.58/4.87 ! [Bs: list_o,N: nat] :
% 8.58/4.87 ( ( bit_se9216721137139052372nteger @ ( groups3417619833198082522nteger @ zero_n356916108424825756nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Bs ) @ N )
% 8.58/4.87 = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 8.58/4.87 & ( nth_o @ Bs @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8404_bit__horner__sum__bit__iff,axiom,
% 8.58/4.87 ! [Bs: list_o,N: nat] :
% 8.58/4.87 ( ( bit_se1148574629649215175it_nat @ ( groups9119017779487936845_o_nat @ zero_n2687167440665602831ol_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Bs ) @ N )
% 8.58/4.87 = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 8.58/4.87 & ( nth_o @ Bs @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8405_bit__horner__sum__bit__iff,axiom,
% 8.58/4.87 ! [Bs: list_o,N: nat] :
% 8.58/4.87 ( ( bit_se1146084159140164899it_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ N )
% 8.58/4.87 = ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
% 8.58/4.87 & ( nth_o @ Bs @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8406_length__subseqs,axiom,
% 8.58/4.87 ! [Xs: list_VEBT_VEBT] :
% 8.58/4.87 ( ( size_s8217280938318005548T_VEBT @ ( subseqs_VEBT_VEBT @ Xs ) )
% 8.58/4.87 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8407_length__subseqs,axiom,
% 8.58/4.87 ! [Xs: list_o] :
% 8.58/4.87 ( ( size_s2710708370519433104list_o @ ( subseqs_o @ Xs ) )
% 8.58/4.87 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_o @ Xs ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8408_length__subseqs,axiom,
% 8.58/4.87 ! [Xs: list_nat] :
% 8.58/4.87 ( ( size_s3023201423986296836st_nat @ ( subseqs_nat @ Xs ) )
% 8.58/4.87 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8409_length__subseqs,axiom,
% 8.58/4.87 ! [Xs: list_int] :
% 8.58/4.87 ( ( size_s533118279054570080st_int @ ( subseqs_int @ Xs ) )
% 8.58/4.87 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_int @ Xs ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8410_drop__bit__numeral__minus__bit1,axiom,
% 8.58/4.87 ! [L: num,K: num] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 8.58/4.87 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8411_drop__bit__rec,axiom,
% 8.58/4.87 ( bit_se8568078237143864401it_int
% 8.58/4.87 = ( ^ [N3: nat,A2: int] : ( if_int @ ( N3 = zero_zero_nat ) @ A2 @ ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8412_drop__bit__rec,axiom,
% 8.58/4.87 ( bit_se8570568707652914677it_nat
% 8.58/4.87 = ( ^ [N3: nat,A2: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ A2 @ ( bit_se8570568707652914677it_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8413_drop__bit__of__0,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ N @ zero_zero_int )
% 8.58/4.87 = zero_zero_int ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8414_drop__bit__of__0,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ N @ zero_zero_nat )
% 8.58/4.87 = zero_zero_nat ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8415_drop__bit__drop__bit,axiom,
% 8.58/4.87 ! [M: nat,N: nat,A: int] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se8568078237143864401it_int @ N @ A ) )
% 8.58/4.87 = ( bit_se8568078237143864401it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8416_drop__bit__drop__bit,axiom,
% 8.58/4.87 ! [M: nat,N: nat,A: nat] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ M @ ( bit_se8570568707652914677it_nat @ N @ A ) )
% 8.58/4.87 = ( bit_se8570568707652914677it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8417_drop__bit__and,axiom,
% 8.58/4.87 ! [N: nat,A: int,B: int] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B ) )
% 8.58/4.87 = ( bit_se725231765392027082nd_int @ ( bit_se8568078237143864401it_int @ N @ A ) @ ( bit_se8568078237143864401it_int @ N @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8418_drop__bit__and,axiom,
% 8.58/4.87 ! [N: nat,A: nat,B: nat] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B ) )
% 8.58/4.87 = ( bit_se727722235901077358nd_nat @ ( bit_se8570568707652914677it_nat @ N @ A ) @ ( bit_se8570568707652914677it_nat @ N @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8419_drop__bit__or,axiom,
% 8.58/4.87 ! [N: nat,A: int,B: int] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ N @ ( bit_se1409905431419307370or_int @ A @ B ) )
% 8.58/4.87 = ( bit_se1409905431419307370or_int @ ( bit_se8568078237143864401it_int @ N @ A ) @ ( bit_se8568078237143864401it_int @ N @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8420_drop__bit__or,axiom,
% 8.58/4.87 ! [N: nat,A: nat,B: nat] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ N @ ( bit_se1412395901928357646or_nat @ A @ B ) )
% 8.58/4.87 = ( bit_se1412395901928357646or_nat @ ( bit_se8570568707652914677it_nat @ N @ A ) @ ( bit_se8570568707652914677it_nat @ N @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8421_drop__bit__xor,axiom,
% 8.58/4.87 ! [N: nat,A: int,B: int] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ N @ ( bit_se6526347334894502574or_int @ A @ B ) )
% 8.58/4.87 = ( bit_se6526347334894502574or_int @ ( bit_se8568078237143864401it_int @ N @ A ) @ ( bit_se8568078237143864401it_int @ N @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8422_drop__bit__xor,axiom,
% 8.58/4.87 ! [N: nat,A: nat,B: nat] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ N @ ( bit_se6528837805403552850or_nat @ A @ B ) )
% 8.58/4.87 = ( bit_se6528837805403552850or_nat @ ( bit_se8570568707652914677it_nat @ N @ A ) @ ( bit_se8570568707652914677it_nat @ N @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8423_drop__bit__of__bool,axiom,
% 8.58/4.87 ! [N: nat,B: $o] :
% 8.58/4.87 ( ( bit_se3928097537394005634nteger @ N @ ( zero_n356916108424825756nteger @ B ) )
% 8.58/4.87 = ( zero_n356916108424825756nteger
% 8.58/4.87 @ ( ( N = zero_zero_nat )
% 8.58/4.87 & B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8424_drop__bit__of__bool,axiom,
% 8.58/4.87 ! [N: nat,B: $o] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ N @ ( zero_n2684676970156552555ol_int @ B ) )
% 8.58/4.87 = ( zero_n2684676970156552555ol_int
% 8.58/4.87 @ ( ( N = zero_zero_nat )
% 8.58/4.87 & B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8425_drop__bit__of__bool,axiom,
% 8.58/4.87 ! [N: nat,B: $o] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ N @ ( zero_n2687167440665602831ol_nat @ B ) )
% 8.58/4.87 = ( zero_n2687167440665602831ol_nat
% 8.58/4.87 @ ( ( N = zero_zero_nat )
% 8.58/4.87 & B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8426_drop__bit__of__Suc__0,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 8.58/4.87 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8427_drop__bit__nonnegative__int__iff,axiom,
% 8.58/4.87 ! [N: nat,K: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
% 8.58/4.87 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8428_drop__bit__negative__int__iff,axiom,
% 8.58/4.87 ! [N: nat,K: int] :
% 8.58/4.87 ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 8.58/4.87 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8429_drop__bit__minus__one,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.87 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8430_drop__bit__Suc__bit0,axiom,
% 8.58/4.87 ! [N: nat,K: num] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 8.58/4.87 = ( bit_se8568078237143864401it_int @ N @ ( numeral_numeral_int @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8431_drop__bit__Suc__bit0,axiom,
% 8.58/4.87 ! [N: nat,K: num] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 8.58/4.87 = ( bit_se8570568707652914677it_nat @ N @ ( numeral_numeral_nat @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8432_drop__bit__Suc__bit1,axiom,
% 8.58/4.87 ! [N: nat,K: num] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 8.58/4.87 = ( bit_se8568078237143864401it_int @ N @ ( numeral_numeral_int @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8433_drop__bit__Suc__bit1,axiom,
% 8.58/4.87 ! [N: nat,K: num] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 8.58/4.87 = ( bit_se8570568707652914677it_nat @ N @ ( numeral_numeral_nat @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8434_drop__bit__of__1,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( bit_se3928097537394005634nteger @ N @ one_one_Code_integer )
% 8.58/4.87 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8435_drop__bit__of__1,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ N @ one_one_int )
% 8.58/4.87 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8436_drop__bit__of__1,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ N @ one_one_nat )
% 8.58/4.87 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8437_drop__bit__numeral__bit0,axiom,
% 8.58/4.87 ! [L: num,K: num] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 8.58/4.87 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8438_drop__bit__numeral__bit0,axiom,
% 8.58/4.87 ! [L: num,K: num] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 8.58/4.87 = ( bit_se8570568707652914677it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8439_drop__bit__numeral__bit1,axiom,
% 8.58/4.87 ! [L: num,K: num] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 8.58/4.87 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8440_drop__bit__numeral__bit1,axiom,
% 8.58/4.87 ! [L: num,K: num] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 8.58/4.87 = ( bit_se8570568707652914677it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8441_drop__bit__Suc__minus__bit0,axiom,
% 8.58/4.87 ! [N: nat,K: num] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 8.58/4.87 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8442_drop__bit__numeral__minus__bit0,axiom,
% 8.58/4.87 ! [L: num,K: num] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 8.58/4.87 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8443_drop__bit__Suc__minus__bit1,axiom,
% 8.58/4.87 ! [N: nat,K: num] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 8.58/4.87 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8444_drop__bit__nat__eq,axiom,
% 8.58/4.87 ! [N: nat,K: int] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
% 8.58/4.87 = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8445_of__nat__drop__bit,axiom,
% 8.58/4.87 ! [M: nat,N: nat] :
% 8.58/4.87 ( ( semiri1314217659103216013at_int @ ( bit_se8570568707652914677it_nat @ M @ N ) )
% 8.58/4.87 = ( bit_se8568078237143864401it_int @ M @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8446_of__nat__drop__bit,axiom,
% 8.58/4.87 ! [M: nat,N: nat] :
% 8.58/4.87 ( ( semiri1316708129612266289at_nat @ ( bit_se8570568707652914677it_nat @ M @ N ) )
% 8.58/4.87 = ( bit_se8570568707652914677it_nat @ M @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8447_drop__bit__of__nat,axiom,
% 8.58/4.87 ! [N: nat,M: nat] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 8.58/4.87 = ( semiri1314217659103216013at_int @ ( bit_se8570568707652914677it_nat @ N @ M ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8448_drop__bit__of__nat,axiom,
% 8.58/4.87 ! [N: nat,M: nat] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 8.58/4.87 = ( semiri1316708129612266289at_nat @ ( bit_se8570568707652914677it_nat @ N @ M ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8449_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
% 8.58/4.87 ! [N: nat,A: int] :
% 8.58/4.87 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 8.58/4.87 = A )
% 8.58/4.87 = ( ( bit_se8568078237143864401it_int @ N @ A )
% 8.58/4.87 = zero_zero_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8450_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
% 8.58/4.87 ! [N: nat,A: nat] :
% 8.58/4.87 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 8.58/4.87 = A )
% 8.58/4.87 = ( ( bit_se8570568707652914677it_nat @ N @ A )
% 8.58/4.87 = zero_zero_nat ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8451_drop__bit__push__bit__int,axiom,
% 8.58/4.87 ! [M: nat,N: nat,K: int] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
% 8.58/4.87 = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8452_take__bit__drop__bit,axiom,
% 8.58/4.87 ! [M: nat,N: nat,A: int] :
% 8.58/4.87 ( ( bit_se2923211474154528505it_int @ M @ ( bit_se8568078237143864401it_int @ N @ A ) )
% 8.58/4.87 = ( bit_se8568078237143864401it_int @ N @ ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8453_take__bit__drop__bit,axiom,
% 8.58/4.87 ! [M: nat,N: nat,A: nat] :
% 8.58/4.87 ( ( bit_se2925701944663578781it_nat @ M @ ( bit_se8570568707652914677it_nat @ N @ A ) )
% 8.58/4.87 = ( bit_se8570568707652914677it_nat @ N @ ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8454_drop__bit__take__bit,axiom,
% 8.58/4.87 ! [M: nat,N: nat,A: int] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 8.58/4.87 = ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N @ M ) @ ( bit_se8568078237143864401it_int @ M @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8455_drop__bit__take__bit,axiom,
% 8.58/4.87 ! [M: nat,N: nat,A: nat] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 8.58/4.87 = ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N @ M ) @ ( bit_se8570568707652914677it_nat @ M @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8456_div__push__bit__of__1__eq__drop__bit,axiom,
% 8.58/4.87 ! [A: nat,N: nat] :
% 8.58/4.87 ( ( divide_divide_nat @ A @ ( bit_se547839408752420682it_nat @ N @ one_one_nat ) )
% 8.58/4.87 = ( bit_se8570568707652914677it_nat @ N @ A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8457_div__push__bit__of__1__eq__drop__bit,axiom,
% 8.58/4.87 ! [A: int,N: nat] :
% 8.58/4.87 ( ( divide_divide_int @ A @ ( bit_se545348938243370406it_int @ N @ one_one_int ) )
% 8.58/4.87 = ( bit_se8568078237143864401it_int @ N @ A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8458_div__push__bit__of__1__eq__drop__bit,axiom,
% 8.58/4.87 ! [A: code_integer,N: nat] :
% 8.58/4.87 ( ( divide6298287555418463151nteger @ A @ ( bit_se7788150548672797655nteger @ N @ one_one_Code_integer ) )
% 8.58/4.87 = ( bit_se3928097537394005634nteger @ N @ A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8459_div__push__bit__of__1__eq__drop__bit,axiom,
% 8.58/4.87 ! [A: code_natural,N: nat] :
% 8.58/4.87 ( ( divide5121882707175180666atural @ A @ ( bit_se6611745700429515170atural @ N @ one_one_Code_natural ) )
% 8.58/4.87 = ( bit_se2751692689150723149atural @ N @ A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8460_bit__iff__and__drop__bit__eq__1,axiom,
% 8.58/4.87 ( bit_se1146084159140164899it_int
% 8.58/4.87 = ( ^ [A2: int,N3: nat] :
% 8.58/4.87 ( ( bit_se725231765392027082nd_int @ ( bit_se8568078237143864401it_int @ N3 @ A2 ) @ one_one_int )
% 8.58/4.87 = one_one_int ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8461_bit__iff__and__drop__bit__eq__1,axiom,
% 8.58/4.87 ( bit_se1148574629649215175it_nat
% 8.58/4.87 = ( ^ [A2: nat,N3: nat] :
% 8.58/4.87 ( ( bit_se727722235901077358nd_nat @ ( bit_se8570568707652914677it_nat @ N3 @ A2 ) @ one_one_nat )
% 8.58/4.87 = one_one_nat ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8462_bits__ident,axiom,
% 8.58/4.87 ! [N: nat,A: code_integer] :
% 8.58/4.87 ( ( plus_p5714425477246183910nteger @ ( bit_se7788150548672797655nteger @ N @ ( bit_se3928097537394005634nteger @ N @ A ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8463_bits__ident,axiom,
% 8.58/4.87 ! [N: nat,A: code_natural] :
% 8.58/4.87 ( ( plus_p4538020629002901425atural @ ( bit_se6611745700429515170atural @ N @ ( bit_se2751692689150723149atural @ N @ A ) ) @ ( bit_se569199155075624693atural @ N @ A ) )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8464_bits__ident,axiom,
% 8.58/4.87 ! [N: nat,A: int] :
% 8.58/4.87 ( ( plus_plus_int @ ( bit_se545348938243370406it_int @ N @ ( bit_se8568078237143864401it_int @ N @ A ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8465_bits__ident,axiom,
% 8.58/4.87 ! [N: nat,A: nat] :
% 8.58/4.87 ( ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N @ ( bit_se8570568707652914677it_nat @ N @ A ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8466_drop__bit__half,axiom,
% 8.58/4.87 ! [N: nat,A: int] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 8.58/4.87 = ( divide_divide_int @ ( bit_se8568078237143864401it_int @ N @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8467_drop__bit__half,axiom,
% 8.58/4.87 ! [N: nat,A: nat] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.87 = ( divide_divide_nat @ ( bit_se8570568707652914677it_nat @ N @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8468_stable__imp__drop__bit__eq,axiom,
% 8.58/4.87 ! [A: int,N: nat] :
% 8.58/4.87 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.87 = A )
% 8.58/4.87 => ( ( bit_se8568078237143864401it_int @ N @ A )
% 8.58/4.87 = A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8469_stable__imp__drop__bit__eq,axiom,
% 8.58/4.87 ! [A: nat,N: nat] :
% 8.58/4.87 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.87 = A )
% 8.58/4.87 => ( ( bit_se8570568707652914677it_nat @ N @ A )
% 8.58/4.87 = A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8470_drop__bit__int__def,axiom,
% 8.58/4.87 ( bit_se8568078237143864401it_int
% 8.58/4.87 = ( ^ [N3: nat,K2: int] : ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8471_drop__bit__nat__def,axiom,
% 8.58/4.87 ( bit_se8570568707652914677it_nat
% 8.58/4.87 = ( ^ [N3: nat,M3: nat] : ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8472_drop__bit__Suc,axiom,
% 8.58/4.87 ! [N: nat,A: int] :
% 8.58/4.87 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ A )
% 8.58/4.87 = ( bit_se8568078237143864401it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8473_drop__bit__Suc,axiom,
% 8.58/4.87 ! [N: nat,A: nat] :
% 8.58/4.87 ( ( bit_se8570568707652914677it_nat @ ( suc @ N ) @ A )
% 8.58/4.87 = ( bit_se8570568707652914677it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8474_drop__bit__eq__div,axiom,
% 8.58/4.87 ( bit_se8568078237143864401it_int
% 8.58/4.87 = ( ^ [N3: nat,A2: int] : ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8475_drop__bit__eq__div,axiom,
% 8.58/4.87 ( bit_se8570568707652914677it_nat
% 8.58/4.87 = ( ^ [N3: nat,A2: nat] : ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8476_bit__iff__odd__drop__bit,axiom,
% 8.58/4.87 ( bit_se9216721137139052372nteger
% 8.58/4.87 = ( ^ [A2: code_integer,N3: nat] :
% 8.58/4.87 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3928097537394005634nteger @ N3 @ A2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8477_bit__iff__odd__drop__bit,axiom,
% 8.58/4.87 ( bit_se1146084159140164899it_int
% 8.58/4.87 = ( ^ [A2: int,N3: nat] :
% 8.58/4.87 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se8568078237143864401it_int @ N3 @ A2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8478_bit__iff__odd__drop__bit,axiom,
% 8.58/4.87 ( bit_se1148574629649215175it_nat
% 8.58/4.87 = ( ^ [A2: nat,N3: nat] :
% 8.58/4.87 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se8570568707652914677it_nat @ N3 @ A2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8479_even__drop__bit__iff__not__bit,axiom,
% 8.58/4.87 ! [N: nat,A: code_integer] :
% 8.58/4.87 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3928097537394005634nteger @ N @ A ) )
% 8.58/4.87 = ( ~ ( bit_se9216721137139052372nteger @ A @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8480_even__drop__bit__iff__not__bit,axiom,
% 8.58/4.87 ! [N: nat,A: int] :
% 8.58/4.87 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se8568078237143864401it_int @ N @ A ) )
% 8.58/4.87 = ( ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8481_even__drop__bit__iff__not__bit,axiom,
% 8.58/4.87 ! [N: nat,A: nat] :
% 8.58/4.87 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se8570568707652914677it_nat @ N @ A ) )
% 8.58/4.87 = ( ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8482_slice__eq__mask,axiom,
% 8.58/4.87 ! [N: nat,M: nat,A: code_integer] :
% 8.58/4.87 ( ( bit_se7788150548672797655nteger @ N @ ( bit_se1745604003318907178nteger @ M @ ( bit_se3928097537394005634nteger @ N @ A ) ) )
% 8.58/4.87 = ( bit_se3949692690581998587nteger @ A @ ( bit_se3949692690581998587nteger @ ( bit_se2119862282449309892nteger @ ( plus_plus_nat @ M @ N ) ) @ ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8483_slice__eq__mask,axiom,
% 8.58/4.87 ! [N: nat,M: nat,A: int] :
% 8.58/4.87 ( ( bit_se545348938243370406it_int @ N @ ( bit_se2923211474154528505it_int @ M @ ( bit_se8568078237143864401it_int @ N @ A ) ) )
% 8.58/4.87 = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ M @ N ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8484_inthall,axiom,
% 8.58/4.87 ! [Xs: list_real,P: real > $o,N: nat] :
% 8.58/4.87 ( ! [X4: real] :
% 8.58/4.87 ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 8.58/4.87 => ( P @ X4 ) )
% 8.58/4.87 => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 8.58/4.87 => ( P @ ( nth_real @ Xs @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8485_inthall,axiom,
% 8.58/4.87 ! [Xs: list_complex,P: complex > $o,N: nat] :
% 8.58/4.87 ( ! [X4: complex] :
% 8.58/4.87 ( ( member_complex @ X4 @ ( set_complex2 @ Xs ) )
% 8.58/4.87 => ( P @ X4 ) )
% 8.58/4.87 => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
% 8.58/4.87 => ( P @ ( nth_complex @ Xs @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8486_inthall,axiom,
% 8.58/4.87 ! [Xs: list_set_nat,P: set_nat > $o,N: nat] :
% 8.58/4.87 ( ! [X4: set_nat] :
% 8.58/4.87 ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
% 8.58/4.87 => ( P @ X4 ) )
% 8.58/4.87 => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 8.58/4.87 => ( P @ ( nth_set_nat @ Xs @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8487_inthall,axiom,
% 8.58/4.87 ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 8.58/4.87 ( ! [X4: vEBT_VEBT] :
% 8.58/4.87 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 8.58/4.87 => ( P @ X4 ) )
% 8.58/4.87 => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 8.58/4.87 => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8488_inthall,axiom,
% 8.58/4.87 ! [Xs: list_o,P: $o > $o,N: nat] :
% 8.58/4.87 ( ! [X4: $o] :
% 8.58/4.87 ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 8.58/4.87 => ( P @ X4 ) )
% 8.58/4.87 => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 8.58/4.87 => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8489_inthall,axiom,
% 8.58/4.87 ! [Xs: list_nat,P: nat > $o,N: nat] :
% 8.58/4.87 ( ! [X4: nat] :
% 8.58/4.87 ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 8.58/4.87 => ( P @ X4 ) )
% 8.58/4.87 => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 8.58/4.87 => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8490_inthall,axiom,
% 8.58/4.87 ! [Xs: list_int,P: int > $o,N: nat] :
% 8.58/4.87 ( ! [X4: int] :
% 8.58/4.87 ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 8.58/4.87 => ( P @ X4 ) )
% 8.58/4.87 => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 8.58/4.87 => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8491_nth__rotate1,axiom,
% 8.58/4.87 ! [N: nat,Xs: list_VEBT_VEBT] :
% 8.58/4.87 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 8.58/4.87 => ( ( nth_VEBT_VEBT @ ( rotate1_VEBT_VEBT @ Xs ) @ N )
% 8.58/4.87 = ( nth_VEBT_VEBT @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8492_nth__rotate1,axiom,
% 8.58/4.87 ! [N: nat,Xs: list_o] :
% 8.58/4.87 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 8.58/4.87 => ( ( nth_o @ ( rotate1_o @ Xs ) @ N )
% 8.58/4.87 = ( nth_o @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_o @ Xs ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8493_nth__rotate1,axiom,
% 8.58/4.87 ! [N: nat,Xs: list_nat] :
% 8.58/4.87 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 8.58/4.87 => ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N )
% 8.58/4.87 = ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8494_nth__rotate1,axiom,
% 8.58/4.87 ! [N: nat,Xs: list_int] :
% 8.58/4.87 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 8.58/4.87 => ( ( nth_int @ ( rotate1_int @ Xs ) @ N )
% 8.58/4.87 = ( nth_int @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8495_root__powr__inverse,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.87 => ( ( root @ N @ X )
% 8.58/4.87 = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8496_xor__minus__numerals_I2_J,axiom,
% 8.58/4.87 ! [K: int,N: num] :
% 8.58/4.87 ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8497_real__root__zero,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( root @ N @ zero_zero_real )
% 8.58/4.87 = zero_zero_real ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8498_real__root__Suc__0,axiom,
% 8.58/4.87 ! [X: real] :
% 8.58/4.87 ( ( root @ ( suc @ zero_zero_nat ) @ X )
% 8.58/4.87 = X ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8499_root__0,axiom,
% 8.58/4.87 ! [X: real] :
% 8.58/4.87 ( ( root @ zero_zero_nat @ X )
% 8.58/4.87 = zero_zero_real ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8500_real__root__eq__iff,axiom,
% 8.58/4.87 ! [N: nat,X: real,Y: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ( root @ N @ X )
% 8.58/4.87 = ( root @ N @ Y ) )
% 8.58/4.87 = ( X = Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8501_sub__num__simps_I1_J,axiom,
% 8.58/4.87 ( ( neg_numeral_sub_real @ one @ one )
% 8.58/4.87 = zero_zero_real ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8502_sub__num__simps_I1_J,axiom,
% 8.58/4.87 ( ( neg_numeral_sub_rat @ one @ one )
% 8.58/4.87 = zero_zero_rat ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8503_sub__num__simps_I1_J,axiom,
% 8.58/4.87 ( ( neg_numeral_sub_int @ one @ one )
% 8.58/4.87 = zero_zero_int ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8504_diff__numeral__simps_I1_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 8.58/4.87 = ( neg_nu8416839295433526191omplex @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8505_diff__numeral__simps_I1_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 8.58/4.87 = ( neg_numeral_sub_real @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8506_diff__numeral__simps_I1_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 8.58/4.87 = ( neg_numeral_sub_rat @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8507_diff__numeral__simps_I1_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.87 = ( neg_numeral_sub_int @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8508_sub__num__simps_I6_J,axiom,
% 8.58/4.87 ! [K: num,L: num] :
% 8.58/4.87 ( ( neg_numeral_sub_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
% 8.58/4.87 = ( neg_numeral_dbl_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8509_sub__num__simps_I9_J,axiom,
% 8.58/4.87 ! [K: num,L: num] :
% 8.58/4.87 ( ( neg_numeral_sub_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
% 8.58/4.87 = ( neg_numeral_dbl_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8510_semiring__norm_I167_J,axiom,
% 8.58/4.87 ! [V: num,W: num,Y: real] :
% 8.58/4.87 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Y ) )
% 8.58/4.87 = ( plus_plus_real @ ( neg_numeral_sub_real @ W @ V ) @ Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8511_semiring__norm_I167_J,axiom,
% 8.58/4.87 ! [V: num,W: num,Y: int] :
% 8.58/4.87 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Y ) )
% 8.58/4.87 = ( plus_plus_int @ ( neg_numeral_sub_int @ W @ V ) @ Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8512_semiring__norm_I167_J,axiom,
% 8.58/4.87 ! [V: num,W: num,Y: complex] :
% 8.58/4.87 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Y ) )
% 8.58/4.87 = ( plus_plus_complex @ ( neg_nu8416839295433526191omplex @ W @ V ) @ Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8513_semiring__norm_I167_J,axiom,
% 8.58/4.87 ! [V: num,W: num,Y: rat] :
% 8.58/4.87 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
% 8.58/4.87 = ( plus_plus_rat @ ( neg_numeral_sub_rat @ W @ V ) @ Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8514_semiring__norm_I167_J,axiom,
% 8.58/4.87 ! [V: num,W: num,Y: code_integer] :
% 8.58/4.87 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
% 8.58/4.87 = ( plus_p5714425477246183910nteger @ ( neg_nu5755505904847501662nteger @ W @ V ) @ Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8515_semiring__norm_I166_J,axiom,
% 8.58/4.87 ! [V: num,W: num,Y: real] :
% 8.58/4.87 ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 8.58/4.87 = ( plus_plus_real @ ( neg_numeral_sub_real @ V @ W ) @ Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8516_semiring__norm_I166_J,axiom,
% 8.58/4.87 ! [V: num,W: num,Y: int] :
% 8.58/4.87 ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 8.58/4.87 = ( plus_plus_int @ ( neg_numeral_sub_int @ V @ W ) @ Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8517_semiring__norm_I166_J,axiom,
% 8.58/4.87 ! [V: num,W: num,Y: complex] :
% 8.58/4.87 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 8.58/4.87 = ( plus_plus_complex @ ( neg_nu8416839295433526191omplex @ V @ W ) @ Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8518_semiring__norm_I166_J,axiom,
% 8.58/4.87 ! [V: num,W: num,Y: rat] :
% 8.58/4.87 ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 8.58/4.87 = ( plus_plus_rat @ ( neg_numeral_sub_rat @ V @ W ) @ Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8519_semiring__norm_I166_J,axiom,
% 8.58/4.87 ! [V: num,W: num,Y: code_integer] :
% 8.58/4.87 ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 8.58/4.87 = ( plus_p5714425477246183910nteger @ ( neg_nu5755505904847501662nteger @ V @ W ) @ Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8520_add__neg__numeral__simps_I2_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 8.58/4.87 = ( neg_numeral_sub_real @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8521_add__neg__numeral__simps_I2_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.87 = ( neg_numeral_sub_int @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8522_add__neg__numeral__simps_I2_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 8.58/4.87 = ( neg_nu8416839295433526191omplex @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8523_add__neg__numeral__simps_I2_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 8.58/4.87 = ( neg_numeral_sub_rat @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8524_add__neg__numeral__simps_I2_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 8.58/4.87 = ( neg_nu5755505904847501662nteger @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8525_add__neg__numeral__simps_I1_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.58/4.87 = ( neg_numeral_sub_real @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8526_add__neg__numeral__simps_I1_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.87 = ( neg_numeral_sub_int @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8527_add__neg__numeral__simps_I1_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.58/4.87 = ( neg_nu8416839295433526191omplex @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8528_add__neg__numeral__simps_I1_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.58/4.87 = ( neg_numeral_sub_rat @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8529_add__neg__numeral__simps_I1_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.58/4.87 = ( neg_nu5755505904847501662nteger @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8530_real__root__eq__0__iff,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ( root @ N @ X )
% 8.58/4.87 = zero_zero_real )
% 8.58/4.87 = ( X = zero_zero_real ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8531_real__root__less__iff,axiom,
% 8.58/4.87 ! [N: nat,X: real,Y: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
% 8.58/4.87 = ( ord_less_real @ X @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8532_diff__numeral__simps_I4_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.58/4.87 = ( neg_numeral_sub_real @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8533_diff__numeral__simps_I4_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.87 = ( neg_numeral_sub_int @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8534_diff__numeral__simps_I4_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.58/4.87 = ( neg_nu8416839295433526191omplex @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8535_diff__numeral__simps_I4_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.58/4.87 = ( neg_numeral_sub_rat @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8536_diff__numeral__simps_I4_J,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.58/4.87 = ( neg_nu5755505904847501662nteger @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8537_real__root__le__iff,axiom,
% 8.58/4.87 ! [N: nat,X: real,Y: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
% 8.58/4.87 = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8538_real__root__eq__1__iff,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ( root @ N @ X )
% 8.58/4.87 = one_one_real )
% 8.58/4.87 = ( X = one_one_real ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8539_real__root__one,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( root @ N @ one_one_real )
% 8.58/4.87 = one_one_real ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8540_rotate1__length01,axiom,
% 8.58/4.87 ! [Xs: list_VEBT_VEBT] :
% 8.58/4.87 ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ one_one_nat )
% 8.58/4.87 => ( ( rotate1_VEBT_VEBT @ Xs )
% 8.58/4.87 = Xs ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8541_rotate1__length01,axiom,
% 8.58/4.87 ! [Xs: list_o] :
% 8.58/4.87 ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ one_one_nat )
% 8.58/4.87 => ( ( rotate1_o @ Xs )
% 8.58/4.87 = Xs ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8542_rotate1__length01,axiom,
% 8.58/4.87 ! [Xs: list_nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
% 8.58/4.87 => ( ( rotate1_nat @ Xs )
% 8.58/4.87 = Xs ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8543_rotate1__length01,axiom,
% 8.58/4.87 ! [Xs: list_int] :
% 8.58/4.87 ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
% 8.58/4.87 => ( ( rotate1_int @ Xs )
% 8.58/4.87 = Xs ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8544_sub__num__simps_I7_J,axiom,
% 8.58/4.87 ! [K: num,L: num] :
% 8.58/4.87 ( ( neg_numeral_sub_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
% 8.58/4.87 = ( neg_nu3811975205180677377ec_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8545_sub__num__simps_I8_J,axiom,
% 8.58/4.87 ! [K: num,L: num] :
% 8.58/4.87 ( ( neg_numeral_sub_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
% 8.58/4.87 = ( neg_nu5851722552734809277nc_int @ ( neg_numeral_sub_int @ K @ L ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8546_diff__numeral__special_I1_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( minus_minus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
% 8.58/4.87 = ( neg_nu8416839295433526191omplex @ one @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8547_diff__numeral__special_I1_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( minus_minus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 8.58/4.87 = ( neg_numeral_sub_real @ one @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8548_diff__numeral__special_I1_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( minus_minus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 8.58/4.87 = ( neg_numeral_sub_rat @ one @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8549_diff__numeral__special_I1_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( minus_minus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 8.58/4.87 = ( neg_numeral_sub_int @ one @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8550_diff__numeral__special_I2_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ one_one_complex )
% 8.58/4.87 = ( neg_nu8416839295433526191omplex @ M @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8551_diff__numeral__special_I2_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ one_one_real )
% 8.58/4.87 = ( neg_numeral_sub_real @ M @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8552_diff__numeral__special_I2_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ one_one_rat )
% 8.58/4.87 = ( neg_numeral_sub_rat @ M @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8553_diff__numeral__special_I2_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int )
% 8.58/4.87 = ( neg_numeral_sub_int @ M @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8554_real__root__gt__0__iff,axiom,
% 8.58/4.87 ! [N: nat,Y: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
% 8.58/4.87 = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8555_real__root__lt__0__iff,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
% 8.58/4.87 = ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8556_real__root__ge__0__iff,axiom,
% 8.58/4.87 ! [N: nat,Y: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
% 8.58/4.87 = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8557_real__root__le__0__iff,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
% 8.58/4.87 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8558_sub__num__simps_I5_J,axiom,
% 8.58/4.87 ! [K: num] :
% 8.58/4.87 ( ( neg_nu8416839295433526191omplex @ ( bit1 @ K ) @ one )
% 8.58/4.87 = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8559_sub__num__simps_I5_J,axiom,
% 8.58/4.87 ! [K: num] :
% 8.58/4.87 ( ( neg_numeral_sub_real @ ( bit1 @ K ) @ one )
% 8.58/4.87 = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8560_sub__num__simps_I5_J,axiom,
% 8.58/4.87 ! [K: num] :
% 8.58/4.87 ( ( neg_numeral_sub_rat @ ( bit1 @ K ) @ one )
% 8.58/4.87 = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8561_sub__num__simps_I5_J,axiom,
% 8.58/4.87 ! [K: num] :
% 8.58/4.87 ( ( neg_numeral_sub_int @ ( bit1 @ K ) @ one )
% 8.58/4.87 = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8562_real__root__gt__1__iff,axiom,
% 8.58/4.87 ! [N: nat,Y: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
% 8.58/4.87 = ( ord_less_real @ one_one_real @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8563_real__root__lt__1__iff,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
% 8.58/4.87 = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8564_real__root__ge__1__iff,axiom,
% 8.58/4.87 ! [N: nat,Y: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
% 8.58/4.87 = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8565_real__root__le__1__iff,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
% 8.58/4.87 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8566_not__minus__numeral__eq,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.58/4.87 = ( neg_nu5755505904847501662nteger @ N @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8567_not__minus__numeral__eq,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.87 = ( neg_numeral_sub_int @ N @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8568_sub__num__simps_I4_J,axiom,
% 8.58/4.87 ! [K: num] :
% 8.58/4.87 ( ( neg_nu8416839295433526191omplex @ ( bit0 @ K ) @ one )
% 8.58/4.87 = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8569_sub__num__simps_I4_J,axiom,
% 8.58/4.87 ! [K: num] :
% 8.58/4.87 ( ( neg_numeral_sub_real @ ( bit0 @ K ) @ one )
% 8.58/4.87 = ( numeral_numeral_real @ ( bitM @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8570_sub__num__simps_I4_J,axiom,
% 8.58/4.87 ! [K: num] :
% 8.58/4.87 ( ( neg_numeral_sub_rat @ ( bit0 @ K ) @ one )
% 8.58/4.87 = ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8571_sub__num__simps_I4_J,axiom,
% 8.58/4.87 ! [K: num] :
% 8.58/4.87 ( ( neg_numeral_sub_int @ ( bit0 @ K ) @ one )
% 8.58/4.87 = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8572_add__neg__numeral__special_I4_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
% 8.58/4.87 = ( neg_numeral_sub_real @ N @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8573_add__neg__numeral__special_I4_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.87 = ( neg_numeral_sub_int @ N @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8574_add__neg__numeral__special_I4_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
% 8.58/4.87 = ( neg_nu8416839295433526191omplex @ N @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8575_add__neg__numeral__special_I4_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
% 8.58/4.87 = ( neg_numeral_sub_rat @ N @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8576_add__neg__numeral__special_I4_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
% 8.58/4.87 = ( neg_nu5755505904847501662nteger @ N @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8577_add__neg__numeral__special_I3_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.87 = ( neg_numeral_sub_real @ M @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8578_add__neg__numeral__special_I3_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.87 = ( neg_numeral_sub_int @ M @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8579_add__neg__numeral__special_I3_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.87 = ( neg_nu8416839295433526191omplex @ M @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8580_add__neg__numeral__special_I3_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.87 = ( neg_numeral_sub_rat @ M @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8581_add__neg__numeral__special_I3_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.87 = ( neg_nu5755505904847501662nteger @ M @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8582_add__neg__numeral__special_I2_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 8.58/4.87 = ( neg_numeral_sub_real @ one @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8583_add__neg__numeral__special_I2_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 8.58/4.87 = ( neg_numeral_sub_int @ one @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8584_add__neg__numeral__special_I2_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 8.58/4.87 = ( neg_nu8416839295433526191omplex @ one @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8585_add__neg__numeral__special_I2_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 8.58/4.87 = ( neg_numeral_sub_rat @ one @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8586_add__neg__numeral__special_I2_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 8.58/4.87 = ( neg_nu5755505904847501662nteger @ one @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8587_add__neg__numeral__special_I1_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) )
% 8.58/4.87 = ( neg_numeral_sub_real @ one @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8588_add__neg__numeral__special_I1_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) )
% 8.58/4.87 = ( neg_numeral_sub_int @ one @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8589_add__neg__numeral__special_I1_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) )
% 8.58/4.87 = ( neg_nu8416839295433526191omplex @ one @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8590_add__neg__numeral__special_I1_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) )
% 8.58/4.87 = ( neg_numeral_sub_rat @ one @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8591_add__neg__numeral__special_I1_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) )
% 8.58/4.87 = ( neg_nu5755505904847501662nteger @ one @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8592_diff__numeral__special_I8_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 8.58/4.87 = ( neg_numeral_sub_real @ one @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8593_diff__numeral__special_I8_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 8.58/4.87 = ( neg_numeral_sub_int @ one @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8594_diff__numeral__special_I8_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.58/4.87 = ( neg_nu8416839295433526191omplex @ one @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8595_diff__numeral__special_I8_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.87 = ( neg_numeral_sub_rat @ one @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8596_diff__numeral__special_I8_J,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.58/4.87 = ( neg_nu5755505904847501662nteger @ one @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8597_diff__numeral__special_I7_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.58/4.87 = ( neg_numeral_sub_real @ N @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8598_diff__numeral__special_I7_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.87 = ( neg_numeral_sub_int @ N @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8599_diff__numeral__special_I7_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.58/4.87 = ( neg_nu8416839295433526191omplex @ N @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8600_diff__numeral__special_I7_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.58/4.87 = ( neg_numeral_sub_rat @ N @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8601_diff__numeral__special_I7_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.58/4.87 = ( neg_nu5755505904847501662nteger @ N @ one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8602_minus__sub__one__diff__one,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( minus_minus_real @ ( uminus_uminus_real @ ( neg_numeral_sub_real @ M @ one ) ) @ one_one_real )
% 8.58/4.87 = ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8603_minus__sub__one__diff__one,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( minus_minus_int @ ( uminus_uminus_int @ ( neg_numeral_sub_int @ M @ one ) ) @ one_one_int )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8604_minus__sub__one__diff__one,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( neg_nu8416839295433526191omplex @ M @ one ) ) @ one_one_complex )
% 8.58/4.87 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8605_minus__sub__one__diff__one,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( neg_numeral_sub_rat @ M @ one ) ) @ one_one_rat )
% 8.58/4.87 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8606_minus__sub__one__diff__one,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( neg_nu5755505904847501662nteger @ M @ one ) ) @ one_one_Code_integer )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8607_sub__num__simps_I3_J,axiom,
% 8.58/4.87 ! [L: num] :
% 8.58/4.87 ( ( neg_numeral_sub_real @ one @ ( bit1 @ L ) )
% 8.58/4.87 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ L ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8608_sub__num__simps_I3_J,axiom,
% 8.58/4.87 ! [L: num] :
% 8.58/4.87 ( ( neg_numeral_sub_int @ one @ ( bit1 @ L ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ L ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8609_sub__num__simps_I3_J,axiom,
% 8.58/4.87 ! [L: num] :
% 8.58/4.87 ( ( neg_nu8416839295433526191omplex @ one @ ( bit1 @ L ) )
% 8.58/4.87 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ L ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8610_sub__num__simps_I3_J,axiom,
% 8.58/4.87 ! [L: num] :
% 8.58/4.87 ( ( neg_numeral_sub_rat @ one @ ( bit1 @ L ) )
% 8.58/4.87 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ L ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8611_sub__num__simps_I3_J,axiom,
% 8.58/4.87 ! [L: num] :
% 8.58/4.87 ( ( neg_nu5755505904847501662nteger @ one @ ( bit1 @ L ) )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ L ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8612_real__root__pow__pos2,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.87 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 8.58/4.87 = X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8613_sub__num__simps_I2_J,axiom,
% 8.58/4.87 ! [L: num] :
% 8.58/4.87 ( ( neg_numeral_sub_real @ one @ ( bit0 @ L ) )
% 8.58/4.87 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bitM @ L ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8614_sub__num__simps_I2_J,axiom,
% 8.58/4.87 ! [L: num] :
% 8.58/4.87 ( ( neg_numeral_sub_int @ one @ ( bit0 @ L ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bitM @ L ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8615_sub__num__simps_I2_J,axiom,
% 8.58/4.87 ! [L: num] :
% 8.58/4.87 ( ( neg_nu8416839295433526191omplex @ one @ ( bit0 @ L ) )
% 8.58/4.87 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bitM @ L ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8616_sub__num__simps_I2_J,axiom,
% 8.58/4.87 ! [L: num] :
% 8.58/4.87 ( ( neg_numeral_sub_rat @ one @ ( bit0 @ L ) )
% 8.58/4.87 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bitM @ L ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8617_sub__num__simps_I2_J,axiom,
% 8.58/4.87 ! [L: num] :
% 8.58/4.87 ( ( neg_nu5755505904847501662nteger @ one @ ( bit0 @ L ) )
% 8.58/4.87 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bitM @ L ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8618_xor__minus__numerals_I1_J,axiom,
% 8.58/4.87 ! [N: num,K: int] :
% 8.58/4.87 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
% 8.58/4.87 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8619_real__root__inverse,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( root @ N @ ( inverse_inverse_real @ X ) )
% 8.58/4.87 = ( inverse_inverse_real @ ( root @ N @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8620_real__root__mult__exp,axiom,
% 8.58/4.87 ! [M: nat,N: nat,X: real] :
% 8.58/4.87 ( ( root @ ( times_times_nat @ M @ N ) @ X )
% 8.58/4.87 = ( root @ M @ ( root @ N @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8621_real__root__mult,axiom,
% 8.58/4.87 ! [N: nat,X: real,Y: real] :
% 8.58/4.87 ( ( root @ N @ ( times_times_real @ X @ Y ) )
% 8.58/4.87 = ( times_times_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8622_real__root__minus,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( root @ N @ ( uminus_uminus_real @ X ) )
% 8.58/4.87 = ( uminus_uminus_real @ ( root @ N @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8623_real__root__commute,axiom,
% 8.58/4.87 ! [M: nat,N: nat,X: real] :
% 8.58/4.87 ( ( root @ M @ ( root @ N @ X ) )
% 8.58/4.87 = ( root @ N @ ( root @ M @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8624_real__root__divide,axiom,
% 8.58/4.87 ! [N: nat,X: real,Y: real] :
% 8.58/4.87 ( ( root @ N @ ( divide_divide_real @ X @ Y ) )
% 8.58/4.87 = ( divide_divide_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8625_real__root__pos__pos__le,axiom,
% 8.58/4.87 ! [X: real,N: nat] :
% 8.58/4.87 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.87 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8626_neg__numeral__class_Osub__def,axiom,
% 8.58/4.87 ( neg_nu8416839295433526191omplex
% 8.58/4.87 = ( ^ [K2: num,L2: num] : ( minus_minus_complex @ ( numera6690914467698888265omplex @ K2 ) @ ( numera6690914467698888265omplex @ L2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8627_neg__numeral__class_Osub__def,axiom,
% 8.58/4.87 ( neg_numeral_sub_real
% 8.58/4.87 = ( ^ [K2: num,L2: num] : ( minus_minus_real @ ( numeral_numeral_real @ K2 ) @ ( numeral_numeral_real @ L2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8628_neg__numeral__class_Osub__def,axiom,
% 8.58/4.87 ( neg_numeral_sub_rat
% 8.58/4.87 = ( ^ [K2: num,L2: num] : ( minus_minus_rat @ ( numeral_numeral_rat @ K2 ) @ ( numeral_numeral_rat @ L2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8629_neg__numeral__class_Osub__def,axiom,
% 8.58/4.87 ( neg_numeral_sub_int
% 8.58/4.87 = ( ^ [K2: num,L2: num] : ( minus_minus_int @ ( numeral_numeral_int @ K2 ) @ ( numeral_numeral_int @ L2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8630_real__root__less__mono,axiom,
% 8.58/4.87 ! [N: nat,X: real,Y: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_real @ X @ Y )
% 8.58/4.87 => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8631_real__root__le__mono,axiom,
% 8.58/4.87 ! [N: nat,X: real,Y: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_eq_real @ X @ Y )
% 8.58/4.87 => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8632_real__root__power,axiom,
% 8.58/4.87 ! [N: nat,X: real,K: nat] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( root @ N @ ( power_power_real @ X @ K ) )
% 8.58/4.87 = ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8633_real__root__abs,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( root @ N @ ( abs_abs_real @ X ) )
% 8.58/4.87 = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8634_length__pos__if__in__set,axiom,
% 8.58/4.87 ! [X: real,Xs: list_real] :
% 8.58/4.87 ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 8.58/4.87 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8635_length__pos__if__in__set,axiom,
% 8.58/4.87 ! [X: complex,Xs: list_complex] :
% 8.58/4.87 ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
% 8.58/4.87 => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8636_length__pos__if__in__set,axiom,
% 8.58/4.87 ! [X: set_nat,Xs: list_set_nat] :
% 8.58/4.87 ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 8.58/4.87 => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8637_length__pos__if__in__set,axiom,
% 8.58/4.87 ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 8.58/4.87 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 8.58/4.87 => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8638_length__pos__if__in__set,axiom,
% 8.58/4.87 ! [X: $o,Xs: list_o] :
% 8.58/4.87 ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 8.58/4.87 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8639_length__pos__if__in__set,axiom,
% 8.58/4.87 ! [X: nat,Xs: list_nat] :
% 8.58/4.87 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 8.58/4.87 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8640_length__pos__if__in__set,axiom,
% 8.58/4.87 ! [X: int,Xs: list_int] :
% 8.58/4.87 ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 8.58/4.87 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8641_sgn__root,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( sgn_sgn_real @ ( root @ N @ X ) )
% 8.58/4.87 = ( sgn_sgn_real @ X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8642_sub__non__negative,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( ord_less_eq_real @ zero_zero_real @ ( neg_numeral_sub_real @ N @ M ) )
% 8.58/4.87 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8643_sub__non__negative,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( ord_less_eq_rat @ zero_zero_rat @ ( neg_numeral_sub_rat @ N @ M ) )
% 8.58/4.87 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8644_sub__non__negative,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( ord_less_eq_int @ zero_zero_int @ ( neg_numeral_sub_int @ N @ M ) )
% 8.58/4.87 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8645_sub__non__positive,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( ord_less_eq_real @ ( neg_numeral_sub_real @ N @ M ) @ zero_zero_real )
% 8.58/4.87 = ( ord_less_eq_num @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8646_sub__non__positive,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( ord_less_eq_rat @ ( neg_numeral_sub_rat @ N @ M ) @ zero_zero_rat )
% 8.58/4.87 = ( ord_less_eq_num @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8647_sub__non__positive,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( ord_less_eq_int @ ( neg_numeral_sub_int @ N @ M ) @ zero_zero_int )
% 8.58/4.87 = ( ord_less_eq_num @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8648_sub__negative,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( ord_less_real @ ( neg_numeral_sub_real @ N @ M ) @ zero_zero_real )
% 8.58/4.87 = ( ord_less_num @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8649_sub__negative,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( ord_less_rat @ ( neg_numeral_sub_rat @ N @ M ) @ zero_zero_rat )
% 8.58/4.87 = ( ord_less_num @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8650_sub__negative,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( ord_less_int @ ( neg_numeral_sub_int @ N @ M ) @ zero_zero_int )
% 8.58/4.87 = ( ord_less_num @ N @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8651_sub__positive,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( ord_less_real @ zero_zero_real @ ( neg_numeral_sub_real @ N @ M ) )
% 8.58/4.87 = ( ord_less_num @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8652_sub__positive,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( ord_less_rat @ zero_zero_rat @ ( neg_numeral_sub_rat @ N @ M ) )
% 8.58/4.87 = ( ord_less_num @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8653_sub__positive,axiom,
% 8.58/4.87 ! [N: num,M: num] :
% 8.58/4.87 ( ( ord_less_int @ zero_zero_int @ ( neg_numeral_sub_int @ N @ M ) )
% 8.58/4.87 = ( ord_less_num @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8654_real__root__gt__zero,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.87 => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8655_real__root__strict__decreasing,axiom,
% 8.58/4.87 ! [N: nat,N5: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_nat @ N @ N5 )
% 8.58/4.87 => ( ( ord_less_real @ one_one_real @ X )
% 8.58/4.87 => ( ord_less_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8656_sqrt__def,axiom,
% 8.58/4.87 ( sqrt
% 8.58/4.87 = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8657_root__abs__power,axiom,
% 8.58/4.87 ! [N: nat,Y: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
% 8.58/4.87 = ( abs_abs_real @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8658_real__root__pos__pos,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.87 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8659_real__root__strict__increasing,axiom,
% 8.58/4.87 ! [N: nat,N5: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_nat @ N @ N5 )
% 8.58/4.87 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.87 => ( ( ord_less_real @ X @ one_one_real )
% 8.58/4.87 => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8660_real__root__decreasing,axiom,
% 8.58/4.87 ! [N: nat,N5: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_eq_nat @ N @ N5 )
% 8.58/4.87 => ( ( ord_less_eq_real @ one_one_real @ X )
% 8.58/4.87 => ( ord_less_eq_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8661_real__root__pow__pos,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.87 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 8.58/4.87 = X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8662_real__root__pos__unique,axiom,
% 8.58/4.87 ! [N: nat,Y: real,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 8.58/4.87 => ( ( ( power_power_real @ Y @ N )
% 8.58/4.87 = X )
% 8.58/4.87 => ( ( root @ N @ X )
% 8.58/4.87 = Y ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8663_real__root__power__cancel,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.87 => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 8.58/4.87 = X ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8664_odd__real__root__power__cancel,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.87 => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 8.58/4.87 = X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8665_odd__real__root__unique,axiom,
% 8.58/4.87 ! [N: nat,Y: real,X: real] :
% 8.58/4.87 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.87 => ( ( ( power_power_real @ Y @ N )
% 8.58/4.87 = X )
% 8.58/4.87 => ( ( root @ N @ X )
% 8.58/4.87 = Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8666_odd__real__root__pow,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.87 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 8.58/4.87 = X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8667_real__root__increasing,axiom,
% 8.58/4.87 ! [N: nat,N5: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_eq_nat @ N @ N5 )
% 8.58/4.87 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.87 => ( ( ord_less_eq_real @ X @ one_one_real )
% 8.58/4.87 => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8668_sub__BitM__One__eq,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
% 8.58/4.87 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8669_root__sgn__power,axiom,
% 8.58/4.87 ! [N: nat,Y: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
% 8.58/4.87 = Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8670_sgn__power__root,axiom,
% 8.58/4.87 ! [N: nat,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
% 8.58/4.87 = X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8671_ln__root,axiom,
% 8.58/4.87 ! [N: nat,B: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.58/4.87 => ( ( ln_ln_real @ ( root @ N @ B ) )
% 8.58/4.87 = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8672_log__root,axiom,
% 8.58/4.87 ! [N: nat,A: real,B: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.58/4.87 => ( ( log2 @ B @ ( root @ N @ A ) )
% 8.58/4.87 = ( divide_divide_real @ ( log2 @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8673_log__base__root,axiom,
% 8.58/4.87 ! [N: nat,B: real,X: real] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.58/4.87 => ( ( log2 @ ( root @ N @ B ) @ X )
% 8.58/4.87 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ B @ X ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8674_split__root,axiom,
% 8.58/4.87 ! [P: real > $o,N: nat,X: real] :
% 8.58/4.87 ( ( P @ ( root @ N @ X ) )
% 8.58/4.87 = ( ( ( N = zero_zero_nat )
% 8.58/4.87 => ( P @ zero_zero_real ) )
% 8.58/4.87 & ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ! [Y6: real] :
% 8.58/4.87 ( ( ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N ) )
% 8.58/4.87 = X )
% 8.58/4.87 => ( P @ Y6 ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8675_num__of__nat_Osimps_I2_J,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( num_of_nat @ ( suc @ N ) )
% 8.58/4.87 = ( inc @ ( num_of_nat @ N ) ) ) )
% 8.58/4.87 & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( num_of_nat @ ( suc @ N ) )
% 8.58/4.87 = one ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8676_pow_Osimps_I3_J,axiom,
% 8.58/4.87 ! [X: num,Y: num] :
% 8.58/4.87 ( ( pow @ X @ ( bit1 @ Y ) )
% 8.58/4.87 = ( times_times_num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8677_num__of__nat__numeral__eq,axiom,
% 8.58/4.87 ! [Q2: num] :
% 8.58/4.87 ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
% 8.58/4.87 = Q2 ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8678_sqr_Osimps_I2_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( sqr @ ( bit0 @ N ) )
% 8.58/4.87 = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8679_sqr_Osimps_I1_J,axiom,
% 8.58/4.87 ( ( sqr @ one )
% 8.58/4.87 = one ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8680_sqr__conv__mult,axiom,
% 8.58/4.87 ( sqr
% 8.58/4.87 = ( ^ [X3: num] : ( times_times_num @ X3 @ X3 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8681_num__of__nat_Osimps_I1_J,axiom,
% 8.58/4.87 ( ( num_of_nat @ zero_zero_nat )
% 8.58/4.87 = one ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8682_numeral__num__of__nat,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 8.58/4.87 = N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8683_num__of__nat__One,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( ord_less_eq_nat @ N @ one_one_nat )
% 8.58/4.87 => ( ( num_of_nat @ N )
% 8.58/4.87 = one ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8684_pow_Osimps_I2_J,axiom,
% 8.58/4.87 ! [X: num,Y: num] :
% 8.58/4.87 ( ( pow @ X @ ( bit0 @ Y ) )
% 8.58/4.87 = ( sqr @ ( pow @ X @ Y ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8685_num__of__nat__double,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 8.58/4.87 = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8686_num__of__nat__plus__distrib,axiom,
% 8.58/4.87 ! [M: nat,N: nat] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.58/4.87 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.87 => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 8.58/4.87 = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8687_sqr_Osimps_I3_J,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( sqr @ ( bit1 @ N ) )
% 8.58/4.87 = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8688_and__int_Oelims,axiom,
% 8.58/4.87 ! [X: int,Xa: int,Y: int] :
% 8.58/4.87 ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 8.58/4.87 = Y )
% 8.58/4.87 => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 8.58/4.87 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 8.58/4.87 => ( Y
% 8.58/4.87 = ( uminus_uminus_int
% 8.58/4.87 @ ( zero_n2684676970156552555ol_int
% 8.58/4.87 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 8.58/4.87 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 8.58/4.87 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 8.58/4.87 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 8.58/4.87 => ( Y
% 8.58/4.87 = ( plus_plus_int
% 8.58/4.87 @ ( zero_n2684676970156552555ol_int
% 8.58/4.87 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 8.58/4.87 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 8.58/4.87 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8689_and__int_Osimps,axiom,
% 8.58/4.87 ( bit_se725231765392027082nd_int
% 8.58/4.87 = ( ^ [K2: int,L2: int] :
% 8.58/4.87 ( if_int
% 8.58/4.87 @ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 8.58/4.87 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 8.58/4.87 @ ( uminus_uminus_int
% 8.58/4.87 @ ( zero_n2684676970156552555ol_int
% 8.58/4.87 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 8.58/4.87 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 8.58/4.87 @ ( plus_plus_int
% 8.58/4.87 @ ( zero_n2684676970156552555ol_int
% 8.58/4.87 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 8.58/4.87 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 8.58/4.87 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8690_min__Suc__Suc,axiom,
% 8.58/4.87 ! [M: nat,N: nat] :
% 8.58/4.87 ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
% 8.58/4.87 = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8691_min__0R,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( ord_min_nat @ N @ zero_zero_nat )
% 8.58/4.87 = zero_zero_nat ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8692_min__0L,axiom,
% 8.58/4.87 ! [N: nat] :
% 8.58/4.87 ( ( ord_min_nat @ zero_zero_nat @ N )
% 8.58/4.87 = zero_zero_nat ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8693_min__Suc__numeral,axiom,
% 8.58/4.87 ! [N: nat,K: num] :
% 8.58/4.87 ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 8.58/4.87 = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8694_min__numeral__Suc,axiom,
% 8.58/4.87 ! [K: num,N: nat] :
% 8.58/4.87 ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 8.58/4.87 = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8695_min__diff,axiom,
% 8.58/4.87 ! [M: nat,I: nat,N: nat] :
% 8.58/4.87 ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
% 8.58/4.87 = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8696_nat__mult__min__left,axiom,
% 8.58/4.87 ! [M: nat,N: nat,Q2: nat] :
% 8.58/4.87 ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
% 8.58/4.87 = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8697_nat__mult__min__right,axiom,
% 8.58/4.87 ! [M: nat,N: nat,Q2: nat] :
% 8.58/4.87 ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
% 8.58/4.87 = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8698_concat__bit__assoc__sym,axiom,
% 8.58/4.87 ! [M: nat,N: nat,K: int,L: int,R: int] :
% 8.58/4.87 ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L ) @ R )
% 8.58/4.87 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L @ R ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8699_take__bit__concat__bit__eq,axiom,
% 8.58/4.87 ! [M: nat,N: nat,K: int,L: int] :
% 8.58/4.87 ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L ) )
% 8.58/4.87 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8700_set__decode__zero,axiom,
% 8.58/4.87 ( ( nat_set_decode @ zero_zero_nat )
% 8.58/4.87 = bot_bot_set_nat ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8701_set__decode__plus__power__2,axiom,
% 8.58/4.87 ! [N: nat,Z2: nat] :
% 8.58/4.87 ( ~ ( member_nat @ N @ ( nat_set_decode @ Z2 ) )
% 8.58/4.87 => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z2 ) )
% 8.58/4.87 = ( insert_nat @ N @ ( nat_set_decode @ Z2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8702_and__int_Opsimps,axiom,
% 8.58/4.87 ! [K: int,L: int] :
% 8.58/4.87 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
% 8.58/4.87 => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 8.58/4.87 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 8.58/4.87 => ( ( bit_se725231765392027082nd_int @ K @ L )
% 8.58/4.87 = ( uminus_uminus_int
% 8.58/4.87 @ ( zero_n2684676970156552555ol_int
% 8.58/4.87 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 8.58/4.87 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 8.58/4.87 & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 8.58/4.87 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 8.58/4.87 => ( ( bit_se725231765392027082nd_int @ K @ L )
% 8.58/4.87 = ( plus_plus_int
% 8.58/4.87 @ ( zero_n2684676970156552555ol_int
% 8.58/4.87 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 8.58/4.87 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 8.58/4.87 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8703_and__int_Opelims,axiom,
% 8.58/4.87 ! [X: int,Xa: int,Y: int] :
% 8.58/4.87 ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 8.58/4.87 = Y )
% 8.58/4.87 => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 8.58/4.87 => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 8.58/4.87 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 8.58/4.87 => ( Y
% 8.58/4.87 = ( uminus_uminus_int
% 8.58/4.87 @ ( zero_n2684676970156552555ol_int
% 8.58/4.87 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 8.58/4.87 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 8.58/4.87 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 8.58/4.87 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 8.58/4.87 => ( Y
% 8.58/4.87 = ( plus_plus_int
% 8.58/4.87 @ ( zero_n2684676970156552555ol_int
% 8.58/4.87 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 8.58/4.87 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 8.58/4.87 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 8.58/4.87 => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8704_bot__nat__def,axiom,
% 8.58/4.87 bot_bot_nat = zero_zero_nat ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8705_and__int_Opinduct,axiom,
% 8.58/4.87 ! [A0: int,A1: int,P: int > int > $o] :
% 8.58/4.87 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 8.58/4.87 => ( ! [K3: int,L3: int] :
% 8.58/4.87 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L3 ) )
% 8.58/4.87 => ( ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 8.58/4.87 & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 8.58/4.87 => ( P @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 8.58/4.87 => ( P @ K3 @ L3 ) ) )
% 8.58/4.87 => ( P @ A0 @ A1 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8706_Divides_Oadjust__div__eq,axiom,
% 8.58/4.87 ! [Q2: int,R: int] :
% 8.58/4.87 ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R ) )
% 8.58/4.87 = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R != zero_zero_int ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8707_neg__eucl__rel__int__mult__2,axiom,
% 8.58/4.87 ! [B: int,A: int,Q2: int,R: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ B @ zero_zero_int )
% 8.58/4.87 => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 8.58/4.87 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) @ one_one_int ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8708_upto_Opinduct,axiom,
% 8.58/4.87 ! [A0: int,A1: int,P: int > int > $o] :
% 8.58/4.87 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 8.58/4.87 => ( ! [I2: int,J2: int] :
% 8.58/4.87 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
% 8.58/4.87 => ( ( ( ord_less_eq_int @ I2 @ J2 )
% 8.58/4.87 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
% 8.58/4.87 => ( P @ I2 @ J2 ) ) )
% 8.58/4.87 => ( P @ A0 @ A1 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8709_unique__remainder,axiom,
% 8.58/4.87 ! [A: int,B: int,Q2: int,R: int,Q4: int,R4: int] :
% 8.58/4.87 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 8.58/4.87 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R4 ) )
% 8.58/4.87 => ( R = R4 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8710_unique__quotient,axiom,
% 8.58/4.87 ! [A: int,B: int,Q2: int,R: int,Q4: int,R4: int] :
% 8.58/4.87 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 8.58/4.87 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R4 ) )
% 8.58/4.87 => ( Q2 = Q4 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8711_xor__num_Ocases,axiom,
% 8.58/4.87 ! [X: product_prod_num_num] :
% 8.58/4.87 ( ( X
% 8.58/4.87 != ( product_Pair_num_num @ one @ one ) )
% 8.58/4.87 => ( ! [N2: num] :
% 8.58/4.87 ( X
% 8.58/4.87 != ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) )
% 8.58/4.87 => ( ! [N2: num] :
% 8.58/4.87 ( X
% 8.58/4.87 != ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) )
% 8.58/4.87 => ( ! [M2: num] :
% 8.58/4.87 ( X
% 8.58/4.87 != ( product_Pair_num_num @ ( bit0 @ M2 ) @ one ) )
% 8.58/4.87 => ( ! [M2: num,N2: num] :
% 8.58/4.87 ( X
% 8.58/4.87 != ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) ) )
% 8.58/4.87 => ( ! [M2: num,N2: num] :
% 8.58/4.87 ( X
% 8.58/4.87 != ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) ) )
% 8.58/4.87 => ( ! [M2: num] :
% 8.58/4.87 ( X
% 8.58/4.87 != ( product_Pair_num_num @ ( bit1 @ M2 ) @ one ) )
% 8.58/4.87 => ( ! [M2: num,N2: num] :
% 8.58/4.87 ( X
% 8.58/4.87 != ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) ) )
% 8.58/4.87 => ~ ! [M2: num,N2: num] :
% 8.58/4.87 ( X
% 8.58/4.87 != ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8712_eucl__rel__int__by0,axiom,
% 8.58/4.87 ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8713_div__int__unique,axiom,
% 8.58/4.87 ! [K: int,L: int,Q2: int,R: int] :
% 8.58/4.87 ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) )
% 8.58/4.87 => ( ( divide_divide_int @ K @ L )
% 8.58/4.87 = Q2 ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8714_mod__int__unique,axiom,
% 8.58/4.87 ! [K: int,L: int,Q2: int,R: int] :
% 8.58/4.87 ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) )
% 8.58/4.87 => ( ( modulo_modulo_int @ K @ L )
% 8.58/4.87 = R ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8715_eucl__rel__int__dividesI,axiom,
% 8.58/4.87 ! [L: int,K: int,Q2: int] :
% 8.58/4.87 ( ( L != zero_zero_int )
% 8.58/4.87 => ( ( K
% 8.58/4.87 = ( times_times_int @ Q2 @ L ) )
% 8.58/4.87 => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8716_eucl__rel__int,axiom,
% 8.58/4.87 ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8717_zminus1__lemma,axiom,
% 8.58/4.87 ! [A: int,B: int,Q2: int,R: int] :
% 8.58/4.87 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 8.58/4.87 => ( ( B != zero_zero_int )
% 8.58/4.87 => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8718_eucl__rel__int__iff,axiom,
% 8.58/4.87 ! [K: int,L: int,Q2: int,R: int] :
% 8.58/4.87 ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) )
% 8.58/4.87 = ( ( K
% 8.58/4.87 = ( plus_plus_int @ ( times_times_int @ L @ Q2 ) @ R ) )
% 8.58/4.87 & ( ( ord_less_int @ zero_zero_int @ L )
% 8.58/4.87 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 8.58/4.87 & ( ord_less_int @ R @ L ) ) )
% 8.58/4.87 & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 8.58/4.87 => ( ( ( ord_less_int @ L @ zero_zero_int )
% 8.58/4.87 => ( ( ord_less_int @ L @ R )
% 8.58/4.87 & ( ord_less_eq_int @ R @ zero_zero_int ) ) )
% 8.58/4.87 & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 8.58/4.87 => ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8719_eucl__rel__int__remainderI,axiom,
% 8.58/4.87 ! [R: int,L: int,K: int,Q2: int] :
% 8.58/4.87 ( ( ( sgn_sgn_int @ R )
% 8.58/4.87 = ( sgn_sgn_int @ L ) )
% 8.58/4.87 => ( ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ L ) )
% 8.58/4.87 => ( ( K
% 8.58/4.87 = ( plus_plus_int @ ( times_times_int @ Q2 @ L ) @ R ) )
% 8.58/4.87 => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8720_eucl__rel__int_Ocases,axiom,
% 8.58/4.87 ! [A1: int,A22: int,A32: product_prod_int_int] :
% 8.58/4.87 ( ( eucl_rel_int @ A1 @ A22 @ A32 )
% 8.58/4.87 => ( ( ( A22 = zero_zero_int )
% 8.58/4.87 => ( A32
% 8.58/4.87 != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
% 8.58/4.87 => ( ! [Q6: int] :
% 8.58/4.87 ( ( A32
% 8.58/4.87 = ( product_Pair_int_int @ Q6 @ zero_zero_int ) )
% 8.58/4.87 => ( ( A22 != zero_zero_int )
% 8.58/4.87 => ( A1
% 8.58/4.87 != ( times_times_int @ Q6 @ A22 ) ) ) )
% 8.58/4.87 => ~ ! [R2: int,Q6: int] :
% 8.58/4.87 ( ( A32
% 8.58/4.87 = ( product_Pair_int_int @ Q6 @ R2 ) )
% 8.58/4.87 => ( ( ( sgn_sgn_int @ R2 )
% 8.58/4.87 = ( sgn_sgn_int @ A22 ) )
% 8.58/4.87 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ A22 ) )
% 8.58/4.87 => ( A1
% 8.58/4.87 != ( plus_plus_int @ ( times_times_int @ Q6 @ A22 ) @ R2 ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8721_eucl__rel__int_Osimps,axiom,
% 8.58/4.87 ( eucl_rel_int
% 8.58/4.87 = ( ^ [A12: int,A23: int,A33: product_prod_int_int] :
% 8.58/4.87 ( ? [K2: int] :
% 8.58/4.87 ( ( A12 = K2 )
% 8.58/4.87 & ( A23 = zero_zero_int )
% 8.58/4.87 & ( A33
% 8.58/4.87 = ( product_Pair_int_int @ zero_zero_int @ K2 ) ) )
% 8.58/4.87 | ? [L2: int,K2: int,Q5: int] :
% 8.58/4.87 ( ( A12 = K2 )
% 8.58/4.87 & ( A23 = L2 )
% 8.58/4.87 & ( A33
% 8.58/4.87 = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
% 8.58/4.87 & ( L2 != zero_zero_int )
% 8.58/4.87 & ( K2
% 8.58/4.87 = ( times_times_int @ Q5 @ L2 ) ) )
% 8.58/4.87 | ? [R5: int,L2: int,K2: int,Q5: int] :
% 8.58/4.87 ( ( A12 = K2 )
% 8.58/4.87 & ( A23 = L2 )
% 8.58/4.87 & ( A33
% 8.58/4.87 = ( product_Pair_int_int @ Q5 @ R5 ) )
% 8.58/4.87 & ( ( sgn_sgn_int @ R5 )
% 8.58/4.87 = ( sgn_sgn_int @ L2 ) )
% 8.58/4.87 & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
% 8.58/4.87 & ( K2
% 8.58/4.87 = ( plus_plus_int @ ( times_times_int @ Q5 @ L2 ) @ R5 ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8722_pos__eucl__rel__int__mult__2,axiom,
% 8.58/4.87 ! [B: int,A: int,Q2: int,R: int] :
% 8.58/4.87 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.58/4.87 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 8.58/4.87 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8723_minus__numeral__div__numeral,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8724_numeral__div__minus__numeral,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8725_minus__one__div__numeral,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8726_one__div__minus__numeral,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8727_divmod__BitM__2__eq,axiom,
% 8.58/4.87 ! [M: num] :
% 8.58/4.87 ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 8.58/4.87 = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8728_divmod_H__nat__def,axiom,
% 8.58/4.87 ( unique5055182867167087721od_nat
% 8.58/4.87 = ( ^ [M3: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8729_divmod__int__def,axiom,
% 8.58/4.87 ( unique5052692396658037445od_int
% 8.58/4.87 = ( ^ [M3: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8730_bezw__0,axiom,
% 8.58/4.87 ! [X: nat] :
% 8.58/4.87 ( ( bezw @ X @ zero_zero_nat )
% 8.58/4.87 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8731_rcis__inverse,axiom,
% 8.58/4.87 ! [R: real,A: real] :
% 8.58/4.87 ( ( invers8013647133539491842omplex @ ( rcis @ R @ A ) )
% 8.58/4.87 = ( rcis @ ( divide_divide_real @ one_one_real @ R ) @ ( uminus_uminus_real @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8732_abs__Gcd__eq,axiom,
% 8.58/4.87 ! [K5: set_int] :
% 8.58/4.87 ( ( abs_abs_int @ ( gcd_Gcd_int @ K5 ) )
% 8.58/4.87 = ( gcd_Gcd_int @ K5 ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8733_Re__rcis,axiom,
% 8.58/4.87 ! [R: real,A: real] :
% 8.58/4.87 ( ( re @ ( rcis @ R @ A ) )
% 8.58/4.87 = ( times_times_real @ R @ ( cos_real @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8734_Im__rcis,axiom,
% 8.58/4.87 ! [R: real,A: real] :
% 8.58/4.87 ( ( im @ ( rcis @ R @ A ) )
% 8.58/4.87 = ( times_times_real @ R @ ( sin_real @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8735_prod__decode__aux_Ocases,axiom,
% 8.58/4.87 ! [X: product_prod_nat_nat] :
% 8.58/4.87 ~ ! [K3: nat,M2: nat] :
% 8.58/4.87 ( X
% 8.58/4.87 != ( product_Pair_nat_nat @ K3 @ M2 ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8736_Gcd__greatest__int,axiom,
% 8.58/4.87 ! [A3: set_int,A: int] :
% 8.58/4.87 ( ! [B4: int] :
% 8.58/4.87 ( ( member_int @ B4 @ A3 )
% 8.58/4.87 => ( dvd_dvd_int @ A @ B4 ) )
% 8.58/4.87 => ( dvd_dvd_int @ A @ ( gcd_Gcd_int @ A3 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8737_Gcd__dvd__int,axiom,
% 8.58/4.87 ! [A: int,A3: set_int] :
% 8.58/4.87 ( ( member_int @ A @ A3 )
% 8.58/4.87 => ( dvd_dvd_int @ ( gcd_Gcd_int @ A3 ) @ A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8738_Gcd__nat__eq__one,axiom,
% 8.58/4.87 ! [N5: set_nat] :
% 8.58/4.87 ( ( member_nat @ one_one_nat @ N5 )
% 8.58/4.87 => ( ( gcd_Gcd_nat @ N5 )
% 8.58/4.87 = one_one_nat ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8739_Gcd__greatest__nat,axiom,
% 8.58/4.87 ! [A3: set_nat,A: nat] :
% 8.58/4.87 ( ! [B4: nat] :
% 8.58/4.87 ( ( member_nat @ B4 @ A3 )
% 8.58/4.87 => ( dvd_dvd_nat @ A @ B4 ) )
% 8.58/4.87 => ( dvd_dvd_nat @ A @ ( gcd_Gcd_nat @ A3 ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8740_Gcd__dvd__nat,axiom,
% 8.58/4.87 ! [A: nat,A3: set_nat] :
% 8.58/4.87 ( ( member_nat @ A @ A3 )
% 8.58/4.87 => ( dvd_dvd_nat @ ( gcd_Gcd_nat @ A3 ) @ A ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8741_Gcd__int__greater__eq__0,axiom,
% 8.58/4.87 ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8742_cis__rcis__eq,axiom,
% 8.58/4.87 ( cis
% 8.58/4.87 = ( rcis @ one_one_real ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8743_rcis__mult,axiom,
% 8.58/4.87 ! [R1: real,A: real,R22: real,B: real] :
% 8.58/4.87 ( ( times_times_complex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B ) )
% 8.58/4.87 = ( rcis @ ( times_times_real @ R1 @ R22 ) @ ( plus_plus_real @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8744_rcis__divide,axiom,
% 8.58/4.87 ! [R1: real,A: real,R22: real,B: real] :
% 8.58/4.87 ( ( divide1717551699836669952omplex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B ) )
% 8.58/4.87 = ( rcis @ ( divide_divide_real @ R1 @ R22 ) @ ( minus_minus_real @ A @ B ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8745_rcis__def,axiom,
% 8.58/4.87 ( rcis
% 8.58/4.87 = ( ^ [R5: real,A2: real] : ( times_times_complex @ ( real_V4546457046886955230omplex @ R5 ) @ ( cis @ A2 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8746_DeMoivre2,axiom,
% 8.58/4.87 ! [R: real,A: real,N: nat] :
% 8.58/4.87 ( ( power_power_complex @ ( rcis @ R @ A ) @ N )
% 8.58/4.87 = ( rcis @ ( power_power_real @ R @ N ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8747_prod__decode__aux_Osimps,axiom,
% 8.58/4.87 ( nat_prod_decode_aux
% 8.58/4.87 = ( ^ [K2: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K2 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K2 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M3 @ ( suc @ K2 ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8748_prod__decode__aux_Oelims,axiom,
% 8.58/4.87 ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
% 8.58/4.87 ( ( ( nat_prod_decode_aux @ X @ Xa )
% 8.58/4.87 = Y )
% 8.58/4.87 => ( ( ( ord_less_eq_nat @ Xa @ X )
% 8.58/4.87 => ( Y
% 8.58/4.87 = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
% 8.58/4.87 & ( ~ ( ord_less_eq_nat @ Xa @ X )
% 8.58/4.87 => ( Y
% 8.58/4.87 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8749_Suc__0__div__numeral,axiom,
% 8.58/4.87 ! [K: num] :
% 8.58/4.87 ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 8.58/4.87 = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8750_Suc__0__mod__numeral,axiom,
% 8.58/4.87 ! [K: num] :
% 8.58/4.87 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 8.58/4.87 = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8751_snd__divmod__nat,axiom,
% 8.58/4.87 ! [M: nat,N: nat] :
% 8.58/4.87 ( ( product_snd_nat_nat @ ( divmod_nat @ M @ N ) )
% 8.58/4.87 = ( modulo_modulo_nat @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8752_fst__divmod__nat,axiom,
% 8.58/4.87 ! [M: nat,N: nat] :
% 8.58/4.87 ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
% 8.58/4.87 = ( divide_divide_nat @ M @ N ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8753_bezw__non__0,axiom,
% 8.58/4.87 ! [Y: nat,X: nat] :
% 8.58/4.87 ( ( ord_less_nat @ zero_zero_nat @ Y )
% 8.58/4.87 => ( ( bezw @ X @ Y )
% 8.58/4.87 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8754_bezw_Osimps,axiom,
% 8.58/4.87 ( bezw
% 8.58/4.87 = ( ^ [X3: nat,Y6: nat] : ( if_Pro3027730157355071871nt_int @ ( Y6 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y6 ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8755_bezw_Oelims,axiom,
% 8.58/4.87 ! [X: nat,Xa: nat,Y: product_prod_int_int] :
% 8.58/4.87 ( ( ( bezw @ X @ Xa )
% 8.58/4.87 = Y )
% 8.58/4.87 => ( ( ( Xa = zero_zero_nat )
% 8.58/4.87 => ( Y
% 8.58/4.87 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 8.58/4.87 & ( ( Xa != zero_zero_nat )
% 8.58/4.87 => ( Y
% 8.58/4.87 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8756_divmod__nat__def,axiom,
% 8.58/4.87 ( divmod_nat
% 8.58/4.87 = ( ^ [M3: nat,N3: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M3 @ N3 ) @ ( modulo_modulo_nat @ M3 @ N3 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8757_one__mod__minus__numeral,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8758_minus__one__mod__numeral,axiom,
% 8.58/4.87 ! [N: num] :
% 8.58/4.87 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.87 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8759_minus__numeral__mod__numeral,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.87 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8760_numeral__mod__minus__numeral,axiom,
% 8.58/4.87 ! [M: num,N: num] :
% 8.58/4.87 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.87 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8761_Divides_Oadjust__mod__def,axiom,
% 8.58/4.87 ( adjust_mod
% 8.58/4.87 = ( ^ [L2: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L2 @ R5 ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8762_bezw_Opelims,axiom,
% 8.58/4.87 ! [X: nat,Xa: nat,Y: product_prod_int_int] :
% 8.58/4.87 ( ( ( bezw @ X @ Xa )
% 8.58/4.87 = Y )
% 8.58/4.87 => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 8.58/4.87 => ~ ( ( ( ( Xa = zero_zero_nat )
% 8.58/4.87 => ( Y
% 8.58/4.87 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 8.58/4.87 & ( ( Xa != zero_zero_nat )
% 8.58/4.87 => ( Y
% 8.58/4.87 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) )
% 8.58/4.87 => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8763_prod__encode__prod__decode__aux,axiom,
% 8.58/4.87 ! [K: nat,M: nat] :
% 8.58/4.87 ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 8.58/4.87 = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8764_bezw__aux,axiom,
% 8.58/4.87 ! [X: nat,Y: nat] :
% 8.58/4.87 ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y ) )
% 8.58/4.87 = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8765_gcd__idem__nat,axiom,
% 8.58/4.87 ! [X: nat] :
% 8.58/4.87 ( ( gcd_gcd_nat @ X @ X )
% 8.58/4.87 = X ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8766_gcd__nat_Oright__idem,axiom,
% 8.58/4.87 ! [A: nat,B: nat] :
% 8.58/4.87 ( ( gcd_gcd_nat @ ( gcd_gcd_nat @ A @ B ) @ B )
% 8.58/4.87 = ( gcd_gcd_nat @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8767_gcd__nat_Oleft__idem,axiom,
% 8.58/4.87 ! [A: nat,B: nat] :
% 8.58/4.87 ( ( gcd_gcd_nat @ A @ ( gcd_gcd_nat @ A @ B ) )
% 8.58/4.87 = ( gcd_gcd_nat @ A @ B ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8768_gcd__nat_Oidem,axiom,
% 8.58/4.87 ! [A: nat] :
% 8.58/4.87 ( ( gcd_gcd_nat @ A @ A )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8769_prod__encode__eq,axiom,
% 8.58/4.87 ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 8.58/4.87 ( ( ( nat_prod_encode @ X )
% 8.58/4.87 = ( nat_prod_encode @ Y ) )
% 8.58/4.87 = ( X = Y ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8770_gcd__0__left__nat,axiom,
% 8.58/4.87 ! [X: nat] :
% 8.58/4.87 ( ( gcd_gcd_nat @ zero_zero_nat @ X )
% 8.58/4.87 = X ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8771_gcd__0__nat,axiom,
% 8.58/4.87 ! [X: nat] :
% 8.58/4.87 ( ( gcd_gcd_nat @ X @ zero_zero_nat )
% 8.58/4.87 = X ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8772_gcd__nat_Oright__neutral,axiom,
% 8.58/4.87 ! [A: nat] :
% 8.58/4.87 ( ( gcd_gcd_nat @ A @ zero_zero_nat )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8773_gcd__nat_Oneutr__eq__iff,axiom,
% 8.58/4.87 ! [A: nat,B: nat] :
% 8.58/4.87 ( ( zero_zero_nat
% 8.58/4.87 = ( gcd_gcd_nat @ A @ B ) )
% 8.58/4.87 = ( ( A = zero_zero_nat )
% 8.58/4.87 & ( B = zero_zero_nat ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8774_gcd__nat_Oleft__neutral,axiom,
% 8.58/4.87 ! [A: nat] :
% 8.58/4.87 ( ( gcd_gcd_nat @ zero_zero_nat @ A )
% 8.58/4.87 = A ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8775_gcd__nat_Oeq__neutr__iff,axiom,
% 8.58/4.87 ! [A: nat,B: nat] :
% 8.58/4.87 ( ( ( gcd_gcd_nat @ A @ B )
% 8.58/4.87 = zero_zero_nat )
% 8.58/4.87 = ( ( A = zero_zero_nat )
% 8.58/4.87 & ( B = zero_zero_nat ) ) ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8776_gcd__1__nat,axiom,
% 8.58/4.87 ! [M: nat] :
% 8.58/4.87 ( ( gcd_gcd_nat @ M @ one_one_nat )
% 8.58/4.87 = one_one_nat ) ).
% 8.58/4.87
% 8.58/4.87 thf(fact_8777_gcd__proj2__if__dvd__nat,axiom,
% 8.58/4.87 ! [Y: nat,X: nat] :
% 8.58/4.87 ( ( dvd_dvd_nat @ Y @ X )
% 8.58/4.87 => ( ( gcd_gcd_nat @ X @ Y )
% 8.58/4.88 = Y ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8778_gcd__proj1__if__dvd__nat,axiom,
% 8.58/4.88 ! [X: nat,Y: nat] :
% 8.58/4.88 ( ( dvd_dvd_nat @ X @ Y )
% 8.58/4.88 => ( ( gcd_gcd_nat @ X @ Y )
% 8.58/4.88 = X ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8779_gcd__nat_Obounded__iff,axiom,
% 8.58/4.88 ! [A: nat,B: nat,C: nat] :
% 8.58/4.88 ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 8.58/4.88 = ( ( dvd_dvd_nat @ A @ B )
% 8.58/4.88 & ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8780_gcd__nat_Oabsorb2,axiom,
% 8.58/4.88 ! [B: nat,A: nat] :
% 8.58/4.88 ( ( dvd_dvd_nat @ B @ A )
% 8.58/4.88 => ( ( gcd_gcd_nat @ A @ B )
% 8.58/4.88 = B ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8781_gcd__nat_Oabsorb1,axiom,
% 8.58/4.88 ! [A: nat,B: nat] :
% 8.58/4.88 ( ( dvd_dvd_nat @ A @ B )
% 8.58/4.88 => ( ( gcd_gcd_nat @ A @ B )
% 8.58/4.88 = A ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8782_gcd__Suc__0,axiom,
% 8.58/4.88 ! [M: nat] :
% 8.58/4.88 ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 8.58/4.88 = ( suc @ zero_zero_nat ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8783_gcd__pos__nat,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 8.58/4.88 = ( ( M != zero_zero_nat )
% 8.58/4.88 | ( N != zero_zero_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8784_gcd__mult__distrib__nat,axiom,
% 8.58/4.88 ! [K: nat,M: nat,N: nat] :
% 8.58/4.88 ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
% 8.58/4.88 = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8785_gcd__unique__nat,axiom,
% 8.58/4.88 ! [D: nat,A: nat,B: nat] :
% 8.58/4.88 ( ( ( dvd_dvd_nat @ D @ A )
% 8.58/4.88 & ( dvd_dvd_nat @ D @ B )
% 8.58/4.88 & ! [E3: nat] :
% 8.58/4.88 ( ( ( dvd_dvd_nat @ E3 @ A )
% 8.58/4.88 & ( dvd_dvd_nat @ E3 @ B ) )
% 8.58/4.88 => ( dvd_dvd_nat @ E3 @ D ) ) )
% 8.58/4.88 = ( D
% 8.58/4.88 = ( gcd_gcd_nat @ A @ B ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8786_gcd__nat_Ostrict__coboundedI2,axiom,
% 8.58/4.88 ! [B: nat,C: nat,A: nat] :
% 8.58/4.88 ( ( ( dvd_dvd_nat @ B @ C )
% 8.58/4.88 & ( B != C ) )
% 8.58/4.88 => ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
% 8.58/4.88 & ( ( gcd_gcd_nat @ A @ B )
% 8.58/4.88 != C ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8787_gcd__nat_Ostrict__coboundedI1,axiom,
% 8.58/4.88 ! [A: nat,C: nat,B: nat] :
% 8.58/4.88 ( ( ( dvd_dvd_nat @ A @ C )
% 8.58/4.88 & ( A != C ) )
% 8.58/4.88 => ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
% 8.58/4.88 & ( ( gcd_gcd_nat @ A @ B )
% 8.58/4.88 != C ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8788_gcd__nat_Ostrict__order__iff,axiom,
% 8.58/4.88 ! [A: nat,B: nat] :
% 8.58/4.88 ( ( ( dvd_dvd_nat @ A @ B )
% 8.58/4.88 & ( A != B ) )
% 8.58/4.88 = ( ( A
% 8.58/4.88 = ( gcd_gcd_nat @ A @ B ) )
% 8.58/4.88 & ( A != B ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8789_gcd__nat_Ostrict__boundedE,axiom,
% 8.58/4.88 ! [A: nat,B: nat,C: nat] :
% 8.58/4.88 ( ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 8.58/4.88 & ( A
% 8.58/4.88 != ( gcd_gcd_nat @ B @ C ) ) )
% 8.58/4.88 => ~ ( ( ( dvd_dvd_nat @ A @ B )
% 8.58/4.88 & ( A != B ) )
% 8.58/4.88 => ~ ( ( dvd_dvd_nat @ A @ C )
% 8.58/4.88 & ( A != C ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8790_gcd__nat_OcoboundedI2,axiom,
% 8.58/4.88 ! [B: nat,C: nat,A: nat] :
% 8.58/4.88 ( ( dvd_dvd_nat @ B @ C )
% 8.58/4.88 => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8791_gcd__nat_OcoboundedI1,axiom,
% 8.58/4.88 ! [A: nat,C: nat,B: nat] :
% 8.58/4.88 ( ( dvd_dvd_nat @ A @ C )
% 8.58/4.88 => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8792_gcd__nat_Oabsorb__iff2,axiom,
% 8.58/4.88 ( dvd_dvd_nat
% 8.58/4.88 = ( ^ [B2: nat,A2: nat] :
% 8.58/4.88 ( ( gcd_gcd_nat @ A2 @ B2 )
% 8.58/4.88 = B2 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8793_gcd__nat_Oabsorb__iff1,axiom,
% 8.58/4.88 ( dvd_dvd_nat
% 8.58/4.88 = ( ^ [A2: nat,B2: nat] :
% 8.58/4.88 ( ( gcd_gcd_nat @ A2 @ B2 )
% 8.58/4.88 = A2 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8794_gcd__nat_Ocobounded2,axiom,
% 8.58/4.88 ! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8795_gcd__nat_Ocobounded1,axiom,
% 8.58/4.88 ! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8796_gcd__nat_Oorder__iff,axiom,
% 8.58/4.88 ( dvd_dvd_nat
% 8.58/4.88 = ( ^ [A2: nat,B2: nat] :
% 8.58/4.88 ( A2
% 8.58/4.88 = ( gcd_gcd_nat @ A2 @ B2 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8797_gcd__nat_OboundedI,axiom,
% 8.58/4.88 ! [A: nat,B: nat,C: nat] :
% 8.58/4.88 ( ( dvd_dvd_nat @ A @ B )
% 8.58/4.88 => ( ( dvd_dvd_nat @ A @ C )
% 8.58/4.88 => ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8798_gcd__nat_OboundedE,axiom,
% 8.58/4.88 ! [A: nat,B: nat,C: nat] :
% 8.58/4.88 ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 8.58/4.88 => ~ ( ( dvd_dvd_nat @ A @ B )
% 8.58/4.88 => ~ ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8799_gcd__nat_Oabsorb4,axiom,
% 8.58/4.88 ! [B: nat,A: nat] :
% 8.58/4.88 ( ( ( dvd_dvd_nat @ B @ A )
% 8.58/4.88 & ( B != A ) )
% 8.58/4.88 => ( ( gcd_gcd_nat @ A @ B )
% 8.58/4.88 = B ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8800_gcd__nat_Oabsorb3,axiom,
% 8.58/4.88 ! [A: nat,B: nat] :
% 8.58/4.88 ( ( ( dvd_dvd_nat @ A @ B )
% 8.58/4.88 & ( A != B ) )
% 8.58/4.88 => ( ( gcd_gcd_nat @ A @ B )
% 8.58/4.88 = A ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8801_gcd__nat_OorderI,axiom,
% 8.58/4.88 ! [A: nat,B: nat] :
% 8.58/4.88 ( ( A
% 8.58/4.88 = ( gcd_gcd_nat @ A @ B ) )
% 8.58/4.88 => ( dvd_dvd_nat @ A @ B ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8802_gcd__nat_OorderE,axiom,
% 8.58/4.88 ! [A: nat,B: nat] :
% 8.58/4.88 ( ( dvd_dvd_nat @ A @ B )
% 8.58/4.88 => ( A
% 8.58/4.88 = ( gcd_gcd_nat @ A @ B ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8803_gcd__nat_Omono,axiom,
% 8.58/4.88 ! [A: nat,C: nat,B: nat,D: nat] :
% 8.58/4.88 ( ( dvd_dvd_nat @ A @ C )
% 8.58/4.88 => ( ( dvd_dvd_nat @ B @ D )
% 8.58/4.88 => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ ( gcd_gcd_nat @ C @ D ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8804_gcd__red__nat,axiom,
% 8.58/4.88 ( gcd_gcd_nat
% 8.58/4.88 = ( ^ [X3: nat,Y6: nat] : ( gcd_gcd_nat @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8805_gcd__le2__nat,axiom,
% 8.58/4.88 ! [B: nat,A: nat] :
% 8.58/4.88 ( ( B != zero_zero_nat )
% 8.58/4.88 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8806_gcd__le1__nat,axiom,
% 8.58/4.88 ! [A: nat,B: nat] :
% 8.58/4.88 ( ( A != zero_zero_nat )
% 8.58/4.88 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8807_gcd__diff2__nat,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.88 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 8.58/4.88 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8808_gcd__diff1__nat,axiom,
% 8.58/4.88 ! [N: nat,M: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.88 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 8.58/4.88 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8809_gcd__non__0__nat,axiom,
% 8.58/4.88 ! [Y: nat,X: nat] :
% 8.58/4.88 ( ( Y != zero_zero_nat )
% 8.58/4.88 => ( ( gcd_gcd_nat @ X @ Y )
% 8.58/4.88 = ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8810_gcd__nat_Osimps,axiom,
% 8.58/4.88 ( gcd_gcd_nat
% 8.58/4.88 = ( ^ [X3: nat,Y6: nat] : ( if_nat @ ( Y6 = zero_zero_nat ) @ X3 @ ( gcd_gcd_nat @ Y6 @ ( modulo_modulo_nat @ X3 @ Y6 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8811_gcd__nat_Oelims,axiom,
% 8.58/4.88 ! [X: nat,Xa: nat,Y: nat] :
% 8.58/4.88 ( ( ( gcd_gcd_nat @ X @ Xa )
% 8.58/4.88 = Y )
% 8.58/4.88 => ( ( ( Xa = zero_zero_nat )
% 8.58/4.88 => ( Y = X ) )
% 8.58/4.88 & ( ( Xa != zero_zero_nat )
% 8.58/4.88 => ( Y
% 8.58/4.88 = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8812_Gcd__in,axiom,
% 8.58/4.88 ! [A3: set_nat] :
% 8.58/4.88 ( ! [A4: nat,B4: nat] :
% 8.58/4.88 ( ( member_nat @ A4 @ A3 )
% 8.58/4.88 => ( ( member_nat @ B4 @ A3 )
% 8.58/4.88 => ( member_nat @ ( gcd_gcd_nat @ A4 @ B4 ) @ A3 ) ) )
% 8.58/4.88 => ( ( A3 != bot_bot_set_nat )
% 8.58/4.88 => ( member_nat @ ( gcd_Gcd_nat @ A3 ) @ A3 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8813_bezout__nat,axiom,
% 8.58/4.88 ! [A: nat,B: nat] :
% 8.58/4.88 ( ( A != zero_zero_nat )
% 8.58/4.88 => ? [X4: nat,Y4: nat] :
% 8.58/4.88 ( ( times_times_nat @ A @ X4 )
% 8.58/4.88 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8814_bezout__gcd__nat_H,axiom,
% 8.58/4.88 ! [B: nat,A: nat] :
% 8.58/4.88 ? [X4: nat,Y4: nat] :
% 8.58/4.88 ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y4 ) @ ( times_times_nat @ A @ X4 ) )
% 8.58/4.88 & ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y4 ) )
% 8.58/4.88 = ( gcd_gcd_nat @ A @ B ) ) )
% 8.58/4.88 | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B @ X4 ) )
% 8.58/4.88 & ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y4 ) )
% 8.58/4.88 = ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8815_le__prod__encode__1,axiom,
% 8.58/4.88 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8816_le__prod__encode__2,axiom,
% 8.58/4.88 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8817_prod__decode__aux_Opelims,axiom,
% 8.58/4.88 ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
% 8.58/4.88 ( ( ( nat_prod_decode_aux @ X @ Xa )
% 8.58/4.88 = Y )
% 8.58/4.88 => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 8.58/4.88 => ~ ( ( ( ( ord_less_eq_nat @ Xa @ X )
% 8.58/4.88 => ( Y
% 8.58/4.88 = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
% 8.58/4.88 & ( ~ ( ord_less_eq_nat @ Xa @ X )
% 8.58/4.88 => ( Y
% 8.58/4.88 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) )
% 8.58/4.88 => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8818_gcd__nat_Opelims,axiom,
% 8.58/4.88 ! [X: nat,Xa: nat,Y: nat] :
% 8.58/4.88 ( ( ( gcd_gcd_nat @ X @ Xa )
% 8.58/4.88 = Y )
% 8.58/4.88 => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 8.58/4.88 => ~ ( ( ( ( Xa = zero_zero_nat )
% 8.58/4.88 => ( Y = X ) )
% 8.58/4.88 & ( ( Xa != zero_zero_nat )
% 8.58/4.88 => ( Y
% 8.58/4.88 = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) )
% 8.58/4.88 => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8819_gcd__1__int,axiom,
% 8.58/4.88 ! [M: int] :
% 8.58/4.88 ( ( gcd_gcd_int @ M @ one_one_int )
% 8.58/4.88 = one_one_int ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8820_gcd__neg1__int,axiom,
% 8.58/4.88 ! [X: int,Y: int] :
% 8.58/4.88 ( ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y )
% 8.58/4.88 = ( gcd_gcd_int @ X @ Y ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8821_gcd__neg2__int,axiom,
% 8.58/4.88 ! [X: int,Y: int] :
% 8.58/4.88 ( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) )
% 8.58/4.88 = ( gcd_gcd_int @ X @ Y ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8822_abs__gcd__int,axiom,
% 8.58/4.88 ! [X: int,Y: int] :
% 8.58/4.88 ( ( abs_abs_int @ ( gcd_gcd_int @ X @ Y ) )
% 8.58/4.88 = ( gcd_gcd_int @ X @ Y ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8823_gcd__abs1__int,axiom,
% 8.58/4.88 ! [X: int,Y: int] :
% 8.58/4.88 ( ( gcd_gcd_int @ ( abs_abs_int @ X ) @ Y )
% 8.58/4.88 = ( gcd_gcd_int @ X @ Y ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8824_gcd__abs2__int,axiom,
% 8.58/4.88 ! [X: int,Y: int] :
% 8.58/4.88 ( ( gcd_gcd_int @ X @ ( abs_abs_int @ Y ) )
% 8.58/4.88 = ( gcd_gcd_int @ X @ Y ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8825_gcd__pos__int,axiom,
% 8.58/4.88 ! [M: int,N: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
% 8.58/4.88 = ( ( M != zero_zero_int )
% 8.58/4.88 | ( N != zero_zero_int ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8826_gcd__neg__numeral__2__int,axiom,
% 8.58/4.88 ! [X: int,N: num] :
% 8.58/4.88 ( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.88 = ( gcd_gcd_int @ X @ ( numeral_numeral_int @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8827_gcd__neg__numeral__1__int,axiom,
% 8.58/4.88 ! [N: num,X: int] :
% 8.58/4.88 ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X )
% 8.58/4.88 = ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8828_gcd__0__int,axiom,
% 8.58/4.88 ! [X: int] :
% 8.58/4.88 ( ( gcd_gcd_int @ X @ zero_zero_int )
% 8.58/4.88 = ( abs_abs_int @ X ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8829_gcd__0__left__int,axiom,
% 8.58/4.88 ! [X: int] :
% 8.58/4.88 ( ( gcd_gcd_int @ zero_zero_int @ X )
% 8.58/4.88 = ( abs_abs_int @ X ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8830_gcd__proj1__if__dvd__int,axiom,
% 8.58/4.88 ! [X: int,Y: int] :
% 8.58/4.88 ( ( dvd_dvd_int @ X @ Y )
% 8.58/4.88 => ( ( gcd_gcd_int @ X @ Y )
% 8.58/4.88 = ( abs_abs_int @ X ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8831_gcd__proj2__if__dvd__int,axiom,
% 8.58/4.88 ! [Y: int,X: int] :
% 8.58/4.88 ( ( dvd_dvd_int @ Y @ X )
% 8.58/4.88 => ( ( gcd_gcd_int @ X @ Y )
% 8.58/4.88 = ( abs_abs_int @ Y ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8832_gcd__int__int__eq,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 8.58/4.88 = ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8833_gcd__nat__abs__right__eq,axiom,
% 8.58/4.88 ! [N: nat,K: int] :
% 8.58/4.88 ( ( gcd_gcd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 8.58/4.88 = ( nat2 @ ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8834_gcd__nat__abs__left__eq,axiom,
% 8.58/4.88 ! [K: int,N: nat] :
% 8.58/4.88 ( ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 8.58/4.88 = ( nat2 @ ( gcd_gcd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8835_gcd__red__int,axiom,
% 8.58/4.88 ( gcd_gcd_int
% 8.58/4.88 = ( ^ [X3: int,Y6: int] : ( gcd_gcd_int @ Y6 @ ( modulo_modulo_int @ X3 @ Y6 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8836_gcd__idem__int,axiom,
% 8.58/4.88 ! [X: int] :
% 8.58/4.88 ( ( gcd_gcd_int @ X @ X )
% 8.58/4.88 = ( abs_abs_int @ X ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8837_gcd__ge__0__int,axiom,
% 8.58/4.88 ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8838_bezout__int,axiom,
% 8.58/4.88 ! [X: int,Y: int] :
% 8.58/4.88 ? [U2: int,V3: int] :
% 8.58/4.88 ( ( plus_plus_int @ ( times_times_int @ U2 @ X ) @ ( times_times_int @ V3 @ Y ) )
% 8.58/4.88 = ( gcd_gcd_int @ X @ Y ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8839_gcd__mult__distrib__int,axiom,
% 8.58/4.88 ! [K: int,M: int,N: int] :
% 8.58/4.88 ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
% 8.58/4.88 = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8840_gcd__le1__int,axiom,
% 8.58/4.88 ! [A: int,B: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ A )
% 8.58/4.88 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8841_gcd__le2__int,axiom,
% 8.58/4.88 ! [B: int,A: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.88 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8842_gcd__cases__int,axiom,
% 8.58/4.88 ! [X: int,Y: int,P: int > $o] :
% 8.58/4.88 ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.58/4.88 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.58/4.88 => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) )
% 8.58/4.88 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.58/4.88 => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 8.58/4.88 => ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
% 8.58/4.88 => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 8.58/4.88 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.58/4.88 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
% 8.58/4.88 => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 8.58/4.88 => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 8.58/4.88 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
% 8.58/4.88 => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8843_gcd__unique__int,axiom,
% 8.58/4.88 ! [D: int,A: int,B: int] :
% 8.58/4.88 ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 8.58/4.88 & ( dvd_dvd_int @ D @ A )
% 8.58/4.88 & ( dvd_dvd_int @ D @ B )
% 8.58/4.88 & ! [E3: int] :
% 8.58/4.88 ( ( ( dvd_dvd_int @ E3 @ A )
% 8.58/4.88 & ( dvd_dvd_int @ E3 @ B ) )
% 8.58/4.88 => ( dvd_dvd_int @ E3 @ D ) ) )
% 8.58/4.88 = ( D
% 8.58/4.88 = ( gcd_gcd_int @ A @ B ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8844_gcd__non__0__int,axiom,
% 8.58/4.88 ! [Y: int,X: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ Y )
% 8.58/4.88 => ( ( gcd_gcd_int @ X @ Y )
% 8.58/4.88 = ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8845_gcd__code__int,axiom,
% 8.58/4.88 ( gcd_gcd_int
% 8.58/4.88 = ( ^ [K2: int,L2: int] : ( abs_abs_int @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( gcd_gcd_int @ L2 @ ( modulo_modulo_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8846_gcd__int__def,axiom,
% 8.58/4.88 ( gcd_gcd_int
% 8.58/4.88 = ( ^ [X3: int,Y6: int] : ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ X3 ) ) @ ( nat2 @ ( abs_abs_int @ Y6 ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8847_upto__aux__rec,axiom,
% 8.58/4.88 ( upto_aux
% 8.58/4.88 = ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8848_powr__real__of__int_H,axiom,
% 8.58/4.88 ! [X: real,N: int] :
% 8.58/4.88 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.88 => ( ( ( X != zero_zero_real )
% 8.58/4.88 | ( ord_less_int @ zero_zero_int @ N ) )
% 8.58/4.88 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 8.58/4.88 = ( power_int_real @ X @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8849_complex__is__Nat__iff,axiom,
% 8.58/4.88 ! [Z2: complex] :
% 8.58/4.88 ( ( member_complex @ Z2 @ semiri3842193898606819883omplex )
% 8.58/4.88 = ( ( ( im @ Z2 )
% 8.58/4.88 = zero_zero_real )
% 8.58/4.88 & ? [I4: nat] :
% 8.58/4.88 ( ( re @ Z2 )
% 8.58/4.88 = ( semiri5074537144036343181t_real @ I4 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8850_cpmi,axiom,
% 8.58/4.88 ! [D3: int,P: int > $o,P4: int > $o,B5: set_int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ D3 )
% 8.58/4.88 => ( ? [Z3: int] :
% 8.58/4.88 ! [X4: int] :
% 8.58/4.88 ( ( ord_less_int @ X4 @ Z3 )
% 8.58/4.88 => ( ( P @ X4 )
% 8.58/4.88 = ( P4 @ X4 ) ) )
% 8.58/4.88 => ( ! [X4: int] :
% 8.58/4.88 ( ! [Xa2: int] :
% 8.58/4.88 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb2: int] :
% 8.58/4.88 ( ( member_int @ Xb2 @ B5 )
% 8.58/4.88 => ( X4
% 8.58/4.88 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 8.58/4.88 => ( ( P @ X4 )
% 8.58/4.88 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 8.58/4.88 => ( ! [X4: int,K3: int] :
% 8.58/4.88 ( ( P4 @ X4 )
% 8.58/4.88 = ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
% 8.58/4.88 => ( ( ? [X8: int] : ( P @ X8 ) )
% 8.58/4.88 = ( ? [X3: int] :
% 8.58/4.88 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 & ( P4 @ X3 ) )
% 8.58/4.88 | ? [X3: int] :
% 8.58/4.88 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 & ? [Y6: int] :
% 8.58/4.88 ( ( member_int @ Y6 @ B5 )
% 8.58/4.88 & ( P @ ( plus_plus_int @ Y6 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8851_cppi,axiom,
% 8.58/4.88 ! [D3: int,P: int > $o,P4: int > $o,A3: set_int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ D3 )
% 8.58/4.88 => ( ? [Z3: int] :
% 8.58/4.88 ! [X4: int] :
% 8.58/4.88 ( ( ord_less_int @ Z3 @ X4 )
% 8.58/4.88 => ( ( P @ X4 )
% 8.58/4.88 = ( P4 @ X4 ) ) )
% 8.58/4.88 => ( ! [X4: int] :
% 8.58/4.88 ( ! [Xa2: int] :
% 8.58/4.88 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb2: int] :
% 8.58/4.88 ( ( member_int @ Xb2 @ A3 )
% 8.58/4.88 => ( X4
% 8.58/4.88 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 8.58/4.88 => ( ( P @ X4 )
% 8.58/4.88 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 8.58/4.88 => ( ! [X4: int,K3: int] :
% 8.58/4.88 ( ( P4 @ X4 )
% 8.58/4.88 = ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
% 8.58/4.88 => ( ( ? [X8: int] : ( P @ X8 ) )
% 8.58/4.88 = ( ? [X3: int] :
% 8.58/4.88 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 & ( P4 @ X3 ) )
% 8.58/4.88 | ? [X3: int] :
% 8.58/4.88 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 & ? [Y6: int] :
% 8.58/4.88 ( ( member_int @ Y6 @ A3 )
% 8.58/4.88 & ( P @ ( minus_minus_int @ Y6 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8852_aset_I2_J,axiom,
% 8.58/4.88 ! [D3: int,A3: set_int,P: int > $o,Q: int > $o] :
% 8.58/4.88 ( ! [X4: int] :
% 8.58/4.88 ( ! [Xa2: int] :
% 8.58/4.88 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb2: int] :
% 8.58/4.88 ( ( member_int @ Xb2 @ A3 )
% 8.58/4.88 => ( X4
% 8.58/4.88 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 8.58/4.88 => ( ( P @ X4 )
% 8.58/4.88 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 8.58/4.88 => ( ! [X4: int] :
% 8.58/4.88 ( ! [Xa2: int] :
% 8.58/4.88 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb2: int] :
% 8.58/4.88 ( ( member_int @ Xb2 @ A3 )
% 8.58/4.88 => ( X4
% 8.58/4.88 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 8.58/4.88 => ( ( Q @ X4 )
% 8.58/4.88 => ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ A3 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( ( P @ X5 )
% 8.58/4.88 | ( Q @ X5 ) )
% 8.58/4.88 => ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
% 8.58/4.88 | ( Q @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8853_aset_I1_J,axiom,
% 8.58/4.88 ! [D3: int,A3: set_int,P: int > $o,Q: int > $o] :
% 8.58/4.88 ( ! [X4: int] :
% 8.58/4.88 ( ! [Xa2: int] :
% 8.58/4.88 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb2: int] :
% 8.58/4.88 ( ( member_int @ Xb2 @ A3 )
% 8.58/4.88 => ( X4
% 8.58/4.88 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 8.58/4.88 => ( ( P @ X4 )
% 8.58/4.88 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 8.58/4.88 => ( ! [X4: int] :
% 8.58/4.88 ( ! [Xa2: int] :
% 8.58/4.88 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb2: int] :
% 8.58/4.88 ( ( member_int @ Xb2 @ A3 )
% 8.58/4.88 => ( X4
% 8.58/4.88 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 8.58/4.88 => ( ( Q @ X4 )
% 8.58/4.88 => ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ A3 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( ( P @ X5 )
% 8.58/4.88 & ( Q @ X5 ) )
% 8.58/4.88 => ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
% 8.58/4.88 & ( Q @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8854_bset_I2_J,axiom,
% 8.58/4.88 ! [D3: int,B5: set_int,P: int > $o,Q: int > $o] :
% 8.58/4.88 ( ! [X4: int] :
% 8.58/4.88 ( ! [Xa2: int] :
% 8.58/4.88 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb2: int] :
% 8.58/4.88 ( ( member_int @ Xb2 @ B5 )
% 8.58/4.88 => ( X4
% 8.58/4.88 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 8.58/4.88 => ( ( P @ X4 )
% 8.58/4.88 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 8.58/4.88 => ( ! [X4: int] :
% 8.58/4.88 ( ! [Xa2: int] :
% 8.58/4.88 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb2: int] :
% 8.58/4.88 ( ( member_int @ Xb2 @ B5 )
% 8.58/4.88 => ( X4
% 8.58/4.88 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 8.58/4.88 => ( ( Q @ X4 )
% 8.58/4.88 => ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ B5 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( ( P @ X5 )
% 8.58/4.88 | ( Q @ X5 ) )
% 8.58/4.88 => ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
% 8.58/4.88 | ( Q @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8855_bset_I1_J,axiom,
% 8.58/4.88 ! [D3: int,B5: set_int,P: int > $o,Q: int > $o] :
% 8.58/4.88 ( ! [X4: int] :
% 8.58/4.88 ( ! [Xa2: int] :
% 8.58/4.88 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb2: int] :
% 8.58/4.88 ( ( member_int @ Xb2 @ B5 )
% 8.58/4.88 => ( X4
% 8.58/4.88 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 8.58/4.88 => ( ( P @ X4 )
% 8.58/4.88 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 8.58/4.88 => ( ! [X4: int] :
% 8.58/4.88 ( ! [Xa2: int] :
% 8.58/4.88 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb2: int] :
% 8.58/4.88 ( ( member_int @ Xb2 @ B5 )
% 8.58/4.88 => ( X4
% 8.58/4.88 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 8.58/4.88 => ( ( Q @ X4 )
% 8.58/4.88 => ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ B5 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( ( P @ X5 )
% 8.58/4.88 & ( Q @ X5 ) )
% 8.58/4.88 => ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
% 8.58/4.88 & ( Q @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8856_aset_I10_J,axiom,
% 8.58/4.88 ! [D: int,D3: int,A3: set_int,T: int] :
% 8.58/4.88 ( ( dvd_dvd_int @ D @ D3 )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ A3 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 8.58/4.88 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8857_aset_I9_J,axiom,
% 8.58/4.88 ! [D: int,D3: int,A3: set_int,T: int] :
% 8.58/4.88 ( ( dvd_dvd_int @ D @ D3 )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ A3 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 8.58/4.88 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8858_bset_I10_J,axiom,
% 8.58/4.88 ! [D: int,D3: int,B5: set_int,T: int] :
% 8.58/4.88 ( ( dvd_dvd_int @ D @ D3 )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ B5 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 8.58/4.88 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8859_bset_I9_J,axiom,
% 8.58/4.88 ! [D: int,D3: int,B5: set_int,T: int] :
% 8.58/4.88 ( ( dvd_dvd_int @ D @ D3 )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ B5 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 8.58/4.88 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8860_atLeastAtMostPlus1__int__conv,axiom,
% 8.58/4.88 ! [M: int,N: int] :
% 8.58/4.88 ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 8.58/4.88 => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 8.58/4.88 = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8861_periodic__finite__ex,axiom,
% 8.58/4.88 ! [D: int,P: int > $o] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ D )
% 8.58/4.88 => ( ! [X4: int,K3: int] :
% 8.58/4.88 ( ( P @ X4 )
% 8.58/4.88 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) )
% 8.58/4.88 => ( ( ? [X8: int] : ( P @ X8 ) )
% 8.58/4.88 = ( ? [X3: int] :
% 8.58/4.88 ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 8.58/4.88 & ( P @ X3 ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8862_bset_I3_J,axiom,
% 8.58/4.88 ! [D3: int,T: int,B5: set_int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ D3 )
% 8.58/4.88 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ B5 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( X5 = T )
% 8.58/4.88 => ( ( minus_minus_int @ X5 @ D3 )
% 8.58/4.88 = T ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8863_bset_I4_J,axiom,
% 8.58/4.88 ! [D3: int,T: int,B5: set_int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ D3 )
% 8.58/4.88 => ( ( member_int @ T @ B5 )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ B5 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( X5 != T )
% 8.58/4.88 => ( ( minus_minus_int @ X5 @ D3 )
% 8.58/4.88 != T ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8864_bset_I5_J,axiom,
% 8.58/4.88 ! [D3: int,B5: set_int,T: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ D3 )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ B5 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( ord_less_int @ X5 @ T )
% 8.58/4.88 => ( ord_less_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8865_bset_I7_J,axiom,
% 8.58/4.88 ! [D3: int,T: int,B5: set_int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ D3 )
% 8.58/4.88 => ( ( member_int @ T @ B5 )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ B5 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( ord_less_int @ T @ X5 )
% 8.58/4.88 => ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8866_aset_I3_J,axiom,
% 8.58/4.88 ! [D3: int,T: int,A3: set_int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ D3 )
% 8.58/4.88 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ A3 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( X5 = T )
% 8.58/4.88 => ( ( plus_plus_int @ X5 @ D3 )
% 8.58/4.88 = T ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8867_aset_I4_J,axiom,
% 8.58/4.88 ! [D3: int,T: int,A3: set_int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ D3 )
% 8.58/4.88 => ( ( member_int @ T @ A3 )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ A3 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( X5 != T )
% 8.58/4.88 => ( ( plus_plus_int @ X5 @ D3 )
% 8.58/4.88 != T ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8868_aset_I5_J,axiom,
% 8.58/4.88 ! [D3: int,T: int,A3: set_int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ D3 )
% 8.58/4.88 => ( ( member_int @ T @ A3 )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ A3 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( ord_less_int @ X5 @ T )
% 8.58/4.88 => ( ord_less_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8869_aset_I7_J,axiom,
% 8.58/4.88 ! [D3: int,A3: set_int,T: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ D3 )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ A3 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( ord_less_int @ T @ X5 )
% 8.58/4.88 => ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8870_simp__from__to,axiom,
% 8.58/4.88 ( set_or1266510415728281911st_int
% 8.58/4.88 = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8871_bset_I6_J,axiom,
% 8.58/4.88 ! [D3: int,B5: set_int,T: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ D3 )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ B5 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( ord_less_eq_int @ X5 @ T )
% 8.58/4.88 => ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8872_bset_I8_J,axiom,
% 8.58/4.88 ! [D3: int,T: int,B5: set_int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ D3 )
% 8.58/4.88 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ B5 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( ord_less_eq_int @ T @ X5 )
% 8.58/4.88 => ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8873_aset_I6_J,axiom,
% 8.58/4.88 ! [D3: int,T: int,A3: set_int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ D3 )
% 8.58/4.88 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ A3 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( ord_less_eq_int @ X5 @ T )
% 8.58/4.88 => ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8874_aset_I8_J,axiom,
% 8.58/4.88 ! [D3: int,A3: set_int,T: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ D3 )
% 8.58/4.88 => ! [X5: int] :
% 8.58/4.88 ( ! [Xa3: int] :
% 8.58/4.88 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 8.58/4.88 => ! [Xb3: int] :
% 8.58/4.88 ( ( member_int @ Xb3 @ A3 )
% 8.58/4.88 => ( X5
% 8.58/4.88 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.58/4.88 => ( ( ord_less_eq_int @ T @ X5 )
% 8.58/4.88 => ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8875_card__atLeastAtMost__int,axiom,
% 8.58/4.88 ! [L: int,U: int] :
% 8.58/4.88 ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
% 8.58/4.88 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8876_dist__real__def,axiom,
% 8.58/4.88 ( real_V975177566351809787t_real
% 8.58/4.88 = ( ^ [X3: real,Y6: real] : ( abs_abs_real @ ( minus_minus_real @ X3 @ Y6 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8877_dist__complex__def,axiom,
% 8.58/4.88 ( real_V3694042436643373181omplex
% 8.58/4.88 = ( ^ [X3: complex,Y6: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y6 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8878_normalize__negative,axiom,
% 8.58/4.88 ! [Q2: int,P2: int] :
% 8.58/4.88 ( ( ord_less_int @ Q2 @ zero_zero_int )
% 8.58/4.88 => ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
% 8.58/4.88 = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P2 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8879_card__greaterThanLessThan__int,axiom,
% 8.58/4.88 ! [L: int,U: int] :
% 8.58/4.88 ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
% 8.58/4.88 = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8880_normalize__denom__zero,axiom,
% 8.58/4.88 ! [P2: int] :
% 8.58/4.88 ( ( normalize @ ( product_Pair_int_int @ P2 @ zero_zero_int ) )
% 8.58/4.88 = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8881_quotient__of__number_I5_J,axiom,
% 8.58/4.88 ! [K: num] :
% 8.58/4.88 ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 8.58/4.88 = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8882_rat__one__code,axiom,
% 8.58/4.88 ( ( quotient_of @ one_one_rat )
% 8.58/4.88 = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8883_rat__zero__code,axiom,
% 8.58/4.88 ( ( quotient_of @ zero_zero_rat )
% 8.58/4.88 = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8884_quotient__of__number_I4_J,axiom,
% 8.58/4.88 ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.58/4.88 = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8885_quotient__of__number_I3_J,axiom,
% 8.58/4.88 ! [K: num] :
% 8.58/4.88 ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 8.58/4.88 = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8886_diff__rat__def,axiom,
% 8.58/4.88 ( minus_minus_rat
% 8.58/4.88 = ( ^ [Q5: rat,R5: rat] : ( plus_plus_rat @ Q5 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8887_sgn__rat__def,axiom,
% 8.58/4.88 ( sgn_sgn_rat
% 8.58/4.88 = ( ^ [A2: rat] : ( if_rat @ ( A2 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A2 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8888_quotient__of__denom__pos,axiom,
% 8.58/4.88 ! [R: rat,P2: int,Q2: int] :
% 8.58/4.88 ( ( ( quotient_of @ R )
% 8.58/4.88 = ( product_Pair_int_int @ P2 @ Q2 ) )
% 8.58/4.88 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8889_quotient__of__denom__pos_H,axiom,
% 8.58/4.88 ! [R: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8890_normalize__denom__pos,axiom,
% 8.58/4.88 ! [R: product_prod_int_int,P2: int,Q2: int] :
% 8.58/4.88 ( ( ( normalize @ R )
% 8.58/4.88 = ( product_Pair_int_int @ P2 @ Q2 ) )
% 8.58/4.88 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8891_normalize__crossproduct,axiom,
% 8.58/4.88 ! [Q2: int,S: int,P2: int,R: int] :
% 8.58/4.88 ( ( Q2 != zero_zero_int )
% 8.58/4.88 => ( ( S != zero_zero_int )
% 8.58/4.88 => ( ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
% 8.58/4.88 = ( normalize @ ( product_Pair_int_int @ R @ S ) ) )
% 8.58/4.88 => ( ( times_times_int @ P2 @ S )
% 8.58/4.88 = ( times_times_int @ R @ Q2 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8892_rat__sgn__code,axiom,
% 8.58/4.88 ! [P2: rat] :
% 8.58/4.88 ( ( quotient_of @ ( sgn_sgn_rat @ P2 ) )
% 8.58/4.88 = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P2 ) ) ) @ one_one_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8893_quotient__of__int,axiom,
% 8.58/4.88 ! [A: int] :
% 8.58/4.88 ( ( quotient_of @ ( of_int @ A ) )
% 8.58/4.88 = ( product_Pair_int_int @ A @ one_one_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8894_Frct__code__post_I5_J,axiom,
% 8.58/4.88 ! [K: num] :
% 8.58/4.88 ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 8.58/4.88 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8895_Frct__code__post_I4_J,axiom,
% 8.58/4.88 ! [K: num] :
% 8.58/4.88 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 8.58/4.88 = ( numeral_numeral_rat @ K ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8896_Frct__code__post_I6_J,axiom,
% 8.58/4.88 ! [K: num,L: num] :
% 8.58/4.88 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
% 8.58/4.88 = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8897_card__atLeastAtMost,axiom,
% 8.58/4.88 ! [L: nat,U: nat] :
% 8.58/4.88 ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 8.58/4.88 = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8898_card__greaterThanLessThan,axiom,
% 8.58/4.88 ! [L: nat,U: nat] :
% 8.58/4.88 ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
% 8.58/4.88 = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8899_divide__rat__def,axiom,
% 8.58/4.88 ( divide_divide_rat
% 8.58/4.88 = ( ^ [Q5: rat,R5: rat] : ( times_times_rat @ Q5 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8900_obtain__pos__sum,axiom,
% 8.58/4.88 ! [R: rat] :
% 8.58/4.88 ( ( ord_less_rat @ zero_zero_rat @ R )
% 8.58/4.88 => ~ ! [S3: rat] :
% 8.58/4.88 ( ( ord_less_rat @ zero_zero_rat @ S3 )
% 8.58/4.88 => ! [T3: rat] :
% 8.58/4.88 ( ( ord_less_rat @ zero_zero_rat @ T3 )
% 8.58/4.88 => ( R
% 8.58/4.88 != ( plus_plus_rat @ S3 @ T3 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8901_all__nat__less,axiom,
% 8.58/4.88 ! [N: nat,P: nat > $o] :
% 8.58/4.88 ( ( ! [M3: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ M3 @ N )
% 8.58/4.88 => ( P @ M3 ) ) )
% 8.58/4.88 = ( ! [X3: nat] :
% 8.58/4.88 ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 8.58/4.88 => ( P @ X3 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8902_ex__nat__less,axiom,
% 8.58/4.88 ! [N: nat,P: nat > $o] :
% 8.58/4.88 ( ( ? [M3: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ M3 @ N )
% 8.58/4.88 & ( P @ M3 ) ) )
% 8.58/4.88 = ( ? [X3: nat] :
% 8.58/4.88 ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 8.58/4.88 & ( P @ X3 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8903_atLeast0__atMost__Suc,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 8.58/4.88 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8904_Icc__eq__insert__lb__nat,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.88 => ( ( set_or1269000886237332187st_nat @ M @ N )
% 8.58/4.88 = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8905_atLeastAtMostSuc__conv,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 8.58/4.88 => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 8.58/4.88 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8906_atLeastAtMost__insertL,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.88 => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 8.58/4.88 = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8907_tanh__real__bounds,axiom,
% 8.58/4.88 ! [X: real] : ( member_real @ ( tanh_real @ X ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8908_Frct__code__post_I2_J,axiom,
% 8.58/4.88 ! [A: int] :
% 8.58/4.88 ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
% 8.58/4.88 = zero_zero_rat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8909_Frct__code__post_I1_J,axiom,
% 8.58/4.88 ! [A: int] :
% 8.58/4.88 ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
% 8.58/4.88 = zero_zero_rat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8910_Frct__code__post_I3_J,axiom,
% 8.58/4.88 ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
% 8.58/4.88 = one_one_rat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8911_both__member__options__def,axiom,
% 8.58/4.88 ( vEBT_V8194947554948674370ptions
% 8.58/4.88 = ( ^ [T2: vEBT_VEBT,X3: nat] :
% 8.58/4.88 ( ( vEBT_V5719532721284313246member @ T2 @ X3 )
% 8.58/4.88 | ( vEBT_VEBT_membermima @ T2 @ X3 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8912_image__Suc__atLeastAtMost,axiom,
% 8.58/4.88 ! [I: nat,J: nat] :
% 8.58/4.88 ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 8.58/4.88 = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8913_Gcd__abs__eq,axiom,
% 8.58/4.88 ! [K5: set_int] :
% 8.58/4.88 ( ( gcd_Gcd_int @ ( image_int_int @ abs_abs_int @ K5 ) )
% 8.58/4.88 = ( gcd_Gcd_int @ K5 ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8914_card__Collect__le__nat,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( finite_card_nat
% 8.58/4.88 @ ( collect_nat
% 8.58/4.88 @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
% 8.58/4.88 = ( suc @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8915_Gcd__int__eq,axiom,
% 8.58/4.88 ! [N5: set_nat] :
% 8.58/4.88 ( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N5 ) )
% 8.58/4.88 = ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N5 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8916_Gcd__nat__abs__eq,axiom,
% 8.58/4.88 ! [K5: set_int] :
% 8.58/4.88 ( ( gcd_Gcd_nat
% 8.58/4.88 @ ( image_int_nat
% 8.58/4.88 @ ^ [K2: int] : ( nat2 @ ( abs_abs_int @ K2 ) )
% 8.58/4.88 @ K5 ) )
% 8.58/4.88 = ( nat2 @ ( gcd_Gcd_int @ K5 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8917_nat__less__as__int,axiom,
% 8.58/4.88 ( ord_less_nat
% 8.58/4.88 = ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8918_card__less,axiom,
% 8.58/4.88 ! [M10: set_nat,I: nat] :
% 8.58/4.88 ( ( member_nat @ zero_zero_nat @ M10 )
% 8.58/4.88 => ( ( finite_card_nat
% 8.58/4.88 @ ( collect_nat
% 8.58/4.88 @ ^ [K2: nat] :
% 8.58/4.88 ( ( member_nat @ K2 @ M10 )
% 8.58/4.88 & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
% 8.58/4.88 != zero_zero_nat ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8919_card__less__Suc,axiom,
% 8.58/4.88 ! [M10: set_nat,I: nat] :
% 8.58/4.88 ( ( member_nat @ zero_zero_nat @ M10 )
% 8.58/4.88 => ( ( suc
% 8.58/4.88 @ ( finite_card_nat
% 8.58/4.88 @ ( collect_nat
% 8.58/4.88 @ ^ [K2: nat] :
% 8.58/4.88 ( ( member_nat @ ( suc @ K2 ) @ M10 )
% 8.58/4.88 & ( ord_less_nat @ K2 @ I ) ) ) ) )
% 8.58/4.88 = ( finite_card_nat
% 8.58/4.88 @ ( collect_nat
% 8.58/4.88 @ ^ [K2: nat] :
% 8.58/4.88 ( ( member_nat @ K2 @ M10 )
% 8.58/4.88 & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8920_card__less__Suc2,axiom,
% 8.58/4.88 ! [M10: set_nat,I: nat] :
% 8.58/4.88 ( ~ ( member_nat @ zero_zero_nat @ M10 )
% 8.58/4.88 => ( ( finite_card_nat
% 8.58/4.88 @ ( collect_nat
% 8.58/4.88 @ ^ [K2: nat] :
% 8.58/4.88 ( ( member_nat @ ( suc @ K2 ) @ M10 )
% 8.58/4.88 & ( ord_less_nat @ K2 @ I ) ) ) )
% 8.58/4.88 = ( finite_card_nat
% 8.58/4.88 @ ( collect_nat
% 8.58/4.88 @ ^ [K2: nat] :
% 8.58/4.88 ( ( member_nat @ K2 @ M10 )
% 8.58/4.88 & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8921_nat__leq__as__int,axiom,
% 8.58/4.88 ( ord_less_eq_nat
% 8.58/4.88 = ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8922_nat__plus__as__int,axiom,
% 8.58/4.88 ( plus_plus_nat
% 8.58/4.88 = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8923_nat__times__as__int,axiom,
% 8.58/4.88 ( times_times_nat
% 8.58/4.88 = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8924_nat__minus__as__int,axiom,
% 8.58/4.88 ( minus_minus_nat
% 8.58/4.88 = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8925_nat__div__as__int,axiom,
% 8.58/4.88 ( divide_divide_nat
% 8.58/4.88 = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8926_nat__mod__as__int,axiom,
% 8.58/4.88 ( modulo_modulo_nat
% 8.58/4.88 = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8927_card__nth__roots,axiom,
% 8.58/4.88 ! [C: complex,N: nat] :
% 8.58/4.88 ( ( C != zero_zero_complex )
% 8.58/4.88 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.88 => ( ( finite_card_complex
% 8.58/4.88 @ ( collect_complex
% 8.58/4.88 @ ^ [Z5: complex] :
% 8.58/4.88 ( ( power_power_complex @ Z5 @ N )
% 8.58/4.88 = C ) ) )
% 8.58/4.88 = N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8928_card__roots__unity__eq,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.88 => ( ( finite_card_complex
% 8.58/4.88 @ ( collect_complex
% 8.58/4.88 @ ^ [Z5: complex] :
% 8.58/4.88 ( ( power_power_complex @ Z5 @ N )
% 8.58/4.88 = one_one_complex ) ) )
% 8.58/4.88 = N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8929_zero__notin__Suc__image,axiom,
% 8.58/4.88 ! [A3: set_nat] :
% 8.58/4.88 ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A3 ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8930_image__int__atLeastAtMost,axiom,
% 8.58/4.88 ! [A: nat,B: nat] :
% 8.58/4.88 ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 8.58/4.88 = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8931_diff__nat__eq__if,axiom,
% 8.58/4.88 ! [Z6: int,Z2: int] :
% 8.58/4.88 ( ( ( ord_less_int @ Z6 @ zero_zero_int )
% 8.58/4.88 => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) )
% 8.58/4.88 = ( nat2 @ Z2 ) ) )
% 8.58/4.88 & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
% 8.58/4.88 => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) )
% 8.58/4.88 = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z2 @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z2 @ Z6 ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8932_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 8.58/4.88 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8933_set__decode__def,axiom,
% 8.58/4.88 ( nat_set_decode
% 8.58/4.88 = ( ^ [X3: nat] :
% 8.58/4.88 ( collect_nat
% 8.58/4.88 @ ^ [N3: nat] :
% 8.58/4.88 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8934_normalize__def,axiom,
% 8.58/4.88 ( normalize
% 8.58/4.88 = ( ^ [P3: product_prod_int_int] :
% 8.58/4.88 ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P3 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P3 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P3 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) )
% 8.58/4.88 @ ( if_Pro3027730157355071871nt_int
% 8.58/4.88 @ ( ( product_snd_int_int @ P3 )
% 8.58/4.88 = zero_zero_int )
% 8.58/4.88 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 8.58/4.88 @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P3 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P3 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8935_in__children__def,axiom,
% 8.58/4.88 ( vEBT_V5917875025757280293ildren
% 8.58/4.88 = ( ^ [N3: nat,TreeList: list_VEBT_VEBT,X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ N3 ) ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8936_monoseq__arctan__series,axiom,
% 8.58/4.88 ! [X: real] :
% 8.58/4.88 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.88 => ( topolo6980174941875973593q_real
% 8.58/4.88 @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8937_fact__eq__fact__times,axiom,
% 8.58/4.88 ! [N: nat,M: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.88 => ( ( semiri1408675320244567234ct_nat @ M )
% 8.58/4.88 = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 8.58/4.88 @ ( groups708209901874060359at_nat
% 8.58/4.88 @ ^ [X3: nat] : X3
% 8.58/4.88 @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8938_monoseq__realpow,axiom,
% 8.58/4.88 ! [X: real] :
% 8.58/4.88 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.88 => ( ( ord_less_eq_real @ X @ one_one_real )
% 8.58/4.88 => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8939_fact__div__fact,axiom,
% 8.58/4.88 ! [N: nat,M: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ N @ M )
% 8.58/4.88 => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 8.58/4.88 = ( groups708209901874060359at_nat
% 8.58/4.88 @ ^ [X3: nat] : X3
% 8.58/4.88 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8940_prod__int__eq,axiom,
% 8.58/4.88 ! [I: nat,J: nat] :
% 8.58/4.88 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 8.58/4.88 = ( groups1705073143266064639nt_int
% 8.58/4.88 @ ^ [X3: int] : X3
% 8.58/4.88 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8941_prod__int__plus__eq,axiom,
% 8.58/4.88 ! [I: nat,J: nat] :
% 8.58/4.88 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
% 8.58/4.88 = ( groups1705073143266064639nt_int
% 8.58/4.88 @ ^ [X3: int] : X3
% 8.58/4.88 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8942_ln__series,axiom,
% 8.58/4.88 ! [X: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.88 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.58/4.88 => ( ( ln_ln_real @ X )
% 8.58/4.88 = ( suminf_real
% 8.58/4.88 @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N3 ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8943_arctan__series,axiom,
% 8.58/4.88 ! [X: real] :
% 8.58/4.88 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.88 => ( ( arctan @ X )
% 8.58/4.88 = ( suminf_real
% 8.58/4.88 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8944_bij__betw__nth__root__unity,axiom,
% 8.58/4.88 ! [C: complex,N: nat] :
% 8.58/4.88 ( ( C != zero_zero_complex )
% 8.58/4.88 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.88 => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 8.58/4.88 @ ( collect_complex
% 8.58/4.88 @ ^ [Z5: complex] :
% 8.58/4.88 ( ( power_power_complex @ Z5 @ N )
% 8.58/4.88 = one_one_complex ) )
% 8.58/4.88 @ ( collect_complex
% 8.58/4.88 @ ^ [Z5: complex] :
% 8.58/4.88 ( ( power_power_complex @ Z5 @ N )
% 8.58/4.88 = C ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8945_pi__series,axiom,
% 8.58/4.88 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.58/4.88 = ( suminf_real
% 8.58/4.88 @ ^ [K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8946_bij__betw__Suc,axiom,
% 8.58/4.88 ! [M10: set_nat,N5: set_nat] :
% 8.58/4.88 ( ( bij_betw_nat_nat @ suc @ M10 @ N5 )
% 8.58/4.88 = ( ( image_nat_nat @ suc @ M10 )
% 8.58/4.88 = N5 ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8947_summable__arctan__series,axiom,
% 8.58/4.88 ! [X: real] :
% 8.58/4.88 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.88 => ( summable_real
% 8.58/4.88 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8948_bij__betw__roots__unity,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.88 => ( bij_betw_nat_complex
% 8.58/4.88 @ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N )
% 8.58/4.88 @ ( collect_complex
% 8.58/4.88 @ ^ [Z5: complex] :
% 8.58/4.88 ( ( power_power_complex @ Z5 @ N )
% 8.58/4.88 = one_one_complex ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8949_sin__paired,axiom,
% 8.58/4.88 ! [X: real] :
% 8.58/4.88 ( sums_real
% 8.58/4.88 @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 8.58/4.88 @ ( sin_real @ X ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8950_lessThan__0,axiom,
% 8.58/4.88 ( ( set_ord_lessThan_nat @ zero_zero_nat )
% 8.58/4.88 = bot_bot_set_nat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8951_lessThan__Suc,axiom,
% 8.58/4.88 ! [K: nat] :
% 8.58/4.88 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 8.58/4.88 = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8952_lessThan__empty__iff,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( ( set_ord_lessThan_nat @ N )
% 8.58/4.88 = bot_bot_set_nat )
% 8.58/4.88 = ( N = zero_zero_nat ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8953_lessThan__nat__numeral,axiom,
% 8.58/4.88 ! [K: num] :
% 8.58/4.88 ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 8.58/4.88 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8954_image__Suc__lessThan,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.88 = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8955_lessThan__Suc__eq__insert__0,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( set_ord_lessThan_nat @ ( suc @ N ) )
% 8.58/4.88 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8956_summable__power__series,axiom,
% 8.58/4.88 ! [F: nat > real,Z2: real] :
% 8.58/4.88 ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
% 8.58/4.88 => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 8.58/4.88 => ( ( ord_less_eq_real @ zero_zero_real @ Z2 )
% 8.58/4.88 => ( ( ord_less_real @ Z2 @ one_one_real )
% 8.58/4.88 => ( summable_real
% 8.58/4.88 @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z2 @ I4 ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8957_power__half__series,axiom,
% 8.58/4.88 ( sums_real
% 8.58/4.88 @ ^ [N3: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N3 ) )
% 8.58/4.88 @ one_one_real ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8958_sums__if_H,axiom,
% 8.58/4.88 ! [G: nat > real,X: real] :
% 8.58/4.88 ( ( sums_real @ G @ X )
% 8.58/4.88 => ( sums_real
% 8.58/4.88 @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.88 @ X ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8959_sums__if,axiom,
% 8.58/4.88 ! [G: nat > real,X: real,F: nat > real,Y: real] :
% 8.58/4.88 ( ( sums_real @ G @ X )
% 8.58/4.88 => ( ( sums_real @ F @ Y )
% 8.58/4.88 => ( sums_real
% 8.58/4.88 @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( F @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.88 @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8960_cos__paired,axiom,
% 8.58/4.88 ! [X: real] :
% 8.58/4.88 ( sums_real
% 8.58/4.88 @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 8.58/4.88 @ ( cos_real @ X ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8961_Maclaurin__minus__cos__expansion,axiom,
% 8.58/4.88 ! [N: nat,X: real] :
% 8.58/4.88 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.88 => ( ( ord_less_real @ X @ zero_zero_real )
% 8.58/4.88 => ? [T3: real] :
% 8.58/4.88 ( ( ord_less_real @ X @ T3 )
% 8.58/4.88 & ( ord_less_real @ T3 @ zero_zero_real )
% 8.58/4.88 & ( ( cos_real @ X )
% 8.58/4.88 = ( plus_plus_real
% 8.58/4.88 @ ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.88 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8962_Maclaurin__cos__expansion2,axiom,
% 8.58/4.88 ! [X: real,N: nat] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.88 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.88 => ? [T3: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ T3 )
% 8.58/4.88 & ( ord_less_real @ T3 @ X )
% 8.58/4.88 & ( ( cos_real @ X )
% 8.58/4.88 = ( plus_plus_real
% 8.58/4.88 @ ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.88 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8963_Maclaurin__sin__expansion3,axiom,
% 8.58/4.88 ! [N: nat,X: real] :
% 8.58/4.88 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.88 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.88 => ? [T3: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ T3 )
% 8.58/4.88 & ( ord_less_real @ T3 @ X )
% 8.58/4.88 & ( ( sin_real @ X )
% 8.58/4.88 = ( plus_plus_real
% 8.58/4.88 @ ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.88 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8964_sumr__cos__zero__one,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ zero_zero_real @ M3 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 8.58/4.88 = one_one_real ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8965_Maclaurin__lemma,axiom,
% 8.58/4.88 ! [H: real,F: real > real,J: nat > real,N: nat] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ H )
% 8.58/4.88 => ? [B7: real] :
% 8.58/4.88 ( ( F @ H )
% 8.58/4.88 = ( plus_plus_real
% 8.58/4.88 @ ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H @ M3 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.88 @ ( times_times_real @ B7 @ ( divide_divide_real @ ( power_power_real @ H @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8966_sum__split__even__odd,axiom,
% 8.58/4.88 ! [F: nat > real,G: nat > real,N: nat] :
% 8.58/4.88 ( ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( F @ I4 ) @ ( G @ I4 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.58/4.88 = ( plus_plus_real
% 8.58/4.88 @ ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [I4: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.88 @ ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ one_one_nat ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8967_Maclaurin__exp__le,axiom,
% 8.58/4.88 ! [X: real,N: nat] :
% 8.58/4.88 ? [T3: real] :
% 8.58/4.88 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 8.58/4.88 & ( ( exp_real @ X )
% 8.58/4.88 = ( plus_plus_real
% 8.58/4.88 @ ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.88 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8968_Maclaurin__sin__bound,axiom,
% 8.58/4.88 ! [X: real,N: nat] :
% 8.58/4.88 ( ord_less_eq_real
% 8.58/4.88 @ ( abs_abs_real
% 8.58/4.88 @ ( minus_minus_real @ ( sin_real @ X )
% 8.58/4.88 @ ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N ) ) ) )
% 8.58/4.88 @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8969_sum__pos__lt__pair,axiom,
% 8.58/4.88 ! [F: nat > real,K: nat] :
% 8.58/4.88 ( ( summable_real @ F )
% 8.58/4.88 => ( ! [D2: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D2 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D2 ) @ one_one_nat ) ) ) ) )
% 8.58/4.88 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8970_Maclaurin__exp__lt,axiom,
% 8.58/4.88 ! [X: real,N: nat] :
% 8.58/4.88 ( ( X != zero_zero_real )
% 8.58/4.88 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.88 => ? [T3: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
% 8.58/4.88 & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 8.58/4.88 & ( ( exp_real @ X )
% 8.58/4.88 = ( plus_plus_real
% 8.58/4.88 @ ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.88 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8971_Maclaurin__sin__expansion,axiom,
% 8.58/4.88 ! [X: real,N: nat] :
% 8.58/4.88 ? [T3: real] :
% 8.58/4.88 ( ( sin_real @ X )
% 8.58/4.88 = ( plus_plus_real
% 8.58/4.88 @ ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.88 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8972_Maclaurin__sin__expansion2,axiom,
% 8.58/4.88 ! [X: real,N: nat] :
% 8.58/4.88 ? [T3: real] :
% 8.58/4.88 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 8.58/4.88 & ( ( sin_real @ X )
% 8.58/4.88 = ( plus_plus_real
% 8.58/4.88 @ ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.88 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8973_Maclaurin__cos__expansion,axiom,
% 8.58/4.88 ! [X: real,N: nat] :
% 8.58/4.88 ? [T3: real] :
% 8.58/4.88 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 8.58/4.88 & ( ( cos_real @ X )
% 8.58/4.88 = ( plus_plus_real
% 8.58/4.88 @ ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.88 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8974_Maclaurin__sin__expansion4,axiom,
% 8.58/4.88 ! [X: real,N: nat] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.88 => ? [T3: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ T3 )
% 8.58/4.88 & ( ord_less_eq_real @ T3 @ X )
% 8.58/4.88 & ( ( sin_real @ X )
% 8.58/4.88 = ( plus_plus_real
% 8.58/4.88 @ ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.88 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8975_card__atMost,axiom,
% 8.58/4.88 ! [U: nat] :
% 8.58/4.88 ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
% 8.58/4.88 = ( suc @ U ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8976_atMost__0,axiom,
% 8.58/4.88 ( ( set_ord_atMost_nat @ zero_zero_nat )
% 8.58/4.88 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8977_sum__choose__upper,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [K2: nat] : ( binomial @ K2 @ M )
% 8.58/4.88 @ ( set_ord_atMost_nat @ N ) )
% 8.58/4.88 = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8978_choose__rising__sum_I2_J,axiom,
% 8.58/4.88 ! [N: nat,M: nat] :
% 8.58/4.88 ( ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 8.58/4.88 @ ( set_ord_atMost_nat @ M ) )
% 8.58/4.88 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8979_choose__rising__sum_I1_J,axiom,
% 8.58/4.88 ! [N: nat,M: nat] :
% 8.58/4.88 ( ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 8.58/4.88 @ ( set_ord_atMost_nat @ M ) )
% 8.58/4.88 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8980_sum__choose__lower,axiom,
% 8.58/4.88 ! [R: nat,N: nat] :
% 8.58/4.88 ( ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R @ K2 ) @ K2 )
% 8.58/4.88 @ ( set_ord_atMost_nat @ N ) )
% 8.58/4.88 = ( binomial @ ( suc @ ( plus_plus_nat @ R @ N ) ) @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8981_atMost__atLeast0,axiom,
% 8.58/4.88 ( set_ord_atMost_nat
% 8.58/4.88 = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8982_lessThan__Suc__atMost,axiom,
% 8.58/4.88 ! [K: nat] :
% 8.58/4.88 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 8.58/4.88 = ( set_ord_atMost_nat @ K ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8983_sum__choose__diagonal,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.88 => ( ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M @ K2 ) )
% 8.58/4.88 @ ( set_ord_atMost_nat @ M ) )
% 8.58/4.88 = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8984_vandermonde,axiom,
% 8.58/4.88 ! [M: nat,N: nat,R: nat] :
% 8.58/4.88 ( ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M @ K2 ) @ ( binomial @ N @ ( minus_minus_nat @ R @ K2 ) ) )
% 8.58/4.88 @ ( set_ord_atMost_nat @ R ) )
% 8.58/4.88 = ( binomial @ ( plus_plus_nat @ M @ N ) @ R ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8985_atMost__Suc,axiom,
% 8.58/4.88 ! [K: nat] :
% 8.58/4.88 ( ( set_ord_atMost_nat @ ( suc @ K ) )
% 8.58/4.88 = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8986_choose__row__sum,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
% 8.58/4.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8987_binomial,axiom,
% 8.58/4.88 ! [A: nat,B: nat,N: nat] :
% 8.58/4.88 ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 8.58/4.88 = ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K2 ) ) )
% 8.58/4.88 @ ( set_ord_atMost_nat @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8988_polynomial__product__nat,axiom,
% 8.58/4.88 ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
% 8.58/4.88 ( ! [I2: nat] :
% 8.58/4.88 ( ( ord_less_nat @ M @ I2 )
% 8.58/4.88 => ( ( A @ I2 )
% 8.58/4.88 = zero_zero_nat ) )
% 8.58/4.88 => ( ! [J2: nat] :
% 8.58/4.88 ( ( ord_less_nat @ N @ J2 )
% 8.58/4.88 => ( ( B @ J2 )
% 8.58/4.88 = zero_zero_nat ) )
% 8.58/4.88 => ( ( times_times_nat
% 8.58/4.88 @ ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( power_power_nat @ X @ I4 ) )
% 8.58/4.88 @ ( set_ord_atMost_nat @ M ) )
% 8.58/4.88 @ ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
% 8.58/4.88 @ ( set_ord_atMost_nat @ N ) ) )
% 8.58/4.88 = ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [R5: nat] :
% 8.58/4.88 ( times_times_nat
% 8.58/4.88 @ ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 8.58/4.88 @ ( set_ord_atMost_nat @ R5 ) )
% 8.58/4.88 @ ( power_power_nat @ X @ R5 ) )
% 8.58/4.88 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8989_choose__square__sum,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [K2: nat] : ( power_power_nat @ ( binomial @ N @ K2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.88 @ ( set_ord_atMost_nat @ N ) )
% 8.58/4.88 = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8990_atMost__nat__numeral,axiom,
% 8.58/4.88 ! [K: num] :
% 8.58/4.88 ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 8.58/4.88 = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8991_binomial__r__part__sum,axiom,
% 8.58/4.88 ! [M: nat] :
% 8.58/4.88 ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 8.58/4.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8992_choose__linear__sum,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N @ I4 ) )
% 8.58/4.88 @ ( set_ord_atMost_nat @ N ) )
% 8.58/4.88 = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8993_image__Suc__atMost,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
% 8.58/4.88 = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8994_atMost__Suc__eq__insert__0,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( set_ord_atMost_nat @ ( suc @ N ) )
% 8.58/4.88 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8995_sum__nth__roots,axiom,
% 8.58/4.88 ! [N: nat,C: complex] :
% 8.58/4.88 ( ( ord_less_nat @ one_one_nat @ N )
% 8.58/4.88 => ( ( groups7754918857620584856omplex
% 8.58/4.88 @ ^ [X3: complex] : X3
% 8.58/4.88 @ ( collect_complex
% 8.58/4.88 @ ^ [Z5: complex] :
% 8.58/4.88 ( ( power_power_complex @ Z5 @ N )
% 8.58/4.88 = C ) ) )
% 8.58/4.88 = zero_zero_complex ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8996_atLeast1__atMost__eq__remove0,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.58/4.88 = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8997_sum__roots__unity,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( ord_less_nat @ one_one_nat @ N )
% 8.58/4.88 => ( ( groups7754918857620584856omplex
% 8.58/4.88 @ ^ [X3: complex] : X3
% 8.58/4.88 @ ( collect_complex
% 8.58/4.88 @ ^ [Z5: complex] :
% 8.58/4.88 ( ( power_power_complex @ Z5 @ N )
% 8.58/4.88 = one_one_complex ) ) )
% 8.58/4.88 = zero_zero_complex ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8998_mask__eq__sum__exp__nat,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 8.58/4.88 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.88 @ ( collect_nat
% 8.58/4.88 @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_8999_gauss__sum__nat,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [X3: nat] : X3
% 8.58/4.88 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 8.58/4.88 = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9000_arith__series__nat,axiom,
% 8.58/4.88 ! [A: nat,D: nat,N: nat] :
% 8.58/4.88 ( ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I4 @ D ) )
% 8.58/4.88 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 8.58/4.88 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9001_Sum__Icc__nat,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [X3: nat] : X3
% 8.58/4.88 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.58/4.88 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9002_Sum__Icc__int,axiom,
% 8.58/4.88 ! [M: int,N: int] :
% 8.58/4.88 ( ( ord_less_eq_int @ M @ N )
% 8.58/4.88 => ( ( groups4538972089207619220nt_int
% 8.58/4.88 @ ^ [X3: int] : X3
% 8.58/4.88 @ ( set_or1266510415728281911st_int @ M @ N ) )
% 8.58/4.88 = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9003_set__vebt__def,axiom,
% 8.58/4.88 ( vEBT_set_vebt
% 8.58/4.88 = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9004_rat__times__code,axiom,
% 8.58/4.88 ! [P2: rat,Q2: rat] :
% 8.58/4.88 ( ( quotient_of @ ( times_times_rat @ P2 @ Q2 ) )
% 8.58/4.88 = ( produc4245557441103728435nt_int
% 8.58/4.88 @ ^ [A2: int,C4: int] :
% 8.58/4.88 ( produc4245557441103728435nt_int
% 8.58/4.88 @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ C4 @ D4 ) ) )
% 8.58/4.88 @ ( quotient_of @ Q2 ) )
% 8.58/4.88 @ ( quotient_of @ P2 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9005_rat__divide__code,axiom,
% 8.58/4.88 ! [P2: rat,Q2: rat] :
% 8.58/4.88 ( ( quotient_of @ ( divide_divide_rat @ P2 @ Q2 ) )
% 8.58/4.88 = ( produc4245557441103728435nt_int
% 8.58/4.88 @ ^ [A2: int,C4: int] :
% 8.58/4.88 ( produc4245557441103728435nt_int
% 8.58/4.88 @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A2 @ D4 ) @ ( times_times_int @ C4 @ B2 ) ) )
% 8.58/4.88 @ ( quotient_of @ Q2 ) )
% 8.58/4.88 @ ( quotient_of @ P2 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9006_divmod__nat__if,axiom,
% 8.58/4.88 ( divmod_nat
% 8.58/4.88 = ( ^ [M3: nat,N3: nat] :
% 8.58/4.88 ( if_Pro6206227464963214023at_nat
% 8.58/4.88 @ ( ( N3 = zero_zero_nat )
% 8.58/4.88 | ( ord_less_nat @ M3 @ N3 ) )
% 8.58/4.88 @ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
% 8.58/4.88 @ ( produc2626176000494625587at_nat
% 8.58/4.88 @ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
% 8.58/4.88 @ ( divmod_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9007_rat__plus__code,axiom,
% 8.58/4.88 ! [P2: rat,Q2: rat] :
% 8.58/4.88 ( ( quotient_of @ ( plus_plus_rat @ P2 @ Q2 ) )
% 8.58/4.88 = ( produc4245557441103728435nt_int
% 8.58/4.88 @ ^ [A2: int,C4: int] :
% 8.58/4.88 ( produc4245557441103728435nt_int
% 8.58/4.88 @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A2 @ D4 ) @ ( times_times_int @ B2 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
% 8.58/4.88 @ ( quotient_of @ Q2 ) )
% 8.58/4.88 @ ( quotient_of @ P2 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9008_rat__minus__code,axiom,
% 8.58/4.88 ! [P2: rat,Q2: rat] :
% 8.58/4.88 ( ( quotient_of @ ( minus_minus_rat @ P2 @ Q2 ) )
% 8.58/4.88 = ( produc4245557441103728435nt_int
% 8.58/4.88 @ ^ [A2: int,C4: int] :
% 8.58/4.88 ( produc4245557441103728435nt_int
% 8.58/4.88 @ ^ [B2: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A2 @ D4 ) @ ( times_times_int @ B2 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
% 8.58/4.88 @ ( quotient_of @ Q2 ) )
% 8.58/4.88 @ ( quotient_of @ P2 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9009_divmod__step__nat__def,axiom,
% 8.58/4.88 ( unique5026877609467782581ep_nat
% 8.58/4.88 = ( ^ [L2: num] :
% 8.58/4.88 ( produc2626176000494625587at_nat
% 8.58/4.88 @ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9010_rat__inverse__code,axiom,
% 8.58/4.88 ! [P2: rat] :
% 8.58/4.88 ( ( quotient_of @ ( inverse_inverse_rat @ P2 ) )
% 8.58/4.88 = ( produc4245557441103728435nt_int
% 8.58/4.88 @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( A2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A2 ) @ B2 ) @ ( abs_abs_int @ A2 ) ) )
% 8.58/4.88 @ ( quotient_of @ P2 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9011_divmod__step__int__def,axiom,
% 8.58/4.88 ( unique5024387138958732305ep_int
% 8.58/4.88 = ( ^ [L2: num] :
% 8.58/4.88 ( produc4245557441103728435nt_int
% 8.58/4.88 @ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9012_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 8.58/4.88 ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
% 8.58/4.88 ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X )
% 8.58/4.88 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.58/4.88 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9013_arctan__def,axiom,
% 8.58/4.88 ( arctan
% 8.58/4.88 = ( ^ [Y6: real] :
% 8.58/4.88 ( the_real
% 8.58/4.88 @ ^ [X3: real] :
% 8.58/4.88 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 8.58/4.88 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.88 & ( ( tan_real @ X3 )
% 8.58/4.88 = Y6 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9014_VEBT_Oinject_I1_J,axiom,
% 8.58/4.88 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
% 8.58/4.88 ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 8.58/4.88 = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
% 8.58/4.88 = ( ( X11 = Y11 )
% 8.58/4.88 & ( X12 = Y12 )
% 8.58/4.88 & ( X13 = Y13 )
% 8.58/4.88 & ( X14 = Y14 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9015_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 8.58/4.88 ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 8.58/4.88 ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9016_pi__half,axiom,
% 8.58/4.88 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.58/4.88 = ( the_real
% 8.58/4.88 @ ^ [X3: real] :
% 8.58/4.88 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 8.58/4.88 & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.58/4.88 & ( ( cos_real @ X3 )
% 8.58/4.88 = zero_zero_real ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9017_pi__def,axiom,
% 8.58/4.88 ( pi
% 8.58/4.88 = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 8.58/4.88 @ ( the_real
% 8.58/4.88 @ ^ [X3: real] :
% 8.58/4.88 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 8.58/4.88 & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.58/4.88 & ( ( cos_real @ X3 )
% 8.58/4.88 = zero_zero_real ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9018_arcsin__def,axiom,
% 8.58/4.88 ( arcsin
% 8.58/4.88 = ( ^ [Y6: real] :
% 8.58/4.88 ( the_real
% 8.58/4.88 @ ^ [X3: real] :
% 8.58/4.88 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 8.58/4.88 & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.88 & ( ( sin_real @ X3 )
% 8.58/4.88 = Y6 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9019_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 8.58/4.88 ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 8.58/4.88 ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 8.58/4.88 = Y )
% 8.58/4.88 => ( ! [A4: $o,B4: $o] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 = ( ~ ( ( ( Xa = zero_zero_nat )
% 8.58/4.88 => A4 )
% 8.58/4.88 & ( ( Xa != zero_zero_nat )
% 8.58/4.88 => ( ( ( Xa = one_one_nat )
% 8.58/4.88 => B4 )
% 8.58/4.88 & ( Xa = one_one_nat ) ) ) ) ) ) )
% 8.58/4.88 => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 8.58/4.88 => Y )
% 8.58/4.88 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 8.58/4.88 ( ? [S3: vEBT_VEBT] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9020_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 8.58/4.88 ! [X: vEBT_VEBT,Xa: nat] :
% 8.58/4.88 ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 8.58/4.88 => ( ! [A4: $o,B4: $o] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.58/4.88 => ~ ( ( ( Xa = zero_zero_nat )
% 8.58/4.88 => A4 )
% 8.58/4.88 & ( ( Xa != zero_zero_nat )
% 8.58/4.88 => ( ( ( Xa = one_one_nat )
% 8.58/4.88 => B4 )
% 8.58/4.88 & ( Xa = one_one_nat ) ) ) ) )
% 8.58/4.88 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 8.58/4.88 ( ? [S3: vEBT_VEBT] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 8.58/4.88 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9021_VEBT_Oinject_I2_J,axiom,
% 8.58/4.88 ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
% 8.58/4.88 ( ( ( vEBT_Leaf @ X21 @ X222 )
% 8.58/4.88 = ( vEBT_Leaf @ Y21 @ Y22 ) )
% 8.58/4.88 = ( ( X21 = Y21 )
% 8.58/4.88 & ( X222 = Y22 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9022_VEBT_Osize_I4_J,axiom,
% 8.58/4.88 ! [X21: $o,X222: $o] :
% 8.58/4.88 ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 8.58/4.88 = zero_zero_nat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9023_VEBT_Oexhaust,axiom,
% 8.58/4.88 ! [Y: vEBT_VEBT] :
% 8.58/4.88 ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
% 8.58/4.88 ( Y
% 8.58/4.88 != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
% 8.58/4.88 => ~ ! [X212: $o,X223: $o] :
% 8.58/4.88 ( Y
% 8.58/4.88 != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9024_VEBT_Odistinct_I1_J,axiom,
% 8.58/4.88 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
% 8.58/4.88 ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 8.58/4.88 != ( vEBT_Leaf @ X21 @ X222 ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9025_VEBT__internal_Ovalid_H_Ocases,axiom,
% 8.58/4.88 ! [X: produc9072475918466114483BT_nat] :
% 8.58/4.88 ( ! [Uu2: $o,Uv2: $o,D2: nat] :
% 8.58/4.88 ( X
% 8.58/4.88 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D2 ) )
% 8.58/4.88 => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg2: nat] :
% 8.58/4.88 ( X
% 8.58/4.88 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) @ Deg2 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9026_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
% 8.58/4.88 ! [Uu: $o,Uv: $o,Uw: nat] :
% 8.58/4.88 ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9027_vebt__buildup_Osimps_I1_J,axiom,
% 8.58/4.88 ( ( vEBT_vebt_buildup @ zero_zero_nat )
% 8.58/4.88 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9028_VEBT__internal_Onaive__member_Ocases,axiom,
% 8.58/4.88 ! [X: produc9072475918466114483BT_nat] :
% 8.58/4.88 ( ! [A4: $o,B4: $o,X4: nat] :
% 8.58/4.88 ( X
% 8.58/4.88 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X4 ) )
% 8.58/4.88 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 8.58/4.88 ( X
% 8.58/4.88 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 8.58/4.88 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
% 8.58/4.88 ( X
% 8.58/4.88 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ X4 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9029_vebt__buildup_Osimps_I2_J,axiom,
% 8.58/4.88 ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 8.58/4.88 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9030_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 8.58/4.88 ! [A: $o,B: $o,X: nat] :
% 8.58/4.88 ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
% 8.58/4.88 = ( ( ( X = zero_zero_nat )
% 8.58/4.88 => A )
% 8.58/4.88 & ( ( X != zero_zero_nat )
% 8.58/4.88 => ( ( ( X = one_one_nat )
% 8.58/4.88 => B )
% 8.58/4.88 & ( X = one_one_nat ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9031_prod__encode__def,axiom,
% 8.58/4.88 ( nat_prod_encode
% 8.58/4.88 = ( produc6842872674320459806at_nat
% 8.58/4.88 @ ^ [M3: nat,N3: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M3 @ N3 ) ) @ M3 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9032_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 8.58/4.88 ! [X: vEBT_VEBT,Xa: nat] :
% 8.58/4.88 ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
% 8.58/4.88 => ( ! [A4: $o,B4: $o] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.58/4.88 => ( ( ( Xa = zero_zero_nat )
% 8.58/4.88 => A4 )
% 8.58/4.88 & ( ( Xa != zero_zero_nat )
% 8.58/4.88 => ( ( ( Xa = one_one_nat )
% 8.58/4.88 => B4 )
% 8.58/4.88 & ( Xa = one_one_nat ) ) ) ) )
% 8.58/4.88 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.58/4.88 ( X
% 8.58/4.88 != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 8.58/4.88 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 8.58/4.88 ( ? [S3: vEBT_VEBT] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 8.58/4.88 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9033_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 8.58/4.88 ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
% 8.58/4.88 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X )
% 8.58/4.88 = ( ( X = Mi )
% 8.58/4.88 | ( X = Ma )
% 8.58/4.88 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.58/4.88 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9034_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 8.58/4.88 ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
% 8.58/4.88 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X )
% 8.58/4.88 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.58/4.88 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9035_finite__interval__int1,axiom,
% 8.58/4.88 ! [A: int,B: int] :
% 8.58/4.88 ( finite_finite_int
% 8.58/4.88 @ ( collect_int
% 8.58/4.88 @ ^ [I4: int] :
% 8.58/4.88 ( ( ord_less_eq_int @ A @ I4 )
% 8.58/4.88 & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9036_finite__interval__int4,axiom,
% 8.58/4.88 ! [A: int,B: int] :
% 8.58/4.88 ( finite_finite_int
% 8.58/4.88 @ ( collect_int
% 8.58/4.88 @ ^ [I4: int] :
% 8.58/4.88 ( ( ord_less_int @ A @ I4 )
% 8.58/4.88 & ( ord_less_int @ I4 @ B ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9037_finite__interval__int2,axiom,
% 8.58/4.88 ! [A: int,B: int] :
% 8.58/4.88 ( finite_finite_int
% 8.58/4.88 @ ( collect_int
% 8.58/4.88 @ ^ [I4: int] :
% 8.58/4.88 ( ( ord_less_eq_int @ A @ I4 )
% 8.58/4.88 & ( ord_less_int @ I4 @ B ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9038_finite__interval__int3,axiom,
% 8.58/4.88 ! [A: int,B: int] :
% 8.58/4.88 ( finite_finite_int
% 8.58/4.88 @ ( collect_int
% 8.58/4.88 @ ^ [I4: int] :
% 8.58/4.88 ( ( ord_less_int @ A @ I4 )
% 8.58/4.88 & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9039_finite__nth__roots,axiom,
% 8.58/4.88 ! [N: nat,C: complex] :
% 8.58/4.88 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.88 => ( finite3207457112153483333omplex
% 8.58/4.88 @ ( collect_complex
% 8.58/4.88 @ ^ [Z5: complex] :
% 8.58/4.88 ( ( power_power_complex @ Z5 @ N )
% 8.58/4.88 = C ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9040_finite__set__decode,axiom,
% 8.58/4.88 ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9041_VEBT__internal_Omembermima_Ocases,axiom,
% 8.58/4.88 ! [X: produc9072475918466114483BT_nat] :
% 8.58/4.88 ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 8.58/4.88 ( X
% 8.58/4.88 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 8.58/4.88 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz: nat] :
% 8.58/4.88 ( X
% 8.58/4.88 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz ) )
% 8.58/4.88 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X4: nat] :
% 8.58/4.88 ( X
% 8.58/4.88 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) @ X4 ) )
% 8.58/4.88 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
% 8.58/4.88 ( X
% 8.58/4.88 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ X4 ) )
% 8.58/4.88 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] :
% 8.58/4.88 ( X
% 8.58/4.88 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ X4 ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9042_finite__divisors__int,axiom,
% 8.58/4.88 ! [I: int] :
% 8.58/4.88 ( ( I != zero_zero_int )
% 8.58/4.88 => ( finite_finite_int
% 8.58/4.88 @ ( collect_int
% 8.58/4.88 @ ^ [D4: int] : ( dvd_dvd_int @ D4 @ I ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9043_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 8.58/4.88 ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz2: nat] :
% 8.58/4.88 ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz2 ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9044_finite__divisors__nat,axiom,
% 8.58/4.88 ! [M: nat] :
% 8.58/4.88 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.58/4.88 => ( finite_finite_nat
% 8.58/4.88 @ ( collect_nat
% 8.58/4.88 @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9045_subset__eq__atLeast0__atMost__finite,axiom,
% 8.58/4.88 ! [N5: set_nat,N: nat] :
% 8.58/4.88 ( ( ord_less_eq_set_nat @ N5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 8.58/4.88 => ( finite_finite_nat @ N5 ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9046_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 8.58/4.88 ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X: nat] :
% 8.58/4.88 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X )
% 8.58/4.88 = ( ( X = Mi )
% 8.58/4.88 | ( X = Ma ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9047_Gcd__remove0__nat,axiom,
% 8.58/4.88 ! [M10: set_nat] :
% 8.58/4.88 ( ( finite_finite_nat @ M10 )
% 8.58/4.88 => ( ( gcd_Gcd_nat @ M10 )
% 8.58/4.88 = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M10 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9048_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 8.58/4.88 ! [X: vEBT_VEBT,Xa: nat] :
% 8.58/4.88 ( ( vEBT_VEBT_membermima @ X @ Xa )
% 8.58/4.88 => ( ! [Mi2: nat,Ma2: nat] :
% 8.58/4.88 ( ? [Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
% 8.58/4.88 => ~ ( ( Xa = Mi2 )
% 8.58/4.88 | ( Xa = Ma2 ) ) )
% 8.58/4.88 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 8.58/4.88 ( ? [Vc2: vEBT_VEBT] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 8.58/4.88 => ~ ( ( Xa = Mi2 )
% 8.58/4.88 | ( Xa = Ma2 )
% 8.58/4.88 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 8.58/4.88 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 8.58/4.88 ( ? [Vd2: vEBT_VEBT] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 8.58/4.88 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9049_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 8.58/4.88 ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 8.58/4.88 ( ( ( vEBT_VEBT_membermima @ X @ Xa )
% 8.58/4.88 = Y )
% 8.58/4.88 => ( ( ? [Uu2: $o,Uv2: $o] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.58/4.88 => Y )
% 8.58/4.88 => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 8.58/4.88 => Y )
% 8.58/4.88 => ( ! [Mi2: nat,Ma2: nat] :
% 8.58/4.88 ( ? [Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 = ( ~ ( ( Xa = Mi2 )
% 8.58/4.88 | ( Xa = Ma2 ) ) ) ) )
% 8.58/4.88 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 8.58/4.88 ( ? [Vc2: vEBT_VEBT] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 = ( ~ ( ( Xa = Mi2 )
% 8.58/4.88 | ( Xa = Ma2 )
% 8.58/4.88 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
% 8.58/4.88 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 8.58/4.88 ( ? [Vd2: vEBT_VEBT] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9050_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 8.58/4.88 ! [X: vEBT_VEBT,Xa: nat] :
% 8.58/4.88 ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
% 8.58/4.88 => ( ! [Uu2: $o,Uv2: $o] :
% 8.58/4.88 ( X
% 8.58/4.88 != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.58/4.88 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 8.58/4.88 ( X
% 8.58/4.88 != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 8.58/4.88 => ( ! [Mi2: nat,Ma2: nat] :
% 8.58/4.88 ( ? [Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
% 8.58/4.88 => ( ( Xa = Mi2 )
% 8.58/4.88 | ( Xa = Ma2 ) ) )
% 8.58/4.88 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT] :
% 8.58/4.88 ( ? [Vc2: vEBT_VEBT] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 8.58/4.88 => ( ( Xa = Mi2 )
% 8.58/4.88 | ( Xa = Ma2 )
% 8.58/4.88 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 8.58/4.88 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT] :
% 8.58/4.88 ( ? [Vd2: vEBT_VEBT] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 8.58/4.88 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9051_invar__vebt_Osimps,axiom,
% 8.58/4.88 ( vEBT_invar_vebt
% 8.58/4.88 = ( ^ [A12: vEBT_VEBT,A23: nat] :
% 8.58/4.88 ( ( ? [A2: $o,B2: $o] :
% 8.58/4.88 ( A12
% 8.58/4.88 = ( vEBT_Leaf @ A2 @ B2 ) )
% 8.58/4.88 & ( A23
% 8.58/4.88 = ( suc @ zero_zero_nat ) ) )
% 8.58/4.88 | ? [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT] :
% 8.58/4.88 ( ( A12
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList @ Summary2 ) )
% 8.58/4.88 & ! [X3: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.58/4.88 => ( vEBT_invar_vebt @ X3 @ N3 ) )
% 8.58/4.88 & ( vEBT_invar_vebt @ Summary2 @ N3 )
% 8.58/4.88 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 8.58/4.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 8.58/4.88 & ( A23
% 8.58/4.88 = ( plus_plus_nat @ N3 @ N3 ) )
% 8.58/4.88 & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 8.58/4.88 & ! [X3: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.58/4.88 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.88 | ? [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT] :
% 8.58/4.88 ( ( A12
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList @ Summary2 ) )
% 8.58/4.88 & ! [X3: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.58/4.88 => ( vEBT_invar_vebt @ X3 @ N3 ) )
% 8.58/4.88 & ( vEBT_invar_vebt @ Summary2 @ ( suc @ N3 ) )
% 8.58/4.88 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 8.58/4.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 8.58/4.88 & ( A23
% 8.58/4.88 = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
% 8.58/4.88 & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 8.58/4.88 & ! [X3: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.58/4.88 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.88 | ? [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 8.58/4.88 ( ( A12
% 8.58/4.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary2 ) )
% 8.58/4.88 & ! [X3: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.58/4.88 => ( vEBT_invar_vebt @ X3 @ N3 ) )
% 8.58/4.88 & ( vEBT_invar_vebt @ Summary2 @ N3 )
% 8.58/4.88 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 8.58/4.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 8.58/4.88 & ( A23
% 8.58/4.88 = ( plus_plus_nat @ N3 @ N3 ) )
% 8.58/4.88 & ! [I4: nat] :
% 8.58/4.88 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 8.58/4.88 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
% 8.58/4.88 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 8.58/4.88 & ( ( Mi3 = Ma3 )
% 8.58/4.88 => ! [X3: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.58/4.88 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.88 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.58/4.88 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 8.58/4.88 & ( ( Mi3 != Ma3 )
% 8.58/4.88 => ! [I4: nat] :
% 8.58/4.88 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 8.58/4.88 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
% 8.58/4.88 = I4 )
% 8.58/4.88 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
% 8.58/4.88 & ! [X3: nat] :
% 8.58/4.88 ( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
% 8.58/4.88 = I4 )
% 8.58/4.88 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
% 8.58/4.88 => ( ( ord_less_nat @ Mi3 @ X3 )
% 8.58/4.88 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) )
% 8.58/4.88 | ? [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 8.58/4.88 ( ( A12
% 8.58/4.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary2 ) )
% 8.58/4.88 & ! [X3: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.58/4.88 => ( vEBT_invar_vebt @ X3 @ N3 ) )
% 8.58/4.88 & ( vEBT_invar_vebt @ Summary2 @ ( suc @ N3 ) )
% 8.58/4.88 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 8.58/4.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 8.58/4.88 & ( A23
% 8.58/4.88 = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
% 8.58/4.88 & ! [I4: nat] :
% 8.58/4.88 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 8.58/4.88 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
% 8.58/4.88 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 8.58/4.88 & ( ( Mi3 = Ma3 )
% 8.58/4.88 => ! [X3: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.58/4.88 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.88 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.58/4.88 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 8.58/4.88 & ( ( Mi3 != Ma3 )
% 8.58/4.88 => ! [I4: nat] :
% 8.58/4.88 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 8.58/4.88 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
% 8.58/4.88 = I4 )
% 8.58/4.88 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
% 8.58/4.88 & ! [X3: nat] :
% 8.58/4.88 ( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
% 8.58/4.88 = I4 )
% 8.58/4.88 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
% 8.58/4.88 => ( ( ord_less_nat @ Mi3 @ X3 )
% 8.58/4.88 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9052_invar__vebt_Ocases,axiom,
% 8.58/4.88 ! [A1: vEBT_VEBT,A22: nat] :
% 8.58/4.88 ( ( vEBT_invar_vebt @ A1 @ A22 )
% 8.58/4.88 => ( ( ? [A4: $o,B4: $o] :
% 8.58/4.88 ( A1
% 8.58/4.88 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.58/4.88 => ( A22
% 8.58/4.88 != ( suc @ zero_zero_nat ) ) )
% 8.58/4.88 => ( ! [TreeList3: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
% 8.58/4.88 ( ( A1
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) )
% 8.58/4.88 => ( ( A22 = Deg )
% 8.58/4.88 => ( ! [X5: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_invar_vebt @ X5 @ N2 ) )
% 8.58/4.88 => ( ( vEBT_invar_vebt @ Summary @ M2 )
% 8.58/4.88 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 8.58/4.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 8.58/4.88 => ( ( M2 = N2 )
% 8.58/4.88 => ( ( Deg
% 8.58/4.88 = ( plus_plus_nat @ N2 @ M2 ) )
% 8.58/4.88 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 8.58/4.88 => ~ ! [X5: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.88 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) ) ) ) ) ) ) )
% 8.58/4.88 => ( ! [TreeList3: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
% 8.58/4.88 ( ( A1
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) )
% 8.58/4.88 => ( ( A22 = Deg )
% 8.58/4.88 => ( ! [X5: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_invar_vebt @ X5 @ N2 ) )
% 8.58/4.88 => ( ( vEBT_invar_vebt @ Summary @ M2 )
% 8.58/4.88 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 8.58/4.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 8.58/4.88 => ( ( M2
% 8.58/4.88 = ( suc @ N2 ) )
% 8.58/4.88 => ( ( Deg
% 8.58/4.88 = ( plus_plus_nat @ N2 @ M2 ) )
% 8.58/4.88 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 8.58/4.88 => ~ ! [X5: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.88 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) ) ) ) ) ) ) )
% 8.58/4.88 => ( ! [TreeList3: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi2: nat,Ma2: nat] :
% 8.58/4.88 ( ( A1
% 8.58/4.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList3 @ Summary ) )
% 8.58/4.88 => ( ( A22 = Deg )
% 8.58/4.88 => ( ! [X5: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_invar_vebt @ X5 @ N2 ) )
% 8.58/4.88 => ( ( vEBT_invar_vebt @ Summary @ M2 )
% 8.58/4.88 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 8.58/4.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 8.58/4.88 => ( ( M2 = N2 )
% 8.58/4.88 => ( ( Deg
% 8.58/4.88 = ( plus_plus_nat @ N2 @ M2 ) )
% 8.58/4.88 => ( ! [I3: nat] :
% 8.58/4.88 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 8.58/4.88 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X8 ) )
% 8.58/4.88 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 8.58/4.88 => ( ( ( Mi2 = Ma2 )
% 8.58/4.88 => ! [X5: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.88 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
% 8.58/4.88 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 8.58/4.88 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.58/4.88 => ~ ( ( Mi2 != Ma2 )
% 8.58/4.88 => ! [I3: nat] :
% 8.58/4.88 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 8.58/4.88 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 8.58/4.88 = I3 )
% 8.58/4.88 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 8.58/4.88 & ! [X5: nat] :
% 8.58/4.88 ( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
% 8.58/4.88 = I3 )
% 8.58/4.88 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
% 8.58/4.88 => ( ( ord_less_nat @ Mi2 @ X5 )
% 8.58/4.88 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 8.58/4.88 => ~ ! [TreeList3: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi2: nat,Ma2: nat] :
% 8.58/4.88 ( ( A1
% 8.58/4.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList3 @ Summary ) )
% 8.58/4.88 => ( ( A22 = Deg )
% 8.58/4.88 => ( ! [X5: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_invar_vebt @ X5 @ N2 ) )
% 8.58/4.88 => ( ( vEBT_invar_vebt @ Summary @ M2 )
% 8.58/4.88 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 8.58/4.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 8.58/4.88 => ( ( M2
% 8.58/4.88 = ( suc @ N2 ) )
% 8.58/4.88 => ( ( Deg
% 8.58/4.88 = ( plus_plus_nat @ N2 @ M2 ) )
% 8.58/4.88 => ( ! [I3: nat] :
% 8.58/4.88 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 8.58/4.88 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X8 ) )
% 8.58/4.88 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 8.58/4.88 => ( ( ( Mi2 = Ma2 )
% 8.58/4.88 => ! [X5: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.88 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
% 8.58/4.88 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 8.58/4.88 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.58/4.88 => ~ ( ( Mi2 != Ma2 )
% 8.58/4.88 => ! [I3: nat] :
% 8.58/4.88 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 8.58/4.88 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 8.58/4.88 = I3 )
% 8.58/4.88 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 8.58/4.88 & ! [X5: nat] :
% 8.58/4.88 ( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
% 8.58/4.88 = I3 )
% 8.58/4.88 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
% 8.58/4.88 => ( ( ord_less_nat @ Mi2 @ X5 )
% 8.58/4.88 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9053_invar__vebt_Ointros_I1_J,axiom,
% 8.58/4.88 ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9054_invar__vebt_Ointros_I2_J,axiom,
% 8.58/4.88 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,M: nat,Deg3: nat] :
% 8.58/4.88 ( ! [X4: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.58/4.88 => ( vEBT_invar_vebt @ X4 @ N ) )
% 8.58/4.88 => ( ( vEBT_invar_vebt @ Summary3 @ M )
% 8.58/4.88 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 8.58/4.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.88 => ( ( M = N )
% 8.58/4.88 => ( ( Deg3
% 8.58/4.88 = ( plus_plus_nat @ N @ M ) )
% 8.58/4.88 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_12 )
% 8.58/4.88 => ( ! [X4: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.58/4.88 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) )
% 8.58/4.88 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg3 @ TreeList2 @ Summary3 ) @ Deg3 ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9055_invar__vebt_Ointros_I3_J,axiom,
% 8.58/4.88 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,M: nat,Deg3: nat] :
% 8.58/4.88 ( ! [X4: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.58/4.88 => ( vEBT_invar_vebt @ X4 @ N ) )
% 8.58/4.88 => ( ( vEBT_invar_vebt @ Summary3 @ M )
% 8.58/4.88 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 8.58/4.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.88 => ( ( M
% 8.58/4.88 = ( suc @ N ) )
% 8.58/4.88 => ( ( Deg3
% 8.58/4.88 = ( plus_plus_nat @ N @ M ) )
% 8.58/4.88 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_12 )
% 8.58/4.88 => ( ! [X4: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.58/4.88 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) )
% 8.58/4.88 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg3 @ TreeList2 @ Summary3 ) @ Deg3 ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9056_invar__vebt_Ointros_I4_J,axiom,
% 8.58/4.88 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,M: nat,Deg3: nat,Mi: nat,Ma: nat] :
% 8.58/4.88 ( ! [X4: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.58/4.88 => ( vEBT_invar_vebt @ X4 @ N ) )
% 8.58/4.88 => ( ( vEBT_invar_vebt @ Summary3 @ M )
% 8.58/4.88 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 8.58/4.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.88 => ( ( M = N )
% 8.58/4.88 => ( ( Deg3
% 8.58/4.88 = ( plus_plus_nat @ N @ M ) )
% 8.58/4.88 => ( ! [I2: nat] :
% 8.58/4.88 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.88 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 8.58/4.88 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
% 8.58/4.88 => ( ( ( Mi = Ma )
% 8.58/4.88 => ! [X4: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.58/4.88 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
% 8.58/4.88 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 8.58/4.88 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg3 ) )
% 8.58/4.88 => ( ( ( Mi != Ma )
% 8.58/4.88 => ! [I2: nat] :
% 8.58/4.88 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.88 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 8.58/4.88 = I2 )
% 8.58/4.88 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 8.58/4.88 & ! [X4: nat] :
% 8.58/4.88 ( ( ( ( vEBT_VEBT_high @ X4 @ N )
% 8.58/4.88 = I2 )
% 8.58/4.88 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
% 8.58/4.88 => ( ( ord_less_nat @ Mi @ X4 )
% 8.58/4.88 & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 8.58/4.88 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg3 @ TreeList2 @ Summary3 ) @ Deg3 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9057_invar__vebt_Ointros_I5_J,axiom,
% 8.58/4.88 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,M: nat,Deg3: nat,Mi: nat,Ma: nat] :
% 8.58/4.88 ( ! [X4: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.58/4.88 => ( vEBT_invar_vebt @ X4 @ N ) )
% 8.58/4.88 => ( ( vEBT_invar_vebt @ Summary3 @ M )
% 8.58/4.88 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 8.58/4.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.88 => ( ( M
% 8.58/4.88 = ( suc @ N ) )
% 8.58/4.88 => ( ( Deg3
% 8.58/4.88 = ( plus_plus_nat @ N @ M ) )
% 8.58/4.88 => ( ! [I2: nat] :
% 8.58/4.88 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.88 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 8.58/4.88 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
% 8.58/4.88 => ( ( ( Mi = Ma )
% 8.58/4.88 => ! [X4: vEBT_VEBT] :
% 8.58/4.88 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.58/4.88 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
% 8.58/4.88 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 8.58/4.88 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg3 ) )
% 8.58/4.88 => ( ( ( Mi != Ma )
% 8.58/4.88 => ! [I2: nat] :
% 8.58/4.88 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.58/4.88 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 8.58/4.88 = I2 )
% 8.58/4.88 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 8.58/4.88 & ! [X4: nat] :
% 8.58/4.88 ( ( ( ( vEBT_VEBT_high @ X4 @ N )
% 8.58/4.88 = I2 )
% 8.58/4.88 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
% 8.58/4.88 => ( ( ord_less_nat @ Mi @ X4 )
% 8.58/4.88 & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 8.58/4.88 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg3 @ TreeList2 @ Summary3 ) @ Deg3 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9058_set__encode__insert,axiom,
% 8.58/4.88 ! [A3: set_nat,N: nat] :
% 8.58/4.88 ( ( finite_finite_nat @ A3 )
% 8.58/4.88 => ( ~ ( member_nat @ N @ A3 )
% 8.58/4.88 => ( ( nat_set_encode @ ( insert_nat @ N @ A3 ) )
% 8.58/4.88 = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9059_vebt__buildup_Oelims,axiom,
% 8.58/4.88 ! [X: nat,Y: vEBT_VEBT] :
% 8.58/4.88 ( ( ( vEBT_vebt_buildup @ X )
% 8.58/4.88 = Y )
% 8.58/4.88 => ( ( ( X = zero_zero_nat )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( vEBT_Leaf @ $false @ $false ) ) )
% 8.58/4.88 => ( ( ( X
% 8.58/4.88 = ( suc @ zero_zero_nat ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( vEBT_Leaf @ $false @ $false ) ) )
% 8.58/4.88 => ~ ! [Va: nat] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( suc @ ( suc @ Va ) ) )
% 8.58/4.88 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 8.58/4.88 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9060_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 8.58/4.88 ! [X: vEBT_VEBT,Xa: nat] :
% 8.58/4.88 ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
% 8.58/4.88 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 8.58/4.88 => ( ! [Uu2: $o,Uv2: $o] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.58/4.88 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
% 8.58/4.88 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 8.58/4.88 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) )
% 8.58/4.88 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
% 8.58/4.88 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) @ Xa ) )
% 8.58/4.88 => ( ( Xa = Mi2 )
% 8.58/4.88 | ( Xa = Ma2 ) ) ) )
% 8.58/4.88 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 8.58/4.88 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa ) )
% 8.58/4.88 => ( ( Xa = Mi2 )
% 8.58/4.88 | ( Xa = Ma2 )
% 8.58/4.88 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 8.58/4.88 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 8.58/4.88 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ Xa ) )
% 8.58/4.88 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9061_set__decode__inverse,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( nat_set_encode @ ( nat_set_decode @ N ) )
% 8.58/4.88 = N ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9062_set__encode__empty,axiom,
% 8.58/4.88 ( ( nat_set_encode @ bot_bot_set_nat )
% 8.58/4.88 = zero_zero_nat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9063_set__encode__inverse,axiom,
% 8.58/4.88 ! [A3: set_nat] :
% 8.58/4.88 ( ( finite_finite_nat @ A3 )
% 8.58/4.88 => ( ( nat_set_decode @ ( nat_set_encode @ A3 ) )
% 8.58/4.88 = A3 ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9064_set__encode__eq,axiom,
% 8.58/4.88 ! [A3: set_nat,B5: set_nat] :
% 8.58/4.88 ( ( finite_finite_nat @ A3 )
% 8.58/4.88 => ( ( finite_finite_nat @ B5 )
% 8.58/4.88 => ( ( ( nat_set_encode @ A3 )
% 8.58/4.88 = ( nat_set_encode @ B5 ) )
% 8.58/4.88 = ( A3 = B5 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9065_set__encode__inf,axiom,
% 8.58/4.88 ! [A3: set_nat] :
% 8.58/4.88 ( ~ ( finite_finite_nat @ A3 )
% 8.58/4.88 => ( ( nat_set_encode @ A3 )
% 8.58/4.88 = zero_zero_nat ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9066_set__encode__def,axiom,
% 8.58/4.88 ( nat_set_encode
% 8.58/4.88 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9067_even__set__encode__iff,axiom,
% 8.58/4.88 ! [A3: set_nat] :
% 8.58/4.88 ( ( finite_finite_nat @ A3 )
% 8.58/4.88 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A3 ) )
% 8.58/4.88 = ( ~ ( member_nat @ zero_zero_nat @ A3 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9068_vebt__buildup_Osimps_I3_J,axiom,
% 8.58/4.88 ! [Va3: nat] :
% 8.58/4.88 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 8.58/4.88 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va3 ) ) )
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 8.58/4.88 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 8.58/4.88 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va3 ) ) )
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9069_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 8.58/4.88 ! [X: vEBT_VEBT,Xa: nat] :
% 8.58/4.88 ( ( vEBT_VEBT_membermima @ X @ Xa )
% 8.58/4.88 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 8.58/4.88 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
% 8.58/4.88 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) @ Xa ) )
% 8.58/4.88 => ~ ( ( Xa = Mi2 )
% 8.58/4.88 | ( Xa = Ma2 ) ) ) )
% 8.58/4.88 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 8.58/4.88 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa ) )
% 8.58/4.88 => ~ ( ( Xa = Mi2 )
% 8.58/4.88 | ( Xa = Ma2 )
% 8.58/4.88 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 8.58/4.88 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 8.58/4.88 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ Xa ) )
% 8.58/4.88 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9070_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 8.58/4.88 ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 8.58/4.88 ( ( ( vEBT_VEBT_membermima @ X @ Xa )
% 8.58/4.88 = Y )
% 8.58/4.88 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 8.58/4.88 => ( ! [Uu2: $o,Uv2: $o] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.58/4.88 => ( ~ Y
% 8.58/4.88 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
% 8.58/4.88 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 8.58/4.88 => ( ~ Y
% 8.58/4.88 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) ) )
% 8.58/4.88 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
% 8.58/4.88 => ( ( Y
% 8.58/4.88 = ( ( Xa = Mi2 )
% 8.58/4.88 | ( Xa = Ma2 ) ) )
% 8.58/4.88 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) @ Xa ) ) ) )
% 8.58/4.88 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
% 8.58/4.88 => ( ( Y
% 8.58/4.88 = ( ( Xa = Mi2 )
% 8.58/4.88 | ( Xa = Ma2 )
% 8.58/4.88 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 8.58/4.88 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa ) ) ) )
% 8.58/4.88 => ~ ! [V3: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) )
% 8.58/4.88 => ( ( Y
% 8.58/4.88 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 8.58/4.88 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList3 @ Vd2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9071_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 8.58/4.88 ! [X: vEBT_VEBT,Xa: nat] :
% 8.58/4.88 ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
% 8.58/4.88 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 8.58/4.88 => ( ! [A4: $o,B4: $o] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.58/4.88 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) )
% 8.58/4.88 => ( ( ( Xa = zero_zero_nat )
% 8.58/4.88 => A4 )
% 8.58/4.88 & ( ( Xa != zero_zero_nat )
% 8.58/4.88 => ( ( ( Xa = one_one_nat )
% 8.58/4.88 => B4 )
% 8.58/4.88 & ( Xa = one_one_nat ) ) ) ) ) )
% 8.58/4.88 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 8.58/4.88 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) )
% 8.58/4.88 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 8.58/4.88 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) )
% 8.58/4.88 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9072_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 8.58/4.88 ! [X: vEBT_VEBT,Xa: nat] :
% 8.58/4.88 ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 8.58/4.88 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 8.58/4.88 => ( ! [A4: $o,B4: $o] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.58/4.88 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) )
% 8.58/4.88 => ~ ( ( ( Xa = zero_zero_nat )
% 8.58/4.88 => A4 )
% 8.58/4.88 & ( ( Xa != zero_zero_nat )
% 8.58/4.88 => ( ( ( Xa = one_one_nat )
% 8.58/4.88 => B4 )
% 8.58/4.88 & ( Xa = one_one_nat ) ) ) ) ) )
% 8.58/4.88 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 8.58/4.88 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) )
% 8.58/4.88 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9073_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 8.58/4.88 ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 8.58/4.88 ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 8.58/4.88 = Y )
% 8.58/4.88 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 8.58/4.88 => ( ! [A4: $o,B4: $o] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.58/4.88 => ( ( Y
% 8.58/4.88 = ( ( ( Xa = zero_zero_nat )
% 8.58/4.88 => A4 )
% 8.58/4.88 & ( ( Xa != zero_zero_nat )
% 8.58/4.88 => ( ( ( Xa = one_one_nat )
% 8.58/4.88 => B4 )
% 8.58/4.88 & ( Xa = one_one_nat ) ) ) ) )
% 8.58/4.88 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa ) ) ) )
% 8.58/4.88 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 8.58/4.88 => ( ~ Y
% 8.58/4.88 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) ) )
% 8.58/4.88 => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
% 8.58/4.88 => ( ( Y
% 8.58/4.88 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 8.58/4.88 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 8.58/4.88 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9074_vebt__buildup_Opelims,axiom,
% 8.58/4.88 ! [X: nat,Y: vEBT_VEBT] :
% 8.58/4.88 ( ( ( vEBT_vebt_buildup @ X )
% 8.58/4.88 = Y )
% 8.58/4.88 => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
% 8.58/4.88 => ( ( ( X = zero_zero_nat )
% 8.58/4.88 => ( ( Y
% 8.58/4.88 = ( vEBT_Leaf @ $false @ $false ) )
% 8.58/4.88 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 8.58/4.88 => ( ( ( X
% 8.58/4.88 = ( suc @ zero_zero_nat ) )
% 8.58/4.88 => ( ( Y
% 8.58/4.88 = ( vEBT_Leaf @ $false @ $false ) )
% 8.58/4.88 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 8.58/4.88 => ~ ! [Va: nat] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( suc @ ( suc @ Va ) ) )
% 8.58/4.88 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 8.58/4.88 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 8.58/4.88 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9075_valid__eq2,axiom,
% 8.58/4.88 ! [T: vEBT_VEBT,D: nat] :
% 8.58/4.88 ( ( vEBT_VEBT_valid @ T @ D )
% 8.58/4.88 => ( vEBT_invar_vebt @ T @ D ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9076_valid__eq1,axiom,
% 8.58/4.88 ! [T: vEBT_VEBT,D: nat] :
% 8.58/4.88 ( ( vEBT_invar_vebt @ T @ D )
% 8.58/4.88 => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9077_valid__eq,axiom,
% 8.58/4.88 vEBT_VEBT_valid = vEBT_invar_vebt ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9078_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 8.58/4.88 ! [Uu: $o,Uv: $o,D: nat] :
% 8.58/4.88 ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 8.58/4.88 = ( D = one_one_nat ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9079_VEBT_Osize_I3_J,axiom,
% 8.58/4.88 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 8.58/4.88 ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 8.58/4.88 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9080_Sum__Ico__nat,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [X3: nat] : X3
% 8.58/4.88 @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 8.58/4.88 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9081_image__Suc__atLeastLessThan,axiom,
% 8.58/4.88 ! [I: nat,J: nat] :
% 8.58/4.88 ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
% 8.58/4.88 = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9082_card__atLeastLessThan,axiom,
% 8.58/4.88 ! [L: nat,U: nat] :
% 8.58/4.88 ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
% 8.58/4.88 = ( minus_minus_nat @ U @ L ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9083_atLeastLessThan__singleton,axiom,
% 8.58/4.88 ! [M: nat] :
% 8.58/4.88 ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 8.58/4.88 = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9084_ex__nat__less__eq,axiom,
% 8.58/4.88 ! [N: nat,P: nat > $o] :
% 8.58/4.88 ( ( ? [M3: nat] :
% 8.58/4.88 ( ( ord_less_nat @ M3 @ N )
% 8.58/4.88 & ( P @ M3 ) ) )
% 8.58/4.88 = ( ? [X3: nat] :
% 8.58/4.88 ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 8.58/4.88 & ( P @ X3 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9085_all__nat__less__eq,axiom,
% 8.58/4.88 ! [N: nat,P: nat > $o] :
% 8.58/4.88 ( ( ! [M3: nat] :
% 8.58/4.88 ( ( ord_less_nat @ M3 @ N )
% 8.58/4.88 => ( P @ M3 ) ) )
% 8.58/4.88 = ( ! [X3: nat] :
% 8.58/4.88 ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 8.58/4.88 => ( P @ X3 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9086_atLeastLessThanSuc__atLeastAtMost,axiom,
% 8.58/4.88 ! [L: nat,U: nat] :
% 8.58/4.88 ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
% 8.58/4.88 = ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9087_lessThan__atLeast0,axiom,
% 8.58/4.88 ( set_ord_lessThan_nat
% 8.58/4.88 = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9088_atLeastLessThan0,axiom,
% 8.58/4.88 ! [M: nat] :
% 8.58/4.88 ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
% 8.58/4.88 = bot_bot_set_nat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9089_atLeastSucLessThan__greaterThanLessThan,axiom,
% 8.58/4.88 ! [L: nat,U: nat] :
% 8.58/4.88 ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
% 8.58/4.88 = ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9090_atLeast0__lessThan__Suc,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 8.58/4.88 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9091_subset__eq__atLeast0__lessThan__finite,axiom,
% 8.58/4.88 ! [N5: set_nat,N: nat] :
% 8.58/4.88 ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 8.58/4.88 => ( finite_finite_nat @ N5 ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9092_subset__card__intvl__is__intvl,axiom,
% 8.58/4.88 ! [A3: set_nat,K: nat] :
% 8.58/4.88 ( ( ord_less_eq_set_nat @ A3 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) )
% 8.58/4.88 => ( A3
% 8.58/4.88 = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9093_atLeastLessThanSuc,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( ( ord_less_eq_nat @ M @ N )
% 8.58/4.88 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 8.58/4.88 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 8.58/4.88 & ( ~ ( ord_less_eq_nat @ M @ N )
% 8.58/4.88 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 8.58/4.88 = bot_bot_set_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9094_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 8.58/4.88 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9095_prod__Suc__fact,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 8.58/4.88 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9096_prod__Suc__Suc__fact,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 8.58/4.88 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9097_subset__eq__atLeast0__lessThan__card,axiom,
% 8.58/4.88 ! [N5: set_nat,N: nat] :
% 8.58/4.88 ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 8.58/4.88 => ( ord_less_eq_nat @ ( finite_card_nat @ N5 ) @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9098_card__sum__le__nat__sum,axiom,
% 8.58/4.88 ! [S2: set_nat] :
% 8.58/4.88 ( ord_less_eq_nat
% 8.58/4.88 @ ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [X3: nat] : X3
% 8.58/4.88 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
% 8.58/4.88 @ ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [X3: nat] : X3
% 8.58/4.88 @ S2 ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9099_atLeastLessThan__nat__numeral,axiom,
% 8.58/4.88 ! [M: nat,K: num] :
% 8.58/4.88 ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 8.58/4.88 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 8.58/4.88 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 8.58/4.88 & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 8.58/4.88 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 8.58/4.88 = bot_bot_set_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9100_image__minus__const__atLeastLessThan__nat,axiom,
% 8.58/4.88 ! [C: nat,Y: nat,X: nat] :
% 8.58/4.88 ( ( ( ord_less_nat @ C @ Y )
% 8.58/4.88 => ( ( image_nat_nat
% 8.58/4.88 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 8.58/4.88 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 8.58/4.88 = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
% 8.58/4.88 & ( ~ ( ord_less_nat @ C @ Y )
% 8.58/4.88 => ( ( ( ord_less_nat @ X @ Y )
% 8.58/4.88 => ( ( image_nat_nat
% 8.58/4.88 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 8.58/4.88 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 8.58/4.88 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 8.58/4.88 & ( ~ ( ord_less_nat @ X @ Y )
% 8.58/4.88 => ( ( image_nat_nat
% 8.58/4.88 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 8.58/4.88 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 8.58/4.88 = bot_bot_set_nat ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9101_atLeast1__lessThan__eq__remove0,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.58/4.88 = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9102_sum__power2,axiom,
% 8.58/4.88 ! [K: nat] :
% 8.58/4.88 ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 8.58/4.88 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9103_Chebyshev__sum__upper__nat,axiom,
% 8.58/4.88 ! [N: nat,A: nat > nat,B: nat > nat] :
% 8.58/4.88 ( ! [I2: nat,J2: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ I2 @ J2 )
% 8.58/4.88 => ( ( ord_less_nat @ J2 @ N )
% 8.58/4.88 => ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J2 ) ) ) )
% 8.58/4.88 => ( ! [I2: nat,J2: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ I2 @ J2 )
% 8.58/4.88 => ( ( ord_less_nat @ J2 @ N )
% 8.58/4.88 => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I2 ) ) ) )
% 8.58/4.88 => ( ord_less_eq_nat
% 8.58/4.88 @ ( times_times_nat @ N
% 8.58/4.88 @ ( groups3542108847815614940at_nat
% 8.58/4.88 @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( B @ I4 ) )
% 8.58/4.88 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 8.58/4.88 @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9104_VEBT_Osize__gen_I1_J,axiom,
% 8.58/4.88 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 8.58/4.88 ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 8.58/4.88 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9105_card__atLeastLessThan__int,axiom,
% 8.58/4.88 ! [L: int,U: int] :
% 8.58/4.88 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
% 8.58/4.88 = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9106_finite__atLeastZeroLessThan__int,axiom,
% 8.58/4.88 ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9107_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 8.58/4.88 ! [L: int,U: int] :
% 8.58/4.88 ( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
% 8.58/4.88 = ( set_or1266510415728281911st_int @ L @ U ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9108_image__int__atLeastLessThan,axiom,
% 8.58/4.88 ! [A: nat,B: nat] :
% 8.58/4.88 ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
% 8.58/4.88 = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9109_card__atLeastZeroLessThan__int,axiom,
% 8.58/4.88 ! [U: int] :
% 8.58/4.88 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
% 8.58/4.88 = ( nat2 @ U ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9110_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 8.58/4.88 ! [L: int,U: int] :
% 8.58/4.88 ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
% 8.58/4.88 = ( set_or5832277885323065728an_int @ L @ U ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9111_image__add__int__atLeastLessThan,axiom,
% 8.58/4.88 ! [L: int,U: int] :
% 8.58/4.88 ( ( image_int_int
% 8.58/4.88 @ ^ [X3: int] : ( plus_plus_int @ X3 @ L )
% 8.58/4.88 @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
% 8.58/4.88 = ( set_or4662586982721622107an_int @ L @ U ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9112_VEBT_Osize__gen_I2_J,axiom,
% 8.58/4.88 ! [X21: $o,X222: $o] :
% 8.58/4.88 ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 8.58/4.88 = zero_zero_nat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9113_image__atLeastZeroLessThan__int,axiom,
% 8.58/4.88 ! [U: int] :
% 8.58/4.88 ( ( ord_less_eq_int @ zero_zero_int @ U )
% 8.58/4.88 => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
% 8.58/4.88 = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9114_int__of__nat__def,axiom,
% 8.58/4.88 code_T6385005292777649522of_nat = semiri1314217659103216013at_int ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9115_Code__Target__Int_Opositive__def,axiom,
% 8.58/4.88 code_Target_positive = numeral_numeral_int ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9116_divmod__step__integer__def,axiom,
% 8.58/4.88 ( unique4921790084139445826nteger
% 8.58/4.88 = ( ^ [L2: num] :
% 8.58/4.88 ( produc6916734918728496179nteger
% 8.58/4.88 @ ^ [Q5: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9117_minus__integer__code_I1_J,axiom,
% 8.58/4.88 ! [K: code_integer] :
% 8.58/4.88 ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
% 8.58/4.88 = K ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9118_times__integer__code_I2_J,axiom,
% 8.58/4.88 ! [L: code_integer] :
% 8.58/4.88 ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
% 8.58/4.88 = zero_z3403309356797280102nteger ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9119_times__integer__code_I1_J,axiom,
% 8.58/4.88 ! [K: code_integer] :
% 8.58/4.88 ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
% 8.58/4.88 = zero_z3403309356797280102nteger ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9120_plus__integer__code_I2_J,axiom,
% 8.58/4.88 ! [L: code_integer] :
% 8.58/4.88 ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
% 8.58/4.88 = L ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9121_plus__integer__code_I1_J,axiom,
% 8.58/4.88 ! [K: code_integer] :
% 8.58/4.88 ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
% 8.58/4.88 = K ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9122_sgn__integer__code,axiom,
% 8.58/4.88 ( sgn_sgn_Code_integer
% 8.58/4.88 = ( ^ [K2: code_integer] : ( if_Code_integer @ ( K2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9123_minus__integer__code_I2_J,axiom,
% 8.58/4.88 ! [L: code_integer] :
% 8.58/4.88 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
% 8.58/4.88 = ( uminus1351360451143612070nteger @ L ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9124_nat_Odisc__eq__case_I2_J,axiom,
% 8.58/4.88 ! [Nat: nat] :
% 8.58/4.88 ( ( Nat != zero_zero_nat )
% 8.58/4.88 = ( case_nat_o @ $false
% 8.58/4.88 @ ^ [Uu3: nat] : $true
% 8.58/4.88 @ Nat ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9125_nat_Odisc__eq__case_I1_J,axiom,
% 8.58/4.88 ! [Nat: nat] :
% 8.58/4.88 ( ( Nat = zero_zero_nat )
% 8.58/4.88 = ( case_nat_o @ $true
% 8.58/4.88 @ ^ [Uu3: nat] : $false
% 8.58/4.88 @ Nat ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9126_zero__natural_Orsp,axiom,
% 8.58/4.88 zero_zero_nat = zero_zero_nat ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9127_zero__integer_Orsp,axiom,
% 8.58/4.88 zero_zero_int = zero_zero_int ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9128_one__integer_Orsp,axiom,
% 8.58/4.88 one_one_int = one_one_int ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9129_one__natural_Orsp,axiom,
% 8.58/4.88 one_one_nat = one_one_nat ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9130_less__eq__nat_Osimps_I2_J,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 8.58/4.88 = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9131_max__Suc2,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( ord_max_nat @ M @ ( suc @ N ) )
% 8.58/4.88 = ( case_nat_nat @ ( suc @ N )
% 8.58/4.88 @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ M5 @ N ) )
% 8.58/4.88 @ M ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9132_max__Suc1,axiom,
% 8.58/4.88 ! [N: nat,M: nat] :
% 8.58/4.88 ( ( ord_max_nat @ ( suc @ N ) @ M )
% 8.58/4.88 = ( case_nat_nat @ ( suc @ N )
% 8.58/4.88 @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ N @ M5 ) )
% 8.58/4.88 @ M ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9133_diff__Suc,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 8.58/4.88 = ( case_nat_nat @ zero_zero_nat
% 8.58/4.88 @ ^ [K2: nat] : K2
% 8.58/4.88 @ ( minus_minus_nat @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9134_min__Suc2,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( ord_min_nat @ M @ ( suc @ N ) )
% 8.58/4.88 = ( case_nat_nat @ zero_zero_nat
% 8.58/4.88 @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ M5 @ N ) )
% 8.58/4.88 @ M ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9135_min__Suc1,axiom,
% 8.58/4.88 ! [N: nat,M: nat] :
% 8.58/4.88 ( ( ord_min_nat @ ( suc @ N ) @ M )
% 8.58/4.88 = ( case_nat_nat @ zero_zero_nat
% 8.58/4.88 @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ N @ M5 ) )
% 8.58/4.88 @ M ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9136_integer__of__int__code,axiom,
% 8.58/4.88 ( code_integer_of_int
% 8.58/4.88 = ( ^ [K2: int] :
% 8.58/4.88 ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K2 ) ) )
% 8.58/4.88 @ ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 8.58/4.88 @ ( if_Code_integer
% 8.58/4.88 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.58/4.88 = zero_zero_int )
% 8.58/4.88 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 8.58/4.88 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9137_integer__of__num_I3_J,axiom,
% 8.58/4.88 ! [N: num] :
% 8.58/4.88 ( ( code_integer_of_num @ ( bit1 @ N ) )
% 8.58/4.88 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9138_push__bit__integer_Oabs__eq,axiom,
% 8.58/4.88 ! [Xa: nat,X: int] :
% 8.58/4.88 ( ( bit_se7788150548672797655nteger @ Xa @ ( code_integer_of_int @ X ) )
% 8.58/4.88 = ( code_integer_of_int @ ( bit_se545348938243370406it_int @ Xa @ X ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9139_zero__integer__def,axiom,
% 8.58/4.88 ( zero_z3403309356797280102nteger
% 8.58/4.88 = ( code_integer_of_int @ zero_zero_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9140_divide__integer_Oabs__eq,axiom,
% 8.58/4.88 ! [Xa: int,X: int] :
% 8.58/4.88 ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 8.58/4.88 = ( code_integer_of_int @ ( divide_divide_int @ Xa @ X ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9141_gcd__code__integer,axiom,
% 8.58/4.88 ( gcd_gcd_Code_integer
% 8.58/4.88 = ( ^ [K2: code_integer,L2: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L2 = zero_z3403309356797280102nteger ) @ K2 @ ( gcd_gcd_Code_integer @ L2 @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9142_gcd__integer_Oabs__eq,axiom,
% 8.58/4.88 ! [Xa: int,X: int] :
% 8.58/4.88 ( ( gcd_gcd_Code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 8.58/4.88 = ( code_integer_of_int @ ( gcd_gcd_int @ Xa @ X ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9143_plus__integer_Oabs__eq,axiom,
% 8.58/4.88 ! [Xa: int,X: int] :
% 8.58/4.88 ( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 8.58/4.88 = ( code_integer_of_int @ ( plus_plus_int @ Xa @ X ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9144_times__integer_Oabs__eq,axiom,
% 8.58/4.88 ! [Xa: int,X: int] :
% 8.58/4.88 ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 8.58/4.88 = ( code_integer_of_int @ ( times_times_int @ Xa @ X ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9145_one__integer__def,axiom,
% 8.58/4.88 ( one_one_Code_integer
% 8.58/4.88 = ( code_integer_of_int @ one_one_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9146_minus__integer_Oabs__eq,axiom,
% 8.58/4.88 ! [Xa: int,X: int] :
% 8.58/4.88 ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 8.58/4.88 = ( code_integer_of_int @ ( minus_minus_int @ Xa @ X ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9147_floor__rat__def,axiom,
% 8.58/4.88 ( archim3151403230148437115or_rat
% 8.58/4.88 = ( ^ [X3: rat] :
% 8.58/4.88 ( the_int
% 8.58/4.88 @ ^ [Z5: int] :
% 8.58/4.88 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X3 )
% 8.58/4.88 & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9148_integer__of__num__triv_I1_J,axiom,
% 8.58/4.88 ( ( code_integer_of_num @ one )
% 8.58/4.88 = one_one_Code_integer ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9149_integer__of__num_I2_J,axiom,
% 8.58/4.88 ! [N: num] :
% 8.58/4.88 ( ( code_integer_of_num @ ( bit0 @ N ) )
% 8.58/4.88 = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9150_integer__of__num__triv_I2_J,axiom,
% 8.58/4.88 ( ( code_integer_of_num @ ( bit0 @ one ) )
% 8.58/4.88 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9151_floor__real__def,axiom,
% 8.58/4.88 ( archim6058952711729229775r_real
% 8.58/4.88 = ( ^ [X3: real] :
% 8.58/4.88 ( the_int
% 8.58/4.88 @ ^ [Z5: int] :
% 8.58/4.88 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X3 )
% 8.58/4.88 & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9152_bit__cut__integer__code,axiom,
% 8.58/4.88 ( code_bit_cut_integer
% 8.58/4.88 = ( ^ [K2: code_integer] :
% 8.58/4.88 ( if_Pro5737122678794959658eger_o @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 8.58/4.88 @ ( produc9125791028180074456eger_o
% 8.58/4.88 @ ^ [R5: code_integer,S4: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S4 ) ) @ ( S4 = one_one_Code_integer ) )
% 8.58/4.88 @ ( code_divmod_abs @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9153_bit__cut__integer__def,axiom,
% 8.58/4.88 ( code_bit_cut_integer
% 8.58/4.88 = ( ^ [K2: code_integer] :
% 8.58/4.88 ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.58/4.88 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K2 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9154_pred__def,axiom,
% 8.58/4.88 ( pred
% 8.58/4.88 = ( case_nat_nat @ zero_zero_nat
% 8.58/4.88 @ ^ [X23: nat] : X23 ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9155_divmod__integer__code,axiom,
% 8.58/4.88 ( code_divmod_integer
% 8.58/4.88 = ( ^ [K2: code_integer,L2: code_integer] :
% 8.58/4.88 ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 8.58/4.88 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
% 8.58/4.88 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ ( code_divmod_abs @ K2 @ L2 )
% 8.58/4.88 @ ( produc6916734918728496179nteger
% 8.58/4.88 @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S4 ) ) )
% 8.58/4.88 @ ( code_divmod_abs @ K2 @ L2 ) ) )
% 8.58/4.88 @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 8.58/4.88 @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 8.58/4.88 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K2 @ L2 )
% 8.58/4.88 @ ( produc6916734918728496179nteger
% 8.58/4.88 @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S4 ) ) )
% 8.58/4.88 @ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9156_Gcd__int__def,axiom,
% 8.58/4.88 ( gcd_Gcd_int
% 8.58/4.88 = ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9157_divmod__integer__eq__cases,axiom,
% 8.58/4.88 ( code_divmod_integer
% 8.58/4.88 = ( ^ [K2: code_integer,L2: code_integer] :
% 8.58/4.88 ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 8.58/4.88 @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 8.58/4.88 @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
% 8.58/4.88 @ ( if_Pro6119634080678213985nteger
% 8.58/4.88 @ ( ( sgn_sgn_Code_integer @ K2 )
% 8.58/4.88 = ( sgn_sgn_Code_integer @ L2 ) )
% 8.58/4.88 @ ( code_divmod_abs @ K2 @ L2 )
% 8.58/4.88 @ ( produc6916734918728496179nteger
% 8.58/4.88 @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S4 ) ) )
% 8.58/4.88 @ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9158_nat__of__integer__non__positive,axiom,
% 8.58/4.88 ! [K: code_integer] :
% 8.58/4.88 ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 8.58/4.88 => ( ( code_nat_of_integer @ K )
% 8.58/4.88 = zero_zero_nat ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9159_Suc__funpow,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( compow_nat_nat @ N @ suc )
% 8.58/4.88 = ( plus_plus_nat @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9160_card_Ocomp__fun__commute__on,axiom,
% 8.58/4.88 ( ( comp_nat_nat_nat @ suc @ suc )
% 8.58/4.88 = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9161_nat__of__integer__code__post_I1_J,axiom,
% 8.58/4.88 ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
% 8.58/4.88 = zero_zero_nat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9162_nat__of__integer__code__post_I3_J,axiom,
% 8.58/4.88 ! [K: num] :
% 8.58/4.88 ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 8.58/4.88 = ( numeral_numeral_nat @ K ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9163_nat__of__integer__code__post_I2_J,axiom,
% 8.58/4.88 ( ( code_nat_of_integer @ one_one_Code_integer )
% 8.58/4.88 = one_one_nat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9164_Code__Target__Int_Onegative__def,axiom,
% 8.58/4.88 ( code_Target_negative
% 8.58/4.88 = ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9165_nat__of__integer__code,axiom,
% 8.58/4.88 ( code_nat_of_integer
% 8.58/4.88 = ( ^ [K2: code_integer] :
% 8.58/4.88 ( if_nat @ ( ord_le3102999989581377725nteger @ K2 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 8.58/4.88 @ ( produc1555791787009142072er_nat
% 8.58/4.88 @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
% 8.58/4.88 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9166_measure__function__int,axiom,
% 8.58/4.88 fun_is_measure_int @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9167_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 8.58/4.88 ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 8.58/4.88 @ ^ [X3: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X3 )
% 8.58/4.88 @ ^ [X3: nat,Y6: nat] : ( ord_less_nat @ Y6 @ X3 ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9168_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
% 8.58/4.88 ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
% 8.58/4.88 @ ^ [M3: nat,N3: nat] :
% 8.58/4.88 ( ( dvd_dvd_nat @ M3 @ N3 )
% 8.58/4.88 & ( M3 != N3 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9169_int__of__integer__code,axiom,
% 8.58/4.88 ( code_int_of_integer
% 8.58/4.88 = ( ^ [K2: code_integer] :
% 8.58/4.88 ( if_int @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K2 ) ) )
% 8.58/4.88 @ ( if_int @ ( K2 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 8.58/4.88 @ ( produc1553301316500091796er_int
% 8.58/4.88 @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
% 8.58/4.88 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9170_times__int_Oabs__eq,axiom,
% 8.58/4.88 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 8.58/4.88 ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 8.58/4.88 = ( abs_Integ
% 8.58/4.88 @ ( produc27273713700761075at_nat
% 8.58/4.88 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.88 ( produc2626176000494625587at_nat
% 8.58/4.88 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) )
% 8.58/4.88 @ Xa
% 8.58/4.88 @ X ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9171_int__of__integer__of__nat,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( code_int_of_integer @ ( semiri4939895301339042750nteger @ N ) )
% 8.58/4.88 = ( semiri1314217659103216013at_int @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9172_zero__integer_Orep__eq,axiom,
% 8.58/4.88 ( ( code_int_of_integer @ zero_z3403309356797280102nteger )
% 8.58/4.88 = zero_zero_int ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9173_int__of__integer__numeral,axiom,
% 8.58/4.88 ! [K: num] :
% 8.58/4.88 ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 8.58/4.88 = ( numeral_numeral_int @ K ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9174_plus__integer_Orep__eq,axiom,
% 8.58/4.88 ! [X: code_integer,Xa: code_integer] :
% 8.58/4.88 ( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X @ Xa ) )
% 8.58/4.88 = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9175_times__integer_Orep__eq,axiom,
% 8.58/4.88 ! [X: code_integer,Xa: code_integer] :
% 8.58/4.88 ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa ) )
% 8.58/4.88 = ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9176_one__integer_Orep__eq,axiom,
% 8.58/4.88 ( ( code_int_of_integer @ one_one_Code_integer )
% 8.58/4.88 = one_one_int ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9177_minus__integer_Orep__eq,axiom,
% 8.58/4.88 ! [X: code_integer,Xa: code_integer] :
% 8.58/4.88 ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X @ Xa ) )
% 8.58/4.88 = ( minus_minus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9178_divide__integer_Orep__eq,axiom,
% 8.58/4.88 ! [X: code_integer,Xa: code_integer] :
% 8.58/4.88 ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa ) )
% 8.58/4.88 = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9179_eq__Abs__Integ,axiom,
% 8.58/4.88 ! [Z2: int] :
% 8.58/4.88 ~ ! [X4: nat,Y4: nat] :
% 8.58/4.88 ( Z2
% 8.58/4.88 != ( abs_Integ @ ( product_Pair_nat_nat @ X4 @ Y4 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9180_int_Oabs__induct,axiom,
% 8.58/4.88 ! [P: int > $o,X: int] :
% 8.58/4.88 ( ! [Y4: product_prod_nat_nat] : ( P @ ( abs_Integ @ Y4 ) )
% 8.58/4.88 => ( P @ X ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9181_gcd__integer_Orep__eq,axiom,
% 8.58/4.88 ! [X: code_integer,Xa: code_integer] :
% 8.58/4.88 ( ( code_int_of_integer @ ( gcd_gcd_Code_integer @ X @ Xa ) )
% 8.58/4.88 = ( gcd_gcd_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9182_push__bit__integer_Orep__eq,axiom,
% 8.58/4.88 ! [X: nat,Xa: code_integer] :
% 8.58/4.88 ( ( code_int_of_integer @ ( bit_se7788150548672797655nteger @ X @ Xa ) )
% 8.58/4.88 = ( bit_se545348938243370406it_int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9183_zero__int__def,axiom,
% 8.58/4.88 ( zero_zero_int
% 8.58/4.88 = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9184_int__def,axiom,
% 8.58/4.88 ( semiri1314217659103216013at_int
% 8.58/4.88 = ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N3 @ zero_zero_nat ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9185_nat_Oabs__eq,axiom,
% 8.58/4.88 ! [X: product_prod_nat_nat] :
% 8.58/4.88 ( ( nat2 @ ( abs_Integ @ X ) )
% 8.58/4.88 = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9186_uminus__int_Oabs__eq,axiom,
% 8.58/4.88 ! [X: product_prod_nat_nat] :
% 8.58/4.88 ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
% 8.58/4.88 = ( abs_Integ
% 8.58/4.88 @ ( produc2626176000494625587at_nat
% 8.58/4.88 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 )
% 8.58/4.88 @ X ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9187_one__int__def,axiom,
% 8.58/4.88 ( one_one_int
% 8.58/4.88 = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9188_less__int_Oabs__eq,axiom,
% 8.58/4.88 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 8.58/4.88 ( ( ord_less_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 8.58/4.88 = ( produc8739625826339149834_nat_o
% 8.58/4.88 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.88 ( produc6081775807080527818_nat_o
% 8.58/4.88 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) )
% 8.58/4.88 @ Xa
% 8.58/4.88 @ X ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9189_less__eq__int_Oabs__eq,axiom,
% 8.58/4.88 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 8.58/4.88 ( ( ord_less_eq_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 8.58/4.88 = ( produc8739625826339149834_nat_o
% 8.58/4.88 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.88 ( produc6081775807080527818_nat_o
% 8.58/4.88 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) )
% 8.58/4.88 @ Xa
% 8.58/4.88 @ X ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9190_plus__int_Oabs__eq,axiom,
% 8.58/4.88 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 8.58/4.88 ( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 8.58/4.88 = ( abs_Integ
% 8.58/4.88 @ ( produc27273713700761075at_nat
% 8.58/4.88 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.88 ( produc2626176000494625587at_nat
% 8.58/4.88 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) )
% 8.58/4.88 @ Xa
% 8.58/4.88 @ X ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9191_minus__int_Oabs__eq,axiom,
% 8.58/4.88 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 8.58/4.88 ( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 8.58/4.88 = ( abs_Integ
% 8.58/4.88 @ ( produc27273713700761075at_nat
% 8.58/4.88 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.88 ( produc2626176000494625587at_nat
% 8.58/4.88 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) )
% 8.58/4.88 @ Xa
% 8.58/4.88 @ X ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9192_rat__floor__lemma,axiom,
% 8.58/4.88 ! [A: int,B: int] :
% 8.58/4.88 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
% 8.58/4.88 & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9193_mult__rat,axiom,
% 8.58/4.88 ! [A: int,B: int,C: int,D: int] :
% 8.58/4.88 ( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 8.58/4.88 = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9194_divide__rat,axiom,
% 8.58/4.88 ! [A: int,B: int,C: int,D: int] :
% 8.58/4.88 ( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 8.58/4.88 = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9195_floor__Fract,axiom,
% 8.58/4.88 ! [A: int,B: int] :
% 8.58/4.88 ( ( archim3151403230148437115or_rat @ ( fract @ A @ B ) )
% 8.58/4.88 = ( divide_divide_int @ A @ B ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9196_less__rat,axiom,
% 8.58/4.88 ! [B: int,D: int,A: int,C: int] :
% 8.58/4.88 ( ( B != zero_zero_int )
% 8.58/4.88 => ( ( D != zero_zero_int )
% 8.58/4.88 => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 8.58/4.88 = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9197_add__rat,axiom,
% 8.58/4.88 ! [B: int,D: int,A: int,C: int] :
% 8.58/4.88 ( ( B != zero_zero_int )
% 8.58/4.88 => ( ( D != zero_zero_int )
% 8.58/4.88 => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 8.58/4.88 = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9198_le__rat,axiom,
% 8.58/4.88 ! [B: int,D: int,A: int,C: int] :
% 8.58/4.88 ( ( B != zero_zero_int )
% 8.58/4.88 => ( ( D != zero_zero_int )
% 8.58/4.88 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 8.58/4.88 = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9199_diff__rat,axiom,
% 8.58/4.88 ! [B: int,D: int,A: int,C: int] :
% 8.58/4.88 ( ( B != zero_zero_int )
% 8.58/4.88 => ( ( D != zero_zero_int )
% 8.58/4.88 => ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 8.58/4.88 = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9200_sgn__rat,axiom,
% 8.58/4.88 ! [A: int,B: int] :
% 8.58/4.88 ( ( sgn_sgn_rat @ ( fract @ A @ B ) )
% 8.58/4.88 = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9201_eq__rat_I3_J,axiom,
% 8.58/4.88 ! [A: int,C: int] :
% 8.58/4.88 ( ( fract @ zero_zero_int @ A )
% 8.58/4.88 = ( fract @ zero_zero_int @ C ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9202_eq__rat_I2_J,axiom,
% 8.58/4.88 ! [A: int] :
% 8.58/4.88 ( ( fract @ A @ zero_zero_int )
% 8.58/4.88 = ( fract @ zero_zero_int @ one_one_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9203_Rat__induct__pos,axiom,
% 8.58/4.88 ! [P: rat > $o,Q2: rat] :
% 8.58/4.88 ( ! [A4: int,B4: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ B4 )
% 8.58/4.88 => ( P @ ( fract @ A4 @ B4 ) ) )
% 8.58/4.88 => ( P @ Q2 ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9204_mult__rat__cancel,axiom,
% 8.58/4.88 ! [C: int,A: int,B: int] :
% 8.58/4.88 ( ( C != zero_zero_int )
% 8.58/4.88 => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.58/4.88 = ( fract @ A @ B ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9205_eq__rat_I1_J,axiom,
% 8.58/4.88 ! [B: int,D: int,A: int,C: int] :
% 8.58/4.88 ( ( B != zero_zero_int )
% 8.58/4.88 => ( ( D != zero_zero_int )
% 8.58/4.88 => ( ( ( fract @ A @ B )
% 8.58/4.88 = ( fract @ C @ D ) )
% 8.58/4.88 = ( ( times_times_int @ A @ D )
% 8.58/4.88 = ( times_times_int @ C @ B ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9206_Fract__of__nat__eq,axiom,
% 8.58/4.88 ! [K: nat] :
% 8.58/4.88 ( ( fract @ ( semiri1314217659103216013at_int @ K ) @ one_one_int )
% 8.58/4.88 = ( semiri681578069525770553at_rat @ K ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9207_rat__number__collapse_I6_J,axiom,
% 8.58/4.88 ! [K: int] :
% 8.58/4.88 ( ( fract @ K @ zero_zero_int )
% 8.58/4.88 = zero_zero_rat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9208_rat__number__collapse_I1_J,axiom,
% 8.58/4.88 ! [K: int] :
% 8.58/4.88 ( ( fract @ zero_zero_int @ K )
% 8.58/4.88 = zero_zero_rat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9209_Fract__coprime,axiom,
% 8.58/4.88 ! [A: int,B: int] :
% 8.58/4.88 ( ( fract @ ( divide_divide_int @ A @ ( gcd_gcd_int @ A @ B ) ) @ ( divide_divide_int @ B @ ( gcd_gcd_int @ A @ B ) ) )
% 8.58/4.88 = ( fract @ A @ B ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9210_One__rat__def,axiom,
% 8.58/4.88 ( one_one_rat
% 8.58/4.88 = ( fract @ one_one_int @ one_one_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9211_Fract__of__int__eq,axiom,
% 8.58/4.88 ! [K: int] :
% 8.58/4.88 ( ( fract @ K @ one_one_int )
% 8.58/4.88 = ( ring_1_of_int_rat @ K ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9212_Zero__rat__def,axiom,
% 8.58/4.88 ( zero_zero_rat
% 8.58/4.88 = ( fract @ zero_zero_int @ one_one_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9213_rat__number__collapse_I3_J,axiom,
% 8.58/4.88 ! [W: num] :
% 8.58/4.88 ( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
% 8.58/4.88 = ( numeral_numeral_rat @ W ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9214_rat__number__expand_I3_J,axiom,
% 8.58/4.88 ( numeral_numeral_rat
% 8.58/4.88 = ( ^ [K2: num] : ( fract @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9215_sorted__list__of__set__greaterThanLessThan,axiom,
% 8.58/4.88 ! [I: nat,J: nat] :
% 8.58/4.88 ( ( ord_less_nat @ ( suc @ I ) @ J )
% 8.58/4.88 => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 8.58/4.88 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9216_zero__less__Fract__iff,axiom,
% 8.58/4.88 ! [B: int,A: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.88 => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 8.58/4.88 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9217_Fract__less__zero__iff,axiom,
% 8.58/4.88 ! [B: int,A: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.88 => ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 8.58/4.88 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9218_Fract__less__one__iff,axiom,
% 8.58/4.88 ! [B: int,A: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.88 => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
% 8.58/4.88 = ( ord_less_int @ A @ B ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9219_one__less__Fract__iff,axiom,
% 8.58/4.88 ! [B: int,A: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.88 => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
% 8.58/4.88 = ( ord_less_int @ B @ A ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9220_rat__number__collapse_I5_J,axiom,
% 8.58/4.88 ( ( fract @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 8.58/4.88 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9221_Fract__add__one,axiom,
% 8.58/4.88 ! [N: int,M: int] :
% 8.58/4.88 ( ( N != zero_zero_int )
% 8.58/4.88 => ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
% 8.58/4.88 = ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9222_zero__le__Fract__iff,axiom,
% 8.58/4.88 ! [B: int,A: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.88 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 8.58/4.88 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9223_Fract__le__zero__iff,axiom,
% 8.58/4.88 ! [B: int,A: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.88 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 8.58/4.88 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9224_Fract__le__one__iff,axiom,
% 8.58/4.88 ! [B: int,A: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.88 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
% 8.58/4.88 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9225_one__le__Fract__iff,axiom,
% 8.58/4.88 ! [B: int,A: int] :
% 8.58/4.88 ( ( ord_less_int @ zero_zero_int @ B )
% 8.58/4.88 => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
% 8.58/4.88 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9226_rat__number__collapse_I4_J,axiom,
% 8.58/4.88 ! [W: num] :
% 8.58/4.88 ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
% 8.58/4.88 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9227_rat__number__expand_I5_J,axiom,
% 8.58/4.88 ! [K: num] :
% 8.58/4.88 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
% 8.58/4.88 = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9228_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 8.58/4.88 ! [N: nat,J: nat,I: nat] :
% 8.58/4.88 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 8.58/4.88 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
% 8.58/4.88 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9229_less__eq__int_Orep__eq,axiom,
% 8.58/4.88 ( ord_less_eq_int
% 8.58/4.88 = ( ^ [X3: int,Xa4: int] :
% 8.58/4.88 ( produc8739625826339149834_nat_o
% 8.58/4.88 @ ^ [Y6: nat,Z5: nat] :
% 8.58/4.88 ( produc6081775807080527818_nat_o
% 8.58/4.88 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y6 @ V4 ) @ ( plus_plus_nat @ U3 @ Z5 ) ) )
% 8.58/4.88 @ ( rep_Integ @ X3 )
% 8.58/4.88 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9230_less__int_Orep__eq,axiom,
% 8.58/4.88 ( ord_less_int
% 8.58/4.88 = ( ^ [X3: int,Xa4: int] :
% 8.58/4.88 ( produc8739625826339149834_nat_o
% 8.58/4.88 @ ^ [Y6: nat,Z5: nat] :
% 8.58/4.88 ( produc6081775807080527818_nat_o
% 8.58/4.88 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y6 @ V4 ) @ ( plus_plus_nat @ U3 @ Z5 ) ) )
% 8.58/4.88 @ ( rep_Integ @ X3 )
% 8.58/4.88 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9231_inf__nat__def,axiom,
% 8.58/4.88 inf_inf_nat = ord_min_nat ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9232_inf__int__def,axiom,
% 8.58/4.88 inf_inf_int = ord_min_int ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9233_nat_Orep__eq,axiom,
% 8.58/4.88 ( nat2
% 8.58/4.88 = ( ^ [X3: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X3 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9234_list__encode_Osimps_I2_J,axiom,
% 8.58/4.88 ! [X: nat,Xs: list_nat] :
% 8.58/4.88 ( ( nat_list_encode @ ( cons_nat @ X @ Xs ) )
% 8.58/4.88 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9235_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 8.58/4.88 ! [N: nat,J: nat,I: nat] :
% 8.58/4.88 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
% 8.58/4.88 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
% 8.58/4.88 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9236_card__greaterThanAtMost,axiom,
% 8.58/4.88 ! [L: nat,U: nat] :
% 8.58/4.88 ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
% 8.58/4.88 = ( minus_minus_nat @ U @ L ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9237_list__encode__eq,axiom,
% 8.58/4.88 ! [X: list_nat,Y: list_nat] :
% 8.58/4.88 ( ( ( nat_list_encode @ X )
% 8.58/4.88 = ( nat_list_encode @ Y ) )
% 8.58/4.88 = ( X = Y ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9238_atLeastSucAtMost__greaterThanAtMost,axiom,
% 8.58/4.88 ! [L: nat,U: nat] :
% 8.58/4.88 ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
% 8.58/4.88 = ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9239_sorted__list__of__set__greaterThanAtMost,axiom,
% 8.58/4.88 ! [I: nat,J: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 8.58/4.88 => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 8.58/4.88 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9240_list__encode_Oelims,axiom,
% 8.58/4.88 ! [X: list_nat,Y: nat] :
% 8.58/4.88 ( ( ( nat_list_encode @ X )
% 8.58/4.88 = Y )
% 8.58/4.88 => ( ( ( X = nil_nat )
% 8.58/4.88 => ( Y != zero_zero_nat ) )
% 8.58/4.88 => ~ ! [X4: nat,Xs2: list_nat] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( cons_nat @ X4 @ Xs2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9241_uminus__int__def,axiom,
% 8.58/4.88 ( uminus_uminus_int
% 8.58/4.88 = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
% 8.58/4.88 @ ( produc2626176000494625587at_nat
% 8.58/4.88 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9242_take__bit__numeral__minus__numeral__int,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.88 = ( case_option_int_num @ zero_zero_int
% 8.58/4.88 @ ^ [Q5: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q5 ) ) )
% 8.58/4.88 @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9243_take__bit__num__simps_I1_J,axiom,
% 8.58/4.88 ! [M: num] :
% 8.58/4.88 ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 8.58/4.88 = none_num ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9244_take__bit__num__simps_I2_J,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( bit_take_bit_num @ ( suc @ N ) @ one )
% 8.58/4.88 = ( some_num @ one ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9245_take__bit__num__simps_I5_J,axiom,
% 8.58/4.88 ! [R: num] :
% 8.58/4.88 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ one )
% 8.58/4.88 = ( some_num @ one ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9246_card__greaterThanAtMost__int,axiom,
% 8.58/4.88 ! [L: int,U: int] :
% 8.58/4.88 ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
% 8.58/4.88 = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9247_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( bit_take_bit_num @ N @ one )
% 8.58/4.88 = ( case_nat_option_num @ none_num
% 8.58/4.88 @ ^ [N3: nat] : ( some_num @ one )
% 8.58/4.88 @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9248_list__encode_Ocases,axiom,
% 8.58/4.88 ! [X: list_nat] :
% 8.58/4.88 ( ( X != nil_nat )
% 8.58/4.88 => ~ ! [X4: nat,Xs2: list_nat] :
% 8.58/4.88 ( X
% 8.58/4.88 != ( cons_nat @ X4 @ Xs2 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9249_list__encode_Osimps_I1_J,axiom,
% 8.58/4.88 ( ( nat_list_encode @ nil_nat )
% 8.58/4.88 = zero_zero_nat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9250_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 8.58/4.88 ! [L: int,U: int] :
% 8.58/4.88 ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
% 8.58/4.88 = ( set_or6656581121297822940st_int @ L @ U ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9251_take__bit__num__def,axiom,
% 8.58/4.88 ( bit_take_bit_num
% 8.58/4.88 = ( ^ [N3: nat,M3: num] :
% 8.58/4.88 ( if_option_num
% 8.58/4.88 @ ( ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M3 ) )
% 8.58/4.88 = zero_zero_nat )
% 8.58/4.88 @ none_num
% 8.58/4.88 @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M3 ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9252_and__minus__numerals_I3_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.58/4.88 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9253_and__minus__numerals_I7_J,axiom,
% 8.58/4.88 ! [N: num,M: num] :
% 8.58/4.88 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 8.58/4.88 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9254_and__minus__numerals_I4_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.58/4.88 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9255_take__bit__num__simps_I4_J,axiom,
% 8.58/4.88 ! [N: nat,M: num] :
% 8.58/4.88 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 8.58/4.88 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9256_take__bit__num__simps_I3_J,axiom,
% 8.58/4.88 ! [N: nat,M: num] :
% 8.58/4.88 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 8.58/4.88 = ( case_o6005452278849405969um_num @ none_num
% 8.58/4.88 @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 8.58/4.88 @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9257_take__bit__num__simps_I7_J,axiom,
% 8.58/4.88 ! [R: num,M: num] :
% 8.58/4.88 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit1 @ M ) )
% 8.58/4.88 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9258_take__bit__num__simps_I6_J,axiom,
% 8.58/4.88 ! [R: num,M: num] :
% 8.58/4.88 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit0 @ M ) )
% 8.58/4.88 = ( case_o6005452278849405969um_num @ none_num
% 8.58/4.88 @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 8.58/4.88 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9259_and__minus__numerals_I8_J,axiom,
% 8.58/4.88 ! [N: num,M: num] :
% 8.58/4.88 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 8.58/4.88 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9260_and__not__num_Osimps_I8_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.58/4.88 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 8.58/4.88 @ ^ [N7: num] : ( some_num @ ( bit1 @ N7 ) )
% 8.58/4.88 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9261_and__not__num_Osimps_I1_J,axiom,
% 8.58/4.88 ( ( bit_and_not_num @ one @ one )
% 8.58/4.88 = none_num ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9262_and__not__num_Osimps_I2_J,axiom,
% 8.58/4.88 ! [N: num] :
% 8.58/4.88 ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
% 8.58/4.88 = ( some_num @ one ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9263_and__not__num_Osimps_I4_J,axiom,
% 8.58/4.88 ! [M: num] :
% 8.58/4.88 ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 8.58/4.88 = ( some_num @ ( bit0 @ M ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9264_and__not__num_Osimps_I3_J,axiom,
% 8.58/4.88 ! [N: num] :
% 8.58/4.88 ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
% 8.58/4.88 = none_num ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9265_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 8.58/4.88 ! [N: nat,M: num] :
% 8.58/4.88 ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
% 8.58/4.88 = ( case_nat_option_num @ none_num
% 8.58/4.88 @ ^ [N3: nat] :
% 8.58/4.88 ( case_o6005452278849405969um_num @ none_num
% 8.58/4.88 @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 8.58/4.88 @ ( bit_take_bit_num @ N3 @ M ) )
% 8.58/4.88 @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9266_and__not__num_Osimps_I7_J,axiom,
% 8.58/4.88 ! [M: num] :
% 8.58/4.88 ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 8.58/4.88 = ( some_num @ ( bit0 @ M ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9267_and__not__num__eq__Some__iff,axiom,
% 8.58/4.88 ! [M: num,N: num,Q2: num] :
% 8.58/4.88 ( ( ( bit_and_not_num @ M @ N )
% 8.58/4.88 = ( some_num @ Q2 ) )
% 8.58/4.88 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.88 = ( numeral_numeral_int @ Q2 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9268_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 8.58/4.88 ! [N: nat,M: num] :
% 8.58/4.88 ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
% 8.58/4.88 = ( case_nat_option_num @ none_num
% 8.58/4.88 @ ^ [N3: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N3 @ M ) ) )
% 8.58/4.88 @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9269_and__not__num__eq__None__iff,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( ( bit_and_not_num @ M @ N )
% 8.58/4.88 = none_num )
% 8.58/4.88 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.88 = zero_zero_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9270_int__numeral__not__and__num,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.88 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9271_int__numeral__and__not__num,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.88 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9272_times__int__def,axiom,
% 8.58/4.88 ( times_times_int
% 8.58/4.88 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 8.58/4.88 @ ( produc27273713700761075at_nat
% 8.58/4.88 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.88 ( produc2626176000494625587at_nat
% 8.58/4.88 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9273_minus__int__def,axiom,
% 8.58/4.88 ( minus_minus_int
% 8.58/4.88 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 8.58/4.88 @ ( produc27273713700761075at_nat
% 8.58/4.88 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.88 ( produc2626176000494625587at_nat
% 8.58/4.88 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9274_plus__int__def,axiom,
% 8.58/4.88 ( plus_plus_int
% 8.58/4.88 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 8.58/4.88 @ ( produc27273713700761075at_nat
% 8.58/4.88 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.88 ( produc2626176000494625587at_nat
% 8.58/4.88 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9275_upto_Opelims,axiom,
% 8.58/4.88 ! [X: int,Xa: int,Y: list_int] :
% 8.58/4.88 ( ( ( upto @ X @ Xa )
% 8.58/4.88 = Y )
% 8.58/4.88 => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 8.58/4.88 => ~ ( ( ( ( ord_less_eq_int @ X @ Xa )
% 8.58/4.88 => ( Y
% 8.58/4.88 = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
% 8.58/4.88 & ( ~ ( ord_less_eq_int @ X @ Xa )
% 8.58/4.88 => ( Y = nil_int ) ) )
% 8.58/4.88 => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9276_upto_Opsimps,axiom,
% 8.58/4.88 ! [I: int,J: int] :
% 8.58/4.88 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 8.58/4.88 => ( ( ( ord_less_eq_int @ I @ J )
% 8.58/4.88 => ( ( upto @ I @ J )
% 8.58/4.88 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 8.58/4.88 & ( ~ ( ord_less_eq_int @ I @ J )
% 8.58/4.88 => ( ( upto @ I @ J )
% 8.58/4.88 = nil_int ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9277_nth__upto,axiom,
% 8.58/4.88 ! [I: int,K: nat,J: int] :
% 8.58/4.88 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 8.58/4.88 => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 8.58/4.88 = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9278_length__upto,axiom,
% 8.58/4.88 ! [I: int,J: int] :
% 8.58/4.88 ( ( size_size_list_int @ ( upto @ I @ J ) )
% 8.58/4.88 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9279_sorted__list__of__set__lessThan__Suc,axiom,
% 8.58/4.88 ! [K: nat] :
% 8.58/4.88 ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
% 8.58/4.88 = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9280_sorted__list__of__set__atMost__Suc,axiom,
% 8.58/4.88 ! [K: nat] :
% 8.58/4.88 ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
% 8.58/4.88 = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9281_upto__rec__numeral_I1_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.88 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.88 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 8.58/4.88 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.88 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.88 = nil_int ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9282_upto__rec__numeral_I2_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.88 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.88 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 8.58/4.88 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.88 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.88 = nil_int ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9283_upto__rec__numeral_I3_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.88 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.88 = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 8.58/4.88 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.88 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.58/4.88 = nil_int ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9284_upto__rec__numeral_I4_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.88 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.88 = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 8.58/4.88 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.88 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.58/4.88 = nil_int ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9285_upto__split1,axiom,
% 8.58/4.88 ! [I: int,J: int,K: int] :
% 8.58/4.88 ( ( ord_less_eq_int @ I @ J )
% 8.58/4.88 => ( ( ord_less_eq_int @ J @ K )
% 8.58/4.88 => ( ( upto @ I @ K )
% 8.58/4.88 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9286_upto__split2,axiom,
% 8.58/4.88 ! [I: int,J: int,K: int] :
% 8.58/4.88 ( ( ord_less_eq_int @ I @ J )
% 8.58/4.88 => ( ( ord_less_eq_int @ J @ K )
% 8.58/4.88 => ( ( upto @ I @ K )
% 8.58/4.88 = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9287_upto__rec2,axiom,
% 8.58/4.88 ! [I: int,J: int] :
% 8.58/4.88 ( ( ord_less_eq_int @ I @ J )
% 8.58/4.88 => ( ( upto @ I @ J )
% 8.58/4.88 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9288_upto__split3,axiom,
% 8.58/4.88 ! [I: int,J: int,K: int] :
% 8.58/4.88 ( ( ord_less_eq_int @ I @ J )
% 8.58/4.88 => ( ( ord_less_eq_int @ J @ K )
% 8.58/4.88 => ( ( upto @ I @ K )
% 8.58/4.88 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9289_atLeastLessThan__upto,axiom,
% 8.58/4.88 ( set_or4662586982721622107an_int
% 8.58/4.88 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9290_greaterThanAtMost__upto,axiom,
% 8.58/4.88 ( set_or6656581121297822940st_int
% 8.58/4.88 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9291_upto_Osimps,axiom,
% 8.58/4.88 ( upto
% 8.58/4.88 = ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9292_upto_Oelims,axiom,
% 8.58/4.88 ! [X: int,Xa: int,Y: list_int] :
% 8.58/4.88 ( ( ( upto @ X @ Xa )
% 8.58/4.88 = Y )
% 8.58/4.88 => ( ( ( ord_less_eq_int @ X @ Xa )
% 8.58/4.88 => ( Y
% 8.58/4.88 = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
% 8.58/4.88 & ( ~ ( ord_less_eq_int @ X @ Xa )
% 8.58/4.88 => ( Y = nil_int ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9293_upto__rec1,axiom,
% 8.58/4.88 ! [I: int,J: int] :
% 8.58/4.88 ( ( ord_less_eq_int @ I @ J )
% 8.58/4.88 => ( ( upto @ I @ J )
% 8.58/4.88 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9294_greaterThanLessThan__upto,axiom,
% 8.58/4.88 ( set_or5832277885323065728an_int
% 8.58/4.88 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9295_and__not__num_Oelims,axiom,
% 8.58/4.88 ! [X: num,Xa: num,Y: option_num] :
% 8.58/4.88 ( ( ( bit_and_not_num @ X @ Xa )
% 8.58/4.88 = Y )
% 8.58/4.88 => ( ( ( X = one )
% 8.58/4.88 => ( ( Xa = one )
% 8.58/4.88 => ( Y != none_num ) ) )
% 8.58/4.88 => ( ( ( X = one )
% 8.58/4.88 => ( ? [N2: num] :
% 8.58/4.88 ( Xa
% 8.58/4.88 = ( bit0 @ N2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( some_num @ one ) ) ) )
% 8.58/4.88 => ( ( ( X = one )
% 8.58/4.88 => ( ? [N2: num] :
% 8.58/4.88 ( Xa
% 8.58/4.88 = ( bit1 @ N2 ) )
% 8.58/4.88 => ( Y != none_num ) ) )
% 8.58/4.88 => ( ! [M2: num] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( bit0 @ M2 ) )
% 8.58/4.88 => ( ( Xa = one )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( some_num @ ( bit0 @ M2 ) ) ) ) )
% 8.58/4.88 => ( ! [M2: num] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( bit0 @ M2 ) )
% 8.58/4.88 => ! [N2: num] :
% 8.58/4.88 ( ( Xa
% 8.58/4.88 = ( bit0 @ N2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N2 ) ) ) ) )
% 8.58/4.88 => ( ! [M2: num] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( bit0 @ M2 ) )
% 8.58/4.88 => ! [N2: num] :
% 8.58/4.88 ( ( Xa
% 8.58/4.88 = ( bit1 @ N2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N2 ) ) ) ) )
% 8.58/4.88 => ( ! [M2: num] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( bit1 @ M2 ) )
% 8.58/4.88 => ( ( Xa = one )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( some_num @ ( bit0 @ M2 ) ) ) ) )
% 8.58/4.88 => ( ! [M2: num] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( bit1 @ M2 ) )
% 8.58/4.88 => ! [N2: num] :
% 8.58/4.88 ( ( Xa
% 8.58/4.88 = ( bit0 @ N2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 8.58/4.88 @ ^ [N7: num] : ( some_num @ ( bit1 @ N7 ) )
% 8.58/4.88 @ ( bit_and_not_num @ M2 @ N2 ) ) ) ) )
% 8.58/4.88 => ~ ! [M2: num] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( bit1 @ M2 ) )
% 8.58/4.88 => ! [N2: num] :
% 8.58/4.88 ( ( Xa
% 8.58/4.88 = ( bit1 @ N2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9296_and__not__num_Osimps_I5_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.58/4.88 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9297_and__not__num_Osimps_I6_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.58/4.88 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9298_and__not__num_Osimps_I9_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.58/4.88 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9299_and__num_Oelims,axiom,
% 8.58/4.88 ! [X: num,Xa: num,Y: option_num] :
% 8.58/4.88 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 8.58/4.88 = Y )
% 8.58/4.88 => ( ( ( X = one )
% 8.58/4.88 => ( ( Xa = one )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( some_num @ one ) ) ) )
% 8.58/4.88 => ( ( ( X = one )
% 8.58/4.88 => ( ? [N2: num] :
% 8.58/4.88 ( Xa
% 8.58/4.88 = ( bit0 @ N2 ) )
% 8.58/4.88 => ( Y != none_num ) ) )
% 8.58/4.88 => ( ( ( X = one )
% 8.58/4.88 => ( ? [N2: num] :
% 8.58/4.88 ( Xa
% 8.58/4.88 = ( bit1 @ N2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( some_num @ one ) ) ) )
% 8.58/4.88 => ( ( ? [M2: num] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( bit0 @ M2 ) )
% 8.58/4.88 => ( ( Xa = one )
% 8.58/4.88 => ( Y != none_num ) ) )
% 8.58/4.88 => ( ! [M2: num] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( bit0 @ M2 ) )
% 8.58/4.88 => ! [N2: num] :
% 8.58/4.88 ( ( Xa
% 8.58/4.88 = ( bit0 @ N2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) ) ) )
% 8.58/4.88 => ( ! [M2: num] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( bit0 @ M2 ) )
% 8.58/4.88 => ! [N2: num] :
% 8.58/4.88 ( ( Xa
% 8.58/4.88 = ( bit1 @ N2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) ) ) )
% 8.58/4.88 => ( ( ? [M2: num] :
% 8.58/4.88 ( X
% 8.58/4.88 = ( bit1 @ M2 ) )
% 8.58/4.88 => ( ( Xa = one )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( some_num @ one ) ) ) )
% 8.58/4.88 => ( ! [M2: num] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( bit1 @ M2 ) )
% 8.58/4.88 => ! [N2: num] :
% 8.58/4.88 ( ( Xa
% 8.58/4.88 = ( bit0 @ N2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) ) ) )
% 8.58/4.88 => ~ ! [M2: num] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( bit1 @ M2 ) )
% 8.58/4.88 => ! [N2: num] :
% 8.58/4.88 ( ( Xa
% 8.58/4.88 = ( bit1 @ N2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 8.58/4.88 @ ^ [N7: num] : ( some_num @ ( bit1 @ N7 ) )
% 8.58/4.88 @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9300_xor__num_Oelims,axiom,
% 8.58/4.88 ! [X: num,Xa: num,Y: option_num] :
% 8.58/4.88 ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 8.58/4.88 = Y )
% 8.58/4.88 => ( ( ( X = one )
% 8.58/4.88 => ( ( Xa = one )
% 8.58/4.88 => ( Y != none_num ) ) )
% 8.58/4.88 => ( ( ( X = one )
% 8.58/4.88 => ! [N2: num] :
% 8.58/4.88 ( ( Xa
% 8.58/4.88 = ( bit0 @ N2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( some_num @ ( bit1 @ N2 ) ) ) ) )
% 8.58/4.88 => ( ( ( X = one )
% 8.58/4.88 => ! [N2: num] :
% 8.58/4.88 ( ( Xa
% 8.58/4.88 = ( bit1 @ N2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( some_num @ ( bit0 @ N2 ) ) ) ) )
% 8.58/4.88 => ( ! [M2: num] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( bit0 @ M2 ) )
% 8.58/4.88 => ( ( Xa = one )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( some_num @ ( bit1 @ M2 ) ) ) ) )
% 8.58/4.88 => ( ! [M2: num] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( bit0 @ M2 ) )
% 8.58/4.88 => ! [N2: num] :
% 8.58/4.88 ( ( Xa
% 8.58/4.88 = ( bit0 @ N2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) ) )
% 8.58/4.88 => ( ! [M2: num] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( bit0 @ M2 ) )
% 8.58/4.88 => ! [N2: num] :
% 8.58/4.88 ( ( Xa
% 8.58/4.88 = ( bit1 @ N2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) ) ) )
% 8.58/4.88 => ( ! [M2: num] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( bit1 @ M2 ) )
% 8.58/4.88 => ( ( Xa = one )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( some_num @ ( bit0 @ M2 ) ) ) ) )
% 8.58/4.88 => ( ! [M2: num] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( bit1 @ M2 ) )
% 8.58/4.88 => ! [N2: num] :
% 8.58/4.88 ( ( Xa
% 8.58/4.88 = ( bit0 @ N2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) ) ) )
% 8.58/4.88 => ~ ! [M2: num] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( bit1 @ M2 ) )
% 8.58/4.88 => ! [N2: num] :
% 8.58/4.88 ( ( Xa
% 8.58/4.88 = ( bit1 @ N2 ) )
% 8.58/4.88 => ( Y
% 8.58/4.88 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9301_and__num_Osimps_I1_J,axiom,
% 8.58/4.88 ( ( bit_un7362597486090784418nd_num @ one @ one )
% 8.58/4.88 = ( some_num @ one ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9302_xor__num_Osimps_I1_J,axiom,
% 8.58/4.88 ( ( bit_un2480387367778600638or_num @ one @ one )
% 8.58/4.88 = none_num ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9303_xor__num_Osimps_I5_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.58/4.88 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9304_and__num_Osimps_I5_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.58/4.88 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9305_and__num_Osimps_I3_J,axiom,
% 8.58/4.88 ! [N: num] :
% 8.58/4.88 ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
% 8.58/4.88 = ( some_num @ one ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9306_and__num_Osimps_I7_J,axiom,
% 8.58/4.88 ! [M: num] :
% 8.58/4.88 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 8.58/4.88 = ( some_num @ one ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9307_and__num_Osimps_I2_J,axiom,
% 8.58/4.88 ! [N: num] :
% 8.58/4.88 ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
% 8.58/4.88 = none_num ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9308_and__num_Osimps_I4_J,axiom,
% 8.58/4.88 ! [M: num] :
% 8.58/4.88 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 8.58/4.88 = none_num ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9309_and__num_Osimps_I8_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.58/4.88 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9310_and__num_Osimps_I6_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.58/4.88 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9311_xor__num_Osimps_I9_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.58/4.88 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9312_xor__num_Osimps_I7_J,axiom,
% 8.58/4.88 ! [M: num] :
% 8.58/4.88 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 8.58/4.88 = ( some_num @ ( bit0 @ M ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9313_xor__num_Osimps_I4_J,axiom,
% 8.58/4.88 ! [M: num] :
% 8.58/4.88 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 8.58/4.88 = ( some_num @ ( bit1 @ M ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9314_xor__num_Osimps_I3_J,axiom,
% 8.58/4.88 ! [N: num] :
% 8.58/4.88 ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
% 8.58/4.88 = ( some_num @ ( bit0 @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9315_xor__num_Osimps_I2_J,axiom,
% 8.58/4.88 ! [N: num] :
% 8.58/4.88 ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
% 8.58/4.88 = ( some_num @ ( bit1 @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9316_and__num_Osimps_I9_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.58/4.88 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 8.58/4.88 @ ^ [N7: num] : ( some_num @ ( bit1 @ N7 ) )
% 8.58/4.88 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9317_xor__num_Osimps_I6_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.58/4.88 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9318_xor__num_Osimps_I8_J,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.58/4.88 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9319_xor__num__dict,axiom,
% 8.58/4.88 bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9320_and__num__dict,axiom,
% 8.58/4.88 bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9321_Bit__Operations_Otake__bit__num__code,axiom,
% 8.58/4.88 ( bit_take_bit_num
% 8.58/4.88 = ( ^ [N3: nat,M3: num] :
% 8.58/4.88 ( produc478579273971653890on_num
% 8.58/4.88 @ ^ [A2: nat,X3: num] :
% 8.58/4.88 ( case_nat_option_num @ none_num
% 8.58/4.88 @ ^ [O: nat] :
% 8.58/4.88 ( case_num_option_num @ ( some_num @ one )
% 8.58/4.88 @ ^ [P3: num] :
% 8.58/4.88 ( case_o6005452278849405969um_num @ none_num
% 8.58/4.88 @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 8.58/4.88 @ ( bit_take_bit_num @ O @ P3 ) )
% 8.58/4.88 @ ^ [P3: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P3 ) ) )
% 8.58/4.88 @ X3 )
% 8.58/4.88 @ A2 )
% 8.58/4.88 @ ( product_Pair_nat_num @ N3 @ M3 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9322_of__nat__eq__id,axiom,
% 8.58/4.88 semiri1316708129612266289at_nat = id_nat ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9323_less__int__def,axiom,
% 8.58/4.88 ( ord_less_int
% 8.58/4.88 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 8.58/4.88 @ ( produc8739625826339149834_nat_o
% 8.58/4.88 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.88 ( produc6081775807080527818_nat_o
% 8.58/4.88 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9324_less__eq__int__def,axiom,
% 8.58/4.88 ( ord_less_eq_int
% 8.58/4.88 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 8.58/4.88 @ ( produc8739625826339149834_nat_o
% 8.58/4.88 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.88 ( produc6081775807080527818_nat_o
% 8.58/4.88 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9325_nat__def,axiom,
% 8.58/4.88 ( nat2
% 8.58/4.88 = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9326_positive__rat,axiom,
% 8.58/4.88 ! [A: int,B: int] :
% 8.58/4.88 ( ( positive @ ( fract @ A @ B ) )
% 8.58/4.88 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9327_list__encode_Opelims,axiom,
% 8.58/4.88 ! [X: list_nat,Y: nat] :
% 8.58/4.88 ( ( ( nat_list_encode @ X )
% 8.58/4.88 = Y )
% 8.58/4.88 => ( ( accp_list_nat @ nat_list_encode_rel @ X )
% 8.58/4.88 => ( ( ( X = nil_nat )
% 8.58/4.88 => ( ( Y = zero_zero_nat )
% 8.58/4.88 => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
% 8.58/4.88 => ~ ! [X4: nat,Xs2: list_nat] :
% 8.58/4.88 ( ( X
% 8.58/4.88 = ( cons_nat @ X4 @ Xs2 ) )
% 8.58/4.88 => ( ( Y
% 8.58/4.88 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) )
% 8.58/4.88 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9328_num__of__integer__code,axiom,
% 8.58/4.88 ( code_num_of_integer
% 8.58/4.88 = ( ^ [K2: code_integer] :
% 8.58/4.88 ( if_num @ ( ord_le3102999989581377725nteger @ K2 @ one_one_Code_integer ) @ one
% 8.58/4.88 @ ( produc7336495610019696514er_num
% 8.58/4.88 @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
% 8.58/4.88 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9329_Rat_Opositive__add,axiom,
% 8.58/4.88 ! [X: rat,Y: rat] :
% 8.58/4.88 ( ( positive @ X )
% 8.58/4.88 => ( ( positive @ Y )
% 8.58/4.88 => ( positive @ ( plus_plus_rat @ X @ Y ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9330_Rat_Opositive__mult,axiom,
% 8.58/4.88 ! [X: rat,Y: rat] :
% 8.58/4.88 ( ( positive @ X )
% 8.58/4.88 => ( ( positive @ Y )
% 8.58/4.88 => ( positive @ ( times_times_rat @ X @ Y ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9331_less__rat__def,axiom,
% 8.58/4.88 ( ord_less_rat
% 8.58/4.88 = ( ^ [X3: rat,Y6: rat] : ( positive @ ( minus_minus_rat @ Y6 @ X3 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9332_Rat_Opositive_Orep__eq,axiom,
% 8.58/4.88 ( positive
% 8.58/4.88 = ( ^ [X3: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X3 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X3 ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9333_Rat_Opositive__def,axiom,
% 8.58/4.88 ( positive
% 8.58/4.88 = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
% 8.58/4.88 @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9334_upt__rec__numeral,axiom,
% 8.58/4.88 ! [M: num,N: num] :
% 8.58/4.88 ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.58/4.88 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.58/4.88 = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
% 8.58/4.88 & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.58/4.88 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.58/4.88 = nil_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9335_tl__upt,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( tl_nat @ ( upt @ M @ N ) )
% 8.58/4.88 = ( upt @ ( suc @ M ) @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9336_drop__upt,axiom,
% 8.58/4.88 ! [M: nat,I: nat,J: nat] :
% 8.58/4.88 ( ( drop_nat @ M @ ( upt @ I @ J ) )
% 8.58/4.88 = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9337_length__upt,axiom,
% 8.58/4.88 ! [I: nat,J: nat] :
% 8.58/4.88 ( ( size_size_list_nat @ ( upt @ I @ J ) )
% 8.58/4.88 = ( minus_minus_nat @ J @ I ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9338_take__upt,axiom,
% 8.58/4.88 ! [I: nat,M: nat,N: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
% 8.58/4.88 => ( ( take_nat @ M @ ( upt @ I @ N ) )
% 8.58/4.88 = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9339_upt__eq__Nil__conv,axiom,
% 8.58/4.88 ! [I: nat,J: nat] :
% 8.58/4.88 ( ( ( upt @ I @ J )
% 8.58/4.88 = nil_nat )
% 8.58/4.88 = ( ( J = zero_zero_nat )
% 8.58/4.88 | ( ord_less_eq_nat @ J @ I ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9340_nth__upt,axiom,
% 8.58/4.88 ! [I: nat,K: nat,J: nat] :
% 8.58/4.88 ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
% 8.58/4.88 => ( ( nth_nat @ ( upt @ I @ J ) @ K )
% 8.58/4.88 = ( plus_plus_nat @ I @ K ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9341_mono__Suc,axiom,
% 8.58/4.88 order_mono_nat_nat @ suc ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9342_atLeastAtMost__upt,axiom,
% 8.58/4.88 ( set_or1269000886237332187st_nat
% 8.58/4.88 = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ N3 @ ( suc @ M3 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9343_atLeast__upt,axiom,
% 8.58/4.88 ( set_ord_lessThan_nat
% 8.58/4.88 = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N3 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9344_upt__conv__Cons__Cons,axiom,
% 8.58/4.88 ! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
% 8.58/4.88 ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
% 8.58/4.88 = ( upt @ M @ Q2 ) )
% 8.58/4.88 = ( ( cons_nat @ N @ Ns )
% 8.58/4.88 = ( upt @ ( suc @ M ) @ Q2 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9345_upt__0,axiom,
% 8.58/4.88 ! [I: nat] :
% 8.58/4.88 ( ( upt @ I @ zero_zero_nat )
% 8.58/4.88 = nil_nat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9346_greaterThanAtMost__upt,axiom,
% 8.58/4.88 ( set_or6659071591806873216st_nat
% 8.58/4.88 = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ ( suc @ M3 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9347_greaterThanLessThan__upt,axiom,
% 8.58/4.88 ( set_or5834768355832116004an_nat
% 8.58/4.88 = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ M3 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9348_mono__times__nat,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.88 => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9349_atMost__upto,axiom,
% 8.58/4.88 ( set_ord_atMost_nat
% 8.58/4.88 = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N3 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9350_upt__conv__Cons,axiom,
% 8.58/4.88 ! [I: nat,J: nat] :
% 8.58/4.88 ( ( ord_less_nat @ I @ J )
% 8.58/4.88 => ( ( upt @ I @ J )
% 8.58/4.88 = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9351_upt__add__eq__append,axiom,
% 8.58/4.88 ! [I: nat,J: nat,K: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ I @ J )
% 8.58/4.88 => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
% 8.58/4.88 = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9352_upt__eq__Cons__conv,axiom,
% 8.58/4.88 ! [I: nat,J: nat,X: nat,Xs: list_nat] :
% 8.58/4.88 ( ( ( upt @ I @ J )
% 8.58/4.88 = ( cons_nat @ X @ Xs ) )
% 8.58/4.88 = ( ( ord_less_nat @ I @ J )
% 8.58/4.88 & ( I = X )
% 8.58/4.88 & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
% 8.58/4.88 = Xs ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9353_upt__rec,axiom,
% 8.58/4.88 ( upt
% 8.58/4.88 = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9354_mono__ge2__power__minus__self,axiom,
% 8.58/4.88 ! [K: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 8.58/4.88 => ( order_mono_nat_nat
% 8.58/4.88 @ ^ [M3: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M3 ) @ M3 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9355_upt__Suc__append,axiom,
% 8.58/4.88 ! [I: nat,J: nat] :
% 8.58/4.88 ( ( ord_less_eq_nat @ I @ J )
% 8.58/4.88 => ( ( upt @ I @ ( suc @ J ) )
% 8.58/4.88 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9356_upt__Suc,axiom,
% 8.58/4.88 ! [I: nat,J: nat] :
% 8.58/4.88 ( ( ( ord_less_eq_nat @ I @ J )
% 8.58/4.88 => ( ( upt @ I @ ( suc @ J ) )
% 8.58/4.88 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 8.58/4.88 & ( ~ ( ord_less_eq_nat @ I @ J )
% 8.58/4.88 => ( ( upt @ I @ ( suc @ J ) )
% 8.58/4.88 = nil_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9357_of__rat__dense,axiom,
% 8.58/4.88 ! [X: real,Y: real] :
% 8.58/4.88 ( ( ord_less_real @ X @ Y )
% 8.58/4.88 => ? [Q6: rat] :
% 8.58/4.88 ( ( ord_less_real @ X @ ( field_7254667332652039916t_real @ Q6 ) )
% 8.58/4.88 & ( ord_less_real @ ( field_7254667332652039916t_real @ Q6 ) @ Y ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9358_range__mod,axiom,
% 8.58/4.88 ! [N: nat] :
% 8.58/4.88 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.88 => ( ( image_nat_nat
% 8.58/4.88 @ ^ [M3: nat] : ( modulo_modulo_nat @ M3 @ N )
% 8.58/4.88 @ top_top_set_nat )
% 8.58/4.88 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9359_UNIV__nat__eq,axiom,
% 8.58/4.88 ( top_top_set_nat
% 8.58/4.88 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9360_plus__rat__def,axiom,
% 8.58/4.88 ( plus_plus_rat
% 8.58/4.88 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 8.58/4.88 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9361_card__UNIV__unit,axiom,
% 8.58/4.88 ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 8.58/4.88 = one_one_nat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9362_range__mult,axiom,
% 8.58/4.88 ! [A: real] :
% 8.58/4.88 ( ( ( A = zero_zero_real )
% 8.58/4.88 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 8.58/4.88 = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 8.58/4.88 & ( ( A != zero_zero_real )
% 8.58/4.88 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 8.58/4.88 = top_top_set_real ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9363_range__abs__Nats,axiom,
% 8.58/4.88 ( ( image_int_int @ abs_abs_int @ top_top_set_int )
% 8.58/4.88 = semiring_1_Nats_int ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9364_card__UNIV__bool,axiom,
% 8.58/4.88 ( ( finite_card_o @ top_top_set_o )
% 8.58/4.88 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9365_map__Suc__upt,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
% 8.58/4.88 = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9366_infinite__UNIV__int,axiom,
% 8.58/4.88 ~ ( finite_finite_int @ top_top_set_int ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9367_surj__list__encode,axiom,
% 8.58/4.88 ( ( image_list_nat_nat @ nat_list_encode @ top_top_set_list_nat )
% 8.58/4.88 = top_top_set_nat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9368_bij__list__encode,axiom,
% 8.58/4.88 bij_be8532844293280997160at_nat @ nat_list_encode @ top_top_set_list_nat @ top_top_set_nat ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9369_surj__prod__encode,axiom,
% 8.58/4.88 ( ( image_2486076414777270412at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat )
% 8.58/4.88 = top_top_set_nat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9370_bij__prod__encode,axiom,
% 8.58/4.88 bij_be5333170631980326235at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat @ top_top_set_nat ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9371_map__add__upt,axiom,
% 8.58/4.88 ! [N: nat,M: nat] :
% 8.58/4.88 ( ( map_nat_nat
% 8.58/4.88 @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
% 8.58/4.88 @ ( upt @ zero_zero_nat @ M ) )
% 8.58/4.88 = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9372_int__in__range__abs,axiom,
% 8.58/4.88 ! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9373_map__decr__upt,axiom,
% 8.58/4.88 ! [M: nat,N: nat] :
% 8.58/4.88 ( ( map_nat_nat
% 8.58/4.88 @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
% 8.58/4.88 @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 8.58/4.88 = ( upt @ M @ N ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9374_one__rat__def,axiom,
% 8.58/4.88 ( one_one_rat
% 8.58/4.88 = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9375_Fract_Oabs__eq,axiom,
% 8.58/4.88 ( fract
% 8.58/4.88 = ( ^ [Xa4: int,X3: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X3 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9376_zero__rat__def,axiom,
% 8.58/4.88 ( zero_zero_rat
% 8.58/4.88 = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9377_times__rat__def,axiom,
% 8.58/4.88 ( times_times_rat
% 8.58/4.88 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 8.58/4.88 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9378_inverse__rat__def,axiom,
% 8.58/4.88 ( inverse_inverse_rat
% 8.58/4.88 = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
% 8.58/4.88 @ ^ [X3: product_prod_int_int] :
% 8.58/4.88 ( if_Pro3027730157355071871nt_int
% 8.58/4.88 @ ( ( product_fst_int_int @ X3 )
% 8.58/4.88 = zero_zero_int )
% 8.58/4.88 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 8.58/4.88 @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9379_root__def,axiom,
% 8.58/4.88 ( root
% 8.58/4.88 = ( ^ [N3: nat,X3: real] :
% 8.58/4.88 ( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
% 8.58/4.88 @ ( the_in5290026491893676941l_real @ top_top_set_real
% 8.58/4.88 @ ^ [Y6: real] : ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N3 ) )
% 8.58/4.88 @ X3 ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9380_card__UNIV__char,axiom,
% 8.58/4.88 ( ( finite_card_char @ top_top_set_char )
% 8.58/4.88 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9381_plus__rat_Oabs__eq,axiom,
% 8.58/4.88 ! [Xa: product_prod_int_int,X: product_prod_int_int] :
% 8.58/4.88 ( ( ratrel @ Xa @ Xa )
% 8.58/4.88 => ( ( ratrel @ X @ X )
% 8.58/4.88 => ( ( plus_plus_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 8.58/4.88 = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9382_ratrel__iff,axiom,
% 8.58/4.88 ( ratrel
% 8.58/4.88 = ( ^ [X3: product_prod_int_int,Y6: product_prod_int_int] :
% 8.58/4.88 ( ( ( product_snd_int_int @ X3 )
% 8.58/4.88 != zero_zero_int )
% 8.58/4.88 & ( ( product_snd_int_int @ Y6 )
% 8.58/4.88 != zero_zero_int )
% 8.58/4.88 & ( ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) )
% 8.58/4.88 = ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9383_one__rat_Orsp,axiom,
% 8.58/4.88 ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9384_zero__rat_Orsp,axiom,
% 8.58/4.88 ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9385_ratrel__def,axiom,
% 8.58/4.88 ( ratrel
% 8.58/4.88 = ( ^ [X3: product_prod_int_int,Y6: product_prod_int_int] :
% 8.58/4.88 ( ( ( product_snd_int_int @ X3 )
% 8.58/4.88 != zero_zero_int )
% 8.58/4.88 & ( ( product_snd_int_int @ Y6 )
% 8.58/4.88 != zero_zero_int )
% 8.58/4.88 & ( ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) )
% 8.58/4.88 = ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9386_times__rat_Oabs__eq,axiom,
% 8.58/4.88 ! [Xa: product_prod_int_int,X: product_prod_int_int] :
% 8.58/4.88 ( ( ratrel @ Xa @ Xa )
% 8.58/4.88 => ( ( ratrel @ X @ X )
% 8.58/4.88 => ( ( times_times_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 8.58/4.88 = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9387_Rat_Opositive_Oabs__eq,axiom,
% 8.58/4.88 ! [X: product_prod_int_int] :
% 8.58/4.88 ( ( ratrel @ X @ X )
% 8.58/4.88 => ( ( positive @ ( abs_Rat @ X ) )
% 8.58/4.88 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9388_inverse__rat_Oabs__eq,axiom,
% 8.58/4.88 ! [X: product_prod_int_int] :
% 8.58/4.88 ( ( ratrel @ X @ X )
% 8.58/4.88 => ( ( inverse_inverse_rat @ ( abs_Rat @ X ) )
% 8.58/4.88 = ( abs_Rat
% 8.58/4.88 @ ( if_Pro3027730157355071871nt_int
% 8.58/4.88 @ ( ( product_fst_int_int @ X )
% 8.58/4.88 = zero_zero_int )
% 8.58/4.88 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 8.58/4.88 @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9389_UNIV__char__of__nat,axiom,
% 8.58/4.88 ( top_top_set_char
% 8.58/4.88 = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9390_char_Osize_I2_J,axiom,
% 8.58/4.88 ! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
% 8.58/4.88 ( ( size_size_char @ ( char2 @ X1 @ X2 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
% 8.58/4.88 = zero_zero_nat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9391_nat__of__char__less__256,axiom,
% 8.58/4.88 ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9392_range__nat__of__char,axiom,
% 8.58/4.88 ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 8.58/4.88 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9393_integer__of__char__code,axiom,
% 8.58/4.88 ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 8.58/4.88 ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 8.58/4.88 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9394_char_Osize__gen,axiom,
% 8.58/4.88 ! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
% 8.58/4.88 ( ( size_char @ ( char2 @ X1 @ X2 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
% 8.58/4.88 = zero_zero_nat ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9395_String_Ochar__of__ascii__of,axiom,
% 8.58/4.88 ! [C: char] :
% 8.58/4.88 ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
% 8.58/4.88 = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9396_DERIV__real__root__generic,axiom,
% 8.58/4.88 ! [N: nat,X: real,D3: real] :
% 8.58/4.88 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.88 => ( ( X != zero_zero_real )
% 8.58/4.88 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.88 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.88 => ( D3
% 8.58/4.88 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 8.58/4.88 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.88 => ( ( ord_less_real @ X @ zero_zero_real )
% 8.58/4.88 => ( D3
% 8.58/4.88 = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 8.58/4.88 => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.88 => ( D3
% 8.58/4.88 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 8.58/4.88 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9397_has__real__derivative__neg__dec__left,axiom,
% 8.58/4.88 ! [F: real > real,L: real,X: real,S2: set_real] :
% 8.58/4.88 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 8.58/4.88 => ( ( ord_less_real @ L @ zero_zero_real )
% 8.58/4.88 => ? [D2: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ D2 )
% 8.58/4.88 & ! [H2: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ H2 )
% 8.58/4.88 => ( ( member_real @ ( minus_minus_real @ X @ H2 ) @ S2 )
% 8.58/4.88 => ( ( ord_less_real @ H2 @ D2 )
% 8.58/4.88 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H2 ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9398_has__real__derivative__pos__inc__left,axiom,
% 8.58/4.88 ! [F: real > real,L: real,X: real,S2: set_real] :
% 8.58/4.88 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 8.58/4.88 => ( ( ord_less_real @ zero_zero_real @ L )
% 8.58/4.88 => ? [D2: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ D2 )
% 8.58/4.88 & ! [H2: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ H2 )
% 8.58/4.88 => ( ( member_real @ ( minus_minus_real @ X @ H2 ) @ S2 )
% 8.58/4.88 => ( ( ord_less_real @ H2 @ D2 )
% 8.58/4.88 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9399_has__real__derivative__pos__inc__right,axiom,
% 8.58/4.88 ! [F: real > real,L: real,X: real,S2: set_real] :
% 8.58/4.88 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 8.58/4.88 => ( ( ord_less_real @ zero_zero_real @ L )
% 8.58/4.88 => ? [D2: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ D2 )
% 8.58/4.88 & ! [H2: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ H2 )
% 8.58/4.88 => ( ( member_real @ ( plus_plus_real @ X @ H2 ) @ S2 )
% 8.58/4.88 => ( ( ord_less_real @ H2 @ D2 )
% 8.58/4.88 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H2 ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9400_has__real__derivative__neg__dec__right,axiom,
% 8.58/4.88 ! [F: real > real,L: real,X: real,S2: set_real] :
% 8.58/4.88 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 8.58/4.88 => ( ( ord_less_real @ L @ zero_zero_real )
% 8.58/4.88 => ? [D2: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ D2 )
% 8.58/4.88 & ! [H2: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ H2 )
% 8.58/4.88 => ( ( member_real @ ( plus_plus_real @ X @ H2 ) @ S2 )
% 8.58/4.88 => ( ( ord_less_real @ H2 @ D2 )
% 8.58/4.88 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9401_DERIV__const__ratio__const2,axiom,
% 8.58/4.88 ! [A: real,B: real,F: real > real,K: real] :
% 8.58/4.88 ( ( A != B )
% 8.58/4.88 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 8.58/4.88 => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
% 8.58/4.88 = K ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9402_DERIV__const__ratio__const,axiom,
% 8.58/4.88 ! [A: real,B: real,F: real > real,K: real] :
% 8.58/4.88 ( ( A != B )
% 8.58/4.88 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 8.58/4.88 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 8.58/4.88 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9403_DERIV__neg__dec__right,axiom,
% 8.58/4.88 ! [F: real > real,L: real,X: real] :
% 8.58/4.88 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.88 => ( ( ord_less_real @ L @ zero_zero_real )
% 8.58/4.88 => ? [D2: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ D2 )
% 8.58/4.88 & ! [H2: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ H2 )
% 8.58/4.88 => ( ( ord_less_real @ H2 @ D2 )
% 8.58/4.88 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9404_DERIV__pos__inc__right,axiom,
% 8.58/4.88 ! [F: real > real,L: real,X: real] :
% 8.58/4.88 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.88 => ( ( ord_less_real @ zero_zero_real @ L )
% 8.58/4.88 => ? [D2: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ D2 )
% 8.58/4.88 & ! [H2: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ H2 )
% 8.58/4.88 => ( ( ord_less_real @ H2 @ D2 )
% 8.58/4.88 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H2 ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9405_DERIV__neg__dec__left,axiom,
% 8.58/4.88 ! [F: real > real,L: real,X: real] :
% 8.58/4.88 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.88 => ( ( ord_less_real @ L @ zero_zero_real )
% 8.58/4.88 => ? [D2: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ D2 )
% 8.58/4.88 & ! [H2: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ H2 )
% 8.58/4.88 => ( ( ord_less_real @ H2 @ D2 )
% 8.58/4.88 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H2 ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9406_DERIV__pos__inc__left,axiom,
% 8.58/4.88 ! [F: real > real,L: real,X: real] :
% 8.58/4.88 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.88 => ( ( ord_less_real @ zero_zero_real @ L )
% 8.58/4.88 => ? [D2: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ D2 )
% 8.58/4.88 & ! [H2: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ H2 )
% 8.58/4.88 => ( ( ord_less_real @ H2 @ D2 )
% 8.58/4.88 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H2 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9407_MVT2,axiom,
% 8.58/4.88 ! [A: real,B: real,F: real > real,F3: real > real] :
% 8.58/4.88 ( ( ord_less_real @ A @ B )
% 8.58/4.88 => ( ! [X4: real] :
% 8.58/4.88 ( ( ord_less_eq_real @ A @ X4 )
% 8.58/4.88 => ( ( ord_less_eq_real @ X4 @ B )
% 8.58/4.88 => ( has_fi5821293074295781190e_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 8.58/4.88 => ? [Z4: real] :
% 8.58/4.88 ( ( ord_less_real @ A @ Z4 )
% 8.58/4.88 & ( ord_less_real @ Z4 @ B )
% 8.58/4.88 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 8.58/4.88 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F3 @ Z4 ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9408_DERIV__local__const,axiom,
% 8.58/4.88 ! [F: real > real,L: real,X: real,D: real] :
% 8.58/4.88 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.88 => ( ( ord_less_real @ zero_zero_real @ D )
% 8.58/4.88 => ( ! [Y4: real] :
% 8.58/4.88 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 8.58/4.88 => ( ( F @ X )
% 8.58/4.88 = ( F @ Y4 ) ) )
% 8.58/4.88 => ( L = zero_zero_real ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9409_DERIV__const__average,axiom,
% 8.58/4.88 ! [A: real,B: real,V: real > real,K: real] :
% 8.58/4.88 ( ( A != B )
% 8.58/4.88 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 8.58/4.88 => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.58/4.88 = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9410_DERIV__local__max,axiom,
% 8.58/4.88 ! [F: real > real,L: real,X: real,D: real] :
% 8.58/4.88 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.88 => ( ( ord_less_real @ zero_zero_real @ D )
% 8.58/4.88 => ( ! [Y4: real] :
% 8.58/4.88 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 8.58/4.88 => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X ) ) )
% 8.58/4.88 => ( L = zero_zero_real ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9411_DERIV__local__min,axiom,
% 8.58/4.88 ! [F: real > real,L: real,X: real,D: real] :
% 8.58/4.88 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.88 => ( ( ord_less_real @ zero_zero_real @ D )
% 8.58/4.88 => ( ! [Y4: real] :
% 8.58/4.88 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 8.58/4.88 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
% 8.58/4.88 => ( L = zero_zero_real ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9412_DERIV__ln__divide,axiom,
% 8.58/4.88 ! [X: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.88 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9413_DERIV__pow,axiom,
% 8.58/4.88 ! [N: nat,X: real,S: set_real] :
% 8.58/4.88 ( has_fi5821293074295781190e_real
% 8.58/4.88 @ ^ [X3: real] : ( power_power_real @ X3 @ N )
% 8.58/4.88 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 8.58/4.88 @ ( topolo2177554685111907308n_real @ X @ S ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9414_DERIV__fun__pow,axiom,
% 8.58/4.88 ! [G: real > real,M: real,X: real,N: nat] :
% 8.58/4.88 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.88 => ( has_fi5821293074295781190e_real
% 8.58/4.88 @ ^ [X3: real] : ( power_power_real @ ( G @ X3 ) @ N )
% 8.58/4.88 @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
% 8.58/4.88 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9415_has__real__derivative__powr,axiom,
% 8.58/4.88 ! [Z2: real,R: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ Z2 )
% 8.58/4.88 => ( has_fi5821293074295781190e_real
% 8.58/4.88 @ ^ [Z5: real] : ( powr_real @ Z5 @ R )
% 8.58/4.88 @ ( times_times_real @ R @ ( powr_real @ Z2 @ ( minus_minus_real @ R @ one_one_real ) ) )
% 8.58/4.88 @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9416_DERIV__log,axiom,
% 8.58/4.88 ! [X: real,B: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.88 => ( has_fi5821293074295781190e_real @ ( log2 @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9417_DERIV__fun__powr,axiom,
% 8.58/4.88 ! [G: real > real,M: real,X: real,R: real] :
% 8.58/4.88 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.88 => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 8.58/4.88 => ( has_fi5821293074295781190e_real
% 8.58/4.88 @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ R )
% 8.58/4.88 @ ( times_times_real @ ( times_times_real @ R @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 8.58/4.88 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9418_DERIV__powr,axiom,
% 8.58/4.88 ! [G: real > real,M: real,X: real,F: real > real,R: real] :
% 8.58/4.88 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.88 => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 8.58/4.88 => ( ( has_fi5821293074295781190e_real @ F @ R @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.88 => ( has_fi5821293074295781190e_real
% 8.58/4.88 @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) )
% 8.58/4.88 @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
% 8.58/4.88 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9419_DERIV__real__sqrt,axiom,
% 8.58/4.88 ! [X: real] :
% 8.58/4.88 ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.88 => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9420_DERIV__series_H,axiom,
% 8.58/4.88 ! [F: real > nat > real,F3: real > nat > real,X0: real,A: real,B: real,L4: nat > real] :
% 8.58/4.88 ( ! [N2: nat] :
% 8.58/4.88 ( has_fi5821293074295781190e_real
% 8.58/4.88 @ ^ [X3: real] : ( F @ X3 @ N2 )
% 8.58/4.88 @ ( F3 @ X0 @ N2 )
% 8.58/4.88 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 8.58/4.88 => ( ! [X4: real] :
% 8.58/4.88 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 8.58/4.88 => ( summable_real @ ( F @ X4 ) ) )
% 8.58/4.88 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 8.58/4.88 => ( ( summable_real @ ( F3 @ X0 ) )
% 8.58/4.88 => ( ( summable_real @ L4 )
% 8.58/4.88 => ( ! [N2: nat,X4: real,Y4: real] :
% 8.58/4.88 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 8.58/4.88 => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 8.58/4.88 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X4 @ N2 ) @ ( F @ Y4 @ N2 ) ) ) @ ( times_times_real @ ( L4 @ N2 ) @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y4 ) ) ) ) ) )
% 8.58/4.88 => ( has_fi5821293074295781190e_real
% 8.58/4.88 @ ^ [X3: real] : ( suminf_real @ ( F @ X3 ) )
% 8.58/4.88 @ ( suminf_real @ ( F3 @ X0 ) )
% 8.58/4.88 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9421_DERIV__arctan,axiom,
% 8.58/4.88 ! [X: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9422_arsinh__real__has__field__derivative,axiom,
% 8.58/4.88 ! [X: real,A3: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A3 ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9423_DERIV__real__sqrt__generic,axiom,
% 8.58/4.88 ! [X: real,D3: real] :
% 8.58/4.88 ( ( X != zero_zero_real )
% 8.58/4.88 => ( ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.88 => ( D3
% 8.58/4.88 = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 8.58/4.88 => ( ( ( ord_less_real @ X @ zero_zero_real )
% 8.58/4.88 => ( D3
% 8.58/4.88 = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 8.58/4.88 => ( has_fi5821293074295781190e_real @ sqrt @ D3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9424_arcosh__real__has__field__derivative,axiom,
% 8.58/4.88 ! [X: real,A3: set_real] :
% 8.58/4.88 ( ( ord_less_real @ one_one_real @ X )
% 8.58/4.88 => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A3 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9425_artanh__real__has__field__derivative,axiom,
% 8.58/4.88 ! [X: real,A3: set_real] :
% 8.58/4.88 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.88 => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A3 ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9426_DERIV__power__series_H,axiom,
% 8.58/4.88 ! [R3: real,F: nat > real,X0: real] :
% 8.58/4.88 ( ! [X4: real] :
% 8.58/4.88 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R3 ) @ R3 ) )
% 8.58/4.88 => ( summable_real
% 8.58/4.88 @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X4 @ N3 ) ) ) )
% 8.58/4.88 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R3 ) @ R3 ) )
% 8.58/4.88 => ( ( ord_less_real @ zero_zero_real @ R3 )
% 8.58/4.88 => ( has_fi5821293074295781190e_real
% 8.58/4.88 @ ^ [X3: real] :
% 8.58/4.88 ( suminf_real
% 8.58/4.88 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X3 @ ( suc @ N3 ) ) ) )
% 8.58/4.88 @ ( suminf_real
% 8.58/4.88 @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X0 @ N3 ) ) )
% 8.58/4.88 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9427_DERIV__real__root,axiom,
% 8.58/4.88 ! [N: nat,X: real] :
% 8.58/4.88 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.88 => ( ( ord_less_real @ zero_zero_real @ X )
% 8.58/4.88 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9428_DERIV__arccos,axiom,
% 8.58/4.88 ! [X: real] :
% 8.58/4.88 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.88 => ( ( ord_less_real @ X @ one_one_real )
% 8.58/4.88 => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9429_DERIV__arcsin,axiom,
% 8.58/4.88 ! [X: real] :
% 8.58/4.88 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.88 => ( ( ord_less_real @ X @ one_one_real )
% 8.58/4.88 => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9430_Maclaurin__all__le__objl,axiom,
% 8.58/4.88 ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 8.58/4.88 ( ( ( ( Diff @ zero_zero_nat )
% 8.58/4.88 = F )
% 8.58/4.88 & ! [M2: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 8.58/4.88 => ? [T3: real] :
% 8.58/4.88 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 8.58/4.88 & ( ( F @ X )
% 8.58/4.88 = ( plus_plus_real
% 8.58/4.88 @ ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.88 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9431_Maclaurin__all__le,axiom,
% 8.58/4.88 ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 8.58/4.88 ( ( ( Diff @ zero_zero_nat )
% 8.58/4.88 = F )
% 8.58/4.88 => ( ! [M2: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 8.58/4.88 => ? [T3: real] :
% 8.58/4.88 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 8.58/4.88 & ( ( F @ X )
% 8.58/4.88 = ( plus_plus_real
% 8.58/4.88 @ ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.88 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9432_DERIV__odd__real__root,axiom,
% 8.58/4.88 ! [N: nat,X: real] :
% 8.58/4.88 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.88 => ( ( X != zero_zero_real )
% 8.58/4.88 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9433_Maclaurin__minus,axiom,
% 8.58/4.88 ! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
% 8.58/4.88 ( ( ord_less_real @ H @ zero_zero_real )
% 8.58/4.88 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.88 => ( ( ( Diff @ zero_zero_nat )
% 8.58/4.88 = F )
% 8.58/4.88 => ( ! [M2: nat,T3: real] :
% 8.58/4.88 ( ( ( ord_less_nat @ M2 @ N )
% 8.58/4.88 & ( ord_less_eq_real @ H @ T3 )
% 8.58/4.88 & ( ord_less_eq_real @ T3 @ zero_zero_real ) )
% 8.58/4.88 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 8.58/4.88 => ? [T3: real] :
% 8.58/4.88 ( ( ord_less_real @ H @ T3 )
% 8.58/4.88 & ( ord_less_real @ T3 @ zero_zero_real )
% 8.58/4.88 & ( ( F @ H )
% 8.58/4.88 = ( plus_plus_real
% 8.58/4.88 @ ( groups6591440286371151544t_real
% 8.58/4.88 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H @ M3 ) )
% 8.58/4.88 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.88 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
% 8.58/4.88
% 8.58/4.88 thf(fact_9434_Maclaurin2,axiom,
% 8.58/4.88 ! [H: real,Diff: nat > real > real,F: real > real,N: nat] :
% 8.58/4.89 ( ( ord_less_real @ zero_zero_real @ H )
% 8.58/4.89 => ( ( ( Diff @ zero_zero_nat )
% 8.58/4.89 = F )
% 8.58/4.89 => ( ! [M2: nat,T3: real] :
% 8.58/4.89 ( ( ( ord_less_nat @ M2 @ N )
% 8.58/4.89 & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 8.58/4.89 & ( ord_less_eq_real @ T3 @ H ) )
% 8.58/4.89 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 8.58/4.89 => ? [T3: real] :
% 8.58/4.89 ( ( ord_less_real @ zero_zero_real @ T3 )
% 8.58/4.89 & ( ord_less_eq_real @ T3 @ H )
% 8.58/4.89 & ( ( F @ H )
% 8.58/4.89 = ( plus_plus_real
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H @ M3 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.89 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9435_Maclaurin,axiom,
% 8.58/4.89 ! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
% 8.58/4.89 ( ( ord_less_real @ zero_zero_real @ H )
% 8.58/4.89 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.89 => ( ( ( Diff @ zero_zero_nat )
% 8.58/4.89 = F )
% 8.58/4.89 => ( ! [M2: nat,T3: real] :
% 8.58/4.89 ( ( ( ord_less_nat @ M2 @ N )
% 8.58/4.89 & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 8.58/4.89 & ( ord_less_eq_real @ T3 @ H ) )
% 8.58/4.89 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 8.58/4.89 => ? [T3: real] :
% 8.58/4.89 ( ( ord_less_real @ zero_zero_real @ T3 )
% 8.58/4.89 & ( ord_less_real @ T3 @ H )
% 8.58/4.89 & ( ( F @ H )
% 8.58/4.89 = ( plus_plus_real
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H @ M3 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.89 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9436_Maclaurin__all__lt,axiom,
% 8.58/4.89 ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 8.58/4.89 ( ( ( Diff @ zero_zero_nat )
% 8.58/4.89 = F )
% 8.58/4.89 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.89 => ( ( X != zero_zero_real )
% 8.58/4.89 => ( ! [M2: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 8.58/4.89 => ? [T3: real] :
% 8.58/4.89 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
% 8.58/4.89 & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 8.58/4.89 & ( ( F @ X )
% 8.58/4.89 = ( plus_plus_real
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.89 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9437_Maclaurin__bi__le,axiom,
% 8.58/4.89 ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 8.58/4.89 ( ( ( Diff @ zero_zero_nat )
% 8.58/4.89 = F )
% 8.58/4.89 => ( ! [M2: nat,T3: real] :
% 8.58/4.89 ( ( ( ord_less_nat @ M2 @ N )
% 8.58/4.89 & ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) ) )
% 8.58/4.89 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 8.58/4.89 => ? [T3: real] :
% 8.58/4.89 ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
% 8.58/4.89 & ( ( F @ X )
% 8.58/4.89 = ( plus_plus_real
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.89 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9438_Taylor__down,axiom,
% 8.58/4.89 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 8.58/4.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.89 => ( ( ( Diff @ zero_zero_nat )
% 8.58/4.89 = F )
% 8.58/4.89 => ( ! [M2: nat,T3: real] :
% 8.58/4.89 ( ( ( ord_less_nat @ M2 @ N )
% 8.58/4.89 & ( ord_less_eq_real @ A @ T3 )
% 8.58/4.89 & ( ord_less_eq_real @ T3 @ B ) )
% 8.58/4.89 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 8.58/4.89 => ( ( ord_less_real @ A @ C )
% 8.58/4.89 => ( ( ord_less_eq_real @ C @ B )
% 8.58/4.89 => ? [T3: real] :
% 8.58/4.89 ( ( ord_less_real @ A @ T3 )
% 8.58/4.89 & ( ord_less_real @ T3 @ C )
% 8.58/4.89 & ( ( F @ A )
% 8.58/4.89 = ( plus_plus_real
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M3 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.89 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9439_Taylor__up,axiom,
% 8.58/4.89 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 8.58/4.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.89 => ( ( ( Diff @ zero_zero_nat )
% 8.58/4.89 = F )
% 8.58/4.89 => ( ! [M2: nat,T3: real] :
% 8.58/4.89 ( ( ( ord_less_nat @ M2 @ N )
% 8.58/4.89 & ( ord_less_eq_real @ A @ T3 )
% 8.58/4.89 & ( ord_less_eq_real @ T3 @ B ) )
% 8.58/4.89 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 8.58/4.89 => ( ( ord_less_eq_real @ A @ C )
% 8.58/4.89 => ( ( ord_less_real @ C @ B )
% 8.58/4.89 => ? [T3: real] :
% 8.58/4.89 ( ( ord_less_real @ C @ T3 )
% 8.58/4.89 & ( ord_less_real @ T3 @ B )
% 8.58/4.89 & ( ( F @ B )
% 8.58/4.89 = ( plus_plus_real
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M3 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.89 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9440_Taylor,axiom,
% 8.58/4.89 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
% 8.58/4.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.89 => ( ( ( Diff @ zero_zero_nat )
% 8.58/4.89 = F )
% 8.58/4.89 => ( ! [M2: nat,T3: real] :
% 8.58/4.89 ( ( ( ord_less_nat @ M2 @ N )
% 8.58/4.89 & ( ord_less_eq_real @ A @ T3 )
% 8.58/4.89 & ( ord_less_eq_real @ T3 @ B ) )
% 8.58/4.89 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 8.58/4.89 => ( ( ord_less_eq_real @ A @ C )
% 8.58/4.89 => ( ( ord_less_eq_real @ C @ B )
% 8.58/4.89 => ( ( ord_less_eq_real @ A @ X )
% 8.58/4.89 => ( ( ord_less_eq_real @ X @ B )
% 8.58/4.89 => ( ( X != C )
% 8.58/4.89 => ? [T3: real] :
% 8.58/4.89 ( ( ( ord_less_real @ X @ C )
% 8.58/4.89 => ( ( ord_less_real @ X @ T3 )
% 8.58/4.89 & ( ord_less_real @ T3 @ C ) ) )
% 8.58/4.89 & ( ~ ( ord_less_real @ X @ C )
% 8.58/4.89 => ( ( ord_less_real @ C @ T3 )
% 8.58/4.89 & ( ord_less_real @ T3 @ X ) ) )
% 8.58/4.89 & ( ( F @ X )
% 8.58/4.89 = ( plus_plus_real
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M3 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ N ) )
% 8.58/4.89 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9441_Maclaurin__lemma2,axiom,
% 8.58/4.89 ! [N: nat,H: real,Diff: nat > real > real,K: nat,B5: real] :
% 8.58/4.89 ( ! [M2: nat,T3: real] :
% 8.58/4.89 ( ( ( ord_less_nat @ M2 @ N )
% 8.58/4.89 & ( ord_less_eq_real @ zero_zero_real @ T3 )
% 8.58/4.89 & ( ord_less_eq_real @ T3 @ H ) )
% 8.58/4.89 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
% 8.58/4.89 => ( ( N
% 8.58/4.89 = ( suc @ K ) )
% 8.58/4.89 => ! [M4: nat,T4: real] :
% 8.58/4.89 ( ( ( ord_less_nat @ M4 @ N )
% 8.58/4.89 & ( ord_less_eq_real @ zero_zero_real @ T4 )
% 8.58/4.89 & ( ord_less_eq_real @ T4 @ H ) )
% 8.58/4.89 => ( has_fi5821293074295781190e_real
% 8.58/4.89 @ ^ [U3: real] :
% 8.58/4.89 ( minus_minus_real @ ( Diff @ M4 @ U3 )
% 8.58/4.89 @ ( plus_plus_real
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [P3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M4 @ P3 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P3 ) ) @ ( power_power_real @ U3 @ P3 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M4 ) ) )
% 8.58/4.89 @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ U3 @ ( minus_minus_nat @ N @ M4 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M4 ) ) ) ) ) )
% 8.58/4.89 @ ( minus_minus_real @ ( Diff @ ( suc @ M4 ) @ T4 )
% 8.58/4.89 @ ( plus_plus_real
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [P3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M4 ) @ P3 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P3 ) ) @ ( power_power_real @ T4 @ P3 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M4 ) ) ) )
% 8.58/4.89 @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ T4 @ ( minus_minus_nat @ N @ ( suc @ M4 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M4 ) ) ) ) ) ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9442_DERIV__arctan__series,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.89 => ( has_fi5821293074295781190e_real
% 8.58/4.89 @ ^ [X10: real] :
% 8.58/4.89 ( suminf_real
% 8.58/4.89 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X10 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 8.58/4.89 @ ( suminf_real
% 8.58/4.89 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9443_DERIV__even__real__root,axiom,
% 8.58/4.89 ! [N: nat,X: real] :
% 8.58/4.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.89 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.89 => ( ( ord_less_real @ X @ zero_zero_real )
% 8.58/4.89 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9444_LIM__fun__gt__zero,axiom,
% 8.58/4.89 ! [F: real > real,L: real,C: real] :
% 8.58/4.89 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 8.58/4.89 => ( ( ord_less_real @ zero_zero_real @ L )
% 8.58/4.89 => ? [R2: real] :
% 8.58/4.89 ( ( ord_less_real @ zero_zero_real @ R2 )
% 8.58/4.89 & ! [X5: real] :
% 8.58/4.89 ( ( ( X5 != C )
% 8.58/4.89 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R2 ) )
% 8.58/4.89 => ( ord_less_real @ zero_zero_real @ ( F @ X5 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9445_LIM__fun__not__zero,axiom,
% 8.58/4.89 ! [F: real > real,L: real,C: real] :
% 8.58/4.89 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 8.58/4.89 => ( ( L != zero_zero_real )
% 8.58/4.89 => ? [R2: real] :
% 8.58/4.89 ( ( ord_less_real @ zero_zero_real @ R2 )
% 8.58/4.89 & ! [X5: real] :
% 8.58/4.89 ( ( ( X5 != C )
% 8.58/4.89 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R2 ) )
% 8.58/4.89 => ( ( F @ X5 )
% 8.58/4.89 != zero_zero_real ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9446_LIM__fun__less__zero,axiom,
% 8.58/4.89 ! [F: real > real,L: real,C: real] :
% 8.58/4.89 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 8.58/4.89 => ( ( ord_less_real @ L @ zero_zero_real )
% 8.58/4.89 => ? [R2: real] :
% 8.58/4.89 ( ( ord_less_real @ zero_zero_real @ R2 )
% 8.58/4.89 & ! [X5: real] :
% 8.58/4.89 ( ( ( X5 != C )
% 8.58/4.89 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R2 ) )
% 8.58/4.89 => ( ord_less_real @ ( F @ X5 ) @ zero_zero_real ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9447_isCont__real__sqrt,axiom,
% 8.58/4.89 ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ sqrt ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9448_isCont__real__root,axiom,
% 8.58/4.89 ! [X: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( root @ N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9449_isCont__arcosh,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ( ( ord_less_real @ one_one_real @ X )
% 8.58/4.89 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9450_LIM__cos__div__sin,axiom,
% 8.58/4.89 ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( cos_real @ X3 ) @ ( sin_real @ X3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9451_isCont__arccos,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.89 => ( ( ord_less_real @ X @ one_one_real )
% 8.58/4.89 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9452_isCont__arcsin,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.89 => ( ( ord_less_real @ X @ one_one_real )
% 8.58/4.89 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9453_isCont__artanh,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 8.58/4.89 => ( ( ord_less_real @ X @ one_one_real )
% 8.58/4.89 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9454_isCont__inverse__function,axiom,
% 8.58/4.89 ! [D: real,X: real,G: real > real,F: real > real] :
% 8.58/4.89 ( ( ord_less_real @ zero_zero_real @ D )
% 8.58/4.89 => ( ! [Z4: real] :
% 8.58/4.89 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X ) ) @ D )
% 8.58/4.89 => ( ( G @ ( F @ Z4 ) )
% 8.58/4.89 = Z4 ) )
% 8.58/4.89 => ( ! [Z4: real] :
% 8.58/4.89 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X ) ) @ D )
% 8.58/4.89 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) )
% 8.58/4.89 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9455_GMVT_H,axiom,
% 8.58/4.89 ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F3: real > real] :
% 8.58/4.89 ( ( ord_less_real @ A @ B )
% 8.58/4.89 => ( ! [Z4: real] :
% 8.58/4.89 ( ( ord_less_eq_real @ A @ Z4 )
% 8.58/4.89 => ( ( ord_less_eq_real @ Z4 @ B )
% 8.58/4.89 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
% 8.58/4.89 => ( ! [Z4: real] :
% 8.58/4.89 ( ( ord_less_eq_real @ A @ Z4 )
% 8.58/4.89 => ( ( ord_less_eq_real @ Z4 @ B )
% 8.58/4.89 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ G ) ) )
% 8.58/4.89 => ( ! [Z4: real] :
% 8.58/4.89 ( ( ord_less_real @ A @ Z4 )
% 8.58/4.89 => ( ( ord_less_real @ Z4 @ B )
% 8.58/4.89 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
% 8.58/4.89 => ( ! [Z4: real] :
% 8.58/4.89 ( ( ord_less_real @ A @ Z4 )
% 8.58/4.89 => ( ( ord_less_real @ Z4 @ B )
% 8.58/4.89 => ( has_fi5821293074295781190e_real @ F @ ( F3 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
% 8.58/4.89 => ? [C3: real] :
% 8.58/4.89 ( ( ord_less_real @ A @ C3 )
% 8.58/4.89 & ( ord_less_real @ C3 @ B )
% 8.58/4.89 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
% 8.58/4.89 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F3 @ C3 ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9456_summable__Leibniz_I3_J,axiom,
% 8.58/4.89 ! [A: nat > real] :
% 8.58/4.89 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.58/4.89 => ( ( topolo6980174941875973593q_real @ A )
% 8.58/4.89 => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 8.58/4.89 => ! [N6: nat] :
% 8.58/4.89 ( member_real
% 8.58/4.89 @ ( suminf_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 8.58/4.89 @ ( set_or1222579329274155063t_real
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) )
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9457_summable__Leibniz_I2_J,axiom,
% 8.58/4.89 ! [A: nat > real] :
% 8.58/4.89 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.58/4.89 => ( ( topolo6980174941875973593q_real @ A )
% 8.58/4.89 => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 8.58/4.89 => ! [N6: nat] :
% 8.58/4.89 ( member_real
% 8.58/4.89 @ ( suminf_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 8.58/4.89 @ ( set_or1222579329274155063t_real
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9458_mult__nat__right__at__top,axiom,
% 8.58/4.89 ! [C: nat] :
% 8.58/4.89 ( ( ord_less_nat @ zero_zero_nat @ C )
% 8.58/4.89 => ( filterlim_nat_nat
% 8.58/4.89 @ ^ [X3: nat] : ( times_times_nat @ X3 @ C )
% 8.58/4.89 @ at_top_nat
% 8.58/4.89 @ at_top_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9459_mult__nat__left__at__top,axiom,
% 8.58/4.89 ! [C: nat] :
% 8.58/4.89 ( ( ord_less_nat @ zero_zero_nat @ C )
% 8.58/4.89 => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9460_LIMSEQ__root,axiom,
% 8.58/4.89 ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] : ( root @ N3 @ ( semiri5074537144036343181t_real @ N3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ one_one_real )
% 8.58/4.89 @ at_top_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9461_nested__sequence__unique,axiom,
% 8.58/4.89 ! [F: nat > real,G: nat > real] :
% 8.58/4.89 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.58/4.89 => ( ! [N2: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
% 8.58/4.89 => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 8.58/4.89 => ( ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.58/4.89 @ at_top_nat )
% 8.58/4.89 => ? [L3: real] :
% 8.58/4.89 ( ! [N6: nat] : ( ord_less_eq_real @ ( F @ N6 ) @ L3 )
% 8.58/4.89 & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat )
% 8.58/4.89 & ! [N6: nat] : ( ord_less_eq_real @ L3 @ ( G @ N6 ) )
% 8.58/4.89 & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9462_lim__inverse__n_H,axiom,
% 8.58/4.89 ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.58/4.89 @ at_top_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9463_LIMSEQ__root__const,axiom,
% 8.58/4.89 ! [C: real] :
% 8.58/4.89 ( ( ord_less_real @ zero_zero_real @ C )
% 8.58/4.89 => ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] : ( root @ N3 @ C )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ one_one_real )
% 8.58/4.89 @ at_top_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9464_LIMSEQ__inverse__real__of__nat,axiom,
% 8.58/4.89 ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.58/4.89 @ at_top_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9465_LIMSEQ__inverse__real__of__nat__add,axiom,
% 8.58/4.89 ! [R: real] :
% 8.58/4.89 ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] : ( plus_plus_real @ R @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ R )
% 8.58/4.89 @ at_top_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9466_increasing__LIMSEQ,axiom,
% 8.58/4.89 ! [F: nat > real,L: real] :
% 8.58/4.89 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.58/4.89 => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ L )
% 8.58/4.89 => ( ! [E2: real] :
% 8.58/4.89 ( ( ord_less_real @ zero_zero_real @ E2 )
% 8.58/4.89 => ? [N6: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N6 ) @ E2 ) ) )
% 8.58/4.89 => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9467_LIMSEQ__realpow__zero,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.89 => ( ( ord_less_real @ X @ one_one_real )
% 8.58/4.89 => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9468_LIMSEQ__divide__realpow__zero,axiom,
% 8.58/4.89 ! [X: real,A: real] :
% 8.58/4.89 ( ( ord_less_real @ one_one_real @ X )
% 8.58/4.89 => ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.58/4.89 @ at_top_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9469_LIMSEQ__abs__realpow__zero2,axiom,
% 8.58/4.89 ! [C: real] :
% 8.58/4.89 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 8.58/4.89 => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9470_LIMSEQ__abs__realpow__zero,axiom,
% 8.58/4.89 ! [C: real] :
% 8.58/4.89 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 8.58/4.89 => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9471_LIMSEQ__inverse__realpow__zero,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ( ( ord_less_real @ one_one_real @ X )
% 8.58/4.89 => ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.58/4.89 @ at_top_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9472_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
% 8.58/4.89 ! [R: real] :
% 8.58/4.89 ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] : ( plus_plus_real @ R @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ R )
% 8.58/4.89 @ at_top_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9473_tendsto__exp__limit__sequentially,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ) @ N3 )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 8.58/4.89 @ at_top_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9474_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 8.58/4.89 ! [R: real] :
% 8.58/4.89 ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] : ( times_times_real @ R @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ R )
% 8.58/4.89 @ at_top_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9475_summable__Leibniz_I1_J,axiom,
% 8.58/4.89 ! [A: nat > real] :
% 8.58/4.89 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.58/4.89 => ( ( topolo6980174941875973593q_real @ A )
% 8.58/4.89 => ( summable_real
% 8.58/4.89 @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9476_summable,axiom,
% 8.58/4.89 ! [A: nat > real] :
% 8.58/4.89 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.58/4.89 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 8.58/4.89 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 8.58/4.89 => ( summable_real
% 8.58/4.89 @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9477_cos__diff__limit__1,axiom,
% 8.58/4.89 ! [Theta: nat > real,Theta2: real] :
% 8.58/4.89 ( ( filterlim_nat_real
% 8.58/4.89 @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ one_one_real )
% 8.58/4.89 @ at_top_nat )
% 8.58/4.89 => ~ ! [K3: nat > int] :
% 8.58/4.89 ~ ( filterlim_nat_real
% 8.58/4.89 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ Theta2 )
% 8.58/4.89 @ at_top_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9478_cos__limit__1,axiom,
% 8.58/4.89 ! [Theta: nat > real] :
% 8.58/4.89 ( ( filterlim_nat_real
% 8.58/4.89 @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ one_one_real )
% 8.58/4.89 @ at_top_nat )
% 8.58/4.89 => ? [K3: nat > int] :
% 8.58/4.89 ( filterlim_nat_real
% 8.58/4.89 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.58/4.89 @ at_top_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9479_summable__Leibniz_I4_J,axiom,
% 8.58/4.89 ! [A: nat > real] :
% 8.58/4.89 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.58/4.89 => ( ( topolo6980174941875973593q_real @ A )
% 8.58/4.89 => ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] :
% 8.58/4.89 ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real
% 8.58/4.89 @ ( suminf_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 8.58/4.89 @ at_top_nat ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9480_zeroseq__arctan__series,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 8.58/4.89 => ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.58/4.89 @ at_top_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9481_summable__Leibniz_H_I2_J,axiom,
% 8.58/4.89 ! [A: nat > real,N: nat] :
% 8.58/4.89 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.58/4.89 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 8.58/4.89 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 8.58/4.89 => ( ord_less_eq_real
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.58/4.89 @ ( suminf_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9482_summable__Leibniz_H_I3_J,axiom,
% 8.58/4.89 ! [A: nat > real] :
% 8.58/4.89 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.58/4.89 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 8.58/4.89 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 8.58/4.89 => ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] :
% 8.58/4.89 ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real
% 8.58/4.89 @ ( suminf_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 8.58/4.89 @ at_top_nat ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9483_sums__alternating__upper__lower,axiom,
% 8.58/4.89 ! [A: nat > real] :
% 8.58/4.89 ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 8.58/4.89 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 8.58/4.89 => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.58/4.89 => ? [L3: real] :
% 8.58/4.89 ( ! [N6: nat] :
% 8.58/4.89 ( ord_less_eq_real
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
% 8.58/4.89 @ L3 )
% 8.58/4.89 & ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] :
% 8.58/4.89 ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ L3 )
% 8.58/4.89 @ at_top_nat )
% 8.58/4.89 & ! [N6: nat] :
% 8.58/4.89 ( ord_less_eq_real @ L3
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) )
% 8.58/4.89 & ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] :
% 8.58/4.89 ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ L3 )
% 8.58/4.89 @ at_top_nat ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9484_summable__Leibniz_I5_J,axiom,
% 8.58/4.89 ! [A: nat > real] :
% 8.58/4.89 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.58/4.89 => ( ( topolo6980174941875973593q_real @ A )
% 8.58/4.89 => ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] :
% 8.58/4.89 ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real
% 8.58/4.89 @ ( suminf_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 8.58/4.89 @ at_top_nat ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9485_summable__Leibniz_H_I5_J,axiom,
% 8.58/4.89 ! [A: nat > real] :
% 8.58/4.89 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.58/4.89 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 8.58/4.89 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 8.58/4.89 => ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] :
% 8.58/4.89 ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real
% 8.58/4.89 @ ( suminf_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 8.58/4.89 @ at_top_nat ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9486_summable__Leibniz_H_I4_J,axiom,
% 8.58/4.89 ! [A: nat > real,N: nat] :
% 8.58/4.89 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.58/4.89 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 8.58/4.89 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 8.58/4.89 => ( ord_less_eq_real
% 8.58/4.89 @ ( suminf_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 8.58/4.89 @ ( groups6591440286371151544t_real
% 8.58/4.89 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.58/4.89 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9487_real__bounded__linear,axiom,
% 8.58/4.89 ( real_V5970128139526366754l_real
% 8.58/4.89 = ( ^ [F2: real > real] :
% 8.58/4.89 ? [C4: real] :
% 8.58/4.89 ( F2
% 8.58/4.89 = ( ^ [X3: real] : ( times_times_real @ X3 @ C4 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9488_filterlim__Suc,axiom,
% 8.58/4.89 filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9489_tendsto__arctan__at__bot,axiom,
% 8.58/4.89 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9490_tendsto__exp__limit__at__right,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ( filterlim_real_real
% 8.58/4.89 @ ^ [Y6: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y6 ) ) @ ( divide_divide_real @ one_one_real @ Y6 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9491_artanh__real__at__right__1,axiom,
% 8.58/4.89 filterlim_real_real @ artanh_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ one_one_real ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9492_filterlim__tan__at__right,axiom,
% 8.58/4.89 filterlim_real_real @ tan_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9493_tanh__real__at__bot,axiom,
% 8.58/4.89 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9494_tendsto__arcosh__at__left__1,axiom,
% 8.58/4.89 filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9495_filterlim__pow__at__bot__odd,axiom,
% 8.58/4.89 ! [N: nat,F: real > real,F4: filter_real] :
% 8.58/4.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.89 => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
% 8.58/4.89 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
% 8.58/4.89 @ at_bot_real
% 8.58/4.89 @ F4 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9496_filterlim__pow__at__bot__even,axiom,
% 8.58/4.89 ! [N: nat,F: real > real,F4: filter_real] :
% 8.58/4.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.89 => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
% 8.58/4.89 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
% 8.58/4.89 @ at_top_real
% 8.58/4.89 @ F4 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9497_sqrt__at__top,axiom,
% 8.58/4.89 filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9498_filterlim__real__sequentially,axiom,
% 8.58/4.89 filterlim_nat_real @ semiri5074537144036343181t_real @ at_top_real @ at_top_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9499_filterlim__real__at__infinity__sequentially,axiom,
% 8.58/4.89 filterlim_nat_real @ semiri5074537144036343181t_real @ at_infinity_real @ at_top_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9500_greaterThan__0,axiom,
% 8.58/4.89 ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 8.58/4.89 = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9501_tanh__real__at__top,axiom,
% 8.58/4.89 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9502_artanh__real__at__left__1,axiom,
% 8.58/4.89 filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9503_greaterThan__Suc,axiom,
% 8.58/4.89 ! [K: nat] :
% 8.58/4.89 ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 8.58/4.89 = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9504_ln__x__over__x__tendsto__0,axiom,
% 8.58/4.89 ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ X3 ) @ X3 )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.58/4.89 @ at_top_real ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9505_tendsto__at__topI__sequentially__real,axiom,
% 8.58/4.89 ! [F: real > real,Y: real] :
% 8.58/4.89 ( ( order_mono_real_real @ F )
% 8.58/4.89 => ( ( filterlim_nat_real
% 8.58/4.89 @ ^ [N3: nat] : ( F @ ( semiri5074537144036343181t_real @ N3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ Y )
% 8.58/4.89 @ at_top_nat )
% 8.58/4.89 => ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Y ) @ at_top_real ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9506_tendsto__power__div__exp__0,axiom,
% 8.58/4.89 ! [K: nat] :
% 8.58/4.89 ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( power_power_real @ X3 @ K ) @ ( exp_real @ X3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.58/4.89 @ at_top_real ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9507_tendsto__exp__limit__at__top,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ( filterlim_real_real
% 8.58/4.89 @ ^ [Y6: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y6 ) ) @ Y6 )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 8.58/4.89 @ at_top_real ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9508_filterlim__tan__at__left,axiom,
% 8.58/4.89 filterlim_real_real @ tan_real @ at_top_real @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( set_or5984915006950818249n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9509_tendsto__arctan__at__top,axiom,
% 8.58/4.89 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9510_lhopital__right__at__top,axiom,
% 8.58/4.89 ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
% 8.58/4.89 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] :
% 8.58/4.89 ( ( G2 @ X3 )
% 8.58/4.89 != zero_zero_real )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 8.58/4.89 => ( ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ Y )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ Y )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9511_eventually__sequentially__Suc,axiom,
% 8.58/4.89 ! [P: nat > $o] :
% 8.58/4.89 ( ( eventually_nat
% 8.58/4.89 @ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
% 8.58/4.89 @ at_top_nat )
% 8.58/4.89 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9512_eventually__sequentially__seg,axiom,
% 8.58/4.89 ! [P: nat > $o,K: nat] :
% 8.58/4.89 ( ( eventually_nat
% 8.58/4.89 @ ^ [N3: nat] : ( P @ ( plus_plus_nat @ N3 @ K ) )
% 8.58/4.89 @ at_top_nat )
% 8.58/4.89 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9513_sequentially__offset,axiom,
% 8.58/4.89 ! [P: nat > $o,K: nat] :
% 8.58/4.89 ( ( eventually_nat @ P @ at_top_nat )
% 8.58/4.89 => ( eventually_nat
% 8.58/4.89 @ ^ [I4: nat] : ( P @ ( plus_plus_nat @ I4 @ K ) )
% 8.58/4.89 @ at_top_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9514_eventually__at__right__to__0,axiom,
% 8.58/4.89 ! [P: real > $o,A: real] :
% 8.58/4.89 ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 8.58/4.89 = ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( P @ ( plus_plus_real @ X3 @ A ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9515_lhopital__left__at__top__at__top,axiom,
% 8.58/4.89 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 8.58/4.89 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 8.58/4.89 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 8.58/4.89 => ( ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 8.58/4.89 @ at_top_real
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 8.58/4.89 @ at_top_real
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9516_lhopital__at__top__at__top,axiom,
% 8.58/4.89 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 8.58/4.89 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 8.58/4.89 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 8.58/4.89 => ( ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 8.58/4.89 @ at_top_real
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 8.58/4.89 @ at_top_real
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9517_lhopital,axiom,
% 8.58/4.89 ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 8.58/4.89 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.89 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] :
% 8.58/4.89 ( ( G @ X3 )
% 8.58/4.89 != zero_zero_real )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] :
% 8.58/4.89 ( ( G2 @ X3 )
% 8.58/4.89 != zero_zero_real )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.89 => ( ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 8.58/4.89 @ F4
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 8.58/4.89 @ F4
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9518_lhopital__left,axiom,
% 8.58/4.89 ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 8.58/4.89 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 8.58/4.89 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] :
% 8.58/4.89 ( ( G @ X3 )
% 8.58/4.89 != zero_zero_real )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] :
% 8.58/4.89 ( ( G2 @ X3 )
% 8.58/4.89 != zero_zero_real )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 8.58/4.89 => ( ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 8.58/4.89 @ F4
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 8.58/4.89 @ F4
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9519_lhopital__right__at__top__at__top,axiom,
% 8.58/4.89 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 8.58/4.89 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 8.58/4.89 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 8.58/4.89 => ( ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 8.58/4.89 @ at_top_real
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 8.58/4.89 @ at_top_real
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9520_lhopital__left__at__top__at__bot,axiom,
% 8.58/4.89 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 8.58/4.89 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 8.58/4.89 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 8.58/4.89 => ( ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 8.58/4.89 @ at_bot_real
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 8.58/4.89 @ at_bot_real
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9521_lhopital__at__top__at__bot,axiom,
% 8.58/4.89 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 8.58/4.89 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 8.58/4.89 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 8.58/4.89 => ( ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 8.58/4.89 @ at_bot_real
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 8.58/4.89 @ at_bot_real
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9522_lhopital__at__top,axiom,
% 8.58/4.89 ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
% 8.58/4.89 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] :
% 8.58/4.89 ( ( G2 @ X3 )
% 8.58/4.89 != zero_zero_real )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.89 => ( ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ Y )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ Y )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9523_lhospital__at__top__at__top,axiom,
% 8.58/4.89 ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
% 8.58/4.89 ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] :
% 8.58/4.89 ( ( G2 @ X3 )
% 8.58/4.89 != zero_zero_real )
% 8.58/4.89 @ at_top_real )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ at_top_real )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ at_top_real )
% 8.58/4.89 => ( ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ X )
% 8.58/4.89 @ at_top_real )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ X )
% 8.58/4.89 @ at_top_real ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9524_lhopital__left__at__top,axiom,
% 8.58/4.89 ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
% 8.58/4.89 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] :
% 8.58/4.89 ( ( G2 @ X3 )
% 8.58/4.89 != zero_zero_real )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 8.58/4.89 => ( ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ Y )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ Y )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9525_lhopital__right,axiom,
% 8.58/4.89 ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 8.58/4.89 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 8.58/4.89 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] :
% 8.58/4.89 ( ( G @ X3 )
% 8.58/4.89 != zero_zero_real )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] :
% 8.58/4.89 ( ( G2 @ X3 )
% 8.58/4.89 != zero_zero_real )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 8.58/4.89 => ( ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 8.58/4.89 @ F4
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 8.58/4.89 @ F4
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9526_lhopital__right__0,axiom,
% 8.58/4.89 ! [F0: real > real,G0: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 8.58/4.89 ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 8.58/4.89 => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] :
% 8.58/4.89 ( ( G0 @ X3 )
% 8.58/4.89 != zero_zero_real )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] :
% 8.58/4.89 ( ( G2 @ X3 )
% 8.58/4.89 != zero_zero_real )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 8.58/4.89 => ( ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 8.58/4.89 @ F4
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F0 @ X3 ) @ ( G0 @ X3 ) )
% 8.58/4.89 @ F4
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9527_lhopital__right__at__top__at__bot,axiom,
% 8.58/4.89 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 8.58/4.89 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 8.58/4.89 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 8.58/4.89 => ( ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 8.58/4.89 @ at_bot_real
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 8.58/4.89 @ at_bot_real
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9528_lhopital__right__0__at__top,axiom,
% 8.58/4.89 ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
% 8.58/4.89 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] :
% 8.58/4.89 ( ( G2 @ X3 )
% 8.58/4.89 != zero_zero_real )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 8.58/4.89 => ( ( eventually_real
% 8.58/4.89 @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 8.58/4.89 => ( ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ X )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 8.58/4.89 => ( filterlim_real_real
% 8.58/4.89 @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
% 8.58/4.89 @ ( topolo2815343760600316023s_real @ X )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9529_GMVT,axiom,
% 8.58/4.89 ! [A: real,B: real,F: real > real,G: real > real] :
% 8.58/4.89 ( ( ord_less_real @ A @ B )
% 8.58/4.89 => ( ! [X4: real] :
% 8.58/4.89 ( ( ( ord_less_eq_real @ A @ X4 )
% 8.58/4.89 & ( ord_less_eq_real @ X4 @ B ) )
% 8.58/4.89 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
% 8.58/4.89 => ( ! [X4: real] :
% 8.58/4.89 ( ( ( ord_less_real @ A @ X4 )
% 8.58/4.89 & ( ord_less_real @ X4 @ B ) )
% 8.58/4.89 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 8.58/4.89 => ( ! [X4: real] :
% 8.58/4.89 ( ( ( ord_less_eq_real @ A @ X4 )
% 8.58/4.89 & ( ord_less_eq_real @ X4 @ B ) )
% 8.58/4.89 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G ) )
% 8.58/4.89 => ( ! [X4: real] :
% 8.58/4.89 ( ( ( ord_less_real @ A @ X4 )
% 8.58/4.89 & ( ord_less_real @ X4 @ B ) )
% 8.58/4.89 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 8.58/4.89 => ? [G_c: real,F_c: real,C3: real] :
% 8.58/4.89 ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 8.58/4.89 & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 8.58/4.89 & ( ord_less_real @ A @ C3 )
% 8.58/4.89 & ( ord_less_real @ C3 @ B )
% 8.58/4.89 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
% 8.58/4.89 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9530_Bseq__realpow,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.89 => ( ( ord_less_eq_real @ X @ one_one_real )
% 8.58/4.89 => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9531_MVT,axiom,
% 8.58/4.89 ! [A: real,B: real,F: real > real] :
% 8.58/4.89 ( ( ord_less_real @ A @ B )
% 8.58/4.89 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 8.58/4.89 => ( ! [X4: real] :
% 8.58/4.89 ( ( ord_less_real @ A @ X4 )
% 8.58/4.89 => ( ( ord_less_real @ X4 @ B )
% 8.58/4.89 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 8.58/4.89 => ? [L3: real,Z4: real] :
% 8.58/4.89 ( ( ord_less_real @ A @ Z4 )
% 8.58/4.89 & ( ord_less_real @ Z4 @ B )
% 8.58/4.89 & ( has_fi5821293074295781190e_real @ F @ L3 @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) )
% 8.58/4.89 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 8.58/4.89 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L3 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9532_continuous__on__arcosh_H,axiom,
% 8.58/4.89 ! [A3: set_real,F: real > real] :
% 8.58/4.89 ( ( topolo5044208981011980120l_real @ A3 @ F )
% 8.58/4.89 => ( ! [X4: real] :
% 8.58/4.89 ( ( member_real @ X4 @ A3 )
% 8.58/4.89 => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 8.58/4.89 => ( topolo5044208981011980120l_real @ A3
% 8.58/4.89 @ ^ [X3: real] : ( arcosh_real @ ( F @ X3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9533_continuous__on__arccos_H,axiom,
% 8.58/4.89 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9534_continuous__on__arcsin_H,axiom,
% 8.58/4.89 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9535_continuous__on__artanh,axiom,
% 8.58/4.89 ! [A3: set_real] :
% 8.58/4.89 ( ( ord_less_eq_set_real @ A3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 8.58/4.89 => ( topolo5044208981011980120l_real @ A3 @ artanh_real ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9536_continuous__on__artanh_H,axiom,
% 8.58/4.89 ! [A3: set_real,F: real > real] :
% 8.58/4.89 ( ( topolo5044208981011980120l_real @ A3 @ F )
% 8.58/4.89 => ( ! [X4: real] :
% 8.58/4.89 ( ( member_real @ X4 @ A3 )
% 8.58/4.89 => ( member_real @ ( F @ X4 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
% 8.58/4.89 => ( topolo5044208981011980120l_real @ A3
% 8.58/4.89 @ ^ [X3: real] : ( artanh_real @ ( F @ X3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9537_mvt,axiom,
% 8.58/4.89 ! [A: real,B: real,F: real > real,F3: real > real > real] :
% 8.58/4.89 ( ( ord_less_real @ A @ B )
% 8.58/4.89 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 8.58/4.89 => ( ! [X4: real] :
% 8.58/4.89 ( ( ord_less_real @ A @ X4 )
% 8.58/4.89 => ( ( ord_less_real @ X4 @ B )
% 8.58/4.89 => ( has_de1759254742604945161l_real @ F @ ( F3 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 8.58/4.89 => ~ ! [Xi: real] :
% 8.58/4.89 ( ( ord_less_real @ A @ Xi )
% 8.58/4.89 => ( ( ord_less_real @ Xi @ B )
% 8.58/4.89 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 8.58/4.89 != ( F3 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9538_card__length__sum__list__rec,axiom,
% 8.58/4.89 ! [M: nat,N5: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 8.58/4.89 => ( ( finite_card_list_nat
% 8.58/4.89 @ ( collect_list_nat
% 8.58/4.89 @ ^ [L2: list_nat] :
% 8.58/4.89 ( ( ( size_size_list_nat @ L2 )
% 8.58/4.89 = M )
% 8.58/4.89 & ( ( groups4561878855575611511st_nat @ L2 )
% 8.58/4.89 = N5 ) ) ) )
% 8.58/4.89 = ( plus_plus_nat
% 8.58/4.89 @ ( finite_card_list_nat
% 8.58/4.89 @ ( collect_list_nat
% 8.58/4.89 @ ^ [L2: list_nat] :
% 8.58/4.89 ( ( ( size_size_list_nat @ L2 )
% 8.58/4.89 = ( minus_minus_nat @ M @ one_one_nat ) )
% 8.58/4.89 & ( ( groups4561878855575611511st_nat @ L2 )
% 8.58/4.89 = N5 ) ) ) )
% 8.58/4.89 @ ( finite_card_list_nat
% 8.58/4.89 @ ( collect_list_nat
% 8.58/4.89 @ ^ [L2: list_nat] :
% 8.58/4.89 ( ( ( size_size_list_nat @ L2 )
% 8.58/4.89 = M )
% 8.58/4.89 & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
% 8.58/4.89 = N5 ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9539_card__length__sum__list,axiom,
% 8.58/4.89 ! [M: nat,N5: nat] :
% 8.58/4.89 ( ( finite_card_list_nat
% 8.58/4.89 @ ( collect_list_nat
% 8.58/4.89 @ ^ [L2: list_nat] :
% 8.58/4.89 ( ( ( size_size_list_nat @ L2 )
% 8.58/4.89 = M )
% 8.58/4.89 & ( ( groups4561878855575611511st_nat @ L2 )
% 8.58/4.89 = N5 ) ) ) )
% 8.58/4.89 = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N5 @ M ) @ one_one_nat ) @ N5 ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9540_surj__int__encode,axiom,
% 8.58/4.89 ( ( image_int_nat @ nat_int_encode @ top_top_set_int )
% 8.58/4.89 = top_top_set_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9541_int__encode__eq,axiom,
% 8.58/4.89 ! [X: int,Y: int] :
% 8.58/4.89 ( ( ( nat_int_encode @ X )
% 8.58/4.89 = ( nat_int_encode @ Y ) )
% 8.58/4.89 = ( X = Y ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9542_bij__int__encode,axiom,
% 8.58/4.89 bij_betw_int_nat @ nat_int_encode @ top_top_set_int @ top_top_set_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9543_Sup__nat__empty,axiom,
% 8.58/4.89 ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
% 8.58/4.89 = zero_zero_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9544_Inf__real__def,axiom,
% 8.58/4.89 ( comple4887499456419720421f_real
% 8.58/4.89 = ( ^ [X8: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X8 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9545_binomial__def,axiom,
% 8.58/4.89 ( binomial
% 8.58/4.89 = ( ^ [N3: nat,K2: nat] :
% 8.58/4.89 ( finite_card_set_nat
% 8.58/4.89 @ ( collect_set_nat
% 8.58/4.89 @ ^ [K7: set_nat] :
% 8.58/4.89 ( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) )
% 8.58/4.89 & ( ( finite_card_nat @ K7 )
% 8.58/4.89 = K2 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9546_inj__sgn__power,axiom,
% 8.58/4.89 ! [N: nat] :
% 8.58/4.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.89 => ( inj_on_real_real
% 8.58/4.89 @ ^ [Y6: real] : ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N ) )
% 8.58/4.89 @ top_top_set_real ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9547_surj__int__decode,axiom,
% 8.58/4.89 ( ( image_nat_int @ nat_int_decode @ top_top_set_nat )
% 8.58/4.89 = top_top_set_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9548_int__encode__inverse,axiom,
% 8.58/4.89 ! [X: int] :
% 8.58/4.89 ( ( nat_int_decode @ ( nat_int_encode @ X ) )
% 8.58/4.89 = X ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9549_int__decode__inverse,axiom,
% 8.58/4.89 ! [N: nat] :
% 8.58/4.89 ( ( nat_int_encode @ ( nat_int_decode @ N ) )
% 8.58/4.89 = N ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9550_int__decode__eq,axiom,
% 8.58/4.89 ! [X: nat,Y: nat] :
% 8.58/4.89 ( ( ( nat_int_decode @ X )
% 8.58/4.89 = ( nat_int_decode @ Y ) )
% 8.58/4.89 = ( X = Y ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9551_bij__int__decode,axiom,
% 8.58/4.89 bij_betw_nat_int @ nat_int_decode @ top_top_set_nat @ top_top_set_int ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9552_log__inj,axiom,
% 8.58/4.89 ! [B: real] :
% 8.58/4.89 ( ( ord_less_real @ one_one_real @ B )
% 8.58/4.89 => ( inj_on_real_real @ ( log2 @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9553_inj__int__decode,axiom,
% 8.58/4.89 ! [A3: set_nat] : ( inj_on_nat_int @ nat_int_decode @ A3 ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9554_inj__Suc,axiom,
% 8.58/4.89 ! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9555_inj__prod__encode,axiom,
% 8.58/4.89 ! [A3: set_Pr1261947904930325089at_nat] : ( inj_on2178005380612969504at_nat @ nat_prod_encode @ A3 ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9556_inj__list__encode,axiom,
% 8.58/4.89 ! [A3: set_list_nat] : ( inj_on_list_nat_nat @ nat_list_encode @ A3 ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9557_inj__int__encode,axiom,
% 8.58/4.89 ! [A3: set_int] : ( inj_on_int_nat @ nat_int_encode @ A3 ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9558_inj__on__set__encode,axiom,
% 8.58/4.89 inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9559_inj__on__diff__nat,axiom,
% 8.58/4.89 ! [N5: set_nat,K: nat] :
% 8.58/4.89 ( ! [N2: nat] :
% 8.58/4.89 ( ( member_nat @ N2 @ N5 )
% 8.58/4.89 => ( ord_less_eq_nat @ K @ N2 ) )
% 8.58/4.89 => ( inj_on_nat_nat
% 8.58/4.89 @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K )
% 8.58/4.89 @ N5 ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9560_inj__on__char__of__nat,axiom,
% 8.58/4.89 inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9561_Max__divisors__self__nat,axiom,
% 8.58/4.89 ! [N: nat] :
% 8.58/4.89 ( ( N != zero_zero_nat )
% 8.58/4.89 => ( ( lattic8265883725875713057ax_nat
% 8.58/4.89 @ ( collect_nat
% 8.58/4.89 @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ N ) ) )
% 8.58/4.89 = N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9562_atLeastLessThan__add__Un,axiom,
% 8.58/4.89 ! [I: nat,J: nat,K: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ I @ J )
% 8.58/4.89 => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 8.58/4.89 = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9563_card__le__Suc__Max,axiom,
% 8.58/4.89 ! [S2: set_nat] :
% 8.58/4.89 ( ( finite_finite_nat @ S2 )
% 8.58/4.89 => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9564_Sup__nat__def,axiom,
% 8.58/4.89 ( complete_Sup_Sup_nat
% 8.58/4.89 = ( ^ [X8: set_nat] : ( if_nat @ ( X8 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X8 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9565_divide__nat__def,axiom,
% 8.58/4.89 ( divide_divide_nat
% 8.58/4.89 = ( ^ [M3: nat,N3: nat] :
% 8.58/4.89 ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
% 8.58/4.89 @ ( lattic8265883725875713057ax_nat
% 8.58/4.89 @ ( collect_nat
% 8.58/4.89 @ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N3 ) @ M3 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9566_gcd__is__Max__divisors__nat,axiom,
% 8.58/4.89 ! [N: nat,M: nat] :
% 8.58/4.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.89 => ( ( gcd_gcd_nat @ M @ N )
% 8.58/4.89 = ( lattic8265883725875713057ax_nat
% 8.58/4.89 @ ( collect_nat
% 8.58/4.89 @ ^ [D4: nat] :
% 8.58/4.89 ( ( dvd_dvd_nat @ D4 @ M )
% 8.58/4.89 & ( dvd_dvd_nat @ D4 @ N ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9567_Gcd__eq__Max,axiom,
% 8.58/4.89 ! [M10: set_nat] :
% 8.58/4.89 ( ( finite_finite_nat @ M10 )
% 8.58/4.89 => ( ( M10 != bot_bot_set_nat )
% 8.58/4.89 => ( ~ ( member_nat @ zero_zero_nat @ M10 )
% 8.58/4.89 => ( ( gcd_Gcd_nat @ M10 )
% 8.58/4.89 = ( lattic8265883725875713057ax_nat
% 8.58/4.89 @ ( comple7806235888213564991et_nat
% 8.58/4.89 @ ( image_nat_set_nat
% 8.58/4.89 @ ^ [M3: nat] :
% 8.58/4.89 ( collect_nat
% 8.58/4.89 @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M3 ) )
% 8.58/4.89 @ M10 ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9568_quotient__of__def,axiom,
% 8.58/4.89 ( quotient_of
% 8.58/4.89 = ( ^ [X3: rat] :
% 8.58/4.89 ( the_Pr4378521158711661632nt_int
% 8.58/4.89 @ ^ [Pair: product_prod_int_int] :
% 8.58/4.89 ( ( X3
% 8.58/4.89 = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
% 8.58/4.89 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
% 8.58/4.89 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9569_coprime__abs__left__iff,axiom,
% 8.58/4.89 ! [K: int,L: int] :
% 8.58/4.89 ( ( algebr932160517623751201me_int @ ( abs_abs_int @ K ) @ L )
% 8.58/4.89 = ( algebr932160517623751201me_int @ K @ L ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9570_coprime__abs__right__iff,axiom,
% 8.58/4.89 ! [K: int,L: int] :
% 8.58/4.89 ( ( algebr932160517623751201me_int @ K @ ( abs_abs_int @ L ) )
% 8.58/4.89 = ( algebr932160517623751201me_int @ K @ L ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9571_Max__divisors__self__int,axiom,
% 8.58/4.89 ! [N: int] :
% 8.58/4.89 ( ( N != zero_zero_int )
% 8.58/4.89 => ( ( lattic8263393255366662781ax_int
% 8.58/4.89 @ ( collect_int
% 8.58/4.89 @ ^ [D4: int] : ( dvd_dvd_int @ D4 @ N ) ) )
% 8.58/4.89 = ( abs_abs_int @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9572_normalize__stable,axiom,
% 8.58/4.89 ! [Q2: int,P2: int] :
% 8.58/4.89 ( ( ord_less_int @ zero_zero_int @ Q2 )
% 8.58/4.89 => ( ( algebr932160517623751201me_int @ P2 @ Q2 )
% 8.58/4.89 => ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
% 8.58/4.89 = ( product_Pair_int_int @ P2 @ Q2 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9573_sup__nat__def,axiom,
% 8.58/4.89 sup_sup_nat = ord_max_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9574_sup__int__def,axiom,
% 8.58/4.89 sup_sup_int = ord_max_int ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9575_coprime__crossproduct__int,axiom,
% 8.58/4.89 ! [A: int,D: int,B: int,C: int] :
% 8.58/4.89 ( ( algebr932160517623751201me_int @ A @ D )
% 8.58/4.89 => ( ( algebr932160517623751201me_int @ B @ C )
% 8.58/4.89 => ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
% 8.58/4.89 = ( times_times_int @ ( abs_abs_int @ B ) @ ( abs_abs_int @ D ) ) )
% 8.58/4.89 = ( ( ( abs_abs_int @ A )
% 8.58/4.89 = ( abs_abs_int @ B ) )
% 8.58/4.89 & ( ( abs_abs_int @ C )
% 8.58/4.89 = ( abs_abs_int @ D ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9576_Rat__cases,axiom,
% 8.58/4.89 ! [Q2: rat] :
% 8.58/4.89 ~ ! [A4: int,B4: int] :
% 8.58/4.89 ( ( Q2
% 8.58/4.89 = ( fract @ A4 @ B4 ) )
% 8.58/4.89 => ( ( ord_less_int @ zero_zero_int @ B4 )
% 8.58/4.89 => ~ ( algebr932160517623751201me_int @ A4 @ B4 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9577_Rat__induct,axiom,
% 8.58/4.89 ! [P: rat > $o,Q2: rat] :
% 8.58/4.89 ( ! [A4: int,B4: int] :
% 8.58/4.89 ( ( ord_less_int @ zero_zero_int @ B4 )
% 8.58/4.89 => ( ( algebr932160517623751201me_int @ A4 @ B4 )
% 8.58/4.89 => ( P @ ( fract @ A4 @ B4 ) ) ) )
% 8.58/4.89 => ( P @ Q2 ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9578_coprime__common__divisor__int,axiom,
% 8.58/4.89 ! [A: int,B: int,X: int] :
% 8.58/4.89 ( ( algebr932160517623751201me_int @ A @ B )
% 8.58/4.89 => ( ( dvd_dvd_int @ X @ A )
% 8.58/4.89 => ( ( dvd_dvd_int @ X @ B )
% 8.58/4.89 => ( ( abs_abs_int @ X )
% 8.58/4.89 = one_one_int ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9579_gcd__is__Max__divisors__int,axiom,
% 8.58/4.89 ! [N: int,M: int] :
% 8.58/4.89 ( ( N != zero_zero_int )
% 8.58/4.89 => ( ( gcd_gcd_int @ M @ N )
% 8.58/4.89 = ( lattic8263393255366662781ax_int
% 8.58/4.89 @ ( collect_int
% 8.58/4.89 @ ^ [D4: int] :
% 8.58/4.89 ( ( dvd_dvd_int @ D4 @ M )
% 8.58/4.89 & ( dvd_dvd_int @ D4 @ N ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9580_Rat__cases__nonzero,axiom,
% 8.58/4.89 ! [Q2: rat] :
% 8.58/4.89 ( ! [A4: int,B4: int] :
% 8.58/4.89 ( ( Q2
% 8.58/4.89 = ( fract @ A4 @ B4 ) )
% 8.58/4.89 => ( ( ord_less_int @ zero_zero_int @ B4 )
% 8.58/4.89 => ( ( A4 != zero_zero_int )
% 8.58/4.89 => ~ ( algebr932160517623751201me_int @ A4 @ B4 ) ) ) )
% 8.58/4.89 => ( Q2 = zero_zero_rat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9581_quotient__of__unique,axiom,
% 8.58/4.89 ! [R: rat] :
% 8.58/4.89 ? [X4: product_prod_int_int] :
% 8.58/4.89 ( ( R
% 8.58/4.89 = ( fract @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) )
% 8.58/4.89 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X4 ) )
% 8.58/4.89 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) )
% 8.58/4.89 & ! [Y5: product_prod_int_int] :
% 8.58/4.89 ( ( ( R
% 8.58/4.89 = ( fract @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 8.58/4.89 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y5 ) )
% 8.58/4.89 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 8.58/4.89 => ( Y5 = X4 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9582_Rat_Opositive_Orsp,axiom,
% 8.58/4.89 ( bNF_re8699439704749558557nt_o_o @ ratrel
% 8.58/4.89 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 8.58/4.89 @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
% 8.58/4.89 @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9583_Rats__abs__iff,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ( ( member_real @ ( abs_abs_real @ X ) @ field_5140801741446780682s_real )
% 8.58/4.89 = ( member_real @ X @ field_5140801741446780682s_real ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9584_coprime__int__iff,axiom,
% 8.58/4.89 ! [M: nat,N: nat] :
% 8.58/4.89 ( ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 8.58/4.89 = ( algebr934650988132801477me_nat @ M @ N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9585_coprime__nat__abs__right__iff,axiom,
% 8.58/4.89 ! [N: nat,K: int] :
% 8.58/4.89 ( ( algebr934650988132801477me_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 8.58/4.89 = ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9586_coprime__nat__abs__left__iff,axiom,
% 8.58/4.89 ! [K: int,N: nat] :
% 8.58/4.89 ( ( algebr934650988132801477me_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 8.58/4.89 = ( algebr932160517623751201me_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9587_coprime__crossproduct__nat,axiom,
% 8.58/4.89 ! [A: nat,D: nat,B: nat,C: nat] :
% 8.58/4.89 ( ( algebr934650988132801477me_nat @ A @ D )
% 8.58/4.89 => ( ( algebr934650988132801477me_nat @ B @ C )
% 8.58/4.89 => ( ( ( times_times_nat @ A @ C )
% 8.58/4.89 = ( times_times_nat @ B @ D ) )
% 8.58/4.89 = ( ( A = B )
% 8.58/4.89 & ( C = D ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9588_coprime__Suc__right__nat,axiom,
% 8.58/4.89 ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9589_coprime__Suc__left__nat,axiom,
% 8.58/4.89 ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ N ) @ N ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9590_coprime__Suc__0__right,axiom,
% 8.58/4.89 ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9591_coprime__Suc__0__left,axiom,
% 8.58/4.89 ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9592_coprime__common__divisor__nat,axiom,
% 8.58/4.89 ! [A: nat,B: nat,X: nat] :
% 8.58/4.89 ( ( algebr934650988132801477me_nat @ A @ B )
% 8.58/4.89 => ( ( dvd_dvd_nat @ X @ A )
% 8.58/4.89 => ( ( dvd_dvd_nat @ X @ B )
% 8.58/4.89 => ( X = one_one_nat ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9593_Rats__abs__nat__div__natE,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ( ( member_real @ X @ field_5140801741446780682s_real )
% 8.58/4.89 => ~ ! [M2: nat,N2: nat] :
% 8.58/4.89 ( ( N2 != zero_zero_nat )
% 8.58/4.89 => ( ( ( abs_abs_real @ X )
% 8.58/4.89 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 8.58/4.89 => ~ ( algebr934650988132801477me_nat @ M2 @ N2 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9594_Rats__no__bot__less,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ? [X4: real] :
% 8.58/4.89 ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 8.58/4.89 & ( ord_less_real @ X4 @ X ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9595_Rats__dense__in__real,axiom,
% 8.58/4.89 ! [X: real,Y: real] :
% 8.58/4.89 ( ( ord_less_real @ X @ Y )
% 8.58/4.89 => ? [X4: real] :
% 8.58/4.89 ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 8.58/4.89 & ( ord_less_real @ X @ X4 )
% 8.58/4.89 & ( ord_less_real @ X4 @ Y ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9596_Rats__no__top__le,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ? [X4: real] :
% 8.58/4.89 ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 8.58/4.89 & ( ord_less_eq_real @ X @ X4 ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9597_coprime__diff__one__right__nat,axiom,
% 8.58/4.89 ! [N: nat] :
% 8.58/4.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.89 => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9598_coprime__diff__one__left__nat,axiom,
% 8.58/4.89 ! [N: nat] :
% 8.58/4.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.89 => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9599_and__not__num_Opelims,axiom,
% 8.58/4.89 ! [X: num,Xa: num,Y: option_num] :
% 8.58/4.89 ( ( ( bit_and_not_num @ X @ Xa )
% 8.58/4.89 = Y )
% 8.58/4.89 => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( ( Y = none_num )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( some_num @ one ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ( ( Y = none_num )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( some_num @ ( bit0 @ M2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ one ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( some_num @ ( bit0 @ M2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ one ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 8.58/4.89 @ ^ [N7: num] : ( some_num @ ( bit1 @ N7 ) )
% 8.58/4.89 @ ( bit_and_not_num @ M2 @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ~ ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9600_times__rat_Orsp,axiom,
% 8.58/4.89 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 8.58/4.89 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) )
% 8.58/4.89 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9601_integer__of__natural_Orsp,axiom,
% 8.58/4.89 ( bNF_re6650684261131312217nt_int
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ semiri1314217659103216013at_int
% 8.58/4.89 @ semiri1314217659103216013at_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9602_sub_Orsp,axiom,
% 8.58/4.89 ( bNF_re8402795839162346335um_int
% 8.58/4.89 @ ^ [Y3: num,Z: num] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re1822329894187522285nt_int
% 8.58/4.89 @ ^ [Y3: num,Z: num] : ( Y3 = Z )
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 8.58/4.89 @ ^ [M3: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) )
% 8.58/4.89 @ ^ [M3: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9603_dup_Orsp,axiom,
% 8.58/4.89 ( bNF_re4712519889275205905nt_int
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ^ [K2: int] : ( plus_plus_int @ K2 @ K2 )
% 8.58/4.89 @ ^ [K2: int] : ( plus_plus_int @ K2 @ K2 ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9604_plus__integer_Orsp,axiom,
% 8.58/4.89 ( bNF_re711492959462206631nt_int
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re4712519889275205905nt_int
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 8.58/4.89 @ plus_plus_int
% 8.58/4.89 @ plus_plus_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9605_plus__natural_Orsp,axiom,
% 8.58/4.89 ( bNF_re1345281282404953727at_nat
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re5653821019739307937at_nat
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 8.58/4.89 @ plus_plus_nat
% 8.58/4.89 @ plus_plus_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9606_Suc_Orsp,axiom,
% 8.58/4.89 ( bNF_re5653821019739307937at_nat
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ suc
% 8.58/4.89 @ suc ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9607_times__integer_Orsp,axiom,
% 8.58/4.89 ( bNF_re711492959462206631nt_int
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re4712519889275205905nt_int
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 8.58/4.89 @ times_times_int
% 8.58/4.89 @ times_times_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9608_times__natural_Orsp,axiom,
% 8.58/4.89 ( bNF_re1345281282404953727at_nat
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re5653821019739307937at_nat
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 8.58/4.89 @ times_times_nat
% 8.58/4.89 @ times_times_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9609_minus__natural_Orsp,axiom,
% 8.58/4.89 ( bNF_re1345281282404953727at_nat
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re5653821019739307937at_nat
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 8.58/4.89 @ minus_minus_nat
% 8.58/4.89 @ minus_minus_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9610_minus__integer_Orsp,axiom,
% 8.58/4.89 ( bNF_re711492959462206631nt_int
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re4712519889275205905nt_int
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 8.58/4.89 @ minus_minus_int
% 8.58/4.89 @ minus_minus_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9611_push__bit__natural_Orsp,axiom,
% 8.58/4.89 ( bNF_re1345281282404953727at_nat
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re5653821019739307937at_nat
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 8.58/4.89 @ bit_se547839408752420682it_nat
% 8.58/4.89 @ bit_se547839408752420682it_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9612_push__bit__integer_Orsp,axiom,
% 8.58/4.89 ( bNF_re4785983289428654063nt_int
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re4712519889275205905nt_int
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 8.58/4.89 @ bit_se545348938243370406it_int
% 8.58/4.89 @ bit_se545348938243370406it_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9613_divide__natural_Orsp,axiom,
% 8.58/4.89 ( bNF_re1345281282404953727at_nat
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re5653821019739307937at_nat
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 8.58/4.89 @ divide_divide_nat
% 8.58/4.89 @ divide_divide_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9614_divide__integer_Orsp,axiom,
% 8.58/4.89 ( bNF_re711492959462206631nt_int
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re4712519889275205905nt_int
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 8.58/4.89 @ divide_divide_int
% 8.58/4.89 @ divide_divide_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9615_gcd__integer_Orsp,axiom,
% 8.58/4.89 ( bNF_re711492959462206631nt_int
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re4712519889275205905nt_int
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 8.58/4.89 @ gcd_gcd_int
% 8.58/4.89 @ gcd_gcd_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9616_Fract_Orsp,axiom,
% 8.58/4.89 ( bNF_re157797125943740599nt_int
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re6250860962936578807nt_int
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ratrel )
% 8.58/4.89 @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) )
% 8.58/4.89 @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9617_plus__rat_Orsp,axiom,
% 8.58/4.89 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 8.58/4.89 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) )
% 8.58/4.89 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9618_inverse__rat_Orsp,axiom,
% 8.58/4.89 ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
% 8.58/4.89 @ ^ [X3: product_prod_int_int] :
% 8.58/4.89 ( if_Pro3027730157355071871nt_int
% 8.58/4.89 @ ( ( product_fst_int_int @ X3 )
% 8.58/4.89 = zero_zero_int )
% 8.58/4.89 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 8.58/4.89 @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
% 8.58/4.89 @ ^ [X3: product_prod_int_int] :
% 8.58/4.89 ( if_Pro3027730157355071871nt_int
% 8.58/4.89 @ ( ( product_fst_int_int @ X3 )
% 8.58/4.89 = zero_zero_int )
% 8.58/4.89 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 8.58/4.89 @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9619_and__num_Opelims,axiom,
% 8.58/4.89 ! [X: num,Xa: num,Y: option_num] :
% 8.58/4.89 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 8.58/4.89 = Y )
% 8.58/4.89 => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( some_num @ one ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ( ( Y = none_num )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( some_num @ one ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( ( Y = none_num )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ one ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( some_num @ one ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ one ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ~ ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 8.58/4.89 @ ^ [N7: num] : ( some_num @ ( bit1 @ N7 ) )
% 8.58/4.89 @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9620_xor__num_Opelims,axiom,
% 8.58/4.89 ! [X: num,Xa: num,Y: option_num] :
% 8.58/4.89 ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 8.58/4.89 = Y )
% 8.58/4.89 => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( ( Y = none_num )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( some_num @ ( bit1 @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( some_num @ ( bit0 @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( some_num @ ( bit1 @ M2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ one ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( some_num @ ( bit0 @ M2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ one ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ~ ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9621_or__not__num__neg_Opelims,axiom,
% 8.58/4.89 ! [X: num,Xa: num,Y: num] :
% 8.58/4.89 ( ( ( bit_or_not_num_neg @ X @ Xa )
% 8.58/4.89 = Y )
% 8.58/4.89 => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( ( Y = one )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ! [M2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M2 ) ) ) ) ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ! [M2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M2 ) ) ) ) ) )
% 8.58/4.89 => ( ! [N2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( bit0 @ one ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ one ) ) ) ) )
% 8.58/4.89 => ( ! [N2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ! [M2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit0 @ M2 ) ) ) ) ) )
% 8.58/4.89 => ( ! [N2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ! [M2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( bit0 @ ( bit_or_not_num_neg @ N2 @ M2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit1 @ M2 ) ) ) ) ) )
% 8.58/4.89 => ( ! [N2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( ( Y = one )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ one ) ) ) ) )
% 8.58/4.89 => ( ! [N2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ! [M2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit0 @ M2 ) ) ) ) ) )
% 8.58/4.89 => ~ ! [N2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ! [M2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit1 @ M2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9622_xor__num__rel__dict,axiom,
% 8.58/4.89 bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9623_and__num__rel__dict,axiom,
% 8.58/4.89 bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9624_plus__rat_Otransfer,axiom,
% 8.58/4.89 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 8.58/4.89 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) )
% 8.58/4.89 @ plus_plus_rat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9625_one__rat_Otransfer,axiom,
% 8.58/4.89 pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9626_zero__rat_Otransfer,axiom,
% 8.58/4.89 pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9627_Fract_Otransfer,axiom,
% 8.58/4.89 ( bNF_re3461391660133120880nt_rat
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re2214769303045360666nt_rat
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ pcr_rat )
% 8.58/4.89 @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) )
% 8.58/4.89 @ fract ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9628_times__rat_Otransfer,axiom,
% 8.58/4.89 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 8.58/4.89 @ ^ [X3: product_prod_int_int,Y6: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y6 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y6 ) ) )
% 8.58/4.89 @ times_times_rat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9629_Rat_Opositive_Otransfer,axiom,
% 8.58/4.89 ( bNF_re1494630372529172596at_o_o @ pcr_rat
% 8.58/4.89 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 8.58/4.89 @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
% 8.58/4.89 @ positive ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9630_inverse__rat_Otransfer,axiom,
% 8.58/4.89 ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
% 8.58/4.89 @ ^ [X3: product_prod_int_int] :
% 8.58/4.89 ( if_Pro3027730157355071871nt_int
% 8.58/4.89 @ ( ( product_fst_int_int @ X3 )
% 8.58/4.89 = zero_zero_int )
% 8.58/4.89 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 8.58/4.89 @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
% 8.58/4.89 @ inverse_inverse_rat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9631_times__int_Otransfer,axiom,
% 8.58/4.89 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 8.58/4.89 @ ( produc27273713700761075at_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.89 ( produc2626176000494625587at_nat
% 8.58/4.89 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) )
% 8.58/4.89 @ times_times_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9632_zero__int_Otransfer,axiom,
% 8.58/4.89 pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9633_int__transfer,axiom,
% 8.58/4.89 ( bNF_re6830278522597306478at_int
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ pcr_int
% 8.58/4.89 @ ^ [N3: nat] : ( product_Pair_nat_nat @ N3 @ zero_zero_nat )
% 8.58/4.89 @ semiri1314217659103216013at_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9634_uminus__int_Otransfer,axiom,
% 8.58/4.89 ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
% 8.58/4.89 @ ( produc2626176000494625587at_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 ) )
% 8.58/4.89 @ uminus_uminus_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9635_nat_Otransfer,axiom,
% 8.58/4.89 ( bNF_re4555766996558763186at_nat @ pcr_int
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 8.58/4.89 @ nat2 ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9636_one__int_Otransfer,axiom,
% 8.58/4.89 pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9637_less__int_Otransfer,axiom,
% 8.58/4.89 ( bNF_re717283939379294677_int_o @ pcr_int
% 8.58/4.89 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 8.58/4.89 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 8.58/4.89 @ ( produc8739625826339149834_nat_o
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.89 ( produc6081775807080527818_nat_o
% 8.58/4.89 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
% 8.58/4.89 @ ord_less_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9638_less__eq__int_Otransfer,axiom,
% 8.58/4.89 ( bNF_re717283939379294677_int_o @ pcr_int
% 8.58/4.89 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 8.58/4.89 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 8.58/4.89 @ ( produc8739625826339149834_nat_o
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.89 ( produc6081775807080527818_nat_o
% 8.58/4.89 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
% 8.58/4.89 @ ord_less_eq_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9639_plus__int_Otransfer,axiom,
% 8.58/4.89 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 8.58/4.89 @ ( produc27273713700761075at_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.89 ( produc2626176000494625587at_nat
% 8.58/4.89 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) )
% 8.58/4.89 @ plus_plus_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9640_minus__int_Otransfer,axiom,
% 8.58/4.89 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 8.58/4.89 @ ( produc27273713700761075at_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.89 ( produc2626176000494625587at_nat
% 8.58/4.89 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) )
% 8.58/4.89 @ minus_minus_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9641_times__int_Orsp,axiom,
% 8.58/4.89 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 8.58/4.89 @ ( produc27273713700761075at_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.89 ( produc2626176000494625587at_nat
% 8.58/4.89 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) )
% 8.58/4.89 @ ( produc27273713700761075at_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.89 ( produc2626176000494625587at_nat
% 8.58/4.89 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U3 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y6 @ U3 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9642_intrel__iff,axiom,
% 8.58/4.89 ! [X: nat,Y: nat,U: nat,V: nat] :
% 8.58/4.89 ( ( intrel @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ U @ V ) )
% 8.58/4.89 = ( ( plus_plus_nat @ X @ V )
% 8.58/4.89 = ( plus_plus_nat @ U @ Y ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9643_int_Orel__eq__transfer,axiom,
% 8.58/4.89 ( bNF_re717283939379294677_int_o @ pcr_int
% 8.58/4.89 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 8.58/4.89 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 8.58/4.89 @ intrel
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9644_int_Oabs__eq__iff,axiom,
% 8.58/4.89 ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 8.58/4.89 ( ( ( abs_Integ @ X )
% 8.58/4.89 = ( abs_Integ @ Y ) )
% 8.58/4.89 = ( intrel @ X @ Y ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9645_zero__int_Orsp,axiom,
% 8.58/4.89 intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9646_uminus__int_Orsp,axiom,
% 8.58/4.89 ( bNF_re2241393799969408733at_nat @ intrel @ intrel
% 8.58/4.89 @ ( produc2626176000494625587at_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 ) )
% 8.58/4.89 @ ( produc2626176000494625587at_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X3 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9647_nat_Orsp,axiom,
% 8.58/4.89 ( bNF_re8246922863344978751at_nat @ intrel
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 8.58/4.89 @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9648_one__int_Orsp,axiom,
% 8.58/4.89 intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9649_product__atMost__eq__Un,axiom,
% 8.58/4.89 ! [A3: set_nat,M: nat] :
% 8.58/4.89 ( ( produc457027306803732586at_nat @ A3
% 8.58/4.89 @ ^ [Uu3: nat] : ( set_ord_atMost_nat @ M ) )
% 8.58/4.89 = ( sup_su6327502436637775413at_nat
% 8.58/4.89 @ ( produc457027306803732586at_nat @ A3
% 8.58/4.89 @ ^ [I4: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ I4 ) ) )
% 8.58/4.89 @ ( produc457027306803732586at_nat @ A3
% 8.58/4.89 @ ^ [I4: nat] : ( set_or6659071591806873216st_nat @ ( minus_minus_nat @ M @ I4 ) @ M ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9650_intrel__def,axiom,
% 8.58/4.89 ( intrel
% 8.58/4.89 = ( produc8739625826339149834_nat_o
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.89 ( produc6081775807080527818_nat_o
% 8.58/4.89 @ ^ [U3: nat,V4: nat] :
% 8.58/4.89 ( ( plus_plus_nat @ X3 @ V4 )
% 8.58/4.89 = ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9651_pairs__le__eq__Sigma,axiom,
% 8.58/4.89 ! [M: nat] :
% 8.58/4.89 ( ( collec3392354462482085612at_nat
% 8.58/4.89 @ ( produc6081775807080527818_nat_o
% 8.58/4.89 @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ M ) ) )
% 8.58/4.89 = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
% 8.58/4.89 @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9652_less__int_Orsp,axiom,
% 8.58/4.89 ( bNF_re4202695980764964119_nat_o @ intrel
% 8.58/4.89 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 8.58/4.89 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 8.58/4.89 @ ( produc8739625826339149834_nat_o
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.89 ( produc6081775807080527818_nat_o
% 8.58/4.89 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
% 8.58/4.89 @ ( produc8739625826339149834_nat_o
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.89 ( produc6081775807080527818_nat_o
% 8.58/4.89 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9653_less__eq__int_Orsp,axiom,
% 8.58/4.89 ( bNF_re4202695980764964119_nat_o @ intrel
% 8.58/4.89 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 8.58/4.89 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 8.58/4.89 @ ( produc8739625826339149834_nat_o
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.89 ( produc6081775807080527818_nat_o
% 8.58/4.89 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) )
% 8.58/4.89 @ ( produc8739625826339149834_nat_o
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.89 ( produc6081775807080527818_nat_o
% 8.58/4.89 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U3 @ Y6 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9654_minus__int_Orsp,axiom,
% 8.58/4.89 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 8.58/4.89 @ ( produc27273713700761075at_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.89 ( produc2626176000494625587at_nat
% 8.58/4.89 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) )
% 8.58/4.89 @ ( produc27273713700761075at_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.89 ( produc2626176000494625587at_nat
% 8.58/4.89 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y6 @ U3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9655_plus__int_Orsp,axiom,
% 8.58/4.89 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 8.58/4.89 @ ( produc27273713700761075at_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.89 ( produc2626176000494625587at_nat
% 8.58/4.89 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) )
% 8.58/4.89 @ ( produc27273713700761075at_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] :
% 8.58/4.89 ( produc2626176000494625587at_nat
% 8.58/4.89 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U3 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9656_Rats__eq__int__div__nat,axiom,
% 8.58/4.89 ( field_5140801741446780682s_real
% 8.58/4.89 = ( collect_real
% 8.58/4.89 @ ^ [Uu3: real] :
% 8.58/4.89 ? [I4: int,N3: nat] :
% 8.58/4.89 ( ( Uu3
% 8.58/4.89 = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 8.58/4.89 & ( N3 != zero_zero_nat ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9657_rat__less__code,axiom,
% 8.58/4.89 ( ord_less_rat
% 8.58/4.89 = ( ^ [P3: rat,Q5: rat] :
% 8.58/4.89 ( produc4947309494688390418_int_o
% 8.58/4.89 @ ^ [A2: int,C4: int] :
% 8.58/4.89 ( produc4947309494688390418_int_o
% 8.58/4.89 @ ^ [B2: int,D4: int] : ( ord_less_int @ ( times_times_int @ A2 @ D4 ) @ ( times_times_int @ C4 @ B2 ) )
% 8.58/4.89 @ ( quotient_of @ Q5 ) )
% 8.58/4.89 @ ( quotient_of @ P3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9658_rat__floor__code,axiom,
% 8.58/4.89 ( archim3151403230148437115or_rat
% 8.58/4.89 = ( ^ [P3: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9659_rat__less__eq__code,axiom,
% 8.58/4.89 ( ord_less_eq_rat
% 8.58/4.89 = ( ^ [P3: rat,Q5: rat] :
% 8.58/4.89 ( produc4947309494688390418_int_o
% 8.58/4.89 @ ^ [A2: int,C4: int] :
% 8.58/4.89 ( produc4947309494688390418_int_o
% 8.58/4.89 @ ^ [B2: int,D4: int] : ( ord_less_eq_int @ ( times_times_int @ A2 @ D4 ) @ ( times_times_int @ C4 @ B2 ) )
% 8.58/4.89 @ ( quotient_of @ Q5 ) )
% 8.58/4.89 @ ( quotient_of @ P3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9660_Rats__eq__int__div__int,axiom,
% 8.58/4.89 ( field_5140801741446780682s_real
% 8.58/4.89 = ( collect_real
% 8.58/4.89 @ ^ [Uu3: real] :
% 8.58/4.89 ? [I4: int,J3: int] :
% 8.58/4.89 ( ( Uu3
% 8.58/4.89 = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( ring_1_of_int_real @ J3 ) ) )
% 8.58/4.89 & ( J3 != zero_zero_int ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9661_Divides_Oadjust__div__def,axiom,
% 8.58/4.89 ( adjust_div
% 8.58/4.89 = ( produc8211389475949308722nt_int
% 8.58/4.89 @ ^ [Q5: int,R5: int] : ( plus_plus_int @ Q5 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9662_int__ge__less__than__def,axiom,
% 8.58/4.89 ( int_ge_less_than
% 8.58/4.89 = ( ^ [D4: int] :
% 8.58/4.89 ( collec213857154873943460nt_int
% 8.58/4.89 @ ( produc4947309494688390418_int_o
% 8.58/4.89 @ ^ [Z7: int,Z5: int] :
% 8.58/4.89 ( ( ord_less_eq_int @ D4 @ Z7 )
% 8.58/4.89 & ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9663_int__ge__less__than2__def,axiom,
% 8.58/4.89 ( int_ge_less_than2
% 8.58/4.89 = ( ^ [D4: int] :
% 8.58/4.89 ( collec213857154873943460nt_int
% 8.58/4.89 @ ( produc4947309494688390418_int_o
% 8.58/4.89 @ ^ [Z7: int,Z5: int] :
% 8.58/4.89 ( ( ord_less_eq_int @ D4 @ Z5 )
% 8.58/4.89 & ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9664_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 8.58/4.89 ! [X: vEBT_VEBT,Xa: nat] :
% 8.58/4.89 ( ~ ( vEBT_VEBT_valid @ X @ Xa )
% 8.58/4.89 => ( ( ? [Uu2: $o,Uv2: $o] :
% 8.58/4.89 ( X
% 8.58/4.89 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.58/4.89 => ( Xa = one_one_nat ) )
% 8.58/4.89 => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) )
% 8.58/4.89 => ( ( Deg = Xa )
% 8.58/4.89 & ! [X4: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.89 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.89 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 8.58/4.89 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.89 & ( case_o184042715313410164at_nat
% 8.58/4.89 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
% 8.58/4.89 & ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.89 @ ( produc6081775807080527818_nat_o
% 8.58/4.89 @ ^ [Mi3: nat,Ma3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.58/4.89 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.58/4.89 & ! [I4: nat] :
% 8.58/4.89 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.89 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
% 8.58/4.89 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 8.58/4.89 & ( ( Mi3 = Ma3 )
% 8.58/4.89 => ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.89 & ( ( Mi3 != Ma3 )
% 8.58/4.89 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 8.58/4.89 & ! [X3: nat] :
% 8.58/4.89 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.58/4.89 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 8.58/4.89 => ( ( ord_less_nat @ Mi3 @ X3 )
% 8.58/4.89 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 8.58/4.89 @ Mima ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9665_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 8.58/4.89 ! [X: vEBT_VEBT,Xa: nat] :
% 8.58/4.89 ( ( vEBT_VEBT_valid @ X @ Xa )
% 8.58/4.89 => ( ( ? [Uu2: $o,Uv2: $o] :
% 8.58/4.89 ( X
% 8.58/4.89 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.58/4.89 => ( Xa != one_one_nat ) )
% 8.58/4.89 => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) )
% 8.58/4.89 => ~ ( ( Deg = Xa )
% 8.58/4.89 & ! [X5: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.89 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.89 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 8.58/4.89 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.89 & ( case_o184042715313410164at_nat
% 8.58/4.89 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
% 8.58/4.89 & ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.89 @ ( produc6081775807080527818_nat_o
% 8.58/4.89 @ ^ [Mi3: nat,Ma3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.58/4.89 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.58/4.89 & ! [I4: nat] :
% 8.58/4.89 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.89 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
% 8.58/4.89 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 8.58/4.89 & ( ( Mi3 = Ma3 )
% 8.58/4.89 => ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.89 & ( ( Mi3 != Ma3 )
% 8.58/4.89 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 8.58/4.89 & ! [X3: nat] :
% 8.58/4.89 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.58/4.89 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 8.58/4.89 => ( ( ord_less_nat @ Mi3 @ X3 )
% 8.58/4.89 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 8.58/4.89 @ Mima ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9666_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 8.58/4.89 ! [Mima2: option4927543243414619207at_nat,Deg3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,Deg4: nat] :
% 8.58/4.89 ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg3 @ TreeList2 @ Summary3 ) @ Deg4 )
% 8.58/4.89 = ( ( Deg3 = Deg4 )
% 8.58/4.89 & ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.58/4.89 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.89 & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg3 @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.89 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 8.58/4.89 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg3 @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.89 & ( case_o184042715313410164at_nat
% 8.58/4.89 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
% 8.58/4.89 & ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.58/4.89 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.89 @ ( produc6081775807080527818_nat_o
% 8.58/4.89 @ ^ [Mi3: nat,Ma3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.58/4.89 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg3 ) )
% 8.58/4.89 & ! [I4: nat] :
% 8.58/4.89 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg3 @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.89 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
% 8.58/4.89 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 8.58/4.89 & ( ( Mi3 = Ma3 )
% 8.58/4.89 => ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.58/4.89 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.89 & ( ( Mi3 != Ma3 )
% 8.58/4.89 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 8.58/4.89 & ! [X3: nat] :
% 8.58/4.89 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg3 ) )
% 8.58/4.89 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
% 8.58/4.89 => ( ( ord_less_nat @ Mi3 @ X3 )
% 8.58/4.89 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 8.58/4.89 @ Mima2 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9667_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 8.58/4.89 ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 8.58/4.89 ( ( ( vEBT_VEBT_valid @ X @ Xa )
% 8.58/4.89 = Y )
% 8.58/4.89 => ( ( ? [Uu2: $o,Uv2: $o] :
% 8.58/4.89 ( X
% 8.58/4.89 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.58/4.89 => ( Y
% 8.58/4.89 = ( Xa != one_one_nat ) ) )
% 8.58/4.89 => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) )
% 8.58/4.89 => ( Y
% 8.58/4.89 = ( ~ ( ( Deg = Xa )
% 8.58/4.89 & ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.89 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.89 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 8.58/4.89 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.89 & ( case_o184042715313410164at_nat
% 8.58/4.89 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
% 8.58/4.89 & ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.89 @ ( produc6081775807080527818_nat_o
% 8.58/4.89 @ ^ [Mi3: nat,Ma3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.58/4.89 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.58/4.89 & ! [I4: nat] :
% 8.58/4.89 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.89 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
% 8.58/4.89 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 8.58/4.89 & ( ( Mi3 = Ma3 )
% 8.58/4.89 => ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.89 & ( ( Mi3 != Ma3 )
% 8.58/4.89 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 8.58/4.89 & ! [X3: nat] :
% 8.58/4.89 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.58/4.89 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 8.58/4.89 => ( ( ord_less_nat @ Mi3 @ X3 )
% 8.58/4.89 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 8.58/4.89 @ Mima ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9668_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 8.58/4.89 ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
% 8.58/4.89 ( ( ( vEBT_VEBT_valid @ X @ Xa )
% 8.58/4.89 = Y )
% 8.58/4.89 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 8.58/4.89 => ( ! [Uu2: $o,Uv2: $o] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( Xa = one_one_nat ) )
% 8.58/4.89 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
% 8.58/4.89 => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( ( Deg = Xa )
% 8.58/4.89 & ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.89 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.89 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 8.58/4.89 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.89 & ( case_o184042715313410164at_nat
% 8.58/4.89 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
% 8.58/4.89 & ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.89 @ ( produc6081775807080527818_nat_o
% 8.58/4.89 @ ^ [Mi3: nat,Ma3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.58/4.89 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.58/4.89 & ! [I4: nat] :
% 8.58/4.89 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.89 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
% 8.58/4.89 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 8.58/4.89 & ( ( Mi3 = Ma3 )
% 8.58/4.89 => ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.89 & ( ( Mi3 != Ma3 )
% 8.58/4.89 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 8.58/4.89 & ! [X3: nat] :
% 8.58/4.89 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.58/4.89 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 8.58/4.89 => ( ( ord_less_nat @ Mi3 @ X3 )
% 8.58/4.89 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 8.58/4.89 @ Mima ) ) )
% 8.58/4.89 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9669_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 8.58/4.89 ! [X: vEBT_VEBT,Xa: nat] :
% 8.58/4.89 ( ( vEBT_VEBT_valid @ X @ Xa )
% 8.58/4.89 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 8.58/4.89 => ( ! [Uu2: $o,Uv2: $o] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.58/4.89 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
% 8.58/4.89 => ( Xa != one_one_nat ) ) )
% 8.58/4.89 => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) )
% 8.58/4.89 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) @ Xa ) )
% 8.58/4.89 => ~ ( ( Deg = Xa )
% 8.58/4.89 & ! [X5: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.89 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.89 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 8.58/4.89 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.89 & ( case_o184042715313410164at_nat
% 8.58/4.89 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
% 8.58/4.89 & ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.89 @ ( produc6081775807080527818_nat_o
% 8.58/4.89 @ ^ [Mi3: nat,Ma3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.58/4.89 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.58/4.89 & ! [I4: nat] :
% 8.58/4.89 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.89 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
% 8.58/4.89 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 8.58/4.89 & ( ( Mi3 = Ma3 )
% 8.58/4.89 => ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.89 & ( ( Mi3 != Ma3 )
% 8.58/4.89 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 8.58/4.89 & ! [X3: nat] :
% 8.58/4.89 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.58/4.89 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 8.58/4.89 => ( ( ord_less_nat @ Mi3 @ X3 )
% 8.58/4.89 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 8.58/4.89 @ Mima ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9670_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 8.58/4.89 ! [X: vEBT_VEBT,Xa: nat] :
% 8.58/4.89 ( ~ ( vEBT_VEBT_valid @ X @ Xa )
% 8.58/4.89 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 8.58/4.89 => ( ! [Uu2: $o,Uv2: $o] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.58/4.89 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
% 8.58/4.89 => ( Xa = one_one_nat ) ) )
% 8.58/4.89 => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) )
% 8.58/4.89 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg @ TreeList3 @ Summary ) @ Xa ) )
% 8.58/4.89 => ( ( Deg = Xa )
% 8.58/4.89 & ! [X4: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.89 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.58/4.89 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 8.58/4.89 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.89 & ( case_o184042715313410164at_nat
% 8.58/4.89 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
% 8.58/4.89 & ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.89 @ ( produc6081775807080527818_nat_o
% 8.58/4.89 @ ^ [Mi3: nat,Ma3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.58/4.89 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.58/4.89 & ! [I4: nat] :
% 8.58/4.89 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.58/4.89 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
% 8.58/4.89 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 8.58/4.89 & ( ( Mi3 = Ma3 )
% 8.58/4.89 => ! [X3: vEBT_VEBT] :
% 8.58/4.89 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.58/4.89 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
% 8.58/4.89 & ( ( Mi3 != Ma3 )
% 8.58/4.89 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 8.58/4.89 & ! [X3: nat] :
% 8.58/4.89 ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.58/4.89 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 8.58/4.89 => ( ( ord_less_nat @ Mi3 @ X3 )
% 8.58/4.89 & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
% 8.58/4.89 @ Mima ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9671_Real_Opositive_Orsp,axiom,
% 8.58/4.89 ( bNF_re728719798268516973at_o_o @ realrel
% 8.58/4.89 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 8.58/4.89 @ ^ [X8: nat > rat] :
% 8.58/4.89 ? [R5: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 8.58/4.89 & ? [K2: nat] :
% 8.58/4.89 ! [N3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K2 @ N3 )
% 8.58/4.89 => ( ord_less_rat @ R5 @ ( X8 @ N3 ) ) ) )
% 8.58/4.89 @ ^ [X8: nat > rat] :
% 8.58/4.89 ? [R5: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 8.58/4.89 & ? [K2: nat] :
% 8.58/4.89 ! [N3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K2 @ N3 )
% 8.58/4.89 => ( ord_less_rat @ R5 @ ( X8 @ N3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9672_zero__real_Orsp,axiom,
% 8.58/4.89 ( realrel
% 8.58/4.89 @ ^ [N3: nat] : zero_zero_rat
% 8.58/4.89 @ ^ [N3: nat] : zero_zero_rat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9673_one__real_Orsp,axiom,
% 8.58/4.89 ( realrel
% 8.58/4.89 @ ^ [N3: nat] : one_one_rat
% 8.58/4.89 @ ^ [N3: nat] : one_one_rat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9674_plus__real_Orsp,axiom,
% 8.58/4.89 ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
% 8.58/4.89 @ ^ [X8: nat > rat,Y7: nat > rat,N3: nat] : ( plus_plus_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) )
% 8.58/4.89 @ ^ [X8: nat > rat,Y7: nat > rat,N3: nat] : ( plus_plus_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9675_times__real_Orsp,axiom,
% 8.58/4.89 ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
% 8.58/4.89 @ ^ [X8: nat > rat,Y7: nat > rat,N3: nat] : ( times_times_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) )
% 8.58/4.89 @ ^ [X8: nat > rat,Y7: nat > rat,N3: nat] : ( times_times_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9676_uminus__real_Orsp,axiom,
% 8.58/4.89 ( bNF_re895249473297799549at_rat @ realrel @ realrel
% 8.58/4.89 @ ^ [X8: nat > rat,N3: nat] : ( uminus_uminus_rat @ ( X8 @ N3 ) )
% 8.58/4.89 @ ^ [X8: nat > rat,N3: nat] : ( uminus_uminus_rat @ ( X8 @ N3 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9677_continuous__on__arcosh,axiom,
% 8.58/4.89 ! [A3: set_real] :
% 8.58/4.89 ( ( ord_less_eq_set_real @ A3 @ ( set_ord_atLeast_real @ one_one_real ) )
% 8.58/4.89 => ( topolo5044208981011980120l_real @ A3 @ arcosh_real ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9678_inverse__real_Orsp,axiom,
% 8.58/4.89 ( bNF_re895249473297799549at_rat @ realrel @ realrel
% 8.58/4.89 @ ^ [X8: nat > rat] :
% 8.58/4.89 ( if_nat_rat @ ( vanishes @ X8 )
% 8.58/4.89 @ ^ [N3: nat] : zero_zero_rat
% 8.58/4.89 @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X8 @ N3 ) ) )
% 8.58/4.89 @ ^ [X8: nat > rat] :
% 8.58/4.89 ( if_nat_rat @ ( vanishes @ X8 )
% 8.58/4.89 @ ^ [N3: nat] : zero_zero_rat
% 8.58/4.89 @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X8 @ N3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9679_atLeast__0,axiom,
% 8.58/4.89 ( ( set_ord_atLeast_nat @ zero_zero_nat )
% 8.58/4.89 = top_top_set_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9680_vanishes__const,axiom,
% 8.58/4.89 ! [C: rat] :
% 8.58/4.89 ( ( vanishes
% 8.58/4.89 @ ^ [N3: nat] : C )
% 8.58/4.89 = ( C = zero_zero_rat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9681_last__upt,axiom,
% 8.58/4.89 ! [I: nat,J: nat] :
% 8.58/4.89 ( ( ord_less_nat @ I @ J )
% 8.58/4.89 => ( ( last_nat @ ( upt @ I @ J ) )
% 8.58/4.89 = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9682_atLeast__Suc__greaterThan,axiom,
% 8.58/4.89 ! [K: nat] :
% 8.58/4.89 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 8.58/4.89 = ( set_or1210151606488870762an_nat @ K ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9683_vanishes__minus,axiom,
% 8.58/4.89 ! [X9: nat > rat] :
% 8.58/4.89 ( ( vanishes @ X9 )
% 8.58/4.89 => ( vanishes
% 8.58/4.89 @ ^ [N3: nat] : ( uminus_uminus_rat @ ( X9 @ N3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9684_vanishes__add,axiom,
% 8.58/4.89 ! [X9: nat > rat,Y8: nat > rat] :
% 8.58/4.89 ( ( vanishes @ X9 )
% 8.58/4.89 => ( ( vanishes @ Y8 )
% 8.58/4.89 => ( vanishes
% 8.58/4.89 @ ^ [N3: nat] : ( plus_plus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9685_vanishes__diff,axiom,
% 8.58/4.89 ! [X9: nat > rat,Y8: nat > rat] :
% 8.58/4.89 ( ( vanishes @ X9 )
% 8.58/4.89 => ( ( vanishes @ Y8 )
% 8.58/4.89 => ( vanishes
% 8.58/4.89 @ ^ [N3: nat] : ( minus_minus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9686_vanishes__def,axiom,
% 8.58/4.89 ( vanishes
% 8.58/4.89 = ( ^ [X8: nat > rat] :
% 8.58/4.89 ! [R5: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 8.58/4.89 => ? [K2: nat] :
% 8.58/4.89 ! [N3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K2 @ N3 )
% 8.58/4.89 => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N3 ) ) @ R5 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9687_vanishesI,axiom,
% 8.58/4.89 ! [X9: nat > rat] :
% 8.58/4.89 ( ! [R2: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 8.58/4.89 => ? [K4: nat] :
% 8.58/4.89 ! [N2: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K4 @ N2 )
% 8.58/4.89 => ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N2 ) ) @ R2 ) ) )
% 8.58/4.89 => ( vanishes @ X9 ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9688_vanishesD,axiom,
% 8.58/4.89 ! [X9: nat > rat,R: rat] :
% 8.58/4.89 ( ( vanishes @ X9 )
% 8.58/4.89 => ( ( ord_less_rat @ zero_zero_rat @ R )
% 8.58/4.89 => ? [K3: nat] :
% 8.58/4.89 ! [N6: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K3 @ N6 )
% 8.58/4.89 => ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N6 ) ) @ R ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9689_vanishes__mult__bounded,axiom,
% 8.58/4.89 ! [X9: nat > rat,Y8: nat > rat] :
% 8.58/4.89 ( ? [A6: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ A6 )
% 8.58/4.89 & ! [N2: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N2 ) ) @ A6 ) )
% 8.58/4.89 => ( ( vanishes @ Y8 )
% 8.58/4.89 => ( vanishes
% 8.58/4.89 @ ^ [N3: nat] : ( times_times_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9690_atLeast__Suc,axiom,
% 8.58/4.89 ! [K: nat] :
% 8.58/4.89 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 8.58/4.89 = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9691_inverse__real_Otransfer,axiom,
% 8.58/4.89 ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
% 8.58/4.89 @ ^ [X8: nat > rat] :
% 8.58/4.89 ( if_nat_rat @ ( vanishes @ X8 )
% 8.58/4.89 @ ^ [N3: nat] : zero_zero_rat
% 8.58/4.89 @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X8 @ N3 ) ) )
% 8.58/4.89 @ inverse_inverse_real ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9692_inverse__real_Oabs__eq,axiom,
% 8.58/4.89 ! [X: nat > rat] :
% 8.58/4.89 ( ( realrel @ X @ X )
% 8.58/4.89 => ( ( inverse_inverse_real @ ( real2 @ X ) )
% 8.58/4.89 = ( real2
% 8.58/4.89 @ ( if_nat_rat @ ( vanishes @ X )
% 8.58/4.89 @ ^ [N3: nat] : zero_zero_rat
% 8.58/4.89 @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X @ N3 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9693_one__real__def,axiom,
% 8.58/4.89 ( one_one_real
% 8.58/4.89 = ( real2
% 8.58/4.89 @ ^ [N3: nat] : one_one_rat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9694_real_Oabs__induct,axiom,
% 8.58/4.89 ! [P: real > $o,X: real] :
% 8.58/4.89 ( ! [Y4: nat > rat] :
% 8.58/4.89 ( ( realrel @ Y4 @ Y4 )
% 8.58/4.89 => ( P @ ( real2 @ Y4 ) ) )
% 8.58/4.89 => ( P @ X ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9695_of__rat__Real,axiom,
% 8.58/4.89 ( field_7254667332652039916t_real
% 8.58/4.89 = ( ^ [X3: rat] :
% 8.58/4.89 ( real2
% 8.58/4.89 @ ^ [N3: nat] : X3 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9696_zero__real__def,axiom,
% 8.58/4.89 ( zero_zero_real
% 8.58/4.89 = ( real2
% 8.58/4.89 @ ^ [N3: nat] : zero_zero_rat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9697_real_Orel__eq__transfer,axiom,
% 8.58/4.89 ( bNF_re4521903465945308077real_o @ pcr_real
% 8.58/4.89 @ ( bNF_re4297313714947099218al_o_o @ pcr_real
% 8.58/4.89 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 8.58/4.89 @ realrel
% 8.58/4.89 @ ^ [Y3: real,Z: real] : ( Y3 = Z ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9698_of__int__Real,axiom,
% 8.58/4.89 ( ring_1_of_int_real
% 8.58/4.89 = ( ^ [X3: int] :
% 8.58/4.89 ( real2
% 8.58/4.89 @ ^ [N3: nat] : ( ring_1_of_int_rat @ X3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9699_zero__real_Otransfer,axiom,
% 8.58/4.89 ( pcr_real
% 8.58/4.89 @ ^ [N3: nat] : zero_zero_rat
% 8.58/4.89 @ zero_zero_real ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9700_one__real_Otransfer,axiom,
% 8.58/4.89 ( pcr_real
% 8.58/4.89 @ ^ [N3: nat] : one_one_rat
% 8.58/4.89 @ one_one_real ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9701_of__nat__Real,axiom,
% 8.58/4.89 ( semiri5074537144036343181t_real
% 8.58/4.89 = ( ^ [X3: nat] :
% 8.58/4.89 ( real2
% 8.58/4.89 @ ^ [N3: nat] : ( semiri681578069525770553at_rat @ X3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9702_uminus__real_Otransfer,axiom,
% 8.58/4.89 ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
% 8.58/4.89 @ ^ [X8: nat > rat,N3: nat] : ( uminus_uminus_rat @ ( X8 @ N3 ) )
% 8.58/4.89 @ uminus_uminus_real ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9703_plus__real_Otransfer,axiom,
% 8.58/4.89 ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
% 8.58/4.89 @ ^ [X8: nat > rat,Y7: nat > rat,N3: nat] : ( plus_plus_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) )
% 8.58/4.89 @ plus_plus_real ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9704_uminus__real_Oabs__eq,axiom,
% 8.58/4.89 ! [X: nat > rat] :
% 8.58/4.89 ( ( realrel @ X @ X )
% 8.58/4.89 => ( ( uminus_uminus_real @ ( real2 @ X ) )
% 8.58/4.89 = ( real2
% 8.58/4.89 @ ^ [N3: nat] : ( uminus_uminus_rat @ ( X @ N3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9705_times__real_Otransfer,axiom,
% 8.58/4.89 ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
% 8.58/4.89 @ ^ [X8: nat > rat,Y7: nat > rat,N3: nat] : ( times_times_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) )
% 8.58/4.89 @ times_times_real ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9706_plus__real_Oabs__eq,axiom,
% 8.58/4.89 ! [Xa: nat > rat,X: nat > rat] :
% 8.58/4.89 ( ( realrel @ Xa @ Xa )
% 8.58/4.89 => ( ( realrel @ X @ X )
% 8.58/4.89 => ( ( plus_plus_real @ ( real2 @ Xa ) @ ( real2 @ X ) )
% 8.58/4.89 = ( real2
% 8.58/4.89 @ ^ [N3: nat] : ( plus_plus_rat @ ( Xa @ N3 ) @ ( X @ N3 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9707_times__real_Oabs__eq,axiom,
% 8.58/4.89 ! [Xa: nat > rat,X: nat > rat] :
% 8.58/4.89 ( ( realrel @ Xa @ Xa )
% 8.58/4.89 => ( ( realrel @ X @ X )
% 8.58/4.89 => ( ( times_times_real @ ( real2 @ Xa ) @ ( real2 @ X ) )
% 8.58/4.89 = ( real2
% 8.58/4.89 @ ^ [N3: nat] : ( times_times_rat @ ( Xa @ N3 ) @ ( X @ N3 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9708_Real_Opositive_Otransfer,axiom,
% 8.58/4.89 ( bNF_re4297313714947099218al_o_o @ pcr_real
% 8.58/4.89 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 8.58/4.89 @ ^ [X8: nat > rat] :
% 8.58/4.89 ? [R5: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 8.58/4.89 & ? [K2: nat] :
% 8.58/4.89 ! [N3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K2 @ N3 )
% 8.58/4.89 => ( ord_less_rat @ R5 @ ( X8 @ N3 ) ) ) )
% 8.58/4.89 @ positive2 ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9709_Real_Opositive_Oabs__eq,axiom,
% 8.58/4.89 ! [X: nat > rat] :
% 8.58/4.89 ( ( realrel @ X @ X )
% 8.58/4.89 => ( ( positive2 @ ( real2 @ X ) )
% 8.58/4.89 = ( ? [R5: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 8.58/4.89 & ? [K2: nat] :
% 8.58/4.89 ! [N3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K2 @ N3 )
% 8.58/4.89 => ( ord_less_rat @ R5 @ ( X @ N3 ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9710_Real_Opositive__mult,axiom,
% 8.58/4.89 ! [X: real,Y: real] :
% 8.58/4.89 ( ( positive2 @ X )
% 8.58/4.89 => ( ( positive2 @ Y )
% 8.58/4.89 => ( positive2 @ ( times_times_real @ X @ Y ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9711_Real_Opositive__add,axiom,
% 8.58/4.89 ! [X: real,Y: real] :
% 8.58/4.89 ( ( positive2 @ X )
% 8.58/4.89 => ( ( positive2 @ Y )
% 8.58/4.89 => ( positive2 @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9712_Real_Opositive__zero,axiom,
% 8.58/4.89 ~ ( positive2 @ zero_zero_real ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9713_Real_Opositive__minus,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ( ~ ( positive2 @ X )
% 8.58/4.89 => ( ( X != zero_zero_real )
% 8.58/4.89 => ( positive2 @ ( uminus_uminus_real @ X ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9714_less__real__def,axiom,
% 8.58/4.89 ( ord_less_real
% 8.58/4.89 = ( ^ [X3: real,Y6: real] : ( positive2 @ ( minus_minus_real @ Y6 @ X3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9715_le__Real,axiom,
% 8.58/4.89 ! [X9: nat > rat,Y8: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ( cauchy @ Y8 )
% 8.58/4.89 => ( ( ord_less_eq_real @ ( real2 @ X9 ) @ ( real2 @ Y8 ) )
% 8.58/4.89 = ( ! [R5: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 8.58/4.89 => ? [K2: nat] :
% 8.58/4.89 ! [N3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K2 @ N3 )
% 8.58/4.89 => ( ord_less_eq_rat @ ( X9 @ N3 ) @ ( plus_plus_rat @ ( Y8 @ N3 ) @ R5 ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9716_Real_Opositive_Orep__eq,axiom,
% 8.58/4.89 ( positive2
% 8.58/4.89 = ( ^ [X3: real] :
% 8.58/4.89 ? [R5: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 8.58/4.89 & ? [K2: nat] :
% 8.58/4.89 ! [N3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K2 @ N3 )
% 8.58/4.89 => ( ord_less_rat @ R5 @ ( rep_real @ X3 @ N3 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9717_realrel__refl,axiom,
% 8.58/4.89 ! [X9: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( realrel @ X9 @ X9 ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9718_cauchy__diff,axiom,
% 8.58/4.89 ! [X9: nat > rat,Y8: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ( cauchy @ Y8 )
% 8.58/4.89 => ( cauchy
% 8.58/4.89 @ ^ [N3: nat] : ( minus_minus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9719_cauchy__mult,axiom,
% 8.58/4.89 ! [X9: nat > rat,Y8: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ( cauchy @ Y8 )
% 8.58/4.89 => ( cauchy
% 8.58/4.89 @ ^ [N3: nat] : ( times_times_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9720_cauchy__add,axiom,
% 8.58/4.89 ! [X9: nat > rat,Y8: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ( cauchy @ Y8 )
% 8.58/4.89 => ( cauchy
% 8.58/4.89 @ ^ [N3: nat] : ( plus_plus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9721_cauchy__minus,axiom,
% 8.58/4.89 ! [X9: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( cauchy
% 8.58/4.89 @ ^ [N3: nat] : ( uminus_uminus_rat @ ( X9 @ N3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9722_cauchy__const,axiom,
% 8.58/4.89 ! [X: rat] :
% 8.58/4.89 ( cauchy
% 8.58/4.89 @ ^ [N3: nat] : X ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9723_Real__induct,axiom,
% 8.58/4.89 ! [P: real > $o,X: real] :
% 8.58/4.89 ( ! [X15: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X15 )
% 8.58/4.89 => ( P @ ( real2 @ X15 ) ) )
% 8.58/4.89 => ( P @ X ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9724_cr__real__eq,axiom,
% 8.58/4.89 ( pcr_real
% 8.58/4.89 = ( ^ [X3: nat > rat,Y6: real] :
% 8.58/4.89 ( ( cauchy @ X3 )
% 8.58/4.89 & ( ( real2 @ X3 )
% 8.58/4.89 = Y6 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9725_cauchy__inverse,axiom,
% 8.58/4.89 ! [X9: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ~ ( vanishes @ X9 )
% 8.58/4.89 => ( cauchy
% 8.58/4.89 @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X9 @ N3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9726_cauchy__imp__bounded,axiom,
% 8.58/4.89 ! [X9: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ? [B4: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ B4 )
% 8.58/4.89 & ! [N6: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N6 ) ) @ B4 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9727_less__RealD,axiom,
% 8.58/4.89 ! [Y8: nat > rat,X: real] :
% 8.58/4.89 ( ( cauchy @ Y8 )
% 8.58/4.89 => ( ( ord_less_real @ X @ ( real2 @ Y8 ) )
% 8.58/4.89 => ? [N2: nat] : ( ord_less_real @ X @ ( field_7254667332652039916t_real @ ( Y8 @ N2 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9728_le__RealI,axiom,
% 8.58/4.89 ! [Y8: nat > rat,X: real] :
% 8.58/4.89 ( ( cauchy @ Y8 )
% 8.58/4.89 => ( ! [N2: nat] : ( ord_less_eq_real @ X @ ( field_7254667332652039916t_real @ ( Y8 @ N2 ) ) )
% 8.58/4.89 => ( ord_less_eq_real @ X @ ( real2 @ Y8 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9729_Real__leI,axiom,
% 8.58/4.89 ! [X9: nat > rat,Y: real] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ! [N2: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X9 @ N2 ) ) @ Y )
% 8.58/4.89 => ( ord_less_eq_real @ ( real2 @ X9 ) @ Y ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9730_minus__Real,axiom,
% 8.58/4.89 ! [X9: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ( uminus_uminus_real @ ( real2 @ X9 ) )
% 8.58/4.89 = ( real2
% 8.58/4.89 @ ^ [N3: nat] : ( uminus_uminus_rat @ ( X9 @ N3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9731_add__Real,axiom,
% 8.58/4.89 ! [X9: nat > rat,Y8: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ( cauchy @ Y8 )
% 8.58/4.89 => ( ( plus_plus_real @ ( real2 @ X9 ) @ ( real2 @ Y8 ) )
% 8.58/4.89 = ( real2
% 8.58/4.89 @ ^ [N3: nat] : ( plus_plus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9732_mult__Real,axiom,
% 8.58/4.89 ! [X9: nat > rat,Y8: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ( cauchy @ Y8 )
% 8.58/4.89 => ( ( times_times_real @ ( real2 @ X9 ) @ ( real2 @ Y8 ) )
% 8.58/4.89 = ( real2
% 8.58/4.89 @ ^ [N3: nat] : ( times_times_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9733_diff__Real,axiom,
% 8.58/4.89 ! [X9: nat > rat,Y8: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ( cauchy @ Y8 )
% 8.58/4.89 => ( ( minus_minus_real @ ( real2 @ X9 ) @ ( real2 @ Y8 ) )
% 8.58/4.89 = ( real2
% 8.58/4.89 @ ^ [N3: nat] : ( minus_minus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9734_realrelI,axiom,
% 8.58/4.89 ! [X9: nat > rat,Y8: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ( cauchy @ Y8 )
% 8.58/4.89 => ( ( vanishes
% 8.58/4.89 @ ^ [N3: nat] : ( minus_minus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) )
% 8.58/4.89 => ( realrel @ X9 @ Y8 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9735_eq__Real,axiom,
% 8.58/4.89 ! [X9: nat > rat,Y8: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ( cauchy @ Y8 )
% 8.58/4.89 => ( ( ( real2 @ X9 )
% 8.58/4.89 = ( real2 @ Y8 ) )
% 8.58/4.89 = ( vanishes
% 8.58/4.89 @ ^ [N3: nat] : ( minus_minus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9736_vanishes__diff__inverse,axiom,
% 8.58/4.89 ! [X9: nat > rat,Y8: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ~ ( vanishes @ X9 )
% 8.58/4.89 => ( ( cauchy @ Y8 )
% 8.58/4.89 => ( ~ ( vanishes @ Y8 )
% 8.58/4.89 => ( ( vanishes
% 8.58/4.89 @ ^ [N3: nat] : ( minus_minus_rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) )
% 8.58/4.89 => ( vanishes
% 8.58/4.89 @ ^ [N3: nat] : ( minus_minus_rat @ ( inverse_inverse_rat @ ( X9 @ N3 ) ) @ ( inverse_inverse_rat @ ( Y8 @ N3 ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9737_realrel__def,axiom,
% 8.58/4.89 ( realrel
% 8.58/4.89 = ( ^ [X8: nat > rat,Y7: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X8 )
% 8.58/4.89 & ( cauchy @ Y7 )
% 8.58/4.89 & ( vanishes
% 8.58/4.89 @ ^ [N3: nat] : ( minus_minus_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9738_cauchy__not__vanishes__cases,axiom,
% 8.58/4.89 ! [X9: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ~ ( vanishes @ X9 )
% 8.58/4.89 => ? [B4: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ B4 )
% 8.58/4.89 & ? [K3: nat] :
% 8.58/4.89 ( ! [N6: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K3 @ N6 )
% 8.58/4.89 => ( ord_less_rat @ B4 @ ( uminus_uminus_rat @ ( X9 @ N6 ) ) ) )
% 8.58/4.89 | ! [N6: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K3 @ N6 )
% 8.58/4.89 => ( ord_less_rat @ B4 @ ( X9 @ N6 ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9739_positive__Real,axiom,
% 8.58/4.89 ! [X9: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ( positive2 @ ( real2 @ X9 ) )
% 8.58/4.89 = ( ? [R5: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 8.58/4.89 & ? [K2: nat] :
% 8.58/4.89 ! [N3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K2 @ N3 )
% 8.58/4.89 => ( ord_less_rat @ R5 @ ( X9 @ N3 ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9740_cauchy__not__vanishes,axiom,
% 8.58/4.89 ! [X9: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ~ ( vanishes @ X9 )
% 8.58/4.89 => ? [B4: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ B4 )
% 8.58/4.89 & ? [K3: nat] :
% 8.58/4.89 ! [N6: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K3 @ N6 )
% 8.58/4.89 => ( ord_less_rat @ B4 @ ( abs_abs_rat @ ( X9 @ N6 ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9741_cauchyD,axiom,
% 8.58/4.89 ! [X9: nat > rat,R: rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ( ord_less_rat @ zero_zero_rat @ R )
% 8.58/4.89 => ? [K3: nat] :
% 8.58/4.89 ! [M4: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K3 @ M4 )
% 8.58/4.89 => ! [N6: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K3 @ N6 )
% 8.58/4.89 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X9 @ M4 ) @ ( X9 @ N6 ) ) ) @ R ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9742_cauchyI,axiom,
% 8.58/4.89 ! [X9: nat > rat] :
% 8.58/4.89 ( ! [R2: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 8.58/4.89 => ? [K4: nat] :
% 8.58/4.89 ! [M2: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K4 @ M2 )
% 8.58/4.89 => ! [N2: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K4 @ N2 )
% 8.58/4.89 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X9 @ M2 ) @ ( X9 @ N2 ) ) ) @ R2 ) ) ) )
% 8.58/4.89 => ( cauchy @ X9 ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9743_cauchy__def,axiom,
% 8.58/4.89 ( cauchy
% 8.58/4.89 = ( ^ [X8: nat > rat] :
% 8.58/4.89 ! [R5: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 8.58/4.89 => ? [K2: nat] :
% 8.58/4.89 ! [M3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K2 @ M3 )
% 8.58/4.89 => ! [N3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K2 @ N3 )
% 8.58/4.89 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) ) @ R5 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9744_inverse__Real,axiom,
% 8.58/4.89 ! [X9: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ( ( vanishes @ X9 )
% 8.58/4.89 => ( ( inverse_inverse_real @ ( real2 @ X9 ) )
% 8.58/4.89 = zero_zero_real ) )
% 8.58/4.89 & ( ~ ( vanishes @ X9 )
% 8.58/4.89 => ( ( inverse_inverse_real @ ( real2 @ X9 ) )
% 8.58/4.89 = ( real2
% 8.58/4.89 @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X9 @ N3 ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9745_not__positive__Real,axiom,
% 8.58/4.89 ! [X9: nat > rat] :
% 8.58/4.89 ( ( cauchy @ X9 )
% 8.58/4.89 => ( ( ~ ( positive2 @ ( real2 @ X9 ) ) )
% 8.58/4.89 = ( ! [R5: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 8.58/4.89 => ? [K2: nat] :
% 8.58/4.89 ! [N3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K2 @ N3 )
% 8.58/4.89 => ( ord_less_eq_rat @ ( X9 @ N3 ) @ R5 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9746_Real_Opositive__def,axiom,
% 8.58/4.89 ( positive2
% 8.58/4.89 = ( map_fu1856342031159181835at_o_o @ rep_real @ id_o
% 8.58/4.89 @ ^ [X8: nat > rat] :
% 8.58/4.89 ? [R5: rat] :
% 8.58/4.89 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 8.58/4.89 & ? [K2: nat] :
% 8.58/4.89 ! [N3: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ K2 @ N3 )
% 8.58/4.89 => ( ord_less_rat @ R5 @ ( X8 @ N3 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9747_inverse__real__def,axiom,
% 8.58/4.89 ( inverse_inverse_real
% 8.58/4.89 = ( map_fu7146612038024189824t_real @ rep_real @ real2
% 8.58/4.89 @ ^ [X8: nat > rat] :
% 8.58/4.89 ( if_nat_rat @ ( vanishes @ X8 )
% 8.58/4.89 @ ^ [N3: nat] : zero_zero_rat
% 8.58/4.89 @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X8 @ N3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9748_uminus__real__def,axiom,
% 8.58/4.89 ( uminus_uminus_real
% 8.58/4.89 = ( map_fu7146612038024189824t_real @ rep_real @ real2
% 8.58/4.89 @ ^ [X8: nat > rat,N3: nat] : ( uminus_uminus_rat @ ( X8 @ N3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9749_times__real__def,axiom,
% 8.58/4.89 ( times_times_real
% 8.58/4.89 = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
% 8.58/4.89 @ ^ [X8: nat > rat,Y7: nat > rat,N3: nat] : ( times_times_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9750_plus__real__def,axiom,
% 8.58/4.89 ( plus_plus_real
% 8.58/4.89 = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
% 8.58/4.89 @ ^ [X8: nat > rat,Y7: nat > rat,N3: nat] : ( plus_plus_rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9751_cr__real__def,axiom,
% 8.58/4.89 ( cr_real
% 8.58/4.89 = ( ^ [X3: nat > rat,Y6: real] :
% 8.58/4.89 ( ( realrel @ X3 @ X3 )
% 8.58/4.89 & ( ( real2 @ X3 )
% 8.58/4.89 = Y6 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9752_real_Opcr__cr__eq,axiom,
% 8.58/4.89 pcr_real = cr_real ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9753_Bseq__monoseq__convergent_H__dec,axiom,
% 8.58/4.89 ! [F: nat > real,M10: nat] :
% 8.58/4.89 ( ( bfun_nat_real
% 8.58/4.89 @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ M10 ) )
% 8.58/4.89 @ at_top_nat )
% 8.58/4.89 => ( ! [M2: nat,N2: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ M10 @ M2 )
% 8.58/4.89 => ( ( ord_less_eq_nat @ M2 @ N2 )
% 8.58/4.89 => ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ M2 ) ) ) )
% 8.58/4.89 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9754_Bseq__monoseq__convergent_H__inc,axiom,
% 8.58/4.89 ! [F: nat > real,M10: nat] :
% 8.58/4.89 ( ( bfun_nat_real
% 8.58/4.89 @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ M10 ) )
% 8.58/4.89 @ at_top_nat )
% 8.58/4.89 => ( ! [M2: nat,N2: nat] :
% 8.58/4.89 ( ( ord_less_eq_nat @ M10 @ M2 )
% 8.58/4.89 => ( ( ord_less_eq_nat @ M2 @ N2 )
% 8.58/4.89 => ( ord_less_eq_real @ ( F @ M2 ) @ ( F @ N2 ) ) ) )
% 8.58/4.89 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9755_convergent__realpow,axiom,
% 8.58/4.89 ! [X: real] :
% 8.58/4.89 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.58/4.89 => ( ( ord_less_eq_real @ X @ one_one_real )
% 8.58/4.89 => ( topolo7531315842566124627t_real @ ( power_power_real @ X ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9756_set__encode__vimage__Suc,axiom,
% 8.58/4.89 ! [A3: set_nat] :
% 8.58/4.89 ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A3 ) )
% 8.58/4.89 = ( divide_divide_nat @ ( nat_set_encode @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9757_vimage__Suc__insert__0,axiom,
% 8.58/4.89 ! [A3: set_nat] :
% 8.58/4.89 ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A3 ) )
% 8.58/4.89 = ( vimage_nat_nat @ suc @ A3 ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9758_vimage__Suc__insert__Suc,axiom,
% 8.58/4.89 ! [N: nat,A3: set_nat] :
% 8.58/4.89 ( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N ) @ A3 ) )
% 8.58/4.89 = ( insert_nat @ N @ ( vimage_nat_nat @ suc @ A3 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9759_finite__vimage__Suc__iff,axiom,
% 8.58/4.89 ! [F4: set_nat] :
% 8.58/4.89 ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F4 ) )
% 8.58/4.89 = ( finite_finite_nat @ F4 ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9760_set__decode__div__2,axiom,
% 8.58/4.89 ! [X: nat] :
% 8.58/4.89 ( ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.58/4.89 = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9761_euclidean__size__nat__def,axiom,
% 8.58/4.89 euclid4777050414544973029ze_nat = id_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9762_euclidean__size__int__def,axiom,
% 8.58/4.89 ( euclid4774559944035922753ze_int
% 8.58/4.89 = ( comp_int_nat_int @ nat2 @ abs_abs_int ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9763_abs__division__segment,axiom,
% 8.58/4.89 ! [K: int] :
% 8.58/4.89 ( ( abs_abs_int @ ( euclid3395696857347342551nt_int @ K ) )
% 8.58/4.89 = one_one_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9764_division__segment__eq__sgn,axiom,
% 8.58/4.89 ! [K: int] :
% 8.58/4.89 ( ( K != zero_zero_int )
% 8.58/4.89 => ( ( euclid3395696857347342551nt_int @ K )
% 8.58/4.89 = ( sgn_sgn_int @ K ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9765_division__segment__nat__def,axiom,
% 8.58/4.89 ( euclid3398187327856392827nt_nat
% 8.58/4.89 = ( ^ [N3: nat] : one_one_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9766_division__segment__int__def,axiom,
% 8.58/4.89 ( euclid3395696857347342551nt_int
% 8.58/4.89 = ( ^ [K2: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9767_transp__realrel,axiom,
% 8.58/4.89 transp_nat_rat @ realrel ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9768_bot__nat__0_Oordering__top__axioms,axiom,
% 8.58/4.89 ( ordering_top_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X3 )
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] : ( ord_less_nat @ Y6 @ X3 )
% 8.58/4.89 @ zero_zero_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9769_strict__mono__imp__increasing,axiom,
% 8.58/4.89 ! [F: nat > nat,N: nat] :
% 8.58/4.89 ( ( order_5726023648592871131at_nat @ F )
% 8.58/4.89 => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9770_gcd__nat_Oordering__top__axioms,axiom,
% 8.58/4.89 ( ordering_top_nat @ dvd_dvd_nat
% 8.58/4.89 @ ^ [M3: nat,N3: nat] :
% 8.58/4.89 ( ( dvd_dvd_nat @ M3 @ N3 )
% 8.58/4.89 & ( M3 != N3 ) )
% 8.58/4.89 @ zero_zero_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9771_prod__decode__triangle__add,axiom,
% 8.58/4.89 ! [K: nat,M: nat] :
% 8.58/4.89 ( ( nat_prod_decode @ ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) )
% 8.58/4.89 = ( nat_prod_decode_aux @ K @ M ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9772_prod__decode__eq,axiom,
% 8.58/4.89 ! [X: nat,Y: nat] :
% 8.58/4.89 ( ( ( nat_prod_decode @ X )
% 8.58/4.89 = ( nat_prod_decode @ Y ) )
% 8.58/4.89 = ( X = Y ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9773_prod__encode__inverse,axiom,
% 8.58/4.89 ! [X: product_prod_nat_nat] :
% 8.58/4.89 ( ( nat_prod_decode @ ( nat_prod_encode @ X ) )
% 8.58/4.89 = X ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9774_prod__decode__inverse,axiom,
% 8.58/4.89 ! [N: nat] :
% 8.58/4.89 ( ( nat_prod_encode @ ( nat_prod_decode @ N ) )
% 8.58/4.89 = N ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9775_inj__prod__decode,axiom,
% 8.58/4.89 ! [A3: set_nat] : ( inj_on5538052773655684606at_nat @ nat_prod_decode @ A3 ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9776_prod__decode__def,axiom,
% 8.58/4.89 ( nat_prod_decode
% 8.58/4.89 = ( nat_prod_decode_aux @ zero_zero_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9777_surj__prod__decode,axiom,
% 8.58/4.89 ( ( image_5846123807819985514at_nat @ nat_prod_decode @ top_top_set_nat )
% 8.58/4.89 = top_to4669805908274784177at_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9778_bij__prod__decode,axiom,
% 8.58/4.89 bij_be8693218025023041337at_nat @ nat_prod_decode @ top_top_set_nat @ top_to4669805908274784177at_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9779_list__decode_Opinduct,axiom,
% 8.58/4.89 ! [A0: nat,P: nat > $o] :
% 8.58/4.89 ( ( accp_nat @ nat_list_decode_rel @ A0 )
% 8.58/4.89 => ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
% 8.58/4.89 => ( P @ zero_zero_nat ) )
% 8.58/4.89 => ( ! [N2: nat] :
% 8.58/4.89 ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N2 ) )
% 8.58/4.89 => ( ! [X5: nat,Y5: nat] :
% 8.58/4.89 ( ( ( product_Pair_nat_nat @ X5 @ Y5 )
% 8.58/4.89 = ( nat_prod_decode @ N2 ) )
% 8.58/4.89 => ( P @ Y5 ) )
% 8.58/4.89 => ( P @ ( suc @ N2 ) ) ) )
% 8.58/4.89 => ( P @ A0 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9780_list__decode_Oelims,axiom,
% 8.58/4.89 ! [X: nat,Y: list_nat] :
% 8.58/4.89 ( ( ( nat_list_decode @ X )
% 8.58/4.89 = Y )
% 8.58/4.89 => ( ( ( X = zero_zero_nat )
% 8.58/4.89 => ( Y != nil_nat ) )
% 8.58/4.89 => ~ ! [N2: nat] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( suc @ N2 ) )
% 8.58/4.89 => ( Y
% 8.58/4.89 != ( produc2761476792215241774st_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y6 ) )
% 8.58/4.89 @ ( nat_prod_decode @ N2 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9781_list__encode__inverse,axiom,
% 8.58/4.89 ! [X: list_nat] :
% 8.58/4.89 ( ( nat_list_decode @ ( nat_list_encode @ X ) )
% 8.58/4.89 = X ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9782_list__decode__inverse,axiom,
% 8.58/4.89 ! [N: nat] :
% 8.58/4.89 ( ( nat_list_encode @ ( nat_list_decode @ N ) )
% 8.58/4.89 = N ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9783_list__decode__eq,axiom,
% 8.58/4.89 ! [X: nat,Y: nat] :
% 8.58/4.89 ( ( ( nat_list_decode @ X )
% 8.58/4.89 = ( nat_list_decode @ Y ) )
% 8.58/4.89 = ( X = Y ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9784_inj__list__decode,axiom,
% 8.58/4.89 ! [A3: set_nat] : ( inj_on_nat_list_nat @ nat_list_decode @ A3 ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9785_list__decode_Opsimps_I1_J,axiom,
% 8.58/4.89 ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
% 8.58/4.89 => ( ( nat_list_decode @ zero_zero_nat )
% 8.58/4.89 = nil_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9786_list__decode_Osimps_I1_J,axiom,
% 8.58/4.89 ( ( nat_list_decode @ zero_zero_nat )
% 8.58/4.89 = nil_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9787_list__decode_Opsimps_I2_J,axiom,
% 8.58/4.89 ! [N: nat] :
% 8.58/4.89 ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N ) )
% 8.58/4.89 => ( ( nat_list_decode @ ( suc @ N ) )
% 8.58/4.89 = ( produc2761476792215241774st_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y6 ) )
% 8.58/4.89 @ ( nat_prod_decode @ N ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9788_bij__list__decode,axiom,
% 8.58/4.89 bij_be6293887246118711976st_nat @ nat_list_decode @ top_top_set_nat @ top_top_set_list_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9789_surj__list__decode,axiom,
% 8.58/4.89 ( ( image_nat_list_nat @ nat_list_decode @ top_top_set_nat )
% 8.58/4.89 = top_top_set_list_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9790_list__decode_Osimps_I2_J,axiom,
% 8.58/4.89 ! [N: nat] :
% 8.58/4.89 ( ( nat_list_decode @ ( suc @ N ) )
% 8.58/4.89 = ( produc2761476792215241774st_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y6 ) )
% 8.58/4.89 @ ( nat_prod_decode @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9791_list__decode_Opelims,axiom,
% 8.58/4.89 ! [X: nat,Y: list_nat] :
% 8.58/4.89 ( ( ( nat_list_decode @ X )
% 8.58/4.89 = Y )
% 8.58/4.89 => ( ( accp_nat @ nat_list_decode_rel @ X )
% 8.58/4.89 => ( ( ( X = zero_zero_nat )
% 8.58/4.89 => ( ( Y = nil_nat )
% 8.58/4.89 => ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
% 8.58/4.89 => ~ ! [N2: nat] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( suc @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( produc2761476792215241774st_nat
% 8.58/4.89 @ ^ [X3: nat,Y6: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y6 ) )
% 8.58/4.89 @ ( nat_prod_decode @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N2 ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9792_compute__powr__real,axiom,
% 8.58/4.89 ( powr_real2
% 8.58/4.89 = ( ^ [B2: real,I4: real] :
% 8.58/4.89 ( if_real @ ( ord_less_eq_real @ B2 @ zero_zero_real )
% 8.58/4.89 @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 8.58/4.89 @ ^ [Uu3: product_unit] : ( powr_real2 @ B2 @ I4 ) )
% 8.58/4.89 @ ( if_real
% 8.58/4.89 @ ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ I4 ) )
% 8.58/4.89 = I4 )
% 8.58/4.89 @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ I4 ) @ ( power_power_real @ B2 @ ( nat2 @ ( archim6058952711729229775r_real @ I4 ) ) ) @ ( divide_divide_real @ one_one_real @ ( power_power_real @ B2 @ ( nat2 @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ I4 ) ) ) ) ) )
% 8.58/4.89 @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 8.58/4.89 @ ^ [Uu3: product_unit] : ( powr_real2 @ B2 @ I4 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9793_int__of__integer__integer__of__nat,axiom,
% 8.58/4.89 ! [N: nat] :
% 8.58/4.89 ( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
% 8.58/4.89 = ( semiri1314217659103216013at_int @ N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9794_integer__of__nat_Orep__eq,axiom,
% 8.58/4.89 ! [X: nat] :
% 8.58/4.89 ( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
% 8.58/4.89 = ( semiri1314217659103216013at_int @ X ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9795_integer__of__nat__0,axiom,
% 8.58/4.89 ( ( code_integer_of_nat @ zero_zero_nat )
% 8.58/4.89 = zero_z3403309356797280102nteger ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9796_integer__of__nat_Oabs__eq,axiom,
% 8.58/4.89 ( code_integer_of_nat
% 8.58/4.89 = ( ^ [X3: nat] : ( code_integer_of_int @ ( semiri1314217659103216013at_int @ X3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9797_integer__of__nat__numeral,axiom,
% 8.58/4.89 ! [N: num] :
% 8.58/4.89 ( ( code_integer_of_nat @ ( numeral_numeral_nat @ N ) )
% 8.58/4.89 = ( numera6620942414471956472nteger @ N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9798_integer__of__nat__1,axiom,
% 8.58/4.89 ( ( code_integer_of_nat @ one_one_nat )
% 8.58/4.89 = one_one_Code_integer ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9799_integer__of__nat__def,axiom,
% 8.58/4.89 ( code_integer_of_nat
% 8.58/4.89 = ( map_fu6290471996055670595nteger @ id_nat @ code_integer_of_int @ semiri1314217659103216013at_int ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9800_Gcd__int__set__eq__fold,axiom,
% 8.58/4.89 ! [Xs: list_int] :
% 8.58/4.89 ( ( gcd_Gcd_int @ ( set_int2 @ Xs ) )
% 8.58/4.89 = ( fold_int_int @ gcd_gcd_int @ Xs @ zero_zero_int ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9801_numeral__le__enat__iff,axiom,
% 8.58/4.89 ! [M: num,N: nat] :
% 8.58/4.89 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
% 8.58/4.89 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9802_plus__enat__simps_I1_J,axiom,
% 8.58/4.89 ! [M: nat,N: nat] :
% 8.58/4.89 ( ( plus_p3455044024723400733d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 8.58/4.89 = ( extended_enat2 @ ( plus_plus_nat @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9803_idiff__enat__0,axiom,
% 8.58/4.89 ! [N: extended_enat] :
% 8.58/4.89 ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ zero_zero_nat ) @ N )
% 8.58/4.89 = ( extended_enat2 @ zero_zero_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9804_idiff__enat__0__right,axiom,
% 8.58/4.89 ! [N: extended_enat] :
% 8.58/4.89 ( ( minus_3235023915231533773d_enat @ N @ ( extended_enat2 @ zero_zero_nat ) )
% 8.58/4.89 = N ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9805_idiff__enat__enat,axiom,
% 8.58/4.89 ! [A: nat,B: nat] :
% 8.58/4.89 ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ A ) @ ( extended_enat2 @ B ) )
% 8.58/4.89 = ( extended_enat2 @ ( minus_minus_nat @ A @ B ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9806_times__enat__simps_I1_J,axiom,
% 8.58/4.89 ! [M: nat,N: nat] :
% 8.58/4.89 ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 8.58/4.89 = ( extended_enat2 @ ( times_times_nat @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9807_numeral__less__enat__iff,axiom,
% 8.58/4.89 ! [M: num,N: nat] :
% 8.58/4.89 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
% 8.58/4.89 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9808_enat__0__iff_I2_J,axiom,
% 8.58/4.89 ! [X: nat] :
% 8.58/4.89 ( ( zero_z5237406670263579293d_enat
% 8.58/4.89 = ( extended_enat2 @ X ) )
% 8.58/4.89 = ( X = zero_zero_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9809_enat__0__iff_I1_J,axiom,
% 8.58/4.89 ! [X: nat] :
% 8.58/4.89 ( ( ( extended_enat2 @ X )
% 8.58/4.89 = zero_z5237406670263579293d_enat )
% 8.58/4.89 = ( X = zero_zero_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9810_zero__enat__def,axiom,
% 8.58/4.89 ( zero_z5237406670263579293d_enat
% 8.58/4.89 = ( extended_enat2 @ zero_zero_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9811_numeral__eq__enat,axiom,
% 8.58/4.89 ( numera1916890842035813515d_enat
% 8.58/4.89 = ( ^ [K2: num] : ( extended_enat2 @ ( numeral_numeral_nat @ K2 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9812_one__enat__def,axiom,
% 8.58/4.89 ( one_on7984719198319812577d_enat
% 8.58/4.89 = ( extended_enat2 @ one_one_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9813_enat__1__iff_I1_J,axiom,
% 8.58/4.89 ! [X: nat] :
% 8.58/4.89 ( ( ( extended_enat2 @ X )
% 8.58/4.89 = one_on7984719198319812577d_enat )
% 8.58/4.89 = ( X = one_one_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9814_enat__1__iff_I2_J,axiom,
% 8.58/4.89 ! [X: nat] :
% 8.58/4.89 ( ( one_on7984719198319812577d_enat
% 8.58/4.89 = ( extended_enat2 @ X ) )
% 8.58/4.89 = ( X = one_one_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9815_Suc__ile__eq,axiom,
% 8.58/4.89 ! [M: nat,N: extended_enat] :
% 8.58/4.89 ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N )
% 8.58/4.89 = ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9816_filtermap__at__right__shift,axiom,
% 8.58/4.89 ! [D: real,A: real] :
% 8.58/4.89 ( ( filtermap_real_real
% 8.58/4.89 @ ^ [X3: real] : ( minus_minus_real @ X3 @ D )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 8.58/4.89 = ( topolo2177554685111907308n_real @ ( minus_minus_real @ A @ D ) @ ( set_or5849166863359141190n_real @ ( minus_minus_real @ A @ D ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9817_iadd__le__enat__iff,axiom,
% 8.58/4.89 ! [X: extended_enat,Y: extended_enat,N: nat] :
% 8.58/4.89 ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ ( extended_enat2 @ N ) )
% 8.58/4.89 = ( ? [Y9: nat,X10: nat] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( extended_enat2 @ X10 ) )
% 8.58/4.89 & ( Y
% 8.58/4.89 = ( extended_enat2 @ Y9 ) )
% 8.58/4.89 & ( ord_less_eq_nat @ ( plus_plus_nat @ X10 @ Y9 ) @ N ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9818_at__right__to__0,axiom,
% 8.58/4.89 ! [A: real] :
% 8.58/4.89 ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
% 8.58/4.89 = ( filtermap_real_real
% 8.58/4.89 @ ^ [X3: real] : ( plus_plus_real @ X3 @ A )
% 8.58/4.89 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9819_Gcd__nat__set__eq__fold,axiom,
% 8.58/4.89 ! [Xs: list_nat] :
% 8.58/4.89 ( ( gcd_Gcd_nat @ ( set_nat2 @ Xs ) )
% 8.58/4.89 = ( fold_nat_nat @ gcd_gcd_nat @ Xs @ zero_zero_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9820_times__enat__simps_I4_J,axiom,
% 8.58/4.89 ! [M: nat] :
% 8.58/4.89 ( ( ( M = zero_zero_nat )
% 8.58/4.89 => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
% 8.58/4.89 = zero_z5237406670263579293d_enat ) )
% 8.58/4.89 & ( ( M != zero_zero_nat )
% 8.58/4.89 => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
% 8.58/4.89 = extend5688581933313929465d_enat ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9821_times__enat__simps_I3_J,axiom,
% 8.58/4.89 ! [N: nat] :
% 8.58/4.89 ( ( ( N = zero_zero_nat )
% 8.58/4.89 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
% 8.58/4.89 = zero_z5237406670263579293d_enat ) )
% 8.58/4.89 & ( ( N != zero_zero_nat )
% 8.58/4.89 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
% 8.58/4.89 = extend5688581933313929465d_enat ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9822_plus__enat__simps_I2_J,axiom,
% 8.58/4.89 ! [Q2: extended_enat] :
% 8.58/4.89 ( ( plus_p3455044024723400733d_enat @ extend5688581933313929465d_enat @ Q2 )
% 8.58/4.89 = extend5688581933313929465d_enat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9823_plus__enat__simps_I3_J,axiom,
% 8.58/4.89 ! [Q2: extended_enat] :
% 8.58/4.89 ( ( plus_p3455044024723400733d_enat @ Q2 @ extend5688581933313929465d_enat )
% 8.58/4.89 = extend5688581933313929465d_enat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9824_idiff__infinity,axiom,
% 8.58/4.89 ! [N: extended_enat] :
% 8.58/4.89 ( ( minus_3235023915231533773d_enat @ extend5688581933313929465d_enat @ N )
% 8.58/4.89 = extend5688581933313929465d_enat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9825_times__enat__simps_I2_J,axiom,
% 8.58/4.89 ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ extend5688581933313929465d_enat )
% 8.58/4.89 = extend5688581933313929465d_enat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9826_idiff__self,axiom,
% 8.58/4.89 ! [N: extended_enat] :
% 8.58/4.89 ( ( N != extend5688581933313929465d_enat )
% 8.58/4.89 => ( ( minus_3235023915231533773d_enat @ N @ N )
% 8.58/4.89 = zero_z5237406670263579293d_enat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9827_add__diff__cancel__enat,axiom,
% 8.58/4.89 ! [X: extended_enat,Y: extended_enat] :
% 8.58/4.89 ( ( X != extend5688581933313929465d_enat )
% 8.58/4.89 => ( ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ X )
% 8.58/4.89 = Y ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9828_idiff__infinity__right,axiom,
% 8.58/4.89 ! [A: nat] :
% 8.58/4.89 ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ A ) @ extend5688581933313929465d_enat )
% 8.58/4.89 = zero_z5237406670263579293d_enat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9829_enat__add__left__cancel__le,axiom,
% 8.58/4.89 ! [A: extended_enat,B: extended_enat,C: extended_enat] :
% 8.58/4.89 ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( plus_p3455044024723400733d_enat @ A @ C ) )
% 8.58/4.89 = ( ( A = extend5688581933313929465d_enat )
% 8.58/4.89 | ( ord_le2932123472753598470d_enat @ B @ C ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9830_enat__add__left__cancel__less,axiom,
% 8.58/4.89 ! [A: extended_enat,B: extended_enat,C: extended_enat] :
% 8.58/4.89 ( ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( plus_p3455044024723400733d_enat @ A @ C ) )
% 8.58/4.89 = ( ( A != extend5688581933313929465d_enat )
% 8.58/4.89 & ( ord_le72135733267957522d_enat @ B @ C ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9831_infinity__ne__i1,axiom,
% 8.58/4.89 extend5688581933313929465d_enat != one_on7984719198319812577d_enat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9832_enat__add__left__cancel,axiom,
% 8.58/4.89 ! [A: extended_enat,B: extended_enat,C: extended_enat] :
% 8.58/4.89 ( ( ( plus_p3455044024723400733d_enat @ A @ B )
% 8.58/4.89 = ( plus_p3455044024723400733d_enat @ A @ C ) )
% 8.58/4.89 = ( ( A = extend5688581933313929465d_enat )
% 8.58/4.89 | ( B = C ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9833_plus__eq__infty__iff__enat,axiom,
% 8.58/4.89 ! [M: extended_enat,N: extended_enat] :
% 8.58/4.89 ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 8.58/4.89 = extend5688581933313929465d_enat )
% 8.58/4.89 = ( ( M = extend5688581933313929465d_enat )
% 8.58/4.89 | ( N = extend5688581933313929465d_enat ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9834_imult__is__infinity,axiom,
% 8.58/4.89 ! [A: extended_enat,B: extended_enat] :
% 8.58/4.89 ( ( ( times_7803423173614009249d_enat @ A @ B )
% 8.58/4.89 = extend5688581933313929465d_enat )
% 8.58/4.89 = ( ( ( A = extend5688581933313929465d_enat )
% 8.58/4.89 & ( B != zero_z5237406670263579293d_enat ) )
% 8.58/4.89 | ( ( B = extend5688581933313929465d_enat )
% 8.58/4.89 & ( A != zero_z5237406670263579293d_enat ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9835_imult__infinity__right,axiom,
% 8.58/4.89 ! [N: extended_enat] :
% 8.58/4.89 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 8.58/4.89 => ( ( times_7803423173614009249d_enat @ N @ extend5688581933313929465d_enat )
% 8.58/4.89 = extend5688581933313929465d_enat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9836_imult__infinity,axiom,
% 8.58/4.89 ! [N: extended_enat] :
% 8.58/4.89 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 8.58/4.89 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ N )
% 8.58/4.89 = extend5688581933313929465d_enat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9837_times__enat__def,axiom,
% 8.58/4.89 ( times_7803423173614009249d_enat
% 8.58/4.89 = ( ^ [M3: extended_enat,N3: extended_enat] :
% 8.58/4.89 ( extend3600170679010898289d_enat
% 8.58/4.89 @ ^ [O: nat] :
% 8.58/4.89 ( extend3600170679010898289d_enat
% 8.58/4.89 @ ^ [P3: nat] : ( extended_enat2 @ ( times_times_nat @ O @ P3 ) )
% 8.58/4.89 @ ( if_Extended_enat @ ( O = zero_zero_nat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
% 8.58/4.89 @ N3 )
% 8.58/4.89 @ ( if_Extended_enat @ ( N3 = zero_z5237406670263579293d_enat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
% 8.58/4.89 @ M3 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9838_diff__enat__def,axiom,
% 8.58/4.89 ( minus_3235023915231533773d_enat
% 8.58/4.89 = ( ^ [A2: extended_enat,B2: extended_enat] :
% 8.58/4.89 ( extend3600170679010898289d_enat
% 8.58/4.89 @ ^ [X3: nat] :
% 8.58/4.89 ( extend3600170679010898289d_enat
% 8.58/4.89 @ ^ [Y6: nat] : ( extended_enat2 @ ( minus_minus_nat @ X3 @ Y6 ) )
% 8.58/4.89 @ zero_z5237406670263579293d_enat
% 8.58/4.89 @ B2 )
% 8.58/4.89 @ extend5688581933313929465d_enat
% 8.58/4.89 @ A2 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9839_plus__enat__def,axiom,
% 8.58/4.89 ( plus_p3455044024723400733d_enat
% 8.58/4.89 = ( ^ [M3: extended_enat,N3: extended_enat] :
% 8.58/4.89 ( extend3600170679010898289d_enat
% 8.58/4.89 @ ^ [O: nat] :
% 8.58/4.89 ( extend3600170679010898289d_enat
% 8.58/4.89 @ ^ [P3: nat] : ( extended_enat2 @ ( plus_plus_nat @ O @ P3 ) )
% 8.58/4.89 @ extend5688581933313929465d_enat
% 8.58/4.89 @ N3 )
% 8.58/4.89 @ extend5688581933313929465d_enat
% 8.58/4.89 @ M3 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9840_eSuc__def,axiom,
% 8.58/4.89 ( extended_eSuc
% 8.58/4.89 = ( extend3600170679010898289d_enat
% 8.58/4.89 @ ^ [N3: nat] : ( extended_enat2 @ ( suc @ N3 ) )
% 8.58/4.89 @ extend5688581933313929465d_enat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9841_eSuc__numeral,axiom,
% 8.58/4.89 ! [K: num] :
% 8.58/4.89 ( ( extended_eSuc @ ( numera1916890842035813515d_enat @ K ) )
% 8.58/4.89 = ( numera1916890842035813515d_enat @ ( plus_plus_num @ K @ one ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9842_eSuc__minus__eSuc,axiom,
% 8.58/4.89 ! [N: extended_enat,M: extended_enat] :
% 8.58/4.89 ( ( minus_3235023915231533773d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 8.58/4.89 = ( minus_3235023915231533773d_enat @ N @ M ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9843_eSuc__minus__1,axiom,
% 8.58/4.89 ! [N: extended_enat] :
% 8.58/4.89 ( ( minus_3235023915231533773d_enat @ ( extended_eSuc @ N ) @ one_on7984719198319812577d_enat )
% 8.58/4.89 = N ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9844_eSuc__enat,axiom,
% 8.58/4.89 ! [N: nat] :
% 8.58/4.89 ( ( extended_eSuc @ ( extended_enat2 @ N ) )
% 8.58/4.89 = ( extended_enat2 @ ( suc @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9845_eSuc__enat__iff,axiom,
% 8.58/4.89 ! [X: extended_enat,Y: nat] :
% 8.58/4.89 ( ( ( extended_eSuc @ X )
% 8.58/4.89 = ( extended_enat2 @ Y ) )
% 8.58/4.89 = ( ? [N3: nat] :
% 8.58/4.89 ( ( Y
% 8.58/4.89 = ( suc @ N3 ) )
% 8.58/4.89 & ( X
% 8.58/4.89 = ( extended_enat2 @ N3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9846_enat__eSuc__iff,axiom,
% 8.58/4.89 ! [Y: nat,X: extended_enat] :
% 8.58/4.89 ( ( ( extended_enat2 @ Y )
% 8.58/4.89 = ( extended_eSuc @ X ) )
% 8.58/4.89 = ( ? [N3: nat] :
% 8.58/4.89 ( ( Y
% 8.58/4.89 = ( suc @ N3 ) )
% 8.58/4.89 & ( ( extended_enat2 @ N3 )
% 8.58/4.89 = X ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9847_iadd__Suc,axiom,
% 8.58/4.89 ! [M: extended_enat,N: extended_enat] :
% 8.58/4.89 ( ( plus_p3455044024723400733d_enat @ ( extended_eSuc @ M ) @ N )
% 8.58/4.89 = ( extended_eSuc @ ( plus_p3455044024723400733d_enat @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9848_iadd__Suc__right,axiom,
% 8.58/4.89 ! [M: extended_enat,N: extended_enat] :
% 8.58/4.89 ( ( plus_p3455044024723400733d_enat @ M @ ( extended_eSuc @ N ) )
% 8.58/4.89 = ( extended_eSuc @ ( plus_p3455044024723400733d_enat @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9849_one__eSuc,axiom,
% 8.58/4.89 ( one_on7984719198319812577d_enat
% 8.58/4.89 = ( extended_eSuc @ zero_z5237406670263579293d_enat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9850_mult__eSuc,axiom,
% 8.58/4.89 ! [M: extended_enat,N: extended_enat] :
% 8.58/4.89 ( ( times_7803423173614009249d_enat @ ( extended_eSuc @ M ) @ N )
% 8.58/4.89 = ( plus_p3455044024723400733d_enat @ N @ ( times_7803423173614009249d_enat @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9851_mult__eSuc__right,axiom,
% 8.58/4.89 ! [M: extended_enat,N: extended_enat] :
% 8.58/4.89 ( ( times_7803423173614009249d_enat @ M @ ( extended_eSuc @ N ) )
% 8.58/4.89 = ( plus_p3455044024723400733d_enat @ M @ ( times_7803423173614009249d_enat @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9852_eSuc__plus__1,axiom,
% 8.58/4.89 ( extended_eSuc
% 8.58/4.89 = ( ^ [N3: extended_enat] : ( plus_p3455044024723400733d_enat @ N3 @ one_on7984719198319812577d_enat ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9853_plus__1__eSuc_I1_J,axiom,
% 8.58/4.89 ! [Q2: extended_enat] :
% 8.58/4.89 ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ Q2 )
% 8.58/4.89 = ( extended_eSuc @ Q2 ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9854_plus__1__eSuc_I2_J,axiom,
% 8.58/4.89 ! [Q2: extended_enat] :
% 8.58/4.89 ( ( plus_p3455044024723400733d_enat @ Q2 @ one_on7984719198319812577d_enat )
% 8.58/4.89 = ( extended_eSuc @ Q2 ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9855_Quotient__real,axiom,
% 8.58/4.89 quotie3684837364556693515t_real @ realrel @ real2 @ rep_real @ cr_real ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9856_has__vector__derivative__id,axiom,
% 8.58/4.89 ! [Net: filter_real] :
% 8.58/4.89 ( has_ve631408500373753343e_real
% 8.58/4.89 @ ^ [X3: real] : X3
% 8.58/4.89 @ one_one_real
% 8.58/4.89 @ Net ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9857_lcm__code__integer,axiom,
% 8.58/4.89 ( gcd_lcm_Code_integer
% 8.58/4.89 = ( ^ [A2: code_integer,B2: code_integer] : ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( gcd_gcd_Code_integer @ A2 @ B2 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9858_lcm__0__iff__nat,axiom,
% 8.58/4.89 ! [M: nat,N: nat] :
% 8.58/4.89 ( ( ( gcd_lcm_nat @ M @ N )
% 8.58/4.89 = zero_zero_nat )
% 8.58/4.89 = ( ( M = zero_zero_nat )
% 8.58/4.89 | ( N = zero_zero_nat ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9859_lcm__0__iff__int,axiom,
% 8.58/4.89 ! [M: int,N: int] :
% 8.58/4.89 ( ( ( gcd_lcm_int @ M @ N )
% 8.58/4.89 = zero_zero_int )
% 8.58/4.89 = ( ( M = zero_zero_int )
% 8.58/4.89 | ( N = zero_zero_int ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9860_lcm__proj1__iff__nat,axiom,
% 8.58/4.89 ! [M: nat,N: nat] :
% 8.58/4.89 ( ( ( gcd_lcm_nat @ M @ N )
% 8.58/4.89 = M )
% 8.58/4.89 = ( dvd_dvd_nat @ N @ M ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9861_lcm__proj2__iff__nat,axiom,
% 8.58/4.89 ! [M: nat,N: nat] :
% 8.58/4.89 ( ( ( gcd_lcm_nat @ M @ N )
% 8.58/4.89 = N )
% 8.58/4.89 = ( dvd_dvd_nat @ M @ N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9862_lcm__proj1__if__dvd__nat,axiom,
% 8.58/4.89 ! [X: nat,Y: nat] :
% 8.58/4.89 ( ( dvd_dvd_nat @ X @ Y )
% 8.58/4.89 => ( ( gcd_lcm_nat @ Y @ X )
% 8.58/4.89 = Y ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9863_lcm__proj2__if__dvd__nat,axiom,
% 8.58/4.89 ! [X: nat,Y: nat] :
% 8.58/4.89 ( ( dvd_dvd_nat @ X @ Y )
% 8.58/4.89 => ( ( gcd_lcm_nat @ X @ Y )
% 8.58/4.89 = Y ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9864_lcm__int__int__eq,axiom,
% 8.58/4.89 ! [M: nat,N: nat] :
% 8.58/4.89 ( ( gcd_lcm_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 8.58/4.89 = ( semiri1314217659103216013at_int @ ( gcd_lcm_nat @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9865_abs__lcm__int,axiom,
% 8.58/4.89 ! [I: int,J: int] :
% 8.58/4.89 ( ( abs_abs_int @ ( gcd_lcm_int @ I @ J ) )
% 8.58/4.89 = ( gcd_lcm_int @ I @ J ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9866_lcm__abs1__int,axiom,
% 8.58/4.89 ! [X: int,Y: int] :
% 8.58/4.89 ( ( gcd_lcm_int @ ( abs_abs_int @ X ) @ Y )
% 8.58/4.89 = ( gcd_lcm_int @ X @ Y ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9867_lcm__abs2__int,axiom,
% 8.58/4.89 ! [X: int,Y: int] :
% 8.58/4.89 ( ( gcd_lcm_int @ X @ ( abs_abs_int @ Y ) )
% 8.58/4.89 = ( gcd_lcm_int @ X @ Y ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9868_lcm__1__iff__nat,axiom,
% 8.58/4.89 ! [M: nat,N: nat] :
% 8.58/4.89 ( ( ( gcd_lcm_nat @ M @ N )
% 8.58/4.89 = ( suc @ zero_zero_nat ) )
% 8.58/4.89 = ( ( M
% 8.58/4.89 = ( suc @ zero_zero_nat ) )
% 8.58/4.89 & ( N
% 8.58/4.89 = ( suc @ zero_zero_nat ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9869_lcm__1__iff__int,axiom,
% 8.58/4.89 ! [M: int,N: int] :
% 8.58/4.89 ( ( ( gcd_lcm_int @ M @ N )
% 8.58/4.89 = one_one_int )
% 8.58/4.89 = ( ( ( M = one_one_int )
% 8.58/4.89 | ( M
% 8.58/4.89 = ( uminus_uminus_int @ one_one_int ) ) )
% 8.58/4.89 & ( ( N = one_one_int )
% 8.58/4.89 | ( N
% 8.58/4.89 = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9870_lcm__proj2__if__dvd__int,axiom,
% 8.58/4.89 ! [X: int,Y: int] :
% 8.58/4.89 ( ( dvd_dvd_int @ X @ Y )
% 8.58/4.89 => ( ( gcd_lcm_int @ X @ Y )
% 8.58/4.89 = ( abs_abs_int @ Y ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9871_lcm__proj1__if__dvd__int,axiom,
% 8.58/4.89 ! [X: int,Y: int] :
% 8.58/4.89 ( ( dvd_dvd_int @ X @ Y )
% 8.58/4.89 => ( ( gcd_lcm_int @ Y @ X )
% 8.58/4.89 = ( abs_abs_int @ Y ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9872_lcm__proj2__iff__int,axiom,
% 8.58/4.89 ! [M: int,N: int] :
% 8.58/4.89 ( ( ( gcd_lcm_int @ M @ N )
% 8.58/4.89 = ( abs_abs_int @ N ) )
% 8.58/4.89 = ( dvd_dvd_int @ M @ N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9873_lcm__proj1__iff__int,axiom,
% 8.58/4.89 ! [M: int,N: int] :
% 8.58/4.89 ( ( ( gcd_lcm_int @ M @ N )
% 8.58/4.89 = ( abs_abs_int @ M ) )
% 8.58/4.89 = ( dvd_dvd_int @ N @ M ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9874_lcm__nat__abs__left__eq,axiom,
% 8.58/4.89 ! [K: int,N: nat] :
% 8.58/4.89 ( ( gcd_lcm_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 8.58/4.89 = ( nat2 @ ( gcd_lcm_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9875_lcm__nat__abs__right__eq,axiom,
% 8.58/4.89 ! [N: nat,K: int] :
% 8.58/4.89 ( ( gcd_lcm_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 8.58/4.89 = ( nat2 @ ( gcd_lcm_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9876_lcm__unique__int,axiom,
% 8.58/4.89 ! [D: int,A: int,B: int] :
% 8.58/4.89 ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 8.58/4.89 & ( dvd_dvd_int @ A @ D )
% 8.58/4.89 & ( dvd_dvd_int @ B @ D )
% 8.58/4.89 & ! [E3: int] :
% 8.58/4.89 ( ( ( dvd_dvd_int @ A @ E3 )
% 8.58/4.89 & ( dvd_dvd_int @ B @ E3 ) )
% 8.58/4.89 => ( dvd_dvd_int @ D @ E3 ) ) )
% 8.58/4.89 = ( D
% 8.58/4.89 = ( gcd_lcm_int @ A @ B ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9877_prod__gcd__lcm__nat,axiom,
% 8.58/4.89 ( times_times_nat
% 8.58/4.89 = ( ^ [M3: nat,N3: nat] : ( times_times_nat @ ( gcd_gcd_nat @ M3 @ N3 ) @ ( gcd_lcm_nat @ M3 @ N3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9878_lcm__integer_Orep__eq,axiom,
% 8.58/4.89 ! [X: code_integer,Xa: code_integer] :
% 8.58/4.89 ( ( code_int_of_integer @ ( gcd_lcm_Code_integer @ X @ Xa ) )
% 8.58/4.89 = ( gcd_lcm_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9879_dvd__lcm__I1__nat,axiom,
% 8.58/4.89 ! [K: nat,M: nat,N: nat] :
% 8.58/4.89 ( ( dvd_dvd_nat @ K @ M )
% 8.58/4.89 => ( dvd_dvd_nat @ K @ ( gcd_lcm_nat @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9880_dvd__lcm__I2__nat,axiom,
% 8.58/4.89 ! [K: nat,N: nat,M: nat] :
% 8.58/4.89 ( ( dvd_dvd_nat @ K @ N )
% 8.58/4.89 => ( dvd_dvd_nat @ K @ ( gcd_lcm_nat @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9881_lcm__unique__nat,axiom,
% 8.58/4.89 ! [A: nat,D: nat,B: nat] :
% 8.58/4.89 ( ( ( dvd_dvd_nat @ A @ D )
% 8.58/4.89 & ( dvd_dvd_nat @ B @ D )
% 8.58/4.89 & ! [E3: nat] :
% 8.58/4.89 ( ( ( dvd_dvd_nat @ A @ E3 )
% 8.58/4.89 & ( dvd_dvd_nat @ B @ E3 ) )
% 8.58/4.89 => ( dvd_dvd_nat @ D @ E3 ) ) )
% 8.58/4.89 = ( D
% 8.58/4.89 = ( gcd_lcm_nat @ A @ B ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9882_dvd__lcm__I1__int,axiom,
% 8.58/4.89 ! [I: int,M: int,N: int] :
% 8.58/4.89 ( ( dvd_dvd_int @ I @ M )
% 8.58/4.89 => ( dvd_dvd_int @ I @ ( gcd_lcm_int @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9883_dvd__lcm__I2__int,axiom,
% 8.58/4.89 ! [I: int,N: int,M: int] :
% 8.58/4.89 ( ( dvd_dvd_int @ I @ N )
% 8.58/4.89 => ( dvd_dvd_int @ I @ ( gcd_lcm_int @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9884_lcm__neg2__int,axiom,
% 8.58/4.89 ! [X: int,Y: int] :
% 8.58/4.89 ( ( gcd_lcm_int @ X @ ( uminus_uminus_int @ Y ) )
% 8.58/4.89 = ( gcd_lcm_int @ X @ Y ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9885_lcm__neg1__int,axiom,
% 8.58/4.89 ! [X: int,Y: int] :
% 8.58/4.89 ( ( gcd_lcm_int @ ( uminus_uminus_int @ X ) @ Y )
% 8.58/4.89 = ( gcd_lcm_int @ X @ Y ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9886_lcm__integer_Oabs__eq,axiom,
% 8.58/4.89 ! [Xa: int,X: int] :
% 8.58/4.89 ( ( gcd_lcm_Code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 8.58/4.89 = ( code_integer_of_int @ ( gcd_lcm_int @ Xa @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9887_lcm__int__def,axiom,
% 8.58/4.89 ( gcd_lcm_int
% 8.58/4.89 = ( ^ [X3: int,Y6: int] : ( semiri1314217659103216013at_int @ ( gcd_lcm_nat @ ( nat2 @ ( abs_abs_int @ X3 ) ) @ ( nat2 @ ( abs_abs_int @ Y6 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9888_lcm__integer_Orsp,axiom,
% 8.58/4.89 ( bNF_re711492959462206631nt_int
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re4712519889275205905nt_int
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ^ [Y3: int,Z: int] : ( Y3 = Z ) )
% 8.58/4.89 @ gcd_lcm_int
% 8.58/4.89 @ gcd_lcm_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9889_lcm__pos__int,axiom,
% 8.58/4.89 ! [M: int,N: int] :
% 8.58/4.89 ( ( M != zero_zero_int )
% 8.58/4.89 => ( ( N != zero_zero_int )
% 8.58/4.89 => ( ord_less_int @ zero_zero_int @ ( gcd_lcm_int @ M @ N ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9890_lcm__pos__nat,axiom,
% 8.58/4.89 ! [M: nat,N: nat] :
% 8.58/4.89 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.58/4.89 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.89 => ( ord_less_nat @ zero_zero_nat @ ( gcd_lcm_nat @ M @ N ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9891_lcm__ge__0__int,axiom,
% 8.58/4.89 ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_lcm_int @ X @ Y ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9892_lcm__cases__int,axiom,
% 8.58/4.89 ! [X: int,Y: int,P: int > $o] :
% 8.58/4.89 ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.58/4.89 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.58/4.89 => ( P @ ( gcd_lcm_int @ X @ Y ) ) ) )
% 8.58/4.89 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.58/4.89 => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 8.58/4.89 => ( P @ ( gcd_lcm_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
% 8.58/4.89 => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 8.58/4.89 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 8.58/4.89 => ( P @ ( gcd_lcm_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
% 8.58/4.89 => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 8.58/4.89 => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 8.58/4.89 => ( P @ ( gcd_lcm_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
% 8.58/4.89 => ( P @ ( gcd_lcm_int @ X @ Y ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9893_lcm__nat__def,axiom,
% 8.58/4.89 ( gcd_lcm_nat
% 8.58/4.89 = ( ^ [X3: nat,Y6: nat] : ( divide_divide_nat @ ( times_times_nat @ X3 @ Y6 ) @ ( gcd_gcd_nat @ X3 @ Y6 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9894_prod__gcd__lcm__int,axiom,
% 8.58/4.89 ! [M: int,N: int] :
% 8.58/4.89 ( ( times_times_int @ ( abs_abs_int @ M ) @ ( abs_abs_int @ N ) )
% 8.58/4.89 = ( times_times_int @ ( gcd_gcd_int @ M @ N ) @ ( gcd_lcm_int @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9895_lcm__altdef__int,axiom,
% 8.58/4.89 ( gcd_lcm_int
% 8.58/4.89 = ( ^ [A2: int,B2: int] : ( divide_divide_int @ ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( gcd_gcd_int @ A2 @ B2 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9896_Lcm__eq__Max__nat,axiom,
% 8.58/4.89 ! [M10: set_nat] :
% 8.58/4.89 ( ( finite_finite_nat @ M10 )
% 8.58/4.89 => ( ( M10 != bot_bot_set_nat )
% 8.58/4.89 => ( ~ ( member_nat @ zero_zero_nat @ M10 )
% 8.58/4.89 => ( ! [M2: nat,N2: nat] :
% 8.58/4.89 ( ( member_nat @ M2 @ M10 )
% 8.58/4.89 => ( ( member_nat @ N2 @ M10 )
% 8.58/4.89 => ( member_nat @ ( gcd_lcm_nat @ M2 @ N2 ) @ M10 ) ) )
% 8.58/4.89 => ( ( gcd_Lcm_nat @ M10 )
% 8.58/4.89 = ( lattic8265883725875713057ax_nat @ M10 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9897_Lcm__int__set__eq__fold,axiom,
% 8.58/4.89 ! [Xs: list_int] :
% 8.58/4.89 ( ( gcd_Lcm_int @ ( set_int2 @ Xs ) )
% 8.58/4.89 = ( fold_int_int @ gcd_lcm_int @ Xs @ one_one_int ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9898_Lcm__eq__0__I__nat,axiom,
% 8.58/4.89 ! [A3: set_nat] :
% 8.58/4.89 ( ( member_nat @ zero_zero_nat @ A3 )
% 8.58/4.89 => ( ( gcd_Lcm_nat @ A3 )
% 8.58/4.89 = zero_zero_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9899_abs__Lcm__eq,axiom,
% 8.58/4.89 ! [K5: set_int] :
% 8.58/4.89 ( ( abs_abs_int @ ( gcd_Lcm_int @ K5 ) )
% 8.58/4.89 = ( gcd_Lcm_int @ K5 ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9900_Lcm__0__iff__nat,axiom,
% 8.58/4.89 ! [A3: set_nat] :
% 8.58/4.89 ( ( finite_finite_nat @ A3 )
% 8.58/4.89 => ( ( ( gcd_Lcm_nat @ A3 )
% 8.58/4.89 = zero_zero_nat )
% 8.58/4.89 = ( member_nat @ zero_zero_nat @ A3 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9901_Lcm__abs__eq,axiom,
% 8.58/4.89 ! [K5: set_int] :
% 8.58/4.89 ( ( gcd_Lcm_int @ ( image_int_int @ abs_abs_int @ K5 ) )
% 8.58/4.89 = ( gcd_Lcm_int @ K5 ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9902_Lcm__int__eq,axiom,
% 8.58/4.89 ! [N5: set_nat] :
% 8.58/4.89 ( ( gcd_Lcm_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N5 ) )
% 8.58/4.89 = ( semiri1314217659103216013at_int @ ( gcd_Lcm_nat @ N5 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9903_Lcm__nat__abs__eq,axiom,
% 8.58/4.89 ! [K5: set_int] :
% 8.58/4.89 ( ( gcd_Lcm_nat
% 8.58/4.89 @ ( image_int_nat
% 8.58/4.89 @ ^ [K2: int] : ( nat2 @ ( abs_abs_int @ K2 ) )
% 8.58/4.89 @ K5 ) )
% 8.58/4.89 = ( nat2 @ ( gcd_Lcm_int @ K5 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9904_Lcm__in__lcm__closed__set__nat,axiom,
% 8.58/4.89 ! [M10: set_nat] :
% 8.58/4.89 ( ( finite_finite_nat @ M10 )
% 8.58/4.89 => ( ( M10 != bot_bot_set_nat )
% 8.58/4.89 => ( ! [M2: nat,N2: nat] :
% 8.58/4.89 ( ( member_nat @ M2 @ M10 )
% 8.58/4.89 => ( ( member_nat @ N2 @ M10 )
% 8.58/4.89 => ( member_nat @ ( gcd_lcm_nat @ M2 @ N2 ) @ M10 ) ) )
% 8.58/4.89 => ( member_nat @ ( gcd_Lcm_nat @ M10 ) @ M10 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9905_Lcm__nat__insert,axiom,
% 8.58/4.89 ! [N: nat,M10: set_nat] :
% 8.58/4.89 ( ( gcd_Lcm_nat @ ( insert_nat @ N @ M10 ) )
% 8.58/4.89 = ( gcd_lcm_nat @ N @ ( gcd_Lcm_nat @ M10 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9906_Lcm__nat__empty,axiom,
% 8.58/4.89 ( ( gcd_Lcm_nat @ bot_bot_set_nat )
% 8.58/4.89 = one_one_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9907_Lcm__dvd__nat,axiom,
% 8.58/4.89 ! [M10: set_nat,N: nat] :
% 8.58/4.89 ( ! [X4: nat] :
% 8.58/4.89 ( ( member_nat @ X4 @ M10 )
% 8.58/4.89 => ( dvd_dvd_nat @ X4 @ N ) )
% 8.58/4.89 => ( dvd_dvd_nat @ ( gcd_Lcm_nat @ M10 ) @ N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9908_dvd__Lcm__nat,axiom,
% 8.58/4.89 ! [M: nat,M10: set_nat] :
% 8.58/4.89 ( ( member_nat @ M @ M10 )
% 8.58/4.89 => ( dvd_dvd_nat @ M @ ( gcd_Lcm_nat @ M10 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9909_dvd__Lcm__int,axiom,
% 8.58/4.89 ! [M: int,M10: set_int] :
% 8.58/4.89 ( ( member_int @ M @ M10 )
% 8.58/4.89 => ( dvd_dvd_int @ M @ ( gcd_Lcm_int @ M10 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9910_Lcm__least__int,axiom,
% 8.58/4.89 ! [A3: set_int,A: int] :
% 8.58/4.89 ( ! [B4: int] :
% 8.58/4.89 ( ( member_int @ B4 @ A3 )
% 8.58/4.89 => ( dvd_dvd_int @ B4 @ A ) )
% 8.58/4.89 => ( dvd_dvd_int @ ( gcd_Lcm_int @ A3 ) @ A ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9911_Lcm__nat__infinite,axiom,
% 8.58/4.89 ! [M10: set_nat] :
% 8.58/4.89 ( ~ ( finite_finite_nat @ M10 )
% 8.58/4.89 => ( ( gcd_Lcm_nat @ M10 )
% 8.58/4.89 = zero_zero_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9912_Lcm__int__greater__eq__0,axiom,
% 8.58/4.89 ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Lcm_int @ K5 ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9913_Gcd__nat__def,axiom,
% 8.58/4.89 ( gcd_Gcd_nat
% 8.58/4.89 = ( ^ [M7: set_nat] :
% 8.58/4.89 ( gcd_Lcm_nat
% 8.58/4.89 @ ( collect_nat
% 8.58/4.89 @ ^ [D4: nat] :
% 8.58/4.89 ! [X3: nat] :
% 8.58/4.89 ( ( member_nat @ X3 @ M7 )
% 8.58/4.89 => ( dvd_dvd_nat @ D4 @ X3 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9914_Lcm__int__def,axiom,
% 8.58/4.89 ( gcd_Lcm_int
% 8.58/4.89 = ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Lcm_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9915_Lcm__nat__set__eq__fold,axiom,
% 8.58/4.89 ! [Xs: list_nat] :
% 8.58/4.89 ( ( gcd_Lcm_nat @ ( set_nat2 @ Xs ) )
% 8.58/4.89 = ( fold_nat_nat @ gcd_lcm_nat @ Xs @ one_one_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9916_Lcm__nat__def,axiom,
% 8.58/4.89 ( gcd_Lcm_nat
% 8.58/4.89 = ( ^ [M7: set_nat] : ( if_nat @ ( finite_finite_nat @ M7 ) @ ( lattic7826324295020591184_F_nat @ gcd_lcm_nat @ one_one_nat @ M7 ) @ zero_zero_nat ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9917_normalize__int__def,axiom,
% 8.58/4.89 normal571682499473568032ze_int = abs_abs_int ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9918_normalize__nat__def,axiom,
% 8.58/4.89 normal574172969982618308ze_nat = id_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9919_cr__int__def,axiom,
% 8.58/4.89 ( cr_int
% 8.58/4.89 = ( ^ [X3: product_prod_nat_nat] :
% 8.58/4.89 ( ^ [Y3: int,Z: int] : ( Y3 = Z )
% 8.58/4.89 @ ( abs_Integ @ X3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9920_int_Opcr__cr__eq,axiom,
% 8.58/4.89 pcr_int = cr_int ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9921_Quotient__int,axiom,
% 8.58/4.89 quotie1194848508323700631at_int @ intrel @ abs_Integ @ rep_Integ @ cr_int ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9922_gcd__nat_Osemilattice__neutr__axioms,axiom,
% 8.58/4.89 semila9081495762789891438tr_nat @ gcd_gcd_nat @ zero_zero_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9923_max__nat_Osemilattice__neutr__axioms,axiom,
% 8.58/4.89 semila9081495762789891438tr_nat @ ord_max_nat @ zero_zero_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9924_linear__scale__real,axiom,
% 8.58/4.89 ! [F: real > real,R: real,B: real] :
% 8.58/4.89 ( ( real_V4572627801940501177l_real @ F )
% 8.58/4.89 => ( ( F @ ( times_times_real @ R @ B ) )
% 8.58/4.89 = ( times_times_real @ R @ ( F @ B ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9925_real__linearD,axiom,
% 8.58/4.89 ! [F: real > real] :
% 8.58/4.89 ( ( real_V4572627801940501177l_real @ F )
% 8.58/4.89 => ~ ! [C3: real] :
% 8.58/4.89 ( F
% 8.58/4.89 != ( times_times_real @ C3 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9926_filterlim__int__sequentially,axiom,
% 8.58/4.89 filterlim_nat_int @ semiri1314217659103216013at_int @ at_top_int @ at_top_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9927_sub_Oabs__eq,axiom,
% 8.58/4.89 ( code_sub
% 8.58/4.89 = ( ^ [Xa4: num,X3: num] : ( code_integer_of_int @ ( minus_minus_int @ ( numeral_numeral_int @ Xa4 ) @ ( numeral_numeral_int @ X3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9928_sub_Orep__eq,axiom,
% 8.58/4.89 ! [X: num,Xa: num] :
% 8.58/4.89 ( ( code_int_of_integer @ ( code_sub @ X @ Xa ) )
% 8.58/4.89 = ( minus_minus_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Xa ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9929_Code__Numeral_Osub__code_I1_J,axiom,
% 8.58/4.89 ( ( code_sub @ one @ one )
% 8.58/4.89 = zero_z3403309356797280102nteger ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9930_Code__Numeral_Osub__code_I9_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( code_sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.58/4.89 = ( minus_8373710615458151222nteger @ ( code_dup @ ( code_sub @ M @ N ) ) @ one_one_Code_integer ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9931_Code__Numeral_Osub__code_I8_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( code_sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.58/4.89 = ( plus_p5714425477246183910nteger @ ( code_dup @ ( code_sub @ M @ N ) ) @ one_one_Code_integer ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9932_dup_Orep__eq,axiom,
% 8.58/4.89 ! [X: code_integer] :
% 8.58/4.89 ( ( code_int_of_integer @ ( code_dup @ X ) )
% 8.58/4.89 = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9933_dup_Oabs__eq,axiom,
% 8.58/4.89 ! [X: int] :
% 8.58/4.89 ( ( code_dup @ ( code_integer_of_int @ X ) )
% 8.58/4.89 = ( code_integer_of_int @ ( plus_plus_int @ X @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9934_Code__Numeral_Osub__code_I6_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( code_sub @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.58/4.89 = ( code_dup @ ( code_sub @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9935_Code__Numeral_Osub__code_I4_J,axiom,
% 8.58/4.89 ! [N: num] :
% 8.58/4.89 ( ( code_sub @ one @ ( bit0 @ N ) )
% 8.58/4.89 = ( code_Neg @ ( bitM @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9936_Code__Numeral_Odup__def,axiom,
% 8.58/4.89 ( code_dup
% 8.58/4.89 = ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int
% 8.58/4.89 @ ^ [K2: int] : ( plus_plus_int @ K2 @ K2 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9937_Code__Numeral_Odup__code_I3_J,axiom,
% 8.58/4.89 ! [N: num] :
% 8.58/4.89 ( ( code_dup @ ( code_Neg @ N ) )
% 8.58/4.89 = ( code_Neg @ ( bit0 @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9938_plus__integer__code_I6_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( plus_p5714425477246183910nteger @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
% 8.58/4.89 = ( code_Neg @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9939_minus__integer__code_I6_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( minus_8373710615458151222nteger @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
% 8.58/4.89 = ( code_sub @ N @ M ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9940_Code__Numeral_Osub__code_I5_J,axiom,
% 8.58/4.89 ! [N: num] :
% 8.58/4.89 ( ( code_sub @ one @ ( bit1 @ N ) )
% 8.58/4.89 = ( code_Neg @ ( bit0 @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9941_push__bit__integer__def,axiom,
% 8.58/4.89 ( bit_se7788150548672797655nteger
% 8.58/4.89 = ( map_fu8224359078269604234nteger @ id_nat @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se545348938243370406it_int ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9942_lcm__integer__def,axiom,
% 8.58/4.89 ( gcd_lcm_Code_integer
% 8.58/4.89 = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ gcd_lcm_int ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9943_plus__integer__def,axiom,
% 8.58/4.89 ( plus_p5714425477246183910nteger
% 8.58/4.89 = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ plus_plus_int ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9944_times__integer__def,axiom,
% 8.58/4.89 ( times_3573771949741848930nteger
% 8.58/4.89 = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ times_times_int ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9945_minus__integer__def,axiom,
% 8.58/4.89 ( minus_8373710615458151222nteger
% 8.58/4.89 = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ minus_minus_int ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9946_divide__integer__def,axiom,
% 8.58/4.89 ( divide6298287555418463151nteger
% 8.58/4.89 = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ divide_divide_int ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9947_gcd__integer__def,axiom,
% 8.58/4.89 ( gcd_gcd_Code_integer
% 8.58/4.89 = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ gcd_gcd_int ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9948_Code__Numeral_Osub__code_I2_J,axiom,
% 8.58/4.89 ! [M: num] :
% 8.58/4.89 ( ( code_sub @ ( bit0 @ M ) @ one )
% 8.58/4.89 = ( code_Pos @ ( bitM @ M ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9949_Code__Numeral_Osub__code_I3_J,axiom,
% 8.58/4.89 ! [M: num] :
% 8.58/4.89 ( ( code_sub @ ( bit1 @ M ) @ one )
% 8.58/4.89 = ( code_Pos @ ( bit0 @ M ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9950_Code__Numeral_Odup__code_I2_J,axiom,
% 8.58/4.89 ! [N: num] :
% 8.58/4.89 ( ( code_dup @ ( code_Pos @ N ) )
% 8.58/4.89 = ( code_Pos @ ( bit0 @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9951_times__integer__code_I3_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( times_3573771949741848930nteger @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
% 8.58/4.89 = ( code_Pos @ ( times_times_num @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9952_one__integer__code,axiom,
% 8.58/4.89 ( one_one_Code_integer
% 8.58/4.89 = ( code_Pos @ one ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9953_Pos__fold_I1_J,axiom,
% 8.58/4.89 ( ( numera6620942414471956472nteger @ one )
% 8.58/4.89 = ( code_Pos @ one ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9954_Pos__fold_I2_J,axiom,
% 8.58/4.89 ! [K: num] :
% 8.58/4.89 ( ( numera6620942414471956472nteger @ ( bit0 @ K ) )
% 8.58/4.89 = ( code_Pos @ ( bit0 @ K ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9955_plus__integer__code_I3_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( plus_p5714425477246183910nteger @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
% 8.58/4.89 = ( code_Pos @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9956_minus__integer__code_I3_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( minus_8373710615458151222nteger @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
% 8.58/4.89 = ( code_sub @ M @ N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9957_minus__integer__code_I4_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( minus_8373710615458151222nteger @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
% 8.58/4.89 = ( code_Pos @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9958_minus__integer__code_I5_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( minus_8373710615458151222nteger @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
% 8.58/4.89 = ( code_Neg @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9959_times__integer__code_I6_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( times_3573771949741848930nteger @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
% 8.58/4.89 = ( code_Pos @ ( times_times_num @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9960_times__integer__code_I5_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( times_3573771949741848930nteger @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
% 8.58/4.89 = ( code_Neg @ ( times_times_num @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9961_times__integer__code_I4_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( times_3573771949741848930nteger @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
% 8.58/4.89 = ( code_Neg @ ( times_times_num @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9962_plus__integer__code_I4_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( plus_p5714425477246183910nteger @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
% 8.58/4.89 = ( code_sub @ M @ N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9963_plus__integer__code_I5_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( plus_p5714425477246183910nteger @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
% 8.58/4.89 = ( code_sub @ N @ M ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9964_max__nat_Omonoid__axioms,axiom,
% 8.58/4.89 monoid_nat @ ord_max_nat @ zero_zero_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9965_gcd__nat_Omonoid__axioms,axiom,
% 8.58/4.89 monoid_nat @ gcd_gcd_nat @ zero_zero_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9966_Sup__real__def,axiom,
% 8.58/4.89 ( comple1385675409528146559p_real
% 8.58/4.89 = ( ^ [X8: set_real] :
% 8.58/4.89 ( ord_Least_real
% 8.58/4.89 @ ^ [Z5: real] :
% 8.58/4.89 ! [X3: real] :
% 8.58/4.89 ( ( member_real @ X3 @ X8 )
% 8.58/4.89 => ( ord_less_eq_real @ X3 @ Z5 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9967_Least__eq__0,axiom,
% 8.58/4.89 ! [P: nat > $o] :
% 8.58/4.89 ( ( P @ zero_zero_nat )
% 8.58/4.89 => ( ( ord_Least_nat @ P )
% 8.58/4.89 = zero_zero_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9968_Least__Suc,axiom,
% 8.58/4.89 ! [P: nat > $o,N: nat] :
% 8.58/4.89 ( ( P @ N )
% 8.58/4.89 => ( ~ ( P @ zero_zero_nat )
% 8.58/4.89 => ( ( ord_Least_nat @ P )
% 8.58/4.89 = ( suc
% 8.58/4.89 @ ( ord_Least_nat
% 8.58/4.89 @ ^ [M3: nat] : ( P @ ( suc @ M3 ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9969_Least__Suc2,axiom,
% 8.58/4.89 ! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
% 8.58/4.89 ( ( P @ N )
% 8.58/4.89 => ( ( Q @ M )
% 8.58/4.89 => ( ~ ( P @ zero_zero_nat )
% 8.58/4.89 => ( ! [K3: nat] :
% 8.58/4.89 ( ( P @ ( suc @ K3 ) )
% 8.58/4.89 = ( Q @ K3 ) )
% 8.58/4.89 => ( ( ord_Least_nat @ P )
% 8.58/4.89 = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9970_unit__factor__simps_I1_J,axiom,
% 8.58/4.89 ( ( unit_f2748546683901255202or_nat @ zero_zero_nat )
% 8.58/4.89 = zero_zero_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9971_unit__factor__simps_I2_J,axiom,
% 8.58/4.89 ! [N: nat] :
% 8.58/4.89 ( ( unit_f2748546683901255202or_nat @ ( suc @ N ) )
% 8.58/4.89 = one_one_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9972_unit__factor__int__def,axiom,
% 8.58/4.89 unit_f2746056213392204926or_int = sgn_sgn_int ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9973_unit__factor__nat__def,axiom,
% 8.58/4.89 ( unit_f2748546683901255202or_nat
% 8.58/4.89 = ( ^ [N3: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9974_gcd__nat_Ocomm__monoid__axioms,axiom,
% 8.58/4.89 comm_monoid_nat @ gcd_gcd_nat @ zero_zero_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9975_max__nat_Ocomm__monoid__axioms,axiom,
% 8.58/4.89 comm_monoid_nat @ ord_max_nat @ zero_zero_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9976_times__num__def,axiom,
% 8.58/4.89 ( times_times_num
% 8.58/4.89 = ( ^ [M3: num,N3: num] : ( num_of_nat @ ( times_times_nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9977_or__num_Osimps_I4_J,axiom,
% 8.58/4.89 ! [M: num] :
% 8.58/4.89 ( ( bit_un6697907153464112080or_num @ ( bit0 @ M ) @ one )
% 8.58/4.89 = ( bit1 @ M ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9978_nat__of__num__sqr,axiom,
% 8.58/4.89 ! [X: num] :
% 8.58/4.89 ( ( nat_of_num @ ( sqr @ X ) )
% 8.58/4.89 = ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9979_nat__of__num__code_I2_J,axiom,
% 8.58/4.89 ! [N: num] :
% 8.58/4.89 ( ( nat_of_num @ ( bit0 @ N ) )
% 8.58/4.89 = ( plus_plus_nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9980_or__num_Osimps_I8_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( bit_un6697907153464112080or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.58/4.89 = ( bit1 @ ( bit_un6697907153464112080or_num @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9981_or__num_Osimps_I6_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( bit_un6697907153464112080or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.58/4.89 = ( bit1 @ ( bit_un6697907153464112080or_num @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9982_or__num_Osimps_I7_J,axiom,
% 8.58/4.89 ! [M: num] :
% 8.58/4.89 ( ( bit_un6697907153464112080or_num @ ( bit1 @ M ) @ one )
% 8.58/4.89 = ( bit1 @ M ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9983_or__num_Osimps_I3_J,axiom,
% 8.58/4.89 ! [N: num] :
% 8.58/4.89 ( ( bit_un6697907153464112080or_num @ one @ ( bit1 @ N ) )
% 8.58/4.89 = ( bit1 @ N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9984_less__eq__num__def,axiom,
% 8.58/4.89 ( ord_less_eq_num
% 8.58/4.89 = ( ^ [M3: num,N3: num] : ( ord_less_eq_nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9985_nat__of__num__add,axiom,
% 8.58/4.89 ! [X: num,Y: num] :
% 8.58/4.89 ( ( nat_of_num @ ( plus_plus_num @ X @ Y ) )
% 8.58/4.89 = ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9986_nat__of__num_Osimps_I2_J,axiom,
% 8.58/4.89 ! [X: num] :
% 8.58/4.89 ( ( nat_of_num @ ( bit0 @ X ) )
% 8.58/4.89 = ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9987_nat__of__num__inverse,axiom,
% 8.58/4.89 ! [X: num] :
% 8.58/4.89 ( ( num_of_nat @ ( nat_of_num @ X ) )
% 8.58/4.89 = X ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9988_or__num_Osimps_I9_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( bit_un6697907153464112080or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.58/4.89 = ( bit1 @ ( bit_un6697907153464112080or_num @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9989_or__num_Osimps_I5_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( bit_un6697907153464112080or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.58/4.89 = ( bit0 @ ( bit_un6697907153464112080or_num @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9990_num__eq__iff,axiom,
% 8.58/4.89 ( ( ^ [Y3: num,Z: num] : ( Y3 = Z ) )
% 8.58/4.89 = ( ^ [X3: num,Y6: num] :
% 8.58/4.89 ( ( nat_of_num @ X3 )
% 8.58/4.89 = ( nat_of_num @ Y6 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9991_or__num_Osimps_I1_J,axiom,
% 8.58/4.89 ( ( bit_un6697907153464112080or_num @ one @ one )
% 8.58/4.89 = one ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9992_nat__of__num__numeral,axiom,
% 8.58/4.89 nat_of_num = numeral_numeral_nat ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9993_nat__of__num__code_I1_J,axiom,
% 8.58/4.89 ( ( nat_of_num @ one )
% 8.58/4.89 = one_one_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9994_less__num__def,axiom,
% 8.58/4.89 ( ord_less_num
% 8.58/4.89 = ( ^ [M3: num,N3: num] : ( ord_less_nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9995_nat__of__num__mult,axiom,
% 8.58/4.89 ! [X: num,Y: num] :
% 8.58/4.89 ( ( nat_of_num @ ( times_times_num @ X @ Y ) )
% 8.58/4.89 = ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9996_nat__of__num__inc,axiom,
% 8.58/4.89 ! [X: num] :
% 8.58/4.89 ( ( nat_of_num @ ( inc @ X ) )
% 8.58/4.89 = ( suc @ ( nat_of_num @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9997_nat__of__num__pos,axiom,
% 8.58/4.89 ! [X: num] : ( ord_less_nat @ zero_zero_nat @ ( nat_of_num @ X ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9998_nat__of__num__neq__0,axiom,
% 8.58/4.89 ! [X: num] :
% 8.58/4.89 ( ( nat_of_num @ X )
% 8.58/4.89 != zero_zero_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_9999_nat__of__num_Osimps_I1_J,axiom,
% 8.58/4.89 ( ( nat_of_num @ one )
% 8.58/4.89 = ( suc @ zero_zero_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10000_nat__of__num_Osimps_I3_J,axiom,
% 8.58/4.89 ! [X: num] :
% 8.58/4.89 ( ( nat_of_num @ ( bit1 @ X ) )
% 8.58/4.89 = ( suc @ ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10001_num__of__nat__inverse,axiom,
% 8.58/4.89 ! [N: nat] :
% 8.58/4.89 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.58/4.89 => ( ( nat_of_num @ ( num_of_nat @ N ) )
% 8.58/4.89 = N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10002_nat__of__num__code_I3_J,axiom,
% 8.58/4.89 ! [N: num] :
% 8.58/4.89 ( ( nat_of_num @ ( bit1 @ N ) )
% 8.58/4.89 = ( suc @ ( plus_plus_nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10003_plus__num__def,axiom,
% 8.58/4.89 ( plus_plus_num
% 8.58/4.89 = ( ^ [M3: num,N3: num] : ( num_of_nat @ ( plus_plus_nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N3 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10004_or__num_Oelims,axiom,
% 8.58/4.89 ! [X: num,Xa: num,Y: num] :
% 8.58/4.89 ( ( ( bit_un6697907153464112080or_num @ X @ Xa )
% 8.58/4.89 = Y )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( Y != one ) ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ( Y
% 8.58/4.89 != ( bit1 @ N2 ) ) ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ( Y
% 8.58/4.89 != ( bit1 @ N2 ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( Y
% 8.58/4.89 != ( bit1 @ M2 ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ( Y
% 8.58/4.89 != ( bit0 @ ( bit_un6697907153464112080or_num @ M2 @ N2 ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ( Y
% 8.58/4.89 != ( bit1 @ ( bit_un6697907153464112080or_num @ M2 @ N2 ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( Y
% 8.58/4.89 != ( bit1 @ M2 ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ( Y
% 8.58/4.89 != ( bit1 @ ( bit_un6697907153464112080or_num @ M2 @ N2 ) ) ) ) )
% 8.58/4.89 => ~ ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ( Y
% 8.58/4.89 != ( bit1 @ ( bit_un6697907153464112080or_num @ M2 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10005_or__num_Osimps_I2_J,axiom,
% 8.58/4.89 ! [N: num] :
% 8.58/4.89 ( ( bit_un6697907153464112080or_num @ one @ ( bit0 @ N ) )
% 8.58/4.89 = ( bit1 @ N ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10006_or__num_Opelims,axiom,
% 8.58/4.89 ! [X: num,Xa: num,Y: num] :
% 8.58/4.89 ( ( ( bit_un6697907153464112080or_num @ X @ Xa )
% 8.58/4.89 = Y )
% 8.58/4.89 => ( ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( ( Y = one )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ( ( ( X = one )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ one ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( bit0 @ ( bit_un6697907153464112080or_num @ M2 @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit0 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( bit1 @ ( bit_un6697907153464112080or_num @ M2 @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ( ( Xa = one )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ one ) ) ) ) )
% 8.58/4.89 => ( ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit0 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( bit1 @ ( bit_un6697907153464112080or_num @ M2 @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 8.58/4.89 => ~ ! [M2: num] :
% 8.58/4.89 ( ( X
% 8.58/4.89 = ( bit1 @ M2 ) )
% 8.58/4.89 => ! [N2: num] :
% 8.58/4.89 ( ( Xa
% 8.58/4.89 = ( bit1 @ N2 ) )
% 8.58/4.89 => ( ( Y
% 8.58/4.89 = ( bit1 @ ( bit_un6697907153464112080or_num @ M2 @ N2 ) ) )
% 8.58/4.89 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10007_or__num__dict,axiom,
% 8.58/4.89 bit_un6697907153464112080or_num = bit_un2785000775030745342or_num ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10008_or__num__rel__dict,axiom,
% 8.58/4.89 bit_un4773296044027857193um_rel = bit_un6909899581280750971um_rel ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10009_real__floor__code,axiom,
% 8.58/4.89 ! [X: rat] :
% 8.58/4.89 ( ( archim6058952711729229775r_real @ ( ratreal @ X ) )
% 8.58/4.89 = ( archim3151403230148437115or_rat @ X ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10010_real__minus__code,axiom,
% 8.58/4.89 ! [X: rat,Y: rat] :
% 8.58/4.89 ( ( minus_minus_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
% 8.58/4.89 = ( ratreal @ ( minus_minus_rat @ X @ Y ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10011_real__inverse__code,axiom,
% 8.58/4.89 ! [X: rat] :
% 8.58/4.89 ( ( inverse_inverse_real @ ( ratreal @ X ) )
% 8.58/4.89 = ( ratreal @ ( inverse_inverse_rat @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10012_real__plus__code,axiom,
% 8.58/4.89 ! [X: rat,Y: rat] :
% 8.58/4.89 ( ( plus_plus_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
% 8.58/4.89 = ( ratreal @ ( plus_plus_rat @ X @ Y ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10013_one__real__code,axiom,
% 8.58/4.89 ( one_one_real
% 8.58/4.89 = ( ratreal @ one_one_rat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10014_zero__real__code,axiom,
% 8.58/4.89 ( zero_zero_real
% 8.58/4.89 = ( ratreal @ zero_zero_rat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10015_real__less__code,axiom,
% 8.58/4.89 ! [X: rat,Y: rat] :
% 8.58/4.89 ( ( ord_less_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
% 8.58/4.89 = ( ord_less_rat @ X @ Y ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10016_Ratreal__def,axiom,
% 8.58/4.89 ratreal = field_7254667332652039916t_real ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10017_real__uminus__code,axiom,
% 8.58/4.89 ! [X: rat] :
% 8.58/4.89 ( ( uminus_uminus_real @ ( ratreal @ X ) )
% 8.58/4.89 = ( ratreal @ ( uminus_uminus_rat @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10018_real__less__eq__code,axiom,
% 8.58/4.89 ! [X: rat,Y: rat] :
% 8.58/4.89 ( ( ord_less_eq_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
% 8.58/4.89 = ( ord_less_eq_rat @ X @ Y ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10019_real__times__code,axiom,
% 8.58/4.89 ! [X: rat,Y: rat] :
% 8.58/4.89 ( ( times_times_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
% 8.58/4.89 = ( ratreal @ ( times_times_rat @ X @ Y ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10020_real__divide__code,axiom,
% 8.58/4.89 ! [X: rat,Y: rat] :
% 8.58/4.89 ( ( divide_divide_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
% 8.58/4.89 = ( ratreal @ ( divide_divide_rat @ X @ Y ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10021_gcd__nat_Osemilattice__order__axioms,axiom,
% 8.58/4.89 ( semila1248733672344298208er_nat @ gcd_gcd_nat @ dvd_dvd_nat
% 8.58/4.89 @ ^ [M3: nat,N3: nat] :
% 8.58/4.89 ( ( dvd_dvd_nat @ M3 @ N3 )
% 8.58/4.89 & ( M3 != N3 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10022_natural__decr,axiom,
% 8.58/4.89 ! [N: code_natural] :
% 8.58/4.89 ( ( N != zero_z2226904508553997617atural )
% 8.58/4.89 => ( ord_less_nat @ ( minus_minus_nat @ ( code_nat_of_natural @ N ) @ ( suc @ zero_zero_nat ) ) @ ( code_nat_of_natural @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10023_nat__of__natural__numeral,axiom,
% 8.58/4.89 ! [K: num] :
% 8.58/4.89 ( ( code_nat_of_natural @ ( numera5444537566228673987atural @ K ) )
% 8.58/4.89 = ( numeral_numeral_nat @ K ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10024_plus__natural_Orep__eq,axiom,
% 8.58/4.89 ! [X: code_natural,Xa: code_natural] :
% 8.58/4.89 ( ( code_nat_of_natural @ ( plus_p4538020629002901425atural @ X @ Xa ) )
% 8.58/4.89 = ( plus_plus_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10025_minus__natural_Orep__eq,axiom,
% 8.58/4.89 ! [X: code_natural,Xa: code_natural] :
% 8.58/4.89 ( ( code_nat_of_natural @ ( minus_7197305767214868737atural @ X @ Xa ) )
% 8.58/4.89 = ( minus_minus_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10026_times__natural_Orep__eq,axiom,
% 8.58/4.89 ! [X: code_natural,Xa: code_natural] :
% 8.58/4.89 ( ( code_nat_of_natural @ ( times_2397367101498566445atural @ X @ Xa ) )
% 8.58/4.89 = ( times_times_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10027_one__natural_Orep__eq,axiom,
% 8.58/4.89 ( ( code_nat_of_natural @ one_one_Code_natural )
% 8.58/4.89 = one_one_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10028_divide__natural_Orep__eq,axiom,
% 8.58/4.89 ! [X: code_natural,Xa: code_natural] :
% 8.58/4.89 ( ( code_nat_of_natural @ ( divide5121882707175180666atural @ X @ Xa ) )
% 8.58/4.89 = ( divide_divide_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10029_zero__natural_Orep__eq,axiom,
% 8.58/4.89 ( ( code_nat_of_natural @ zero_z2226904508553997617atural )
% 8.58/4.89 = zero_zero_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10030_push__bit__natural_Orep__eq,axiom,
% 8.58/4.89 ! [X: nat,Xa: code_natural] :
% 8.58/4.89 ( ( code_nat_of_natural @ ( bit_se6611745700429515170atural @ X @ Xa ) )
% 8.58/4.89 = ( bit_se547839408752420682it_nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10031_natural__zero__minus__one,axiom,
% 8.58/4.89 ( ( minus_7197305767214868737atural @ zero_z2226904508553997617atural @ one_one_Code_natural )
% 8.58/4.89 = zero_z2226904508553997617atural ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10032_integer__of__natural_Orep__eq,axiom,
% 8.58/4.89 ! [X: code_natural] :
% 8.58/4.89 ( ( code_int_of_integer @ ( code_i5400310926305786745atural @ X ) )
% 8.58/4.89 = ( semiri1314217659103216013at_int @ ( code_nat_of_natural @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10033_int__of__integer__of__natural,axiom,
% 8.58/4.89 ! [N: code_natural] :
% 8.58/4.89 ( ( code_int_of_integer @ ( code_i5400310926305786745atural @ N ) )
% 8.58/4.89 = ( semiri1314217659103216013at_int @ ( code_nat_of_natural @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10034_next_Osimps,axiom,
% 8.58/4.89 ! [V: code_natural,W: code_natural] :
% 8.58/4.89 ( ( next @ ( produc3574140220909816553atural @ V @ W ) )
% 8.58/4.89 = ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ one_one_Code_natural ) ) @ one_one_Code_natural ) @ ( produc3574140220909816553atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10035_log_Osimps,axiom,
% 8.58/4.89 ( log
% 8.58/4.89 = ( ^ [B2: code_natural,I4: code_natural] :
% 8.58/4.89 ( if_Code_natural
% 8.58/4.89 @ ( ( ord_le1926595141338095240atural @ B2 @ one_one_Code_natural )
% 8.58/4.89 | ( ord_le5570908160329646204atural @ I4 @ B2 ) )
% 8.58/4.89 @ one_one_Code_natural
% 8.58/4.89 @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ B2 @ ( divide5121882707175180666atural @ I4 @ B2 ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10036_minus__shift__def,axiom,
% 8.58/4.89 ( minus_shift
% 8.58/4.89 = ( ^ [R5: code_natural,K2: code_natural,L2: code_natural] : ( if_Code_natural @ ( ord_le5570908160329646204atural @ K2 @ L2 ) @ ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ R5 @ K2 ) @ L2 ) @ ( minus_7197305767214868737atural @ K2 @ L2 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10037_log_Oelims,axiom,
% 8.58/4.89 ! [X: code_natural,Xa: code_natural,Y: code_natural] :
% 8.58/4.89 ( ( ( log @ X @ Xa )
% 8.58/4.89 = Y )
% 8.58/4.89 => ( ( ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
% 8.58/4.89 | ( ord_le5570908160329646204atural @ Xa @ X ) )
% 8.58/4.89 => ( Y = one_one_Code_natural ) )
% 8.58/4.89 & ( ~ ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
% 8.58/4.89 | ( ord_le5570908160329646204atural @ Xa @ X ) )
% 8.58/4.89 => ( Y
% 8.58/4.89 = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ X @ ( divide5121882707175180666atural @ Xa @ X ) ) ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10038_split__seed__def,axiom,
% 8.58/4.89 ( split_seed
% 8.58/4.89 = ( ^ [S4: produc7822875418678951345atural] :
% 8.58/4.89 ( produc8282080750456430313atural
% 8.58/4.89 @ ^ [V4: code_natural,W2: code_natural] :
% 8.58/4.89 ( produc8282080750456430313atural
% 8.58/4.89 @ ^ [V5: code_natural,W3: code_natural] : ( produc4480994950612372183atural @ ( produc3574140220909816553atural @ ( inc_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ V4 ) @ W3 ) @ ( produc3574140220909816553atural @ V5 @ ( inc_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ W2 ) ) )
% 8.58/4.89 @ ( produc6591912806276919810atural @ ( next @ S4 ) ) )
% 8.58/4.89 @ S4 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10039_Random_Orange__def,axiom,
% 8.58/4.89 ( range
% 8.58/4.89 = ( ^ [K2: code_natural] :
% 8.58/4.89 ( produc5538323210962509403atural
% 8.58/4.89 @ ( iterat8892046348760725948atural @ ( log @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K2 )
% 8.58/4.89 @ ^ [L2: code_natural] :
% 8.58/4.89 ( produc5538323210962509403atural @ next
% 8.58/4.89 @ ^ [V4: code_natural] : ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ V4 @ ( times_2397367101498566445atural @ L2 @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 8.58/4.89 @ one_one_Code_natural )
% 8.58/4.89 @ ^ [V4: code_natural] : ( produc6639722614265839536atural @ ( modulo8411746178871703098atural @ V4 @ K2 ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10040_inc__shift__def,axiom,
% 8.58/4.89 ( inc_shift
% 8.58/4.89 = ( ^ [V4: code_natural,K2: code_natural] : ( if_Code_natural @ ( V4 = K2 ) @ one_one_Code_natural @ ( plus_p4538020629002901425atural @ K2 @ one_one_Code_natural ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10041_log_Opelims,axiom,
% 8.58/4.89 ! [X: code_natural,Xa: code_natural,Y: code_natural] :
% 8.58/4.89 ( ( ( log @ X @ Xa )
% 8.58/4.89 = Y )
% 8.58/4.89 => ( ( accp_P8126237942716283194atural @ log_rel @ ( produc3574140220909816553atural @ X @ Xa ) )
% 8.58/4.89 => ~ ( ( ( ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
% 8.58/4.89 | ( ord_le5570908160329646204atural @ Xa @ X ) )
% 8.58/4.89 => ( Y = one_one_Code_natural ) )
% 8.58/4.89 & ( ~ ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
% 8.58/4.89 | ( ord_le5570908160329646204atural @ Xa @ X ) )
% 8.58/4.89 => ( Y
% 8.58/4.89 = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ X @ ( divide5121882707175180666atural @ Xa @ X ) ) ) ) ) )
% 8.58/4.89 => ~ ( accp_P8126237942716283194atural @ log_rel @ ( produc3574140220909816553atural @ X @ Xa ) ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10042_divide__natural_Oabs__eq,axiom,
% 8.58/4.89 ! [Xa: nat,X: nat] :
% 8.58/4.89 ( ( divide5121882707175180666atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 8.58/4.89 = ( code_natural_of_nat @ ( divide_divide_nat @ Xa @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10043_push__bit__natural_Oabs__eq,axiom,
% 8.58/4.89 ! [Xa: nat,X: nat] :
% 8.58/4.89 ( ( bit_se6611745700429515170atural @ Xa @ ( code_natural_of_nat @ X ) )
% 8.58/4.89 = ( code_natural_of_nat @ ( bit_se547839408752420682it_nat @ Xa @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10044_times__natural_Oabs__eq,axiom,
% 8.58/4.89 ! [Xa: nat,X: nat] :
% 8.58/4.89 ( ( times_2397367101498566445atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 8.58/4.89 = ( code_natural_of_nat @ ( times_times_nat @ Xa @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10045_one__natural__def,axiom,
% 8.58/4.89 ( one_one_Code_natural
% 8.58/4.89 = ( code_natural_of_nat @ one_one_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10046_plus__natural_Oabs__eq,axiom,
% 8.58/4.89 ! [Xa: nat,X: nat] :
% 8.58/4.89 ( ( plus_p4538020629002901425atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 8.58/4.89 = ( code_natural_of_nat @ ( plus_plus_nat @ Xa @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10047_minus__natural_Oabs__eq,axiom,
% 8.58/4.89 ! [Xa: nat,X: nat] :
% 8.58/4.89 ( ( minus_7197305767214868737atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 8.58/4.89 = ( code_natural_of_nat @ ( minus_minus_nat @ Xa @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10048_zero__natural__def,axiom,
% 8.58/4.89 ( zero_z2226904508553997617atural
% 8.58/4.89 = ( code_natural_of_nat @ zero_zero_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10049_integer__of__natural_Oabs__eq,axiom,
% 8.58/4.89 ! [X: nat] :
% 8.58/4.89 ( ( code_i5400310926305786745atural @ ( code_natural_of_nat @ X ) )
% 8.58/4.89 = ( code_integer_of_int @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10050_integer__of__natural__def,axiom,
% 8.58/4.89 ( code_i5400310926305786745atural
% 8.58/4.89 = ( map_fu2787874002554666395nteger @ code_nat_of_natural @ code_integer_of_int @ semiri1314217659103216013at_int ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10051_Suc_Orep__eq,axiom,
% 8.58/4.89 ! [X: code_natural] :
% 8.58/4.89 ( ( code_nat_of_natural @ ( code_Suc @ X ) )
% 8.58/4.89 = ( suc @ ( code_nat_of_natural @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10052_Suc__natural__minus__one,axiom,
% 8.58/4.89 ! [N: code_natural] :
% 8.58/4.89 ( ( minus_7197305767214868737atural @ ( code_Suc @ N ) @ one_one_Code_natural )
% 8.58/4.89 = N ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10053_Suc_Oabs__eq,axiom,
% 8.58/4.89 ! [X: nat] :
% 8.58/4.89 ( ( code_Suc @ ( code_natural_of_nat @ X ) )
% 8.58/4.89 = ( code_natural_of_nat @ ( suc @ X ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10054_Code__Numeral_OSuc__def,axiom,
% 8.58/4.89 ( code_Suc
% 8.58/4.89 = ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat @ suc ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10055_Quotient3__int,axiom,
% 8.58/4.89 quotie6776551016481293500at_int @ intrel @ abs_Integ @ rep_Integ ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10056_push__bit__natural__def,axiom,
% 8.58/4.89 ( bit_se6611745700429515170atural
% 8.58/4.89 = ( map_fu6981160461238011112atural @ id_nat @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se547839408752420682it_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10057_divide__natural__def,axiom,
% 8.58/4.89 ( divide5121882707175180666atural
% 8.58/4.89 = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ divide_divide_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10058_plus__natural__def,axiom,
% 8.58/4.89 ( plus_p4538020629002901425atural
% 8.58/4.89 = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ plus_plus_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10059_minus__natural__def,axiom,
% 8.58/4.89 ( minus_7197305767214868737atural
% 8.58/4.89 = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ minus_minus_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10060_times__natural__def,axiom,
% 8.58/4.89 ( times_2397367101498566445atural
% 8.58/4.89 = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ times_times_nat ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10061_sub_Otransfer,axiom,
% 8.58/4.89 ( bNF_re7876454716742015248nteger
% 8.58/4.89 @ ^ [Y3: num,Z: num] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re6501075790457514782nteger
% 8.58/4.89 @ ^ [Y3: num,Z: num] : ( Y3 = Z )
% 8.58/4.89 @ code_pcr_integer )
% 8.58/4.89 @ ^ [M3: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) )
% 8.58/4.89 @ code_sub ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10062_push__bit__integer_Otransfer,axiom,
% 8.58/4.89 ( bNF_re4935368626670024657nteger
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
% 8.58/4.89 @ bit_se545348938243370406it_int
% 8.58/4.89 @ bit_se7788150548672797655nteger ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10063_divide__integer_Otransfer,axiom,
% 8.58/4.89 bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ divide_divide_int @ divide6298287555418463151nteger ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10064_minus__integer_Otransfer,axiom,
% 8.58/4.89 bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ minus_minus_int @ minus_8373710615458151222nteger ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10065_times__integer_Otransfer,axiom,
% 8.58/4.89 bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ times_times_int @ times_3573771949741848930nteger ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10066_plus__integer_Otransfer,axiom,
% 8.58/4.89 bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ plus_plus_int @ plus_p5714425477246183910nteger ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10067_one__integer_Otransfer,axiom,
% 8.58/4.89 code_pcr_integer @ one_one_int @ one_one_Code_integer ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10068_dup_Otransfer,axiom,
% 8.58/4.89 ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer
% 8.58/4.89 @ ^ [K2: int] : ( plus_plus_int @ K2 @ K2 )
% 8.58/4.89 @ code_dup ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10069_integer__of__nat_Otransfer,axiom,
% 8.58/4.89 ( bNF_re4153400068438556298nteger
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ code_pcr_integer
% 8.58/4.89 @ semiri1314217659103216013at_int
% 8.58/4.89 @ code_integer_of_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10070_gcd__integer_Otransfer,axiom,
% 8.58/4.89 bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ gcd_gcd_int @ gcd_gcd_Code_integer ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10071_lcm__integer_Otransfer,axiom,
% 8.58/4.89 bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ gcd_lcm_int @ gcd_lcm_Code_integer ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10072_zero__integer_Otransfer,axiom,
% 8.58/4.89 code_pcr_integer @ zero_zero_int @ zero_z3403309356797280102nteger ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10073_integer__of__natural_Otransfer,axiom,
% 8.58/4.89 bNF_re5252274238750452962nteger @ code_pcr_natural @ code_pcr_integer @ semiri1314217659103216013at_int @ code_i5400310926305786745atural ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10074_Suc_Otransfer,axiom,
% 8.58/4.89 bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural @ suc @ code_Suc ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10075_divide__natural_Otransfer,axiom,
% 8.58/4.89 bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ divide_divide_nat @ divide5121882707175180666atural ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10076_push__bit__natural_Otransfer,axiom,
% 8.58/4.89 ( bNF_re3692170009638431535atural
% 8.58/4.89 @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
% 8.58/4.89 @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural )
% 8.58/4.89 @ bit_se547839408752420682it_nat
% 8.58/4.89 @ bit_se6611745700429515170atural ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10077_minus__natural_Otransfer,axiom,
% 8.58/4.89 bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ minus_minus_nat @ minus_7197305767214868737atural ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10078_times__natural_Otransfer,axiom,
% 8.58/4.89 bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ times_times_nat @ times_2397367101498566445atural ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10079_plus__natural_Otransfer,axiom,
% 8.58/4.89 bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ plus_plus_nat @ plus_p4538020629002901425atural ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10080_less__eq__int__code_I9_J,axiom,
% 8.58/4.89 ! [K: num,L: num] :
% 8.58/4.89 ( ( ord_less_eq_int @ ( neg @ K ) @ ( neg @ L ) )
% 8.58/4.89 = ( ord_less_eq_num @ L @ K ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10081_less__eq__int__code_I3_J,axiom,
% 8.58/4.89 ! [L: num] :
% 8.58/4.89 ~ ( ord_less_eq_int @ zero_zero_int @ ( neg @ L ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10082_less__eq__int__code_I7_J,axiom,
% 8.58/4.89 ! [K: num] : ( ord_less_eq_int @ ( neg @ K ) @ zero_zero_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10083_zero__natural_Otransfer,axiom,
% 8.58/4.89 code_pcr_natural @ zero_zero_nat @ zero_z2226904508553997617atural ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10084_nat__code_I1_J,axiom,
% 8.58/4.89 ! [K: num] :
% 8.58/4.89 ( ( nat2 @ ( neg @ K ) )
% 8.58/4.89 = zero_zero_nat ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10085_less__int__code_I9_J,axiom,
% 8.58/4.89 ! [K: num,L: num] :
% 8.58/4.89 ( ( ord_less_int @ ( neg @ K ) @ ( neg @ L ) )
% 8.58/4.89 = ( ord_less_num @ L @ K ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10086_one__natural_Otransfer,axiom,
% 8.58/4.89 code_pcr_natural @ one_one_nat @ one_one_Code_natural ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10087_plus__int__code_I6_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( plus_plus_int @ ( neg @ M ) @ ( neg @ N ) )
% 8.58/4.89 = ( neg @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10088_less__int__code_I3_J,axiom,
% 8.58/4.89 ! [L: num] :
% 8.58/4.89 ~ ( ord_less_int @ zero_zero_int @ ( neg @ L ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10089_less__int__code_I7_J,axiom,
% 8.58/4.89 ! [K: num] : ( ord_less_int @ ( neg @ K ) @ zero_zero_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10090_Int_Osub__code_I4_J,axiom,
% 8.58/4.89 ! [N: num] :
% 8.58/4.89 ( ( sub @ one @ ( bit0 @ N ) )
% 8.58/4.89 = ( neg @ ( bitM @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10091_Int_Osub__code_I5_J,axiom,
% 8.58/4.89 ! [N: num] :
% 8.58/4.89 ( ( sub @ one @ ( bit1 @ N ) )
% 8.58/4.89 = ( neg @ ( bit0 @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10092_Int_Osub__def,axiom,
% 8.58/4.89 ( sub
% 8.58/4.89 = ( ^ [M3: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10093_Int_Osub__code_I1_J,axiom,
% 8.58/4.89 ( ( sub @ one @ one )
% 8.58/4.89 = zero_zero_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10094_minus__int__code_I6_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( minus_minus_int @ ( neg @ M ) @ ( neg @ N ) )
% 8.58/4.89 = ( sub @ N @ M ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10095_Int_Osub__code_I9_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.58/4.89 = ( minus_minus_int @ ( dup @ ( sub @ M @ N ) ) @ one_one_int ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10096_Int_Osub__code_I8_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.58/4.89 = ( plus_plus_int @ ( dup @ ( sub @ M @ N ) ) @ one_one_int ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10097_Int_Odup__code_I1_J,axiom,
% 8.58/4.89 ( ( dup @ zero_zero_int )
% 8.58/4.89 = zero_zero_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10098_Int_Odup__def,axiom,
% 8.58/4.89 ( dup
% 8.58/4.89 = ( ^ [K2: int] : ( plus_plus_int @ K2 @ K2 ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10099_Int_Odup__code_I3_J,axiom,
% 8.58/4.89 ! [N: num] :
% 8.58/4.89 ( ( dup @ ( neg @ N ) )
% 8.58/4.89 = ( neg @ ( bit0 @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10100_Int_Osub__code_I6_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( sub @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.58/4.89 = ( dup @ ( sub @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10101_Int_Osub__code_I7_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.89 ( ( sub @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.58/4.89 = ( dup @ ( sub @ M @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10102_Int_Osub__code_I2_J,axiom,
% 8.58/4.89 ! [M: num] :
% 8.58/4.89 ( ( sub @ ( bit0 @ M ) @ one )
% 8.58/4.89 = ( pos @ ( bitM @ M ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10103_Int_Osub__code_I3_J,axiom,
% 8.58/4.89 ! [M: num] :
% 8.58/4.89 ( ( sub @ ( bit1 @ M ) @ one )
% 8.58/4.89 = ( pos @ ( bit0 @ M ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10104_Int_Odup__code_I2_J,axiom,
% 8.58/4.89 ! [N: num] :
% 8.58/4.89 ( ( dup @ ( pos @ N ) )
% 8.58/4.89 = ( pos @ ( bit0 @ N ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10105_less__eq__int__code_I6_J,axiom,
% 8.58/4.89 ! [K: num,L: num] :
% 8.58/4.89 ~ ( ord_less_eq_int @ ( pos @ K ) @ ( neg @ L ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10106_less__eq__int__code_I8_J,axiom,
% 8.58/4.89 ! [K: num,L: num] : ( ord_less_eq_int @ ( neg @ K ) @ ( pos @ L ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10107_Int_ONeg__def,axiom,
% 8.58/4.89 ( neg
% 8.58/4.89 = ( ^ [N3: num] : ( uminus_uminus_int @ ( pos @ N3 ) ) ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10108_uminus__int__code_I2_J,axiom,
% 8.58/4.89 ! [M: num] :
% 8.58/4.89 ( ( uminus_uminus_int @ ( pos @ M ) )
% 8.58/4.89 = ( neg @ M ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10109_uminus__int__code_I3_J,axiom,
% 8.58/4.89 ! [M: num] :
% 8.58/4.89 ( ( uminus_uminus_int @ ( neg @ M ) )
% 8.58/4.89 = ( pos @ M ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10110_less__int__code_I8_J,axiom,
% 8.58/4.89 ! [K: num,L: num] : ( ord_less_int @ ( neg @ K ) @ ( pos @ L ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10111_less__int__code_I6_J,axiom,
% 8.58/4.89 ! [K: num,L: num] :
% 8.58/4.89 ~ ( ord_less_int @ ( pos @ K ) @ ( neg @ L ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10112_less__int__code_I4_J,axiom,
% 8.58/4.89 ! [K: num] :
% 8.58/4.89 ~ ( ord_less_int @ ( pos @ K ) @ zero_zero_int ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10113_less__int__code_I2_J,axiom,
% 8.58/4.89 ! [L: num] : ( ord_less_int @ zero_zero_int @ ( pos @ L ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10114_Int_OPos__def,axiom,
% 8.58/4.89 pos = numeral_numeral_int ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10115_one__int__code,axiom,
% 8.58/4.89 ( one_one_int
% 8.58/4.89 = ( pos @ one ) ) ).
% 8.58/4.89
% 8.58/4.89 thf(fact_10116_plus__int__code_I3_J,axiom,
% 8.58/4.89 ! [M: num,N: num] :
% 8.58/4.90 ( ( plus_plus_int @ ( pos @ M ) @ ( pos @ N ) )
% 8.58/4.90 = ( pos @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10117_times__int__code_I3_J,axiom,
% 8.58/4.90 ! [M: num,N: num] :
% 8.58/4.90 ( ( times_times_int @ ( pos @ M ) @ ( pos @ N ) )
% 8.58/4.90 = ( pos @ ( times_times_num @ M @ N ) ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10118_less__int__code_I5_J,axiom,
% 8.58/4.90 ! [K: num,L: num] :
% 8.58/4.90 ( ( ord_less_int @ ( pos @ K ) @ ( pos @ L ) )
% 8.58/4.90 = ( ord_less_num @ K @ L ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10119_nat__code_I3_J,axiom,
% 8.58/4.90 ! [K: num] :
% 8.58/4.90 ( ( nat2 @ ( pos @ K ) )
% 8.58/4.90 = ( nat_of_num @ K ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10120_less__eq__int__code_I2_J,axiom,
% 8.58/4.90 ! [L: num] : ( ord_less_eq_int @ zero_zero_int @ ( pos @ L ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10121_less__eq__int__code_I4_J,axiom,
% 8.58/4.90 ! [K: num] :
% 8.58/4.90 ~ ( ord_less_eq_int @ ( pos @ K ) @ zero_zero_int ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10122_less__eq__int__code_I5_J,axiom,
% 8.58/4.90 ! [K: num,L: num] :
% 8.58/4.90 ( ( ord_less_eq_int @ ( pos @ K ) @ ( pos @ L ) )
% 8.58/4.90 = ( ord_less_eq_num @ K @ L ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10123_minus__int__code_I3_J,axiom,
% 8.58/4.90 ! [M: num,N: num] :
% 8.58/4.90 ( ( minus_minus_int @ ( pos @ M ) @ ( pos @ N ) )
% 8.58/4.90 = ( sub @ M @ N ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10124_minus__int__code_I5_J,axiom,
% 8.58/4.90 ! [M: num,N: num] :
% 8.58/4.90 ( ( minus_minus_int @ ( neg @ M ) @ ( pos @ N ) )
% 8.58/4.90 = ( neg @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10125_minus__int__code_I4_J,axiom,
% 8.58/4.90 ! [M: num,N: num] :
% 8.58/4.90 ( ( minus_minus_int @ ( pos @ M ) @ ( neg @ N ) )
% 8.58/4.90 = ( pos @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10126_times__int__code_I6_J,axiom,
% 8.58/4.90 ! [M: num,N: num] :
% 8.58/4.90 ( ( times_times_int @ ( neg @ M ) @ ( neg @ N ) )
% 8.58/4.90 = ( pos @ ( times_times_num @ M @ N ) ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10127_times__int__code_I5_J,axiom,
% 8.58/4.90 ! [M: num,N: num] :
% 8.58/4.90 ( ( times_times_int @ ( neg @ M ) @ ( pos @ N ) )
% 8.58/4.90 = ( neg @ ( times_times_num @ M @ N ) ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10128_times__int__code_I4_J,axiom,
% 8.58/4.90 ! [M: num,N: num] :
% 8.58/4.90 ( ( times_times_int @ ( pos @ M ) @ ( neg @ N ) )
% 8.58/4.90 = ( neg @ ( times_times_num @ M @ N ) ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10129_plus__int__code_I4_J,axiom,
% 8.58/4.90 ! [M: num,N: num] :
% 8.58/4.90 ( ( plus_plus_int @ ( pos @ M ) @ ( neg @ N ) )
% 8.58/4.90 = ( sub @ M @ N ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10130_plus__int__code_I5_J,axiom,
% 8.58/4.90 ! [M: num,N: num] :
% 8.58/4.90 ( ( plus_plus_int @ ( neg @ M ) @ ( pos @ N ) )
% 8.58/4.90 = ( sub @ N @ M ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10131_sum__encode__def,axiom,
% 8.58/4.90 ( nat_sum_encode
% 8.58/4.90 = ( sum_ca6763686470577984908at_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.58/4.90 @ ^ [B2: nat] : ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10132_Quotient3__real,axiom,
% 8.58/4.90 quotie8700032322157300518t_real @ realrel @ real2 @ rep_real ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10133_sum__encode__eq,axiom,
% 8.58/4.90 ! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] :
% 8.58/4.90 ( ( ( nat_sum_encode @ X )
% 8.58/4.90 = ( nat_sum_encode @ Y ) )
% 8.58/4.90 = ( X = Y ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10134_bij__sum__encode,axiom,
% 8.58/4.90 bij_be5432664580149595207at_nat @ nat_sum_encode @ top_to6661820994512907621at_nat @ top_top_set_nat ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10135_surj__sum__encode,axiom,
% 8.58/4.90 ( ( image_1320371278474632150at_nat @ nat_sum_encode @ top_to6661820994512907621at_nat )
% 8.58/4.90 = top_top_set_nat ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10136_inj__sum__encode,axiom,
% 8.58/4.90 ! [A3: set_Sum_sum_nat_nat] : ( inj_on6343450744447823682at_nat @ nat_sum_encode @ A3 ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10137_int__encode__def,axiom,
% 8.58/4.90 ( nat_int_encode
% 8.58/4.90 = ( ^ [I4: int] : ( nat_sum_encode @ ( if_Sum_sum_nat_nat @ ( ord_less_eq_int @ zero_zero_int @ I4 ) @ ( sum_Inl_nat_nat @ ( nat2 @ I4 ) ) @ ( sum_Inr_nat_nat @ ( nat2 @ ( minus_minus_int @ ( uminus_uminus_int @ I4 ) @ one_one_int ) ) ) ) ) ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10138_sum__decode__def,axiom,
% 8.58/4.90 ( nat_sum_decode
% 8.58/4.90 = ( ^ [N3: nat] : ( if_Sum_sum_nat_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( sum_Inl_nat_nat @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( sum_Inr_nat_nat @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10139_sum__encode__inverse,axiom,
% 8.58/4.90 ! [X: sum_sum_nat_nat] :
% 8.58/4.90 ( ( nat_sum_decode @ ( nat_sum_encode @ X ) )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10140_sum__decode__inverse,axiom,
% 8.58/4.90 ! [N: nat] :
% 8.58/4.90 ( ( nat_sum_encode @ ( nat_sum_decode @ N ) )
% 8.58/4.90 = N ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10141_inj__sum__decode,axiom,
% 8.58/4.90 ! [A3: set_nat] : ( inj_on5701776251185195458at_nat @ nat_sum_decode @ A3 ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10142_surj__sum__decode,axiom,
% 8.58/4.90 ( ( image_678696785212003926at_nat @ nat_sum_decode @ top_top_set_nat )
% 8.58/4.90 = top_to6661820994512907621at_nat ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10143_bij__sum__decode,axiom,
% 8.58/4.90 bij_be4790990086886966983at_nat @ nat_sum_decode @ top_top_set_nat @ top_to6661820994512907621at_nat ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10144_sum__decode__eq,axiom,
% 8.58/4.90 ! [X: nat,Y: nat] :
% 8.58/4.90 ( ( ( nat_sum_decode @ X )
% 8.58/4.90 = ( nat_sum_decode @ Y ) )
% 8.58/4.90 = ( X = Y ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10145_nth__item_Opinduct,axiom,
% 8.58/4.90 ! [A0: nat,P: nat > $o] :
% 8.58/4.90 ( ( accp_nat @ nth_item_rel @ A0 )
% 8.58/4.90 => ( ( ( accp_nat @ nth_item_rel @ zero_zero_nat )
% 8.58/4.90 => ( P @ zero_zero_nat ) )
% 8.58/4.90 => ( ! [N2: nat] :
% 8.58/4.90 ( ( accp_nat @ nth_item_rel @ ( suc @ N2 ) )
% 8.58/4.90 => ( ! [A6: nat,Aa: nat] :
% 8.58/4.90 ( ( ( nat_sum_decode @ N2 )
% 8.58/4.90 = ( sum_Inl_nat_nat @ A6 ) )
% 8.58/4.90 => ( ( ( nat_sum_decode @ A6 )
% 8.58/4.90 = ( sum_Inl_nat_nat @ Aa ) )
% 8.58/4.90 => ( P @ Aa ) ) )
% 8.58/4.90 => ( ! [A6: nat,B8: nat] :
% 8.58/4.90 ( ( ( nat_sum_decode @ N2 )
% 8.58/4.90 = ( sum_Inl_nat_nat @ A6 ) )
% 8.58/4.90 => ( ( ( nat_sum_decode @ A6 )
% 8.58/4.90 = ( sum_Inr_nat_nat @ B8 ) )
% 8.58/4.90 => ( P @ B8 ) ) )
% 8.58/4.90 => ( ! [B8: nat,Ba: nat,X5: nat,Y5: nat] :
% 8.58/4.90 ( ( ( nat_sum_decode @ N2 )
% 8.58/4.90 = ( sum_Inr_nat_nat @ B8 ) )
% 8.58/4.90 => ( ( ( nat_sum_decode @ B8 )
% 8.58/4.90 = ( sum_Inr_nat_nat @ Ba ) )
% 8.58/4.90 => ( ( ( product_Pair_nat_nat @ X5 @ Y5 )
% 8.58/4.90 = ( nat_prod_decode @ Ba ) )
% 8.58/4.90 => ( P @ X5 ) ) ) )
% 8.58/4.90 => ( ! [B8: nat,Ba: nat,X5: nat,Y5: nat] :
% 8.58/4.90 ( ( ( nat_sum_decode @ N2 )
% 8.58/4.90 = ( sum_Inr_nat_nat @ B8 ) )
% 8.58/4.90 => ( ( ( nat_sum_decode @ B8 )
% 8.58/4.90 = ( sum_Inr_nat_nat @ Ba ) )
% 8.58/4.90 => ( ( ( product_Pair_nat_nat @ X5 @ Y5 )
% 8.58/4.90 = ( nat_prod_decode @ Ba ) )
% 8.58/4.90 => ( P @ Y5 ) ) ) )
% 8.58/4.90 => ( P @ ( suc @ N2 ) ) ) ) ) ) )
% 8.58/4.90 => ( P @ A0 ) ) ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(fact_10146_int__decode__def,axiom,
% 8.58/4.90 ( nat_int_decode
% 8.58/4.90 = ( ^ [N3: nat] :
% 8.58/4.90 ( sum_ca7763040182479039464nt_nat @ semiri1314217659103216013at_int
% 8.58/4.90 @ ^ [B2: nat] : ( minus_minus_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ B2 ) ) @ one_one_int )
% 8.58/4.90 @ ( nat_sum_decode @ N3 ) ) ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 8.58/4.90 ! [X: int,Y: int] :
% 8.58/4.90 ( ( if_int @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 8.58/4.90 ! [X: int,Y: int] :
% 8.58/4.90 ( ( if_int @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 8.58/4.90 ! [X: nat,Y: nat] :
% 8.58/4.90 ( ( if_nat @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 8.58/4.90 ! [X: nat,Y: nat] :
% 8.58/4.90 ( ( if_nat @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 8.58/4.90 ! [X: num,Y: num] :
% 8.58/4.90 ( ( if_num @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 8.58/4.90 ! [X: num,Y: num] :
% 8.58/4.90 ( ( if_num @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 8.58/4.90 ! [X: rat,Y: rat] :
% 8.58/4.90 ( ( if_rat @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 8.58/4.90 ! [X: rat,Y: rat] :
% 8.58/4.90 ( ( if_rat @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 8.58/4.90 ! [X: real,Y: real] :
% 8.58/4.90 ( ( if_real @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 8.58/4.90 ! [X: real,Y: real] :
% 8.58/4.90 ( ( if_real @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 8.58/4.90 ! [X: complex,Y: complex] :
% 8.58/4.90 ( ( if_complex @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 8.58/4.90 ! [X: complex,Y: complex] :
% 8.58/4.90 ( ( if_complex @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 8.58/4.90 ! [X: extended_enat,Y: extended_enat] :
% 8.58/4.90 ( ( if_Extended_enat @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 8.58/4.90 ! [X: extended_enat,Y: extended_enat] :
% 8.58/4.90 ( ( if_Extended_enat @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 8.58/4.90 ! [X: code_integer,Y: code_integer] :
% 8.58/4.90 ( ( if_Code_integer @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 8.58/4.90 ! [X: code_integer,Y: code_integer] :
% 8.58/4.90 ( ( if_Code_integer @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__Code____Numeral__Onatural_T,axiom,
% 8.58/4.90 ! [X: code_natural,Y: code_natural] :
% 8.58/4.90 ( ( if_Code_natural @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__Code____Numeral__Onatural_T,axiom,
% 8.58/4.90 ! [X: code_natural,Y: code_natural] :
% 8.58/4.90 ( ( if_Code_natural @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 8.58/4.90 ! [X: set_int,Y: set_int] :
% 8.58/4.90 ( ( if_set_int @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 8.58/4.90 ! [X: set_int,Y: set_int] :
% 8.58/4.90 ( ( if_set_int @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 8.58/4.90 ! [X: list_int,Y: list_int] :
% 8.58/4.90 ( ( if_list_int @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 8.58/4.90 ! [X: list_int,Y: list_int] :
% 8.58/4.90 ( ( if_list_int @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 8.58/4.90 ! [X: list_nat,Y: list_nat] :
% 8.58/4.90 ( ( if_list_nat @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 8.58/4.90 ! [X: list_nat,Y: list_nat] :
% 8.58/4.90 ( ( if_list_nat @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 8.58/4.90 ! [X: int > int,Y: int > int] :
% 8.58/4.90 ( ( if_int_int @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 8.58/4.90 ! [X: int > int,Y: int > int] :
% 8.58/4.90 ( ( if_int_int @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
% 8.58/4.90 ! [X: nat > rat,Y: nat > rat] :
% 8.58/4.90 ( ( if_nat_rat @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
% 8.58/4.90 ! [X: nat > rat,Y: nat > rat] :
% 8.58/4.90 ( ( if_nat_rat @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 8.58/4.90 ! [X: option_num,Y: option_num] :
% 8.58/4.90 ( ( if_option_num @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 8.58/4.90 ! [X: option_num,Y: option_num] :
% 8.58/4.90 ( ( if_option_num @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 8.58/4.90 ! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] :
% 8.58/4.90 ( ( if_Sum_sum_nat_nat @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 8.58/4.90 ! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] :
% 8.58/4.90 ( ( if_Sum_sum_nat_nat @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 8.58/4.90 ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 8.58/4.90 ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 8.58/4.90 ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 8.58/4.90 ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 8.58/4.90 ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 8.58/4.90 ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 8.58/4.90 ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 8.58/4.90 ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
% 8.58/4.90 ! [X: nat > int > int,Y: nat > int > int] :
% 8.58/4.90 ( ( if_nat_int_int @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
% 8.58/4.90 ! [X: nat > int > int,Y: nat > int > int] :
% 8.58/4.90 ( ( if_nat_int_int @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
% 8.58/4.90 ! [X: nat > nat > nat,Y: nat > nat > nat] :
% 8.58/4.90 ( ( if_nat_nat_nat @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
% 8.58/4.90 ! [X: nat > nat > nat,Y: nat > nat > nat] :
% 8.58/4.90 ( ( if_nat_nat_nat @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 8.58/4.90 ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
% 8.58/4.90 ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 8.58/4.90 ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
% 8.58/4.90 ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 8.58/4.90 ! [P: $o] :
% 8.58/4.90 ( ( P = $true )
% 8.58/4.90 | ( P = $false ) ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 8.58/4.90 ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 8.58/4.90 ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
% 8.58/4.90 = Y ) ).
% 8.58/4.90
% 8.58/4.90 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 8.58/4.90 ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 8.58/4.90 ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
% 8.58/4.90 = X ) ).
% 8.58/4.90
% 8.58/4.90 thf(conj_0,conjecture,
% 8.58/4.90 ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ y ) ).
% 8.58/4.90 ------------------------
% 8.99/5.37 % (27375)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_27261 for (2995ds/18Mi)
% 8.99/5.38 % (27370)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_27261 for (2995ds/4Mi)
% 8.99/5.38 % (27369)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_27261 for (2995ds/183Mi)
% 8.99/5.38 % (27371)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_27261 for (2995ds/27Mi)
% 8.99/5.38 % (27372)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_27261 for (2995ds/2Mi)
% 8.99/5.38 % (27373)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_27261 for (2995ds/2Mi)
% 8.99/5.38 % (27374)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_27261 for (2995ds/275Mi)
% 8.99/5.38 % (27372)Instruction limit reached!
% 8.99/5.38 % (27372)------------------------------
% 8.99/5.38 % (27372)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.99/5.38 % (27372)Termination reason: Unknown
% 8.99/5.38 % (27372)Termination phase: shuffling
% 8.99/5.38
% 8.99/5.38 % (27372)Memory used [KB]: 10618
% 8.99/5.38 % (27372)Time elapsed: 0.003 s
% 8.99/5.38 % (27372)Instructions burned: 2 (million)
% 8.99/5.38 % (27372)------------------------------
% 8.99/5.38 % (27372)------------------------------
% 8.99/5.38 % (27373)Instruction limit reached!
% 8.99/5.38 % (27373)------------------------------
% 8.99/5.38 % (27373)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.99/5.38 % (27373)Termination reason: Unknown
% 8.99/5.38 % (27373)Termination phase: shuffling
% 8.99/5.38
% 8.99/5.38 % (27373)Memory used [KB]: 10618
% 8.99/5.38 % (27373)Time elapsed: 0.003 s
% 8.99/5.38 % (27373)Instructions burned: 2 (million)
% 8.99/5.38 % (27373)------------------------------
% 8.99/5.38 % (27373)------------------------------
% 8.99/5.38 % (27370)Instruction limit reached!
% 8.99/5.38 % (27370)------------------------------
% 8.99/5.38 % (27370)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.99/5.38 % (27370)Termination reason: Unknown
% 8.99/5.38 % (27370)Termination phase: shuffling
% 8.99/5.38
% 8.99/5.38 % (27370)Memory used [KB]: 10746
% 8.99/5.38 % (27370)Time elapsed: 0.006 s
% 8.99/5.38 % (27370)Instructions burned: 4 (million)
% 8.99/5.38 % (27370)------------------------------
% 8.99/5.38 % (27370)------------------------------
% 8.99/5.39 % (27375)Instruction limit reached!
% 8.99/5.39 % (27375)------------------------------
% 8.99/5.39 % (27375)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.99/5.39 % (27375)Termination reason: Unknown
% 8.99/5.39 % (27375)Termination phase: shuffling
% 8.99/5.39
% 8.99/5.39 % (27375)Memory used [KB]: 12409
% 8.99/5.39 % (27375)Time elapsed: 0.015 s
% 8.99/5.39 % (27375)Instructions burned: 18 (million)
% 8.99/5.39 % (27375)------------------------------
% 8.99/5.39 % (27375)------------------------------
% 9.13/5.40 % (27379)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_27261 for (2995ds/3Mi)
% 9.13/5.40 % (27376)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_27261 for (2995ds/3Mi)
% 9.13/5.40 % (27377)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_27261 for (2995ds/37Mi)
% 9.13/5.40 % (27379)Instruction limit reached!
% 9.13/5.40 % (27379)------------------------------
% 9.13/5.40 % (27379)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.13/5.40 % (27379)Termination reason: Unknown
% 9.13/5.40 % (27379)Termination phase: shuffling
% 9.13/5.40
% 9.13/5.40 % (27379)Memory used [KB]: 10746
% 9.13/5.40 % (27379)Time elapsed: 0.004 s
% 9.13/5.40 % (27379)Instructions burned: 3 (million)
% 9.13/5.40 % (27379)------------------------------
% 9.13/5.40 % (27379)------------------------------
% 9.13/5.40 % (27378)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_27261 for (2995ds/15Mi)
% 9.13/5.40 % (27376)Instruction limit reached!
% 9.13/5.40 % (27376)------------------------------
% 9.13/5.40 % (27376)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.13/5.40 % (27376)Termination reason: Unknown
% 9.13/5.40 % (27376)Termination phase: shuffling
% 9.13/5.40
% 9.13/5.40 % (27376)Memory used [KB]: 10746
% 9.13/5.40 % (27376)Time elapsed: 0.005 s
% 9.13/5.40 % (27376)Instructions burned: 3 (million)
% 9.13/5.40 % (27376)------------------------------
% 9.13/5.40 % (27376)------------------------------
% 9.13/5.41 % (27371)Instruction limit reached!
% 9.13/5.41 % (27371)------------------------------
% 9.13/5.41 % (27371)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.13/5.41 % (27371)Termination reason: Unknown
% 9.13/5.41 % (27371)Termination phase: shuffling
% 9.13/5.41
% 9.13/5.41 % (27371)Memory used [KB]: 12537
% 9.13/5.41 % (27371)Time elapsed: 0.029 s
% 9.13/5.41 % (27371)Instructions burned: 28 (million)
% 9.13/5.41 % (27371)------------------------------
% 9.13/5.41 % (27371)------------------------------
% 9.13/5.41 % (27380)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_27261 for (2995ds/1041Mi)
% 9.13/5.42 % (27378)Instruction limit reached!
% 9.13/5.42 % (27378)------------------------------
% 9.13/5.42 % (27378)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.13/5.42 % (27378)Termination reason: Unknown
% 9.13/5.42 % (27378)Termination phase: shuffling
% 9.13/5.42
% 9.13/5.42 % (27378)Memory used [KB]: 12409
% 9.13/5.42 % (27378)Time elapsed: 0.021 s
% 9.13/5.42 % (27378)Instructions burned: 16 (million)
% 9.13/5.42 % (27378)------------------------------
% 9.13/5.42 % (27378)------------------------------
% 9.13/5.42 % (27381)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_27261 for (2995ds/7Mi)
% 9.13/5.43 % (27382)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_27261 for (2994ds/16Mi)
% 9.13/5.43 % (27377)Instruction limit reached!
% 9.13/5.43 % (27377)------------------------------
% 9.13/5.43 % (27377)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.13/5.43 % (27377)Termination reason: Unknown
% 9.13/5.43 % (27377)Termination phase: shuffling
% 9.13/5.43
% 9.13/5.43 % (27377)Memory used [KB]: 12665
% 9.13/5.43 % (27377)Time elapsed: 0.032 s
% 9.13/5.43 % (27377)Instructions burned: 38 (million)
% 9.13/5.43 % (27377)------------------------------
% 9.13/5.43 % (27377)------------------------------
% 9.13/5.43 % (27381)Instruction limit reached!
% 9.13/5.43 % (27381)------------------------------
% 9.13/5.43 % (27381)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.13/5.43 % (27381)Termination reason: Unknown
% 9.13/5.43 % (27381)Termination phase: shuffling
% 9.13/5.43
% 9.13/5.43 % (27381)Memory used [KB]: 11001
% 9.13/5.43 % (27381)Time elapsed: 0.012 s
% 9.13/5.43 % (27381)Instructions burned: 7 (million)
% 9.13/5.43 % (27381)------------------------------
% 9.13/5.43 % (27381)------------------------------
% 9.13/5.44 % (27383)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_27261 for (2994ds/3Mi)
% 9.13/5.44 % (27383)Instruction limit reached!
% 9.13/5.44 % (27383)------------------------------
% 9.13/5.44 % (27383)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.13/5.44 % (27383)Termination reason: Unknown
% 9.13/5.44 % (27383)Termination phase: shuffling
% 9.13/5.44
% 9.13/5.44 % (27383)Memory used [KB]: 10618
% 9.13/5.44 % (27383)Time elapsed: 0.005 s
% 9.13/5.44 % (27383)Instructions burned: 3 (million)
% 9.13/5.44 % (27383)------------------------------
% 9.13/5.44 % (27383)------------------------------
% 9.13/5.45 % (27382)Instruction limit reached!
% 9.13/5.45 % (27382)------------------------------
% 9.13/5.45 % (27382)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.13/5.45 % (27382)Termination reason: Unknown
% 9.13/5.45 % (27382)Termination phase: shuffling
% 9.13/5.45
% 9.13/5.45 % (27382)Memory used [KB]: 12409
% 9.13/5.45 % (27382)Time elapsed: 0.022 s
% 9.13/5.45 % (27382)Instructions burned: 16 (million)
% 9.13/5.45 % (27382)------------------------------
% 9.13/5.45 % (27382)------------------------------
% 9.13/5.45 % (27384)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_27261 for (2994ds/3Mi)
% 9.13/5.45 % (27385)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_27261 for (2994ds/7Mi)
% 9.13/5.45 % (27384)Instruction limit reached!
% 9.13/5.45 % (27384)------------------------------
% 9.13/5.45 % (27384)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.13/5.45 % (27384)Termination reason: Unknown
% 9.13/5.45 % (27384)Termination phase: shuffling
% 9.13/5.45
% 9.13/5.45 % (27384)Memory used [KB]: 10746
% 9.13/5.45 % (27384)Time elapsed: 0.005 s
% 9.13/5.45 % (27384)Instructions burned: 3 (million)
% 9.13/5.45 % (27384)------------------------------
% 9.13/5.45 % (27384)------------------------------
% 9.13/5.46 % (27385)Instruction limit reached!
% 9.13/5.46 % (27385)------------------------------
% 9.13/5.46 % (27385)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.13/5.46 % (27385)Termination reason: Unknown
% 9.13/5.46 % (27385)Termination phase: shuffling
% 9.13/5.46
% 9.13/5.46 % (27385)Memory used [KB]: 11001
% 9.13/5.46 % (27385)Time elapsed: 0.011 s
% 9.13/5.46 % (27385)Instructions burned: 7 (million)
% 9.13/5.46 % (27385)------------------------------
% 9.13/5.46 % (27385)------------------------------
% 9.13/5.46 % (27386)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_27261 for (2994ds/3Mi)
% 9.13/5.46 % (27387)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_27261 for (2994ds/4Mi)
% 9.13/5.47 % (27386)Instruction limit reached!
% 9.13/5.47 % (27386)------------------------------
% 9.13/5.47 % (27386)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.13/5.47 % (27386)Termination reason: Unknown
% 9.13/5.47 % (27386)Termination phase: shuffling
% 9.13/5.47
% 9.13/5.47 % (27386)Memory used [KB]: 10746
% 9.13/5.47 % (27386)Time elapsed: 0.005 s
% 9.13/5.47 % (27386)Instructions burned: 3 (million)
% 9.13/5.47 % (27386)------------------------------
% 9.13/5.47 % (27386)------------------------------
% 9.13/5.47 % (27387)Instruction limit reached!
% 9.13/5.47 % (27387)------------------------------
% 9.13/5.47 % (27387)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.13/5.47 % (27387)Termination reason: Unknown
% 9.13/5.47 % (27387)Termination phase: shuffling
% 9.13/5.47
% 9.13/5.47 % (27387)Memory used [KB]: 10746
% 9.13/5.47 % (27387)Time elapsed: 0.005 s
% 9.13/5.47 % (27387)Instructions burned: 4 (million)
% 9.13/5.47 % (27387)------------------------------
% 9.13/5.47 % (27387)------------------------------
% 9.13/5.47 % (27388)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_27261 for (2994ds/18Mi)
% 9.13/5.48 % (27389)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_27261 for (2994ds/710Mi)
% 9.13/5.48 % (27390)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_27261 for (2994ds/6Mi)
% 9.13/5.48 % (27391)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_27261 for (2994ds/902Mi)
% 9.13/5.48 % (27390)Instruction limit reached!
% 9.13/5.48 % (27390)------------------------------
% 9.13/5.48 % (27390)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.13/5.48 % (27390)Termination reason: Unknown
% 9.13/5.48 % (27390)Termination phase: shuffling
% 9.13/5.48
% 9.13/5.48 % (27390)Memory used [KB]: 10874
% 9.13/5.48 % (27390)Time elapsed: 0.007 s
% 9.13/5.48 % (27390)Instructions burned: 6 (million)
% 9.13/5.48 % (27390)------------------------------
% 9.13/5.48 % (27390)------------------------------
% 9.13/5.48 % (27388)Instruction limit reached!
% 9.13/5.48 % (27388)------------------------------
% 9.13/5.48 % (27388)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.13/5.48 % (27388)Termination reason: Unknown
% 9.13/5.48 % (27388)Termination phase: shuffling
% 9.13/5.48
% 9.13/5.48 % (27388)Memory used [KB]: 12409
% 9.13/5.48 % (27388)Time elapsed: 0.016 s
% 9.13/5.48 % (27388)Instructions burned: 18 (million)
% 9.13/5.48 % (27388)------------------------------
% 9.13/5.48 % (27388)------------------------------
% 9.13/5.48 % (27369)Instruction limit reached!
% 9.13/5.48 % (27369)------------------------------
% 9.13/5.48 % (27369)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.13/5.48 % (27369)Termination reason: Unknown
% 9.13/5.48 % (27369)Termination phase: shuffling
% 9.13/5.48
% 9.13/5.48 % (27369)Memory used [KB]: 14583
% 9.13/5.48 % (27369)Time elapsed: 0.108 s
% 9.13/5.48 % (27369)Instructions burned: 183 (million)
% 9.13/5.48 % (27369)------------------------------
% 9.13/5.48 % (27369)------------------------------
% 9.13/5.49 % (27392)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_27261 for (2994ds/21Mi)
% 9.13/5.50 % (27393)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_27261 for (2994ds/5Mi)
% 9.65/5.50 % (27394)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_27261 for (2994ds/6Mi)
% 9.65/5.50 % (27393)Instruction limit reached!
% 9.65/5.50 % (27393)------------------------------
% 9.65/5.50 % (27393)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.65/5.50 % (27393)Termination reason: Unknown
% 9.65/5.50 % (27393)Termination phase: shuffling
% 9.65/5.50
% 9.65/5.50 % (27393)Memory used [KB]: 10874
% 9.65/5.50 % (27393)Time elapsed: 0.005 s
% 9.65/5.50 % (27393)Instructions burned: 5 (million)
% 9.65/5.50 % (27393)------------------------------
% 9.65/5.50 % (27393)------------------------------
% 9.65/5.50 % (27394)Instruction limit reached!
% 9.65/5.50 % (27394)------------------------------
% 9.65/5.50 % (27394)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.65/5.50 % (27394)Termination reason: Unknown
% 9.65/5.50 % (27394)Termination phase: shuffling
% 9.65/5.50
% 9.65/5.50 % (27394)Memory used [KB]: 10874
% 9.65/5.50 % (27394)Time elapsed: 0.006 s
% 9.65/5.50 % (27394)Instructions burned: 6 (million)
% 9.65/5.50 % (27394)------------------------------
% 9.65/5.50 % (27394)------------------------------
% 9.65/5.51 % (27392)Instruction limit reached!
% 9.65/5.51 % (27392)------------------------------
% 9.65/5.51 % (27392)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.65/5.51 % (27392)Termination reason: Unknown
% 9.65/5.51 % (27392)Termination phase: shuffling
% 9.65/5.51
% 9.65/5.51 % (27392)Memory used [KB]: 12409
% 9.65/5.51 % (27392)Time elapsed: 0.015 s
% 9.65/5.51 % (27392)Instructions burned: 21 (million)
% 9.65/5.51 % (27392)------------------------------
% 9.65/5.51 % (27392)------------------------------
% 9.65/5.51 % (27395)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_27261 for (2994ds/377Mi)
% 9.65/5.51 % (27396)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_27261 for (2994ds/779Mi)
% 9.65/5.52 % (27397)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_27261 for (2994ds/19Mi)
% 9.65/5.53 % (27374)Instruction limit reached!
% 9.65/5.53 % (27374)------------------------------
% 9.65/5.53 % (27374)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.65/5.53 % (27374)Termination reason: Unknown
% 9.65/5.53 % (27374)Termination phase: shuffling
% 9.65/5.53
% 9.65/5.53 % (27374)Memory used [KB]: 15863
% 9.65/5.53 % (27374)Time elapsed: 0.149 s
% 9.65/5.53 % (27374)Instructions burned: 276 (million)
% 9.65/5.53 % (27374)------------------------------
% 9.65/5.53 % (27374)------------------------------
% 9.65/5.53 % (27397)Instruction limit reached!
% 9.65/5.53 % (27397)------------------------------
% 9.65/5.53 % (27397)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.65/5.53 % (27397)Termination reason: Unknown
% 9.65/5.53 % (27397)Termination phase: shuffling
% 9.65/5.53
% 9.65/5.53 % (27397)Memory used [KB]: 12409
% 9.65/5.53 % (27397)Time elapsed: 0.014 s
% 9.65/5.53 % (27397)Instructions burned: 20 (million)
% 9.65/5.53 % (27397)------------------------------
% 9.65/5.53 % (27397)------------------------------
% 9.65/5.54 % (27398)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_27261 for (2993ds/879Mi)
% 9.65/5.54 % (27399)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_27261 for (2993ds/17Mi)
% 9.65/5.55 % (27399)Instruction limit reached!
% 9.65/5.55 % (27399)------------------------------
% 9.65/5.55 % (27399)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.65/5.55 % (27399)Termination reason: Unknown
% 9.65/5.55 % (27399)Termination phase: shuffling
% 9.65/5.55
% 9.65/5.55 % (27399)Memory used [KB]: 12409
% 9.65/5.55 % (27399)Time elapsed: 0.014 s
% 9.65/5.55 % (27399)Instructions burned: 19 (million)
% 9.65/5.55 % (27399)------------------------------
% 9.65/5.55 % (27399)------------------------------
% 9.65/5.57 % (27400)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_27261 for (2993ds/3Mi)
% 9.65/5.57 % (27400)Instruction limit reached!
% 9.65/5.57 % (27400)------------------------------
% 9.65/5.57 % (27400)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.65/5.57 % (27400)Termination reason: Unknown
% 9.65/5.57 % (27400)Termination phase: shuffling
% 9.65/5.57
% 9.65/5.57 % (27400)Memory used [KB]: 10746
% 9.65/5.57 % (27400)Time elapsed: 0.004 s
% 9.65/5.57 % (27400)Instructions burned: 3 (million)
% 9.65/5.57 % (27400)------------------------------
% 9.65/5.57 % (27400)------------------------------
% 9.65/5.58 % (27401)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_27261 for (2993ds/30Mi)
% 9.65/5.59 % (27401)Instruction limit reached!
% 9.65/5.59 % (27401)------------------------------
% 9.65/5.59 % (27401)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.65/5.59 % (27401)Termination reason: Unknown
% 9.65/5.59 % (27401)Termination phase: shuffling
% 9.65/5.59
% 9.65/5.59 % (27401)Memory used [KB]: 12537
% 9.65/5.59 % (27401)Time elapsed: 0.018 s
% 9.65/5.59 % (27401)Instructions burned: 32 (million)
% 9.65/5.59 % (27401)------------------------------
% 9.65/5.59 % (27401)------------------------------
% 9.96/5.61 % (27402)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_27261 for (2993ds/127Mi)
% 9.96/5.65 % (27395)Instruction limit reached!
% 9.96/5.65 % (27395)------------------------------
% 9.96/5.65 % (27395)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.96/5.65 % (27395)Termination reason: Unknown
% 9.96/5.65 % (27395)Termination phase: shuffling
% 9.96/5.65
% 9.96/5.65 % (27395)Memory used [KB]: 17014
% 9.96/5.65 % (27395)Time elapsed: 0.143 s
% 9.96/5.65 % (27395)Instructions burned: 378 (million)
% 9.96/5.65 % (27395)------------------------------
% 9.96/5.65 % (27395)------------------------------
% 9.96/5.65 % (27402)Instruction limit reached!
% 9.96/5.65 % (27402)------------------------------
% 9.96/5.65 % (27402)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.96/5.65 % (27402)Termination reason: Unknown
% 9.96/5.65 % (27402)Termination phase: shuffling
% 9.96/5.65
% 9.96/5.65 % (27402)Memory used [KB]: 13944
% 9.96/5.65 % (27402)Time elapsed: 0.050 s
% 9.96/5.65 % (27402)Instructions burned: 128 (million)
% 9.96/5.65 % (27402)------------------------------
% 9.96/5.65 % (27402)------------------------------
% 9.96/5.66 % (27404)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_27261 for (2992ds/3Mi)
% 9.96/5.67 % (27403)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_27261 for (2992ds/100Mi)
% 9.96/5.67 % (27404)Instruction limit reached!
% 9.96/5.67 % (27404)------------------------------
% 9.96/5.67 % (27404)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.96/5.67 % (27404)Termination reason: Unknown
% 9.96/5.67 % (27404)Termination phase: shuffling
% 9.96/5.67
% 9.96/5.67 % (27404)Memory used [KB]: 10746
% 9.96/5.67 % (27404)Time elapsed: 0.004 s
% 9.96/5.67 % (27404)Instructions burned: 3 (million)
% 9.96/5.67 % (27404)------------------------------
% 9.96/5.67 % (27404)------------------------------
% 9.96/5.68 % (27405)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_27261 for (2992ds/20Mi)
% 9.96/5.69 % (27405)Instruction limit reached!
% 9.96/5.69 % (27405)------------------------------
% 9.96/5.69 % (27405)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.96/5.69 % (27405)Termination reason: Unknown
% 9.96/5.69 % (27405)Termination phase: shuffling
% 9.96/5.69
% 9.96/5.69 % (27405)Memory used [KB]: 12409
% 9.96/5.69 % (27405)Time elapsed: 0.015 s
% 9.96/5.69 % (27405)Instructions burned: 21 (million)
% 9.96/5.69 % (27405)------------------------------
% 9.96/5.69 % (27405)------------------------------
% 10.22/5.70 % (27406)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_27261 for (2992ds/86Mi)
% 10.22/5.70 % (27403)Instruction limit reached!
% 10.22/5.70 % (27403)------------------------------
% 10.22/5.70 % (27403)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.22/5.70 % (27403)Termination reason: Unknown
% 10.22/5.70 % (27403)Termination phase: shuffling
% 10.22/5.70
% 10.22/5.70 % (27403)Memory used [KB]: 13560
% 10.22/5.70 % (27403)Time elapsed: 0.039 s
% 10.22/5.70 % (27403)Instructions burned: 103 (million)
% 10.22/5.70 % (27403)------------------------------
% 10.22/5.70 % (27403)------------------------------
% 10.22/5.71 % (27380)Instruction limit reached!
% 10.22/5.71 % (27380)------------------------------
% 10.22/5.71 % (27380)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.22/5.71 % (27380)Termination reason: Unknown
% 10.22/5.71 % (27380)Termination phase: shuffling
% 10.22/5.71
% 10.22/5.71 % (27380)Memory used [KB]: 32494
% 10.22/5.71 % (27380)Time elapsed: 0.305 s
% 10.22/5.71 % (27380)Instructions burned: 1044 (million)
% 10.22/5.71 % (27380)------------------------------
% 10.22/5.71 % (27380)------------------------------
% 10.22/5.71 % (27407)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_27261 for (2991ds/107Mi)
% 10.22/5.73 % (27408)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_27261 for (2991ds/39Mi)
% 10.22/5.74 % (27406)Instruction limit reached!
% 10.22/5.74 % (27406)------------------------------
% 10.22/5.74 % (27406)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.22/5.74 % (27406)Termination reason: Unknown
% 10.22/5.74 % (27406)Termination phase: shuffling
% 10.22/5.74
% 10.22/5.74 % (27406)Memory used [KB]: 13304
% 10.22/5.74 % (27406)Time elapsed: 0.036 s
% 10.22/5.74 % (27406)Instructions burned: 87 (million)
% 10.22/5.74 % (27406)------------------------------
% 10.22/5.74 % (27406)------------------------------
% 10.22/5.74 % (27389)Instruction limit reached!
% 10.22/5.74 % (27389)------------------------------
% 10.22/5.74 % (27389)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.22/5.74 % (27389)Termination reason: Unknown
% 10.22/5.74 % (27389)Termination phase: Property scanning
% 10.22/5.74
% 10.22/5.74 % (27389)Memory used [KB]: 19957
% 10.22/5.74 % (27389)Time elapsed: 0.261 s
% 10.22/5.74 % (27389)Instructions burned: 710 (million)
% 10.22/5.74 % (27389)------------------------------
% 10.22/5.74 % (27389)------------------------------
% 10.22/5.74 % (27408)Instruction limit reached!
% 10.22/5.74 % (27408)------------------------------
% 10.22/5.74 % (27408)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.22/5.74 % (27408)Termination reason: Unknown
% 10.22/5.74 % (27408)Termination phase: shuffling
% 10.22/5.74
% 10.22/5.74 % (27408)Memory used [KB]: 12665
% 10.22/5.74 % (27408)Time elapsed: 0.020 s
% 10.22/5.74 % (27408)Instructions burned: 42 (million)
% 10.22/5.74 % (27408)------------------------------
% 10.22/5.74 % (27408)------------------------------
% 10.22/5.75 % (27409)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_27261 for (2991ds/448Mi)
% 10.22/5.75 % (27410)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_27261 for (2991ds/46Mi)
% 10.22/5.76 % (27411)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_27261 for (2991ds/98Mi)
% 10.22/5.76 % (27396)Instruction limit reached!
% 10.22/5.76 % (27396)------------------------------
% 10.22/5.76 % (27396)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.22/5.76 % (27396)Termination reason: Unknown
% 10.22/5.76 % (27396)Termination phase: shuffling
% 10.22/5.76
% 10.22/5.76 % (27396)Memory used [KB]: 24818
% 10.22/5.76 % (27396)Time elapsed: 0.247 s
% 10.22/5.76 % (27396)Instructions burned: 779 (million)
% 10.22/5.76 % (27396)------------------------------
% 10.22/5.76 % (27396)------------------------------
% 10.22/5.76 % (27407)Instruction limit reached!
% 10.22/5.76 % (27407)------------------------------
% 10.22/5.76 % (27407)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.22/5.76 % (27407)Termination reason: Unknown
% 10.22/5.76 % (27407)Termination phase: shuffling
% 10.22/5.76
% 10.22/5.76 % (27407)Memory used [KB]: 13560
% 10.22/5.76 % (27407)Time elapsed: 0.046 s
% 10.22/5.76 % (27407)Instructions burned: 107 (million)
% 10.22/5.76 % (27407)------------------------------
% 10.22/5.76 % (27407)------------------------------
% 10.22/5.77 % (27412)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_27261 for (2991ds/507Mi)
% 10.22/5.77 % (27413)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_27261 for (2991ds/149Mi)
% 10.22/5.77 % (27410)Instruction limit reached!
% 10.22/5.77 % (27410)------------------------------
% 10.22/5.77 % (27410)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.22/5.77 % (27410)Termination reason: Unknown
% 10.22/5.77 % (27410)Termination phase: shuffling
% 10.22/5.77
% 10.22/5.77 % (27410)Memory used [KB]: 12792
% 10.22/5.77 % (27410)Time elapsed: 0.025 s
% 10.22/5.77 % (27410)Instructions burned: 48 (million)
% 10.22/5.77 % (27410)------------------------------
% 10.22/5.77 % (27410)------------------------------
% 10.22/5.79 % (27414)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_27261 for (2991ds/39Mi)
% 10.22/5.79 % (27411)Instruction limit reached!
% 10.22/5.79 % (27411)------------------------------
% 10.22/5.79 % (27411)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.22/5.79 % (27411)Termination reason: Unknown
% 10.22/5.79 % (27411)Termination phase: shuffling
% 10.22/5.79
% 10.22/5.79 % (27411)Memory used [KB]: 13560
% 10.22/5.79 % (27411)Time elapsed: 0.036 s
% 10.22/5.79 % (27411)Instructions burned: 100 (million)
% 10.22/5.79 % (27411)------------------------------
% 10.22/5.79 % (27411)------------------------------
% 11.32/5.80 % (27391)Instruction limit reached!
% 11.32/5.80 % (27391)------------------------------
% 11.32/5.80 % (27391)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.32/5.80 % (27391)Termination reason: Unknown
% 11.32/5.80 % (27391)Termination phase: Property scanning
% 11.32/5.80
% 11.32/5.80 % (27391)Memory used [KB]: 19957
% 11.32/5.80 % (27391)Time elapsed: 0.318 s
% 11.32/5.80 % (27391)Instructions burned: 904 (million)
% 11.32/5.80 % (27391)------------------------------
% 11.32/5.80 % (27391)------------------------------
% 11.32/5.80 % (27415)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_27261 for (2991ds/6Mi)
% 11.32/5.81 % (27414)Instruction limit reached!
% 11.32/5.81 % (27414)------------------------------
% 11.32/5.81 % (27414)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.32/5.81 % (27414)Termination reason: Unknown
% 11.32/5.81 % (27414)Termination phase: shuffling
% 11.32/5.81
% 11.32/5.81 % (27414)Memory used [KB]: 12665
% 11.32/5.81 % (27414)Time elapsed: 0.022 s
% 11.32/5.81 % (27414)Instructions burned: 39 (million)
% 11.32/5.81 % (27414)------------------------------
% 11.32/5.81 % (27414)------------------------------
% 11.32/5.81 % (27415)Instruction limit reached!
% 11.32/5.81 % (27415)------------------------------
% 11.32/5.81 % (27415)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.32/5.81 % (27415)Termination reason: Unknown
% 11.32/5.81 % (27415)Termination phase: shuffling
% 11.32/5.81
% 11.32/5.81 % (27415)Memory used [KB]: 10874
% 11.32/5.81 % (27415)Time elapsed: 0.007 s
% 11.32/5.81 % (27415)Instructions burned: 6 (million)
% 11.32/5.81 % (27415)------------------------------
% 11.32/5.81 % (27415)------------------------------
% 11.32/5.81 % (27416)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_27261 for (2991ds/1087Mi)
% 11.32/5.82 % (27418)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_27261 for (2991ds/44Mi)
% 11.32/5.82 % (27417)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_27261 for (2991ds/47Mi)
% 11.32/5.83 % (27413)Instruction limit reached!
% 11.32/5.83 % (27413)------------------------------
% 11.32/5.83 % (27413)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.32/5.83 % (27413)Termination reason: Unknown
% 11.32/5.83 % (27413)Termination phase: shuffling
% 11.32/5.83
% 11.32/5.83 % (27413)Memory used [KB]: 14200
% 11.32/5.83 % (27413)Time elapsed: 0.062 s
% 11.32/5.83 % (27413)Instructions burned: 151 (million)
% 11.32/5.83 % (27413)------------------------------
% 11.32/5.83 % (27413)------------------------------
% 11.32/5.84 % (27418)Instruction limit reached!
% 11.32/5.84 % (27418)------------------------------
% 11.32/5.84 % (27418)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.32/5.84 % (27418)Termination reason: Unknown
% 11.32/5.84 % (27418)Termination phase: shuffling
% 11.32/5.84
% 11.32/5.84 % (27418)Memory used [KB]: 12792
% 11.32/5.84 % (27418)Time elapsed: 0.021 s
% 11.32/5.84 % (27418)Instructions burned: 45 (million)
% 11.32/5.84 % (27418)------------------------------
% 11.32/5.84 % (27418)------------------------------
% 11.32/5.84 % (27417)Instruction limit reached!
% 11.32/5.84 % (27417)------------------------------
% 11.32/5.84 % (27417)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.32/5.84 % (27417)Termination reason: Unknown
% 11.32/5.84 % (27417)Termination phase: shuffling
% 11.32/5.84
% 11.32/5.84 % (27417)Memory used [KB]: 12792
% 11.32/5.84 % (27417)Time elapsed: 0.024 s
% 11.32/5.84 % (27417)Instructions burned: 49 (million)
% 11.32/5.84 % (27417)------------------------------
% 11.32/5.84 % (27417)------------------------------
% 11.32/5.84 % (27398)Instruction limit reached!
% 11.32/5.84 % (27398)------------------------------
% 11.32/5.84 % (27398)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.32/5.84 % (27398)Termination reason: Unknown
% 11.32/5.84 % (27398)Termination phase: Property scanning
% 11.32/5.84
% 11.32/5.84 % (27398)Memory used [KB]: 19829
% 11.32/5.84 % (27398)Time elapsed: 0.304 s
% 11.32/5.84 % (27398)Instructions burned: 881 (million)
% 11.32/5.84 % (27398)------------------------------
% 11.32/5.84 % (27398)------------------------------
% 11.32/5.84 % (27419)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_27261 for (2990ds/432Mi)
% 11.32/5.85 % (27420)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_27261 for (2990ds/62Mi)
% 11.32/5.85 % (27421)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_27261 for (2990ds/31Mi)
% 11.32/5.85 % (27422)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_27261 for (2990ds/96Mi)
% 11.32/5.87 % (27421)Instruction limit reached!
% 11.32/5.87 % (27421)------------------------------
% 11.32/5.87 % (27421)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.32/5.87 % (27421)Termination reason: Unknown
% 11.32/5.87 % (27421)Termination phase: shuffling
% 11.32/5.87
% 11.32/5.87 % (27421)Memory used [KB]: 12537
% 11.32/5.87 % (27421)Time elapsed: 0.018 s
% 11.32/5.87 % (27421)Instructions burned: 31 (million)
% 11.32/5.87 % (27421)------------------------------
% 11.32/5.87 % (27421)------------------------------
% 11.32/5.87 % (27420)Instruction limit reached!
% 11.32/5.87 % (27420)------------------------------
% 11.32/5.87 % (27420)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.32/5.87 % (27420)Termination reason: Unknown
% 11.32/5.87 % (27420)Termination phase: shuffling
% 11.32/5.87
% 11.32/5.87 % (27420)Memory used [KB]: 13048
% 11.32/5.87 % (27420)Time elapsed: 0.026 s
% 11.32/5.87 % (27420)Instructions burned: 63 (million)
% 11.32/5.87 % (27420)------------------------------
% 11.32/5.87 % (27420)------------------------------
% 11.32/5.88 % (27423)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_27261 for (2990ds/453Mi)
% 11.32/5.88 % (27409)Instruction limit reached!
% 11.32/5.88 % (27409)------------------------------
% 11.32/5.88 % (27409)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.32/5.88 % (27409)Termination reason: Unknown
% 11.32/5.88 % (27409)Termination phase: shuffling
% 11.32/5.88
% 11.32/5.88 % (27409)Memory used [KB]: 20468
% 11.32/5.88 % (27409)Time elapsed: 0.135 s
% 11.32/5.88 % (27409)Instructions burned: 448 (million)
% 11.32/5.88 % (27409)------------------------------
% 11.32/5.88 % (27409)------------------------------
% 11.32/5.88 % (27424)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_27261 for (2990ds/71Mi)
% 11.32/5.89 % (27422)Instruction limit reached!
% 11.32/5.89 % (27422)------------------------------
% 11.32/5.89 % (27422)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.32/5.89 % (27422)Termination reason: Unknown
% 11.32/5.89 % (27422)Termination phase: shuffling
% 11.32/5.89
% 11.32/5.89 % (27422)Memory used [KB]: 13432
% 11.32/5.89 % (27422)Time elapsed: 0.040 s
% 11.32/5.89 % (27422)Instructions burned: 98 (million)
% 11.32/5.89 % (27422)------------------------------
% 11.32/5.89 % (27422)------------------------------
% 11.32/5.89 % (27425)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_27261 for (2990ds/83Mi)
% 12.41/5.90 % (27426)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_27261 for (2990ds/11Mi)
% 12.41/5.91 % (27424)Instruction limit reached!
% 12.41/5.91 % (27424)------------------------------
% 12.41/5.91 % (27424)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.41/5.91 % (27424)Termination reason: Unknown
% 12.41/5.91 % (27424)Termination phase: shuffling
% 12.41/5.91
% 12.41/5.91 % (27424)Memory used [KB]: 13048
% 12.41/5.91 % (27424)Time elapsed: 0.031 s
% 12.41/5.91 % (27424)Instructions burned: 71 (million)
% 12.41/5.91 % (27424)------------------------------
% 12.41/5.91 % (27424)------------------------------
% 12.41/5.91 % (27426)Instruction limit reached!
% 12.41/5.91 % (27426)------------------------------
% 12.41/5.91 % (27426)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.41/5.91 % (27426)Termination reason: Unknown
% 12.41/5.91 % (27426)Termination phase: shuffling
% 12.41/5.91
% 12.41/5.91 % (27426)Memory used [KB]: 12409
% 12.41/5.91 % (27426)Time elapsed: 0.012 s
% 12.41/5.91 % (27426)Instructions burned: 12 (million)
% 12.41/5.91 % (27426)------------------------------
% 12.41/5.91 % (27426)------------------------------
% 12.41/5.92 % (27427)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_27261 for (2990ds/41Mi)
% 12.41/5.92 % (27428)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_27261 for (2990ds/6Mi)
% 12.41/5.93 % (27425)Instruction limit reached!
% 12.41/5.93 % (27425)------------------------------
% 12.41/5.93 % (27425)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.41/5.93 % (27425)Termination reason: Unknown
% 12.41/5.93 % (27425)Termination phase: shuffling
% 12.41/5.93
% 12.41/5.93 % (27425)Memory used [KB]: 13176
% 12.41/5.93 % (27425)Time elapsed: 0.035 s
% 12.41/5.93 % (27425)Instructions burned: 83 (million)
% 12.41/5.93 % (27425)------------------------------
% 12.41/5.93 % (27425)------------------------------
% 12.41/5.93 % (27428)Instruction limit reached!
% 12.41/5.93 % (27428)------------------------------
% 12.41/5.93 % (27428)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.41/5.93 % (27428)Termination reason: Unknown
% 12.41/5.93 % (27428)Termination phase: shuffling
% 12.41/5.93
% 12.41/5.93 % (27428)Memory used [KB]: 10874
% 12.41/5.93 % (27428)Time elapsed: 0.006 s
% 12.41/5.93 % (27428)Instructions burned: 6 (million)
% 12.41/5.93 % (27428)------------------------------
% 12.41/5.93 % (27428)------------------------------
% 12.41/5.94 % (27429)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_27261 for (2989ds/150Mi)
% 12.41/5.94 % (27430)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_27261 for (2989ds/343Mi)
% 12.41/5.94 % (27427)Instruction limit reached!
% 12.41/5.94 % (27427)------------------------------
% 12.41/5.94 % (27427)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.41/5.94 % (27427)Termination reason: Unknown
% 12.41/5.94 % (27427)Termination phase: shuffling
% 12.41/5.94
% 12.41/5.94 % (27427)Memory used [KB]: 12665
% 12.41/5.94 % (27427)Time elapsed: 0.020 s
% 12.41/5.94 % (27427)Instructions burned: 42 (million)
% 12.41/5.94 % (27427)------------------------------
% 12.41/5.94 % (27427)------------------------------
% 12.41/5.95 % (27412)Instruction limit reached!
% 12.41/5.95 % (27412)------------------------------
% 12.41/5.95 % (27412)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.41/5.95 % (27412)Termination reason: Unknown
% 12.41/5.95 % (27412)Termination phase: shuffling
% 12.41/5.95
% 12.41/5.95 % (27412)Memory used [KB]: 18293
% 12.41/5.95 % (27412)Time elapsed: 0.179 s
% 12.41/5.95 % (27412)Instructions burned: 508 (million)
% 12.41/5.95 % (27412)------------------------------
% 12.41/5.95 % (27412)------------------------------
% 12.41/5.95 % (27431)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_27261 for (2989ds/109Mi)
% 12.41/5.96 % (27432)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_27261 for (2989ds/253Mi)
% 12.41/5.97 % (27419)Instruction limit reached!
% 12.41/5.97 % (27419)------------------------------
% 12.41/5.97 % (27419)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.41/5.97 % (27419)Termination reason: Unknown
% 12.41/5.97 % (27419)Termination phase: shuffling
% 12.41/5.97
% 12.41/5.97 % (27419)Memory used [KB]: 20212
% 12.41/5.97 % (27419)Time elapsed: 0.131 s
% 12.41/5.97 % (27419)Instructions burned: 433 (million)
% 12.41/5.97 % (27419)------------------------------
% 12.41/5.97 % (27419)------------------------------
% 12.41/5.98 % (27433)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_27261 for (2989ds/140Mi)
% 12.41/5.99 % (27429)Instruction limit reached!
% 12.41/5.99 % (27429)------------------------------
% 12.41/5.99 % (27429)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.41/5.99 % (27429)Termination reason: Unknown
% 12.41/5.99 % (27429)Termination phase: shuffling
% 12.41/5.99
% 12.41/5.99 % (27429)Memory used [KB]: 14072
% 12.41/5.99 % (27429)Time elapsed: 0.058 s
% 12.41/5.99 % (27429)Instructions burned: 152 (million)
% 12.41/5.99 % (27429)------------------------------
% 12.41/5.99 % (27429)------------------------------
% 12.41/6.00 % (27431)Instruction limit reached!
% 12.41/6.00 % (27431)------------------------------
% 12.41/6.00 % (27431)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.41/6.00 % (27431)Termination reason: Unknown
% 12.41/6.00 % (27431)Termination phase: shuffling
% 12.41/6.00
% 12.41/6.00 % (27431)Memory used [KB]: 13688
% 12.41/6.00 % (27431)Time elapsed: 0.044 s
% 12.41/6.00 % (27431)Instructions burned: 110 (million)
% 12.41/6.00 % (27431)------------------------------
% 12.41/6.00 % (27431)------------------------------
% 12.41/6.00 % (27434)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_27261 for (2989ds/614Mi)
% 12.41/6.01 % (27435)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_27261 for (2989ds/202Mi)
% 12.41/6.01 % (27423)Instruction limit reached!
% 12.41/6.01 % (27423)------------------------------
% 12.41/6.01 % (27423)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.41/6.01 % (27423)Termination reason: Unknown
% 12.41/6.01 % (27423)Termination phase: shuffling
% 13.51/6.02
% 13.51/6.02 % (27423)Memory used [KB]: 20596
% 13.51/6.02 % (27423)Time elapsed: 0.137 s
% 13.51/6.02 % (27423)Instructions burned: 455 (million)
% 13.51/6.02 % (27423)------------------------------
% 13.51/6.02 % (27423)------------------------------
% 13.51/6.03 % (27436)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_27261 for (2988ds/339Mi)
% 13.51/6.04 % (27433)Instruction limit reached!
% 13.51/6.04 % (27433)------------------------------
% 13.51/6.04 % (27433)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.51/6.04 % (27433)Termination reason: Unknown
% 13.51/6.04 % (27433)Termination phase: shuffling
% 13.51/6.04
% 13.51/6.04 % (27433)Memory used [KB]: 13944
% 13.51/6.04 % (27433)Time elapsed: 0.054 s
% 13.51/6.04 % (27433)Instructions burned: 142 (million)
% 13.51/6.04 % (27433)------------------------------
% 13.51/6.04 % (27433)------------------------------
% 13.51/6.05 % (27437)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=283:si=on:rtra=on_0 on DTF2THF_27261 for (2988ds/283Mi)
% 13.51/6.05 % (27432)Instruction limit reached!
% 13.51/6.05 % (27432)------------------------------
% 13.51/6.05 % (27432)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.51/6.05 % (27432)Termination reason: Unknown
% 13.51/6.05 % (27432)Termination phase: shuffling
% 13.51/6.05
% 13.51/6.05 % (27432)Memory used [KB]: 15607
% 13.51/6.05 % (27432)Time elapsed: 0.094 s
% 13.51/6.05 % (27432)Instructions burned: 255 (million)
% 13.51/6.05 % (27432)------------------------------
% 13.51/6.05 % (27432)------------------------------
% 13.51/6.06 % (27430)Instruction limit reached!
% 13.51/6.06 % (27430)------------------------------
% 13.51/6.06 % (27430)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.51/6.06 % (27430)Termination reason: Unknown
% 13.51/6.06 % (27430)Termination phase: shuffling
% 13.51/6.06
% 13.51/6.06 % (27430)Memory used [KB]: 16502
% 13.51/6.06 % (27430)Time elapsed: 0.121 s
% 13.51/6.06 % (27430)Instructions burned: 343 (million)
% 13.51/6.06 % (27430)------------------------------
% 13.51/6.06 % (27430)------------------------------
% 13.51/6.06 % (27438)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_27261 for (2988ds/134Mi)
% 13.51/6.07 % (27439)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_27261 for (2988ds/298Mi)
% 13.51/6.08 % (27435)Instruction limit reached!
% 13.51/6.08 % (27435)------------------------------
% 13.51/6.08 % (27435)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.51/6.08 % (27435)Termination reason: Unknown
% 13.51/6.08 % (27435)Termination phase: shuffling
% 13.51/6.08
% 13.51/6.08 % (27435)Memory used [KB]: 14839
% 13.51/6.08 % (27435)Time elapsed: 0.076 s
% 13.51/6.08 % (27435)Instructions burned: 203 (million)
% 13.51/6.08 % (27435)------------------------------
% 13.51/6.08 % (27435)------------------------------
% 13.51/6.09 % (27440)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_27261 for (2988ds/192Mi)
% 13.51/6.11 % (27438)Instruction limit reached!
% 13.51/6.11 % (27438)------------------------------
% 13.51/6.11 % (27438)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.51/6.11 % (27438)Termination reason: Unknown
% 13.51/6.11 % (27438)Termination phase: shuffling
% 13.51/6.11
% 13.51/6.11 % (27438)Memory used [KB]: 13944
% 13.51/6.11 % (27438)Time elapsed: 0.052 s
% 13.51/6.11 % (27438)Instructions burned: 135 (million)
% 13.51/6.11 % (27438)------------------------------
% 13.51/6.11 % (27438)------------------------------
% 14.01/6.13 % (27441)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_27261 for (2987ds/258Mi)
% 14.01/6.13 % (27437)Instruction limit reached!
% 14.01/6.13 % (27437)------------------------------
% 14.01/6.13 % (27437)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.01/6.13 % (27437)Termination reason: Unknown
% 14.01/6.13 % (27437)Termination phase: shuffling
% 14.01/6.13
% 14.01/6.13 % (27437)Memory used [KB]: 18166
% 14.01/6.13 % (27437)Time elapsed: 0.087 s
% 14.01/6.13 % (27437)Instructions burned: 283 (million)
% 14.01/6.13 % (27437)------------------------------
% 14.01/6.13 % (27437)------------------------------
% 14.01/6.14 % (27442)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_27261 for (2987ds/90Mi)
% 14.01/6.16 % (27439)Instruction limit reached!
% 14.01/6.16 % (27439)------------------------------
% 14.01/6.16 % (27439)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.01/6.16 % (27439)Termination reason: Unknown
% 14.01/6.16 % (27439)Termination phase: shuffling
% 14.01/6.16
% 14.01/6.16 % (27439)Memory used [KB]: 18421
% 14.01/6.16 % (27439)Time elapsed: 0.093 s
% 14.01/6.16 % (27439)Instructions burned: 300 (million)
% 14.01/6.16 % (27439)------------------------------
% 14.01/6.16 % (27439)------------------------------
% 14.01/6.16 % (27440)Instruction limit reached!
% 14.01/6.16 % (27440)------------------------------
% 14.01/6.16 % (27440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.01/6.16 % (27440)Termination reason: Unknown
% 14.01/6.16 % (27440)Termination phase: shuffling
% 14.01/6.16
% 14.01/6.16 % (27440)Memory used [KB]: 14711
% 14.01/6.16 % (27440)Time elapsed: 0.072 s
% 14.01/6.16 % (27440)Instructions burned: 193 (million)
% 14.01/6.16 % (27440)------------------------------
% 14.01/6.16 % (27440)------------------------------
% 14.01/6.17 % (27416)Instruction limit reached!
% 14.01/6.17 % (27416)------------------------------
% 14.01/6.17 % (27416)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.01/6.17 % (27416)Termination reason: Unknown
% 14.01/6.17 % (27416)Termination phase: Property scanning
% 14.01/6.17
% 14.01/6.17 % (27416)Memory used [KB]: 19957
% 14.01/6.17 % (27416)Time elapsed: 0.344 s
% 14.01/6.17 % (27416)Instructions burned: 1088 (million)
% 14.01/6.17 % (27416)------------------------------
% 14.01/6.17 % (27416)------------------------------
% 14.01/6.17 % (27443)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_27261 for (2987ds/231Mi)
% 14.01/6.17 % (27444)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_27261 for (2987ds/458Mi)
% 14.01/6.18 % (27445)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_27261 for (2987ds/31Mi)
% 14.01/6.18 % (27442)Instruction limit reached!
% 14.01/6.18 % (27442)------------------------------
% 14.01/6.18 % (27442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.01/6.18 % (27442)Termination reason: Unknown
% 14.01/6.18 % (27442)Termination phase: shuffling
% 14.01/6.18
% 14.01/6.18 % (27442)Memory used [KB]: 13432
% 14.01/6.18 % (27442)Time elapsed: 0.039 s
% 14.01/6.18 % (27442)Instructions burned: 92 (million)
% 14.01/6.18 % (27442)------------------------------
% 14.01/6.18 % (27442)------------------------------
% 14.01/6.18 % (27436)Instruction limit reached!
% 14.01/6.18 % (27436)------------------------------
% 14.01/6.18 % (27436)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.01/6.18 % (27436)Termination reason: Unknown
% 14.01/6.18 % (27436)Termination phase: shuffling
% 14.01/6.18
% 14.01/6.18 % (27436)Memory used [KB]: 16630
% 14.01/6.18 % (27436)Time elapsed: 0.157 s
% 14.01/6.18 % (27436)Instructions burned: 339 (million)
% 14.01/6.18 % (27436)------------------------------
% 14.01/6.18 % (27436)------------------------------
% 14.01/6.19 % (27446)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_27261 for (2987ds/297Mi)
% 14.01/6.19 % (27445)Instruction limit reached!
% 14.01/6.19 % (27445)------------------------------
% 14.01/6.19 % (27445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.01/6.19 % (27445)Termination reason: Unknown
% 14.01/6.19 % (27445)Termination phase: shuffling
% 14.01/6.19
% 14.01/6.19 % (27445)Memory used [KB]: 12537
% 14.01/6.19 % (27445)Time elapsed: 0.019 s
% 14.01/6.19 % (27445)Instructions burned: 33 (million)
% 14.01/6.19 % (27445)------------------------------
% 14.01/6.19 % (27445)------------------------------
% 14.01/6.20 % (27447)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_27261 for (2987ds/987Mi)
% 14.68/6.21 % (27448)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_27261 for (2987ds/182Mi)
% 14.68/6.22 % (27434)Instruction limit reached!
% 14.68/6.22 % (27434)------------------------------
% 14.68/6.22 % (27434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.68/6.22 % (27434)Termination reason: Unknown
% 14.68/6.22 % (27434)Termination phase: Property scanning
% 14.68/6.22
% 14.68/6.22 % (27434)Memory used [KB]: 19829
% 14.68/6.22 % (27434)Time elapsed: 0.214 s
% 14.68/6.22 % (27434)Instructions burned: 614 (million)
% 14.68/6.22 % (27434)------------------------------
% 14.68/6.22 % (27434)------------------------------
% 14.68/6.22 % (27441)Instruction limit reached!
% 14.68/6.22 % (27441)------------------------------
% 14.68/6.22 % (27441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.68/6.22 % (27441)Termination reason: Unknown
% 14.68/6.22 % (27441)Termination phase: shuffling
% 14.68/6.22
% 14.68/6.22 % (27441)Memory used [KB]: 15479
% 14.68/6.22 % (27441)Time elapsed: 0.094 s
% 14.68/6.22 % (27441)Instructions burned: 258 (million)
% 14.68/6.22 % (27441)------------------------------
% 14.68/6.22 % (27441)------------------------------
% 14.68/6.23 % (27449)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_27261 for (2986ds/182Mi)
% 14.68/6.23 % (27450)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_27261 for (2986ds/154Mi)
% 14.68/6.24 % (27443)Instruction limit reached!
% 14.68/6.24 % (27443)------------------------------
% 14.68/6.24 % (27443)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.68/6.24 % (27443)Termination reason: Unknown
% 14.68/6.24 % (27443)Termination phase: shuffling
% 14.68/6.24
% 14.68/6.24 % (27443)Memory used [KB]: 15479
% 14.68/6.24 % (27443)Time elapsed: 0.074 s
% 14.68/6.24 % (27443)Instructions burned: 233 (million)
% 14.68/6.24 % (27443)------------------------------
% 14.68/6.24 % (27443)------------------------------
% 14.68/6.26 % (27451)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_27261 for (2986ds/79Mi)
% 14.68/6.27 % (27448)Instruction limit reached!
% 14.68/6.27 % (27448)------------------------------
% 14.68/6.27 % (27448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.68/6.27 % (27448)Termination reason: Unknown
% 14.68/6.27 % (27448)Termination phase: shuffling
% 14.68/6.27
% 14.68/6.27 % (27448)Memory used [KB]: 14456
% 14.68/6.27 % (27448)Time elapsed: 0.068 s
% 14.68/6.27 % (27448)Instructions burned: 183 (million)
% 14.68/6.27 % (27448)------------------------------
% 14.68/6.27 % (27448)------------------------------
% 14.68/6.28 % (27450)Instruction limit reached!
% 14.68/6.28 % (27450)------------------------------
% 14.68/6.28 % (27450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.68/6.28 % (27450)Termination reason: Unknown
% 14.68/6.28 % (27450)Termination phase: shuffling
% 14.68/6.28
% 14.68/6.28 % (27450)Memory used [KB]: 14328
% 14.68/6.28 % (27450)Time elapsed: 0.051 s
% 14.68/6.28 % (27450)Instructions burned: 156 (million)
% 14.68/6.28 % (27450)------------------------------
% 14.68/6.28 % (27450)------------------------------
% 14.68/6.28 % (27452)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_27261 for (2986ds/216Mi)
% 14.68/6.28 % (27446)Instruction limit reached!
% 14.68/6.28 % (27446)------------------------------
% 14.68/6.28 % (27446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.68/6.28 % (27446)Termination reason: Unknown
% 14.68/6.28 % (27446)Termination phase: shuffling
% 14.68/6.28
% 14.68/6.28 % (27446)Memory used [KB]: 18421
% 14.68/6.28 % (27446)Time elapsed: 0.092 s
% 14.68/6.28 % (27446)Instructions burned: 298 (million)
% 14.68/6.28 % (27446)------------------------------
% 14.68/6.28 % (27446)------------------------------
% 14.68/6.29 % (27451)Instruction limit reached!
% 14.68/6.29 % (27451)------------------------------
% 14.68/6.29 % (27451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.68/6.29 % (27451)Termination reason: Unknown
% 14.68/6.29 % (27451)Termination phase: shuffling
% 14.68/6.29
% 14.68/6.29 % (27451)Memory used [KB]: 13176
% 14.68/6.29 % (27451)Time elapsed: 0.034 s
% 14.68/6.29 % (27451)Instructions burned: 79 (million)
% 14.68/6.29 % (27451)------------------------------
% 14.68/6.29 % (27451)------------------------------
% 14.68/6.29 % (27453)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_27261 for (2986ds/114Mi)
% 14.68/6.30 % (27454)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_27261 for (2986ds/922Mi)
% 15.18/6.30 % (27449)Instruction limit reached!
% 15.18/6.30 % (27449)------------------------------
% 15.18/6.30 % (27449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.18/6.30 % (27449)Termination reason: Unknown
% 15.18/6.30 % (27449)Termination phase: shuffling
% 15.18/6.30
% 15.18/6.30 % (27449)Memory used [KB]: 14583
% 15.18/6.30 % (27449)Time elapsed: 0.069 s
% 15.18/6.30 % (27449)Instructions burned: 182 (million)
% 15.18/6.30 % (27449)------------------------------
% 15.18/6.30 % (27449)------------------------------
% 15.18/6.30 % (27455)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_27261 for (2986ds/182Mi)
% 15.18/6.31 % (27456)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_27261 for (2986ds/164Mi)
% 15.18/6.33 % (27444)Instruction limit reached!
% 15.18/6.33 % (27444)------------------------------
% 15.18/6.33 % (27444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.18/6.33 % (27444)Termination reason: Unknown
% 15.18/6.33 % (27444)Termination phase: shuffling
% 15.18/6.33
% 15.18/6.33 % (27444)Memory used [KB]: 18038
% 15.18/6.33 % (27444)Time elapsed: 0.161 s
% 15.18/6.33 % (27444)Instructions burned: 459 (million)
% 15.18/6.33 % (27444)------------------------------
% 15.18/6.33 % (27444)------------------------------
% 15.18/6.33 % (27453)Instruction limit reached!
% 15.18/6.33 % (27453)------------------------------
% 15.18/6.33 % (27453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.18/6.33 % (27453)Termination reason: Unknown
% 15.18/6.33 % (27453)Termination phase: shuffling
% 15.18/6.33
% 15.18/6.33 % (27453)Memory used [KB]: 13688
% 15.18/6.33 % (27453)Time elapsed: 0.046 s
% 15.18/6.33 % (27453)Instructions burned: 116 (million)
% 15.18/6.33 % (27453)------------------------------
% 15.18/6.33 % (27453)------------------------------
% 15.18/6.34 % (27457)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_27261 for (2985ds/50Mi)
% 15.18/6.35 % (27458)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_27261 for (2985ds/368Mi)
% 15.18/6.35 % (27452)Instruction limit reached!
% 15.18/6.35 % (27452)------------------------------
% 15.18/6.35 % (27452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.18/6.35 % (27452)Termination reason: Unknown
% 15.18/6.35 % (27452)Termination phase: shuffling
% 15.18/6.35
% 15.18/6.35 % (27452)Memory used [KB]: 15223
% 15.18/6.35 % (27452)Time elapsed: 0.069 s
% 15.18/6.35 % (27452)Instructions burned: 218 (million)
% 15.18/6.35 % (27452)------------------------------
% 15.18/6.35 % (27452)------------------------------
% 15.18/6.36 % (27455)Instruction limit reached!
% 15.18/6.36 % (27455)------------------------------
% 15.18/6.36 % (27455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.18/6.36 % (27455)Termination reason: Unknown
% 15.18/6.36 % (27455)Termination phase: shuffling
% 15.18/6.36
% 15.18/6.36 % (27455)Memory used [KB]: 14839
% 15.18/6.36 % (27455)Time elapsed: 0.059 s
% 15.18/6.36 % (27455)Instructions burned: 184 (million)
% 15.18/6.36 % (27455)------------------------------
% 15.18/6.36 % (27455)------------------------------
% 15.18/6.36 % (27456)Instruction limit reached!
% 15.18/6.36 % (27456)------------------------------
% 15.18/6.36 % (27456)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.18/6.36 % (27456)Termination reason: Unknown
% 15.18/6.36 % (27456)Termination phase: shuffling
% 15.18/6.36
% 15.18/6.36 % (27456)Memory used [KB]: 14456
% 15.18/6.36 % (27456)Time elapsed: 0.054 s
% 15.18/6.36 % (27456)Instructions burned: 165 (million)
% 15.18/6.36 % (27456)------------------------------
% 15.18/6.36 % (27456)------------------------------
% 15.18/6.36 % (27459)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_27261 for (2985ds/1021Mi)
% 15.18/6.37 % (27457)Instruction limit reached!
% 15.18/6.37 % (27457)------------------------------
% 15.18/6.37 % (27457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.18/6.37 % (27457)Termination reason: Unknown
% 15.18/6.37 % (27457)Termination phase: shuffling
% 15.18/6.37
% 15.18/6.37 % (27457)Memory used [KB]: 12920
% 15.18/6.37 % (27457)Time elapsed: 0.023 s
% 15.18/6.37 % (27457)Instructions burned: 52 (million)
% 15.18/6.37 % (27457)------------------------------
% 15.18/6.37 % (27457)------------------------------
% 15.18/6.37 % (27460)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_27261 for (2985ds/267Mi)
% 15.18/6.37 % (27461)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_27261 for (2985ds/204Mi)
% 15.18/6.38 % (27462)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_27261 for (2985ds/26Mi)
% 15.18/6.39 % (27462)Instruction limit reached!
% 15.18/6.39 % (27462)------------------------------
% 15.18/6.39 % (27462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.18/6.39 % (27462)Termination reason: Unknown
% 15.18/6.39 % (27462)Termination phase: shuffling
% 15.18/6.39
% 15.18/6.39 % (27462)Memory used [KB]: 12537
% 15.18/6.39 % (27462)Time elapsed: 0.016 s
% 15.18/6.39 % (27462)Instructions burned: 27 (million)
% 15.18/6.39 % (27462)------------------------------
% 15.18/6.39 % (27462)------------------------------
% 15.75/6.40 % (27463)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_27261 for (2985ds/53Mi)
% 15.75/6.42 % (27463)Instruction limit reached!
% 15.75/6.42 % (27463)------------------------------
% 15.75/6.42 % (27463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.75/6.42 % (27463)Termination reason: Unknown
% 15.75/6.42 % (27463)Termination phase: shuffling
% 15.75/6.42
% 15.75/6.42 % (27463)Memory used [KB]: 12920
% 15.75/6.42 % (27463)Time elapsed: 0.024 s
% 15.75/6.42 % (27463)Instructions burned: 56 (million)
% 15.75/6.42 % (27463)------------------------------
% 15.75/6.42 % (27463)------------------------------
% 15.75/6.44 % (27464)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_27261 for (2984ds/4436Mi)
% 15.75/6.44 % (27461)Instruction limit reached!
% 15.75/6.44 % (27461)------------------------------
% 15.75/6.44 % (27461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.75/6.44 % (27461)Termination reason: Unknown
% 15.75/6.44 % (27461)Termination phase: shuffling
% 15.75/6.44
% 15.75/6.44 % (27461)Memory used [KB]: 14839
% 15.75/6.44 % (27461)Time elapsed: 0.075 s
% 15.75/6.44 % (27461)Instructions burned: 205 (million)
% 15.75/6.44 % (27461)------------------------------
% 15.75/6.44 % (27461)------------------------------
% 15.75/6.45 % (27460)Instruction limit reached!
% 15.75/6.45 % (27460)------------------------------
% 15.75/6.45 % (27460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.75/6.45 % (27460)Termination reason: Unknown
% 15.75/6.45 % (27460)Termination phase: shuffling
% 15.75/6.45
% 15.75/6.45 % (27460)Memory used [KB]: 15991
% 15.75/6.45 % (27460)Time elapsed: 0.082 s
% 15.75/6.45 % (27460)Instructions burned: 267 (million)
% 15.75/6.45 % (27460)------------------------------
% 15.75/6.45 % (27460)------------------------------
% 15.75/6.46 % (27465)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_27261 for (2984ds/1539Mi)
% 15.75/6.46 % (27466)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_27261 for (2984ds/408Mi)
% 15.75/6.47 % (27458)Instruction limit reached!
% 15.75/6.47 % (27458)------------------------------
% 15.75/6.47 % (27458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.75/6.47 % (27458)Termination reason: Unknown
% 15.75/6.47 % (27458)Termination phase: shuffling
% 15.75/6.47
% 15.75/6.47 % (27458)Memory used [KB]: 16886
% 15.75/6.47 % (27458)Time elapsed: 0.130 s
% 15.75/6.47 % (27458)Instructions burned: 371 (million)
% 15.75/6.47 % (27458)------------------------------
% 15.75/6.47 % (27458)------------------------------
% 15.75/6.48 % (27447)Instruction limit reached!
% 15.75/6.48 % (27447)------------------------------
% 15.75/6.48 % (27447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.75/6.48 % (27447)Termination reason: Unknown
% 15.75/6.48 % (27447)Termination phase: shuffling
% 15.75/6.48
% 15.75/6.48 % (27447)Memory used [KB]: 31726
% 15.75/6.48 % (27447)Time elapsed: 0.290 s
% 15.75/6.48 % (27447)Instructions burned: 988 (million)
% 15.75/6.48 % (27447)------------------------------
% 15.75/6.48 % (27447)------------------------------
% 15.75/6.49 % (27467)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_27261 for (2984ds/5205Mi)
% 16.41/6.50 % (27468)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_27261 for (2984ds/1913Mi)
% 16.41/6.59 % (27454)Instruction limit reached!
% 16.41/6.59 % (27454)------------------------------
% 16.41/6.59 % (27454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.41/6.59 % (27454)Termination reason: Unknown
% 16.41/6.59 % (27454)Termination phase: Property scanning
% 16.41/6.59
% 16.41/6.59 % (27454)Memory used [KB]: 19957
% 16.41/6.59 % (27454)Time elapsed: 0.297 s
% 16.41/6.59 % (27454)Instructions burned: 924 (million)
% 16.41/6.59 % (27454)------------------------------
% 16.41/6.59 % (27454)------------------------------
% 16.72/6.60 % (27469)lrs+2_1:1_bd=off:cbe=off:piset=equals:sd=1:sgt=20:sos=all:ss=axioms:i=4350:si=on:rtra=on_0 on DTF2THF_27261 for (2983ds/4350Mi)
% 16.72/6.60 % (27466)Instruction limit reached!
% 16.72/6.60 % (27466)------------------------------
% 16.72/6.60 % (27466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.72/6.60 % (27466)Termination reason: Unknown
% 16.72/6.60 % (27466)Termination phase: shuffling
% 16.72/6.60
% 16.72/6.60 % (27466)Memory used [KB]: 17654
% 16.72/6.60 % (27466)Time elapsed: 0.146 s
% 16.72/6.60 % (27466)Instructions burned: 410 (million)
% 16.72/6.60 % (27466)------------------------------
% 16.72/6.60 % (27466)------------------------------
% 16.72/6.62 % (27470)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=68:si=on:rtra=on_0 on DTF2THF_27261 for (2982ds/68Mi)
% 16.72/6.64 % (27470)Instruction limit reached!
% 16.72/6.64 % (27470)------------------------------
% 16.72/6.64 % (27470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.72/6.64 % (27470)Termination reason: Unknown
% 16.72/6.64 % (27470)Termination phase: shuffling
% 16.72/6.64
% 16.72/6.64 % (27470)Memory used [KB]: 13048
% 16.72/6.64 % (27470)Time elapsed: 0.031 s
% 16.72/6.64 % (27470)Instructions burned: 70 (million)
% 16.72/6.64 % (27470)------------------------------
% 16.72/6.64 % (27470)------------------------------
% 16.72/6.66 % (27471)dis+10_1:1_au=on:nwc=5.0:s2a=on:s2at=2.0:sac=on:sd=7:ss=axioms:st=2.0:i=358:si=on:rtra=on_0 on DTF2THF_27261 for (2982ds/358Mi)
% 16.72/6.68 % (27459)Instruction limit reached!
% 16.72/6.68 % (27459)------------------------------
% 16.72/6.68 % (27459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.72/6.68 % (27459)Termination reason: Unknown
% 16.72/6.68 % (27459)Termination phase: Property scanning
% 16.72/6.68
% 16.72/6.68 % (27459)Memory used [KB]: 19957
% 16.72/6.68 % (27459)Time elapsed: 0.320 s
% 16.72/6.68 % (27459)Instructions burned: 1024 (million)
% 16.72/6.68 % (27459)------------------------------
% 16.72/6.68 % (27459)------------------------------
% 16.72/6.69 % (27472)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_27261 for (2982ds/31Mi)
% 17.24/6.71 % (27472)Instruction limit reached!
% 17.24/6.71 % (27472)------------------------------
% 17.24/6.71 % (27472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.24/6.71 % (27472)Termination reason: Unknown
% 17.24/6.71 % (27472)Termination phase: shuffling
% 17.24/6.71
% 17.24/6.71 % (27472)Memory used [KB]: 12537
% 17.24/6.71 % (27472)Time elapsed: 0.019 s
% 17.24/6.71 % (27472)Instructions burned: 33 (million)
% 17.24/6.71 % (27472)------------------------------
% 17.24/6.71 % (27472)------------------------------
% 17.24/6.72 % (27473)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_27261 for (2981ds/279Mi)
% 17.24/6.78 % (27471)Instruction limit reached!
% 17.24/6.78 % (27471)------------------------------
% 17.24/6.78 % (27471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.24/6.78 % (27471)Termination reason: Unknown
% 17.24/6.78 % (27471)Termination phase: shuffling
% 17.24/6.78
% 17.24/6.78 % (27471)Memory used [KB]: 16758
% 17.24/6.78 % (27471)Time elapsed: 0.126 s
% 17.24/6.78 % (27471)Instructions burned: 359 (million)
% 17.24/6.78 % (27471)------------------------------
% 17.24/6.78 % (27471)------------------------------
% 17.24/6.79 % (27474)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_27261 for (2981ds/258Mi)
% 17.40/6.80 % (27473)Instruction limit reached!
% 17.40/6.80 % (27473)------------------------------
% 17.40/6.80 % (27473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.40/6.80 % (27473)Termination reason: Unknown
% 17.40/6.80 % (27473)Termination phase: shuffling
% 17.40/6.80
% 17.40/6.81 % (27473)Memory used [KB]: 18166
% 17.40/6.81 % (27473)Time elapsed: 0.086 s
% 17.40/6.81 % (27473)Instructions burned: 279 (million)
% 17.40/6.81 % (27473)------------------------------
% 17.40/6.81 % (27473)------------------------------
% 17.40/6.82 % (27475)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_27261 for (2980ds/26Mi)
% 17.40/6.83 % (27475)Instruction limit reached!
% 17.40/6.83 % (27475)------------------------------
% 17.40/6.83 % (27475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.40/6.83 % (27475)Termination reason: Unknown
% 17.40/6.83 % (27475)Termination phase: shuffling
% 17.40/6.83
% 17.40/6.83 % (27475)Memory used [KB]: 12409
% 17.40/6.83 % (27475)Time elapsed: 0.016 s
% 17.40/6.83 % (27475)Instructions burned: 26 (million)
% 17.40/6.83 % (27475)------------------------------
% 17.40/6.83 % (27475)------------------------------
% 17.40/6.84 % (27476)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_be_gen:ntd=on:pe=on:i=995:si=on:rtra=on_0 on DTF2THF_27261 for (2980ds/995Mi)
% 17.40/6.88 % (27474)Instruction limit reached!
% 17.40/6.88 % (27474)------------------------------
% 17.40/6.88 % (27474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.40/6.88 % (27474)Termination reason: Unknown
% 17.40/6.88 % (27474)Termination phase: shuffling
% 17.40/6.88
% 17.40/6.88 % (27474)Memory used [KB]: 15479
% 17.40/6.88 % (27474)Time elapsed: 0.093 s
% 17.40/6.88 % (27474)Instructions burned: 258 (million)
% 17.40/6.88 % (27474)------------------------------
% 17.40/6.88 % (27474)------------------------------
% 17.40/6.89 % (27477)dis+10_1:8_amm=sco:cbe=off:hflw=5:hfsq=on:hfsqc=14:hfsql=off:hfsqr=4,1:hud=5:ile=on:lma=on:nm=2:ntd=on:prag=on:sac=on:sp=frequency:i=373:si=on:rtra=on_0 on DTF2THF_27261 for (2980ds/373Mi)
% 17.62/6.94 % (27465)Instruction limit reached!
% 17.62/6.94 % (27465)------------------------------
% 17.62/6.94 % (27465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.62/6.94 % (27465)Termination reason: Unknown
% 17.62/6.94 % (27465)Termination phase: Preprocessing 3
% 17.62/6.94
% 17.62/6.94 % (27465)Memory used [KB]: 26737
% 17.62/6.94 % (27465)Time elapsed: 0.489 s
% 17.62/6.94 % (27465)Instructions burned: 1540 (million)
% 17.62/6.94 % (27465)------------------------------
% 17.62/6.94 % (27465)------------------------------
% 17.62/6.96 % (27478)lrs+1010_1:128_au=on:cnfonf=lazy_pi_sigma_gen:hflw=4:hfsq=on:hfsqc=20:hfsql=off:i=253:si=on:rtra=on_0 on DTF2THF_27261 for (2979ds/253Mi)
% 20.45/7.02 % (27477)Instruction limit reached!
% 20.45/7.02 % (27477)------------------------------
% 20.45/7.02 % (27477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.45/7.02 % (27477)Termination reason: Unknown
% 20.45/7.02 % (27477)Termination phase: shuffling
% 20.45/7.02
% 20.45/7.02 % (27477)Memory used [KB]: 17014
% 20.45/7.03 % (27477)Time elapsed: 0.132 s
% 20.45/7.03 % (27477)Instructions burned: 373 (million)
% 20.45/7.03 % (27477)------------------------------
% 20.45/7.03 % (27477)------------------------------
% 20.45/7.03 % (27478)Instruction limit reached!
% 20.45/7.03 % (27478)------------------------------
% 20.45/7.03 % (27478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.45/7.03 % (27478)Termination reason: Unknown
% 20.45/7.03 % (27478)Termination phase: shuffling
% 20.45/7.03
% 20.45/7.03 % (27478)Memory used [KB]: 15735
% 20.45/7.03 % (27478)Time elapsed: 0.080 s
% 20.45/7.03 % (27478)Instructions burned: 253 (million)
% 20.45/7.03 % (27478)------------------------------
% 20.45/7.03 % (27478)------------------------------
% 20.45/7.04 % (27479)lrs+1002_1:14_au=on:bd=off:e2e=on:fde=unused:sac=on:sos=on:i=832:si=on:rtra=on_0 on DTF2THF_27261 for (2978ds/832Mi)
% 20.45/7.05 % (27480)lrs+1010_1:1_chr=on:cnfonf=lazy_not_be_gen:gs=on:hud=5:plsq=on:plsqc=1:plsqr=5,1:prag=on:rawr=on:i=668:si=on:rtra=on_0 on DTF2THF_27261 for (2978ds/668Mi)
% 20.45/7.09 % (27468)Instruction limit reached!
% 20.45/7.09 % (27468)------------------------------
% 20.45/7.09 % (27468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.45/7.09 % (27468)Termination reason: Unknown
% 20.45/7.09 % (27468)Termination phase: Preprocessing 3
% 20.45/7.09
% 20.45/7.09 % (27468)Memory used [KB]: 25458
% 20.45/7.09 % (27468)Time elapsed: 0.597 s
% 20.45/7.09 % (27468)Instructions burned: 1913 (million)
% 20.45/7.09 % (27468)------------------------------
% 20.45/7.09 % (27468)------------------------------
% 20.45/7.10 % (27481)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=40:si=on:rtra=on_0 on DTF2THF_27261 for (2978ds/40Mi)
% 21.03/7.12 % (27481)Instruction limit reached!
% 21.03/7.12 % (27481)------------------------------
% 21.03/7.12 % (27481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.03/7.12 % (27481)Termination reason: Unknown
% 21.03/7.12 % (27481)Termination phase: shuffling
% 21.03/7.12
% 21.03/7.12 % (27481)Memory used [KB]: 12665
% 21.03/7.12 % (27481)Time elapsed: 0.020 s
% 21.03/7.12 % (27481)Instructions burned: 40 (million)
% 21.03/7.12 % (27481)------------------------------
% 21.03/7.12 % (27481)------------------------------
% 21.03/7.13 % (27482)lrs+1010_1:1_cnfonf=lazy_pi_sigma_gen:fd=off:ntd=on:sd=1:sgt=32:sos=on:ss=axioms:i=294:si=on:rtra=on_0 on DTF2THF_27261 for (2977ds/294Mi)
% 21.03/7.14 % (27476)Instruction limit reached!
% 21.03/7.14 % (27476)------------------------------
% 21.03/7.14 % (27476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.03/7.14 % (27476)Termination reason: Unknown
% 21.03/7.14 % (27476)Termination phase: shuffling
% 21.03/7.14
% 21.03/7.14 % (27476)Memory used [KB]: 31854
% 21.03/7.14 % (27476)Time elapsed: 0.298 s
% 21.03/7.14 % (27476)Instructions burned: 995 (million)
% 21.03/7.14 % (27476)------------------------------
% 21.03/7.14 % (27476)------------------------------
% 21.03/7.16 % (27483)dis+1010_1:1_avsq=on:fe=off:fsr=off:hud=10:plsq=on:plsqr=32,1:prag=on:i=187:si=on:rtra=on_0 on DTF2THF_27261 for (2977ds/187Mi)
% 21.34/7.22 % (27482)Instruction limit reached!
% 21.34/7.22 % (27482)------------------------------
% 21.34/7.22 % (27482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.34/7.22 % (27482)Termination reason: Unknown
% 21.34/7.22 % (27482)Termination phase: shuffling
% 21.34/7.22
% 21.34/7.22 % (27482)Memory used [KB]: 18293
% 21.34/7.22 % (27482)Time elapsed: 0.090 s
% 21.34/7.22 % (27482)Instructions burned: 294 (million)
% 21.34/7.22 % (27482)------------------------------
% 21.34/7.22 % (27482)------------------------------
% 21.34/7.23 % (27483)Instruction limit reached!
% 21.34/7.23 % (27483)------------------------------
% 21.34/7.23 % (27483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.34/7.23 % (27483)Termination reason: Unknown
% 21.34/7.23 % (27483)Termination phase: shuffling
% 21.34/7.23
% 21.34/7.23 % (27483)Memory used [KB]: 14711
% 21.34/7.23 % (27483)Time elapsed: 0.070 s
% 21.34/7.23 % (27483)Instructions burned: 189 (million)
% 21.34/7.23 % (27483)------------------------------
% 21.34/7.23 % (27483)------------------------------
% 21.34/7.23 % (27484)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_27261 for (2976ds/371Mi)
% 21.34/7.24 % (27480)Instruction limit reached!
% 21.34/7.24 % (27480)------------------------------
% 21.34/7.24 % (27480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.34/7.24 % (27480)Termination reason: Unknown
% 21.34/7.24 % (27480)Termination phase: shuffling
% 21.34/7.24
% 21.34/7.24 % (27480)Memory used [KB]: 23411
% 21.34/7.24 % (27480)Time elapsed: 0.196 s
% 21.34/7.24 % (27480)Instructions burned: 668 (million)
% 21.34/7.24 % (27480)------------------------------
% 21.34/7.24 % (27480)------------------------------
% 21.34/7.24 % (27485)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1827:si=on:rtra=on_0 on DTF2THF_27261 for (2976ds/1827Mi)
% 21.34/7.25 % (27486)lrs+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:tnu=1:i=70:si=on:rtra=on_0 on DTF2THF_27261 for (2976ds/70Mi)
% 21.34/7.28 % (27486)Instruction limit reached!
% 21.34/7.28 % (27486)------------------------------
% 21.34/7.28 % (27486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.34/7.28 % (27486)Termination reason: Unknown
% 21.34/7.28 % (27486)Termination phase: shuffling
% 21.34/7.28
% 21.34/7.28 % (27486)Memory used [KB]: 13176
% 21.34/7.28 % (27486)Time elapsed: 0.028 s
% 21.34/7.28 % (27486)Instructions burned: 71 (million)
% 21.34/7.28 % (27486)------------------------------
% 21.34/7.28 % (27486)------------------------------
% 21.34/7.29 % (27487)lrs+10_1:20_bd=off:cnfonf=lazy_not_be_gen:fe=axiom:hud=10:ins=1:nm=20:prag=on:sos=on:i=316:si=on:rtra=on_0 on DTF2THF_27261 for (2976ds/316Mi)
% 22.08/7.31 % (27479)Instruction limit reached!
% 22.08/7.31 % (27479)------------------------------
% 22.08/7.31 % (27479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.08/7.31 % (27479)Termination reason: Unknown
% 22.08/7.31 % (27479)Termination phase: Property scanning
% 22.08/7.31
% 22.08/7.31 % (27479)Memory used [KB]: 19957
% 22.08/7.31 % (27479)Time elapsed: 0.271 s
% 22.08/7.31 % (27479)Instructions burned: 835 (million)
% 22.08/7.31 % (27479)------------------------------
% 22.08/7.31 % (27479)------------------------------
% 22.08/7.32 % (27488)ott+1010_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=3548551,65536:s2a=on:i=932:si=on:rtra=on_0 on DTF2THF_27261 for (2976ds/932Mi)
% 22.08/7.36 % (27484)Instruction limit reached!
% 22.08/7.36 % (27484)------------------------------
% 22.08/7.36 % (27484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.08/7.36 % (27484)Termination reason: Unknown
% 22.08/7.36 % (27484)Termination phase: shuffling
% 22.08/7.36
% 22.08/7.36 % (27484)Memory used [KB]: 16886
% 22.08/7.36 % (27484)Time elapsed: 0.130 s
% 22.08/7.36 % (27484)Instructions burned: 371 (million)
% 22.08/7.36 % (27484)------------------------------
% 22.08/7.36 % (27484)------------------------------
% 22.08/7.37 % (27489)lrs+1010_1:128_au=on:bd=preordered:cnfonf=lazy_pi_sigma_gen:plsq=on:plsqc=1:plsqr=32,1:sos=all:sp=occurrence:i=4714:si=on:rtra=on_0 on DTF2THF_27261 for (2975ds/4714Mi)
% 22.08/7.38 % (27487)Instruction limit reached!
% 22.08/7.38 % (27487)------------------------------
% 22.08/7.38 % (27487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.08/7.38 % (27487)Termination reason: Unknown
% 22.08/7.38 % (27487)Termination phase: shuffling
% 22.08/7.38
% 22.08/7.38 % (27487)Memory used [KB]: 18677
% 22.08/7.38 % (27487)Time elapsed: 0.097 s
% 22.08/7.38 % (27487)Instructions burned: 318 (million)
% 22.08/7.38 % (27487)------------------------------
% 22.08/7.38 % (27487)------------------------------
% 22.08/7.40 % (27490)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=8250:si=on:rtra=on_0 on DTF2THF_27261 for (2975ds/8250Mi)
% 23.21/7.59 % (27488)Instruction limit reached!
% 23.21/7.59 % (27488)------------------------------
% 23.21/7.59 % (27488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.21/7.59 % (27488)Termination reason: Unknown
% 23.21/7.59 % (27488)Termination phase: shuffling
% 23.21/7.59
% 23.21/7.59 % (27488)Memory used [KB]: 30958
% 23.21/7.59 % (27488)Time elapsed: 0.271 s
% 23.21/7.59 % (27488)Instructions burned: 932 (million)
% 23.21/7.59 % (27488)------------------------------
% 23.21/7.59 % (27488)------------------------------
% 23.80/7.60 % (27491)lrs+1002_1:1_apa=on:cnfonf=off:fd=off:flr=on:hud=1:lcm=reverse:prag=on:sd=1:sos=on:ss=axioms:i=770:si=on:rtra=on_0 on DTF2THF_27261 for (2973ds/770Mi)
% 24.10/7.79 % (27467)Instruction limit reached!
% 24.10/7.79 % (27467)------------------------------
% 24.10/7.79 % (27467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.10/7.79 % (27467)Termination reason: Unknown
% 24.10/7.79 % (27467)Termination phase: Function definition elimination
% 24.10/7.79
% 24.10/7.79 % (27467)Memory used [KB]: 43240
% 24.10/7.79 % (27467)Time elapsed: 1.309 s
% 24.10/7.79 % (27467)Instructions burned: 5207 (million)
% 24.10/7.79 % (27467)------------------------------
% 24.10/7.79 % (27467)------------------------------
% 24.10/7.80 % (27485)Instruction limit reached!
% 24.10/7.80 % (27485)------------------------------
% 24.10/7.80 % (27485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.10/7.80 % (27485)Termination reason: Unknown
% 24.10/7.80 % (27485)Termination phase: Property scanning
% 24.10/7.80
% 24.10/7.80 % (27485)Memory used [KB]: 29679
% 24.10/7.80 % (27485)Time elapsed: 0.556 s
% 24.10/7.80 % (27485)Instructions burned: 1828 (million)
% 24.10/7.80 % (27485)------------------------------
% 24.10/7.80 % (27485)------------------------------
% 23.09/7.81 % (27492)ott+1010_5:1_av=off:fe=off:ins=1:kws=precedence:sp=reverse_arity:i=794:si=on:rtra=on_0 on DTF2THF_27261 for (2971ds/794Mi)
% 23.09/7.81 % (27493)dis+1002_1:1_au=on:avsq=on:avsqr=1,16:cond=on:sp=frequency:i=464:si=on:rtra=on_0 on DTF2THF_27261 for (2971ds/464Mi)
% 23.09/7.83 % (27464)Instruction limit reached!
% 23.09/7.83 % (27464)------------------------------
% 23.09/7.83 % (27464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.09/7.83 % (27464)Termination reason: Unknown
% 23.09/7.83 % (27464)Termination phase: Saturation
% 23.09/7.83
% 23.09/7.83 % (27464)Memory used [KB]: 31598
% 23.09/7.83 % (27464)Time elapsed: 1.420 s
% 23.09/7.83 % (27464)Instructions burned: 4437 (million)
% 23.09/7.83 % (27464)------------------------------
% 23.09/7.83 % (27464)------------------------------
% 23.09/7.84 % (27494)ott+21_1:3_avsq=on:bd=off:cnfonf=lazy_gen:fe=off:hud=10:ixr=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=1376:si=on:rtra=on_0 on DTF2THF_27261 for (2970ds/1376Mi)
% 23.09/7.86 % (27491)Instruction limit reached!
% 23.09/7.86 % (27491)------------------------------
% 23.09/7.86 % (27491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.09/7.86 % (27491)Termination reason: Unknown
% 23.09/7.86 % (27491)Termination phase: shuffling
% 23.09/7.86
% 23.09/7.86 % (27491)Memory used [KB]: 23411
% 23.09/7.86 % (27491)Time elapsed: 0.263 s
% 23.09/7.86 % (27491)Instructions burned: 771 (million)
% 23.09/7.86 % (27491)------------------------------
% 23.09/7.86 % (27491)------------------------------
% 23.09/7.87 % (27495)dis+10_1:1_etr=on:plsq=on:plsqc=1:plsqr=32,1:i=906:si=on:rtra=on_0 on DTF2THF_27261 for (2970ds/906Mi)
% 25.38/7.92 % (27469)Instruction limit reached!
% 25.38/7.92 % (27469)------------------------------
% 25.38/7.92 % (27469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.38/7.92 % (27469)Termination reason: Unknown
% 25.38/7.92 % (27469)Termination phase: Saturation
% 25.38/7.92
% 25.38/7.92 % (27469)Memory used [KB]: 30958
% 25.38/7.92 % (27469)Time elapsed: 1.311 s
% 25.38/7.92 % (27469)Instructions burned: 4351 (million)
% 25.38/7.92 % (27469)------------------------------
% 25.38/7.92 % (27469)------------------------------
% 25.38/7.93 % (27496)lrs+1002_1:16_bd=off:e2e=on:fde=unused:kws=precedence:sac=on:sos=on:ss=axioms:i=434:si=on:rtra=on_0 on DTF2THF_27261 for (2969ds/434Mi)
% 25.38/7.97 % (27493)Instruction limit reached!
% 25.38/7.97 % (27493)------------------------------
% 25.38/7.97 % (27493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.38/7.97 % (27493)Termination reason: Unknown
% 25.38/7.97 % (27493)Termination phase: shuffling
% 25.38/7.97
% 25.38/7.97 % (27493)Memory used [KB]: 18038
% 25.38/7.97 % (27493)Time elapsed: 0.162 s
% 25.38/7.97 % (27493)Instructions burned: 464 (million)
% 25.38/7.97 % (27493)------------------------------
% 25.38/7.97 % (27493)------------------------------
% 25.38/7.98 % (27497)lrs+10_5:4_cnfonf=conj_eager:fe=axiom:hflw=5:hfsq=on:hfsqc=10:hfsql=off:hud=10:plsq=on:sgt=5:spb=goal_then_units:ss=axioms:i=865:si=on:rtra=on_0 on DTF2THF_27261 for (2969ds/865Mi)
% 27.51/8.06 % (27492)Instruction limit reached!
% 27.51/8.06 % (27492)------------------------------
% 27.51/8.06 % (27492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.51/8.06 % (27492)Termination reason: Unknown
% 27.51/8.06 % (27492)Termination phase: Property scanning
% 27.51/8.06
% 27.51/8.06 % (27492)Memory used [KB]: 19829
% 27.51/8.06 % (27492)Time elapsed: 0.258 s
% 27.51/8.06 % (27492)Instructions burned: 795 (million)
% 27.51/8.06 % (27492)------------------------------
% 27.51/8.06 % (27492)------------------------------
% 27.51/8.08 % (27498)dis+1002_1:1_cnfonf=lazy_simp:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=593:si=on:rtra=on_0 on DTF2THF_27261 for (2968ds/593Mi)
% 27.51/8.09 % (27496)Instruction limit reached!
% 27.51/8.09 % (27496)------------------------------
% 27.51/8.09 % (27496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.51/8.09 % (27496)Termination reason: Unknown
% 27.51/8.09 % (27496)Termination phase: shuffling
% 27.51/8.09
% 27.51/8.09 % (27496)Memory used [KB]: 17910
% 27.51/8.09 % (27496)Time elapsed: 0.156 s
% 27.51/8.09 % (27496)Instructions burned: 434 (million)
% 27.51/8.09 % (27496)------------------------------
% 27.51/8.09 % (27496)------------------------------
% 27.51/8.10 % (27499)dis+10_1:1_sd=2:sgt=20:slsq=on:ss=axioms:st=6.0:i=443:si=on:rtra=on_0 on DTF2THF_27261 for (2968ds/443Mi)
% 28.14/8.16 % (27495)Instruction limit reached!
% 28.14/8.16 % (27495)------------------------------
% 28.14/8.16 % (27495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.14/8.16 % (27495)Termination reason: Unknown
% 28.14/8.16 % (27495)Termination phase: Property scanning
% 28.14/8.16
% 28.14/8.16 % (27495)Memory used [KB]: 19829
% 28.14/8.16 % (27495)Time elapsed: 0.286 s
% 28.14/8.16 % (27495)Instructions burned: 906 (million)
% 28.14/8.16 % (27495)------------------------------
% 28.14/8.16 % (27495)------------------------------
% 28.14/8.17 % (27500)dis+10_1:1_atotf=0.1:cbe=off:sd=3:ss=axioms:st=2.0:i=594:si=on:rtra=on_0 on DTF2THF_27261 for (2967ds/594Mi)
% 28.48/8.24 % (27497)Instruction limit reached!
% 28.48/8.24 % (27497)------------------------------
% 28.48/8.24 % (27497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.48/8.24 % (27497)Termination reason: Unknown
% 28.48/8.24 % (27497)Termination phase: shuffling
% 28.48/8.24
% 28.48/8.24 % (27497)Memory used [KB]: 26097
% 28.48/8.24 % (27497)Time elapsed: 0.253 s
% 28.48/8.24 % (27497)Instructions burned: 867 (million)
% 28.48/8.24 % (27497)------------------------------
% 28.48/8.24 % (27497)------------------------------
% 28.48/8.24 % (27494)Instruction limit reached!
% 28.48/8.24 % (27494)------------------------------
% 28.48/8.24 % (27494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.48/8.24 % (27494)Termination reason: Unknown
% 28.48/8.24 % (27494)Termination phase: Property scanning
% 28.48/8.24
% 28.48/8.24 % (27494)Memory used [KB]: 36715
% 28.48/8.24 % (27494)Time elapsed: 0.398 s
% 28.48/8.24 % (27494)Instructions burned: 1378 (million)
% 28.48/8.24 % (27494)------------------------------
% 28.48/8.24 % (27494)------------------------------
% 28.48/8.25 % (27501)ott+10_1:1024_anc=none:bd=off:e2e=on:fde=none:fsr=off:hud=10:ixr=off:nm=4:nwc=5.0:spb=goal:i=327:si=on:rtra=on_0 on DTF2THF_27261 for (2966ds/327Mi)
% 28.48/8.25 % (27498)Instruction limit reached!
% 28.48/8.25 % (27498)------------------------------
% 28.48/8.25 % (27498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.48/8.25 % (27498)Termination reason: Unknown
% 28.48/8.25 % (27498)Termination phase: shuffling
% 28.48/8.25
% 28.48/8.25 % (27498)Memory used [KB]: 22387
% 28.48/8.25 % (27498)Time elapsed: 0.174 s
% 28.48/8.25 % (27498)Instructions burned: 593 (million)
% 28.48/8.25 % (27498)------------------------------
% 28.48/8.25 % (27498)------------------------------
% 28.48/8.25 % (27499)Instruction limit reached!
% 28.48/8.25 % (27499)------------------------------
% 28.48/8.25 % (27499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.48/8.25 % (27499)Termination reason: Unknown
% 28.48/8.25 % (27499)Termination phase: shuffling
% 28.48/8.25
% 28.48/8.25 % (27499)Memory used [KB]: 17782
% 28.48/8.25 % (27499)Time elapsed: 0.155 s
% 28.48/8.25 % (27499)Instructions burned: 444 (million)
% 28.48/8.25 % (27499)------------------------------
% 28.48/8.25 % (27499)------------------------------
% 28.48/8.25 % (27502)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=5411:si=on:rtra=on_0 on DTF2THF_27261 for (2966ds/5411Mi)
% 28.48/8.26 % (27503)lrs+10_1:20_au=on:c=on:cbe=off:cnfonf=lazy_gen:ins=1:plsq=on:plsqc=1:plsqr=46,31:sac=on:i=1762:si=on:rtra=on_0 on DTF2THF_27261 for (2966ds/1762Mi)
% 28.48/8.26 % (27504)dis+1003_1:64_cha=on:cnfonf=lazy_gen:hud=9:nwc=8.0:prag=on:i=96:si=on:rtra=on_0 on DTF2THF_27261 for (2966ds/96Mi)
% 29.15/8.30 % (27504)Instruction limit reached!
% 29.15/8.30 % (27504)------------------------------
% 29.15/8.30 % (27504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.15/8.30 % (27504)Termination reason: Unknown
% 29.15/8.30 % (27504)Termination phase: shuffling
% 29.15/8.30
% 29.15/8.30 % (27504)Memory used [KB]: 13560
% 29.15/8.30 % (27504)Time elapsed: 0.035 s
% 29.15/8.30 % (27504)Instructions burned: 97 (million)
% 29.15/8.30 % (27504)------------------------------
% 29.15/8.30 % (27504)------------------------------
% 29.15/8.31 % (27505)lrs+21_1:1_hud=5:kws=inv_frequency:nm=16:nwc=2.0:prag=on:sos=on:ss=axioms:i=503:si=on:rtra=on_0 on DTF2THF_27261 for (2966ds/503Mi)
% 29.15/8.36 % (27501)Instruction limit reached!
% 29.15/8.36 % (27501)------------------------------
% 29.15/8.36 % (27501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.15/8.36 % (27501)Termination reason: Unknown
% 29.15/8.36 % (27501)Termination phase: shuffling
% 29.15/8.36
% 29.15/8.36 % (27501)Memory used [KB]: 16502
% 29.15/8.36 % (27501)Time elapsed: 0.118 s
% 29.15/8.36 % (27501)Instructions burned: 328 (million)
% 29.15/8.36 % (27501)------------------------------
% 29.15/8.36 % (27501)------------------------------
% 29.15/8.37 % (27500)Instruction limit reached!
% 29.15/8.37 % (27500)------------------------------
% 29.15/8.37 % (27500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.15/8.37 % (27500)Termination reason: Unknown
% 29.15/8.37 % (27500)Termination phase: shuffling
% 29.15/8.37
% 29.15/8.37 % (27500)Memory used [KB]: 19701
% 29.15/8.37 % (27500)Time elapsed: 0.206 s
% 29.15/8.37 % (27500)Instructions burned: 596 (million)
% 29.15/8.37 % (27500)------------------------------
% 29.15/8.37 % (27500)------------------------------
% 29.15/8.38 % (27506)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_27261 for (2965ds/90Mi)
% 29.15/8.39 % (27507)lrs+10_1:64_hud=5:plsq=on:plsqr=2011171,524288:i=593:si=on:rtra=on_0 on DTF2THF_27261 for (2965ds/593Mi)
% 29.73/8.41 % (27506)Instruction limit reached!
% 29.73/8.41 % (27506)------------------------------
% 29.73/8.41 % (27506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.73/8.41 % (27506)Termination reason: Unknown
% 29.73/8.41 % (27506)Termination phase: shuffling
% 29.73/8.41
% 29.73/8.41 % (27506)Memory used [KB]: 13304
% 29.73/8.41 % (27506)Time elapsed: 0.037 s
% 29.73/8.41 % (27506)Instructions burned: 90 (million)
% 29.73/8.41 % (27506)------------------------------
% 29.73/8.41 % (27506)------------------------------
% 29.73/8.42 % (27508)dis-1002_1:1_alpa=false:apa=on:au=on:bd=off:cnfonf=off:e2e=on:er=filter:fde=unused:plsq=on:plsqr=9699663,1048576:s2a=on:sos=on:sp=unary_first:i=1512:si=on:rtra=on_0 on DTF2THF_27261 for (2964ds/1512Mi)
% 29.73/8.48 % (27505)Instruction limit reached!
% 29.73/8.48 % (27505)------------------------------
% 29.73/8.48 % (27505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.73/8.48 % (27505)Termination reason: Unknown
% 29.73/8.48 % (27505)Termination phase: shuffling
% 29.73/8.48
% 29.73/8.48 % (27505)Memory used [KB]: 18549
% 29.73/8.48 % (27505)Time elapsed: 0.175 s
% 29.73/8.48 % (27505)Instructions burned: 505 (million)
% 29.73/8.48 % (27505)------------------------------
% 29.73/8.48 % (27505)------------------------------
% 29.73/8.49 % (27509)lrs+1002_1:128_cbe=off:cnfonf=lazy_not_gen_be_off:ins=1:i=569:si=on:rtra=on_0 on DTF2THF_27261 for (2964ds/569Mi)
% 30.17/8.53 % (27489)Instruction limit reached!
% 30.17/8.53 % (27489)------------------------------
% 30.17/8.53 % (27489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.17/8.53 % (27489)Termination reason: Unknown
% 30.17/8.53 % (27489)Termination phase: Function definition elimination
% 30.17/8.53
% 30.17/8.53 % (27489)Memory used [KB]: 39274
% 30.17/8.53 % (27489)Time elapsed: 1.161 s
% 30.17/8.53 % (27489)Instructions burned: 4718 (million)
% 30.17/8.53 % (27489)------------------------------
% 30.17/8.53 % (27489)------------------------------
% 30.17/8.55 % (27510)ott+1002_1:1_bd=off:cnfonf=off:e2e=on:fd=off:fe=axiom:hud=5:prag=on:sos=on:i=501:si=on:rtra=on_0 on DTF2THF_27261 for (2963ds/501Mi)
% 30.17/8.59 % (27507)Instruction limit reached!
% 30.17/8.59 % (27507)------------------------------
% 30.17/8.59 % (27507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.17/8.59 % (27507)Termination reason: Unknown
% 30.17/8.59 % (27507)Termination phase: shuffling
% 30.17/8.59
% 30.17/8.59 % (27507)Memory used [KB]: 19573
% 30.17/8.59 % (27507)Time elapsed: 0.204 s
% 30.17/8.59 % (27507)Instructions burned: 593 (million)
% 30.17/8.59 % (27507)------------------------------
% 30.17/8.59 % (27507)------------------------------
% 30.55/8.60 % (27511)lrs+1010_1:3_cnfonf=conj_eager:fsr=off:prag=on:sd=4:sgt=10:ss=axioms:i=562:si=on:rtra=on_0 on DTF2THF_27261 for (2963ds/562Mi)
% 30.55/8.66 % (27509)Instruction limit reached!
% 30.55/8.66 % (27509)------------------------------
% 30.55/8.66 % (27509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.55/8.66 % (27509)Termination reason: Unknown
% 30.55/8.66 % (27509)Termination phase: shuffling
% 30.55/8.66
% 30.55/8.66 % (27509)Memory used [KB]: 22131
% 30.55/8.66 % (27509)Time elapsed: 0.169 s
% 30.55/8.66 % (27509)Instructions burned: 570 (million)
% 30.55/8.66 % (27509)------------------------------
% 30.55/8.66 % (27509)------------------------------
% 30.55/8.67 % (27512)lrs+10_1:1_au=on:cnfonf=off:fde=unused:kws=precedence:tgt=full:i=406:si=on:rtra=on_0 on DTF2THF_27261 for (2962ds/406Mi)
% 30.82/8.72 % (27510)Instruction limit reached!
% 30.82/8.72 % (27510)------------------------------
% 30.82/8.72 % (27510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.82/8.72 % (27510)Termination reason: Unknown
% 30.82/8.72 % (27510)Termination phase: shuffling
% 30.82/8.72
% 30.82/8.72 % (27510)Memory used [KB]: 18421
% 30.82/8.72 % (27510)Time elapsed: 0.174 s
% 30.82/8.72 % (27510)Instructions burned: 503 (million)
% 30.82/8.72 % (27510)------------------------------
% 30.82/8.72 % (27510)------------------------------
% 30.82/8.73 % (27513)dis+1004_1:128_au=on:cbe=off:e2e=on:sac=on:i=582:si=on:rtra=on_0 on DTF2THF_27261 for (2961ds/582Mi)
% 30.82/8.75 % (27503)Instruction limit reached!
% 30.82/8.75 % (27503)------------------------------
% 30.82/8.75 % (27503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.82/8.75 % (27503)Termination reason: Unknown
% 30.82/8.75 % (27503)Termination phase: Property scanning
% 30.82/8.75
% 30.82/8.75 % (27503)Memory used [KB]: 36715
% 30.82/8.75 % (27503)Time elapsed: 0.493 s
% 30.82/8.75 % (27503)Instructions burned: 1765 (million)
% 30.82/8.75 % (27503)------------------------------
% 30.82/8.75 % (27503)------------------------------
% 30.82/8.76 % (27511)Instruction limit reached!
% 30.82/8.76 % (27511)------------------------------
% 30.82/8.76 % (27511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.82/8.76 % (27511)Termination reason: Unknown
% 30.82/8.76 % (27511)Termination phase: shuffling
% 30.82/8.76
% 30.82/8.76 % (27511)Memory used [KB]: 22003
% 30.82/8.76 % (27511)Time elapsed: 0.166 s
% 30.82/8.76 % (27511)Instructions burned: 563 (million)
% 30.82/8.76 % (27511)------------------------------
% 30.82/8.76 % (27511)------------------------------
% 30.82/8.76 % (27514)lrs+2_1:1_au=on:bd=off:bsr=unit_only:sd=2:sgt=32:sos=all:ss=axioms:i=456:si=on:rtra=on_0 on DTF2THF_27261 for (2961ds/456Mi)
% 30.82/8.78 % (27515)lrs+1010_1:1_au=on:cbe=off:cnfonf=conj_eager:tgt=ground:i=393:si=on:rtra=on_0 on DTF2THF_27261 for (2961ds/393Mi)
% 31.79/8.81 % (27512)Instruction limit reached!
% 31.79/8.81 % (27512)------------------------------
% 31.79/8.81 % (27512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.79/8.81 % (27512)Termination reason: Unknown
% 31.79/8.81 % (27512)Termination phase: shuffling
% 31.79/8.81
% 31.79/8.81 % (27512)Memory used [KB]: 17142
% 31.79/8.81 % (27512)Time elapsed: 0.140 s
% 31.79/8.81 % (27512)Instructions burned: 408 (million)
% 31.79/8.81 % (27512)------------------------------
% 31.79/8.81 % (27512)------------------------------
% 31.79/8.82 % (27516)lrs+1010_1:6_au=on:cnfonf=conj_eager:nwc=10.0:pe=on:prag=on:s2a=on:ss=axioms:st=1.5:i=1335:si=on:rtra=on_0 on DTF2THF_27261 for (2960ds/1335Mi)
% 31.79/8.89 % (27515)Instruction limit reached!
% 31.79/8.89 % (27515)------------------------------
% 31.79/8.89 % (27515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.79/8.89 % (27515)Termination reason: Unknown
% 31.79/8.89 % (27515)Termination phase: shuffling
% 31.79/8.89 % (27508)Instruction limit reached!
% 31.79/8.89 % (27508)------------------------------
% 31.79/8.89 % (27508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.79/8.89 % (27508)Termination reason: Unknown
% 31.79/8.89 % (27508)Termination phase: SInE selection
% 31.79/8.89
% 31.79/8.89 % (27508)Memory used [KB]: 25330
% 31.79/8.89 % (27508)Time elapsed: 0.471 s
% 31.79/8.89 % (27508)Instructions burned: 1514 (million)
% 31.79/8.89 % (27508)------------------------------
% 31.79/8.89 % (27508)------------------------------
% 31.79/8.89
% 31.79/8.89 % (27515)Memory used [KB]: 19701
% 31.79/8.89 % (27515)Time elapsed: 0.118 s
% 31.79/8.89 % (27515)Instructions burned: 393 (million)
% 31.79/8.89 % (27515)------------------------------
% 31.79/8.89 % (27515)------------------------------
% 32.64/8.90 % (27517)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=3373:si=on:rtra=on_0 on DTF2THF_27261 for (2960ds/3373Mi)
% 32.64/8.90 % (27518)dis+10_1:1024_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=32,1:spb=goal_then_units:i=474:si=on:rtra=on_0 on DTF2THF_27261 for (2960ds/474Mi)
% 32.64/8.92 % (27514)Instruction limit reached!
% 32.64/8.92 % (27514)------------------------------
% 32.64/8.92 % (27514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.64/8.92 % (27514)Termination reason: Unknown
% 32.64/8.92 % (27514)Termination phase: shuffling
% 32.64/8.92
% 32.64/8.92 % (27514)Memory used [KB]: 17910
% 32.64/8.92 % (27514)Time elapsed: 0.158 s
% 32.64/8.92 % (27514)Instructions burned: 457 (million)
% 32.64/8.92 % (27514)------------------------------
% 32.64/8.92 % (27514)------------------------------
% 32.64/8.92 % (27490)First to succeed.
% 32.64/8.93 % (27519)lrs+1002_1:16_amm=sco:cnfonf=lazy_pi_sigma_gen:fe=off:nm=60:piset=all_but_not_eq:plsq=on:plsqc=1:plsqr=2,1:rawr=on:sp=occurrence:ss=included:st=-1.0:i=157:si=on:rtra=on_0 on DTF2THF_27261 for (2959ds/157Mi)
% 32.64/8.93 % (27513)Instruction limit reached!
% 32.64/8.93 % (27513)------------------------------
% 32.64/8.93 % (27513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.64/8.93 % (27513)Termination reason: Unknown
% 32.64/8.93 % (27513)Termination phase: shuffling
% 32.64/8.93
% 32.64/8.93 % (27513)Memory used [KB]: 19701
% 32.64/8.93 % (27513)Time elapsed: 0.205 s
% 32.64/8.93 % (27513)Instructions burned: 583 (million)
% 32.64/8.93 % (27513)------------------------------
% 32.64/8.93 % (27513)------------------------------
% 32.64/8.94 % (27490)Refutation found. Thanks to Tanya!
% 32.64/8.94 % SZS status Theorem for DTF2THF_27261
% 32.64/8.94 % SZS output start Proof for DTF2THF_27261
% 32.64/8.94 thf(type_def_5, type, rat: $tType).
% 32.64/8.94 thf(type_def_6, type, int: $tType).
% 32.64/8.94 thf(type_def_8, type, real: $tType).
% 32.64/8.94 thf(type_def_9, type, nat: $tType).
% 32.64/8.94 thf(type_def_10, type, code_integer: $tType).
% 32.64/8.94 thf(type_def_11, type, product_prod_int_int: $tType).
% 32.64/8.94 thf(type_def_12, type, code_natural: $tType).
% 32.64/8.94 thf(type_def_13, type, product_prod_nat_nat: $tType).
% 32.64/8.94 thf(type_def_14, type, num: $tType).
% 32.64/8.94 thf(type_def_15, type, complex: $tType).
% 32.64/8.94 thf(type_def_16, type, option_num: $tType).
% 32.64/8.94 thf(type_def_17, type, product_prod_num_num: $tType).
% 32.64/8.94 thf(type_def_18, type, produc6271795597528267376eger_o: $tType).
% 32.64/8.94 thf(type_def_19, type, produc8923325533196201883nteger: $tType).
% 32.64/8.94 thf(type_def_20, type, set_real: $tType).
% 32.64/8.94 thf(type_def_21, type, set_set_nat: $tType).
% 32.64/8.94 thf(type_def_22, type, set_nat: $tType).
% 32.64/8.94 thf(type_def_23, type, filter_real: $tType).
% 32.64/8.94 thf(type_def_24, type, extended_enat: $tType).
% 32.64/8.94 thf(type_def_25, type, filter_int: $tType).
% 32.64/8.94 thf(type_def_26, type, filter_nat: $tType).
% 32.64/8.94 thf(type_def_27, type, set_o: $tType).
% 32.64/8.94 thf(type_def_28, type, set_complex: $tType).
% 32.64/8.94 thf(type_def_29, type, set_int: $tType).
% 32.64/8.94 thf(type_def_30, type, set_list_nat: $tType).
% 32.64/8.94 thf(type_def_31, type, set_Product_unit: $tType).
% 32.64/8.94 thf(type_def_32, type, set_char: $tType).
% 32.64/8.94 thf(type_def_33, type, list_nat: $tType).
% 32.64/8.94 thf(type_def_34, type, set_Pr1261947904930325089at_nat: $tType).
% 32.64/8.94 thf(type_def_35, type, sum_sum_nat_nat: $tType).
% 32.64/8.94 thf(type_def_36, type, set_Sum_sum_nat_nat: $tType).
% 32.64/8.94 thf(type_def_37, type, char: $tType).
% 32.64/8.94 thf(type_def_38, type, literal: $tType).
% 32.64/8.94 thf(type_def_39, type, list_o: $tType).
% 32.64/8.94 thf(type_def_40, type, list_int: $tType).
% 32.64/8.94 thf(type_def_41, type, set_Pr958786334691620121nt_int: $tType).
% 32.64/8.94 thf(type_def_42, type, set_Code_integer: $tType).
% 32.64/8.94 thf(type_def_43, type, set_rat: $tType).
% 32.64/8.94 thf(type_def_44, type, list_complex: $tType).
% 32.64/8.94 thf(type_def_45, type, list_real: $tType).
% 32.64/8.94 thf(type_def_46, type, list_set_nat: $tType).
% 32.64/8.94 thf(type_def_47, type, list_VEBT_VEBT: $tType).
% 32.64/8.94 thf(type_def_48, type, set_VEBT_VEBT: $tType).
% 32.64/8.94 thf(type_def_49, type, vEBT_VEBT: $tType).
% 32.64/8.94 thf(type_def_50, type, list_list_o: $tType).
% 32.64/8.94 thf(type_def_51, type, list_list_int: $tType).
% 32.64/8.94 thf(type_def_52, type, list_list_nat: $tType).
% 32.64/8.94 thf(type_def_53, type, list_list_VEBT_VEBT: $tType).
% 32.64/8.94 thf(type_def_54, type, option4927543243414619207at_nat: $tType).
% 32.64/8.94 thf(type_def_55, type, produc7822875418678951345atural: $tType).
% 32.64/8.94 thf(type_def_56, type, produc5835291356934675326atural: $tType).
% 32.64/8.94 thf(type_def_57, type, product_prod_nat_num: $tType).
% 32.64/8.94 thf(type_def_58, type, produc6756470014200349799atural: $tType).
% 32.64/8.94 thf(type_def_59, type, produc9072475918466114483BT_nat: $tType).
% 32.64/8.94 thf(type_def_60, type, product_unit: $tType).
% 32.64/8.94 thf(func_def_0, type, produc6756470014200349799atural: $tType).
% 32.64/8.94 thf(func_def_1, type, produc5835291356934675326atural: $tType).
% 32.64/8.94 thf(func_def_2, type, produc7822875418678951345atural: $tType).
% 32.64/8.94 thf(func_def_3, type, produc8923325533196201883nteger: $tType).
% 32.64/8.94 thf(func_def_4, type, option4927543243414619207at_nat: $tType).
% 32.64/8.94 thf(func_def_5, type, produc9072475918466114483BT_nat: $tType).
% 32.64/8.94 thf(func_def_6, type, set_Pr1261947904930325089at_nat: $tType).
% 32.64/8.94 thf(func_def_7, type, set_Pr958786334691620121nt_int: $tType).
% 32.64/8.94 thf(func_def_8, type, set_Sum_sum_nat_nat: $tType).
% 32.64/8.94 thf(func_def_9, type, list_list_VEBT_VEBT: $tType).
% 32.64/8.94 thf(func_def_10, type, produc6271795597528267376eger_o: $tType).
% 32.64/8.94 thf(func_def_11, type, product_prod_num_num: $tType).
% 32.64/8.94 thf(func_def_12, type, product_prod_nat_num: $tType).
% 32.64/8.94 thf(func_def_13, type, product_prod_nat_nat: $tType).
% 32.64/8.94 thf(func_def_14, type, product_prod_int_int: $tType).
% 32.64/8.94 thf(func_def_15, type, sum_sum_nat_nat: $tType).
% 32.64/8.94 thf(func_def_16, type, list_list_nat: $tType).
% 32.64/8.94 thf(func_def_17, type, list_list_int: $tType).
% 32.64/8.94 thf(func_def_18, type, list_VEBT_VEBT: $tType).
% 32.64/8.94 thf(func_def_19, type, set_list_nat: $tType).
% 32.64/8.94 thf(func_def_20, type, list_set_nat: $tType).
% 32.64/8.94 thf(func_def_21, type, set_VEBT_VEBT: $tType).
% 32.64/8.94 thf(func_def_22, type, set_set_nat: $tType).
% 32.64/8.94 thf(func_def_23, type, set_Code_integer: $tType).
% 32.64/8.94 thf(func_def_24, type, set_Product_unit: $tType).
% 32.64/8.94 thf(func_def_25, type, list_list_o: $tType).
% 32.64/8.94 thf(func_def_26, type, list_complex: $tType).
% 32.64/8.94 thf(func_def_27, type, set_complex: $tType).
% 32.64/8.94 thf(func_def_28, type, filter_real: $tType).
% 32.64/8.94 thf(func_def_29, type, option_num: $tType).
% 32.64/8.94 thf(func_def_30, type, filter_nat: $tType).
% 32.64/8.94 thf(func_def_31, type, filter_int: $tType).
% 32.64/8.94 thf(func_def_32, type, set_char: $tType).
% 32.64/8.94 thf(func_def_33, type, list_real: $tType).
% 32.64/8.94 thf(func_def_34, type, set_real: $tType).
% 32.64/8.94 thf(func_def_35, type, list_nat: $tType).
% 32.64/8.94 thf(func_def_36, type, list_int: $tType).
% 32.64/8.94 thf(func_def_37, type, vEBT_VEBT: $tType).
% 32.64/8.94 thf(func_def_38, type, set_rat: $tType).
% 32.64/8.94 thf(func_def_39, type, set_nat: $tType).
% 32.64/8.94 thf(func_def_40, type, set_int: $tType).
% 32.64/8.94 thf(func_def_41, type, code_natural: $tType).
% 32.64/8.94 thf(func_def_42, type, code_integer: $tType).
% 32.64/8.94 thf(func_def_43, type, product_unit: $tType).
% 32.64/8.94 thf(func_def_44, type, extended_enat: $tType).
% 32.64/8.94 thf(func_def_45, type, list_o: $tType).
% 32.64/8.94 thf(func_def_46, type, complex: $tType).
% 32.64/8.94 thf(func_def_47, type, literal: $tType).
% 32.64/8.94 thf(func_def_48, type, set_o: $tType).
% 32.64/8.94 thf(func_def_49, type, char: $tType).
% 32.64/8.94 thf(func_def_50, type, real: $tType).
% 32.64/8.94 thf(func_def_51, type, rat: $tType).
% 32.64/8.94 thf(func_def_52, type, num: $tType).
% 32.64/8.94 thf(func_def_53, type, nat: $tType).
% 32.64/8.94 thf(func_def_54, type, int: $tType).
% 32.64/8.94 thf(func_def_55, type, archim2889992004027027881ng_rat: rat > int).
% 32.64/8.94 thf(func_def_56, type, archim7802044766580827645g_real: real > int).
% 32.64/8.94 thf(func_def_57, type, archim3151403230148437115or_rat: rat > int).
% 32.64/8.94 thf(func_def_58, type, archim6058952711729229775r_real: real > int).
% 32.64/8.94 thf(func_def_59, type, archimedean_frac_rat: rat > rat).
% 32.64/8.94 thf(func_def_60, type, archim2898591450579166408c_real: real > real).
% 32.64/8.94 thf(func_def_61, type, archim7778729529865785530nd_rat: rat > int).
% 32.64/8.94 thf(func_def_62, type, archim8280529875227126926d_real: real > int).
% 32.64/8.94 thf(func_def_63, type, bNF_re1962705104956426057at_rat: ((nat > rat) > (nat > rat) > $o) > (((nat > rat) > nat > rat) > ((nat > rat) > nat > rat) > $o) > ((nat > rat) > (nat > rat) > nat > rat) > ((nat > rat) > (nat > rat) > nat > rat) > $o).
% 32.64/8.94 thf(func_def_64, type, bNF_re895249473297799549at_rat: ((nat > rat) > (nat > rat) > $o) > ((nat > rat) > (nat > rat) > $o) > ((nat > rat) > nat > rat) > ((nat > rat) > nat > rat) > $o).
% 32.64/8.94 thf(func_def_65, type, bNF_re728719798268516973at_o_o: ((nat > rat) > (nat > rat) > $o) > ($o > $o > $o) > ((nat > rat) > $o) > ((nat > rat) > $o) > $o).
% 32.64/8.94 thf(func_def_66, type, bNF_re4695409256820837752l_real: ((nat > rat) > real > $o) > (((nat > rat) > nat > rat) > (real > real) > $o) > ((nat > rat) > (nat > rat) > nat > rat) > (real > real > real) > $o).
% 32.64/8.94 thf(func_def_67, type, bNF_re4521903465945308077real_o: ((nat > rat) > real > $o) > (((nat > rat) > $o) > (real > $o) > $o) > ((nat > rat) > (nat > rat) > $o) > (real > real > $o) > $o).
% 32.64/8.94 thf(func_def_68, type, bNF_re3023117138289059399t_real: ((nat > rat) > real > $o) > ((nat > rat) > real > $o) > ((nat > rat) > nat > rat) > (real > real) > $o).
% 32.64/8.94 thf(func_def_69, type, bNF_re4297313714947099218al_o_o: ((nat > rat) > real > $o) > ($o > $o > $o) > ((nat > rat) > $o) > (real > $o) > $o).
% 32.64/8.94 thf(func_def_70, type, bNF_re398004352372739002nteger: (int > code_integer > $o) > ((int > int) > (code_integer > code_integer) > $o) > (int > int > int) > (code_integer > code_integer > code_integer) > $o).
% 32.64/8.94 thf(func_def_71, type, bNF_re3379532845092657523nteger: (int > code_integer > $o) > (int > code_integer > $o) > (int > int) > (code_integer > code_integer) > $o).
% 32.64/8.94 thf(func_def_72, type, bNF_re711492959462206631nt_int: (int > int > $o) > ((int > int) > (int > int) > $o) > (int > int > int) > (int > int > int) > $o).
% 32.64/8.94 thf(func_def_73, type, bNF_re157797125943740599nt_int: (int > int > $o) > ((int > product_prod_int_int) > (int > product_prod_int_int) > $o) > (int > int > product_prod_int_int) > (int > int > product_prod_int_int) > $o).
% 32.64/8.94 thf(func_def_74, type, bNF_re3461391660133120880nt_rat: (int > int > $o) > ((int > product_prod_int_int) > (int > rat) > $o) > (int > int > product_prod_int_int) > (int > int > rat) > $o).
% 32.64/8.94 thf(func_def_75, type, bNF_re4712519889275205905nt_int: (int > int > $o) > (int > int > $o) > (int > int) > (int > int) > $o).
% 32.64/8.94 thf(func_def_76, type, bNF_re6250860962936578807nt_int: (int > int > $o) > (product_prod_int_int > product_prod_int_int > $o) > (int > product_prod_int_int) > (int > product_prod_int_int) > $o).
% 32.64/8.94 thf(func_def_77, type, bNF_re2214769303045360666nt_rat: (int > int > $o) > (product_prod_int_int > rat > $o) > (int > product_prod_int_int) > (int > rat) > $o).
% 32.64/8.94 thf(func_def_78, type, bNF_re88643428490162567atural: (nat > code_natural > $o) > ((nat > nat) > (code_natural > code_natural) > $o) > (nat > nat > nat) > (code_natural > code_natural > code_natural) > $o).
% 32.64/8.94 thf(func_def_79, type, bNF_re5252274238750452962nteger: (nat > code_natural > $o) > (int > code_integer > $o) > (nat > int) > (code_natural > code_integer) > $o).
% 32.64/8.94 thf(func_def_80, type, bNF_re3704215830270325841atural: (nat > code_natural > $o) > (nat > code_natural > $o) > (nat > nat) > (code_natural > code_natural) > $o).
% 32.64/8.94 thf(func_def_81, type, bNF_re4935368626670024657nteger: (nat > nat > $o) > ((int > int) > (code_integer > code_integer) > $o) > (nat > int > int) > (nat > code_integer > code_integer) > $o).
% 32.64/8.94 thf(func_def_82, type, bNF_re4785983289428654063nt_int: (nat > nat > $o) > ((int > int) > (int > int) > $o) > (nat > int > int) > (nat > int > int) > $o).
% 32.64/8.94 thf(func_def_83, type, bNF_re3692170009638431535atural: (nat > nat > $o) > ((nat > nat) > (code_natural > code_natural) > $o) > (nat > nat > nat) > (nat > code_natural > code_natural) > $o).
% 32.64/8.94 thf(func_def_84, type, bNF_re1345281282404953727at_nat: (nat > nat > $o) > ((nat > nat) > (nat > nat) > $o) > (nat > nat > nat) > (nat > nat > nat) > $o).
% 32.64/8.94 thf(func_def_85, type, bNF_re4153400068438556298nteger: (nat > nat > $o) > (int > code_integer > $o) > (nat > int) > (nat > code_integer) > $o).
% 32.64/8.94 thf(func_def_86, type, bNF_re6650684261131312217nt_int: (nat > nat > $o) > (int > int > $o) > (nat > int) > (nat > int) > $o).
% 32.64/8.94 thf(func_def_87, type, bNF_re5653821019739307937at_nat: (nat > nat > $o) > (nat > nat > $o) > (nat > nat) > (nat > nat) > $o).
% 32.64/8.94 thf(func_def_88, type, bNF_re6830278522597306478at_int: (nat > nat > $o) > (product_prod_nat_nat > int > $o) > (nat > product_prod_nat_nat) > (nat > int) > $o).
% 32.64/8.94 thf(func_def_89, type, bNF_re7876454716742015248nteger: (num > num > $o) > ((num > int) > (num > code_integer) > $o) > (num > num > int) > (num > num > code_integer) > $o).
% 32.64/8.94 thf(func_def_90, type, bNF_re8402795839162346335um_int: (num > num > $o) > ((num > int) > (num > int) > $o) > (num > num > int) > (num > num > int) > $o).
% 32.64/8.94 thf(func_def_91, type, bNF_re6501075790457514782nteger: (num > num > $o) > (int > code_integer > $o) > (num > int) > (num > code_integer) > $o).
% 32.64/8.94 thf(func_def_92, type, bNF_re1822329894187522285nt_int: (num > num > $o) > (int > int > $o) > (num > int) > (num > int) > $o).
% 32.64/8.94 thf(func_def_93, type, bNF_re5228765855967844073nt_int: (product_prod_int_int > product_prod_int_int > $o) > ((product_prod_int_int > product_prod_int_int) > (product_prod_int_int > product_prod_int_int) > $o) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > $o).
% 32.64/8.94 thf(func_def_94, type, bNF_re8699439704749558557nt_o_o: (product_prod_int_int > product_prod_int_int > $o) > ($o > $o > $o) > (product_prod_int_int > $o) > (product_prod_int_int > $o) > $o).
% 32.64/8.94 thf(func_def_95, type, bNF_re7145576690424134365nt_int: (product_prod_int_int > product_prod_int_int > $o) > (product_prod_int_int > product_prod_int_int > $o) > (product_prod_int_int > product_prod_int_int) > (product_prod_int_int > product_prod_int_int) > $o).
% 32.64/8.94 thf(func_def_96, type, bNF_re7627151682743391978at_rat: (product_prod_int_int > rat > $o) > ((product_prod_int_int > product_prod_int_int) > (rat > rat) > $o) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > (rat > rat > rat) > $o).
% 32.64/8.94 thf(func_def_97, type, bNF_re1494630372529172596at_o_o: (product_prod_int_int > rat > $o) > ($o > $o > $o) > (product_prod_int_int > $o) > (rat > $o) > $o).
% 32.64/8.94 thf(func_def_98, type, bNF_re8279943556446156061nt_rat: (product_prod_int_int > rat > $o) > (product_prod_int_int > rat > $o) > (product_prod_int_int > product_prod_int_int) > (rat > rat) > $o).
% 32.64/8.94 thf(func_def_99, type, bNF_re717283939379294677_int_o: (product_prod_nat_nat > int > $o) > ((product_prod_nat_nat > $o) > (int > $o) > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > (int > int > $o) > $o).
% 32.64/8.94 thf(func_def_100, type, bNF_re7408651293131936558nt_int: (product_prod_nat_nat > int > $o) > ((product_prod_nat_nat > product_prod_nat_nat) > (int > int) > $o) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > (int > int > int) > $o).
% 32.64/8.94 thf(func_def_101, type, bNF_re6644619430987730960nt_o_o: (product_prod_nat_nat > int > $o) > ($o > $o > $o) > (product_prod_nat_nat > $o) > (int > $o) > $o).
% 32.64/8.94 thf(func_def_102, type, bNF_re4555766996558763186at_nat: (product_prod_nat_nat > int > $o) > (nat > nat > $o) > (product_prod_nat_nat > nat) > (int > nat) > $o).
% 32.64/8.94 thf(func_def_103, type, bNF_re7400052026677387805at_int: (product_prod_nat_nat > int > $o) > (product_prod_nat_nat > int > $o) > (product_prod_nat_nat > product_prod_nat_nat) > (int > int) > $o).
% 32.64/8.94 thf(func_def_104, type, bNF_re4202695980764964119_nat_o: (product_prod_nat_nat > product_prod_nat_nat > $o) > ((product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > $o).
% 32.64/8.94 thf(func_def_105, type, bNF_re3099431351363272937at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > ((product_prod_nat_nat > product_prod_nat_nat) > (product_prod_nat_nat > product_prod_nat_nat) > $o) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > $o).
% 32.64/8.94 thf(func_def_106, type, bNF_re3666534408544137501at_o_o: (product_prod_nat_nat > product_prod_nat_nat > $o) > ($o > $o > $o) > (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o).
% 32.64/8.94 thf(func_def_107, type, bNF_re8246922863344978751at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > (nat > nat > $o) > (product_prod_nat_nat > nat) > (product_prod_nat_nat > nat) > $o).
% 32.64/8.94 thf(func_def_108, type, bNF_re2241393799969408733at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat) > (product_prod_nat_nat > product_prod_nat_nat) > $o).
% 32.64/8.94 thf(func_def_109, type, binomial: nat > nat > nat).
% 32.64/8.94 thf(func_def_110, type, gbinomial_complex: complex > nat > complex).
% 32.64/8.94 thf(func_def_111, type, gbinomial_int: int > nat > int).
% 32.64/8.94 thf(func_def_112, type, gbinomial_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_113, type, gbinomial_rat: rat > nat > rat).
% 32.64/8.94 thf(func_def_114, type, gbinomial_real: real > nat > real).
% 32.64/8.94 thf(func_def_115, type, bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o).
% 32.64/8.94 thf(func_def_116, type, bit_and_not_num: num > num > option_num).
% 32.64/8.94 thf(func_def_117, type, bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o).
% 32.64/8.94 thf(func_def_118, type, bit_concat_bit: nat > int > int > int).
% 32.64/8.94 thf(func_def_119, type, bit_or_not_num_neg: num > num > num).
% 32.64/8.94 thf(func_def_120, type, bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o).
% 32.64/8.94 thf(func_def_121, type, bit_ri7632146776885996613nteger: code_integer > code_integer).
% 32.64/8.94 thf(func_def_122, type, bit_ri7919022796975470100ot_int: int > int).
% 32.64/8.94 thf(func_def_123, type, bit_ri6519982836138164636nteger: nat > code_integer > code_integer).
% 32.64/8.94 thf(func_def_124, type, bit_ri631733984087533419it_int: nat > int > int).
% 32.64/8.94 thf(func_def_125, type, bit_se3949692690581998587nteger: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_126, type, bit_se2773287842338716102atural: code_natural > code_natural > code_natural).
% 32.64/8.94 thf(func_def_127, type, bit_se725231765392027082nd_int: int > int > int).
% 32.64/8.94 thf(func_def_128, type, bit_se727722235901077358nd_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_129, type, bit_se3928097537394005634nteger: nat > code_integer > code_integer).
% 32.64/8.94 thf(func_def_130, type, bit_se2751692689150723149atural: nat > code_natural > code_natural).
% 32.64/8.94 thf(func_def_131, type, bit_se8568078237143864401it_int: nat > int > int).
% 32.64/8.94 thf(func_def_132, type, bit_se8570568707652914677it_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_133, type, bit_se1345352211410354436nteger: nat > code_integer > code_integer).
% 32.64/8.94 thf(func_def_134, type, bit_se168947363167071951atural: nat > code_natural > code_natural).
% 32.64/8.94 thf(func_def_135, type, bit_se2159334234014336723it_int: nat > int > int).
% 32.64/8.94 thf(func_def_136, type, bit_se2161824704523386999it_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_137, type, bit_se2119862282449309892nteger: nat > code_integer).
% 32.64/8.94 thf(func_def_138, type, bit_se2000444600071755411sk_int: nat > int).
% 32.64/8.94 thf(func_def_139, type, bit_se2002935070580805687sk_nat: nat > nat).
% 32.64/8.94 thf(func_def_140, type, bit_se1080825931792720795nteger: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_141, type, bit_se9127793120404214118atural: code_natural > code_natural > code_natural).
% 32.64/8.94 thf(func_def_142, type, bit_se1409905431419307370or_int: int > int > int).
% 32.64/8.94 thf(func_def_143, type, bit_se1412395901928357646or_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_144, type, bit_se7788150548672797655nteger: nat > code_integer > code_integer).
% 32.64/8.94 thf(func_def_145, type, bit_se6611745700429515170atural: nat > code_natural > code_natural).
% 32.64/8.94 thf(func_def_146, type, bit_se545348938243370406it_int: nat > int > int).
% 32.64/8.94 thf(func_def_147, type, bit_se547839408752420682it_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_148, type, bit_se2793503036327961859nteger: nat > code_integer > code_integer).
% 32.64/8.94 thf(func_def_149, type, bit_se1617098188084679374atural: nat > code_natural > code_natural).
% 32.64/8.94 thf(func_def_150, type, bit_se7879613467334960850it_int: nat > int > int).
% 32.64/8.94 thf(func_def_151, type, bit_se7882103937844011126it_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_152, type, bit_se1745604003318907178nteger: nat > code_integer > code_integer).
% 32.64/8.94 thf(func_def_153, type, bit_se569199155075624693atural: nat > code_natural > code_natural).
% 32.64/8.94 thf(func_def_154, type, bit_se2923211474154528505it_int: nat > int > int).
% 32.64/8.94 thf(func_def_155, type, bit_se2925701944663578781it_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_156, type, bit_se8260200283734997820nteger: nat > code_integer > code_integer).
% 32.64/8.94 thf(func_def_157, type, bit_se7083795435491715335atural: nat > code_natural > code_natural).
% 32.64/8.94 thf(func_def_158, type, bit_se4203085406695923979it_int: nat > int > int).
% 32.64/8.94 thf(func_def_159, type, bit_se4205575877204974255it_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_160, type, bit_se3222712562003087583nteger: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_161, type, bit_se2046307713759805098atural: code_natural > code_natural > code_natural).
% 32.64/8.94 thf(func_def_162, type, bit_se6526347334894502574or_int: int > int > int).
% 32.64/8.94 thf(func_def_163, type, bit_se6528837805403552850or_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_164, type, bit_se9216721137139052372nteger: code_integer > nat > $o).
% 32.64/8.94 thf(func_def_165, type, bit_se8040316288895769887atural: code_natural > nat > $o).
% 32.64/8.94 thf(func_def_166, type, bit_se1146084159140164899it_int: int > nat > $o).
% 32.64/8.94 thf(func_def_167, type, bit_se1148574629649215175it_nat: nat > nat > $o).
% 32.64/8.94 thf(func_def_168, type, bit_take_bit_num: nat > num > option_num).
% 32.64/8.94 thf(func_def_169, type, bit_un1837492267222099188nd_num: num > num > option_num).
% 32.64/8.94 thf(func_def_170, type, bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o).
% 32.64/8.94 thf(func_def_171, type, bit_un2785000775030745342or_num: num > num > num).
% 32.64/8.94 thf(func_def_172, type, bit_un6909899581280750971um_rel: product_prod_num_num > product_prod_num_num > $o).
% 32.64/8.94 thf(func_def_173, type, bit_un6178654185764691216or_num: num > num > option_num).
% 32.64/8.94 thf(func_def_174, type, bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o).
% 32.64/8.94 thf(func_def_175, type, bit_un7362597486090784418nd_num: num > num > option_num).
% 32.64/8.94 thf(func_def_176, type, bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o).
% 32.64/8.94 thf(func_def_177, type, bit_un6697907153464112080or_num: num > num > num).
% 32.64/8.94 thf(func_def_178, type, bit_un4773296044027857193um_rel: product_prod_num_num > product_prod_num_num > $o).
% 32.64/8.94 thf(func_def_179, type, bit_un2480387367778600638or_num: num > num > option_num).
% 32.64/8.94 thf(func_def_180, type, bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o).
% 32.64/8.94 thf(func_def_181, type, boolea2445317508997433345nteger: (code_integer > code_integer > code_integer) > (code_integer > code_integer > code_integer) > (code_integer > code_integer) > code_integer > code_integer > (code_integer > code_integer > code_integer) > $o).
% 32.64/8.94 thf(func_def_182, type, boolea8527374999097803216ff_int: (int > int > int) > (int > int > int) > (int > int) > int > int > (int > int > int) > $o).
% 32.64/8.94 thf(func_def_183, type, code_Neg: num > code_integer).
% 32.64/8.94 thf(func_def_184, type, code_Pos: num > code_integer).
% 32.64/8.94 thf(func_def_185, type, code_Suc: code_natural > code_natural).
% 32.64/8.94 thf(func_def_186, type, code_bit_cut_integer: code_integer > produc6271795597528267376eger_o).
% 32.64/8.94 thf(func_def_187, type, code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger).
% 32.64/8.94 thf(func_def_188, type, code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger).
% 32.64/8.94 thf(func_def_189, type, code_dup: code_integer > code_integer).
% 32.64/8.94 thf(func_def_190, type, code_int_of_integer: code_integer > int).
% 32.64/8.94 thf(func_def_191, type, code_integer_of_int: int > code_integer).
% 32.64/8.94 thf(func_def_192, type, code_integer_of_nat: nat > code_integer).
% 32.64/8.94 thf(func_def_193, type, code_i5400310926305786745atural: code_natural > code_integer).
% 32.64/8.94 thf(func_def_194, type, code_integer_of_num: num > code_integer).
% 32.64/8.94 thf(func_def_195, type, code_nat_of_integer: code_integer > nat).
% 32.64/8.94 thf(func_def_196, type, code_nat_of_natural: code_natural > nat).
% 32.64/8.94 thf(func_def_197, type, code_natural_of_nat: nat > code_natural).
% 32.64/8.94 thf(func_def_198, type, code_num_of_integer: code_integer > num).
% 32.64/8.94 thf(func_def_199, type, code_pcr_integer: int > code_integer > $o).
% 32.64/8.94 thf(func_def_200, type, code_pcr_natural: nat > code_natural > $o).
% 32.64/8.94 thf(func_def_201, type, code_sub: num > num > code_integer).
% 32.64/8.94 thf(func_def_202, type, code_Target_negative: num > int).
% 32.64/8.94 thf(func_def_203, type, code_Target_positive: num > int).
% 32.64/8.94 thf(func_def_204, type, code_T6385005292777649522of_nat: nat > int).
% 32.64/8.94 thf(func_def_205, type, comple4887499456419720421f_real: set_real > real).
% 32.64/8.94 thf(func_def_206, type, comple7806235888213564991et_nat: set_set_nat > set_nat).
% 32.64/8.94 thf(func_def_207, type, complete_Sup_Sup_nat: set_nat > nat).
% 32.64/8.94 thf(func_def_208, type, comple1385675409528146559p_real: set_real > real).
% 32.64/8.94 thf(func_def_209, type, arg: complex > real).
% 32.64/8.94 thf(func_def_210, type, cis: real > complex).
% 32.64/8.94 thf(func_def_211, type, cnj: complex > complex).
% 32.64/8.94 thf(func_def_212, type, complex2: real > real > complex).
% 32.64/8.94 thf(func_def_213, type, im: complex > real).
% 32.64/8.94 thf(func_def_214, type, re: complex > real).
% 32.64/8.94 thf(func_def_215, type, csqrt: complex > complex).
% 32.64/8.94 thf(func_def_216, type, imaginary_unit: complex).
% 32.64/8.94 thf(func_def_217, type, rcis: real > real > complex).
% 32.64/8.94 thf(func_def_218, type, nth_item_rel: nat > nat > $o).
% 32.64/8.94 thf(func_def_219, type, differ6690327859849518006l_real: (real > real) > filter_real > $o).
% 32.64/8.94 thf(func_def_220, type, has_de1759254742604945161l_real: (real > real) > (real > real) > filter_real > $o).
% 32.64/8.94 thf(func_def_221, type, has_fi5821293074295781190e_real: (real > real) > real > filter_real > $o).
% 32.64/8.94 thf(func_def_222, type, has_ve631408500373753343e_real: (real > real) > real > filter_real > $o).
% 32.64/8.94 thf(func_def_223, type, adjust_div: product_prod_int_int > int).
% 32.64/8.94 thf(func_def_224, type, adjust_mod: int > int > int).
% 32.64/8.94 thf(func_def_225, type, divmod_nat: nat > nat > product_prod_nat_nat).
% 32.64/8.94 thf(func_def_226, type, eucl_rel_int: int > int > product_prod_int_int > $o).
% 32.64/8.94 thf(func_def_227, type, unique5052692396658037445od_int: num > num > product_prod_int_int).
% 32.64/8.94 thf(func_def_228, type, unique5055182867167087721od_nat: num > num > product_prod_nat_nat).
% 32.64/8.94 thf(func_def_229, type, unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 32.64/8.94 thf(func_def_230, type, unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int).
% 32.64/8.94 thf(func_def_231, type, unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat).
% 32.64/8.94 thf(func_def_232, type, euclid4774559944035922753ze_int: int > nat).
% 32.64/8.94 thf(func_def_233, type, euclid4777050414544973029ze_nat: nat > nat).
% 32.64/8.94 thf(func_def_234, type, euclid3395696857347342551nt_int: int > int).
% 32.64/8.94 thf(func_def_235, type, euclid3398187327856392827nt_nat: nat > nat).
% 32.64/8.94 thf(func_def_236, type, extended_eSuc: extended_enat > extended_enat).
% 32.64/8.94 thf(func_def_237, type, extended_enat2: nat > extended_enat).
% 32.64/8.94 thf(func_def_238, type, extend3600170679010898289d_enat: (nat > extended_enat) > extended_enat > extended_enat > extended_enat).
% 32.64/8.94 thf(func_def_239, type, extend5688581933313929465d_enat: extended_enat).
% 32.64/8.94 thf(func_def_240, type, comm_s8582702949713902594nteger: code_integer > nat > code_integer).
% 32.64/8.94 thf(func_def_241, type, comm_s2602460028002588243omplex: complex > nat > complex).
% 32.64/8.94 thf(func_def_242, type, comm_s4660882817536571857er_int: int > nat > int).
% 32.64/8.94 thf(func_def_243, type, comm_s4663373288045622133er_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_244, type, comm_s4028243227959126397er_rat: rat > nat > rat).
% 32.64/8.94 thf(func_def_245, type, comm_s7457072308508201937r_real: real > nat > real).
% 32.64/8.94 thf(func_def_246, type, semiri3624122377584611663nteger: nat > code_integer).
% 32.64/8.94 thf(func_def_247, type, semiri2447717529341329178atural: nat > code_natural).
% 32.64/8.94 thf(func_def_248, type, semiri5044797733671781792omplex: nat > complex).
% 32.64/8.94 thf(func_def_249, type, semiri1406184849735516958ct_int: nat > int).
% 32.64/8.94 thf(func_def_250, type, semiri1408675320244567234ct_nat: nat > nat).
% 32.64/8.94 thf(func_def_251, type, semiri773545260158071498ct_rat: nat > rat).
% 32.64/8.94 thf(func_def_252, type, semiri2265585572941072030t_real: nat > real).
% 32.64/8.94 thf(func_def_253, type, invers8013647133539491842omplex: complex > complex).
% 32.64/8.94 thf(func_def_254, type, inverse_inverse_rat: rat > rat).
% 32.64/8.94 thf(func_def_255, type, inverse_inverse_real: real > real).
% 32.64/8.94 thf(func_def_256, type, at_bot_real: filter_real).
% 32.64/8.94 thf(func_def_257, type, at_top_int: filter_int).
% 32.64/8.94 thf(func_def_258, type, at_top_nat: filter_nat).
% 32.64/8.94 thf(func_def_259, type, at_top_real: filter_real).
% 32.64/8.94 thf(func_def_260, type, eventually_nat: (nat > $o) > filter_nat > $o).
% 32.64/8.94 thf(func_def_261, type, eventually_real: (real > $o) > filter_real > $o).
% 32.64/8.94 thf(func_def_262, type, filterlim_nat_int: (nat > int) > filter_int > filter_nat > $o).
% 32.64/8.94 thf(func_def_263, type, filterlim_nat_nat: (nat > nat) > filter_nat > filter_nat > $o).
% 32.64/8.94 thf(func_def_264, type, filterlim_nat_real: (nat > real) > filter_real > filter_nat > $o).
% 32.64/8.94 thf(func_def_265, type, filterlim_real_real: (real > real) > filter_real > filter_real > $o).
% 32.64/8.94 thf(func_def_266, type, filtermap_real_real: (real > real) > filter_real > filter_real).
% 32.64/8.94 thf(func_def_267, type, finite_card_o: set_o > nat).
% 32.64/8.94 thf(func_def_268, type, finite_card_complex: set_complex > nat).
% 32.64/8.94 thf(func_def_269, type, finite_card_int: set_int > nat).
% 32.64/8.94 thf(func_def_270, type, finite_card_list_nat: set_list_nat > nat).
% 32.64/8.94 thf(func_def_271, type, finite_card_nat: set_nat > nat).
% 32.64/8.94 thf(func_def_272, type, finite410649719033368117t_unit: set_Product_unit > nat).
% 32.64/8.94 thf(func_def_273, type, finite_card_set_nat: set_set_nat > nat).
% 32.64/8.94 thf(func_def_274, type, finite_card_char: set_char > nat).
% 32.64/8.94 thf(func_def_275, type, finite3207457112153483333omplex: set_complex > $o).
% 32.64/8.94 thf(func_def_276, type, finite_finite_int: set_int > $o).
% 32.64/8.94 thf(func_def_277, type, finite_finite_nat: set_nat > $o).
% 32.64/8.94 thf(func_def_278, type, bij_be1856998921033663316omplex: (complex > complex) > set_complex > set_complex > $o).
% 32.64/8.94 thf(func_def_279, type, bij_betw_int_nat: (int > nat) > set_int > set_nat > $o).
% 32.64/8.94 thf(func_def_280, type, bij_be8532844293280997160at_nat: (list_nat > nat) > set_list_nat > set_nat > $o).
% 32.64/8.94 thf(func_def_281, type, bij_betw_nat_complex: (nat > complex) > set_nat > set_complex > $o).
% 32.64/8.94 thf(func_def_282, type, bij_betw_nat_int: (nat > int) > set_nat > set_int > $o).
% 32.64/8.94 thf(func_def_283, type, bij_be6293887246118711976st_nat: (nat > list_nat) > set_nat > set_list_nat > $o).
% 32.64/8.94 thf(func_def_284, type, bij_betw_nat_nat: (nat > nat) > set_nat > set_nat > $o).
% 32.64/8.94 thf(func_def_285, type, bij_be8693218025023041337at_nat: (nat > product_prod_nat_nat) > set_nat > set_Pr1261947904930325089at_nat > $o).
% 32.64/8.94 thf(func_def_286, type, bij_be4790990086886966983at_nat: (nat > sum_sum_nat_nat) > set_nat > set_Sum_sum_nat_nat > $o).
% 32.64/8.94 thf(func_def_287, type, bij_be5333170631980326235at_nat: (product_prod_nat_nat > nat) > set_Pr1261947904930325089at_nat > set_nat > $o).
% 32.64/8.94 thf(func_def_288, type, bij_be5432664580149595207at_nat: (sum_sum_nat_nat > nat) > set_Sum_sum_nat_nat > set_nat > $o).
% 32.64/8.94 thf(func_def_289, type, comp_C8797469213163452608nteger: ((code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 32.64/8.94 thf(func_def_290, type, comp_C1593894019821074884nteger: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 32.64/8.94 thf(func_def_291, type, comp_int_int_num: (int > int) > (num > int) > num > int).
% 32.64/8.94 thf(func_def_292, type, comp_int_nat_int: (int > nat) > (int > int) > int > nat).
% 32.64/8.94 thf(func_def_293, type, comp_nat_nat_nat: (nat > nat) > (nat > nat) > nat > nat).
% 32.64/8.94 thf(func_def_294, type, id_o: $o > $o).
% 32.64/8.94 thf(func_def_295, type, id_nat: nat > nat).
% 32.64/8.94 thf(func_def_296, type, inj_on_int_nat: (int > nat) > set_int > $o).
% 32.64/8.94 thf(func_def_297, type, inj_on_list_nat_nat: (list_nat > nat) > set_list_nat > $o).
% 32.64/8.94 thf(func_def_298, type, inj_on_nat_int: (nat > int) > set_nat > $o).
% 32.64/8.94 thf(func_def_299, type, inj_on_nat_list_nat: (nat > list_nat) > set_nat > $o).
% 32.64/8.94 thf(func_def_300, type, inj_on_nat_nat: (nat > nat) > set_nat > $o).
% 32.64/8.94 thf(func_def_301, type, inj_on5538052773655684606at_nat: (nat > product_prod_nat_nat) > set_nat > $o).
% 32.64/8.94 thf(func_def_302, type, inj_on_nat_char: (nat > char) > set_nat > $o).
% 32.64/8.94 thf(func_def_303, type, inj_on5701776251185195458at_nat: (nat > sum_sum_nat_nat) > set_nat > $o).
% 32.64/8.94 thf(func_def_304, type, inj_on2178005380612969504at_nat: (product_prod_nat_nat > nat) > set_Pr1261947904930325089at_nat > $o).
% 32.64/8.94 thf(func_def_305, type, inj_on_real_real: (real > real) > set_real > $o).
% 32.64/8.94 thf(func_def_306, type, inj_on_set_nat_nat: (set_nat > nat) > set_set_nat > $o).
% 32.64/8.94 thf(func_def_307, type, inj_on6343450744447823682at_nat: (sum_sum_nat_nat > nat) > set_Sum_sum_nat_nat > $o).
% 32.64/8.94 thf(func_def_308, type, map_fu8272188784021352819nteger: (code_integer > int) > ((int > int) > code_integer > code_integer) > (int > int > int) > code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_309, type, map_fu2599414010547811884nteger: (code_integer > int) > (int > code_integer) > (int > int) > code_integer > code_integer).
% 32.64/8.94 thf(func_def_310, type, map_fu6549440983881763648atural: (code_natural > nat) > ((nat > nat) > code_natural > code_natural) > (nat > nat > nat) > code_natural > code_natural > code_natural).
% 32.64/8.94 thf(func_def_311, type, map_fu2787874002554666395nteger: (code_natural > nat) > (int > code_integer) > (nat > int) > code_natural > code_integer).
% 32.64/8.94 thf(func_def_312, type, map_fu1239815594074539274atural: (code_natural > nat) > (nat > code_natural) > (nat > nat) > code_natural > code_natural).
% 32.64/8.94 thf(func_def_313, type, map_fu434086159418415080_int_o: (int > product_prod_nat_nat) > ((product_prod_nat_nat > $o) > int > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > int > int > $o).
% 32.64/8.94 thf(func_def_314, type, map_fu4960017516451851995nt_int: (int > product_prod_nat_nat) > ((product_prod_nat_nat > product_prod_nat_nat) > int > int) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > int > int > int).
% 32.64/8.94 thf(func_def_315, type, map_fu4826362097070443709at_o_o: (int > product_prod_nat_nat) > ($o > $o) > (product_prod_nat_nat > $o) > int > $o).
% 32.64/8.94 thf(func_def_316, type, map_fu2345160673673942751at_nat: (int > product_prod_nat_nat) > (nat > nat) > (product_prod_nat_nat > nat) > int > nat).
% 32.64/8.94 thf(func_def_317, type, map_fu3667384564859982768at_int: (int > product_prod_nat_nat) > (product_prod_nat_nat > int) > (product_prod_nat_nat > product_prod_nat_nat) > int > int).
% 32.64/8.94 thf(func_def_318, type, map_fu8224359078269604234nteger: (nat > nat) > ((int > int) > code_integer > code_integer) > (nat > int > int) > nat > code_integer > code_integer).
% 32.64/8.94 thf(func_def_319, type, map_fu6981160461238011112atural: (nat > nat) > ((nat > nat) > code_natural > code_natural) > (nat > nat > nat) > nat > code_natural > code_natural).
% 32.64/8.94 thf(func_def_320, type, map_fu6290471996055670595nteger: (nat > nat) > (int > code_integer) > (nat > int) > nat > code_integer).
% 32.64/8.94 thf(func_def_321, type, map_fu4333342158222067775at_rat: (rat > product_prod_int_int) > ((product_prod_int_int > product_prod_int_int) > rat > rat) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > rat > rat > rat).
% 32.64/8.94 thf(func_def_322, type, map_fu898904425404107465nt_o_o: (rat > product_prod_int_int) > ($o > $o) > (product_prod_int_int > $o) > rat > $o).
% 32.64/8.94 thf(func_def_323, type, map_fu5673905371560938248nt_rat: (rat > product_prod_int_int) > (product_prod_int_int > rat) > (product_prod_int_int > product_prod_int_int) > rat > rat).
% 32.64/8.94 thf(func_def_324, type, map_fu1532550112467129777l_real: (real > nat > rat) > (((nat > rat) > nat > rat) > real > real) > ((nat > rat) > (nat > rat) > nat > rat) > real > real > real).
% 32.64/8.94 thf(func_def_325, type, map_fu7146612038024189824t_real: (real > nat > rat) > ((nat > rat) > real) > ((nat > rat) > nat > rat) > real > real).
% 32.64/8.94 thf(func_def_326, type, map_fu1856342031159181835at_o_o: (real > nat > rat) > ($o > $o) > ((nat > rat) > $o) > real > $o).
% 32.64/8.94 thf(func_def_327, type, the_in5290026491893676941l_real: set_real > (real > real) > real > real).
% 32.64/8.94 thf(func_def_328, type, fun_is_measure_int: (int > nat) > $o).
% 32.64/8.94 thf(func_def_329, type, gcd_Gcd_int: set_int > int).
% 32.64/8.94 thf(func_def_330, type, gcd_Gcd_nat: set_nat > nat).
% 32.64/8.94 thf(func_def_331, type, gcd_Lcm_int: set_int > int).
% 32.64/8.94 thf(func_def_332, type, gcd_Lcm_nat: set_nat > nat).
% 32.64/8.94 thf(func_def_333, type, bezw: nat > nat > product_prod_int_int).
% 32.64/8.94 thf(func_def_334, type, bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 32.64/8.94 thf(func_def_335, type, gcd_gcd_Code_integer: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_336, type, gcd_gcd_int: int > int > int).
% 32.64/8.94 thf(func_def_337, type, gcd_gcd_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_338, type, gcd_lcm_Code_integer: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_339, type, gcd_lcm_int: int > int > int).
% 32.64/8.94 thf(func_def_340, type, gcd_lcm_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_341, type, gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 32.64/8.94 thf(func_def_342, type, abs_abs_Code_integer: code_integer > code_integer).
% 32.64/8.94 thf(func_def_343, type, abs_abs_complex: complex > complex).
% 32.64/8.94 thf(func_def_344, type, abs_abs_int: int > int).
% 32.64/8.94 thf(func_def_345, type, abs_abs_rat: rat > rat).
% 32.64/8.94 thf(func_def_346, type, abs_abs_real: real > real).
% 32.64/8.94 thf(func_def_347, type, comm_monoid_nat: (nat > nat > nat) > nat > $o).
% 32.64/8.94 thf(func_def_348, type, minus_8373710615458151222nteger: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_349, type, minus_7197305767214868737atural: code_natural > code_natural > code_natural).
% 32.64/8.94 thf(func_def_350, type, minus_minus_complex: complex > complex > complex).
% 32.64/8.94 thf(func_def_351, type, minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat).
% 32.64/8.94 thf(func_def_352, type, minus_minus_int: int > int > int).
% 32.64/8.94 thf(func_def_353, type, minus_minus_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_354, type, minus_minus_rat: rat > rat > rat).
% 32.64/8.94 thf(func_def_355, type, minus_minus_real: real > real > real).
% 32.64/8.94 thf(func_def_356, type, minus_minus_set_nat: set_nat > set_nat > set_nat).
% 32.64/8.94 thf(func_def_357, type, monoid_nat: (nat > nat > nat) > nat > $o).
% 32.64/8.94 thf(func_def_358, type, one_one_Code_integer: code_integer).
% 32.64/8.94 thf(func_def_359, type, one_one_Code_natural: code_natural).
% 32.64/8.94 thf(func_def_360, type, one_one_complex: complex).
% 32.64/8.94 thf(func_def_361, type, one_on7984719198319812577d_enat: extended_enat).
% 32.64/8.94 thf(func_def_362, type, one_one_int: int).
% 32.64/8.94 thf(func_def_363, type, one_one_nat: nat).
% 32.64/8.94 thf(func_def_364, type, one_one_rat: rat).
% 32.64/8.94 thf(func_def_365, type, one_one_real: real).
% 32.64/8.94 thf(func_def_366, type, plus_p5714425477246183910nteger: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_367, type, plus_p4538020629002901425atural: code_natural > code_natural > code_natural).
% 32.64/8.94 thf(func_def_368, type, plus_plus_complex: complex > complex > complex).
% 32.64/8.94 thf(func_def_369, type, plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat).
% 32.64/8.94 thf(func_def_370, type, plus_plus_int: int > int > int).
% 32.64/8.94 thf(func_def_371, type, plus_plus_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_372, type, plus_plus_num: num > num > num).
% 32.64/8.94 thf(func_def_373, type, plus_plus_rat: rat > rat > rat).
% 32.64/8.94 thf(func_def_374, type, plus_plus_real: real > real > real).
% 32.64/8.94 thf(func_def_375, type, plus_plus_literal: literal > literal > literal).
% 32.64/8.94 thf(func_def_376, type, sgn_sgn_Code_integer: code_integer > code_integer).
% 32.64/8.94 thf(func_def_377, type, sgn_sgn_complex: complex > complex).
% 32.64/8.94 thf(func_def_378, type, sgn_sgn_int: int > int).
% 32.64/8.94 thf(func_def_379, type, sgn_sgn_rat: rat > rat).
% 32.64/8.94 thf(func_def_380, type, sgn_sgn_real: real > real).
% 32.64/8.94 thf(func_def_381, type, times_3573771949741848930nteger: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_382, type, times_2397367101498566445atural: code_natural > code_natural > code_natural).
% 32.64/8.94 thf(func_def_383, type, times_times_complex: complex > complex > complex).
% 32.64/8.94 thf(func_def_384, type, times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat).
% 32.64/8.94 thf(func_def_385, type, times_times_int: int > int > int).
% 32.64/8.94 thf(func_def_386, type, times_times_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_387, type, times_times_num: num > num > num).
% 32.64/8.94 thf(func_def_388, type, times_times_rat: rat > rat > rat).
% 32.64/8.94 thf(func_def_389, type, times_times_real: real > real > real).
% 32.64/8.94 thf(func_def_390, type, uminus1351360451143612070nteger: code_integer > code_integer).
% 32.64/8.94 thf(func_def_391, type, uminus1482373934393186551omplex: complex > complex).
% 32.64/8.94 thf(func_def_392, type, uminus_uminus_int: int > int).
% 32.64/8.94 thf(func_def_393, type, uminus_uminus_rat: rat > rat).
% 32.64/8.94 thf(func_def_394, type, uminus_uminus_real: real > real).
% 32.64/8.94 thf(func_def_395, type, zero_z3403309356797280102nteger: code_integer).
% 32.64/8.94 thf(func_def_396, type, zero_z2226904508553997617atural: code_natural).
% 32.64/8.94 thf(func_def_397, type, zero_zero_complex: complex).
% 32.64/8.94 thf(func_def_398, type, zero_z5237406670263579293d_enat: extended_enat).
% 32.64/8.94 thf(func_def_399, type, zero_zero_int: int).
% 32.64/8.94 thf(func_def_400, type, zero_zero_nat: nat).
% 32.64/8.94 thf(func_def_401, type, zero_zero_rat: rat).
% 32.64/8.94 thf(func_def_402, type, zero_zero_real: real).
% 32.64/8.94 thf(func_def_403, type, zero_zero_literal: literal).
% 32.64/8.94 thf(func_def_404, type, groups7754918857620584856omplex: (complex > complex) > set_complex > complex).
% 32.64/8.94 thf(func_def_405, type, groups4538972089207619220nt_int: (int > int) > set_int > int).
% 32.64/8.94 thf(func_def_406, type, groups3542108847815614940at_nat: (nat > nat) > set_nat > nat).
% 32.64/8.94 thf(func_def_407, type, groups6591440286371151544t_real: (nat > real) > set_nat > real).
% 32.64/8.94 thf(func_def_408, type, groups1705073143266064639nt_int: (int > int) > set_int > int).
% 32.64/8.94 thf(func_def_409, type, groups705719431365010083at_int: (nat > int) > set_nat > int).
% 32.64/8.94 thf(func_def_410, type, groups708209901874060359at_nat: (nat > nat) > set_nat > nat).
% 32.64/8.94 thf(func_def_411, type, groups3417619833198082522nteger: ($o > code_integer) > code_integer > list_o > code_integer).
% 32.64/8.94 thf(func_def_412, type, groups9116527308978886569_o_int: ($o > int) > int > list_o > int).
% 32.64/8.94 thf(func_def_413, type, groups9119017779487936845_o_nat: ($o > nat) > nat > list_o > nat).
% 32.64/8.94 thf(func_def_414, type, groups4561878855575611511st_nat: list_nat > nat).
% 32.64/8.94 thf(func_def_415, type, the_int: (int > $o) > int).
% 32.64/8.94 thf(func_def_416, type, the_Pr4378521158711661632nt_int: (product_prod_int_int > $o) > product_prod_int_int).
% 32.64/8.94 thf(func_def_417, type, the_real: (real > $o) > real).
% 32.64/8.94 thf(func_def_418, type, if_int_int: $o > (int > int) > (int > int) > int > int).
% 32.64/8.94 thf(func_def_419, type, if_nat_int_int: $o > (nat > int > int) > (nat > int > int) > nat > int > int).
% 32.64/8.94 thf(func_def_420, type, if_nat_nat_nat: $o > (nat > nat > nat) > (nat > nat > nat) > nat > nat > nat).
% 32.64/8.94 thf(func_def_421, type, if_nat_rat: $o > (nat > rat) > (nat > rat) > nat > rat).
% 32.64/8.94 thf(func_def_422, type, if_Code_integer: $o > code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_423, type, if_Code_natural: $o > code_natural > code_natural > code_natural).
% 32.64/8.94 thf(func_def_424, type, if_complex: $o > complex > complex > complex).
% 32.64/8.94 thf(func_def_425, type, if_Extended_enat: $o > extended_enat > extended_enat > extended_enat).
% 32.64/8.94 thf(func_def_426, type, if_int: $o > int > int > int).
% 32.64/8.94 thf(func_def_427, type, if_list_int: $o > list_int > list_int > list_int).
% 32.64/8.94 thf(func_def_428, type, if_list_nat: $o > list_nat > list_nat > list_nat).
% 32.64/8.94 thf(func_def_429, type, if_nat: $o > nat > nat > nat).
% 32.64/8.94 thf(func_def_430, type, if_num: $o > num > num > num).
% 32.64/8.94 thf(func_def_431, type, if_option_num: $o > option_num > option_num > option_num).
% 32.64/8.94 thf(func_def_432, type, if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o).
% 32.64/8.94 thf(func_def_433, type, if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 32.64/8.94 thf(func_def_434, type, if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int).
% 32.64/8.94 thf(func_def_435, type, if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat).
% 32.64/8.94 thf(func_def_436, type, if_rat: $o > rat > rat > rat).
% 32.64/8.94 thf(func_def_437, type, if_real: $o > real > real > real).
% 32.64/8.94 thf(func_def_438, type, if_set_int: $o > set_int > set_int > set_int).
% 32.64/8.94 thf(func_def_439, type, if_Sum_sum_nat_nat: $o > sum_sum_nat_nat > sum_sum_nat_nat > sum_sum_nat_nat).
% 32.64/8.94 thf(func_def_440, type, abs_Integ: product_prod_nat_nat > int).
% 32.64/8.94 thf(func_def_441, type, neg: num > int).
% 32.64/8.94 thf(func_def_442, type, pos: num > int).
% 32.64/8.94 thf(func_def_443, type, rep_Integ: int > product_prod_nat_nat).
% 32.64/8.94 thf(func_def_444, type, cr_int: product_prod_nat_nat > int > $o).
% 32.64/8.94 thf(func_def_445, type, dup: int > int).
% 32.64/8.94 thf(func_def_446, type, int_ge_less_than: int > set_Pr958786334691620121nt_int).
% 32.64/8.94 thf(func_def_447, type, int_ge_less_than2: int > set_Pr958786334691620121nt_int).
% 32.64/8.94 thf(func_def_448, type, intrel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 32.64/8.94 thf(func_def_449, type, nat2: int > nat).
% 32.64/8.94 thf(func_def_450, type, pcr_int: product_prod_nat_nat > int > $o).
% 32.64/8.94 thf(func_def_451, type, power_int_real: real > int > real).
% 32.64/8.94 thf(func_def_452, type, ring_11222124179247155820nteger: set_Code_integer).
% 32.64/8.94 thf(func_def_453, type, ring_1_Ints_complex: set_complex).
% 32.64/8.94 thf(func_def_454, type, ring_1_Ints_int: set_int).
% 32.64/8.94 thf(func_def_455, type, ring_1_Ints_rat: set_rat).
% 32.64/8.94 thf(func_def_456, type, ring_1_Ints_real: set_real).
% 32.64/8.94 thf(func_def_457, type, ring_18347121197199848620nteger: int > code_integer).
% 32.64/8.94 thf(func_def_458, type, ring_17405671764205052669omplex: int > complex).
% 32.64/8.94 thf(func_def_459, type, ring_1_of_int_int: int > int).
% 32.64/8.94 thf(func_def_460, type, ring_1_of_int_rat: int > rat).
% 32.64/8.94 thf(func_def_461, type, ring_1_of_int_real: int > real).
% 32.64/8.94 thf(func_def_462, type, sub: num > num > int).
% 32.64/8.94 thf(func_def_463, type, inf_inf_int: int > int > int).
% 32.64/8.94 thf(func_def_464, type, inf_inf_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_465, type, semila9081495762789891438tr_nat: (nat > nat > nat) > nat > $o).
% 32.64/8.94 thf(func_def_466, type, semila1623282765462674594er_nat: (nat > nat > nat) > nat > (nat > nat > $o) > (nat > nat > $o) > $o).
% 32.64/8.94 thf(func_def_467, type, semila1248733672344298208er_nat: (nat > nat > nat) > (nat > nat > $o) > (nat > nat > $o) > $o).
% 32.64/8.94 thf(func_def_468, type, sup_sup_int: int > int > int).
% 32.64/8.94 thf(func_def_469, type, sup_sup_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_470, type, sup_sup_set_nat: set_nat > set_nat > set_nat).
% 32.64/8.94 thf(func_def_471, type, sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 32.64/8.94 thf(func_def_472, type, lattic8263393255366662781ax_int: set_int > int).
% 32.64/8.94 thf(func_def_473, type, lattic8265883725875713057ax_nat: set_nat > nat).
% 32.64/8.94 thf(func_def_474, type, lattic7826324295020591184_F_nat: (nat > nat > nat) > nat > set_nat > nat).
% 32.64/8.94 thf(func_def_475, type, quotie3684837364556693515t_real: ((nat > rat) > (nat > rat) > $o) > ((nat > rat) > real) > (real > nat > rat) > ((nat > rat) > real > $o) > $o).
% 32.64/8.94 thf(func_def_476, type, quotie1194848508323700631at_int: (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > int) > (int > product_prod_nat_nat) > (product_prod_nat_nat > int > $o) > $o).
% 32.64/8.94 thf(func_def_477, type, bfun_nat_real: (nat > real) > filter_nat > $o).
% 32.64/8.94 thf(func_def_478, type, at_infinity_real: filter_real).
% 32.64/8.94 thf(func_def_479, type, append_int: list_int > list_int > list_int).
% 32.64/8.94 thf(func_def_480, type, append_nat: list_nat > list_nat > list_nat).
% 32.64/8.94 thf(func_def_481, type, drop_nat: nat > list_nat > list_nat).
% 32.64/8.94 thf(func_def_482, type, fold_int_int: (int > int > int) > list_int > int > int).
% 32.64/8.94 thf(func_def_483, type, fold_nat_nat: (nat > nat > nat) > list_nat > nat > nat).
% 32.64/8.94 thf(func_def_484, type, last_nat: list_nat > nat).
% 32.64/8.94 thf(func_def_485, type, linord2614967742042102400et_nat: set_nat > list_nat).
% 32.64/8.94 thf(func_def_486, type, cons_int: int > list_int > list_int).
% 32.64/8.94 thf(func_def_487, type, cons_nat: nat > list_nat > list_nat).
% 32.64/8.94 thf(func_def_488, type, nil_int: list_int).
% 32.64/8.94 thf(func_def_489, type, nil_nat: list_nat).
% 32.64/8.94 thf(func_def_490, type, map_nat_nat: (nat > nat) > list_nat > list_nat).
% 32.64/8.94 thf(func_def_491, type, set_o2: list_o > set_o).
% 32.64/8.94 thf(func_def_492, type, set_complex2: list_complex > set_complex).
% 32.64/8.94 thf(func_def_493, type, set_int2: list_int > set_int).
% 32.64/8.94 thf(func_def_494, type, set_nat2: list_nat > set_nat).
% 32.64/8.94 thf(func_def_495, type, set_real2: list_real > set_real).
% 32.64/8.94 thf(func_def_496, type, set_set_nat2: list_set_nat > set_set_nat).
% 32.64/8.94 thf(func_def_497, type, set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT).
% 32.64/8.94 thf(func_def_498, type, size_list_VEBT_VEBT: (vEBT_VEBT > nat) > list_VEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_499, type, tl_nat: list_nat > list_nat).
% 32.64/8.94 thf(func_def_500, type, nth_o: list_o > nat > $o).
% 32.64/8.94 thf(func_def_501, type, nth_complex: list_complex > nat > complex).
% 32.64/8.94 thf(func_def_502, type, nth_int: list_int > nat > int).
% 32.64/8.94 thf(func_def_503, type, nth_nat: list_nat > nat > nat).
% 32.64/8.94 thf(func_def_504, type, nth_real: list_real > nat > real).
% 32.64/8.94 thf(func_def_505, type, nth_set_nat: list_set_nat > nat > set_nat).
% 32.64/8.94 thf(func_def_506, type, nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_507, type, replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_508, type, rotate1_o: list_o > list_o).
% 32.64/8.94 thf(func_def_509, type, rotate1_int: list_int > list_int).
% 32.64/8.94 thf(func_def_510, type, rotate1_nat: list_nat > list_nat).
% 32.64/8.94 thf(func_def_511, type, rotate1_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_512, type, subseqs_o: list_o > list_list_o).
% 32.64/8.94 thf(func_def_513, type, subseqs_int: list_int > list_list_int).
% 32.64/8.94 thf(func_def_514, type, subseqs_nat: list_nat > list_list_nat).
% 32.64/8.94 thf(func_def_515, type, subseqs_VEBT_VEBT: list_VEBT_VEBT > list_list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_516, type, take_nat: nat > list_nat > list_nat).
% 32.64/8.94 thf(func_def_517, type, upt: nat > nat > list_nat).
% 32.64/8.94 thf(func_def_518, type, upto: int > int > list_int).
% 32.64/8.94 thf(func_def_519, type, upto_aux: int > int > list_int > list_int).
% 32.64/8.94 thf(func_def_520, type, upto_rel: product_prod_int_int > product_prod_int_int > $o).
% 32.64/8.94 thf(func_def_521, type, suc: nat > nat).
% 32.64/8.94 thf(func_def_522, type, compow_nat_nat: nat > (nat > nat) > nat > nat).
% 32.64/8.94 thf(func_def_523, type, case_nat_o: $o > (nat > $o) > nat > $o).
% 32.64/8.94 thf(func_def_524, type, case_nat_nat: nat > (nat > nat) > nat > nat).
% 32.64/8.94 thf(func_def_525, type, case_nat_option_num: option_num > (nat > option_num) > nat > option_num).
% 32.64/8.94 thf(func_def_526, type, pred: nat > nat).
% 32.64/8.94 thf(func_def_527, type, semiri3842193898606819883omplex: set_complex).
% 32.64/8.94 thf(func_def_528, type, semiring_1_Nats_int: set_int).
% 32.64/8.94 thf(func_def_529, type, semiri4939895301339042750nteger: nat > code_integer).
% 32.64/8.94 thf(func_def_530, type, semiri3763490453095760265atural: nat > code_natural).
% 32.64/8.94 thf(func_def_531, type, semiri8010041392384452111omplex: nat > complex).
% 32.64/8.94 thf(func_def_532, type, semiri1314217659103216013at_int: nat > int).
% 32.64/8.94 thf(func_def_533, type, semiri1316708129612266289at_nat: nat > nat).
% 32.64/8.94 thf(func_def_534, type, semiri681578069525770553at_rat: nat > rat).
% 32.64/8.94 thf(func_def_535, type, semiri5074537144036343181t_real: nat > real).
% 32.64/8.94 thf(func_def_536, type, size_size_list_o: list_o > nat).
% 32.64/8.94 thf(func_def_537, type, size_s3451745648224563538omplex: list_complex > nat).
% 32.64/8.94 thf(func_def_538, type, size_size_list_int: list_int > nat).
% 32.64/8.94 thf(func_def_539, type, size_s2710708370519433104list_o: list_list_o > nat).
% 32.64/8.94 thf(func_def_540, type, size_s533118279054570080st_int: list_list_int > nat).
% 32.64/8.94 thf(func_def_541, type, size_s3023201423986296836st_nat: list_list_nat > nat).
% 32.64/8.94 thf(func_def_542, type, size_s8217280938318005548T_VEBT: list_list_VEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_543, type, size_size_list_nat: list_nat > nat).
% 32.64/8.94 thf(func_def_544, type, size_size_list_real: list_real > nat).
% 32.64/8.94 thf(func_def_545, type, size_s3254054031482475050et_nat: list_set_nat > nat).
% 32.64/8.94 thf(func_def_546, type, size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_547, type, size_size_num: num > nat).
% 32.64/8.94 thf(func_def_548, type, size_size_char: char > nat).
% 32.64/8.94 thf(func_def_549, type, size_size_VEBT_VEBT: vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_550, type, nat_int_decode: nat > int).
% 32.64/8.94 thf(func_def_551, type, nat_int_encode: int > nat).
% 32.64/8.94 thf(func_def_552, type, nat_list_decode: nat > list_nat).
% 32.64/8.94 thf(func_def_553, type, nat_list_decode_rel: nat > nat > $o).
% 32.64/8.94 thf(func_def_554, type, nat_list_encode: list_nat > nat).
% 32.64/8.94 thf(func_def_555, type, nat_list_encode_rel: list_nat > list_nat > $o).
% 32.64/8.94 thf(func_def_556, type, nat_prod_decode: nat > product_prod_nat_nat).
% 32.64/8.94 thf(func_def_557, type, nat_prod_decode_aux: nat > nat > product_prod_nat_nat).
% 32.64/8.94 thf(func_def_558, type, nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 32.64/8.94 thf(func_def_559, type, nat_prod_encode: product_prod_nat_nat > nat).
% 32.64/8.94 thf(func_def_560, type, nat_set_decode: nat > set_nat).
% 32.64/8.94 thf(func_def_561, type, nat_set_encode: set_nat > nat).
% 32.64/8.94 thf(func_def_562, type, nat_sum_decode: nat > sum_sum_nat_nat).
% 32.64/8.94 thf(func_def_563, type, nat_sum_encode: sum_sum_nat_nat > nat).
% 32.64/8.94 thf(func_def_564, type, nat_triangle: nat > nat).
% 32.64/8.94 thf(func_def_565, type, root: nat > real > real).
% 32.64/8.94 thf(func_def_566, type, sqrt: real > real).
% 32.64/8.94 thf(func_def_567, type, bitM: num > num).
% 32.64/8.94 thf(func_def_568, type, inc: num > num).
% 32.64/8.94 thf(func_def_569, type, nat_of_num: num > nat).
% 32.64/8.94 thf(func_def_570, type, neg_nu8804712462038260780nteger: code_integer > code_integer).
% 32.64/8.94 thf(func_def_571, type, neg_nu7009210354673126013omplex: complex > complex).
% 32.64/8.94 thf(func_def_572, type, neg_numeral_dbl_int: int > int).
% 32.64/8.94 thf(func_def_573, type, neg_numeral_dbl_rat: rat > rat).
% 32.64/8.94 thf(func_def_574, type, neg_numeral_dbl_real: real > real).
% 32.64/8.94 thf(func_def_575, type, neg_nu7757733837767384882nteger: code_integer > code_integer).
% 32.64/8.94 thf(func_def_576, type, neg_nu6511756317524482435omplex: complex > complex).
% 32.64/8.94 thf(func_def_577, type, neg_nu3811975205180677377ec_int: int > int).
% 32.64/8.94 thf(func_def_578, type, neg_nu3179335615603231917ec_rat: rat > rat).
% 32.64/8.94 thf(func_def_579, type, neg_nu6075765906172075777c_real: real > real).
% 32.64/8.94 thf(func_def_580, type, neg_nu5831290666863070958nteger: code_integer > code_integer).
% 32.64/8.94 thf(func_def_581, type, neg_nu8557863876264182079omplex: complex > complex).
% 32.64/8.94 thf(func_def_582, type, neg_nu5851722552734809277nc_int: int > int).
% 32.64/8.94 thf(func_def_583, type, neg_nu5219082963157363817nc_rat: rat > rat).
% 32.64/8.94 thf(func_def_584, type, neg_nu8295874005876285629c_real: real > real).
% 32.64/8.94 thf(func_def_585, type, neg_nu5755505904847501662nteger: num > num > code_integer).
% 32.64/8.94 thf(func_def_586, type, neg_nu8416839295433526191omplex: num > num > complex).
% 32.64/8.94 thf(func_def_587, type, neg_numeral_sub_int: num > num > int).
% 32.64/8.94 thf(func_def_588, type, neg_numeral_sub_rat: num > num > rat).
% 32.64/8.94 thf(func_def_589, type, neg_numeral_sub_real: num > num > real).
% 32.64/8.94 thf(func_def_590, type, bit0: num > num).
% 32.64/8.94 thf(func_def_591, type, bit1: num > num).
% 32.64/8.94 thf(func_def_592, type, one: num).
% 32.64/8.94 thf(func_def_593, type, case_num_option_num: option_num > (num > option_num) > (num > option_num) > num > option_num).
% 32.64/8.94 thf(func_def_594, type, size_num: num > nat).
% 32.64/8.94 thf(func_def_595, type, num_of_nat: nat > num).
% 32.64/8.94 thf(func_def_596, type, numera6620942414471956472nteger: num > code_integer).
% 32.64/8.94 thf(func_def_597, type, numera5444537566228673987atural: num > code_natural).
% 32.64/8.94 thf(func_def_598, type, numera6690914467698888265omplex: num > complex).
% 32.64/8.94 thf(func_def_599, type, numera1916890842035813515d_enat: num > extended_enat).
% 32.64/8.94 thf(func_def_600, type, numeral_numeral_int: num > int).
% 32.64/8.94 thf(func_def_601, type, numeral_numeral_nat: num > nat).
% 32.64/8.94 thf(func_def_602, type, numeral_numeral_rat: num > rat).
% 32.64/8.94 thf(func_def_603, type, numeral_numeral_real: num > real).
% 32.64/8.94 thf(func_def_604, type, pow: num > num > num).
% 32.64/8.94 thf(func_def_605, type, pred_numeral: num > nat).
% 32.64/8.94 thf(func_def_606, type, sqr: num > num).
% 32.64/8.94 thf(func_def_607, type, none_num: option_num).
% 32.64/8.94 thf(func_def_608, type, none_P5556105721700978146at_nat: option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_609, type, some_num: num > option_num).
% 32.64/8.94 thf(func_def_610, type, some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_611, type, case_o184042715313410164at_nat: $o > (product_prod_nat_nat > $o) > option4927543243414619207at_nat > $o).
% 32.64/8.94 thf(func_def_612, type, case_option_int_num: int > (num > int) > option_num > int).
% 32.64/8.94 thf(func_def_613, type, case_option_num_num: num > (num > num) > option_num > num).
% 32.64/8.94 thf(func_def_614, type, case_o6005452278849405969um_num: option_num > (num > option_num) > option_num > option_num).
% 32.64/8.94 thf(func_def_615, type, map_option_num_num: (num > num) > option_num > option_num).
% 32.64/8.94 thf(func_def_616, type, bot_bot_nat: nat).
% 32.64/8.94 thf(func_def_617, type, bot_bot_set_int: set_int).
% 32.64/8.94 thf(func_def_618, type, bot_bot_set_nat: set_nat).
% 32.64/8.94 thf(func_def_619, type, bot_bot_set_real: set_real).
% 32.64/8.94 thf(func_def_620, type, ord_Least_nat: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_621, type, ord_Least_real: (real > $o) > real).
% 32.64/8.94 thf(func_def_622, type, ord_le6747313008572928689nteger: code_integer > code_integer > $o).
% 32.64/8.94 thf(func_def_623, type, ord_le5570908160329646204atural: code_natural > code_natural > $o).
% 32.64/8.94 thf(func_def_624, type, ord_le72135733267957522d_enat: extended_enat > extended_enat > $o).
% 32.64/8.94 thf(func_def_625, type, ord_less_int: int > int > $o).
% 32.64/8.94 thf(func_def_626, type, ord_less_nat: nat > nat > $o).
% 32.64/8.94 thf(func_def_627, type, ord_less_num: num > num > $o).
% 32.64/8.94 thf(func_def_628, type, ord_less_rat: rat > rat > $o).
% 32.64/8.94 thf(func_def_629, type, ord_less_real: real > real > $o).
% 32.64/8.94 thf(func_def_630, type, ord_le3102999989581377725nteger: code_integer > code_integer > $o).
% 32.64/8.94 thf(func_def_631, type, ord_le1926595141338095240atural: code_natural > code_natural > $o).
% 32.64/8.94 thf(func_def_632, type, ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o).
% 32.64/8.94 thf(func_def_633, type, ord_less_eq_int: int > int > $o).
% 32.64/8.94 thf(func_def_634, type, ord_less_eq_nat: nat > nat > $o).
% 32.64/8.94 thf(func_def_635, type, ord_less_eq_num: num > num > $o).
% 32.64/8.94 thf(func_def_636, type, ord_less_eq_rat: rat > rat > $o).
% 32.64/8.94 thf(func_def_637, type, ord_less_eq_real: real > real > $o).
% 32.64/8.94 thf(func_def_638, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 32.64/8.94 thf(func_def_639, type, ord_less_eq_set_real: set_real > set_real > $o).
% 32.64/8.94 thf(func_def_640, type, ord_max_Code_integer: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_641, type, ord_max_int: int > int > int).
% 32.64/8.94 thf(func_def_642, type, ord_max_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_643, type, ord_max_num: num > num > num).
% 32.64/8.94 thf(func_def_644, type, ord_max_rat: rat > rat > rat).
% 32.64/8.94 thf(func_def_645, type, ord_max_real: real > real > real).
% 32.64/8.94 thf(func_def_646, type, ord_min_int: int > int > int).
% 32.64/8.94 thf(func_def_647, type, ord_min_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_648, type, order_mono_nat_nat: (nat > nat) > $o).
% 32.64/8.94 thf(func_def_649, type, order_mono_real_real: (real > real) > $o).
% 32.64/8.94 thf(func_def_650, type, order_5726023648592871131at_nat: (nat > nat) > $o).
% 32.64/8.94 thf(func_def_651, type, ordering_top_nat: (nat > nat > $o) > (nat > nat > $o) > nat > $o).
% 32.64/8.94 thf(func_def_652, type, top_top_set_o: set_o).
% 32.64/8.94 thf(func_def_653, type, top_top_set_int: set_int).
% 32.64/8.94 thf(func_def_654, type, top_top_set_list_nat: set_list_nat).
% 32.64/8.94 thf(func_def_655, type, top_top_set_nat: set_nat).
% 32.64/8.94 thf(func_def_656, type, top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat).
% 32.64/8.94 thf(func_def_657, type, top_to1996260823553986621t_unit: set_Product_unit).
% 32.64/8.94 thf(func_def_658, type, top_top_set_real: set_real).
% 32.64/8.94 thf(func_def_659, type, top_top_set_char: set_char).
% 32.64/8.94 thf(func_def_660, type, top_to6661820994512907621at_nat: set_Sum_sum_nat_nat).
% 32.64/8.94 thf(func_def_661, type, power_8256067586552552935nteger: code_integer > nat > code_integer).
% 32.64/8.94 thf(func_def_662, type, power_7079662738309270450atural: code_natural > nat > code_natural).
% 32.64/8.94 thf(func_def_663, type, power_power_complex: complex > nat > complex).
% 32.64/8.94 thf(func_def_664, type, power_power_int: int > nat > int).
% 32.64/8.94 thf(func_def_665, type, power_power_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_666, type, power_power_rat: rat > nat > rat).
% 32.64/8.94 thf(func_def_667, type, power_power_real: real > nat > real).
% 32.64/8.94 thf(func_def_668, type, produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o).
% 32.64/8.94 thf(func_def_669, type, produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger).
% 32.64/8.94 thf(func_def_670, type, produc3574140220909816553atural: code_natural > code_natural > produc7822875418678951345atural).
% 32.64/8.94 thf(func_def_671, type, produc6639722614265839536atural: code_natural > produc7822875418678951345atural > produc5835291356934675326atural).
% 32.64/8.94 thf(func_def_672, type, product_Pair_int_int: int > int > product_prod_int_int).
% 32.64/8.94 thf(func_def_673, type, product_Pair_nat_nat: nat > nat > product_prod_nat_nat).
% 32.64/8.94 thf(func_def_674, type, product_Pair_nat_num: nat > num > product_prod_nat_num).
% 32.64/8.94 thf(func_def_675, type, product_Pair_num_num: num > num > product_prod_num_num).
% 32.64/8.94 thf(func_def_676, type, produc4480994950612372183atural: produc7822875418678951345atural > produc7822875418678951345atural > produc6756470014200349799atural).
% 32.64/8.94 thf(func_def_677, type, produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat).
% 32.64/8.94 thf(func_def_678, type, produc457027306803732586at_nat: set_nat > (nat > set_nat) > set_Pr1261947904930325089at_nat).
% 32.64/8.94 thf(func_def_679, type, produc6499014454317279255nteger: (code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 32.64/8.94 thf(func_def_680, type, produc1553301316500091796er_int: (code_integer > code_integer > int) > produc8923325533196201883nteger > int).
% 32.64/8.94 thf(func_def_681, type, produc1555791787009142072er_nat: (code_integer > code_integer > nat) > produc8923325533196201883nteger > nat).
% 32.64/8.94 thf(func_def_682, type, produc7336495610019696514er_num: (code_integer > code_integer > num) > produc8923325533196201883nteger > num).
% 32.64/8.94 thf(func_def_683, type, produc9125791028180074456eger_o: (code_integer > code_integer > produc6271795597528267376eger_o) > produc8923325533196201883nteger > produc6271795597528267376eger_o).
% 32.64/8.94 thf(func_def_684, type, produc6916734918728496179nteger: (code_integer > code_integer > produc8923325533196201883nteger) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 32.64/8.94 thf(func_def_685, type, produc8282080750456430313atural: (code_natural > code_natural > produc6756470014200349799atural) > produc7822875418678951345atural > produc6756470014200349799atural).
% 32.64/8.94 thf(func_def_686, type, produc4947309494688390418_int_o: (int > int > $o) > product_prod_int_int > $o).
% 32.64/8.94 thf(func_def_687, type, produc8211389475949308722nt_int: (int > int > int) > product_prod_int_int > int).
% 32.64/8.94 thf(func_def_688, type, produc4245557441103728435nt_int: (int > int > product_prod_int_int) > product_prod_int_int > product_prod_int_int).
% 32.64/8.94 thf(func_def_689, type, produc8739625826339149834_nat_o: (nat > nat > product_prod_nat_nat > $o) > product_prod_nat_nat > product_prod_nat_nat > $o).
% 32.64/8.94 thf(func_def_690, type, produc27273713700761075at_nat: (nat > nat > product_prod_nat_nat > product_prod_nat_nat) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat).
% 32.64/8.94 thf(func_def_691, type, produc6081775807080527818_nat_o: (nat > nat > $o) > product_prod_nat_nat > $o).
% 32.64/8.94 thf(func_def_692, type, produc2761476792215241774st_nat: (nat > nat > list_nat) > product_prod_nat_nat > list_nat).
% 32.64/8.94 thf(func_def_693, type, produc6842872674320459806at_nat: (nat > nat > nat) > product_prod_nat_nat > nat).
% 32.64/8.94 thf(func_def_694, type, produc2626176000494625587at_nat: (nat > nat > product_prod_nat_nat) > product_prod_nat_nat > product_prod_nat_nat).
% 32.64/8.94 thf(func_def_695, type, produc478579273971653890on_num: (nat > num > option_num) > product_prod_nat_num > option_num).
% 32.64/8.94 thf(func_def_696, type, product_fst_int_int: product_prod_int_int > int).
% 32.64/8.94 thf(func_def_697, type, product_fst_nat_nat: product_prod_nat_nat > nat).
% 32.64/8.94 thf(func_def_698, type, produc6591912806276919810atural: produc5835291356934675326atural > produc7822875418678951345atural).
% 32.64/8.94 thf(func_def_699, type, product_snd_int_int: product_prod_int_int > int).
% 32.64/8.94 thf(func_def_700, type, product_snd_nat_nat: product_prod_nat_nat > nat).
% 32.64/8.94 thf(func_def_701, type, produc5538323210962509403atural: (produc7822875418678951345atural > produc5835291356934675326atural) > (code_natural > produc7822875418678951345atural > produc5835291356934675326atural) > produc7822875418678951345atural > produc5835291356934675326atural).
% 32.64/8.94 thf(func_def_702, type, quotie8700032322157300518t_real: ((nat > rat) > (nat > rat) > $o) > ((nat > rat) > real) > (real > nat > rat) > $o).
% 32.64/8.94 thf(func_def_703, type, quotie6776551016481293500at_int: (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > int) > (int > product_prod_nat_nat) > $o).
% 32.64/8.94 thf(func_def_704, type, inc_shift: code_natural > code_natural > code_natural).
% 32.64/8.94 thf(func_def_705, type, iterat8892046348760725948atural: code_natural > (code_natural > produc7822875418678951345atural > produc5835291356934675326atural) > code_natural > produc7822875418678951345atural > produc5835291356934675326atural).
% 32.64/8.94 thf(func_def_706, type, log: code_natural > code_natural > code_natural).
% 32.64/8.94 thf(func_def_707, type, log_rel: produc7822875418678951345atural > produc7822875418678951345atural > $o).
% 32.64/8.94 thf(func_def_708, type, minus_shift: code_natural > code_natural > code_natural > code_natural).
% 32.64/8.94 thf(func_def_709, type, next: produc7822875418678951345atural > produc5835291356934675326atural).
% 32.64/8.94 thf(func_def_710, type, range: code_natural > produc7822875418678951345atural > produc5835291356934675326atural).
% 32.64/8.94 thf(func_def_711, type, split_seed: produc7822875418678951345atural > produc6756470014200349799atural).
% 32.64/8.94 thf(func_def_712, type, abs_Rat: product_prod_int_int > rat).
% 32.64/8.94 thf(func_def_713, type, fract: int > int > rat).
% 32.64/8.94 thf(func_def_714, type, frct: product_prod_int_int > rat).
% 32.64/8.94 thf(func_def_715, type, rep_Rat: rat > product_prod_int_int).
% 32.64/8.94 thf(func_def_716, type, field_5140801741446780682s_real: set_real).
% 32.64/8.94 thf(func_def_717, type, field_7254667332652039916t_real: rat > real).
% 32.64/8.94 thf(func_def_718, type, normalize: product_prod_int_int > product_prod_int_int).
% 32.64/8.94 thf(func_def_719, type, of_int: int > rat).
% 32.64/8.94 thf(func_def_720, type, pcr_rat: product_prod_int_int > rat > $o).
% 32.64/8.94 thf(func_def_721, type, positive: rat > $o).
% 32.64/8.94 thf(func_def_722, type, quotient_of: rat > product_prod_int_int).
% 32.64/8.94 thf(func_def_723, type, ratrel: product_prod_int_int > product_prod_int_int > $o).
% 32.64/8.94 thf(func_def_724, type, ratreal: rat > real).
% 32.64/8.94 thf(func_def_725, type, real2: (nat > rat) > real).
% 32.64/8.94 thf(func_def_726, type, cauchy: (nat > rat) > $o).
% 32.64/8.94 thf(func_def_727, type, cr_real: (nat > rat) > real > $o).
% 32.64/8.94 thf(func_def_728, type, pcr_real: (nat > rat) > real > $o).
% 32.64/8.94 thf(func_def_729, type, positive2: real > $o).
% 32.64/8.94 thf(func_def_730, type, realrel: (nat > rat) > (nat > rat) > $o).
% 32.64/8.94 thf(func_def_731, type, rep_real: real > nat > rat).
% 32.64/8.94 thf(func_def_732, type, vanishes: (nat > rat) > $o).
% 32.64/8.94 thf(func_def_733, type, real_V2521375963428798218omplex: set_complex).
% 32.64/8.94 thf(func_def_734, type, real_V470468836141973256s_real: set_real).
% 32.64/8.94 thf(func_def_735, type, real_V5970128139526366754l_real: (real > real) > $o).
% 32.64/8.94 thf(func_def_736, type, real_V3694042436643373181omplex: complex > complex > real).
% 32.64/8.94 thf(func_def_737, type, real_V975177566351809787t_real: real > real > real).
% 32.64/8.94 thf(func_def_738, type, real_V4572627801940501177l_real: (real > real) > $o).
% 32.64/8.94 thf(func_def_739, type, real_V1022390504157884413omplex: complex > real).
% 32.64/8.94 thf(func_def_740, type, real_V7735802525324610683m_real: real > real).
% 32.64/8.94 thf(func_def_741, type, real_V4546457046886955230omplex: real > complex).
% 32.64/8.94 thf(func_def_742, type, real_V1803761363581548252l_real: real > real).
% 32.64/8.94 thf(func_def_743, type, real_V2046097035970521341omplex: real > complex > complex).
% 32.64/8.94 thf(func_def_744, type, real_V1485227260804924795R_real: real > real > real).
% 32.64/8.94 thf(func_def_745, type, transp_nat_rat: ((nat > rat) > (nat > rat) > $o) > $o).
% 32.64/8.94 thf(func_def_746, type, algebr932160517623751201me_int: int > int > $o).
% 32.64/8.94 thf(func_def_747, type, algebr934650988132801477me_nat: nat > nat > $o).
% 32.64/8.94 thf(func_def_748, type, divide6298287555418463151nteger: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_749, type, divide5121882707175180666atural: code_natural > code_natural > code_natural).
% 32.64/8.94 thf(func_def_750, type, divide1717551699836669952omplex: complex > complex > complex).
% 32.64/8.94 thf(func_def_751, type, divide_divide_int: int > int > int).
% 32.64/8.94 thf(func_def_752, type, divide_divide_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_753, type, divide_divide_rat: rat > rat > rat).
% 32.64/8.94 thf(func_def_754, type, divide_divide_real: real > real > real).
% 32.64/8.94 thf(func_def_755, type, dvd_dvd_Code_integer: code_integer > code_integer > $o).
% 32.64/8.94 thf(func_def_756, type, dvd_dvd_Code_natural: code_natural > code_natural > $o).
% 32.64/8.94 thf(func_def_757, type, dvd_dvd_complex: complex > complex > $o).
% 32.64/8.94 thf(func_def_758, type, dvd_dvd_int: int > int > $o).
% 32.64/8.94 thf(func_def_759, type, dvd_dvd_nat: nat > nat > $o).
% 32.64/8.94 thf(func_def_760, type, dvd_dvd_rat: rat > rat > $o).
% 32.64/8.94 thf(func_def_761, type, dvd_dvd_real: real > real > $o).
% 32.64/8.94 thf(func_def_762, type, modulo364778990260209775nteger: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_763, type, modulo8411746178871703098atural: code_natural > code_natural > code_natural).
% 32.64/8.94 thf(func_def_764, type, modulo_modulo_int: int > int > int).
% 32.64/8.94 thf(func_def_765, type, modulo_modulo_nat: nat > nat > nat).
% 32.64/8.94 thf(func_def_766, type, normal571682499473568032ze_int: int > int).
% 32.64/8.94 thf(func_def_767, type, normal574172969982618308ze_nat: nat > nat).
% 32.64/8.94 thf(func_def_768, type, unit_f2746056213392204926or_int: int > int).
% 32.64/8.94 thf(func_def_769, type, unit_f2748546683901255202or_nat: nat > nat).
% 32.64/8.94 thf(func_def_770, type, zero_n356916108424825756nteger: $o > code_integer).
% 32.64/8.94 thf(func_def_771, type, zero_n8403883297036319079atural: $o > code_natural).
% 32.64/8.94 thf(func_def_772, type, zero_n1201886186963655149omplex: $o > complex).
% 32.64/8.94 thf(func_def_773, type, zero_n2684676970156552555ol_int: $o > int).
% 32.64/8.94 thf(func_def_774, type, zero_n2687167440665602831ol_nat: $o > nat).
% 32.64/8.94 thf(func_def_775, type, zero_n2052037380579107095ol_rat: $o > rat).
% 32.64/8.94 thf(func_def_776, type, zero_n3304061248610475627l_real: $o > real).
% 32.64/8.94 thf(func_def_777, type, suminf_real: (nat > real) > real).
% 32.64/8.94 thf(func_def_778, type, summable_real: (nat > real) > $o).
% 32.64/8.94 thf(func_def_779, type, sums_real: (nat > real) > real > $o).
% 32.64/8.94 thf(func_def_780, type, collect_complex: (complex > $o) > set_complex).
% 32.64/8.94 thf(func_def_781, type, collect_int: (int > $o) > set_int).
% 32.64/8.94 thf(func_def_782, type, collect_list_nat: (list_nat > $o) > set_list_nat).
% 32.64/8.94 thf(func_def_783, type, collect_nat: (nat > $o) > set_nat).
% 32.64/8.94 thf(func_def_784, type, collec213857154873943460nt_int: (product_prod_int_int > $o) > set_Pr958786334691620121nt_int).
% 32.64/8.94 thf(func_def_785, type, collec3392354462482085612at_nat: (product_prod_nat_nat > $o) > set_Pr1261947904930325089at_nat).
% 32.64/8.94 thf(func_def_786, type, collect_real: (real > $o) > set_real).
% 32.64/8.94 thf(func_def_787, type, collect_set_nat: (set_nat > $o) > set_set_nat).
% 32.64/8.94 thf(func_def_788, type, pow_nat: set_nat > set_set_nat).
% 32.64/8.94 thf(func_def_789, type, image_int_int: (int > int) > set_int > set_int).
% 32.64/8.94 thf(func_def_790, type, image_int_nat: (int > nat) > set_int > set_nat).
% 32.64/8.94 thf(func_def_791, type, image_list_nat_nat: (list_nat > nat) > set_list_nat > set_nat).
% 32.64/8.94 thf(func_def_792, type, image_nat_int: (nat > int) > set_nat > set_int).
% 32.64/8.94 thf(func_def_793, type, image_nat_list_nat: (nat > list_nat) > set_nat > set_list_nat).
% 32.64/8.94 thf(func_def_794, type, image_nat_nat: (nat > nat) > set_nat > set_nat).
% 32.64/8.94 thf(func_def_795, type, image_5846123807819985514at_nat: (nat > product_prod_nat_nat) > set_nat > set_Pr1261947904930325089at_nat).
% 32.64/8.94 thf(func_def_796, type, image_nat_set_nat: (nat > set_nat) > set_nat > set_set_nat).
% 32.64/8.94 thf(func_def_797, type, image_nat_char: (nat > char) > set_nat > set_char).
% 32.64/8.94 thf(func_def_798, type, image_678696785212003926at_nat: (nat > sum_sum_nat_nat) > set_nat > set_Sum_sum_nat_nat).
% 32.64/8.94 thf(func_def_799, type, image_2486076414777270412at_nat: (product_prod_nat_nat > nat) > set_Pr1261947904930325089at_nat > set_nat).
% 32.64/8.94 thf(func_def_800, type, image_real_real: (real > real) > set_real > set_real).
% 32.64/8.94 thf(func_def_801, type, image_char_nat: (char > nat) > set_char > set_nat).
% 32.64/8.94 thf(func_def_802, type, image_1320371278474632150at_nat: (sum_sum_nat_nat > nat) > set_Sum_sum_nat_nat > set_nat).
% 32.64/8.94 thf(func_def_803, type, insert_int: int > set_int > set_int).
% 32.64/8.94 thf(func_def_804, type, insert_nat: nat > set_nat > set_nat).
% 32.64/8.94 thf(func_def_805, type, insert_real: real > set_real > set_real).
% 32.64/8.94 thf(func_def_806, type, vimage_nat_nat: (nat > nat) > set_nat > set_nat).
% 32.64/8.94 thf(func_def_807, type, set_fo2584398358068434914at_nat: (nat > nat > nat) > nat > nat > nat > nat).
% 32.64/8.94 thf(func_def_808, type, set_or1266510415728281911st_int: int > int > set_int).
% 32.64/8.94 thf(func_def_809, type, set_or1269000886237332187st_nat: nat > nat > set_nat).
% 32.64/8.94 thf(func_def_810, type, set_or1222579329274155063t_real: real > real > set_real).
% 32.64/8.94 thf(func_def_811, type, set_or4662586982721622107an_int: int > int > set_int).
% 32.64/8.94 thf(func_def_812, type, set_or4665077453230672383an_nat: nat > nat > set_nat).
% 32.64/8.94 thf(func_def_813, type, set_ord_atLeast_nat: nat > set_nat).
% 32.64/8.94 thf(func_def_814, type, set_ord_atLeast_real: real > set_real).
% 32.64/8.94 thf(func_def_815, type, set_ord_atMost_nat: nat > set_nat).
% 32.64/8.94 thf(func_def_816, type, set_or6656581121297822940st_int: int > int > set_int).
% 32.64/8.94 thf(func_def_817, type, set_or6659071591806873216st_nat: nat > nat > set_nat).
% 32.64/8.94 thf(func_def_818, type, set_or5832277885323065728an_int: int > int > set_int).
% 32.64/8.94 thf(func_def_819, type, set_or5834768355832116004an_nat: nat > nat > set_nat).
% 32.64/8.94 thf(func_def_820, type, set_or1633881224788618240n_real: real > real > set_real).
% 32.64/8.94 thf(func_def_821, type, set_or1210151606488870762an_nat: nat > set_nat).
% 32.64/8.94 thf(func_def_822, type, set_or5849166863359141190n_real: real > set_real).
% 32.64/8.94 thf(func_def_823, type, set_ord_lessThan_nat: nat > set_nat).
% 32.64/8.94 thf(func_def_824, type, set_or5984915006950818249n_real: real > set_real).
% 32.64/8.94 thf(func_def_825, type, abort_real: literal > (product_unit > real) > real).
% 32.64/8.94 thf(func_def_826, type, literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal).
% 32.64/8.94 thf(func_def_827, type, ascii_of: char > char).
% 32.64/8.94 thf(func_def_828, type, char2: $o > $o > $o > $o > $o > $o > $o > $o > char).
% 32.64/8.94 thf(func_def_829, type, size_char: char > nat).
% 32.64/8.94 thf(func_def_830, type, comm_s629917340098488124ar_nat: char > nat).
% 32.64/8.94 thf(func_def_831, type, integer_of_char: char > code_integer).
% 32.64/8.94 thf(func_def_832, type, unique3096191561947761185of_nat: nat > char).
% 32.64/8.94 thf(func_def_833, type, sum_Inl_nat_nat: nat > sum_sum_nat_nat).
% 32.64/8.94 thf(func_def_834, type, sum_Inr_nat_nat: nat > sum_sum_nat_nat).
% 32.64/8.94 thf(func_def_835, type, sum_ca7763040182479039464nt_nat: (nat > int) > (nat > int) > sum_sum_nat_nat > int).
% 32.64/8.94 thf(func_def_836, type, sum_ca6763686470577984908at_nat: (nat > nat) > (nat > nat) > sum_sum_nat_nat > nat).
% 32.64/8.94 thf(func_def_837, type, topolo4422821103128117721l_real: filter_real > (real > real) > $o).
% 32.64/8.94 thf(func_def_838, type, topolo5044208981011980120l_real: set_real > (real > real) > $o).
% 32.64/8.94 thf(func_def_839, type, topolo6980174941875973593q_real: (nat > real) > $o).
% 32.64/8.94 thf(func_def_840, type, topolo2177554685111907308n_real: real > set_real > filter_real).
% 32.64/8.94 thf(func_def_841, type, topolo7531315842566124627t_real: (nat > real) > $o).
% 32.64/8.94 thf(func_def_842, type, topolo2815343760600316023s_real: real > filter_real).
% 32.64/8.94 thf(func_def_843, type, topolo6517432010174082258omplex: (nat > complex) > $o).
% 32.64/8.94 thf(func_def_844, type, topolo4055970368930404560y_real: (nat > real) > $o).
% 32.64/8.94 thf(func_def_845, type, arccos: real > real).
% 32.64/8.94 thf(func_def_846, type, arcosh_real: real > real).
% 32.64/8.94 thf(func_def_847, type, arcsin: real > real).
% 32.64/8.94 thf(func_def_848, type, arctan: real > real).
% 32.64/8.94 thf(func_def_849, type, arsinh_real: real > real).
% 32.64/8.94 thf(func_def_850, type, artanh_real: real > real).
% 32.64/8.94 thf(func_def_851, type, cos_complex: complex > complex).
% 32.64/8.94 thf(func_def_852, type, cos_real: real > real).
% 32.64/8.94 thf(func_def_853, type, cos_coeff: nat > real).
% 32.64/8.94 thf(func_def_854, type, cosh_complex: complex > complex).
% 32.64/8.94 thf(func_def_855, type, cosh_real: real > real).
% 32.64/8.94 thf(func_def_856, type, cot_complex: complex > complex).
% 32.64/8.94 thf(func_def_857, type, cot_real: real > real).
% 32.64/8.94 thf(func_def_858, type, exp_complex: complex > complex).
% 32.64/8.94 thf(func_def_859, type, exp_real: real > real).
% 32.64/8.94 thf(func_def_860, type, ln_ln_real: real > real).
% 32.64/8.94 thf(func_def_861, type, log2: real > real > real).
% 32.64/8.94 thf(func_def_862, type, pi: real).
% 32.64/8.94 thf(func_def_863, type, powr_real: real > real > real).
% 32.64/8.94 thf(func_def_864, type, powr_real2: real > real > real).
% 32.64/8.94 thf(func_def_865, type, sin_complex: complex > complex).
% 32.64/8.94 thf(func_def_866, type, sin_real: real > real).
% 32.64/8.94 thf(func_def_867, type, sin_coeff: nat > real).
% 32.64/8.94 thf(func_def_868, type, sinh_complex: complex > complex).
% 32.64/8.94 thf(func_def_869, type, sinh_real: real > real).
% 32.64/8.94 thf(func_def_870, type, tan_complex: complex > complex).
% 32.64/8.94 thf(func_def_871, type, tan_real: real > real).
% 32.64/8.94 thf(func_def_872, type, tanh_complex: complex > complex).
% 32.64/8.94 thf(func_def_873, type, tanh_real: real > real).
% 32.64/8.94 thf(func_def_874, type, vEBT_Leaf: $o > $o > vEBT_VEBT).
% 32.64/8.94 thf(func_def_875, type, vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_876, type, vEBT_size_VEBT: vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_877, type, vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_878, type, vEBT_VEBT_high: nat > nat > nat).
% 32.64/8.94 thf(func_def_879, type, vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_880, type, vEBT_VEBT_low: nat > nat > nat).
% 32.64/8.94 thf(func_def_881, type, vEBT_VEBT_membermima: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_882, type, vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 32.64/8.94 thf(func_def_883, type, vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_884, type, vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 32.64/8.94 thf(func_def_885, type, vEBT_VEBT_valid: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_886, type, vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 32.64/8.94 thf(func_def_887, type, vEBT_invar_vebt: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_888, type, vEBT_set_vebt: vEBT_VEBT > set_nat).
% 32.64/8.94 thf(func_def_889, type, vEBT_vebt_buildup: nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_890, type, vEBT_v4011308405150292612up_rel: nat > nat > $o).
% 32.64/8.94 thf(func_def_891, type, accp_list_nat: (list_nat > list_nat > $o) > list_nat > $o).
% 32.64/8.94 thf(func_def_892, type, accp_nat: (nat > nat > $o) > nat > $o).
% 32.64/8.94 thf(func_def_893, type, accp_P8126237942716283194atural: (produc7822875418678951345atural > produc7822875418678951345atural > $o) > produc7822875418678951345atural > $o).
% 32.64/8.94 thf(func_def_894, type, accp_P1096762738010456898nt_int: (product_prod_int_int > product_prod_int_int > $o) > product_prod_int_int > $o).
% 32.64/8.94 thf(func_def_895, type, accp_P4275260045618599050at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 32.64/8.94 thf(func_def_896, type, accp_P3113834385874906142um_num: (product_prod_num_num > product_prod_num_num > $o) > product_prod_num_num > $o).
% 32.64/8.94 thf(func_def_897, type, accp_P2887432264394892906BT_nat: (produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o) > produc9072475918466114483BT_nat > $o).
% 32.64/8.94 thf(func_def_898, type, member_o: $o > set_o > $o).
% 32.64/8.94 thf(func_def_899, type, member_Code_integer: code_integer > set_Code_integer > $o).
% 32.64/8.94 thf(func_def_900, type, member_complex: complex > set_complex > $o).
% 32.64/8.94 thf(func_def_901, type, member_int: int > set_int > $o).
% 32.64/8.94 thf(func_def_902, type, member_list_nat: list_nat > set_list_nat > $o).
% 32.64/8.94 thf(func_def_903, type, member_nat: nat > set_nat > $o).
% 32.64/8.94 thf(func_def_904, type, member_rat: rat > set_rat > $o).
% 32.64/8.94 thf(func_def_905, type, member_real: real > set_real > $o).
% 32.64/8.94 thf(func_def_906, type, member_set_nat: set_nat > set_set_nat > $o).
% 32.64/8.94 thf(func_def_907, type, member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_908, type, va: nat).
% 32.64/8.94 thf(func_def_909, type, y: nat).
% 32.64/8.94 thf(func_def_928, type, sP0: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_929, type, sP1: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_930, type, sP2: real > real > real > $o).
% 32.64/8.94 thf(func_def_931, type, sP3: nat > (nat > nat) > $o).
% 32.64/8.94 thf(func_def_932, type, sP4: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_933, type, sP5: num > num > option_num > $o).
% 32.64/8.94 thf(func_def_934, type, sP6: num > option_num > num > $o).
% 32.64/8.94 thf(func_def_935, type, sP7: num > option_num > num > $o).
% 32.64/8.94 thf(func_def_936, type, sP8: num > num > option_num > $o).
% 32.64/8.94 thf(func_def_937, type, sP9: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_938, type, sP10: num > option_num > num > $o).
% 32.64/8.94 thf(func_def_939, type, sP11: num > option_num > num > $o).
% 32.64/8.94 thf(func_def_940, type, sP12: (real > real) > (nat > real > real) > nat > real > real > $o).
% 32.64/8.94 thf(func_def_941, type, sP13: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 32.64/8.94 thf(func_def_942, type, sP14: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_943, type, sP15: nat > nat > list_VEBT_VEBT > nat > nat > $o).
% 32.64/8.94 thf(func_def_944, type, sP16: list_VEBT_VEBT > vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_945, type, sP17: real > nat > (nat > real > real) > $o).
% 32.64/8.94 thf(func_def_946, type, sP18: $o > nat > vEBT_VEBT > option4927543243414619207at_nat > list_VEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_947, type, sP19: vEBT_VEBT > $o > nat > $o).
% 32.64/8.94 thf(func_def_948, type, sP20: real > real > real > $o).
% 32.64/8.94 thf(func_def_949, type, sP21: nat > nat > (nat > int) > int > $o).
% 32.64/8.94 thf(func_def_950, type, sP22: nat > list_nat > $o).
% 32.64/8.94 thf(func_def_951, type, sP23: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_952, type, sP24: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 32.64/8.94 thf(func_def_953, type, sP25: rat > rat > rat > $o).
% 32.64/8.94 thf(func_def_954, type, sP26: real > num > real > $o).
% 32.64/8.94 thf(func_def_955, type, sP27: rat > num > rat > $o).
% 32.64/8.94 thf(func_def_956, type, sP28: num > real > real > $o).
% 32.64/8.94 thf(func_def_957, type, sP29: nat > list_VEBT_VEBT > nat > nat > nat > $o).
% 32.64/8.94 thf(func_def_958, type, sP30: nat > list_VEBT_VEBT > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_959, type, sP31: nat > nat > nat > list_VEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_960, type, sP32: list_VEBT_VEBT > vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_961, type, sP33: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_962, type, sP34: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_963, type, sP35: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_964, type, sP36: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_965, type, sP37: int > int > (int > $o) > $o).
% 32.64/8.94 thf(func_def_966, type, sP38: int > int > (int > $o) > $o).
% 32.64/8.94 thf(func_def_967, type, sP39: num > option_num > num > $o).
% 32.64/8.94 thf(func_def_968, type, sP40: num > num > option_num > $o).
% 32.64/8.94 thf(func_def_969, type, sP41: num > num > option_num > $o).
% 32.64/8.94 thf(func_def_970, type, sP42: num > option_num > num > $o).
% 32.64/8.94 thf(func_def_971, type, sP43: num > option_num > num > $o).
% 32.64/8.94 thf(func_def_972, type, sP44: num > num > option_num > $o).
% 32.64/8.94 thf(func_def_973, type, sP45: num > option_num > num > $o).
% 32.64/8.94 thf(func_def_974, type, sP46: num > num > option_num > $o).
% 32.64/8.94 thf(func_def_975, type, sP47: rat > rat > rat > $o).
% 32.64/8.94 thf(func_def_976, type, sP48: rat > rat > rat > $o).
% 32.64/8.94 thf(func_def_977, type, sP49: int > int > int > $o).
% 32.64/8.94 thf(func_def_978, type, sP50: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_979, type, sP51: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_980, type, sP52: (rat > $o) > (rat > $o) > (rat > $o) > (rat > $o) > $o).
% 32.64/8.94 thf(func_def_981, type, sP53: rat > rat > rat > $o).
% 32.64/8.94 thf(func_def_982, type, sP54: (int > $o) > int > int > $o).
% 32.64/8.94 thf(func_def_983, type, sP55: (int > $o) > int > int > $o).
% 32.64/8.94 thf(func_def_984, type, sP56: (int > $o) > int > int > $o).
% 32.64/8.94 thf(func_def_985, type, sP57: rat > num > rat > $o).
% 32.64/8.94 thf(func_def_986, type, sP58: set_int > int > (int > $o) > (int > $o) > $o).
% 32.64/8.94 thf(func_def_987, type, sP59: set_int > int > (int > $o) > $o).
% 32.64/8.94 thf(func_def_988, type, sP60: (int > $o) > set_int > int > (int > $o) > $o).
% 32.64/8.94 thf(func_def_989, type, sP61: set_int > int > (int > $o) > $o).
% 32.64/8.94 thf(func_def_990, type, sP62: real > real > real > $o).
% 32.64/8.94 thf(func_def_991, type, sP63: real > real > real > $o).
% 32.64/8.94 thf(func_def_992, type, sP64: (real > $o) > (real > $o) > (real > $o) > (real > $o) > $o).
% 32.64/8.94 thf(func_def_993, type, sP65: int > product_prod_int_int > int > $o).
% 32.64/8.94 thf(func_def_994, type, sP66: real > (nat > real > real) > real > nat > $o).
% 32.64/8.94 thf(func_def_995, type, sP67: (rat > $o) > (rat > $o) > (rat > $o) > (rat > $o) > $o).
% 32.64/8.94 thf(func_def_996, type, sP68: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_997, type, sP69: real > real > real > $o).
% 32.64/8.94 thf(func_def_998, type, sP70: rat > rat > rat > $o).
% 32.64/8.94 thf(func_def_999, type, sP71: real > num > real > $o).
% 32.64/8.94 thf(func_def_1000, type, sP72: real > real > (real > real > $o) > $o).
% 32.64/8.94 thf(func_def_1001, type, sP73: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_1002, type, sP74: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_1003, type, sP75: list_nat > nat > $o).
% 32.64/8.94 thf(func_def_1004, type, sP76: real > num > real > $o).
% 32.64/8.94 thf(func_def_1005, type, sP77: real > real > real > $o).
% 32.64/8.94 thf(func_def_1006, type, sP78: real > real > real > $o).
% 32.64/8.94 thf(func_def_1007, type, sP79: (real > $o) > (real > $o) > (real > $o) > (real > $o) > $o).
% 32.64/8.94 thf(func_def_1008, type, sP80: rat > num > rat > $o).
% 32.64/8.94 thf(func_def_1009, type, sP81: num > num > num > $o).
% 32.64/8.94 thf(func_def_1010, type, sP82: num > num > num > $o).
% 32.64/8.94 thf(func_def_1011, type, sP83: num > num > num > $o).
% 32.64/8.94 thf(func_def_1012, type, sP84: num > num > num > $o).
% 32.64/8.94 thf(func_def_1013, type, sP85: num > num > num > $o).
% 32.64/8.94 thf(func_def_1014, type, sP86: num > num > num > $o).
% 32.64/8.94 thf(func_def_1015, type, sP87: num > num > num > $o).
% 32.64/8.94 thf(func_def_1016, type, sP88: num > num > num > $o).
% 32.64/8.94 thf(func_def_1017, type, sP89: num > num > num > $o).
% 32.64/8.94 thf(func_def_1018, type, sP90: num > num > num > $o).
% 32.64/8.94 thf(func_def_1019, type, sP91: num > num > num > $o).
% 32.64/8.94 thf(func_def_1020, type, sP92: num > num > num > $o).
% 32.64/8.94 thf(func_def_1021, type, sP93: num > num > num > $o).
% 32.64/8.94 thf(func_def_1022, type, sP94: num > num > num > $o).
% 32.64/8.94 thf(func_def_1023, type, sP95: num > num > num > $o).
% 32.64/8.94 thf(func_def_1024, type, sP96: num > num > num > $o).
% 32.64/8.94 thf(func_def_1025, type, sP97: (int > $o) > (int > $o) > (int > $o) > (int > $o) > $o).
% 32.64/8.94 thf(func_def_1026, type, sP98: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1027, type, sP99: nat > list_VEBT_VEBT > nat > nat > nat > $o).
% 32.64/8.94 thf(func_def_1028, type, sP100: nat > nat > list_VEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1029, type, sP101: $o > $o > nat > $o).
% 32.64/8.94 thf(func_def_1030, type, sP102: $o > $o > $o > nat > $o).
% 32.64/8.94 thf(func_def_1031, type, sP103: vEBT_VEBT > $o > nat > $o).
% 32.64/8.94 thf(func_def_1032, type, sP104: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 32.64/8.94 thf(func_def_1033, type, sP105: real > nat > real > $o).
% 32.64/8.94 thf(func_def_1034, type, sP106: (rat > $o) > (rat > $o) > (rat > $o) > (rat > $o) > $o).
% 32.64/8.94 thf(func_def_1035, type, sP107: (num > $o) > (num > $o) > (num > $o) > (num > $o) > $o).
% 32.64/8.94 thf(func_def_1036, type, sP108: real > (real > real) > (real > real > real) > real > $o).
% 32.64/8.94 thf(func_def_1037, type, sP109: real > num > real > $o).
% 32.64/8.94 thf(func_def_1038, type, sP110: $o > nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1039, type, sP111: $o > nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1040, type, sP112: $o > nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1041, type, sP113: real > real > real > $o).
% 32.64/8.94 thf(func_def_1042, type, sP114: rat > num > rat > $o).
% 32.64/8.94 thf(func_def_1043, type, sP115: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_1044, type, sP116: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_1045, type, sP117: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_1046, type, sP118: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_1047, type, sP119: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1048, type, sP120: num > rat > rat > $o).
% 32.64/8.94 thf(func_def_1049, type, sP121: (int > $o) > int > int > $o).
% 32.64/8.94 thf(func_def_1050, type, sP122: int > (int > $o) > int > $o).
% 32.64/8.94 thf(func_def_1051, type, sP123: real > num > real > $o).
% 32.64/8.94 thf(func_def_1052, type, sP124: real > (real > real) > real > $o).
% 32.64/8.94 thf(func_def_1053, type, sP125: real > real > nat > (nat > real > real) > $o).
% 32.64/8.94 thf(func_def_1054, type, sP126: num > num > num > $o).
% 32.64/8.94 thf(func_def_1055, type, sP127: num > num > num > $o).
% 32.64/8.94 thf(func_def_1056, type, sP128: num > num > num > $o).
% 32.64/8.94 thf(func_def_1057, type, sP129: num > num > num > $o).
% 32.64/8.94 thf(func_def_1058, type, sP130: num > num > num > $o).
% 32.64/8.94 thf(func_def_1059, type, sP131: num > num > num > $o).
% 32.64/8.94 thf(func_def_1060, type, sP132: num > num > num > $o).
% 32.64/8.94 thf(func_def_1061, type, sP133: num > num > num > $o).
% 32.64/8.94 thf(func_def_1062, type, sP134: (num > $o) > (num > $o) > (num > $o) > (num > $o) > $o).
% 32.64/8.94 thf(func_def_1063, type, sP135: set_int > (int > $o) > (int > $o) > int > $o).
% 32.64/8.94 thf(func_def_1064, type, sP136: rat > rat > rat > $o).
% 32.64/8.94 thf(func_def_1065, type, sP137: (real > $o) > (real > $o) > (real > $o) > (real > $o) > $o).
% 32.64/8.94 thf(func_def_1066, type, sP138: rat > num > rat > $o).
% 32.64/8.94 thf(func_def_1067, type, sP139: $o > $o > nat > $o).
% 32.64/8.94 thf(func_def_1068, type, sP140: (real > $o) > (real > $o) > (real > $o) > (real > $o) > $o).
% 32.64/8.94 thf(func_def_1069, type, sP141: nat > nat > (nat > $o) > $o).
% 32.64/8.94 thf(func_def_1070, type, sP142: num > num > num > $o).
% 32.64/8.94 thf(func_def_1071, type, sP143: num > num > num > $o).
% 32.64/8.94 thf(func_def_1072, type, sP144: num > num > num > $o).
% 32.64/8.94 thf(func_def_1073, type, sP145: num > num > num > $o).
% 32.64/8.94 thf(func_def_1074, type, sP146: num > num > num > $o).
% 32.64/8.94 thf(func_def_1075, type, sP147: num > num > num > $o).
% 32.64/8.94 thf(func_def_1076, type, sP148: num > num > num > $o).
% 32.64/8.94 thf(func_def_1077, type, sP149: num > num > num > $o).
% 32.64/8.94 thf(func_def_1078, type, sP150: $o > $o > nat > $o).
% 32.64/8.94 thf(func_def_1079, type, sP151: $o > $o > nat > $o > $o).
% 32.64/8.94 thf(func_def_1080, type, sP152: $o > nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1081, type, sP153: vEBT_VEBT > nat > $o > $o).
% 32.64/8.94 thf(func_def_1082, type, sP154: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_1083, type, sP155: real > nat > (nat > real > real) > $o).
% 32.64/8.94 thf(func_def_1084, type, sP156: real > (nat > real > real) > nat > $o).
% 32.64/8.94 thf(func_def_1085, type, sP157: vEBT_VEBT > nat > $o > $o).
% 32.64/8.94 thf(func_def_1086, type, sP158: (int > $o) > int > int > $o).
% 32.64/8.94 thf(func_def_1087, type, sP159: (int > $o) > int > int > $o).
% 32.64/8.94 thf(func_def_1088, type, sP160: (int > $o) > int > int > $o).
% 32.64/8.94 thf(func_def_1089, type, sP161: rat > num > rat > $o).
% 32.64/8.94 thf(func_def_1090, type, sP162: num > rat > rat > $o).
% 32.64/8.94 thf(func_def_1091, type, sP163: (rat > $o) > (rat > $o) > (rat > $o) > (rat > $o) > $o).
% 32.64/8.94 thf(func_def_1092, type, sP164: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1093, type, sP165: real > num > real > $o).
% 32.64/8.94 thf(func_def_1094, type, sP166: (int > $o) > (int > $o) > (int > $o) > (int > $o) > $o).
% 32.64/8.94 thf(func_def_1095, type, sP167: set_int > (int > $o) > (int > $o) > int > $o).
% 32.64/8.94 thf(func_def_1096, type, sP168: rat > rat > rat > $o).
% 32.64/8.94 thf(func_def_1097, type, sP169: (int > $o) > (int > $o) > (int > $o) > (int > $o) > $o).
% 32.64/8.94 thf(func_def_1098, type, sP170: (num > $o) > (num > $o) > (num > $o) > (num > $o) > $o).
% 32.64/8.94 thf(func_def_1099, type, sP171: int > (int > $o) > $o).
% 32.64/8.94 thf(func_def_1100, type, sP172: nat > nat > (nat > $o) > $o).
% 32.64/8.94 thf(func_def_1101, type, sP173: real > num > real > $o).
% 32.64/8.94 thf(func_def_1102, type, sP174: (int > $o) > $o).
% 32.64/8.94 thf(func_def_1103, type, sP175: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_1104, type, sP176: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_1105, type, sP177: num > num > option_num > $o).
% 32.64/8.94 thf(func_def_1106, type, sP178: num > num > option_num > $o).
% 32.64/8.94 thf(func_def_1107, type, sP179: num > num > option_num > $o).
% 32.64/8.94 thf(func_def_1108, type, sP180: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_1109, type, sP181: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_1110, type, sP182: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_1111, type, sP183: (real > real) > real > (real > real) > real > $o).
% 32.64/8.94 thf(func_def_1112, type, sP184: (real > real) > real > real > $o).
% 32.64/8.94 thf(func_def_1113, type, sP185: real > real > (real > real) > $o).
% 32.64/8.94 thf(func_def_1114, type, sP186: real > (real > real) > real > $o).
% 32.64/8.94 thf(func_def_1115, type, sP187: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1116, type, sP188: (num > $o) > (num > $o) > (num > $o) > (num > $o) > $o).
% 32.64/8.94 thf(func_def_1117, type, sP189: real > (real > real) > (real > real) > real > $o).
% 32.64/8.94 thf(func_def_1118, type, sP190: real > real > (real > real) > $o).
% 32.64/8.94 thf(func_def_1119, type, sP191: (real > real) > (real > real) > real > real > $o).
% 32.64/8.94 thf(func_def_1120, type, sP192: (real > real) > real > real > $o).
% 32.64/8.94 thf(func_def_1121, type, sP193: (nat > $o) > $o).
% 32.64/8.94 thf(func_def_1122, type, sP194: (int > $o) > (int > $o) > (int > $o) > (int > $o) > $o).
% 32.64/8.94 thf(func_def_1123, type, sP195: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_1124, type, sP196: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_1125, type, sP197: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_1126, type, sP198: num > option_num > num > $o).
% 32.64/8.94 thf(func_def_1127, type, sP199: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_1128, type, sP200: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_1129, type, sP201: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_1130, type, sP202: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_1131, type, sP203: num > option_num > num > $o).
% 32.64/8.94 thf(func_def_1132, type, sP204: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_1133, type, sP205: num > option_num > num > $o).
% 32.64/8.94 thf(func_def_1134, type, sP206: num > num > option_num > $o).
% 32.64/8.94 thf(func_def_1135, type, sP207: num > num > option_num > $o).
% 32.64/8.94 thf(func_def_1136, type, sP208: num > option_num > num > $o).
% 32.64/8.94 thf(func_def_1137, type, sP209: num > num > option_num > $o).
% 32.64/8.94 thf(func_def_1138, type, sP210: num > option_num > num > $o).
% 32.64/8.94 thf(func_def_1139, type, sP211: (int > $o) > (int > $o) > int > set_int > $o).
% 32.64/8.94 thf(func_def_1140, type, sP212: set_int > int > (int > $o) > $o).
% 32.64/8.94 thf(func_def_1141, type, sP213: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_1142, type, sP214: num > num > option_num > $o).
% 32.64/8.94 thf(func_def_1143, type, sP215: option_num > num > num > $o).
% 32.64/8.94 thf(func_def_1144, type, sP216: num > option_num > num > $o).
% 32.64/8.94 thf(func_def_1145, type, sP217: num > num > option_num > $o).
% 32.64/8.94 thf(func_def_1146, type, sP218: num > option_num > num > $o).
% 32.64/8.94 thf(func_def_1147, type, sP219: num > option_num > num > $o).
% 32.64/8.94 thf(func_def_1148, type, sP220: num > num > option_num > $o).
% 32.64/8.94 thf(func_def_1149, type, sP221: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 32.64/8.94 thf(func_def_1150, type, sP222: vEBT_VEBT > $o > nat > $o).
% 32.64/8.94 thf(func_def_1151, type, sP223: vEBT_VEBT > $o > nat > $o).
% 32.64/8.94 thf(func_def_1152, type, sP224: nat > $o > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1153, type, sP225: nat > $o > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1154, type, sP226: (real > real) > (real > real) > real > real > $o).
% 32.64/8.94 thf(func_def_1155, type, sP227: set_int > (int > $o) > (int > $o) > int > $o).
% 32.64/8.94 thf(func_def_1156, type, sP228: set_int > (int > $o) > int > $o).
% 32.64/8.94 thf(func_def_1157, type, sP229: rat > rat > rat > $o).
% 32.64/8.94 thf(func_def_1158, type, sK230: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1159, type, sK231: nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1160, type, sK232: nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1161, type, sK233: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1162, type, sK234: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1163, type, sK235: vEBT_VEBT > nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1164, type, sK236: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1165, type, sK237: vEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1166, type, sK238: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1167, type, sK239: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1168, type, sK240: nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1169, type, sK241: nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1170, type, sK242: vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1171, type, sK243: vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1172, type, sK244: vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1173, type, sK245: vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1174, type, sK246: (code_integer > $o) > code_integer > code_integer).
% 32.64/8.94 thf(func_def_1175, type, sK247: (code_integer > $o) > code_integer > code_integer).
% 32.64/8.94 thf(func_def_1176, type, sK248: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1177, type, sK249: (nat > $o) > nat > nat).
% 32.64/8.94 thf(func_def_1178, type, sK250: (nat > $o) > nat > nat).
% 32.64/8.94 thf(func_def_1179, type, sK251: rat > rat > rat).
% 32.64/8.94 thf(func_def_1180, type, sK252: real > real > real).
% 32.64/8.94 thf(func_def_1181, type, sK253: real > real > nat).
% 32.64/8.94 thf(func_def_1182, type, sK254: nat > set_nat > nat).
% 32.64/8.94 thf(func_def_1183, type, sK255: real > nat).
% 32.64/8.94 thf(func_def_1184, type, sK256: int > (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_1185, type, sK257: (int > $o) > int).
% 32.64/8.94 thf(func_def_1186, type, sK258: (int > $o) > int > int).
% 32.64/8.94 thf(func_def_1187, type, sK259: (int > $o) > int > int).
% 32.64/8.94 thf(func_def_1188, type, sK260: int > int > int).
% 32.64/8.94 thf(func_def_1189, type, sK261: int > set_int > int).
% 32.64/8.94 thf(func_def_1190, type, sK262: nat > nat).
% 32.64/8.94 thf(func_def_1191, type, sK263: nat > nat).
% 32.64/8.94 thf(func_def_1192, type, sK264: real > (real > real) > set_real > real).
% 32.64/8.94 thf(func_def_1193, type, sK265: int > int).
% 32.64/8.94 thf(func_def_1194, type, sK266: nat > nat > nat > nat).
% 32.64/8.94 thf(func_def_1195, type, sK267: nat > (nat > nat) > nat).
% 32.64/8.94 thf(func_def_1196, type, sK268: nat > (nat > nat) > nat).
% 32.64/8.94 thf(func_def_1197, type, sK269: (nat > nat) > nat > nat).
% 32.64/8.94 thf(func_def_1198, type, sK270: (nat > nat) > nat > nat).
% 32.64/8.94 thf(func_def_1199, type, sK271: rat > int).
% 32.64/8.94 thf(func_def_1200, type, sK272: int > nat).
% 32.64/8.94 thf(func_def_1201, type, sK273: int > nat).
% 32.64/8.94 thf(func_def_1202, type, sK274: rat > nat).
% 32.64/8.94 thf(func_def_1203, type, sK275: int > int).
% 32.64/8.94 thf(func_def_1204, type, sK276: int > int).
% 32.64/8.94 thf(func_def_1205, type, sK277: int > nat).
% 32.64/8.94 thf(func_def_1206, type, sK278: int > nat).
% 32.64/8.94 thf(func_def_1207, type, sK279: rat > product_prod_int_int).
% 32.64/8.94 thf(func_def_1208, type, sK280: rat > rat).
% 32.64/8.94 thf(func_def_1209, type, sK281: num > num).
% 32.64/8.94 thf(func_def_1210, type, sK282: num > option_num > num > num).
% 32.64/8.94 thf(func_def_1211, type, sK283: num > option_num > num > num).
% 32.64/8.94 thf(func_def_1212, type, sK284: num > num).
% 32.64/8.94 thf(func_def_1213, type, sK285: num > num > option_num > num).
% 32.64/8.94 thf(func_def_1214, type, sK286: num > num > option_num > num).
% 32.64/8.94 thf(func_def_1215, type, sK287: num > option_num > num > num).
% 32.64/8.94 thf(func_def_1216, type, sK288: num > option_num > num > num).
% 32.64/8.94 thf(func_def_1217, type, sK289: num > option_num > num > num).
% 32.64/8.94 thf(func_def_1218, type, sK290: num > num > option_num > num).
% 32.64/8.94 thf(func_def_1219, type, sK291: num > num > option_num > num).
% 32.64/8.94 thf(func_def_1220, type, sK292: num > num > option_num > num).
% 32.64/8.94 thf(func_def_1221, type, sK293: int > int > int > int).
% 32.64/8.94 thf(func_def_1222, type, sK294: int > int > int > int).
% 32.64/8.94 thf(func_def_1223, type, sK295: nat > nat > nat).
% 32.64/8.94 thf(func_def_1224, type, sK296: real > real > real).
% 32.64/8.94 thf(func_def_1225, type, sK297: (real > real) > (nat > real > real) > nat > real > real > real).
% 32.64/8.94 thf(func_def_1226, type, sK298: real > (nat > real > real) > nat > real > real).
% 32.64/8.94 thf(func_def_1227, type, sK299: real > (nat > real > real) > nat > real > nat).
% 32.64/8.94 thf(func_def_1228, type, sK300: int > int > set_int > int).
% 32.64/8.94 thf(func_def_1229, type, sK301: int > int > set_int > int).
% 32.64/8.94 thf(func_def_1230, type, sK302: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1231, type, sK303: nat > (nat > $o) > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1232, type, sK304: nat > (nat > $o) > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1233, type, sK305: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_1234, type, sK306: complex > int).
% 32.64/8.94 thf(func_def_1235, type, sK307: num > num).
% 32.64/8.94 thf(func_def_1236, type, sK308: real > real > real).
% 32.64/8.94 thf(func_def_1237, type, sK309: (nat > int) > nat).
% 32.64/8.94 thf(func_def_1238, type, sK310: (nat > $o) > nat > nat).
% 32.64/8.94 thf(func_def_1239, type, sK311: nat > nat > nat).
% 32.64/8.94 thf(func_def_1240, type, sK312: nat > nat > nat).
% 32.64/8.94 thf(func_def_1241, type, sK313: nat > nat > nat).
% 32.64/8.94 thf(func_def_1242, type, sK314: nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1243, type, sK315: nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1244, type, sK316: nat > nat > nat).
% 32.64/8.94 thf(func_def_1245, type, sK317: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1246, type, sK318: (int > int > $o) > int).
% 32.64/8.94 thf(func_def_1247, type, sK319: (int > int > $o) > int).
% 32.64/8.94 thf(func_def_1248, type, sK320: (real > real > $o) > real).
% 32.64/8.94 thf(func_def_1249, type, sK321: (nat > rat) > rat).
% 32.64/8.94 thf(func_def_1250, type, sK322: nat > (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1251, type, sK323: real > real > real).
% 32.64/8.94 thf(func_def_1252, type, sK324: rat > rat).
% 32.64/8.94 thf(func_def_1253, type, sK325: real > int).
% 32.64/8.94 thf(func_def_1254, type, sK326: int > int > set_int > int).
% 32.64/8.94 thf(func_def_1255, type, sK327: int > int > set_int > int).
% 32.64/8.94 thf(func_def_1256, type, sK328: (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1257, type, sK329: real > real).
% 32.64/8.94 thf(func_def_1258, type, sK330: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1259, type, sK331: real > real).
% 32.64/8.94 thf(func_def_1260, type, sK332: nat > complex > nat).
% 32.64/8.94 thf(func_def_1261, type, sK333: int > int > int).
% 32.64/8.94 thf(func_def_1262, type, sK334: nat > rat > nat).
% 32.64/8.94 thf(func_def_1263, type, sK335: (nat > real) > nat).
% 32.64/8.94 thf(func_def_1264, type, sK336: nat > nat > nat > nat).
% 32.64/8.94 thf(func_def_1265, type, sK337: nat > nat > nat > nat).
% 32.64/8.94 thf(func_def_1266, type, sK338: nat > nat > nat > nat).
% 32.64/8.94 thf(func_def_1267, type, sK339: rat > (nat > rat) > (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1268, type, sK340: (nat > rat) > (nat > rat) > rat).
% 32.64/8.94 thf(func_def_1269, type, sK341: nat > (nat > rat) > (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1270, type, sK342: rat > rat).
% 32.64/8.94 thf(func_def_1271, type, sK343: nat > nat > nat).
% 32.64/8.94 thf(func_def_1272, type, sK344: rat > int).
% 32.64/8.94 thf(func_def_1273, type, sK345: rat > int).
% 32.64/8.94 thf(func_def_1274, type, sK346: set_real > real > (real > real) > real).
% 32.64/8.94 thf(func_def_1275, type, sK347: num > num).
% 32.64/8.94 thf(func_def_1276, type, sK348: rat > (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1277, type, sK349: list_o > ($o > $o) > $o).
% 32.64/8.94 thf(func_def_1278, type, sK350: int > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1279, type, sK351: (nat > real) > nat).
% 32.64/8.94 thf(func_def_1280, type, sK352: (nat > real) > nat).
% 32.64/8.94 thf(func_def_1281, type, sK353: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_1282, type, sK354: nat > nat > nat).
% 32.64/8.94 thf(func_def_1283, type, sK355: list_VEBT_VEBT > vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1284, type, sK356: list_VEBT_VEBT > vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1285, type, sK357: nat > nat > list_VEBT_VEBT > nat > nat > nat).
% 32.64/8.94 thf(func_def_1286, type, sK358: nat > nat > list_VEBT_VEBT > nat > nat > nat).
% 32.64/8.94 thf(func_def_1287, type, sK359: list_VEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1288, type, sK360: list_VEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1289, type, sK361: list_VEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1290, type, sK362: rat > rat).
% 32.64/8.94 thf(func_def_1291, type, sK363: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1292, type, sK364: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1293, type, sK365: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1294, type, sK366: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1295, type, sK367: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1296, type, sK368: real > nat > (nat > real > real) > real).
% 32.64/8.94 thf(func_def_1297, type, sK369: real > nat > (nat > real > real) > nat).
% 32.64/8.94 thf(func_def_1298, type, sK370: nat > (real > real) > (nat > real > real) > real > real).
% 32.64/8.94 thf(func_def_1299, type, sK371: real > real).
% 32.64/8.94 thf(func_def_1300, type, sK372: nat > nat).
% 32.64/8.94 thf(func_def_1301, type, sK373: real > real).
% 32.64/8.94 thf(func_def_1302, type, sK374: real > real > real).
% 32.64/8.94 thf(func_def_1303, type, sK375: real > nat).
% 32.64/8.94 thf(func_def_1304, type, sK376: rat > (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1305, type, sK377: rat > int).
% 32.64/8.94 thf(func_def_1306, type, sK378: nat > nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1307, type, sK379: vEBT_VEBT > $o > nat > $o).
% 32.64/8.94 thf(func_def_1308, type, sK380: vEBT_VEBT > $o > nat > $o).
% 32.64/8.94 thf(func_def_1309, type, sK381: list_VEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1310, type, sK382: nat > vEBT_VEBT > $o > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_1311, type, sK383: nat > vEBT_VEBT > $o > nat).
% 32.64/8.94 thf(func_def_1312, type, sK384: nat > vEBT_VEBT > $o > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1313, type, sK385: nat > vEBT_VEBT > $o > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1314, type, sK386: real > real).
% 32.64/8.94 thf(func_def_1315, type, sK387: (int > int > $o) > int > int > int).
% 32.64/8.94 thf(func_def_1316, type, sK388: (int > int > $o) > int > int > int).
% 32.64/8.94 thf(func_def_1317, type, sK389: (int > int > $o) > int).
% 32.64/8.94 thf(func_def_1318, type, sK390: rat > rat).
% 32.64/8.94 thf(func_def_1319, type, sK391: real > nat).
% 32.64/8.94 thf(func_def_1320, type, sK392: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_1321, type, sK393: set_real > real).
% 32.64/8.94 thf(func_def_1322, type, sK394: real > set_real > real).
% 32.64/8.94 thf(func_def_1323, type, sK395: real > set_real > real).
% 32.64/8.94 thf(func_def_1324, type, sK396: rat > rat).
% 32.64/8.94 thf(func_def_1325, type, sK397: nat > nat).
% 32.64/8.94 thf(func_def_1326, type, sK398: nat > nat > (nat > int) > int > nat).
% 32.64/8.94 thf(func_def_1327, type, sK399: nat > (nat > int) > nat > nat).
% 32.64/8.94 thf(func_def_1328, type, sK400: (nat > nat) > nat).
% 32.64/8.94 thf(func_def_1329, type, sK401: (nat > nat) > nat).
% 32.64/8.94 thf(func_def_1330, type, sK402: nat > (nat > int) > nat).
% 32.64/8.94 thf(func_def_1331, type, sK403: nat > int > (nat > int) > nat).
% 32.64/8.94 thf(func_def_1332, type, sK404: int > nat).
% 32.64/8.94 thf(func_def_1333, type, sK405: int > nat).
% 32.64/8.94 thf(func_def_1334, type, sK406: (nat > real) > nat > int).
% 32.64/8.94 thf(func_def_1335, type, sK407: nat > list_nat > nat).
% 32.64/8.94 thf(func_def_1336, type, sK408: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_1337, type, sK409: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_1338, type, sK410: vEBT_VEBT > nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1339, type, sK411: vEBT_VEBT > nat > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_1340, type, sK412: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1341, type, sK413: vEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1342, type, sK414: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1343, type, sK415: nat > (nat > $o) > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1344, type, sK416: nat > (nat > $o) > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1345, type, sK417: real > real > real).
% 32.64/8.94 thf(func_def_1346, type, sK418: nat > nat).
% 32.64/8.94 thf(func_def_1347, type, sK419: (num > $o) > num).
% 32.64/8.94 thf(func_def_1348, type, sK420: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1349, type, sK421: (nat > $o) > $o).
% 32.64/8.94 thf(func_def_1350, type, sK422: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1351, type, sK423: (nat > real > real) > real > nat > nat).
% 32.64/8.94 thf(func_def_1352, type, sK424: (nat > real > real) > real > nat > real).
% 32.64/8.94 thf(func_def_1353, type, sK425: int > int > set_int > int).
% 32.64/8.94 thf(func_def_1354, type, sK426: int > int > set_int > int).
% 32.64/8.94 thf(func_def_1355, type, sK427: nat > nat > nat).
% 32.64/8.94 thf(func_def_1356, type, sK428: nat > nat > nat).
% 32.64/8.94 thf(func_def_1357, type, sK429: rat > nat).
% 32.64/8.94 thf(func_def_1358, type, sK430: nat > nat > nat > nat).
% 32.64/8.94 thf(func_def_1359, type, sK431: nat > nat > nat > nat).
% 32.64/8.94 thf(func_def_1360, type, sK432: (rat > $o) > rat > rat).
% 32.64/8.94 thf(func_def_1361, type, sK433: (rat > $o) > rat > rat).
% 32.64/8.94 thf(func_def_1362, type, sK434: (rat > $o) > rat > rat).
% 32.64/8.94 thf(func_def_1363, type, sK435: (rat > $o) > rat > rat).
% 32.64/8.94 thf(func_def_1364, type, sK436: real > real > real > real).
% 32.64/8.94 thf(func_def_1365, type, sK437: set_nat > nat > nat).
% 32.64/8.94 thf(func_def_1366, type, sK438: int > int).
% 32.64/8.94 thf(func_def_1367, type, sK439: rat > rat).
% 32.64/8.94 thf(func_def_1368, type, sK440: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_1369, type, sK441: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1370, type, sK442: nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1371, type, sK443: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1372, type, sK444: nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1373, type, sK445: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1374, type, sK446: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1375, type, sK447: nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1376, type, sK448: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1377, type, sK449: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1378, type, sK450: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1379, type, sK451: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1380, type, sK452: nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1381, type, sK453: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1382, type, sK454: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1383, type, sK455: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1384, type, sK456: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1385, type, sK457: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1386, type, sK458: vEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1387, type, sK459: vEBT_VEBT > nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1388, type, sK460: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1389, type, sK461: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1390, type, sK462: nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1391, type, sK463: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1392, type, sK464: nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1393, type, sK465: nat > list_VEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1394, type, sK466: nat > list_VEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1395, type, sK467: vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1396, type, sK468: vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1397, type, sK469: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_1398, type, sK470: produc9072475918466114483BT_nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1399, type, sK471: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_1400, type, sK472: produc9072475918466114483BT_nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1401, type, sK473: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_1402, type, sK474: produc9072475918466114483BT_nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1403, type, sK475: produc9072475918466114483BT_nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1404, type, sK476: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_1405, type, sK477: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_1406, type, sK478: produc9072475918466114483BT_nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1407, type, sK479: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_1408, type, sK480: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_1409, type, sK481: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_1410, type, sK482: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_1411, type, sK483: produc9072475918466114483BT_nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1412, type, sK484: produc9072475918466114483BT_nat > $o).
% 32.64/8.94 thf(func_def_1413, type, sK485: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_1414, type, sK486: produc9072475918466114483BT_nat > $o).
% 32.64/8.94 thf(func_def_1415, type, sK487: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_1416, type, sK488: produc9072475918466114483BT_nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1417, type, sK489: produc9072475918466114483BT_nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1418, type, sK490: int > int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1419, type, sK491: int > int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1420, type, sK492: int > int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1421, type, sK493: int > int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1422, type, sK494: num > option_num > num).
% 32.64/8.94 thf(func_def_1423, type, sK495: num > option_num > num > num).
% 32.64/8.94 thf(func_def_1424, type, sK496: num > option_num > num > num).
% 32.64/8.94 thf(func_def_1425, type, sK497: num > num > option_num > num).
% 32.64/8.94 thf(func_def_1426, type, sK498: num > option_num > num).
% 32.64/8.94 thf(func_def_1427, type, sK499: num > option_num > num > num).
% 32.64/8.94 thf(func_def_1428, type, sK500: num > option_num > num > num).
% 32.64/8.94 thf(func_def_1429, type, sK501: num > num > option_num > num).
% 32.64/8.94 thf(func_def_1430, type, sK502: num > num > option_num > num).
% 32.64/8.94 thf(func_def_1431, type, sK503: num > num > option_num > num).
% 32.64/8.94 thf(func_def_1432, type, sK504: num > num > option_num > num).
% 32.64/8.94 thf(func_def_1433, type, sK505: num > option_num > num > num).
% 32.64/8.94 thf(func_def_1434, type, sK506: (nat > rat) > rat).
% 32.64/8.94 thf(func_def_1435, type, sK507: nat > (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1436, type, sK508: nat > (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1437, type, sK509: (nat > real) > nat).
% 32.64/8.94 thf(func_def_1438, type, sK510: (nat > $o) > nat > nat).
% 32.64/8.94 thf(func_def_1439, type, sK511: (nat > $o) > nat > nat).
% 32.64/8.94 thf(func_def_1440, type, sK512: int > nat).
% 32.64/8.94 thf(func_def_1441, type, sK513: list_int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1442, type, sK514: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1443, type, sK515: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1444, type, sK516: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1445, type, sK517: (real > $o) > int).
% 32.64/8.94 thf(func_def_1446, type, sK518: rat > nat).
% 32.64/8.94 thf(func_def_1447, type, sK519: rat > rat).
% 32.64/8.94 thf(func_def_1448, type, sK520: nat > real > real).
% 32.64/8.94 thf(func_def_1449, type, sK521: real > nat).
% 32.64/8.94 thf(func_def_1450, type, sK522: real > nat).
% 32.64/8.94 thf(func_def_1451, type, sK523: int > nat).
% 32.64/8.94 thf(func_def_1452, type, sK524: int > nat).
% 32.64/8.94 thf(func_def_1453, type, sK525: rat > (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1454, type, sK526: (nat > rat) > rat).
% 32.64/8.94 thf(func_def_1455, type, sK527: nat > (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1456, type, sK528: num > num).
% 32.64/8.94 thf(func_def_1457, type, sK529: real > real > real).
% 32.64/8.94 thf(func_def_1458, type, sK530: nat > nat > nat > nat).
% 32.64/8.94 thf(func_def_1459, type, sK531: (rat > $o) > int).
% 32.64/8.94 thf(func_def_1460, type, sK532: rat > rat > rat).
% 32.64/8.94 thf(func_def_1461, type, sK533: (int > $o) > real > int).
% 32.64/8.94 thf(func_def_1462, type, sK534: real > (nat > real) > nat).
% 32.64/8.94 thf(func_def_1463, type, sK535: real > (nat > real) > real).
% 32.64/8.94 thf(func_def_1464, type, sK536: (nat > real) > nat).
% 32.64/8.94 thf(func_def_1465, type, sK537: (real > real) > real > real).
% 32.64/8.94 thf(func_def_1466, type, sK538: nat > code_natural > code_natural).
% 32.64/8.94 thf(func_def_1467, type, sK539: int > int > int > int).
% 32.64/8.94 thf(func_def_1468, type, sK540: real > real).
% 32.64/8.94 thf(func_def_1469, type, sK541: nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1470, type, sK542: nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1471, type, sK543: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1472, type, sK544: nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1473, type, sK545: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1474, type, sK546: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1475, type, sK547: nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1476, type, sK548: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1477, type, sK549: nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1478, type, sK550: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1479, type, sK551: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1480, type, sK552: nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1481, type, sK553: real > real).
% 32.64/8.94 thf(func_def_1482, type, sK554: (rat > $o) > (rat > $o) > (rat > $o) > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_1483, type, sK555: rat > (rat > $o) > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_1484, type, sK556: rat > (rat > $o) > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_1485, type, sK557: int > nat).
% 32.64/8.94 thf(func_def_1486, type, sK558: real > int).
% 32.64/8.94 thf(func_def_1487, type, sK559: list_VEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1488, type, sK560: int > nat).
% 32.64/8.94 thf(func_def_1489, type, sK561: int > nat).
% 32.64/8.94 thf(func_def_1490, type, sK562: nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1491, type, sK563: nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1492, type, sK564: complex > real).
% 32.64/8.94 thf(func_def_1493, type, sK565: num > num).
% 32.64/8.94 thf(func_def_1494, type, sK566: real > real > nat).
% 32.64/8.94 thf(func_def_1495, type, sK567: rat > int).
% 32.64/8.94 thf(func_def_1496, type, sK568: rat > int).
% 32.64/8.94 thf(func_def_1497, type, sK569: (nat > rat) > real > nat).
% 32.64/8.94 thf(func_def_1498, type, sK570: (nat > set_nat) > nat).
% 32.64/8.94 thf(func_def_1499, type, sK571: set_nat > nat).
% 32.64/8.94 thf(func_def_1500, type, sK572: set_nat > nat).
% 32.64/8.94 thf(func_def_1501, type, sK573: nat > nat > nat > nat).
% 32.64/8.94 thf(func_def_1502, type, sK574: set_int > int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1503, type, sK575: int > set_int > int > int).
% 32.64/8.94 thf(func_def_1504, type, sK576: int > set_int > int > int).
% 32.64/8.94 thf(func_def_1505, type, sK577: (int > $o) > set_int > int > int).
% 32.64/8.94 thf(func_def_1506, type, sK578: set_int > int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1507, type, sK579: int > set_int > int > int).
% 32.64/8.94 thf(func_def_1508, type, sK580: int > set_int > int > int).
% 32.64/8.94 thf(func_def_1509, type, sK581: (int > $o) > set_int > int > int).
% 32.64/8.94 thf(func_def_1510, type, sK582: nat > nat).
% 32.64/8.94 thf(func_def_1511, type, sK583: real > real).
% 32.64/8.94 thf(func_def_1512, type, sK584: list_nat > nat).
% 32.64/8.94 thf(func_def_1513, type, sK585: list_nat > list_nat).
% 32.64/8.94 thf(func_def_1514, type, sK586: real > (real > $o) > real).
% 32.64/8.94 thf(func_def_1515, type, sK587: real > (real > $o) > real).
% 32.64/8.94 thf(func_def_1516, type, sK588: code_integer > nat).
% 32.64/8.94 thf(func_def_1517, type, sK589: real > nat).
% 32.64/8.94 thf(func_def_1518, type, sK590: real > int).
% 32.64/8.94 thf(func_def_1519, type, sK591: rat > int).
% 32.64/8.94 thf(func_def_1520, type, sK592: num > num).
% 32.64/8.94 thf(func_def_1521, type, sK593: (nat > $o) > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1522, type, sK594: rat > int).
% 32.64/8.94 thf(func_def_1523, type, sK595: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1524, type, sK596: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1525, type, sK597: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1526, type, sK598: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1527, type, sK599: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1528, type, sK600: vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1529, type, sK601: list_VEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1530, type, sK602: list_VEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1531, type, sK603: list_VEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1532, type, sK604: real > (real > $o) > real).
% 32.64/8.94 thf(func_def_1533, type, sK605: real > (real > $o) > real).
% 32.64/8.94 thf(func_def_1534, type, sK606: (real > $o) > real > real).
% 32.64/8.94 thf(func_def_1535, type, sK607: (real > $o) > real > real).
% 32.64/8.94 thf(func_def_1536, type, sK608: nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1537, type, sK609: real > int).
% 32.64/8.94 thf(func_def_1538, type, sK610: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_1539, type, sK611: real > nat > real).
% 32.64/8.94 thf(func_def_1540, type, sK612: int > int > set_int > int).
% 32.64/8.94 thf(func_def_1541, type, sK613: int > int > set_int > int).
% 32.64/8.94 thf(func_def_1542, type, sK614: nat > nat > nat).
% 32.64/8.94 thf(func_def_1543, type, sK615: real > real > real).
% 32.64/8.94 thf(func_def_1544, type, sK616: real > real > real).
% 32.64/8.94 thf(func_def_1545, type, sK617: extended_enat > extended_enat > nat > nat).
% 32.64/8.94 thf(func_def_1546, type, sK618: extended_enat > extended_enat > nat > nat).
% 32.64/8.94 thf(func_def_1547, type, sK619: (vEBT_VEBT > $o) > list_VEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1548, type, sK620: (nat > nat) > nat).
% 32.64/8.94 thf(func_def_1549, type, sK621: rat > nat).
% 32.64/8.94 thf(func_def_1550, type, sK622: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1551, type, sK623: num > num).
% 32.64/8.94 thf(func_def_1552, type, sK624: (real > $o) > (real > $o) > (real > $o) > (real > $o) > real).
% 32.64/8.94 thf(func_def_1553, type, sK625: real > (real > $o) > (real > $o) > real).
% 32.64/8.94 thf(func_def_1554, type, sK626: real > (real > $o) > (real > $o) > real).
% 32.64/8.94 thf(func_def_1555, type, sK627: (nat > $o) > nat > nat).
% 32.64/8.94 thf(func_def_1556, type, sK628: nat > nat > nat).
% 32.64/8.94 thf(func_def_1557, type, sK629: int > product_prod_int_int > int > int).
% 32.64/8.94 thf(func_def_1558, type, sK630: int > product_prod_int_int > int > int).
% 32.64/8.94 thf(func_def_1559, type, sK631: product_prod_int_int > int > int > int).
% 32.64/8.94 thf(func_def_1560, type, sK632: real > (nat > real > real) > real > nat > nat).
% 32.64/8.94 thf(func_def_1561, type, sK633: real > (nat > real > real) > real > nat > real).
% 32.64/8.94 thf(func_def_1562, type, sK634: nat > real > real > (real > real) > (nat > real > real) > real).
% 32.64/8.94 thf(func_def_1563, type, sK635: list_nat > nat > nat).
% 32.64/8.94 thf(func_def_1564, type, sK636: list_nat > nat > list_nat).
% 32.64/8.94 thf(func_def_1565, type, sK637: (nat > rat) > rat).
% 32.64/8.94 thf(func_def_1566, type, sK638: (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1567, type, sK639: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1568, type, sK640: (rat > $o) > (rat > $o) > (rat > $o) > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_1569, type, sK641: rat > (rat > $o) > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_1570, type, sK642: rat > (rat > $o) > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_1571, type, sK643: rat > rat > rat).
% 32.64/8.94 thf(func_def_1572, type, sK644: int > nat).
% 32.64/8.94 thf(func_def_1573, type, sK645: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_1574, type, sK646: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1575, type, sK647: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1576, type, sK648: (nat > $o) > nat > nat > nat).
% 32.64/8.94 thf(func_def_1577, type, sK649: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1578, type, sK650: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1579, type, sK651: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1580, type, sK652: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1581, type, sK653: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1582, type, sK654: (complex > $o) > list_complex > complex).
% 32.64/8.94 thf(func_def_1583, type, sK655: int > int).
% 32.64/8.94 thf(func_def_1584, type, sK656: (real > real) > real > set_real > real).
% 32.64/8.94 thf(func_def_1585, type, sK657: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1586, type, sK658: (nat > real) > real > nat).
% 32.64/8.94 thf(func_def_1587, type, sK659: rat > rat > rat).
% 32.64/8.94 thf(func_def_1588, type, sK660: nat > nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1589, type, sK661: real > real).
% 32.64/8.94 thf(func_def_1590, type, sK662: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1591, type, sK663: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1592, type, sK664: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1593, type, sK665: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1594, type, sK666: (nat > real) > real).
% 32.64/8.94 thf(func_def_1595, type, sK667: nat > (nat > real) > nat).
% 32.64/8.94 thf(func_def_1596, type, sK668: nat > (nat > real) > nat).
% 32.64/8.94 thf(func_def_1597, type, sK669: (set_nat > $o) > list_set_nat > set_nat).
% 32.64/8.94 thf(func_def_1598, type, sK670: rat > int).
% 32.64/8.94 thf(func_def_1599, type, sK671: code_integer > code_integer).
% 32.64/8.94 thf(func_def_1600, type, sK672: nat > nat > nat > nat).
% 32.64/8.94 thf(func_def_1601, type, sK673: nat > nat > nat > nat).
% 32.64/8.94 thf(func_def_1602, type, sK674: nat > nat > nat).
% 32.64/8.94 thf(func_def_1603, type, sK675: vEBT_VEBT > nat > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_1604, type, sK676: vEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1605, type, sK677: vEBT_VEBT > nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1606, type, sK678: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1607, type, sK679: vEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1608, type, sK680: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1609, type, sK681: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1610, type, sK682: (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1611, type, sK683: (nat > $o) > nat > nat).
% 32.64/8.94 thf(func_def_1612, type, sK684: (nat > $o) > nat > nat).
% 32.64/8.94 thf(func_def_1613, type, sK685: nat > nat).
% 32.64/8.94 thf(func_def_1614, type, sK686: int > int).
% 32.64/8.94 thf(func_def_1615, type, sK687: real > real > (real > real > $o) > real).
% 32.64/8.94 thf(func_def_1616, type, sK688: real > real > real > (real > real > $o) > real).
% 32.64/8.94 thf(func_def_1617, type, sK689: real > real > real > (real > real > $o) > real).
% 32.64/8.94 thf(func_def_1618, type, sK690: (real > real > $o) > real).
% 32.64/8.94 thf(func_def_1619, type, sK691: (real > real > $o) > real).
% 32.64/8.94 thf(func_def_1620, type, sK692: (real > real > $o) > real).
% 32.64/8.94 thf(func_def_1621, type, sK693: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1622, type, sK694: (nat > $o) > nat > nat).
% 32.64/8.94 thf(func_def_1623, type, sK695: (nat > $o) > nat > nat).
% 32.64/8.94 thf(func_def_1624, type, sK696: (nat > real) > nat).
% 32.64/8.94 thf(func_def_1625, type, sK697: (nat > real) > nat).
% 32.64/8.94 thf(func_def_1626, type, sK698: (real > real) > set_real > real).
% 32.64/8.94 thf(func_def_1627, type, sK699: list_nat > nat > nat).
% 32.64/8.94 thf(func_def_1628, type, sK700: list_nat > nat > list_nat).
% 32.64/8.94 thf(func_def_1629, type, sK701: nat > nat).
% 32.64/8.94 thf(func_def_1630, type, sK702: nat > (nat > $o) > nat > nat).
% 32.64/8.94 thf(func_def_1631, type, sK703: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1632, type, sK704: nat > nat).
% 32.64/8.94 thf(func_def_1633, type, sK705: real > real).
% 32.64/8.94 thf(func_def_1634, type, sK706: code_integer > code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_1635, type, sK707: nat > nat).
% 32.64/8.94 thf(func_def_1636, type, sK708: nat > real > real).
% 32.64/8.94 thf(func_def_1637, type, sK709: real > real > real).
% 32.64/8.94 thf(func_def_1638, type, sK710: (nat > $o) > nat > nat).
% 32.64/8.94 thf(func_def_1639, type, sK711: (nat > $o) > nat > nat).
% 32.64/8.94 thf(func_def_1640, type, sK712: int > int > nat).
% 32.64/8.94 thf(func_def_1641, type, sK713: nat > nat > nat).
% 32.64/8.94 thf(func_def_1642, type, sK714: real > nat).
% 32.64/8.94 thf(func_def_1643, type, sK715: real > nat).
% 32.64/8.94 thf(func_def_1644, type, sK716: (real > $o) > (real > $o) > (real > $o) > (real > $o) > real).
% 32.64/8.94 thf(func_def_1645, type, sK717: real > (real > $o) > (real > $o) > real).
% 32.64/8.94 thf(func_def_1646, type, sK718: real > (real > $o) > (real > $o) > real).
% 32.64/8.94 thf(func_def_1647, type, sK719: int > nat).
% 32.64/8.94 thf(func_def_1648, type, sK720: nat > nat).
% 32.64/8.94 thf(func_def_1649, type, sK721: (real > $o) > real > nat > real).
% 32.64/8.94 thf(func_def_1650, type, sK722: int > int > int).
% 32.64/8.94 thf(func_def_1651, type, sK723: num > num > num).
% 32.64/8.94 thf(func_def_1652, type, sK724: num > num > num > num).
% 32.64/8.94 thf(func_def_1653, type, sK725: num > num > num > num).
% 32.64/8.94 thf(func_def_1654, type, sK726: num > num > num > num).
% 32.64/8.94 thf(func_def_1655, type, sK727: num > num > num > num).
% 32.64/8.94 thf(func_def_1656, type, sK728: num > num > num > num).
% 32.64/8.94 thf(func_def_1657, type, sK729: num > num > num > num).
% 32.64/8.94 thf(func_def_1658, type, sK730: num > num > num > num).
% 32.64/8.94 thf(func_def_1659, type, sK731: num > num > num).
% 32.64/8.94 thf(func_def_1660, type, sK732: num > num > num > num).
% 32.64/8.94 thf(func_def_1661, type, sK733: num > num > num > num).
% 32.64/8.94 thf(func_def_1662, type, sK734: num > num > num > num).
% 32.64/8.94 thf(func_def_1663, type, sK735: real > int).
% 32.64/8.94 thf(func_def_1664, type, sK736: nat > real > real).
% 32.64/8.94 thf(func_def_1665, type, sK737: list_real > (real > $o) > real).
% 32.64/8.94 thf(func_def_1666, type, sK738: num > num > num > num).
% 32.64/8.94 thf(func_def_1667, type, sK739: num > num > num > num).
% 32.64/8.94 thf(func_def_1668, type, sK740: num > num > num).
% 32.64/8.94 thf(func_def_1669, type, sK741: num > num).
% 32.64/8.94 thf(func_def_1670, type, sK742: num > num > num > num).
% 32.64/8.94 thf(func_def_1671, type, sK743: num > num > num > num).
% 32.64/8.94 thf(func_def_1672, type, sK744: num > num > num).
% 32.64/8.94 thf(func_def_1673, type, sK745: num > num).
% 32.64/8.94 thf(func_def_1674, type, sK746: num > num > num > num).
% 32.64/8.94 thf(func_def_1675, type, sK747: num > num > num > num).
% 32.64/8.94 thf(func_def_1676, type, sK748: num > num > num > num).
% 32.64/8.94 thf(func_def_1677, type, sK749: num > num > num > num).
% 32.64/8.94 thf(func_def_1678, type, sK750: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_1679, type, sK751: int > (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_1680, type, sK752: int > (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_1681, type, sK753: rat > rat).
% 32.64/8.94 thf(func_def_1682, type, sK754: nat > real > (nat > real) > (real > real) > real).
% 32.64/8.94 thf(func_def_1683, type, sK755: product_prod_nat_nat > nat).
% 32.64/8.94 thf(func_def_1684, type, sK756: product_prod_nat_nat > nat).
% 32.64/8.94 thf(func_def_1685, type, sK757: nat > real > real).
% 32.64/8.94 thf(func_def_1686, type, sK758: (rat > $o) > int).
% 32.64/8.94 thf(func_def_1687, type, sK759: (rat > $o) > int).
% 32.64/8.94 thf(func_def_1688, type, sK760: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1689, type, sK761: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1690, type, sK762: vEBT_VEBT > nat > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_1691, type, sK763: vEBT_VEBT > nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1692, type, sK764: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1693, type, sK765: vEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1694, type, sK766: num > num).
% 32.64/8.94 thf(func_def_1695, type, sK767: nat > nat).
% 32.64/8.94 thf(func_def_1696, type, sK768: (int > int > $o) > int).
% 32.64/8.94 thf(func_def_1697, type, sK769: (int > int > $o) > int).
% 32.64/8.94 thf(func_def_1698, type, sK770: set_nat > nat).
% 32.64/8.94 thf(func_def_1699, type, sK771: set_nat > nat).
% 32.64/8.94 thf(func_def_1700, type, sK772: list_VEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1701, type, sK773: list_VEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1702, type, sK774: nat > list_VEBT_VEBT > nat > nat > nat > nat).
% 32.64/8.94 thf(func_def_1703, type, sK775: nat > list_VEBT_VEBT > nat > nat > nat > nat).
% 32.64/8.94 thf(func_def_1704, type, sK776: vEBT_VEBT > nat > list_VEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1705, type, sK777: vEBT_VEBT > nat > list_VEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1706, type, sK778: nat > list_VEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1707, type, sK779: real > nat > real).
% 32.64/8.94 thf(func_def_1708, type, sK780: num > num).
% 32.64/8.94 thf(func_def_1709, type, sK781: vEBT_VEBT > $o > nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1710, type, sK782: vEBT_VEBT > $o > nat > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_1711, type, sK783: vEBT_VEBT > $o > nat > nat).
% 32.64/8.94 thf(func_def_1712, type, sK784: vEBT_VEBT > $o > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1713, type, sK785: vEBT_VEBT > $o > nat > $o).
% 32.64/8.94 thf(func_def_1714, type, sK786: vEBT_VEBT > $o > nat > $o).
% 32.64/8.94 thf(func_def_1715, type, sK787: vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1716, type, sK788: vEBT_VEBT > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_1717, type, sK789: vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1718, type, sK790: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1719, type, sK791: nat > (nat > $o) > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1720, type, sK792: nat > (nat > $o) > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1721, type, sK793: int > nat).
% 32.64/8.94 thf(func_def_1722, type, sK794: nat > nat).
% 32.64/8.94 thf(func_def_1723, type, sK795: real > (real > real) > real).
% 32.64/8.94 thf(func_def_1724, type, sK796: (real > real) > real > real > real).
% 32.64/8.94 thf(func_def_1725, type, sK797: (nat > rat) > rat).
% 32.64/8.94 thf(func_def_1726, type, sK798: real > (int > $o) > int).
% 32.64/8.94 thf(func_def_1727, type, sK799: rat > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_1728, type, sK800: rat > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_1729, type, sK801: int > nat).
% 32.64/8.94 thf(func_def_1730, type, sK802: int > int).
% 32.64/8.94 thf(func_def_1731, type, sK803: (nat > num) > nat).
% 32.64/8.94 thf(func_def_1732, type, sK804: (real > real) > real > real).
% 32.64/8.94 thf(func_def_1733, type, sK805: (rat > $o) > (rat > $o) > (rat > $o) > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_1734, type, sK806: rat > (rat > $o) > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_1735, type, sK807: rat > (rat > $o) > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_1736, type, sK808: (num > $o) > (num > $o) > (num > $o) > (num > $o) > num).
% 32.64/8.94 thf(func_def_1737, type, sK809: num > (num > $o) > (num > $o) > num).
% 32.64/8.94 thf(func_def_1738, type, sK810: num > (num > $o) > (num > $o) > num).
% 32.64/8.94 thf(func_def_1739, type, sK811: int > nat).
% 32.64/8.94 thf(func_def_1740, type, sK812: int > int > int).
% 32.64/8.94 thf(func_def_1741, type, sK813: int > int > nat).
% 32.64/8.94 thf(func_def_1742, type, sK814: real > (real > real) > (real > real > real) > real > real).
% 32.64/8.94 thf(func_def_1743, type, sK815: real > real > (real > real > real) > (real > real) > real).
% 32.64/8.94 thf(func_def_1744, type, sK816: (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1745, type, sK817: rat > rat).
% 32.64/8.94 thf(func_def_1746, type, sK818: real > nat > real).
% 32.64/8.94 thf(func_def_1747, type, sK819: nat > nat).
% 32.64/8.94 thf(func_def_1748, type, sK820: (nat > real) > nat).
% 32.64/8.94 thf(func_def_1749, type, sK821: (nat > real) > nat).
% 32.64/8.94 thf(func_def_1750, type, sK822: (nat > real) > real).
% 32.64/8.94 thf(func_def_1751, type, sK823: num > num).
% 32.64/8.94 thf(func_def_1752, type, sK824: (nat > complex) > real).
% 32.64/8.94 thf(func_def_1753, type, sK825: nat > (nat > complex) > nat).
% 32.64/8.94 thf(func_def_1754, type, sK826: nat > (nat > complex) > nat).
% 32.64/8.94 thf(func_def_1755, type, sK827: int > int).
% 32.64/8.94 thf(func_def_1756, type, sK828: real > (nat > real) > real).
% 32.64/8.94 thf(func_def_1757, type, sK829: int > int > int).
% 32.64/8.94 thf(func_def_1758, type, sK830: int > int > int).
% 32.64/8.94 thf(func_def_1759, type, sK831: real > real).
% 32.64/8.94 thf(func_def_1760, type, sK832: (code_integer > code_integer > $o) > code_integer).
% 32.64/8.94 thf(func_def_1761, type, sK833: $o > nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1762, type, sK834: $o > nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1763, type, sK835: $o > nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1764, type, sK836: $o > nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1765, type, sK837: $o > nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1766, type, sK838: $o > nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1767, type, sK839: $o > nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1768, type, sK840: $o > nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1769, type, sK841: $o > nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1770, type, sK842: $o > nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1771, type, sK843: $o > nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1772, type, sK844: $o > nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1773, type, sK845: vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1774, type, sK846: vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1775, type, sK847: vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1776, type, sK848: vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1777, type, sK849: num > num).
% 32.64/8.94 thf(func_def_1778, type, sK850: (nat > $o) > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1779, type, sK851: (real > real) > real).
% 32.64/8.94 thf(func_def_1780, type, sK852: nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1781, type, sK853: real > int).
% 32.64/8.94 thf(func_def_1782, type, sK854: rat > rat > rat).
% 32.64/8.94 thf(func_def_1783, type, sK855: int > set_int > int > int).
% 32.64/8.94 thf(func_def_1784, type, sK856: int > set_int > int > int).
% 32.64/8.94 thf(func_def_1785, type, sK857: nat > nat > nat).
% 32.64/8.94 thf(func_def_1786, type, sK858: int > set_int > int > int).
% 32.64/8.94 thf(func_def_1787, type, sK859: int > set_int > int > int).
% 32.64/8.94 thf(func_def_1788, type, sK860: real > real).
% 32.64/8.94 thf(func_def_1789, type, sK861: real > nat > real).
% 32.64/8.94 thf(func_def_1790, type, sK862: vEBT_VEBT > nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1791, type, sK863: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1792, type, sK864: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1793, type, sK865: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1794, type, sK866: vEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1795, type, sK867: nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1796, type, sK868: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1797, type, sK869: nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1798, type, sK870: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1799, type, sK871: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1800, type, sK872: nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1801, type, sK873: nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1802, type, sK874: (nat > real > real) > nat > (real > real) > real > real).
% 32.64/8.94 thf(func_def_1803, type, sK875: (nat > real > real) > nat).
% 32.64/8.94 thf(func_def_1804, type, sK876: (nat > real > real) > real).
% 32.64/8.94 thf(func_def_1805, type, sK877: (int > $o) > rat > int).
% 32.64/8.94 thf(func_def_1806, type, sK878: vEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1807, type, sK879: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1808, type, sK880: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1809, type, sK881: vEBT_VEBT > nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1810, type, sK882: vEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1811, type, sK883: vEBT_VEBT > nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1812, type, sK884: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1813, type, sK885: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1814, type, sK886: vEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1815, type, sK887: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1816, type, sK888: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1817, type, sK889: vEBT_VEBT > nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1818, type, sK890: nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1819, type, sK891: nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1820, type, sK892: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1821, type, sK893: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1822, type, sK894: real > real > nat).
% 32.64/8.94 thf(func_def_1823, type, sK895: vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1824, type, sK896: vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1825, type, sK897: nat > vEBT_VEBT > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_1826, type, sK898: nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1827, type, sK899: nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1828, type, sK900: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1829, type, sK901: nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1830, type, sK902: num > num).
% 32.64/8.94 thf(func_def_1831, type, sK903: real > nat).
% 32.64/8.94 thf(func_def_1832, type, sK904: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1833, type, sK905: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1834, type, sK906: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_1835, type, sK907: vEBT_VEBT > nat > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_1836, type, sK908: vEBT_VEBT > nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1837, type, sK909: vEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1838, type, sK910: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1839, type, sK911: vEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1840, type, sK912: vEBT_VEBT > nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1841, type, sK913: vEBT_VEBT > nat > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_1842, type, sK914: real > (real > real) > real).
% 32.64/8.94 thf(func_def_1843, type, sK915: rat > rat > rat).
% 32.64/8.94 thf(func_def_1844, type, sK916: real > (real > $o) > real).
% 32.64/8.94 thf(func_def_1845, type, sK917: real > (real > $o) > real).
% 32.64/8.94 thf(func_def_1846, type, sK918: real > (real > $o) > real).
% 32.64/8.94 thf(func_def_1847, type, sK919: real > (real > $o) > real).
% 32.64/8.94 thf(func_def_1848, type, sK920: int > set_int > int).
% 32.64/8.94 thf(func_def_1849, type, sK921: rat > int).
% 32.64/8.94 thf(func_def_1850, type, sK922: int > (int > $o) > int > int).
% 32.64/8.94 thf(func_def_1851, type, sK923: int > (int > $o) > int > int).
% 32.64/8.94 thf(func_def_1852, type, sK924: (int > $o) > int > int > int).
% 32.64/8.94 thf(func_def_1853, type, sK925: (int > $o) > int > int > int).
% 32.64/8.94 thf(func_def_1854, type, sK926: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1855, type, sK927: nat > rat > (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1856, type, sK928: (nat > rat) > rat).
% 32.64/8.94 thf(func_def_1857, type, sK929: (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1858, type, sK930: nat > nat > nat).
% 32.64/8.94 thf(func_def_1859, type, sK931: int > int > int > int).
% 32.64/8.94 thf(func_def_1860, type, sK932: rat > rat > nat).
% 32.64/8.94 thf(func_def_1861, type, sK933: num > num).
% 32.64/8.94 thf(func_def_1862, type, sK934: num > num).
% 32.64/8.94 thf(func_def_1863, type, sK935: real > int).
% 32.64/8.94 thf(func_def_1864, type, sK936: (real > $o) > (real > $o) > real).
% 32.64/8.94 thf(func_def_1865, type, sK937: real > real > nat).
% 32.64/8.94 thf(func_def_1866, type, sK938: set_nat > nat).
% 32.64/8.94 thf(func_def_1867, type, sK939: set_nat > nat).
% 32.64/8.94 thf(func_def_1868, type, sK940: real > real).
% 32.64/8.94 thf(func_def_1869, type, sK941: real > (real > real) > real > real).
% 32.64/8.94 thf(func_def_1870, type, sK942: real > (real > real) > real > real).
% 32.64/8.94 thf(func_def_1871, type, sK943: (real > real) > real > real > real).
% 32.64/8.94 thf(func_def_1872, type, sK944: real > real > nat > (nat > real > real) > real).
% 32.64/8.94 thf(func_def_1873, type, sK945: real > real > nat > (nat > real > real) > nat).
% 32.64/8.94 thf(func_def_1874, type, sK946: real > nat > real > (real > real) > (nat > real > real) > real).
% 32.64/8.94 thf(func_def_1875, type, sK947: (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_1876, type, sK948: (int > $o) > product_prod_nat_nat).
% 32.64/8.94 thf(func_def_1877, type, sK949: int > int > int > int).
% 32.64/8.94 thf(func_def_1878, type, sK950: int > int > int > int).
% 32.64/8.94 thf(func_def_1879, type, sK951: num > num > num).
% 32.64/8.94 thf(func_def_1880, type, sK952: num > num > num > num).
% 32.64/8.94 thf(func_def_1881, type, sK953: num > num > num > num).
% 32.64/8.94 thf(func_def_1882, type, sK954: num > num > num > num).
% 32.64/8.94 thf(func_def_1883, type, sK955: num > num > num > num).
% 32.64/8.94 thf(func_def_1884, type, sK956: num > num > num > num).
% 32.64/8.94 thf(func_def_1885, type, sK957: num > num > num > num).
% 32.64/8.94 thf(func_def_1886, type, sK958: num > num > num).
% 32.64/8.94 thf(func_def_1887, type, sK959: num > num > num > num).
% 32.64/8.94 thf(func_def_1888, type, sK960: num > num > num > num).
% 32.64/8.94 thf(func_def_1889, type, sK961: num > num > num > num).
% 32.64/8.94 thf(func_def_1890, type, sK962: num > num > num > num).
% 32.64/8.94 thf(func_def_1891, type, sK963: complex > nat).
% 32.64/8.94 thf(func_def_1892, type, sK964: int > set_int > int > int).
% 32.64/8.94 thf(func_def_1893, type, sK965: int > set_int > int > int).
% 32.64/8.94 thf(func_def_1894, type, sK966: (nat > num) > nat).
% 32.64/8.94 thf(func_def_1895, type, sK967: (num > $o) > (num > $o) > (num > $o) > (num > $o) > num).
% 32.64/8.94 thf(func_def_1896, type, sK968: num > (num > $o) > (num > $o) > num).
% 32.64/8.94 thf(func_def_1897, type, sK969: num > (num > $o) > (num > $o) > num).
% 32.64/8.94 thf(func_def_1898, type, sK970: nat > nat).
% 32.64/8.94 thf(func_def_1899, type, sK971: real > set_real > (real > real) > real).
% 32.64/8.94 thf(func_def_1900, type, sK972: nat > extended_enat > nat).
% 32.64/8.94 thf(func_def_1901, type, sK973: int > int > set_int > int).
% 32.64/8.94 thf(func_def_1902, type, sK974: int > int > set_int > int).
% 32.64/8.94 thf(func_def_1903, type, sK975: (complex > $o) > int).
% 32.64/8.94 thf(func_def_1904, type, sK976: rat > rat).
% 32.64/8.94 thf(func_def_1905, type, sK977: int > nat).
% 32.64/8.94 thf(func_def_1906, type, sK978: (int > $o) > int).
% 32.64/8.94 thf(func_def_1907, type, sK979: (int > $o) > int > int).
% 32.64/8.94 thf(func_def_1908, type, sK980: set_int > (int > $o) > int > int).
% 32.64/8.94 thf(func_def_1909, type, sK981: set_int > (int > $o) > int > int).
% 32.64/8.94 thf(func_def_1910, type, sK982: int > set_int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1911, type, sK983: int > (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_1912, type, sK984: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1913, type, sK985: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1914, type, sK986: real > nat).
% 32.64/8.94 thf(func_def_1915, type, sK987: (int > $o) > int > int).
% 32.64/8.94 thf(func_def_1916, type, sK988: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1917, type, sK989: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1918, type, sK990: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1919, type, sK991: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1920, type, sK992: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1921, type, sK993: (nat > nat > $o) > nat).
% 32.64/8.94 thf(func_def_1922, type, sK994: (real > real) > set_real > real).
% 32.64/8.94 thf(func_def_1923, type, sK995: (real > $o) > (real > $o) > (real > $o) > (real > $o) > real).
% 32.64/8.94 thf(func_def_1924, type, sK996: real > (real > $o) > (real > $o) > real).
% 32.64/8.94 thf(func_def_1925, type, sK997: real > (real > $o) > (real > $o) > real).
% 32.64/8.94 thf(func_def_1926, type, sK998: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1927, type, sK999: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_1928, type, sK1000: int > int).
% 32.64/8.94 thf(func_def_1929, type, sK1001: (real > $o) > nat).
% 32.64/8.94 thf(func_def_1930, type, sK1002: (real > $o) > real).
% 32.64/8.94 thf(func_def_1931, type, sK1003: (real > $o) > real).
% 32.64/8.94 thf(func_def_1932, type, sK1004: real > int).
% 32.64/8.94 thf(func_def_1933, type, sK1005: real > int).
% 32.64/8.94 thf(func_def_1934, type, sK1006: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1935, type, sK1007: real > nat > real).
% 32.64/8.94 thf(func_def_1936, type, sK1008: real > (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1937, type, sK1009: rat > rat).
% 32.64/8.94 thf(func_def_1938, type, sK1010: nat > (nat > nat) > nat).
% 32.64/8.94 thf(func_def_1939, type, sK1011: (nat > nat) > nat > nat).
% 32.64/8.94 thf(func_def_1940, type, sK1012: int > int > nat).
% 32.64/8.94 thf(func_def_1941, type, sK1013: nat > real > nat).
% 32.64/8.94 thf(func_def_1942, type, sK1014: real > real).
% 32.64/8.94 thf(func_def_1943, type, sK1015: (list_nat > $o) > (list_nat > $o) > list_nat).
% 32.64/8.94 thf(func_def_1944, type, sK1016: (rat > rat > $o) > rat).
% 32.64/8.94 thf(func_def_1945, type, sK1017: (real > $o) > (real > $o) > (real > $o) > (real > $o) > real).
% 32.64/8.94 thf(func_def_1946, type, sK1018: real > (real > $o) > (real > $o) > real).
% 32.64/8.94 thf(func_def_1947, type, sK1019: real > (real > $o) > (real > $o) > real).
% 32.64/8.94 thf(func_def_1948, type, sK1020: (real > real) > real > real).
% 32.64/8.94 thf(func_def_1949, type, sK1021: nat > nat).
% 32.64/8.94 thf(func_def_1950, type, sK1022: nat > nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1951, type, sK1023: nat > nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_1952, type, sK1024: real > real > (real > real) > real).
% 32.64/8.94 thf(func_def_1953, type, sK1025: (real > real) > real > real > (real > real) > real).
% 32.64/8.94 thf(func_def_1954, type, sK1026: real > real).
% 32.64/8.94 thf(func_def_1955, type, sK1027: num > num > num > num).
% 32.64/8.94 thf(func_def_1956, type, sK1028: num > num > num > num).
% 32.64/8.94 thf(func_def_1957, type, sK1029: num > num > num > num).
% 32.64/8.94 thf(func_def_1958, type, sK1030: num > num > num > num).
% 32.64/8.94 thf(func_def_1959, type, sK1031: num > num > num).
% 32.64/8.94 thf(func_def_1960, type, sK1032: num > num > num > num).
% 32.64/8.94 thf(func_def_1961, type, sK1033: num > num > num > num).
% 32.64/8.94 thf(func_def_1962, type, sK1034: num > num > num > num).
% 32.64/8.94 thf(func_def_1963, type, sK1035: num > num > num > num).
% 32.64/8.94 thf(func_def_1964, type, sK1036: num > num > num).
% 32.64/8.94 thf(func_def_1965, type, sK1037: num > num > num > num).
% 32.64/8.94 thf(func_def_1966, type, sK1038: num > num > num > num).
% 32.64/8.94 thf(func_def_1967, type, sK1039: nat > nat > nat).
% 32.64/8.94 thf(func_def_1968, type, sK1040: nat > nat > nat).
% 32.64/8.94 thf(func_def_1969, type, sK1041: real > nat > real).
% 32.64/8.94 thf(func_def_1970, type, sK1042: vEBT_VEBT > nat > $o > $o).
% 32.64/8.94 thf(func_def_1971, type, sK1043: vEBT_VEBT > nat > $o > $o).
% 32.64/8.94 thf(func_def_1972, type, sK1044: $o > nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_1973, type, sK1045: $o > nat > vEBT_VEBT > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_1974, type, sK1046: $o > nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1975, type, sK1047: $o > nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1976, type, sK1048: nat > $o > vEBT_VEBT > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_1977, type, sK1049: nat > $o > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1978, type, sK1050: nat > $o > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1979, type, sK1051: (int > $o) > nat).
% 32.64/8.94 thf(func_def_1980, type, sK1052: (int > $o) > nat).
% 32.64/8.94 thf(func_def_1981, type, sK1053: real > real).
% 32.64/8.94 thf(func_def_1982, type, sK1054: vEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_1983, type, sK1055: vEBT_VEBT > nat > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_1984, type, sK1056: vEBT_VEBT > nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_1985, type, sK1057: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_1986, type, sK1058: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_1987, type, sK1059: vEBT_VEBT > nat > $o).
% 32.64/8.94 thf(func_def_1988, type, sK1060: real > real).
% 32.64/8.94 thf(func_def_1989, type, sK1061: nat > nat > nat).
% 32.64/8.94 thf(func_def_1990, type, sK1062: nat > nat > nat).
% 32.64/8.94 thf(func_def_1991, type, sK1063: int > int > set_int > int).
% 32.64/8.94 thf(func_def_1992, type, sK1064: int > int > set_int > int).
% 32.64/8.94 thf(func_def_1993, type, sK1065: real > real > rat).
% 32.64/8.94 thf(func_def_1994, type, sK1066: (nat > rat) > nat).
% 32.64/8.94 thf(func_def_1995, type, sK1067: int > int > set_int > int).
% 32.64/8.94 thf(func_def_1996, type, sK1068: int > int > set_int > int).
% 32.64/8.94 thf(func_def_1997, type, sK1069: (code_integer > $o) > code_integer).
% 32.64/8.94 thf(func_def_1998, type, sK1070: (code_integer > $o) > code_integer).
% 32.64/8.94 thf(func_def_1999, type, sK1071: (code_integer > $o) > $o).
% 32.64/8.94 thf(func_def_2000, type, sK1072: nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2001, type, sK1073: nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2002, type, sK1074: (nat > nat) > nat).
% 32.64/8.94 thf(func_def_2003, type, sK1075: (nat > nat) > nat).
% 32.64/8.94 thf(func_def_2004, type, sK1076: (nat > num) > nat).
% 32.64/8.94 thf(func_def_2005, type, sK1077: real > real > real).
% 32.64/8.94 thf(func_def_2006, type, sK1078: (nat > int) > nat).
% 32.64/8.94 thf(func_def_2007, type, sK1079: int > int > int).
% 32.64/8.94 thf(func_def_2008, type, sK1080: real > nat > (nat > real > real) > real).
% 32.64/8.94 thf(func_def_2009, type, sK1081: real > nat > (nat > real > real) > nat).
% 32.64/8.94 thf(func_def_2010, type, sK1082: (nat > real > real) > nat > real > (real > real) > real).
% 32.64/8.94 thf(func_def_2011, type, sK1083: nat > nat).
% 32.64/8.94 thf(func_def_2012, type, sK1084: real > (nat > real > real) > nat > nat).
% 32.64/8.94 thf(func_def_2013, type, sK1085: real > (nat > real > real) > nat > real).
% 32.64/8.94 thf(func_def_2014, type, sK1086: (real > real) > (nat > real > real) > real > nat > real).
% 32.64/8.94 thf(func_def_2015, type, sK1087: real > nat).
% 32.64/8.94 thf(func_def_2016, type, sK1088: real > nat).
% 32.64/8.94 thf(func_def_2017, type, sK1089: nat > nat).
% 32.64/8.94 thf(func_def_2018, type, sK1090: vEBT_VEBT > nat > $o > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_2019, type, sK1091: vEBT_VEBT > nat > $o > nat).
% 32.64/8.94 thf(func_def_2020, type, sK1092: vEBT_VEBT > nat > $o > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_2021, type, sK1093: vEBT_VEBT > nat > $o > vEBT_VEBT).
% 32.64/8.94 thf(func_def_2022, type, sK1094: vEBT_VEBT > nat > $o > vEBT_VEBT).
% 32.64/8.94 thf(func_def_2023, type, sK1095: vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_2024, type, sK1096: vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_2025, type, sK1097: int > int).
% 32.64/8.94 thf(func_def_2026, type, sK1098: nat > nat > nat).
% 32.64/8.94 thf(func_def_2027, type, sK1099: real > real).
% 32.64/8.94 thf(func_def_2028, type, sK1100: int > int).
% 32.64/8.94 thf(func_def_2029, type, sK1101: (real > $o) > nat).
% 32.64/8.94 thf(func_def_2030, type, sK1102: (real > $o) > real).
% 32.64/8.94 thf(func_def_2031, type, sK1103: (real > $o) > real).
% 32.64/8.94 thf(func_def_2032, type, sK1104: extended_enat > nat > nat).
% 32.64/8.94 thf(func_def_2033, type, sK1105: int > nat).
% 32.64/8.94 thf(func_def_2034, type, sK1106: int > nat).
% 32.64/8.94 thf(func_def_2035, type, sK1107: (nat > int) > nat).
% 32.64/8.94 thf(func_def_2036, type, sK1108: nat > nat > nat).
% 32.64/8.94 thf(func_def_2037, type, sK1109: nat > nat > nat).
% 32.64/8.94 thf(func_def_2038, type, sK1110: nat > nat > nat).
% 32.64/8.94 thf(func_def_2039, type, sK1111: real > real > real).
% 32.64/8.94 thf(func_def_2040, type, sK1112: rat > int).
% 32.64/8.94 thf(func_def_2041, type, sK1113: real > int).
% 32.64/8.94 thf(func_def_2042, type, sK1114: (nat > real) > real > nat > int).
% 32.64/8.94 thf(func_def_2043, type, sK1115: real > real > nat).
% 32.64/8.94 thf(func_def_2044, type, sK1116: (rat > $o) > (rat > $o) > (rat > $o) > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_2045, type, sK1117: rat > (rat > $o) > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_2046, type, sK1118: rat > (rat > $o) > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_2047, type, sK1119: int > (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_2048, type, sK1120: (int > $o) > int).
% 32.64/8.94 thf(func_def_2049, type, sK1121: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_2050, type, sK1122: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_2051, type, sK1123: rat > rat > rat).
% 32.64/8.94 thf(func_def_2052, type, sK1124: int > int > int).
% 32.64/8.94 thf(func_def_2053, type, sK1125: real > int).
% 32.64/8.94 thf(func_def_2054, type, sK1126: real > real).
% 32.64/8.94 thf(func_def_2055, type, sK1127: (real > nat > real) > real > (real > nat > real) > nat).
% 32.64/8.94 thf(func_def_2056, type, sK1128: (real > nat > real) > real > real > real).
% 32.64/8.94 thf(func_def_2057, type, sK1129: (nat > real) > (real > nat > real) > real > real > real).
% 32.64/8.94 thf(func_def_2058, type, sK1130: (nat > real) > (real > nat > real) > real > real > real).
% 32.64/8.94 thf(func_def_2059, type, sK1131: (nat > real) > (real > nat > real) > real > real > nat).
% 32.64/8.94 thf(func_def_2060, type, sK1132: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_2061, type, sK1133: nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_2062, type, sK1134: vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_2063, type, sK1135: vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_2064, type, sK1136: vEBT_VEBT > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_2065, type, sK1137: vEBT_VEBT > nat > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_2066, type, sK1138: vEBT_VEBT > nat > nat).
% 32.64/8.94 thf(func_def_2067, type, sK1139: vEBT_VEBT > nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_2068, type, sK1140: vEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_2069, type, sK1141: rat > rat).
% 32.64/8.94 thf(func_def_2070, type, sK1142: rat > rat > rat).
% 32.64/8.94 thf(func_def_2071, type, sK1143: int > int).
% 32.64/8.94 thf(func_def_2072, type, sK1144: code_integer > code_integer).
% 32.64/8.94 thf(func_def_2073, type, sK1145: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_2074, type, sK1146: int > (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_2075, type, sK1147: int > (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_2076, type, sK1148: int > int > set_int > int).
% 32.64/8.94 thf(func_def_2077, type, sK1149: int > int > set_int > int).
% 32.64/8.94 thf(func_def_2078, type, sK1150: nat > nat > nat).
% 32.64/8.94 thf(func_def_2079, type, sK1151: int > int > int).
% 32.64/8.94 thf(func_def_2080, type, sK1152: real > int).
% 32.64/8.94 thf(func_def_2081, type, sK1153: nat > nat > nat).
% 32.64/8.94 thf(func_def_2082, type, sK1154: list_nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2083, type, sK1155: int > int > int > int).
% 32.64/8.94 thf(func_def_2084, type, sK1156: (int > $o) > int).
% 32.64/8.94 thf(func_def_2085, type, sK1157: (int > $o) > int > int).
% 32.64/8.94 thf(func_def_2086, type, sK1158: set_int > (int > $o) > int > int).
% 32.64/8.94 thf(func_def_2087, type, sK1159: set_int > (int > $o) > int > int).
% 32.64/8.94 thf(func_def_2088, type, sK1160: int > (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_2089, type, sK1161: int > (int > $o) > set_int > int).
% 32.64/8.94 thf(func_def_2090, type, sK1162: (int > $o) > int > int).
% 32.64/8.94 thf(func_def_2091, type, sK1163: (int > $o) > int > int).
% 32.64/8.94 thf(func_def_2092, type, sK1164: (rat > $o) > int).
% 32.64/8.94 thf(func_def_2093, type, sK1165: (rat > $o) > int).
% 32.64/8.94 thf(func_def_2094, type, sK1166: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_2095, type, sK1167: int > (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_2096, type, sK1168: int > (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_2097, type, sK1169: int > nat).
% 32.64/8.94 thf(func_def_2098, type, sK1170: nat > nat).
% 32.64/8.94 thf(func_def_2099, type, sK1171: nat > nat > nat).
% 32.64/8.94 thf(func_def_2100, type, sK1172: nat > nat > nat).
% 32.64/8.94 thf(func_def_2101, type, sK1173: nat > nat > nat).
% 32.64/8.94 thf(func_def_2102, type, sK1174: real > nat > real).
% 32.64/8.94 thf(func_def_2103, type, sK1175: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_2104, type, sK1176: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_2105, type, sK1177: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_2106, type, sK1178: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_2107, type, sK1179: real > real).
% 32.64/8.94 thf(func_def_2108, type, sK1180: real > int).
% 32.64/8.94 thf(func_def_2109, type, sK1181: nat > rat > (nat > rat) > nat).
% 32.64/8.94 thf(func_def_2110, type, sK1182: (nat > rat) > rat).
% 32.64/8.94 thf(func_def_2111, type, sK1183: (nat > rat) > nat).
% 32.64/8.94 thf(func_def_2112, type, sK1184: rat > nat).
% 32.64/8.94 thf(func_def_2113, type, sK1185: int > nat).
% 32.64/8.94 thf(func_def_2114, type, sK1186: nat > nat > nat).
% 32.64/8.94 thf(func_def_2115, type, sK1187: (num > $o) > (num > $o) > (num > $o) > (num > $o) > num).
% 32.64/8.94 thf(func_def_2116, type, sK1188: num > (num > $o) > (num > $o) > num).
% 32.64/8.94 thf(func_def_2117, type, sK1189: num > (num > $o) > (num > $o) > num).
% 32.64/8.94 thf(func_def_2118, type, sK1190: complex > complex).
% 32.64/8.94 thf(func_def_2119, type, sK1191: complex > real).
% 32.64/8.94 thf(func_def_2120, type, sK1192: product_prod_num_num > num).
% 32.64/8.94 thf(func_def_2121, type, sK1193: product_prod_num_num > num).
% 32.64/8.94 thf(func_def_2122, type, sK1194: product_prod_num_num > num).
% 32.64/8.94 thf(func_def_2123, type, sK1195: product_prod_num_num > num).
% 32.64/8.94 thf(func_def_2124, type, sK1196: product_prod_num_num > num).
% 32.64/8.94 thf(func_def_2125, type, sK1197: product_prod_num_num > num).
% 32.64/8.94 thf(func_def_2126, type, sK1198: product_prod_num_num > num).
% 32.64/8.94 thf(func_def_2127, type, sK1199: product_prod_num_num > num).
% 32.64/8.94 thf(func_def_2128, type, sK1200: product_prod_num_num > num).
% 32.64/8.94 thf(func_def_2129, type, sK1201: product_prod_num_num > num).
% 32.64/8.94 thf(func_def_2130, type, sK1202: product_prod_num_num > num).
% 32.64/8.94 thf(func_def_2131, type, sK1203: product_prod_num_num > num).
% 32.64/8.94 thf(func_def_2132, type, sK1204: int > int > int).
% 32.64/8.94 thf(func_def_2133, type, sK1205: vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_2134, type, sK1206: vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_2135, type, sK1207: vEBT_VEBT > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_2136, type, sK1208: vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_2137, type, sK1209: vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_2138, type, sK1210: vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_2139, type, sK1211: nat > code_integer > code_integer).
% 32.64/8.94 thf(func_def_2140, type, sK1212: int > (int > $o) > int).
% 32.64/8.94 thf(func_def_2141, type, sK1213: (int > $o) > int > int).
% 32.64/8.94 thf(func_def_2142, type, sK1214: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2143, type, sK1215: nat > nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2144, type, sK1216: nat > nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2145, type, sK1217: (nat > int) > nat).
% 32.64/8.94 thf(func_def_2146, type, sK1218: code_integer > code_integer > code_integer).
% 32.64/8.94 thf(func_def_2147, type, sK1219: (int > $o) > int).
% 32.64/8.94 thf(func_def_2148, type, sK1220: (int > $o) > int).
% 32.64/8.94 thf(func_def_2149, type, sK1221: nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2150, type, sK1222: nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2151, type, sK1223: nat > nat > nat).
% 32.64/8.94 thf(func_def_2152, type, sK1224: (nat > set_nat) > nat).
% 32.64/8.94 thf(func_def_2153, type, sK1225: (nat > real) > nat).
% 32.64/8.94 thf(func_def_2154, type, sK1226: (nat > real) > nat).
% 32.64/8.94 thf(func_def_2155, type, sK1227: (int > $o) > int).
% 32.64/8.94 thf(func_def_2156, type, sK1228: (int > $o) > int > int).
% 32.64/8.94 thf(func_def_2157, type, sK1229: (int > $o) > int > int).
% 32.64/8.94 thf(func_def_2158, type, sK1230: (int > $o) > int > int).
% 32.64/8.94 thf(func_def_2159, type, sK1231: int > nat > (nat > int) > nat).
% 32.64/8.94 thf(func_def_2160, type, sK1232: nat > (nat > int) > nat).
% 32.64/8.94 thf(func_def_2161, type, sK1233: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2162, type, sK1234: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2163, type, sK1235: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2164, type, sK1236: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2165, type, sK1237: num > num > option_num > num).
% 32.64/8.94 thf(func_def_2166, type, sK1238: num > num > option_num > num).
% 32.64/8.94 thf(func_def_2167, type, sK1239: num > num > option_num > num).
% 32.64/8.94 thf(func_def_2168, type, sK1240: num > option_num > num).
% 32.64/8.94 thf(func_def_2169, type, sK1241: option_num > num > num).
% 32.64/8.94 thf(func_def_2170, type, sK1242: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2171, type, sK1243: num > num > option_num > num).
% 32.64/8.94 thf(func_def_2172, type, sK1244: num > num > option_num > num).
% 32.64/8.94 thf(func_def_2173, type, sK1245: int > int > set_int > int).
% 32.64/8.94 thf(func_def_2174, type, sK1246: int > int > set_int > int).
% 32.64/8.94 thf(func_def_2175, type, sK1247: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2176, type, sK1248: (nat > $o) > nat > nat).
% 32.64/8.94 thf(func_def_2177, type, sK1249: real > real).
% 32.64/8.94 thf(func_def_2178, type, sK1250: int > int).
% 32.64/8.94 thf(func_def_2179, type, sK1251: (nat > complex) > real > nat).
% 32.64/8.94 thf(func_def_2180, type, sK1252: real > (real > real) > real > real).
% 32.64/8.94 thf(func_def_2181, type, sK1253: real > real > (real > real) > real).
% 32.64/8.94 thf(func_def_2182, type, sK1254: (real > real) > real > real > real).
% 32.64/8.94 thf(func_def_2183, type, sK1255: (real > real) > real > (real > real) > real > real).
% 32.64/8.94 thf(func_def_2184, type, sK1256: (real > real) > real > (real > real) > real > real).
% 32.64/8.94 thf(func_def_2185, type, sK1257: (real > real) > real > (real > real) > real > real).
% 32.64/8.94 thf(func_def_2186, type, sK1258: (real > real) > real > real > real).
% 32.64/8.94 thf(func_def_2187, type, sK1259: int > int > set_int > int).
% 32.64/8.94 thf(func_def_2188, type, sK1260: int > int > set_int > int).
% 32.64/8.94 thf(func_def_2189, type, sK1261: nat > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_2190, type, sK1262: nat > vEBT_VEBT > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_2191, type, sK1263: nat > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_2192, type, sK1264: nat > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_2193, type, sK1265: vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_2194, type, sK1266: vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_2195, type, sK1267: (num > $o) > (num > $o) > (num > $o) > (num > $o) > num).
% 32.64/8.94 thf(func_def_2196, type, sK1268: num > (num > $o) > (num > $o) > num).
% 32.64/8.94 thf(func_def_2197, type, sK1269: num > (num > $o) > (num > $o) > num).
% 32.64/8.94 thf(func_def_2198, type, sK1270: (int > $o) > int).
% 32.64/8.94 thf(func_def_2199, type, sK1271: (int > $o) > $o).
% 32.64/8.94 thf(func_def_2200, type, sK1272: (int > $o) > int).
% 32.64/8.94 thf(func_def_2201, type, sK1273: nat > nat).
% 32.64/8.94 thf(func_def_2202, type, sK1274: code_integer > nat).
% 32.64/8.94 thf(func_def_2203, type, sK1275: real > (real > real) > (nat > real > real) > nat > real).
% 32.64/8.94 thf(func_def_2204, type, sK1276: real > (nat > real > real) > nat > real).
% 32.64/8.94 thf(func_def_2205, type, sK1277: real > (nat > real > real) > nat > nat).
% 32.64/8.94 thf(func_def_2206, type, sK1278: (real > real) > real > real > real).
% 32.64/8.94 thf(func_def_2207, type, sK1279: (real > real) > (real > real) > real > real > real).
% 32.64/8.94 thf(func_def_2208, type, sK1280: real > real > (real > real) > real).
% 32.64/8.94 thf(func_def_2209, type, sK1281: real > (real > real) > (real > real) > real > real).
% 32.64/8.94 thf(func_def_2210, type, sK1282: (real > real) > (real > real) > (real > real) > (real > real) > real > real > real).
% 32.64/8.94 thf(func_def_2211, type, sK1283: real > nat).
% 32.64/8.94 thf(func_def_2212, type, sK1284: nat > nat > nat).
% 32.64/8.94 thf(func_def_2213, type, sK1285: complex > real).
% 32.64/8.94 thf(func_def_2214, type, sK1286: complex > real).
% 32.64/8.94 thf(func_def_2215, type, sK1287: int > nat).
% 32.64/8.94 thf(func_def_2216, type, sK1288: real).
% 32.64/8.94 thf(func_def_2217, type, sK1289: real > (real > real) > real).
% 32.64/8.94 thf(func_def_2218, type, sK1290: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_2219, type, sK1291: produc9072475918466114483BT_nat > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_2220, type, sK1292: produc9072475918466114483BT_nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_2221, type, sK1293: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_2222, type, sK1294: produc9072475918466114483BT_nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_2223, type, sK1295: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_2224, type, sK1296: produc9072475918466114483BT_nat > $o).
% 32.64/8.94 thf(func_def_2225, type, sK1297: produc9072475918466114483BT_nat > $o).
% 32.64/8.94 thf(func_def_2226, type, sK1298: produc9072475918466114483BT_nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_2227, type, sK1299: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_2228, type, sK1300: produc9072475918466114483BT_nat > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_2229, type, sK1301: produc9072475918466114483BT_nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_2230, type, sK1302: set_nat > nat > nat).
% 32.64/8.94 thf(func_def_2231, type, sK1303: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2232, type, sK1304: rat > (nat > rat) > nat).
% 32.64/8.94 thf(func_def_2233, type, sK1305: nat > nat > nat > nat).
% 32.64/8.94 thf(func_def_2234, type, sK1306: real > real > real).
% 32.64/8.94 thf(func_def_2235, type, sK1307: (real > real) > real > real).
% 32.64/8.94 thf(func_def_2236, type, sK1308: (int > $o) > int > int).
% 32.64/8.94 thf(func_def_2237, type, sK1309: rat > rat).
% 32.64/8.94 thf(func_def_2238, type, sK1310: rat > rat).
% 32.64/8.94 thf(func_def_2239, type, sK1311: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_2240, type, sK1312: int > (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_2241, type, sK1313: int > (int > $o) > (int > $o) > int).
% 32.64/8.94 thf(func_def_2242, type, sK1314: rat > (int > $o) > int).
% 32.64/8.94 thf(func_def_2243, type, sK1315: vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_2244, type, sK1316: list_VEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_2245, type, sK1317: list_VEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_2246, type, sK1318: list_VEBT_VEBT > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_2247, type, sK1319: nat > nat > nat).
% 32.64/8.94 thf(func_def_2248, type, sK1320: (nat > real > real) > real).
% 32.64/8.94 thf(func_def_2249, type, sK1321: (nat > real > real) > nat).
% 32.64/8.94 thf(func_def_2250, type, sK1322: (nat > real > real) > nat > (real > real) > real > real).
% 32.64/8.94 thf(func_def_2251, type, sK1323: (nat > real) > nat).
% 32.64/8.94 thf(func_def_2252, type, sK1324: (nat > real) > nat).
% 32.64/8.94 thf(func_def_2253, type, sK1325: nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2254, type, sK1326: nat > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2255, type, sK1327: real > (nat > rat) > nat).
% 32.64/8.94 thf(func_def_2256, type, sK1328: real > real).
% 32.64/8.94 thf(func_def_2257, type, sK1329: (nat > real > real) > real).
% 32.64/8.94 thf(func_def_2258, type, sK1330: (nat > real > real) > nat).
% 32.64/8.94 thf(func_def_2259, type, sK1331: (nat > real > real) > (real > real) > real > nat > real).
% 32.64/8.94 thf(func_def_2260, type, sK1332: (nat > real) > nat > nat).
% 32.64/8.94 thf(func_def_2261, type, sK1333: (nat > real) > nat > nat).
% 32.64/8.94 thf(func_def_2262, type, sK1334: (real > real) > real > real).
% 32.64/8.94 thf(func_def_2263, type, sK1335: nat > nat).
% 32.64/8.94 thf(func_def_2264, type, sK1336: nat > nat).
% 32.64/8.94 thf(func_def_2265, type, sK1337: (nat > real) > nat > nat).
% 32.64/8.94 thf(func_def_2266, type, sK1338: (nat > real) > nat > nat).
% 32.64/8.94 thf(func_def_2267, type, sK1339: int > set_int > int > int).
% 32.64/8.94 thf(func_def_2268, type, sK1340: int > set_int > int > int).
% 32.64/8.94 thf(func_def_2269, type, sK1341: nat > int > int).
% 32.64/8.94 thf(func_def_2270, type, sK1342: num > num).
% 32.64/8.94 thf(func_def_2271, type, sK1343: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2272, type, sK1344: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2273, type, sK1345: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2274, type, sK1346: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2275, type, sK1347: num > num).
% 32.64/8.94 thf(func_def_2276, type, sK1348: num > option_num > num > num).
% 32.64/8.94 thf(func_def_2277, type, sK1349: num > option_num > num > num).
% 32.64/8.94 thf(func_def_2278, type, sK1350: num > num).
% 32.64/8.94 thf(func_def_2279, type, sK1351: num > num).
% 32.64/8.94 thf(func_def_2280, type, sK1352: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2281, type, sK1353: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2282, type, sK1354: (real > $o) > nat > rat).
% 32.64/8.94 thf(func_def_2283, type, sK1355: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2284, type, sK1356: nat > nat > nat > nat).
% 32.64/8.94 thf(func_def_2285, type, sK1357: nat > nat > nat > nat).
% 32.64/8.94 thf(func_def_2286, type, sK1358: int > int).
% 32.64/8.94 thf(func_def_2287, type, sK1359: num > option_num > num > num).
% 32.64/8.94 thf(func_def_2288, type, sK1360: num > option_num > num > num).
% 32.64/8.94 thf(func_def_2289, type, sK1361: num > num > option_num > num).
% 32.64/8.94 thf(func_def_2290, type, sK1362: option_num > num > num).
% 32.64/8.94 thf(func_def_2291, type, sK1363: num > option_num > num).
% 32.64/8.94 thf(func_def_2292, type, sK1364: num > num > option_num > num).
% 32.64/8.94 thf(func_def_2293, type, sK1365: num > option_num > num > num).
% 32.64/8.94 thf(func_def_2294, type, sK1366: num > option_num > num > num).
% 32.64/8.94 thf(func_def_2295, type, sK1367: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2296, type, sK1368: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2297, type, sK1369: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2298, type, sK1370: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2299, type, sK1371: (int > $o) > int).
% 32.64/8.94 thf(func_def_2300, type, sK1372: (nat > rat) > rat).
% 32.64/8.94 thf(func_def_2301, type, sK1373: (nat > rat) > nat).
% 32.64/8.94 thf(func_def_2302, type, sK1374: int > int > int).
% 32.64/8.94 thf(func_def_2303, type, sK1375: set_int > int > (int > $o) > int).
% 32.64/8.94 thf(func_def_2304, type, sK1376: int > int > set_int > int).
% 32.64/8.94 thf(func_def_2305, type, sK1377: int > int > set_int > int).
% 32.64/8.94 thf(func_def_2306, type, sK1378: int > set_int > (int > $o) > int).
% 32.64/8.94 thf(func_def_2307, type, sK1379: int > (int > int > $o) > int > int).
% 32.64/8.94 thf(func_def_2308, type, sK1380: int > (int > int > $o) > int > int).
% 32.64/8.94 thf(func_def_2309, type, sK1381: int > int).
% 32.64/8.94 thf(func_def_2310, type, sK1382: (nat > real) > nat > nat).
% 32.64/8.94 thf(func_def_2311, type, sK1383: (nat > nat) > nat).
% 32.64/8.94 thf(func_def_2312, type, sK1384: nat > nat).
% 32.64/8.94 thf(func_def_2313, type, sK1385: nat > nat).
% 32.64/8.94 thf(func_def_2314, type, sK1386: real > real).
% 32.64/8.94 thf(func_def_2315, type, sK1387: (nat > real) > (nat > real) > real).
% 32.64/8.94 thf(func_def_2316, type, sK1388: (nat > real) > (nat > real) > nat).
% 32.64/8.94 thf(func_def_2317, type, sK1389: (nat > real) > nat).
% 32.64/8.94 thf(func_def_2318, type, sK1390: (nat > real) > nat).
% 32.64/8.94 thf(func_def_2319, type, sK1391: int > set_int > int > int).
% 32.64/8.94 thf(func_def_2320, type, sK1392: int > set_int > int > int).
% 32.64/8.94 thf(func_def_2321, type, sK1393: (set_nat > $o) > (set_nat > $o) > set_nat).
% 32.64/8.94 thf(func_def_2322, type, sK1394: rat > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_2323, type, sK1395: rat > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_2324, type, sK1396: rat > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_2325, type, sK1397: rat > (rat > $o) > rat).
% 32.64/8.94 thf(func_def_2326, type, sK1398: (nat > real) > nat).
% 32.64/8.94 thf(func_def_2327, type, sK1399: (nat > real) > nat).
% 32.64/8.94 thf(func_def_2328, type, sK1400: real > real).
% 32.64/8.94 thf(func_def_2329, type, sK1401: real > (real > real) > real).
% 32.64/8.94 thf(func_def_2330, type, sK1402: num > option_num > num > num).
% 32.64/8.94 thf(func_def_2331, type, sK1403: num > option_num > num > num).
% 32.64/8.94 thf(func_def_2332, type, sK1404: num > option_num > num > num).
% 32.64/8.94 thf(func_def_2333, type, sK1405: num > option_num > num > num).
% 32.64/8.94 thf(func_def_2334, type, sK1406: num > option_num > num).
% 32.64/8.94 thf(func_def_2335, type, sK1407: num > option_num > num > num).
% 32.64/8.94 thf(func_def_2336, type, sK1408: option_num > num > num).
% 32.64/8.94 thf(func_def_2337, type, sK1409: num > num > option_num > num).
% 32.64/8.94 thf(func_def_2338, type, sK1410: num > num > option_num > num).
% 32.64/8.94 thf(func_def_2339, type, sK1411: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2340, type, sK1412: option_num > num > num > num).
% 32.64/8.94 thf(func_def_2341, type, sK1413: num > num > option_num > num).
% 32.64/8.94 thf(func_def_2342, type, sK1414: real > real).
% 32.64/8.94 thf(func_def_2343, type, sK1415: nat > (nat > $o) > nat > nat).
% 32.64/8.94 thf(func_def_2344, type, sK1416: rat > int).
% 32.64/8.94 thf(func_def_2345, type, sK1417: rat > rat).
% 32.64/8.94 thf(func_def_2346, type, sK1418: rat > rat).
% 32.64/8.94 thf(func_def_2347, type, sK1419: num > num).
% 32.64/8.94 thf(func_def_2348, type, sK1420: (real > real) > real > real).
% 32.64/8.94 thf(func_def_2349, type, sK1421: real > real > real > real).
% 32.64/8.94 thf(func_def_2350, type, sK1422: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2351, type, sK1423: nat > (nat > $o) > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2352, type, sK1424: nat > (nat > $o) > (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2353, type, sK1425: produc9072475918466114483BT_nat > $o).
% 32.64/8.94 thf(func_def_2354, type, sK1426: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_2355, type, sK1427: produc9072475918466114483BT_nat > $o).
% 32.64/8.94 thf(func_def_2356, type, sK1428: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_2357, type, sK1429: produc9072475918466114483BT_nat > option4927543243414619207at_nat).
% 32.64/8.94 thf(func_def_2358, type, sK1430: produc9072475918466114483BT_nat > nat).
% 32.64/8.94 thf(func_def_2359, type, sK1431: produc9072475918466114483BT_nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_2360, type, sK1432: produc9072475918466114483BT_nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_2361, type, sK1433: nat > $o > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_2362, type, sK1434: nat > $o > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_2363, type, sK1435: nat > $o > vEBT_VEBT > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_2364, type, sK1436: nat > $o > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_2365, type, sK1437: nat > $o > vEBT_VEBT > nat).
% 32.64/8.94 thf(func_def_2366, type, sK1438: nat > $o > vEBT_VEBT > vEBT_VEBT).
% 32.64/8.94 thf(func_def_2367, type, sK1439: vEBT_VEBT > $o > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_2368, type, sK1440: vEBT_VEBT > $o > nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_2369, type, sK1441: vEBT_VEBT > $o > nat > nat).
% 32.64/8.94 thf(func_def_2370, type, sK1442: vEBT_VEBT > $o > nat > nat).
% 32.64/8.94 thf(func_def_2371, type, sK1443: vEBT_VEBT > $o > nat > nat).
% 32.64/8.94 thf(func_def_2372, type, sK1444: vEBT_VEBT > $o > nat > list_VEBT_VEBT).
% 32.64/8.94 thf(func_def_2373, type, sK1445: vEBT_VEBT > $o > nat > vEBT_VEBT).
% 32.64/8.94 thf(func_def_2374, type, sK1446: vEBT_VEBT > $o > nat > nat).
% 32.64/8.94 thf(func_def_2375, type, sK1447: $o > nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_2376, type, sK1448: $o > nat > vEBT_VEBT > $o).
% 32.64/8.94 thf(func_def_2377, type, sK1449: int > int).
% 32.64/8.94 thf(func_def_2378, type, sK1450: (real > real) > real > real > real).
% 32.64/8.94 thf(func_def_2379, type, sK1451: nat > nat).
% 32.64/8.94 thf(func_def_2380, type, sK1452: nat > nat > nat).
% 32.64/8.94 thf(func_def_2381, type, sK1453: (nat > real) > nat).
% 32.64/8.94 thf(func_def_2382, type, sK1454: (nat > real) > nat).
% 32.64/8.94 thf(func_def_2383, type, sK1455: real > nat).
% 32.64/8.94 thf(func_def_2384, type, sK1456: real > nat).
% 32.64/8.94 thf(func_def_2385, type, sK1457: real > real > real).
% 32.64/8.94 thf(func_def_2386, type, sK1458: (nat > num) > nat).
% 32.64/8.94 thf(func_def_2387, type, sK1459: (real > real) > (real > real) > real > real > real).
% 32.64/8.94 thf(func_def_2388, type, sK1460: (real > real) > real > real > (real > real) > real).
% 32.64/8.94 thf(func_def_2389, type, sK1461: set_int > (int > $o) > int > int).
% 32.64/8.94 thf(func_def_2390, type, sK1462: int > set_int > int > int).
% 32.64/8.94 thf(func_def_2391, type, sK1463: int > set_int > int > int).
% 32.64/8.94 thf(func_def_2392, type, sK1464: (int > $o) > int > set_int > int).
% 32.64/8.94 thf(func_def_2393, type, sK1465: (nat > $o) > nat).
% 32.64/8.94 thf(func_def_2394, type, sK1466: real > real > real).
% 32.64/8.94 thf(func_def_2395, type, sK1467: int > set_int > int > int).
% 32.64/8.94 thf(func_def_2396, type, sK1468: int > set_int > int > int).
% 32.64/8.94 thf(func_def_2397, type, sK1469: (real > $o) > nat > rat).
% 32.64/8.94 thf(func_def_2398, type, sK1470: (nat > nat) > nat).
% 32.64/8.94 thf(func_def_2399, type, sK1471: nat > nat > nat).
% 32.64/8.94 thf(func_def_2400, type, sK1472: real > nat).
% 32.64/8.94 thf(func_def_2401, type, sK1473: nat > nat > nat).
% 32.64/8.94 thf(func_def_2402, type, sK1474: nat > nat > nat).
% 32.64/8.94 thf(func_def_2403, type, sK1475: (nat > nat) > nat).
% 32.64/8.94 thf(func_def_2404, type, sK1476: real > nat > real).
% 32.64/8.94 thf(func_def_2405, type, sK1477: nat > list_nat > nat).
% 32.64/8.94 thf(func_def_2406, type, sK1478: real > real > (real > real) > real).
% 32.64/8.94 thf(func_def_2408, type, ph1480: !>[X0: $tType]:(X0)).
% 32.64/8.94 thf(f58174,plain,(
% 32.64/8.94 $false),
% 32.64/8.94 inference(avatar_sat_refutation,[status(thm)],[f55010,f55104,f58173])).
% 32.64/8.94 thf(f58173,plain,(
% 32.64/8.94 ~spl1479_44),
% 32.64/8.94 inference(avatar_contradiction_clause,[status(thm)],[f58172])).
% 32.64/8.94 thf(f58172,plain,(
% 32.64/8.94 $false | ~spl1479_44),
% 32.64/8.94 inference(trivial_inequality_removal,[status(thm)],[f58171])).
% 32.64/8.94 thf(f58171,plain,(
% 32.64/8.94 ($true != $true) | ~spl1479_44),
% 32.64/8.94 inference(superposition,[status(thm)],[f58101,f55163])).
% 32.64/8.94 thf(f55163,plain,(
% 32.64/8.94 ((vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ (divide_divide_nat @ (suc @ (suc @ va)) @ (suc @ one_one_nat))) @ y) = $true)),
% 32.64/8.94 inference(backward_demodulation,[status(thm)],[f41975,f49513])).
% 32.64/8.94 thf(f49513,plain,(
% 32.64/8.94 ((numeral_numeral_nat @ (bit0 @ one)) = (suc @ one_one_nat))),
% 32.64/8.94 inference(cnf_transformation,[status(thm)],[f526])).
% 32.64/8.94 thf(f526,axiom,(
% 32.64/8.94 ((numeral_numeral_nat @ (bit0 @ one)) = (suc @ one_one_nat))),
% 32.64/8.94 file('/export/starexec/sandbox/tmp/tmp.iMRIUser6P/DTF2THF_27261.p',fact_525_Suc__1)).
% 32.64/8.94 thf(f41975,plain,(
% 32.64/8.94 ((vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ (divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one)))) @ y) = $true)),
% 32.64/8.94 inference(cnf_transformation,[status(thm)],[f23686])).
% 32.64/8.94 thf(f23686,plain,(
% 32.64/8.94 ((vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ (divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one)))) @ y) = $true)),
% 32.64/8.94 inference(flattening,[status(thm)],[f16165])).
% 32.64/8.94 thf(f16165,plain,(
% 32.64/8.94 ~~((vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ (divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one)))) @ y) = $true)),
% 32.64/8.94 inference(fool_elimination,[status(thm)],[f16164])).
% 32.64/8.94 thf(f16164,plain,(
% 32.64/8.94 ~~(vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ (divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one)))) @ y)),
% 32.64/8.94 inference(rectify,[status(thm)],[f10194])).
% 32.64/8.94 thf(f10194,negated_conjecture,(
% 32.64/8.94 ~~(vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ (divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one)))) @ y)),
% 32.64/8.94 inference(negated_conjecture,[status(cth)],[f10193])).
% 32.64/8.94 thf(f10193,conjecture,(
% 32.64/8.94 ~(vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ (divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one)))) @ y)),
% 32.64/8.94 file('/export/starexec/sandbox/tmp/tmp.iMRIUser6P/DTF2THF_27261.p',conj_0)).
% 32.64/8.94 thf(f58101,plain,(
% 32.64/8.94 ( ! [X1 : nat] : (((vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ (divide_divide_nat @ (suc @ (suc @ va)) @ (suc @ one_one_nat))) @ X1) != $true)) ) | ~spl1479_44),
% 32.64/8.94 inference(forward_demodulation,[status(thm)],[f55009,f49513])).
% 32.64/8.94 thf(f55009,plain,(
% 32.64/8.94 ( ! [X1 : nat] : (((vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ (divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one)))) @ X1) != $true)) ) | ~spl1479_44),
% 32.64/8.94 inference(avatar_component_clause,[status(thm)],[f55008])).
% 32.64/8.94 thf(f55008,definition,(
% 32.64/8.94 spl1479_44 <=> ! [X1 : nat] : ((vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ (divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one)))) @ X1) != $true)),
% 32.64/8.94 introduced(definition,[new_symbols(naming,[spl1479_44])],[avatar_definition])).
% 32.64/8.94 thf(f55104,plain,(
% 32.64/8.94 spl1479_44 | ~spl1479_43),
% 32.64/8.94 inference(avatar_split_clause,[status(thm)],[f51020,f55004,f55008])).
% 32.64/8.94 thf(f55004,definition,(
% 32.64/8.94 spl1479_43 <=> ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) = $true)),
% 32.64/8.94 introduced(definition,[new_symbols(naming,[spl1479_43])],[avatar_definition])).
% 32.64/8.94 thf(f51020,plain,(
% 32.64/8.94 ( ! [X0 : nat] : (((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) != $true) | ((vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ (divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one)))) @ X0) != $true)) )),
% 32.64/8.94 inference(equality_resolution,[status(thm)],[f40894])).
% 32.64/8.94 thf(f40894,plain,(
% 32.64/8.94 ( ! [X0 : nat,X1 : nat] : (((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) != $true) | ($true != (vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ X1) @ X0)) | ((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) != X1)) )),
% 32.64/8.94 inference(cnf_transformation,[status(thm)],[f26632])).
% 32.64/8.94 thf(f26632,plain,(
% 32.64/8.94 ! [X0 : nat,X1 : nat] : (((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) != $true) | ($true != (vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ X1) @ X0)) | ((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) != X1))),
% 32.64/8.94 inference(flattening,[status(thm)],[f26631])).
% 32.64/8.94 thf(f26631,plain,(
% 32.64/8.94 ! [X0 : nat,X1 : nat] : ((($true != (vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ X1) @ X0)) | ((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) != X1)) | ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) != $true))),
% 32.64/8.94 inference(ennf_transformation,[status(thm)],[f24942])).
% 32.64/8.94 thf(f24942,plain,(
% 32.64/8.94 ! [X0 : nat,X1 : nat] : (((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) = $true) => (((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) = X1) => ($true != (vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ X1) @ X0))))),
% 32.64/8.94 inference(flattening,[status(thm)],[f19337])).
% 32.64/8.94 thf(f19337,plain,(
% 32.64/8.94 ! [X0 : nat,X1 : nat] : (((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) = $true) => (((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) = X1) => ~($true = (vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ X1) @ X0))))),
% 32.64/8.94 inference(fool_elimination,[status(thm)],[f19336])).
% 32.64/8.94 thf(f19336,plain,(
% 32.64/8.94 ! [X0 : nat,X1 : nat] : ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) => (((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) = X1) => ~(vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ X1) @ X0)))),
% 32.64/8.94 inference(rectify,[status(thm)],[f2])).
% 32.64/8.94 thf(f2,axiom,(
% 32.64/8.94 ! [X1 : nat,X0 : nat] : ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) => (((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) = X0) => ~(vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ X0) @ X1)))),
% 32.64/8.94 file('/export/starexec/sandbox/tmp/tmp.iMRIUser6P/DTF2THF_27261.p',fact_1__C3_OIH_C_I1_J)).
% 32.64/8.94 thf(f55010,plain,(
% 32.64/8.94 spl1479_43 | spl1479_44),
% 32.64/8.94 inference(avatar_split_clause,[status(thm)],[f51563,f55008,f55004])).
% 32.64/8.94 thf(f51563,plain,(
% 32.64/8.94 ( ! [X1 : nat] : (((vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ (divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one)))) @ X1) != $true) | ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) = $true)) )),
% 32.64/8.94 inference(equality_resolution,[status(thm)],[f44576])).
% 32.64/8.94 thf(f44576,plain,(
% 32.64/8.94 ( ! [X0 : nat,X1 : nat] : (((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) != X0) | ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) = $true) | ((vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ X0) @ X1) != $true)) )),
% 32.64/8.94 inference(cnf_transformation,[status(thm)],[f26380])).
% 32.64/8.94 thf(f26380,plain,(
% 32.64/8.94 ! [X0 : nat,X1 : nat] : (((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) != X0) | ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) = $true) | ((vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ X0) @ X1) != $true))),
% 32.64/8.94 inference(flattening,[status(thm)],[f26379])).
% 32.64/8.94 thf(f26379,plain,(
% 32.64/8.94 ! [X0 : nat,X1 : nat] : ((((vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ X0) @ X1) != $true) | ((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) != X0)) | ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) = $true))),
% 32.64/8.94 inference(ennf_transformation,[status(thm)],[f25716])).
% 32.64/8.94 thf(f25716,plain,(
% 32.64/8.94 ! [X0 : nat,X1 : nat] : (((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) != $true) => (((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) = X0) => ((vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ X0) @ X1) != $true)))),
% 32.64/8.94 inference(flattening,[status(thm)],[f21360])).
% 32.64/8.94 thf(f21360,plain,(
% 32.64/8.94 ! [X0 : nat,X1 : nat] : (~((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) = $true) => (((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) = X0) => ~((vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ X0) @ X1) = $true)))),
% 32.64/8.94 inference(fool_elimination,[status(thm)],[f21359])).
% 32.64/8.94 thf(f21359,plain,(
% 32.64/8.94 ! [X0 : nat,X1 : nat] : (~(dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) => (((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) = X0) => ~(vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ X0) @ X1)))),
% 32.64/8.94 inference(rectify,[status(thm)],[f4])).
% 32.64/8.94 thf(f4,axiom,(
% 32.64/8.94 ! [X0 : nat,X1 : nat] : (~(dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ va))) => (((divide_divide_nat @ (suc @ (suc @ va)) @ (numeral_numeral_nat @ (bit0 @ one))) = X0) => ~(vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ X0) @ X1)))),
% 32.64/8.94 file('/export/starexec/sandbox/tmp/tmp.iMRIUser6P/DTF2THF_27261.p',fact_3__C3_OIH_C_I3_J)).
% 32.64/8.94 % SZS output end Proof for DTF2THF_27261
% 32.64/8.94 % (27490)------------------------------
% 32.64/8.94 % (27490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.64/8.94 % (27490)Termination reason: Refutation
% 32.64/8.94
% 32.64/8.94 % (27490)Memory used [KB]: 43240
% 32.64/8.94 % (27490)Time elapsed: 1.552 s
% 32.64/8.94 % (27490)Instructions burned: 4866 (million)
% 32.64/8.94 % (27490)------------------------------
% 32.64/8.94 % (27490)------------------------------
% 32.64/8.94 % (27368)Success in time 4.027 s
%------------------------------------------------------------------------------